Radiative Transfer in Curved Media
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Basic Mathematical Methods for Radiative Transfer and Transport Problems in Participating Media of Spherical and Cylindrical Geometry
Radiative Transfer in Curved Media KKSen Formerly Professor of Mathematics, University of Singapore
S J Wilson Associate Professor of Mathematics, National University of Singapore
World Scientific S/ngapo6P1oWi@bt0&Mfiteimlndon • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farter Road, Singapore 9128 USA office: 687 Hartwell Street, Teaneck, NJ 07666 UK office: 73 Lynton Mead, Totteridge, London N20 8DH
RADIATIVE TRANSFER IN CURVED MEDIA Copyright © 1990 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photo copying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
ISBN 981-02-0184-2 ISBN 981-02-0185-0 (pbk)
Printed in Singapore by JBW Printers and Binders Pte. Ltd.
V
Preface
The subject of radiative transfer or transport deals with the interaction of radiation and matter.
In stellar atmospheres, it
describes the radiative mode of passage of energy from the interior of the star to the surface.
Heat transfer here deals with the radiative
transport of heat energy in participating medium.
There is also the
mathematically analogous and physically different case of neutron transport.
As our aim is to illustrate the basic mathematical
features for solving transfer problems in curved geometry, we have chosen mainly the simple examples of one dimensional radiative transfer.
The physical processes of interaction between radiation and
matter have been confined to absorption, emission and scattering occuring singly or collectively. Hathematically one is confronted with the task of seeking solution to boundary value problems through integro—differential and integral equations for transfer or transport processes.
Exact solutions are rarely available in most cases.
Consequently, a wide variety of methods have been developed which give approximate solutions of such problems.
Often boundary value problems
are converted into initial value problems and appropriate methods are designed to tackle them. For plane parallel participating media, the methods for solving transfer and transport problems are in an advanced state of development.
Several excellent treatise are available in the field
dealing with transfer problems in stellar atmospheres, radiative heat transfer and neutron transport.
The books of Chandrasekhar,
Kourganoff, Sobolev, Busbridge and Mihalas on stellar atmospheres, those of Ozisik and Viskanta on heat transfer and the books of Davison and Sykes and Case and Zweifel for neutron transport are classics in their fields. However, there exists a number of problems in radiative transfer where the curvature of the medium cannot be ignored.
To solve these
VI
problems innovations are sometimes required for extending the methods of plane medium to spherical and cylindrical medium.
Some new
techniques for approximate and exact solutions of such problems have also been developed for this purpose.
Broadly speaking the methods
employed up to the present can be classified into four main categories: (i) moment or moment like methods, (ii) schemes converting two point boundary value problems to initial value problems, (iii) methods involving solution of integral equations of transfer; (iv) strictly numerical methods.
Two publications edited by Kalkofen (1984,1987)
give excellent accounts of some of the numerical techniques available for transfer problems in plane and spherical geometry.
An overview of
these methods has been given in Chapter VII of this book. In this treatise, we have attempted to introduce the basic mathematical methods which can be used to solve transfer problems in spherical and cylindrical media.
We believe that most of the methods
described in this book can be used with proper modifications to solve transfer problems of greater complexity. to make this book self—contained. wherever necessary.
All attempts have been made
Cross references have been given
Bibliographical references are appended
at the end of each chapter for the convenience of the readers.
This
book is addressed to the graduate students and researchers on the subjects of radiative transfer, heat and neutron transport.
They will
find under the same cover an introduction to the basic mathematical methodologies known for solving transfer equations in spherical and cylindrical media. We acknowledge with gratitude our debt to our teachers, and research associates who have influenced us in our work.
It is also a
pleasure to record our thanks to the members of our family for their constant support during the preparation of this book.
We also wish to
thank Madam Tay of National University of Singapore for kindly typing the manuscript. K K Sen S J Wilson Singapore 0511 December, 1989
VII
CONTENTS
PART I
Chapter I
Physical preliminaries
3
1.1
Basic notions of radiation matter interaction
1.2
Equation of radiative transfer
12
3
1.3
Equations of transfer in integro-differential form
18
in different geometries 1.4
Boundary conditions
30
1.5
Integral equations for source function
36
1.6
Neutron transport equations
51
References
52
Mathematical Preliminaries
53
Basic methods of solving integral equation of
53
Chapter II 2.1
transfer in spherical and cylindrical geometries 2.2
Basic methods of solving the integro-differential
63
equations of transfer in spherical geometry 2.3
Ambarzumian's method
70
2.4
Integral transform methods in curved geometry
84
References
90
PART
Chapter III
Spherical harmonic and discrete ordlnate methods
95
3.1
Introduction
95
3.2
Spherical harmonic methods in plane—parallel medium
96
3.3
The Spherical harmonic methods in spherical geometry: 106 Single interval spherical harmonic method
3.4
Method of discrete ordinates in spherical geometry
117
3.5
Discrete ordlnate matrix method
126
References
127
VIII
Chapter IV
Moment method
129
4.1
Introduction
129
4.2
Methods of solution using variable Eddington factor,
132
4.3
Variable Eddington factor methods
133
4.4
Generalised Eddington relations in radiative
137
f(r)
transfer in spherically symmetric medium 4.5
Light scattering by planetary atmosphere
146
4.6
The Generalised Eddington approximation method in
158
solving problems of light scattering by an optically thin, inhomogeneous spherically symmetric planetary atmosphere
Chapter V
References
168
Ambarzumian's physical method : Principle of
170
invariance 5.1
Introduction
170
5.2
Invariant imbedding (particle counting) method
171
5.3
Invariant imbedding (particle counting) method in spherical geometry
175
5.4
Invariant imbedding in cylindrical geometry
184
5.5
Probabilistic method in spherical and cylindrical
190
geometry
Chapter VI
References
226
Solution of integral equation of transfer in curved
228
geometry 6.1
Neumann series method for homogeneous medium in
228
curved geometry 6.2
Integral transform methods in curved geometry
235
6.3
The F„ method N The Pincherle-Goursat kernel method for solving The Pincherle-Goursat kernel method for solving
251
6.4 6.4 6.5 6.5
integral equation of transfer in curved geometry Ambarzumian's mathematical method Ambarzumian's mathematical method
282 282
References
319
258 258
IX
Chapter VII
Numerical methods for transfer problems in
321
spherical geometry 7.1
Introduction
321
7.2
Direct method using impact parameter
321
7.3
Method using moments and variable Eddington factors
325
7.4
The cell method using reflection and transmission
for isotropic scattering 328
functions 7.5
Direct method using polar coordinates
333
7.6
Operator-perturbation methods
334
References
339
Glossary of symbols
341
General References
347
3 CHAPTER I PHYSICAL PRELIMINARIES
1.1
BASIC NOTIONS OF RADIATION MATTER INTERACTION The classical theory of radiative transfer in stellar
atmospheres and that of heat transfer in participating media concern the interaction between radiation and matter namely absorption, emission and scattering.
Absorption is the portion of energy of
certain frequency lost by the radiation field due to interaction with matter.
This portion of energy may, however, be re—emitted in other
frequencies or may be transformed into other forms of energy. Emission is the energy generated within the bulk of the interacting medium due to excitation and de—excitation at a certain frequency. Scattering is a process in which photon after encountering a scatterer in the medium emerges in a new direction, occasionally with a slightly modified frequency.
However, its energy is not converted
into kinetic energy of the particles.
When the redistribution of
energy following scattering is in the same frequency we call the process to be one of "coherent scattering".
Otherwise it is
considered non—coherent.
The specific
intensity
:
We focus our attention on a particular pencil of radiation of frequency v at a. point r in a direction s in a medium.
The radiation
field is usually defined by specific intensity I(r,s,i/) and is given by dE == = I(r,s,i/)cos(n,s)dZdfidi/, dt — — — —
(1.1.1)
dE where -j- is the rate of flow of radiant energy in the frequency interval (i/,i/+di/) across an infinitesimal area dE within an element of solid angle dfi. n is unit normal to d2 and s is the direction of the radiation field.
4
Fig. 1.1.1 A
The frequency integrated intensity of radiation I(r,s) is given by A
A
«>
I(r,s) -
I(r,s,i/)di/.
(1.1.2)
J0 A radiation field at a point is said to be isotropic if the intensity of radiation at that point is independent of direction.
The net flux
:
From (1.1.1), the net rate of flow of energy in the frequency interval (v,i/+di/) across the elementary area dS (<j> I(r,s,i')cos(nls)dn)d2di/ = (*F )dSdi/, where the net flux wF
is given by »
A
A
jrF^ = A I(r,s,i/)cos(n,s)dn.
(1.1.3)
It represents the rate of flow of energy per unit area across dS per unit frequency interval at frequency v.
5 In spherical polar coordinates, dn = sinSdSdtp,
(1.1.4)
where 8 is the angle made by the pencil of radiation with the outward drawn normal to dS and
where cos 0
=
J
l(r,0,
J A J' 0A ^ 0
(1.1.5)
= cos (n,s).
For axially symmetric radiation field, we have „+l F = 2 I(r,/i,i/)/id/i, v j -1 _ — where /z = cos S.
Absorption
coefficient
(1.1.6)
(1.1.7)
:
This involves the portion of energy lost from a pencil of radiation which in its passage through a. medium does not reappear as scattered radiation.
It is sometimes said to be "truly" absorbed in
the sense that it is transformed into other forms of energy.
It is a
fraction of the total loss of energy from the pencil of radiation. There may be losses due to scattering as well. The loss of radiant energy from a pencil due to "true" absorption of radiation by the medium per unit volume, per unit time, per unit solid angle and frequency interval is A
k (r)I(r,s.i/),
(1-1.8)
where k (r) is the absorption coefficient or strictly speaking a volume absorption coefficient.
k(r>.
Mass absorption coefficient is
6
Emission
coefficient
:
The gain of radiation in a pencil may be due to true emission as a result of excitation and de-excitation of atoms and contribution from other forms of generation of energy in the bulk of the medium. The gain in energy due to radiation emitted by the medium per unit time, unit volume, unit solid angle and frequency interval is equal to £e(r,l/).
(1.1.9)
In addition there may be gain of energy in the pencil as a result of radiation received from other pencils by way of scattering.
e (£>")
is called emission coefficient.
Redistribution
function,
Phase function,
Scattering
coefficient
:
To describe the scattering process, we may introduce after Hummer
the idea of a redistribution function.
If the scattering
process is non—coherent, it results in the change of both direction and frequency, while in the case of coherent scattering the change is A
A
If the quantity R(i/',s';v,s)
only in direction.
is a redistribution
function, the probability that a photon will be scattered from A
direction s' in the solid angle dfl' dfi' and frequency interval (_ur ,v'+d>/')
to a direction s into a solid angle df Cl 3 and frequency
interval (v,v+di/)
is given by
*»■:•■;>:*>*>■* ((j £S '' ] [ £s ] • *»■:•■;>:*>*>■* A
A
The redistribution function R(v',s'\v The redistribution function R(v',s'\v
,s) is normalised such that ,s) is normalised such that
2^ ,2 <j> dfl'(tdO I di/' I R(i/' ,s' \v,z)&v
<j> dn'cj>dn A
A
'o
J
= 1.
(1.1.10)
o
A Through R(i/' ,A s' \u, s) , we can define both the normalised absorption Through R(v',s';v,s), we can define both the normalised absorption
profile and the normalised emission profile for the process of scattering.
7 Thus if we integrate over all the emitted frequencies and angles, we obtain the probability for absorption from frequency range di/' and solid angle dO'.
This is given by 0(i/')di/'dn74w,
where the
absorption
profile
(v') i-s given by CO
A
A
= ]- & dO [ R(f' ,s' ;y,s)aV
(1.1.11)
Then from (1.1.10), we have CO
^(j/')dy' = 1. J
(1.1.12)
0
That is the absorption profile is normalised to unity. scattering
coefficient
The
total
a (r,i/') can be written as aT(r,^') ff a(r)«J(i/'). T (£,./')= =a(r)«J(i/').
cr(r) is usually termed the scattering
(1.1.13) (1.1.13)
coefficient.
Then the amount of energy removed from a beam of specific intensity I(r,s';i/') in solid angle dfi' at frequency interval (i>',v'+dv') scattered into the solid angle dQ in the frequency interval
and (v,v+dv)
is
(1.1.14)
Integrating over all the incoming frequencies and angles, we obtain the total amount of energy emitted in frequency (v,u+du) angle dO.
into a solid
It is given by
eS(r,s,j/)di/(dn/4?r) = a(r)di/(dn/47r)<£(dn'/^T) [ R(i/',s';v,s)I(r,s';«/' )df .
(1.1.15)
(1 .1.15) gives the full angle and frequency dependence of emission emissivity or emission coefficient of profile. < S (£ , S ,1/) is the scattering.
8 It is difficult to deal with emissivity of such general nature in studying transfer problems. of it serve the purpose.
In some problems, simplified versions
For example, one may be interested in
redistribution of frequency alone or only in coherent redistribution of angles.
It is much simpler to consider these special cases some
of which are given below.
Angle-averaged
redistribution
function
:
The angle—averaged redistribution function can be defined as
R(v',v)
= i- & R(i/' ,s' ;t/,s)dfl' = i- <£ R(i/' ,s' ;
(1.1.16)
With this redistribution function, emissivity due to scattering can be written as CO
eS(r,y) = a(r) I R(u" ,i/)J(r,i/')di/' , J 0
(1.1.17)
where J(r,i/') is the mean intensity defined by
J(r,u')
= j> l(v,s',u')
^ ' .
(1.1.18)
R(y'.^) gives the measure of the probability of redistribution of a photon from frequency range (v" ,v'+dv')
to (i/,]/+di/) .
It is
normalised such that CO
CO
(Kv')dv' J
0
CO
= | di/' I R(i/' ,v)Av J J 0 0
= 1.
(1.1.19)
The angle-averaged emission profile is given by CO
V>(") = eS(r,i^)/j eS(r,i/)dt/
CXI
= J* R(i/' ,i/)J(r,i/')di/'/f
CO
«K«/')J(r,i>')di/'.
(1.1.20)
9 This implies that the distribution of photons emitted depends on the profile of the incident radiation. In the particular case of frequency independent intensity V>(i/) = P" R(i/' ,i/)dv . J
(1.1.21)
0
If further
R(i/,i/') = R(i/' ,u) ,
(1.1.22)
*(«0 - (")•
(1.1.23)
One example of equality of emission and absorption profiles occurs in the case of thermodynamic equilibrium.
When the scattering is
essentially coherent (i/' = i/) A
A
A
A
R(i/',s';i/,s) = p(s' ,s)(^(i/')S(v - i/'). A
(1.1.24)
A
Here 5 is the Dirac's delta—function and p(s',s) is the phase function.p(s',s) is normalised as
~- I p(s',s)dO' = 1.
(1.1.25)
In the case of isotropic scattering, the phase function A
A
p(s',s) = 1.
(1.1.26)
Another special case is that of Rayleigh's phase function given by p(s',s) = | (1 + cos2 6),
(1.1.27)
8 being the angle between the incident and the scattered pencil. coherent scattering, emissivity or the emission coefficient can be written as
For
10 .28) eS(r,s,i/) - a(r)^(i>)(j> I(r, s',i/)p(s ' , s) j^' ■ ( 1 . 1(1.1.28)
Attenuation or extinction
coefficient
:
The total loss of intensity from a pencil of radiation is usually expressed in terms of an attenuation or extinction coefficient defined as follows :
Fig. 1.1.2
Let the radiant energy pass through an elementary volume dv = dSds. The rate at which the energy from the pencil of incident radiation I(r,s,i/) is removed or attenuated by the volume is proportional to (i)
ds, the length traversed,
(ii)
I(r, s ,i/)dSdf)di/, the rate at which the radiant energy in the frequency interval (v.^+di/) is transmitted across the elementary area dS at r along s within the solid angle dQ.
Then the rate of loss or attenuation of energy from the incident beam in its passage through the volume element dv is
a(r,i/) [I(r,s,i/)d2dfidi/]ds. The factor of proportionality a(r,i/) is called
extinction
coefficient.
(1.1.29) the attenuation
or
11 This rate of loss of energy from the incident beam per unit volume, per unit solid angle and frequency interval may be (a)
due to true absorption by the medium given by k(r,i/)I(r,s,v), where k(r,f) is the volume coefficient of absorption and
(b)
due to scattering from the beam to other beams given by
ff(r,i/)I(r,s,i/) ,
where o(x,v)
is the volume coefficient of scattering. a(v,v)
= k(r,i/) + o(x,i>).
Then
(1.1.30)
Source function : The term source function ?(r,i/) at a particular point r in the medium is the ratio of total emissivity to extinction or attenuation coefficient. If the medium is in local thermodynamic equilibrium we can assign at each point a local temperature T(r).
Then Kirchoff's law
is valid and the source function ?(r,i/) is given by
*<£•"> = ^ y = V T ) -
(1 1 31)
- -
where B (T), the Planck function is given by
V T ) - ^Tc
h,/kT e
;
t 1 - 1 - 32 )
- 1
Here h is the Planck constant and k, the Boltzmann constant and T, the temperature. For scattering atmosphere involving coherent scattering ?(r,i/) = w(r)lp(s' ,s)I(r,s' ,u) ^ ' ,
(1.1.33)
12 assuming the emission and absorption profiles are the same.
In
general, we are required to deal with media which involves simultaneously absorption, emission and scattering.
In some special
cases, it is relatively easy to find simple expressions for the source function.
In (1.1.33), w(r) is the albedo for single
scattering and is given by cr(r)/a(r) . Note
u(r) = 1
represents the case of perfect scattering,
and
w(r) = 0
represents the case of true absorption.
When the phase function is independent of the directions, scattering is said to isotropic.
When the absorption and scattering
coefficients are independent of position, the medium is called "homogeneous" medium. 1.2.
Equation of radiative transfer The general equation of radiative transfer is the fundamental
equation for studying the variation of intensity from point to point of a medium due to interaction between radiation and matter. establish it, we adopt an Eulerian point of view.
To
The interaction is
studied through the attenuation and emission coefficients. We consider the passage of radiation through a cylindrical element of volume dSds at a point r along a direction s.
dS is an
infinitesimal area about the point r normal to the direction s. is the length of the cylinder [cf Fig. 1.2.1].
Fig. 1,.2. 1
ds
13 A pencil of radiation of frequency v is incident at the point r in a direction s.
The rate at which the radiant energy is attenuated
in passing through the volume element dSds is given by a(r ,v)ds[I(r,s,y)dSdndi/] , where we recall that a(r,i/) is the attenuation coefficient, I(r,s,i/), the specific intensity represents the rate of flow of radiant energy of frequency u per unit frequency interval within unit solid angle across unit area normal to the direction s at a point r. A portion of attenuated radiation may be re—emitted as scattered radiation in other directions, sometimes with slightly altered frequency.
The measure of this fraction is given by the albedo for
single scattering u>, which is the ratio of scattering to attenuation coefficient.
In fact the process of scattering may contribute to
both the loss and the gain of energy in the radiation—matter interaction.
The contribution due to scattering is expressed in
terms of the redistribution or phase function. The gain in radiant energy from the volume element dSds can be given in terms of emission coefficient e(r,i/).
The rate at which
radiant energy is emitted within the volume element dEds is e(r,t/)dSdsd£3di/. Hence the balance equation can be written as A
A
d l ( r , s , v) 62.6D.Av = - a ( r , i / ) d s I ( r , s ,i/)dfid£di/ + e ( r ,i/)d2dsdfidi/. Where d l ( r , s , i / )
(1.2.1)
i s t h e change i n s p e c i f i c i n t e n s i t y i n p a s s a g e
t h r o u g h t h e element of t h e medium. I n t h e l i m i t ds —> 0, and we o b t a i n %- I ( r , s , i / ) = - a ( r , i / ) I ( r , s , i / ) + e ( r , i / ) .
(1.2.2)
14 It has already been stated that the attenuation coefficient a(r,v) = k(r,v) + «r(r,f).
[cf (1.1.30)]
The emissivity or emission coefficient e(r,i/) may contain contribution from true emission which is the radiation emitted by the medium per unit time, per unit volume, per unit solid angle and unit frequency interval.
This we label as e (r,i/).
It may also contain
gain in energy by the beam by scattering of radiation from all other directions and we label this part of emissivity as e (r,v) .
Hence
equation (1.2.2) can be explicitly written as A
dl(r,s,i/)
-[k(r,i/) +CT(r,i/)]I(r,s,i/)+ £e(r,j/) + eS(r,i/).
^
(1.2.3)
Equations (1.2.2) or (1.2.3) is the equation of transfer in the general form. In terms of source function, we can state this equation of transfer as dl(r,s,i/) g=
a(r,i/)[I(r,s,i/) -»(r,i/)],
where 5(r,f) is the source function.
(1.2.4)
We may consider some special
cases. (a)
If a condition of local thermodynamic equilibrium prevails in the medium and there is no scattering, and Kirchoff's law holds, the source function 5((r,y) is given by ?<£,!/) = e
(1.2.5)
where B (T), Planck function, is given by 3
VT)
=
\ -
exp(h,/kT) - 1 •
C1'2-6)
15 h is Planck constant, and T is the characteristic temperature at r.
The equation of transfer is
dl(r,s,i/) = -a(r,i/)[I(r,s,i/) - B ^ T ) ] .
s
(b)
(1.2.7)
Purely scattering medium is described by u(r) = 1, k(r) = 0, a(r) = a(r).
The transfer equation in this case reads as
dl(r,s,i/)
1
a(r,i/)
I(r,s,i/) + 7 —
ds
(1.2.8) (c)
In the case of a coherent scattering medium in local thermodynamic equilibrium, the transfer equation can be written as —,
dI(r,s,K) ^-r = -I(r,s,i/) + y(r,i/),
c
a(r ,i/)
where where
x
ds
v
~'~'
~'
-
A
A
A
? ( r , i / ) = (1 - u ( r , i / ) ) B (T) + (w(r , y ) / 4 i r )-< b l ( rA , s ' , i / ) p A( s ' ,As ) d n ' . ?(r,i/) = (1 - u(r,i/))B (T) + (w(r,y)/4ff)cbl(r,s' ,i/)p(s' ,s)dn' . (1.2.9) Here a(r,v)/a(r,i>) k(r,i/)/a(r,»/)
=
= w(r,f), the albedo for single scattering. 1 — "(r.i/).
In addition to the generation of internal energy, if there is presence of external radiation incident at the bounding surfaces, the source function ?(r,i/) can be written as w(r,i/) ?(
£,l/)
=
A
A
— i ~ — Ol(r,s,i/)p(s',s)dn' + B(r),
(1.2.10)
where B(r) - B (r) + B Q (r).
(1.2.11)
16 B (r) is the contribution to the source function from the reduced o external incident radiation and B..(r) is the contribution from internal sources other than scattering.
B (r) and B (r) are specified in
particular problems. (d)
Grey medium : If the medium possesses radiative properties independent of frequency, the medium is termed a "grey medium".
In radiative
transfer in stellar atmosphere and heat transfer, there are many problems where grey approximation is quite valid.
In this
case, we may use frequency integrated quantities in the equation of transfer.
For example, we may write
:(r,s) =
J* B^T)d „
I(r,s,i/)div,
2-T4 „ E_£J_
(1.2.12)
(1.2.13)
(
where n is the refractive index of the medium and a is the Stefan constant. The transfer equation in this case (frequently used in heat transfer) is given by A
,
dl(r,s)
^7y —idr- =
(1
w
2-4
- <*» - r -
+
W (r)
*
*
*
iHr f ^ s ' W r •*)**' ■ (1.2.14)
In most cases, we obtain the equation of transfer as an Integro—differential equation. A few examples of the above constructions of source functions have been given.
But in some problems the source function may be much
more complex, when scattering is anisotroplc, medium is non-homogeneous, redistribution functions are simultaneously angle and
17 frequency dependent or the axial symmetry of the radiation field is not present.
Such source functions will be deduced whenever we meet
them in subsequent chapters.
Formal solution
of transfer
equation
:
We have seen that the general form of transfer equation can be written as dl(r,s) a(r)
ds
I(r,s' ,1/) +
V(r,v).
(1.2.15)
When the source function ?(r,i/) is spelt out, (1.2.15) turns out to be an integro—differential equation.
It is in fact a partial
integro—differential equation when the space derivative is written in terms of specific coordinate systems. may be quite involved in many cases.
The solution of this equation However, a formal integration of A
this equation along the path s in the direction s as shown in the figure 1.2.2 can be written out.
Fig. 1.2.2
Integration of equation (1.2.15) under the formal boundary condition I(s , s ,i/) = I ov yields
at s - s o
(1.2.16)
18
I(s,s,i/) = I
exp[-
a(s',f)ds'] s o
s + J
s
» s a(s',i/)?(s',s,»/)exp[a(s" ,i>) ]ds"]ds' . J ~ s'
(1.2.17)
o (1.2.17) is the integral form of equation of radiative transfer.
The
first term on the right hand side of the equation accounts for the contribution of the reduced incident radiation from the surface s = s and the second term that of diffuse radiation. and the second term that of diffuse radiation.
o One cannot consider it One cannot consider it
to be the solution of transfer equation as the source function is dependent on intensity. 1.3.
Equations of transfer in integro—differential form in different geometries
(a)
Plane-parallel
system :
We consider a medium stratified in planes perpendicular to oz—axis(say).
The radiative properties in each plane layer is
uniform. In studying radiative transfer problems in plane parallel medium, linear distances are measured normal to the plane of stratification.
If z is this distance and 8,cp are the angles shown in
Fig 1.3.1, the equation of radiative transfer in plane geometry can be written as dI Z ,ip) C O S 99 d a^( z^ cos ) d z y) == a(z)dz
1 \9(z,9,a) ,,t-/ - -l(z,0,
z '>9 T^ / ++g(z.*.-P)
d.3.1)
19
Fig. 1.3.1
Let s denote the direction of a pencil of radiation of specific intensity l(z,0,
In this, we suppress the variable v for
simplicity or the model considered is grey or the radiation monochromatic or the terms mentioned in (1.3.1) are frequency integrated. The radiation field is supposed to have axial symmetry about z—axis.
That is, the specific intensity is cp independent.
Now then d S d z . 3 - r - = ■=- • - j - =■ c o s 8 ^ ds dz ds oz
,, „ „. (1.3.2)
We d e f i n e o p t i c a l depth r by dr = — a ( z ) d z .
(1.3.3.)
CO
Hence the optical depth T =
a(z)dz,
measured from the boundary inward.
(1.3.4)
Then the standard form of
radiative transfer equation in plane parallel axially symmetric system in the grey case can be written as
l ft ^ —M AT where p = cos 0.
^ = I
(1.3.5)
20 In general, two types of problems are considered.
One involves
semi—infinite atmospheres bounded at r = 0 and extends to infinity (T —> c°) in the other direction.
The other is the case of finite
atmosphere bounded by r = 0 and T = T.. . A class of problems in which one can have a very satisfying mathematical solution is that in a semi—infinite plane—parallel medium with constant net flux and no external radiation incident at the boundary.
Scattering
atmosphere :
We consider the case of non—absorbing, non—emitting medium where we suppose that the interaction of radiation with matter takes place only by way of coherent scattering.
We also assume that the source
function and specific intensity exhibit axial symmetry about the z—axis. The equation of radiative transfer for diffuse radiation in the grey case can be written as dl(r,/i) ' 'l = I(r,M) - | | P°(/i>/i')I(r,M')dAl', M - dr . ^
(1.3.6)
1 r^n P°(A»./*') = ■?p(/J,
(1.3.7)
where
In the isotropic case, the phase function
p°(/i,/i') = 1,
(1.3.8)
and the equation takes the simple form
^
dr
= K T . A O - 9(T),
(1.3.9)
21
i r +1 where 9(r) - J(r) = j
I
I(r,/t)d/i.
Local themodynamic equilibrium
and Milne problem :
We have also the physically different but mathematically analogous case of a medium in local thermodynamic equilibrium or LTE as it is generally termed.
In this,as first pointed by Eddington, one
can assign a temperature T at each point of the medium.
The
properties of the medium in the immediate neighbourhood of the point is similar to that confined to an ideal adiabatic enclosure at temperature T.
Then the elementary portion of the medium at that
point is said to be in LTE at temperature T and in this case absorption coefficient k(r,i>) and emission coefficient e(r,y) are connected by Kirchoff's law, e(v,v)
= k(r,v)B (T),
(1.3.10)
where B (T), the Planck function is given by
VT) =^-^7kr-T ' c
e
(1 3
- "n)
-1
In plane—parallel media in LTE each plane—parallel layer of the medium can be assigned a temperature T and the source function from (1.3.10) is Planck function B (T). The equation of radiative transfer in LTE in the non—grey case can be written as p dI(r,M,„) _
I(TtMjy)
-B^T),
(1.3.12)
It is a first order differential equation with v and p. as parameters.
Liaison
condition
:
It is often assumed that strict radiative equilibrium exists in the medium.
This implies that the integrated energy of radiation is
conserved and it is simply transferred from layer to layer.
22 00
J
00
a{r
o
,V)J(T
,v)&v
=
[ J
B
o "
i r where t h e mean i n t e n s i t y J ( T , I / ) ■■ = ^ 1
(T)a(T,i/)di/,
+1
I(T,/*,i/)d/j.
(1.3.13)
(1.3.14)
The liaison condition (1.3.13) and transfer equation (1.3.12) imply that flux F given by I(T ,fi,v)fidn
F = 2 J
= constant.
(1.3.15)
-l
For g r e y medium, CO
CO
I ( T , / X ) = [ I(T,/i,i/)di/ J 0
and B ( r ) = [ B (T)di/. J 0 "
(1.3.16)
In the grey medium the equation of transfer reads as H d I ( ^ ; M ) - Kr.M) ~ B(r)
(1.3.17)
and the liaison condition as J(r) = B(r).
(1.3.18)
The problem described by (1.3.11) — (1.3.18) (or its purely scattering analogue) was first initiated by Milne to explore and analyse the solar spectrum.
It is known as Milne problem when the boundary
conditions under which it is solved are (a)
absence of any external incident radiation at r = 0 1(0,-fi)
= 0 for 0 < n & 1,
(1.3.19)
and and (b) (b)
absence of singularity as r —> «J absence of singularity as T —> =o B(r)e~TT —> 0 as r —> m B(r)e~ —> 0 as T —> «
(1.3.20) (1.3.20)
23
Milne problem is characterised (a)
by the following
properties
:
The medium concerned for which the radiative properties are to be studied must be in strict radiative equilibrium
(b)
The state of local thermodynamic equilibrium is supposed to prevail.
The source function 3(T ,v) = B (T), the Planck
function. Formal solution of the Milne type of equation for the semi—infinite, plane—parallel,grey medium in LTE (or for azimuth independent isotropic scattering medium) is CO
iI+
= [
II_ ((T,-H) r,-/i)
= [ ?(t)e
; 0 < /a < 1 ; 0 < n < 1
(1.3.21) (1.3.21)
1. ;; 00 << n/i << 1.
(1.3.22) (1.3.22)
?(t)e_(t_T)/M —
and JJ
o0
(r
t)/fi
—
M M
For purely scattering atmosphere
i r +1 >(r) = J(r) = i J I(r>/i)dp. For medium in local thermodynamic equilibrium * ( 0 = B(r). The radiative transfer equations stated in this article are to be solved for specific intensity and source function under appropriate boundary conditions.
In fact, the mathematical solutions of radiative
transfer problems in participating medium of plane parallel geometry, are in a very advanced state of development.
Excellent accounts of
these methods are given in the books of Chandrasekhar, Kourganoff, Sobolev, Busbridge, Mihalas, Ozisik, Viskanta, Davison and Sykes, Case and Zweifel.
24 In this book we are mainly interested in radiative transfer problems in participating media in spherical and cylindrical geometries.
(b)
We shall principally focus our attention on them.
Spherical
medium :
Certain problems like the illumination of a medium by a central Isotropic point source or a uniformly emitting core occur in the context of radiative transfer in stellar atmospheres or in neutron transport in spherical reactors.
These involve the study of spherical
atmospheres with axial symmetric source functions. In this section, we establish the equation of transfer in an integro—differential form for spherically symmetric medium.
To
express the directional derivatives -=— in the transfer equation in dscoordinates we use the spherical terms of partial derivatives of space terms of partial derivatives of space coordinates we use the spherical polar coordinate system. In a radiation field with axial symmetry, the specific intensity of radiation at any point depends on the radius vector of the point from the centre and the angle between the direction of the beam s and the direction of the radius vector r where the radiation field is supposed to be azimuth independent [cf Fig. 1.3.2]
Fig.
1.3.2
25 The specific intensity for frequency integrated radiation can be written as I(r,s) = l(r,0).
dI(£
'g) ds
Then
dl(r,fl) ds
31 dr Jr ' ds
SI d£ 3J ' ds '
dr = ds cos 8, rdfl = - ds sin 8.
Now
„. l
"
;
(1.3.24)
„, . , dl(r,0) . 3I(r,0) . . SI(r,0) These imply — ' = cos 8 — ,' - sin 8 — v ' . ds Sr 99
,n Q „c. (1.3.25)
Writing cos 8 = \i and the specific intensity as I = I(r,/x), we have dI
ds
=
gl . d£ Sr ds
SI . dM S/i ds
(1 3 26) '"'
u -
and d/i
3/i
ds
ae
A9 ds
- sin 6
sin S 1 r
2 sin $ r
r
(1.3.271
The equation of radiative transfer for diffuse radition in spherically symmetric medium can be written as M
ai(r^o
+
i z Z ai(r^o _ _ Q(r)
[I(r/i)
_,(r)]>
3. 28) (1 ( 1 3.28)
where the source function ?(r) for a coherent scattering, frequency independent radiation field in a participating medium could be written as . . +1 f(r) - 2£± J I(r,/i')p(/i',M)d^' + BQ(r) + B 1 (r).
(1.3.29)
Here u(r) , the albedo for single scattering = <j(r)/a(r) , 0 < a> < 1.
26 B (r) : Contribution to the source function from radiation reduced o from any incident flux at the bounding surface or surfaces. B 1 (r) : Contribution from internal sources other than scattering. In case the source function is frequency dependent, the distribution function R(i/',i/) defined in (1.1.16) is used [cf. Hummer
].
The radiative transfer equation in this case reads as
4 - Kr./i.i/) + or
1
2 ~ ** |- I(r,/*,
The attenuation coefficient, a(r,i/) is usually considered separable as a(r,i/) = a(r)0(i/),
(1.3.31)
where <^(") . the absorption profile, is defined as
IoR CO
R(»/' ,i/)di/' =
J
tf(v)
(1.3.32)
0
The source function ^(r,i/) comprises of two parts, one arising from diffuse radiation field and other from given distribution of sources internal or external other than scattering.
The diffuse
radiation field is the contribution to radiation beam of frequency v and direction \i from radiation in other frequencies and other directions. For the frequency dependent case, the source function is given by so
*
J
+1
R (
^)
} d
"' J
Kr^/Jdn + g M ,
(1.3.33)
where g(r,//) is the contribution from the source internal or external other than scattering.
27
(c)
Anisotropic
scattering
in spherical
medium :
In studying transfer problems in planetary atmospheres, one has to consider a spherical shell medium illuminated by uniform solar parallel beam.
The problem no longer retains spherical symmetry.
To
determine the basic equations for this case, we follow a scheme sugested by Sobolev (16) which is fairly general. An anisotropically scattering spherical atmosphere surrounding a planet is considered.
The radiation field is assumed grey.
Fig. 1.3.3
Let the coordinates of a point P in the atmosphere of the planet be denoted by (r,ip). angles (6,ip)
The direction of radiation at P is identified by
6 is the angle between the radius vector at the point
and the direction of radiation and tp, the azimuthal angle in a horizontal plane. The specific intensity, I = I(r ,tp,6 ,
(1.3.34)
Hence the space derivative of specific intensity can be written as dI(r,V»,g,
Slctyai^dflai^d^ dip ds 88 ds 8
(1.3.35)
in the general form of transfer equation
g--[I-*]. as
(1.3.36)
28 Here [cf. Sobolev (16) ,
equation (11.27) p.219]
dr „ d6 sin 8 -r- = cos 8,-7- = — ds ds r (1.3.37)
dtp _ s i n 8 cos
c o t \j> s i n 8 s i n
The equation of radiative transfer can be written explictly as , 31 31 sin 8 cos r 3(9 3r (1.3.38) where the source function 9(T,$",8 V{T,f;8,
,tp) can be expressed as
= ^i^\ dtpI(r,y>;0,
+ B (r,tf;0,
where p(7') is the phase function for scattering between (9',
and
(«,*>). (1.3.40)
and
cos 7' «= cosflcosfl' + sinflsinfl ' cos((p — cp' ) .
B (r,ip;8,
arises out of the scattering of direct solar radiation and
is given by
Bo(r,y>;0)ip) = ^ p - F i P ( 7 ) e
(1.3.41)
where cos 7 = - cos0cosi/> + sin0sinV>cos
(1.3.42)
JTF. is the flux of solar radiation through an area perpendicular to the direction towards the Sun and T is the optical distance along a ray from the Sun to the given point of the atmosphere.
Let z be the
distance along such a ray taking z = 0 at ip = -= , the terminator co
T(r,V) = |
CO
a(r')dz'=J z
a[ ( r 2 sin2V> + z ' V rcosV>
/ 2
]dz' .
(1.3.43)
29 This equation is valid both for V > ? and ij> <, -z. The radiative transfer equation (1.3.38) - (1.3.43) is solved under the appropriate boundary conditions.
Transfer
equation
in cylindrical
medium :
In heat transfer and neutron transport, we meet with a number of problems where cylindrical symmetry exists and it is advantageous to use the radiative transfer or transport equations in cylindrical coordinates. We consider a cylindrical coordinate system with z—axis as the axis of the cylinder and assume that the specific intensity does not A
vary in the z—direction.
Let s denote the direction of the pencil of
radiation at a point P, radial distance r from the z-axis inclined to it at an angle $ and azimuth angle
s is not
necessarily on the vertical plane through P. We choose an element of length ds from P to Q along the direction of the pencil of radiation and project ds on the horizontal plane through P normal to z—axis. The projected point Q' of Q is located at the radial distance r + dr from the z—axis and azimuth angle cp + dtp. the coordinates.
The Fig. 1.3.4 illustrates
The cylindrical symmetry requires that intensity of
radiation is unchanged due to rotation about the z—axis.
A change in
ds will result in a change in dr and dip, but the angle 0 will remain the same.
Fig.
1.3.4
30 d_ ds
d_ =
3r
>
dr
3_ #
ds
Scp
(1.3.44)
ds
From Fig. 1.3.4, we have
dr
(1.3.45)
cos cp,
ds sin
rdy ds sin 9
(1.3.46)
sin cp,
and I = l(r,0,
sin J
cos
a(r)[*(r) -
3I(r,g ,p) dr
sin
I(r,9,
where
*(r)
"(r)
^
Lit
[
_7T
;
I(r,«',?»•)sin
9'd9'dtp'
+ B(r) .
(1.3.48)
'
4 I ( r ' " » + ^fel
?
- V
- a(r)
(r) - I(r,r/,M) (1.3.49)
where
*(r) -5?g) JJi(r,,'.M') ^1^1 + B(r). Jl-M'
(1.3.50)
2
These equations of transfer are solved under appropriate boundary conditions.
1.4. Boundary conditions In solving radiative transfer equations in plane parallel, spherical and cylindrical geometry, it is necessary to spell out the appropriate forms of boundary conditions for every special problem.
31 Attempt is also made to develop techniques to accommodate them exactly in any mathematical models designed to solve them.
Boundary value
problems of varying complexity arise as a result.
A good number of
them do not yield exact solutions.
Approximate solutions of
integro—differential equations of transfer have been proposed and often equivalent boundary conditions were used due to inability of the mathematical techniques to take care of exact boundary conditions. In what follows, we state some common boundary conditions used in radiative transfer, heat transfer and neutron transport problems in different geometries.
Though our main interest is transfer and
transport problems in spherical and cylindrical geometry, for completeness we have mentioned some familiar boundary conditions used in plane—parallel geometry, all of whose equivalents have not yet been explored in spherical and cylindrical geometry.
Boundary conditions
for plane-parallel
medium :
In the case of plane-parallel system distinction is usually made between two cases : (i)
The semi—infinite atmosphere bounded on one side by r = 0 and extending on the other side to r —> <=,
and
(ii) the finite atmosphere bounded at r = 0 and T = T .
For semi—infinite, plane—parallel, grey medium, if no radiation is incident on the outer surface r = 0 from outside, the boundary conditions could be stated as follows. I_(0,-/0 = 0 , 0 < n < 1
(1.4.1)
(b) ?(T)e~r —> 0 as r —> ».
(1.4.2)
(a)
(a) suggests that there is no inward radiation at the outer surface and (b) ensures the convergence of intensities.
For scattering
atmosphere ^ ( T ) = J(r) and for medium in LTE ? ( T ) = B(r). medium, the boundary condition can be written as
For finite
32
I(T,n,v)\T=Q
= I_(0,-M,i/), 0 < fi < 1.
I(r,/*,i>) , _
= I AT
T= T
(1.4.3)
+II,U). O
+
O
(1.4.3) implies that external incident radiations at r = 0 and r -> f are given.
Fig.
1.4.1
We consider some special cases. A)
Transparent
boundaries
:
By this we mean that the space surrounding the plane parallel slab does not interact with radiation. boundary we have vacuum.
That is, adjacent to the
In this case, the external incident
radiation at the bounding surfaces are given functions of n and u and are of azimuthal symmetry.
I_(0,~H,u)
= f^.f), ►
I"(T O ,+M.»0
0 < fi < 1.
(1.4.4)
= f20*.»0.
A special case of transparent boundaries is one in which f.. and f„ are constants.
33
B)
Black boundaries
:
In this the bounding surfaces T = 0 and r = r constant temperatures T.. and T„ respectively.
are maintained at
The boundary conditions
can be written as I°(0,-M,i/) = Bi/(T1), for 11 > 0,
>
(1.4.5)
I+(ro,/i,i/) = B v (T 2 ), for p > 0, where B (T ) and B (T ) are Planck functions.
C)
Opaque, diffusely
emitting
and reflecting
boundaries
:
Let T, and T. be temperatures, e, and e0 , the spectral 2 li> 2i/ hemispherical emissivities and p. , p0 , the spectral hemispherical diffuse reflectivities at r = 0 and T = T
respectively.
Then the
intensity of radiation I_(0,—n) for Q < ft < 1 leaving the surface r = 0 in the negative fi direction -2* -27T |
I > , - ^ ) - <1A(T1)
+
p\v
°
11 I l+(.0,n>
° !
I
J
J
I
= e l l / B v
,u)n'
0
.. i J . . * J . _ . f M'd^'dp'
0
I+(0tp',i»)/*'dj»'.
(1.4.6)
Similarly d I°(r ,/*,!/) = e_ B (T,) + 2p,
r1
I (0,-M',./)/*'^'.
(1.4.7)
The boundary conditions (1.4.5) - (1.4.7) are relevant to the problems of heat transfer. In the context of neutron transport in plane-parallel geometry, free surface boundary condition is mathematically identical with that in radiative transfer which can be stated as
34 A o , - f O = F(/i), 0 < fi < 1,
(1.4.8)
where ^,o,(0,—/i) , the neutron angular density at the free surface in the inward direction is equal to a given incident distribution of neutrons.
Boundary conditions
in spherical
medium :
In radiative transfer problems in spherical geometry one generally deals with spherical media with point source, with spherical shells or spherical media with absorbing and emitting cores etc.
In
all these cases statements of boundary conditions are fairly straight forward.
The relevant transfer equations for these problems are of
the form (1.3.28) and (1.3.30).
Fig. 1.4.2
(A)
A sphere of finite
radius with a point
source of energy :
The boundary condition that is generally used is the absence of angular distribution of intensity at free surface. and I (r,—fi,v)
Writing I (r,+/i,i/)
to be outward and inward intensites, for a sphere of
radius R with a point source, the boundary condition at free surface is given by I_(R,-/i,»/) = 0 for 0 < /i < 1.
(1.4.9)
In any mathematical scheme of solutions of these problems care must be taken to avoid singularity at r = 0.
35 (B)
In a spherical shell medium bounded by surfaces of radius r.. and
R(0 < r.. < R) , the transfer equation for diffuse radiation in spherical geometry are generally solved for source function and intensity under the boundary conditions
I(BL,-H,v) - 0 • I(r1,+M,^) =
(1.4.10)
ylix^-n.v)
for 0 < i/ < =>, 0 < jj. < 1, 0 < 7 < 1. Note
7 = 0 = > perfectly absorbing core
and
7 = 1 ==> perfectly reflecting core.
It may, however, be mentioned that the source function in the transfer equation may contain contributions from internal and external sources (other than scattering) in addition to that from diffuse radiation.
(1.4.10) is the boundary condition for diffuse radiation.
The total intensity at any point is calculated taking into account the contribution from all sources.
Boundary conditions
for cylindrical
medium :
The transfer problems in cylindrical geometry are important in the study of heat transfer and neutron transport. For a homogeneous, isotropically scattering, infinite cylindrical medium of radius R, with an incident flux in a specific direction at the boundary surface, the transfer equation for diffuse radiation is written as in (1.3.49).
They are usually solved under the boundary
condition I(R,77,-AO = 0, 0 < n S
1, -1 < rj <
1,
(1.4.11)
where cos 0 = f? and cos
The boundary condition (1.4.11) implies the
absence of diffuse radiation at the outer boundary surface.
36 For a cylindrical shell medium bounded by r^ and R (0 < r^ < R ) , which may be inhomogeneous, non-coherent scattering, one can write down the equation of transfer for diffuse radiation.
A sample of boundary
conditions under which the transfer equation is solved can be written as I(R,»7,-M) = 0, 0 < ft < 1, -1 < v ^ 1,
(1.4.12)
I(tvri,tt)
(1.4.13)
= yl(r1,v,-ti),
0 < p < 1, -1 < r, < 1,
0 < 7 < 1. (1.4.12) implies absence of incident external radiation at the surface of the outer shell and (1.4.13) gives the fractional reflection at the surface of a inner shell. For heat transfer problems in spherical and cylindrical medium, one may have to deal with more involved boundary conditions.
Presence
of opaque, diffuesly reflecting, diffusely emitting or specularly reflecting boundary surfaces cannot be ruled out.
They can usually be
accommodated within judicious mathematical formulation of the problems. 1.5. Integral equations for source function It is sometimes convenient to deal with transfer and transport equations in integral form.
In most cases they lead to Fredholm type
of integral equations with kernels of varying complexity. methods have been developed to solve them.
Numerous
In plane geometry, the
various methods have been very well documented in the books of Kourganoff, Busbridge, Case, Sobolev and others.
However, in
spherical and cylindrical geometry use of equation of transfer in integral equation form is rare.
But attempts that have been made to
solve them are rather novel in character. Chapter VI of this book.
They will be dealt with in
We shall derive here the integral equations
for source function in spherical and cylindrical geometry.
For
completeness, we shall deduce the integral equation for slab geometry as well.
37 For simplicity we consider, a homogeneous grey medium with isotropic scattering.
Let B ^ r ) be the internal source present,
B Q (r), the contribution from external radiation to the source function, and let B(r) = Bn(r) + B (r). 1 o
Fig. 1.5.1
The intensity of diffuse radiation at the point r in a direction s can be found by a formal integration of equation (1.2.15) which in this simplified form leads to s I(r,s) =
-a(|r-r'|) *(r')e
a ds',
(1.5.1)
'0 where the boundary condition used is I(R,s) - 0,
(1.5.2)
a being attenuation coefficient. The path of integration is along a ray extending from any point r inside the medium to a point R at the bounding surface along s, the point r = R being s = 0 [cf. Fig. 1.5.1] The source function ?(r) for this model can be written as
*(r) = 4^ I(r,s')dtr + B(r).
(1.5.3)
38 <j is the albedo for single scattering, o From (1.5.3) §t(r) = u J(r) + B(r),
(1-5.4)
where the mean intensity, J(r) = 7— 9 I(r,s')dfJ'. Now inserting (1.5.1) in (1.5.3) and remembering that
(|r - r' |)2ds'dTi = dv'
(1.5.5)
is the volume element at the point r' and that the integration over s' in all directions is equivalent to an integration over the whole volume of the medium, we obtain
*(r) = ^ | ? ( r ' ) exp[- a(|r - r'|)]
°
g
+ B(r).
(1.5.6)
I r — r' I This is a general form of the integral equation for the source function.
Integral
equation
of transfer
in plane-parallel
medium :
Let us consider a plane—parallel medium of thickness T axially symmetric diffuse radiation field.
having
A flux of radiation 7rF. is
supposed to be incident on either of the bounding surfaces r = 0 and T = r1 at an angle cos
/i
to the outward drawn normal.
distribution of internal sources may also be present.
A
For simplicity
we take the medium to be homogeneous and isotropic scattering. Referring to Fig. 1.3.1 and remembering equations (1.3.3) — (1.3.5) we can write o|r - r'j = a|s — s'| = \r — T' \/fi.
(1.5.7)
It is convenient to distinguish between the reduced incident radiation due to 7rF. and the diffused radiation field.
The specific intensity
I(T,^) characterises the diffused radiation field.
39 1
dn - 2TT Ir . . . d/i, and from ( 1 . 3 . 2 1 ) and ( 1 . 3 . 2 2 ) f o r 0
Now
0 < T < T1 ,
Kr
rri ,M) = I + ( T , / I ) = J
*
-(t-r)//i dt > 0 < u < 1.
►
: ( r . - / i ) = I _ ( T . - M ) - [ ?(t)e" K J 0
-(r-t)//x dt A*
We can w r i t e
c
i r+1 J(r) = | J
i r r1 I(T,/i)d/i = |
J
_ 1 2
1
J
r1 I(r,ju)dp + J
5 ( t ) e - ( t - T ) / ' i d t *f + r V J0
ii
J0
I(r,-M)d/i
»(t)e- (r-t)//i
i r1^f1»
J
0
]i
d(_
d/i d/j
(1.5.8)
JQ
Inverting the order of integrations we have
J(T) = [
1
?(r')dT'
I
1
e
_,
|T
r
\/fi
d/j
A»
(1.5.9)
Putting /i = -
J(r) - f S c O d , . f §5 e~x( I " ' I > /I J
where the exponential
0
* ( r ' ) E (|r -
r'|)dr\
(1.5.10)
integral CO
E-^x) = j
exp(-xu) - £ .
(1.5.11)
40 From (1.5.4) »(,)__£[ Z
J
*(r') E <|r - r'|)dr' + B(r),
(1.5.12)
0
where B(r) = B (r) + B..(T). o 1
(1.5.13)
B (T) is the part of the source function contributed by the reduced incident radiation. co F. —T/H B (r) = -5-i e ° o 4
(1.5.14)
if the reduced incident radiation is due to net flux nF
incident on
the upper surface T = 0, whereas it is u> F B (O = - ^ e o 4
-(T 1
-r)/u ° ,
(1.5.15)
if it is incident at the surface r = T.. B..(T) is the contribution to source function from internal sources, if any. In the case of semi—infinite plane—parallel medium, the integral equation for source function takes the form U>
?(T) =
eo
T~ J
9(T,) E ( f
l l ~ r ' | ) d T ' + »( T ).
(1.5.16)
with u F.
o I B(r) = B 1 ( 0 + - ~ e
Integral
equation for radiative
—T/U
transfer
' 'o "
in spherical
(1.5.17)
geometry :
The starting point of our consideration is again the general form of integral equation for source function derived in equation (1.5.6).
We express the quantites involved in (1.5.6) in spherical
polar coordinates (r,0,cp).
We assume axial symmetry of the radiation
field unless otherwise mentioned.
Then [cf. Fig. 1.5.1]
41
|r - r'| - |s - s'| = (r2 + r'2 - 2rr'cos B')1,
(1.5.18)
dv' = r'2sin0'dr'd0'd
(1.5.19)
Then the integral equation for source function can be written after integration over
W
o
(r)
=—
R
,2_, .. , ,f r' ?(r')adr'
0
J
f J
2
0
2
2
exp[-a(r + r' - 2rr'cos 0') ]sin B' dB' — ^ * = = '—1 r + r' - 2rr'cos B' + B(r).
(1.5.20)
Now changing the variable B' to u through the relation r 2 + r'2 - 2rr'cos B' - u 2 ,
(1.5.21)
we obtain u_(r) ^R r _, i ,.|r+r'| _-au (r) - — — | f*(r»)adr' ' ^ — du + B(r) . 2
J
0 Ir
J
J
|r-r'|
u
(1.5.22)
Now the exponential integral _ , .
r00 -xt dt
r°° -t dt dt
(1.5.23)
Hence the integral equation for source function in homogeneous (a and a; are constants) spherical medium can be written as o r?(r) - -^- [ r'?(r')k(r,r')dr' + rB(r),
where [cf. Cuperman, Engelmann and Oxenius
(1.5.24)
, p.Ill equation (3.13)]
k(r.r') - a[E1(a|r - r'|) - ^(ajr + r'|)] and B(r) - B (r) + B (r).
(1.5.25)
42 B (r) is the contribution to source function from reduced incident o radiation and B.(r) is that from internal sources other than scattering. The piart of the source function BQ(r) arising from inci dent flux l
at the outer boundary r = R inciident at an angl e cos-1
to the
inward normal could be calculated
Fig.
1.5.2
At r the angle between the direction of flux and inward drawn normal -1 * IS
COS
fl
Thus
2 2 2 *2 R*<1 - //) - r (1 - /xo ),
(1.5.26)
or
v{-7"-rf
(1.5.27)
Since 7rF. is the incident flux per unit area normal to the direction of r
l
incidence, the flux incident per unit area of the surface at r = R is then TTF.U .
to The reduced incident flux at any point r is obtained by multiplying TTF.U by the reduction factor iro J R/x exp
I * a(r')ds' , r
where ru = s. o
"o
43 Then the illumination of the spherical surface at r due to the incident flux on the surface r = R is given by
2
f
*
(4?rr )2n-
*
R/x
2
I ^5(/i - /i )dji = n nF. (4TTR
a(r')ds'
)exp r
^,(1.5.28)
assuming axial symmetry of the radiation field. The reduced incident intensity at r is given by
* * Io (r.-uo ) =
. -* ] [r j r
exp
[ - J * Q(r )ds
o
(1.5.29)
ru *>„
Then the contribution to source function from the reduced incident radiation B (r) is given by
B
o(r) "
+1 * * w(r) r ^ ! I0(r,M')5(M0+M')dM'
u(r)F. r M I I o
- - 4 —
[TF o
R
HIT
exp
^r,
[-J >
(1.5.30)
)ds'
r
M„
Furthermore, if o> and a are constants, that is when the medium is homogeneous,
ti>F.
r
V > - -IT
[cf.
p
o
*
R r
,2
exp [-(aBji
o
Leong and SerT
, p.472].
- ar/i ) ] , - 1 <, n
< 1
(1.5.31)
44 I f 0 < fi
< 1, we have
UJF. 2 "R " r) = - ^ B (r) rr Vo - -IT 4 (_ :
uF. +
-4
l
«
r
—
[I] 2
| exp
[^(RMQ -
r Mo ) ]
H(r - R Jl - /J; 2 )
* i o exp [—a (R/i + r/i )] H(r - R U ~* ^ o '
^[If(^)
" 2 - «
)
—aR/j
° 2 cosh (arji*) H(r - R \1 - M Q )
where H is the Heaviside unit step function given by 1 H(r - R \:
[cf. Kho and Sen
# -
(9)
ir^?
when r > R 41
(1.5.33) 0
■i 1 ——l*u when r < R 41
1971, page 225].
Bn(r) is the contribution to the source function from internal sources other than scattering. for every case.
The distribution of these sources are given
It is common to have axially symmetric distribution
of internal sources. In the above, we have considered the medium to be homogeneous and scattering to be isotropic just to avoid complexity in the deduction of basic integral equation for source function in spherical geometry. However, more complex cases will be considered and basic deductions made when we take up special problems in Chapter VI.
Integral
equation
for source function
in cylindrical
geometry
:
We consider an infinite cylindrical medium of radius R with a net flux TTF. incident on the bounding surface at an angle cos vertical and cos
rj with the
\x with the outward drawn normal to the surface.
The distribution of internal sources other than scattering may be given by B ^ r ) , r being the distance measured from the axis of the
45 cylinder.
We introduce the cylindrical coordinates (r,0,z) relative
to the centre of the horizontal section of symmetry of the cylinder.
Fig. 1.5.3 Then we have
and
dv' = r'dfl'dr'dz',
(1.5.34)
|r - r'l = Is - s'l = (r2 + r'2 - 2rr'cos 9' + z ' 2 ) 2 .
(1.5.35)
Now recall that the general form of the integral equation for source function is given by (1.5.6), i.e.
*(r) = fr ]
- d V ' 2 + B(r). Ir — r'l
(1.5.6)
Substituting (1.5.34) and (1.5.35) in (1.5.6), we obtain the integral equation of radiative transfer in cylindrical geometry as
?(r) = £- f r'?(r')o dr'f d$' 47T J Q JQ 1 ,2,2, r°° exp(-a(r + r' - 2rr'cos 9' + z'")"}dz x lJ — E ^ 2 2 0 r + r' - 2rr'cos tf' + z' 2
where B(r) = BQ(r) +
2
Z^r).
+ B
(1.5.36)
(1.5.37)
46 B (r) is the contribution to the source function due to reduced o incident radiation and B.. (r) that due to cylindrical distribution of internal sources, if any. 2 2 2 Now putting r + r' — 2rr'cos 8' = u ,
(1.5.38)
and u
(1.5.39)
+ z
= t ,
we obtain
?f(r) (i
,R = r'y(r')a dr "J0
' J.0 - I. J
e
dt
+ B(r).
(1.5.40)
v/72 ty t —u2
If further the cylindrical medium is homogeneous (a and u> are constants), the integral equation for transfer in cylindrical geometry can be written as
4T 7 ?(r')k(r,r')dr' + -[r B(r)
■fr ?(r) = u
(1.5.41)
where
t
1
2
k(r,r') - —
22
(rr')
T -°° f dfl' I K (auy)dy J p,
and K
J
T
(1.5.42)
O
is the modified Bessel function of the second kind.
[cf. ° Heaslet and Warming^(6) , equations (9), (10), (lib); [cf. Heaslet and warmingv
, equations (9), (10), (lib);
Also note that in (1.5.42) k(r,r') = ak(ar,ar') of (lib)]. Now remembering that TTK (ary)I (ar'y), 71
r' < r
K (Quy)dS' Jn
(1.5.43)
° 71 o(ar'y)Io(ary), ^K
r < r'
47 where IQ(x) and K (x) are the zeroth order modified Bessel functions of the first and second kinds respectively, we have 1 2 r°° a (rr') I K (ary)I (ar'y)dy, 2
•
k(r.r')
r' < r (1.5.44)
1 a 2 (rr') 2 J KQ(ar'y)Io(ary)dy,
[cf. Heaslet and Warming^
r < r' .
, equations (14), (15b)].
B (r) can be calculated for cylindrical medium following more or less analogous arguments as in spherical geometry for establishing (1.5.39) and (1.5.41).
B (r) for homogeneous cylindrical medium when
a uniform incident flux TTF. per unit area normal to its direction of incidence is incident at the outer surface in the direction (—6 ,—V> ) o o [cf. Fig. 1.5.4], [cos n and cos i/> = u ] o o o
Fig. 1.5.4 is given by
taF.
V
r)
- -IT [I](
arfi
aR/i
- /*
o exp p * "o '
2 cosh
—
1
Jl - ri
\l
;4 - r,
V-R\ ' - » : •
(1.5.45) i
-l
where the angles cos
y. , cos
reference to Fig. 1.5.2.
■%
/iQ, cos
i
»JQ can be understood with
48 Here
0 < n , fi* < 1, -1 < t) < 1, o o o
(1.5.46)
and
R2(l - n2) o
(1.5.47)
- r2(l - /i*2) , o
and H(r — R.J 1 — /i ) is the Heaviside unit step function. Incidentally it may be mentioned that sometimes it is convenient to use (r,0,
In that case, the source function ?(r) is given by
[cf. equation (1.3.50)]
»
dr? dM>
'
•Jl - M '
+B(r),
(1.5.48)
2
where I(r,/),/i) denotes the intensity at r along the direction 6 with the vertical and i/> with the outward radius [cf. Fig. 1.3.4]. rj = cos 6 and /i - cos ip.
?(r), a(r) and u(r) are source function,
attenuation coefficient and albedo for single scattering respectively. Let us consider a medium in the form of an infinite cylindrical shell bounded by surfaces of radii r1 and R(0 < r. < R ) .
We use the
boundary condition for diffuse radiation as
K R . ^ . - A O - 0, -1 < r, < 1, 0 < /i < 1;
(1.5.49)
I
(1.5.50)
-1 S r? < 1, 0 < n < 1;
where 0 < 7 < 1. (1.5.49) indicates the absence of diffuse inward radiation at r = R and (1.5.50) accounts for the fractional reflectivity of the surface at r — r. .
Let s denote the length along the direction of the pencil of radiation, s = 0 being at r = R. From the formal integration of equation of transfer under the boundary condition (1.5.49), we have
49 s s I(r,r),/i) = [ exp[- [ a(r")ds" ]?(r') ds'. J J 0 s'
(1.5.51)
Substitution of equation (1.5.51) in (1.5.48) and with some mathematical manipulation, we are led to the integral equation of transfer in cylindrical geometry in the following form [cf. Leong and (12) Senv , 1971, note in particular pages 249 - 253] ■Fc ?(r) <= A[Jr»(r>] + Fr B(r),
(1.5.52)
where A[Fr 5(r)] = ^^-
\
[k(r,r') + 7k (r ,r') ]*(r' )fr'dr'
(1.5.53)
with the kernels k(r,r') and k* (r,r') given by
I I
dr> k(r.r') = (rr') 2 f ( exp[-N(r ,r' ,
X
I
/
1
(1.5.54)
2
(r +r' -2rr'cos
and
i .1 2 k (r,r') = (rr') f
r,
^ d>?
f exp[-N*(r ,r' ,
dip' dip' ^ ■
x
(1.5.55)
/ 2 2 (r +r' —2rr'cos cp')
/< with N(r,r',
t
a(r")di",
(1.5.56)
and
* e
i
J
N*(r,r',
(1.5.57)
50 The limits of integration 2's
are the line elements arising from the
projection of the line elements of the pencil of radiation on the (12) horizontal plane. Leong and Sen (1971) deduced the integral equation for transfer in the cylindrical shell medium by dividing the horizontal section of the cylindrical shell into three separate zones [cf. Fig. 1.5.5] depending on the range of p.
I- 0
Fig. 1.5.5
A HORIZONTAL PROJECTION AB is the projection of pencil of radiation on a horizontal plane. i = 0 corresponds to s = 0, i and SL' corresponds to s and s' . Z
and
i.. refer to the projections in the region (3a) and (3b) respectively. The integration limit 4>, satisfies the relation r~2 2 ,2 -Jr + r' - 2rr'cos
(1.5.58)
The deduction of integral equation (1.5.53) with the equations (1.5.54) - (1.5.58) is rather involved.
Similar relations have been
deduced for spherical geometry in Chapter VI section 6.6, where we demonstrate the use of Pincherle - Goursat Kernel method for solving integral equation of transfer in curved geometry.
51 1.6. Neutron transport equations The phenomenon of neutron transport involves interaction between neutrons and atomic nuclei.
Although this process is absolutely
distinct from interaction between radiation and matter, they are both subject to similar mathematical treatment.
One group theory of neutron
transport for example, deals separately with fast neutrons, thermal neutrons etc.
They are described in terms of integrated quantities
integration being on energy groups.
In point of terminology, neutron
flux corresponds to the mean integrated intensity in radiative transfer except for possibly a multiplicative constant.
The "constant
cross—section approximation" corresponds to the grey case in radiative transfer.
Similarly some of the other notions used in neutron
transport can be tied down to analogous notions in the theory of radiative transfer. In plane—parallel medium, most of the methodologies used in radiative transfer could be extended to neutron transport problems with sometimes only cosmetic modifications.
Similarly most of the
mathematical techniques described in the book for solving radiative transfer problems are adaptable to the solution of neutron transport problems.
52 References 1.
Ambarzumian, V. A., Theoretical Astrophysics, Pergamon Press, London 1958.
2.
Busbridge, I. W., The Mathematics of Radiative transfer, Cambridge University Press, 1970.
3.
Case K. M and Zweifel P. F., Linear Transport Theory, Addison Wesley, 1967.
4.
Chandrasekhar, S., Radiative Transfer, Clarendon Press, Oxford, 1958; also Dover Publications Inc., New York, 1960.
5.
Cuperman, S., Engleman F. and Oxenius J., Physics of Fluids ^, 108, 1963.
6.
Heaslet, M. A. and Warming R. F., J. Quant. Spectro. Radiative Transfer 6, 751, 1966.
7.
Hopf, E., Mathematical problems of radiative equilibrium, Cambridge Tracts, no. 31, 1934.
8.
Hummer D. G., Mon. Not. Roy. Astro. Soc. 12^, 21, 1962 145, 95, 1969.
9.
Kho, T. H. and Sen K. K., Astrophys. Sp. Sc. 14, 223, 1971
10.
Kourganoff, V. and Busbridge, I. W., Basic methods in transfer
16, 151, 1972.
problems, Oxford University Press, 1952; also Dover Publications Inc., New York, 1963. 11.
Leong, T. K. and Sen K. K., Ann. Astrophys. 31, 467. 1968.
12.
Leong, T. K. and Sen K. K., Publ. Astro. Soc. Japan, 21, 167, 1969. ^2, 57. 1970. 23, 247, 1971.
13.
Wilson, S. J., Ph. D. Thesis, University of Singapore, 1966.
14.
Menguc, M. P. and Viskanta, R., J. Quant. Spectro. Radiative Transfer 29, 381, 1983.
15. 15.
Ozisik, M. N., Radiative transfer and interactions with conduction and convection, Wiley Interscience Publications, 1972.
16. 16.
Sobolev, V. V., Light scattering in planetary atmospheres, Pergamon Press, London, 1975.
53 CHAPTER II MATHEMATICAL PRELIMINARIES
2.1
BASIC METHODS OF SOLVING INTEGRAL EQUATION OF TRANSFER IN SPHERICAL AND CYLINDRICAL GEOMETRIES In the first chapter, we have stated the basic physical notions
of radiative transfer.
We established the transfer and transport
equations in the integro—differential and integral equation forms in different geometries.
Some of the boundary conditions used commonly
in different models have also been listed.
The use of the transfer
equations along with the appropriate boundary conditions will be demonstrated in Part II of the book. In the present chapter, we shall introduce the fundamental basis of the mathematical methods employed to solve the transfer and transp problems in curved geometry.
To discuss the different available methods, let us look at a simple model of a homogeneous spherical medium scattering isotropically with spherically symmetric internal source B(p). If ?(p) is the source function, then we know that the integral equation satisfied by ? is given by
P9(p)
= pB(p)
» ."2 + -f [ P'Z(P') J 0
(E (|p - p'\)
- E (p +
p'))dp-. (2.1.1)
where E.. (x) is the exponential integral given by „ / v E 1 (x)
and u
x t dt rr°° -- x t dt - ; ■ -
is the albedo of single scattering. o Unlike most problems in spherical medium, this simple problem can be
54 solved exactly by first converting it to an equivalent transfer problem in a plane medium and invoking all the standard results of plane geometry.
Hence this problem can also be the basis to test
approximate methods proposed to solve the more complex spherical medium problems. To convert this to an equivalent plane problem we rewrite equation (2.1.1) as follows :
pV{p) = pB(p) + -| J
p'?(p')E1(|p-p'|)dp'
-P-.
- -f f c
J
p'Sf(-p')E |p - p'|dp'
0
Since the source function "9{p) was defined only for p > 0, we can extend its definition so that S(p) = ?(-p), p < 0.
T
o " >2
-
rQ - p,
we have [cf. Heaslet and Warming
If we now set
F(p)/B(p) = 0(r),
]. IT
a> o 0(r) - T - To + -I [ 0(r')E,(IT 2 JQ I I - r'I)dr'. This is the integral equation satisfied by the source function for a plane medium of thickness 2T . Thus we can utilize all the techniques for the plane case.
In particular we can obtain the values of the
source function at the surface of the sphere using Chandrasekhar — Ambarzumian functions. We have [cf. Heaslet and Warming (6).
| r\^x fi(2To)
- fij + | ro(«2 - p2)
+
\ («3 - ^ ) ,
' 'o^ 1
Tt".-',»-r i - id
( Q
i-^ o
WQ
_ 1,
55 where .1 _ a n = | X(/*,2r X ( M , 2 r )/i%, o)M%,
^ B
,1 = | Y(/i, 2r Q )/i%, =
X, Y being the Chandrasekhar — Ambarzumian functions, [cf. Chandrasekhar
] The above analysis was possible as the kernel
of the integral equation took a special form in the homogeneous case. If the attenuation coefficient a(r) is not a constant, then this simple reduction to the plane case is not possible and we have to solve an integral equation of the type W
K.
r£(r) = rB(r) + -^ [ k(r,r')r'5(r')dr'
(2.1.2)
We shall now indicate the different approaches adopted to solve integral equations of this type.
These integral equations belong to
the general class called the Fredholm type of the second kind and take the form y(r) - g(r) + A
k(r,r')y(r')dr'. J
(a)
Neumann series
solution
:
By introducing the m—iterated kernel k (T,T')
k (T.T') m
=
(2.1.3)
a
by the equation
rb k(r,t)k ,(t,T')dt, m > 1 J . m—1 a
k^r.r') = k(T.T'),
the solution of equation (2.1.3) can be written as 00
y(r) - g(r) +
where Am{g(r)) = 1 k^r
Y A m Am{g(r)}. m=l
, r ' )g(r ' )dr ' ,
(2.1.4)
56 .b „b j | |k(r, r' ).| drdr' < 1
provided
[see Miklin (15) , p.13] In our context, we shall show that the kernels satisfy the inequality |AL{X}| < 1 and hence the requirement for the infinite series representation of the solution.
(b)
Degenerate kernels
(Pincherle
Goursat kernels)
:
If the kernel in equation (2.1.3) is an L„—kernel i.e.
. .
rb rb 2
norm k = ||k|| =
a
a
k (r,r')dTdr'
< =>,
then it is known that we can decompose the kernel (in a non—unique way) as follows. k(r,T') = k(r,T') + T(T,T-')
where
k(r,r') =
N V 5C (r)Y, (r') k k=l k
is degenerate and the norm |T| can be made arbitrarily small.
As the
norm of T is small, the Neumann series solution for the kernel T converges fast.
In view of this we first obtain a solution of the
integral equation with kernel T(r,r') and then use standard methods to obtain the solution of the integral equation with degenerate kernels. The details follow. The basic equation (2.1.3) can be written as rb y(r) = G(r) + A J T(r,r')y(r')dr'
(2.1.5)
57 D r
with
G(r) = g(r) + A J
_ k(r,r')y(r')dr'.
a
If r(r.r') is the resolvent kernel of equation (2.1.5) then by definition its solution is b
y ( 0 = G(r) + A
r
r(r,r')G(r')dr', a
= g(r) + J
r(T.T') (g(r') + A J
i.e.
k(r',r")y(r")dr"; df'
* r y ( 0 - g (r) + A J
[
* Xk(r)Yk(r")y(r")dr"
(2.1.6)
with
g (r) = g(r) + A J" r(r,r')g(r')dr', and .b X ^ r ) = J T(r,r')Xk(r')dr'
The solution of the integral equation (2.1.6) with degenerate kernels can be expressed as [cf. Miklin (15) p.19]
£
y(r) - g (r) + [ ^ k \ (r) ' k=l
where £, = A
Y, (r)y(r)dr and satisfies the algebraic equations
*h-\L W k ^ h ' h-1,2,...• ,N k=l
58
with
a^-J
xJ(r>Yh(r>dr.
and b
h = I V r ) g* ( T ) d r a
The resolvent
kernel
and the iterated
kernels
:
If r(r,r') is the resolvent kernel of the integral equation (2.1.3) then it's solution can be written as b r
y(r) = g(r) + A
r(r,r')g(r')dr'.
(2.1.7)
J
a But from equation (2.1.4) we also have But from equation (2.1.4) we also have n
co
y ( 0 = g(r) +
Am J
[
km(r,r')g(r')dr';
-1 a
in'--l
where k (T,T') are the iterated kernels, m Assuming that the series is uniformly convergent, we obtain, by interchanging the order and comparing
T(T.T')
=
A"
f
Li-
m=l
1 -
^ (T,T') m
.
From this we can deduce that the resolvent kernel satisfies the following integral equation :
r(r.r') = k(r.r') + A
rb J
k(r,t)T(t,r')dt,
a
on using the relation on using the relation k (T,T') = m j
k(r,t)k (t.T')dt. m— 1
(2.1.8)
59
(c)
Reduction
to Cauchy initial
value problem [Goldberg
, p.20U] :
Let us denote the solution of the equation (2.1.3) by y(r;A) and its resolvent kernel by r(r,r';A) to emphasize its dependence on the parameter A.
From equation (2.1.8) we know that the resolvent kernel
satisfies the equation
r(r,r';A) = k(T.r') + A
k(r,t)T(t,T';A)dt.
(2.1.9)
Differentiating both sides with respect to A, we have
r (r,T';A) = A
rb J
k(r,t)r(t,r';A)dt
a
+ A
rb
k(r,t)T (t,r';A)dt.
Regarding this as an integral equation for the function T (T,T';A) with the same kernel k(r,t), its solution can be expressed in terms of the resolvent kernel r(r,r';A) as follows :
I\(r,r';A)=J
+A
k(r,t)r(t,T';A)dt
b b I r(r,t;A){[ k(t,t')r(t',T-;A)dt'}dt.
Using equation (2.1.9) this becomes „b r x (r,r',A)-[ r(r,t;A)r(t,T';A)dt.
(2.1.10)
Also from equation (2.1.9) when A = 0, we have r(T,r';0) = k(r,r').
(2.1.11)
60 Thus we obtain a Cauchy integro—differential system to solve for the resolvent kernel.
Following a similar procedure, we can obtain a
Cauchy integro-differential system for y(r;A).
r
We obtain
b
yA(r;A) = j
r(r,t;A)y(t;A)dt,
(2.1.12a)
y(r;0) = g(r).
(2.1.12b)
The above Cauchy systems can be solved numerically using a quadrature formula for the integrals with N quadrature points r.
and weights w..
Thus if r(r.,T.;A) = r..(A) then the Cauchy system for the resolvent kernel becomes N -jf L[r..(A)] = ) r. (A)r . ( A ) W , J dA ljJ L. lm mnJ m m=l T..(0) = k(r.,T.). tj i j The desired solution y(r) can be obtained either from the representation y('"i;A) == g(''i) + A j
r(r1,t;A)g(t)dt a
g(r ) + A l
N V T (A)g(r )w Li, im m m m=l
or by solving the Cauchy system (2.1.12) after computing the resolvent kernel.
(d)
Goldberg's point
method for semi-degenerate
boundary value problem [Goldberg
kernels p.307]
reduction
to two-
:
We assume that the kernel of the integral equation (2.1.3) is of the following degenerate form
61 IN
)
a.(r)b • (s) i
a < s
<
k(r,s) =
£ c1(r)d1(s). i=l
Let
<XL(T)
=
= J
r < s < b.
b.(s)y(s)ds, i = 1,2, . . . ,N;
a
PL(.T)
= J
dj.(s)y(s)ds, i = 1,2
M;
then N
y ( 0 = g(r) + A
M
i
V
a
1-1
i^l
L
L
x
l
J
and the functions a . ( r ) , / 3 . ( T ) satisfy the following systems of d i f f e r e n t i a l equations : d _,, [ Q i ( r ) ] = b . ( r ) y ( r )
= bt(r)
_d dr
fl3CO
j- N g(r) + A | [
aj(7-)aj(r)
M + [ c,(r)0,(r)
r N M = dt(r) g(r) + A | [ aj(r)aj(r) + [ cj(r)j9j(r)
}]■
>}]•
The defining equations for a.(r) a±(.a)
=0,
3.(b) = 0 ,
and /3.(r) clearly indicate
i = 1,2,3,...,N
i = 1,2
M.
62 Thus solving the above two—point boundary value problem will give us the necessary functions a AT),
/?.(T) to obtain the solution y(r) .
(e)
methods
Collocation
and Galerkin
[Golddberg]
In these methods, we represent the solution y(r) in terms of some chosen basis functions & AT) as follows 1 N T(T)
(2.1.13)
= V « A AT) j-1 ] J
where c. are constants to be determined.
Substituting equation (2.1.13)
into equation (2.1.3), we obtain
I e A AT) = g(r) + \ I k(r,r') I crf(r')dr', j J J j J J a
i.e.
N N b ]T C.^.(T) = g(r) + A [ c j k(T,T')^ (T')dr' J j=lJ-1 j=l -1 a
(2.1.14)
The collocation and Galerkin methods differ in their strategy to obtain N algebraic equations from (2.1.14) to determine the N unknown constants «i..
In the collocation method, equation (2.1.14) is satisfied at N
selected points T. called the collocation points. In the Galerkin J method the equation (2.1.14) is multiplied by the basis functions 4>AT) a n d the resulting equation is integrated over (a,b) .
r |
T 4>±(T)
1 [
r
c rf (r) dr - j
g(r)^i(r)dr
i N + A )
„b , J | k(7-,T')4> (r')4. (OdT-dr-
I '.1 I.= I jb
c
i = 1,2
J
a
N.
!
63 These moment equations yield the necessary N algebraic equations for the N unknown constants c .
The solution is clearly approximate as the
equation (2.1.14) is not satisfied completely.
The degree of accuracy
depends not only on the number of terms but also on the choice of the basis function.
A judicious choice will increase the accuracy even
with limited number of terms.
The above two methods belong to the
general class of me methods called projection methods [Goldberg
P-8] for
integral equations.
2.2
BASIC METHODS OF SOLVING THE INTEGRO-DIFFERENTIAL EQUATIONS OF TRANSFER IN SPHERICAL GEOMETRY Let us illustrate the methods by considering a simple model.
The
integro—differential equation of transfer for a spherically symmetric finite homogeneous medium under conservative, Isotropic scattering is [cf. equation (1.3.28) with a = 1, p = 1, B = 0.]
3I(r,/0 ^ l V 3I(r,/0 + ^ ^ ^ + I ( r , M ) 3r
M -
=f J
I(r,M<)dM'
(2.2.1)
with boundary condition I(R,-Ai) = 0 ,
0 < n < 1,
(2.2.2)
where I(r,/0 is the diffuse intensity. Integrating (2.2.1) over (—1,1) we find that the flux F satisfies the simple relation r+l
F = 2
F I(r,/*)/*d/i = -~ . -1 r
(2.2.3)
The arbitrariness in the solution of the homogeneous system (2.2.1), (2.2.2) is fixed by the choice of the constant F . We shall use the above model to discuss the different methods used to solve the integro—differential equation of transfer of spherical systems.
The main idea behind all these methods is to reduce the
integro—differential equation to a system of ordinary differential
64 equations in the space coordinate r by first removing the dependence on the angle variable p.
(a)
Discrete
ordinate
method : (Chandresakhar
(2)
p.366)
:
In this method the integral is replaced by a quadrature sum as follows : r
i
I(r,M)dM -
£
WjICr,^):
fi. being the quadrature points and w. the corresponding weights.
Thus
the integro—differential equation reduces to a system of 2n ordinary differential equations for I. = I(r,/i.),
dI i . XA v* AZT +
i dr
r
r ai ^
[gj
- »
*H-\l
^1=/^
r« 1
provided we have a scheme to express
l—n
Vi-
in terms of I..
In the
l
"="i discrete ordinate method one uses the Gaussian quadrature rule because of the natural interval (—1,1) for p. zeros of the Legendre polynomial P
The quadrature points p. are the
. (/J) . If we set
V"> - jmi) tW") - W ^ - Mi) *i^< ^ (2) where P' = -;— , then we find [see Chandrasekhar p.365]
,+1
„,
.+1
!_x V " > g d" - J_x PjjC^Kr,/*)^. Replacing the integrals by their Gaussian sum, we have
65
»iW
I
L a" .
M-M-,
E vwv
l=-n
The above expression can be used to determine the derivative
£1
at
da
Using this we obtain [see Chandrasekhar (2)
the quadrature points /i..
p.366] the following system of ordinary differential equations
d_ dr
L
y
i—n
i .1 j—^n ("b^
w ./i.p'(/i.)i.
w z
jj I
J
J
Spherical
+ ±H±i> y w.p.ooi. + y w.p'(/i.)i.
V W
i=—n harmonic
method
:
The basic idea of this method is to express the diffused specific intensity I(r,/i) in the form of a series of Legendre polynomials P.(/i) , i.e. N
I(r,M) -
J
(.22+1) I (r)P.(M).
(2.2.4)
The Legendre polynomials are selected as basis functions since they form a complete set of orthogonal functions in the interval (—1,1), the exact range through which /J varies in these problems.
The
representation (2.2.4) is substituted in equation (2.2.1) and the resulting equation is multiplied by P.(/i) and integrated over the interval (—1,1).
Using the orthogonality relation we obtain a set of
(N+l) differential equations in the (N+l) variables I.(r) provided we assume I
1
(r) - 0.
This assumption is called the P
approximation.
Kofink (9) has discussed the error introduced by this assumption.
Here
again the exact boundary condition at the free surface cannot be used. One uses either Mark's or Marshak's equivalent boundary conditions which are as follows in the present model.
66 Mark's boundary condition is I(R,-/j.) = 0, i = 1,2
N+l
l
where fi. are the positive roots of P
- (/i) = 0.
Marshak's boundary condition is .1 I(R,-/OP2i+10Od/i - 0, i - 0,1,2
N.
Davidson and Sykes (4) discussed the merits and demerits of the different types of approximate boundary conditions and they concluded that the odd order in the Legendre polynomial, as suggested in the above approximations, yield better results than even order Legendre (19) polynomials. Viskanta in a separate study has concluded that in lower order approximations, P. approximation is desirable. To overcome this difficulty of using the exact boundary condition a modification of the series representation (2.2.4) has been suggested.
We use a double interval representation as follows :
I.(r.M) = A(r) + +
I (r,/i) = A(r) +
N V JPO
<2i + 1)1 *
(r)pP-(2/i - 1 ) , 0 < *
M
< 1
N V (2£ + 1)1 „(r)/iP.(2/1+1), -1 < p < 0. J J i^0
(2.2.5a)
(2.2.5b)
This representation is prompted by the orthogonality properties of the Legendre polynomials P.(2^-l) and P.(2/i+l) in the intervals (0,1) and (—1,0) respectively and the continuity of I (r,/j) and I (r,/j) at /i - 0. In this representation, the exact boundary condition can be used to yield
I^(R) - 0, 1 - 0,1,2
N.
A(R) = 0. As in the single-interval method, the differential equations for + I.(r) are obtained by first multiplying the transfer equation by
67 either P.(2/J-1) or P.(2/J+1) and integrating it over the corresponding interval where it has the orthogonality property.
The only draw back
of this method is that the unknown function A(r) which appears in the representation had to be given a suitable form depending on the problem at hand.
The form of A(r) is often suggested by the nature of the
boundary condition and the expected nature of the radiation in the direction fi = 0.
This method has the advantage of using the exact
boundary condition of the problem.
(c)
Moment method
:
In the moment method, as the name suggests, we eliminate the angle variable \L by multiplying the transfer equation by \? , 3 = 0 , 1
N
and integrating the resulting equation over the interval (—1,1).
Thus
we get a system of ordinary differential equations for the different moments J, F, K etc. where
J = | J
I(r,A»)dA»; F = 2 j
I(r,/i)/id/i, K = | j
I(r,|i)/djt.
The main difficulty in this method is that, because of the nature of the transfer equation, we always have one less equation than the number of unknown moments introduced.
As a consequence one has to introduce
a relationship between the moments in order to balance the number of equations with the number of unknowns.
The famous Eddington relation
J = 3K
(2.2.6)
was introduced with this in mind. The above relation is valid in the deep interior of the medium where the intensity is nearly isotropic and hence independent of the angle variable /J.
Eddington's assumption is that this relation
(2.2.6) is valid throughout the medium, even at the outer surface. This relationship can also be obtained from the spherical harmonic representation (2.2.4) provided we retain just two terms (N = 1).
68 Since
1
Ux.it)
r-i = V
(2i+l)I,(r)P ,00
,8=0
*
*
= I (r) + 3/iI-C/i), o i we have J - IQ(r) J = 3K.
■
(2.2.6)
K - 3 Io(r)
It was noted that the solution to the spherical problem of transfer by other methods did not satisfy the Eddington assumption (2.2.6).
In
fact the factor K/J became nearly unity as one approached the surface of the finite spherical medium.
This is sometimes referred to as the
peaking effect in the context of spherical medium. Hummer and Rybicki
In view of this
introduced the variable Eddington factor f(r) = K/J.
As the first two moments of the transfer equation (2.2.1) read
V* ™ + 1 (3K _ J} . _ «F dr
r
4
using the variable y(r) = f(r)r J(r), we obtain, for u constant,
dr
r
1 -
f(r)
uF y(r) + - ^
= 0.
(2.2.7)
We use the above differential equation together with the boundary condition F
y(R) =
of 4g
(2.2.8a)
69 where g
F(R) 4JCR)
(2.2.8b)
to obtain the solution. The procedure is to start with the standard Eddington approximation (2.2.6) and hence knowing f(r) we solve equation (2.2.7).
Initially we
assume that there is peaking and take f(R) and g to be unity.
To obtain
the second iterate we update both f(r) and g from the previously computed values.
This process is continued until consistency in the
solution is obtained.
Unno and Kondo
generalizing the Eddington relation.
(181
proposed a new way of
As we noted earlier, the Eddington
relation is equivalent to retaining just two terms in the spherical * harmonic method. This is equivalent to assuming that the diffuse radiation field has the piecewise constant representation I 1 (r), Kr,/i)
0 < /i < 1
<
(2.2.9) I 2 (r),
-1 < M < 0.
This representation gives
J=2 Ilx-V K
=6 ^ 1 - V-
which yields the standard Eddington approximation.
Unno amd Kondo
(18)
suggested that we divide the region as follows :
I.. ,
1'
I(r,M)
fj.
r
< p, < 1
•
(2.2.10)
V
-1 < li < MT
where u is an unknown function of r. Although they have introduced r an extra unknown function y. , using the above representation, one is
70 able to obtain the extra relation between the moments thus balancing the number of the equations with the number of unknowns. and Sen
Wilson, Wan
suggested a modification in the above scheme with a view
to use the exact boundary condition.
The two stream representation
(2.2.10) was replaced by the following three stream representation
I(r,/i) =
•
I 1 (r),
Mr < M < 1
I 2 (r),
0 < ii <
I 3 (r),
-1 < ix < 0.
Mr
(2.2.11)
This l e d t o t h e f o l l o w i n g r e l a t i o n s between t h e moments. 2L - F = 2/i r (3K - J ) 5M - 3K = n (3K - J ) , +1 where L = 2
I(r,II)II
d/j and M
J
-l higher moments.
-if1 2
J
-i
I ( r , / i ) / x dp, a r e t h e n e x t
To solve the resulting set of ordinary differential equations involving the moments and ii , one also needs a boundary condition for the unknown function ii .
This is usually dictated by the problem at hand.
In the case of a spherically symmetric shell, r1 < r < R with an opaque core of radius a, we may take 1
^rW
= f1 1
2
a !2
I*• " ~2r
J
indicating the shadowing of the photons by the core.
2.3
AMBARZUMIAN'S METHOD Ambarzumian's method for solving transfer problems suggests schemes
for converting two point boundary value problems into initial value
71
problems.
The method was first proposed by Ambarzumian
for solving
transfer problems in slab geometry, Ambarzumian's technique could be subdivided into two broad divisions of (a) Ambarzumian'« physical method and (b) Ambarzumian's mathematical method. Both schemes were designed to obtain the law of darkening 1(0,p) directly without going through the knowledge of source function.
For
this suitable scattering and transmission functions were defined, which were functions of the total depth of the medium and the directions of incident and emergent intensities at the free surface. The expressions for l(0,/i) were obtained in terms of the scattering and transmission functions. Though our look out is to utilise the Ambarzumian's techniques for transfer problems in curved geometry, we shall give in this chapter the main features of the techniques as they are applied to transfer problems in plane— parallel medium.
The physical and mathematical
basis of the method is better understood when the use of the method in slab geometry is examined.
With judicious adaptations and changes,
the techniques can be successfully applied to solve transfer and transport problems in spherical and cylindrical geometries.
The
illustration of such applications will be given in Chapters V and VI.
Ambarzumian's
physical
method
:
Ambarzumian's physical method is based on the "Principle of invariance".
Ambarzumian's original argument in putting forward this
principle related to transfer problems in homogeneous, semi—infinite slab medium.
It expressed the invariance of emergent intensity from a
semi—infinite, homogeneous slab to the addition or subtraction of a layer of infinitesimal optical thickness, both the layers and the slab having the same physical proporties.
Sobolev
[p.54] states the
principle of invariance in the following form, "The reflecting property of the medium does not change if one adds a layer of infinitely small thickness to the given medium".
72 What is true for reflection property is also true for transmission property.
Sobolev described the process as the "methpd of addition of
layers". The main feature of the method is to study the changes in the intensity of diffuse radiation field due to addition or subtraction of an infinitesimal thin layer of the same physical property to the slab and set the net contribution to zero.
Ambarziumian defined the
reflection coefficient r(/i,^ ) as l(lt.pe)
- r(M,M0)Fi
(2.3.1)
where I(/X,AI ) is the specific emergent intensity at the free surface
° flux irT. is incident at an angle cos-1y. at that surface when the net when the net flux irT. is incident at an angle cos /i at that surface and cos
/i is the angle of emergence of l{fi,/i
function p(n,n
). He defined a
) , given by p(A»,A«o)M0 = r(M,M 0 ),
(2.3.2)
and obtained a non—linear integral equation in Chandrasekhar's H—function in terms of P(M.M )• The uniqueness of the solution of the integral equation satisfied by p(n,n
) required that
p(M.M0) = P<.P0,»)Chandrasekhar
(2)
(2.3.3)
extended Ambarzumian's physical technique to transfer
problems in finite slab and put the principle of invariance on a firm mathematical foundation of great flexibility.
A short account of the
basis of the method of Ambarzumian—Chandrasekhar is discribed through the following simple example. We consider the problem of diffuse reflection and transmission of a parallel beam of radiation in a plane—prallel medium. A parallel beam of radiation of net flux TTF. is incident on a plane-parallel slab of optical thickness T
C-VV-
in some direction
73 Our aim is to find the angular distribution of intensities diffusely reflected at r = 0 and transmitted through r =
r..
Fig 2.3.1
The resulting laws of diffuse reflection and transmission are expressed in terms of scattering and transmission functions S(T
n,tp,ii
,
F. Id+(0,/,,^)
idJfv-n,4)
(2.3.4)
2^ H*X\». + *0>VJ
0
T±Urx;n,,n0,
(2.3.5)
The subscript d dencffe6s that the radiation considered is diffuse radiation.
That is L and tA refer to the radiation which has a+ u"=
suffered one or more scattering processes.
I T does d_( 1.-/*>*>)
not
Include the directly transmitted flux ^ Fj exp(-T//iQ) in the direction
HVV The factor l/l* is introduced for securing the symmetry of S and T in the pair of variables (>*,?). (/V'V principle of reciprocity.
aS re(
5 u i r e d by
Helmholtz
's
74 S(T S(T
,i*,tp,(i S((kTrl;fio,<po,n,tp) ,M .?»:•**„ ,
•"J
T(.T T ( r1,n,
More generally, if I.
.** .*)
,
■
(2.3.6)
(2.3.7)
(—p' ,
radiation in the direction (—/i' ,
I
d + (refl)<°^'^
=
4 ^ I n IJn
S(, i; ^ V ,^^')I lnc (-M',^)dM'^',
0 0
(2.3.8)
I
,1 -2* ^ - ( t r a n s ) ^ ! - - ^ " 4 ^ J n JJ n 0 0
^ v " ' ™ '
*' ' W ^ ' ■*' >d"'**' (2.3.9)
Equations (2.3.8) and (2.3.9) are consistent with (2.3.4) and (2.3.5) provided, we define I. (—u'.oo') as (-/I1,
I.
mc r l where S is Dirac's delta function.
o
(2.3.10)
^ ^o
where 6 is Dirac's delta function. For semi—infinite atmosphere, we are interested only in the law of reflection For semi—infinite atmosphere, we are interested only in the law of F. reflection l (2.3.11) id+(0,/.,*,) = j - S ( M , V ; / * 0 , V 0 ) . F. Id+<0,/*,,») = - ^ S(M,V;/* 0 .V 0 )-
Invariance of the law of diffuse
reflection
(2.3.11)
:
At any depth r, taking into account the reflection of reduced incident radiation, we have for the outward intensity at r in the direction (/J,
1
.1
+ 4^]
Z7T
j
S(/i,<po)Id_(r.-M',v')d/i'd
(2.3.12)
75 This is the statement of invariance of S(u,
Law of darkening
and its
invariance
:
The invariance of 1(0,/J), the emergent intensity, for the addition (or subtraction) of layer of arbitrary optical thickness to (or from) the atmosphere is equivalent to the statement that I can differ from 1(0,^) in that at r there exists I
(r,fi)
at depth r
(T ,—fj.) which will
be reflected by the atmosphere below r by the law of diffuse reflection. If the layers above r are removed I
(r,/i) will be equal to
1 ( 0 , ft).
I
„1 „2TT
Id+(r,M) - K 0 , M ) + ~
J | 0 J0
S(/i,
(2.3.13)
For axially symmetric case 1 Id+(r,Ai) = 1(0,(U) + j - j S°(/i,/i')Id_(T,-M')d^'d
(2.3.14)
1 r2" S°(M,M')=?S(>i,
(2.3.15)
where
It is the azimuth independent term in the Fourier expansion of S(n,
The integral
equation
of scattering
function
:
We differentiate equation (2.3.12) with respect to r and then pass to the limit r = 0.
76 We have
rdId+(T
M
F.
S(u,ip;u ,f> )
= — ■;
dr
4 T=0
+
^o rdld_(r.-/x'
w r j J,
so..*;/*',*')L
'0 0
dT
,
r=0 (2.3.16)
The derivatives in (2.3.16) are found from the transfer equation
/ . ^ - I ( r , M ) -*(r.;*,*0.
(2.3.17)
with the source function, 1 ~T/v i A - r^ir f(r,/i,")d/*»
(2.3.18) where the phase function, 2 1/2 2 1/2 p(/.,(p;/j' ,
(2.3.19) 8 is the angle between the incident and the scattered ray. From (2.3.17), we have dI
( j + ( T >/*>*)■ AT
=
1" Id+(0,/*,
(2.3.20)
r=0
dld_(r.-/i',
dr
f(0,-M' ,
(2.3.21)
^
In writing (2.3.21), we have made use of the boundary condition
I_(0,-H,
= 0 for 0 < fi < 1.
Now inserting (2.3.20) and (2.3.21) in (2.3.16), we have
(2.3.22)
77 I
d+(0,/i,
- ?(0,/,,
F, - ^ - S(n,
Then from ( 2 . 3 . 1 1 ) ,
+ _ J
j 0 J0
S ( / i , , p ; M ' , ¥ , ' ) ? ( 0 , - / i ' , 1 p ' ) ^ T dp(2.3.23)
( 2 . 3 . 2 3 ) and ( 2 . 3 . 2 2 ) , we g e t
1 1 1 — + — S ( M . < P ; J * .*> ) 4Fi o o . 1 -2TT
XO.M.V) + T - f | ** J 0 J 0
S(M,
(2.3.24)
where 9(0,H,
= ^ F i P (/i, o ) F.
„1
jgj J
„2JT
|
p(/i,V;M",
(2.3.25)
F i n a l l y t h e i n t e g r a l e q u a t i o n f o r t h e s c a t t e r i n g f u n c t i o n S(Ai,
.1 ,2w i^ J j p(/i,
.1 T— [ W J'0n
,2TT
I
S(/i,
Jn J 0
O O / J
i r 1 r2* r 1 r2* -^-5 ! S(/i>?»;/.',»)')p(-|*',v',M",*n) 16JT J0 •'O •'O " 0 x S(/i",,p";M0,?'0) ^ ! d p ' £j£ dp1'. Equation ( 2 . 3 . 2 6 ) , i f s o l v a b l e should give us the s c a t t e r i n g For t h i s , of c o u r s e , p(/i,
And once
(2.3.26) function.
78 S(^,(p;^j ,
can be calculated
from (2.3.11). Incidentally it may be mentioned that the symmetry property and reciprocity of S(n,(p;n
,
inferred from the equations (2.3.26).
However, the complete proof of
reciprocity and symmetry properties can be obtained [cf. Chandrasekhar(2) p.171]. For a finite atmosphere one has to deal with transmission function T as well.
In some problems, S and T both satisfy an integro—
differential equation which has to be solved under appropriate initial conditions. One particular case of interest is that of a semi—infinite axially symmetric, isotropic scattering (p = w medium.
= constant, to
i
In this case, equation (2.3.26) takes the simple form
oMJ>.-'>^Hi + H > - ^
(i + i-)s(M,M0) - o
(2.3.27) Remembering t h a t S ( ^ , / x ' ) i s s y m m e t r i c a l , we may d e f i n e
H00 - 1 + i
[ "S(M.H') s O ^^'Mr ^=- 1i .+iiJfJ 1
f
'o
o
S(A.',/,)^-.
(2.3.28)
and we can express (2.3.27) as
(
u
+
T)S(,i'^o)
' "0H(A«)H(/i ) •
(2.3.29)
Substituting (2.3.29) in (2.3.28), we have .1
H(M) - 1 + \
UQH(MV
J ^ X
d/i'.
(2.3.30)
79 This is the integral equation for Chandrasekhar's H-function (2) [cf. Chandrasekhar , p.97]. Thus we have seen above that the salient features of Ambarzumian's physical method are the introduction of the global quantities of scattering (or reflection) and transmission functions and the deduction of integral or integro—differential equations for these function which could be solved under appropriate initial conditions.
Once the scattering and/or transmission functions
are known, the law of darkening l(0,/i,cp) could be determined.
Though
the scheme has been developed and demonstrated for a rather simple model in plane—parallel geometry, it can be used to solve transfer problems in much more complex systems in plane—parallel medium and in spherical and cylindrical medium without any drastic change in physical and mathematical basis.
In Chapter V, we shall illustrate
the use of Ambarzumian's physical method for solving some transfer problems in spherical and cylindrical geometries.
As an off—shoot of
Ambarzumian's physical method were developed two powerful methods. They are (i) Invariant imbedding or particle counting and (ii) Probabilistic methods.
In Chapter V, we deal with some examples of
these methods. Ambarzumian's
mathematical
method
:
The aim of this method is also to obtain the law of darkening l(0,/x) without going through the formal integration of equation of transfer after determining the source function.
It is essentially an integral
equation method. To demonstrate this, we shall utilise the simple model described by Milne's first integral equation.
The basic feature of this method is
to develop an inhomogeneous integral equation auxiliary to Milne's first integral equation, both having the same kernel.
For this, he
considered the kernel to be built up of a linear aggregation of terms of exponential type.
The solution of the auxiliary equation was
obtained by the transform method leading to the law of darkening.
The
end result was a relation between the emergent intensity at free surface 1(0,/i), the incident flux F. and Chandrasekhar's H-function. This is consistent with the relation obtained by Ambarzumian's
80 physical method.
For finite atmospheres, Chandrasekhar's X— and
Y-functions were invoked. We give below an outline of the method for solving Milne problem [cf. Kourganoff (12) p.164-174)].
Auxiliary
equation
:
Milne's first integral equation for transfer problems In planeparallel, semi—infinite medium could be written as [cf. equation (1.5.16) with B(r) = 0]. 00
»(r) - i f *(t)E ((|t-r|)dt, * J0
(2.3.31)
where ? ( T ) is the source function and E.(x) is the exponential integral given by E l( x) - \
.
(2.3.32)
We may rewrite (2.3.31) as ? ( T ) = AT(?(r)} where A
(2.3.33)
is the Hopf's operator given by T
A {?(»■))-■*[
*(t)E (|t-r|)dt,
(2.3.34)
satisfying the relation A ; { * ( 0 ) =A r (5'(T)} + \ ?(0)E1(r) [cf. Kourganoff(12),p.42]
(2.3.35)
Differentiating (2.3.31) with respect to T, denoting the differential with a prime and using (2.3.34) and (2.3.35), we have 00
*'(r> = j J *'(t)E1(|t-r|)dt + \ »(0)E1(r)
- Afl*'(r)J + \ *(0)E1(r).
(2.3.36)
81 Thus we see that S(r) satisfies the homogeneous equation (2.3.31) and $' (T) , the inhomogeneous equation (2.3.36) with the same kernel. Remembering the definition for E.(T) given by (2.3.32), the second term on the right hand side of (2.3.36) can be considered to be sum of terms of the type A(cr)exp(-7C7) where
A(a) = \ *(0)/a.
(2.3.37)
This idea of "linear aggregation" led Ambarzumian to introduce an auxiliary equation of the form CO
9(r,o)
= ~ j
8f(t,a)E1(|t-r|)dt + exp(-ra)
= A (5(T,CT)) + exp(-ra).
(2.3.38)
T
By virtue of the linear character of (2.3.36) and (2.3.38), we can express 9'(r) as a sum of the solutions of auxiliary equation.
The method of solution
:
Multiplying (2.3.38) by 1/tr, using (2.3.37) and integrating with respect to a from 1 to », we have C rm ; A(CT)?(T,a)d<7 = A j k{a)9{r,a)Aa\ 00
I f 0 + \
0
A(o)e~Tada,
(2.3.39)
and hence comparing with (2.3.36), we obtain CO
CO
?'(r) = [ A(a)*(T,a)da - i »(0)f 9{T,a) J J l l
—
.
(2.3.40)
Taking Laplace transform of both s i d e s ,
L s [r
(2.3.41)
82
L (f(t)J = s f f(t)e S t dt. s JQ
where
(2.3.42)
Integrating (2.3.41) by parts and writing f(s) = L (9(T))
and
{9(r,a)).,
f(s,a) = L
(2.3.43)
we have f(s) = *(0) 1 +
= »(0)
1 T f(s,g) 6a 2 J s
^H/K
(2.3.44)
00
where
R(S,CT)
f(s,ff)
= f J 0
9(T,a)e~ST dr.
(2.3.45)
From (2.3.38), it can be shown that R(s,a)
(2.3.46)
= R(CT,S)
and CO
0,a) - 1 + \ J R(s,«7) ^j .
(2.3.47)
We use (2.3.46), interchange the roles of n and s and write CO
9(0,a)
=1 + \ J R(s,<7) ^
.-. f(s) = 9(0)9(0, So if 9(0) and 9(0,s) 9(0,s)
(2.3.48)
s).
(2.3.49)
are known, f(s) can be determined.
^(0) and
axe source functions at r = 0 for the fundamental problem
(2.3.31) and the auxiliary problem (2.3.38) respectively.
To find 9(0,s)
and 9(0),
we treat the auxiliary equation (2.3.38) in
the same way as the fundamental equation (2.3.31). integral equation in 9(0,s)
v
[cf. Kourganoff
We are led to the
', p.169-170].
83 »(0,a) - 1 + i*(0,a) I l
9< 0 a
}'l
da.
(2.3.50)
J . (7(S+(7)
This integral equation is solved by successive approximations to obtain 9(0,s). We now write s = l//i, cr =1/^' and 9(0, s) = H(/i). s = 1/n, a =1/M' and 5(0,s) = H(/i). Then Then
f(l//0 = L f(l//0 = L
(*(r)) = 1(0,/,). (*(r)) = 1(0,/,).
(2.3.51) (2.3.51)
[cf. ( 1 . 3 . 2 1 ) ] [cf. (1.3.21)]
(2.3.52) (2.3.52)
Hence from (2.3.49), we have the law of darkening Hence from (2.3.49), we have the law of darkening 1(0, n) = Sf(0)H(M),
(2.3.53)
and from (2.3.50), we have
HOi) = 1 + i M H(M) J ~^Td/i'.
(2.3.54)
Now integrating (2.3.53) over n between the limits 0 and 1 and using the principle of conservation of flux for plane—parallel medium, we (121 can show that [Kourganoffv "p.171-172]. /3 5(0) = if F . 4 o Hence I(0,/J) can be determined from (2.3.53) as 1(0,ju) - L* FoH(/i).
(2.3.55)
Ambarzumian's mathematical method which has been applied to the solution of Milne problem could be used to solve transfer problems in much more complex situations in plane and curved geometries.
In
extending this scheme, the basic mathematical features remain more or less the same.
These are the setting up of auxiliary equation
corresponding to integral equation of transfer, seeking a solution of
84 auxiliary equation in terms of which the scattering and transmission functions are expressed (equivalent to H(/i) in the Milne problem), establishing integro—differential equations for scattering and transmission functions (with Initial conditions given) and finally expressing the law of darkening in terms of these scattering and transmission functions.
Combined operations method, first proposed by
Busbrldge, is a good example of Ambarzumian's mathematical technique. It has been used extensively for solving transfer problems In plane geometry.
It has also been extended to spherical and cylindrical
geometry and these extensions will be considered in Chapter VI, §6.5.
2.4
INTEGRAL TRANSFORM METHODS IN CURVED GEOMETRY Certain simple problems of transfer in both spherical and cylindrical
medium can be resolved using transform methods.
These solutions, like
the one discussed in §1, can provide the necessary mathematical guidance to compare algorithms proposed to solve the more complex transfer problems in curved geometry.
The starting point for the
transform methods is the integral equation for the source function of the problem.
We shall consider a simple system of a homogeneous
medium with an infinite extension and having a source with either a spherical or cylindrical symmetry.
(a)
Cylindrical
symmetry : /ON
We consider a homogeneous infinite cylindrical medium [Hunt p.1256] which has a line source distribution B(r) = e the cylindrical coordinates.
:
/r; r, $ being
The Integral equation for the source
function ?(r) is given by Hunt [cf. Hunt
, equations (2.3) and (2.4)]
CO
/r ?(r) = /r B(r) + u> I k(r,u)du, o Jx
(2.4.1)
where k(r,u) = /r K (ur) /t I (ut)/t ?(t)dt + I (ur)f /t K (ut)/t S(t)dt , O Jn O O J O J 0 "r (2.4.2)
85 K (z), I (z) are the Bessel functions of imaginary arguments and co (0 < co < 1) is the albedo of single scattering.
Let us first
consider the corresponding homogeneous integral equation
Q(r) = co I k(r,u) du, o Jt
(2.4.3)
where Q(r) = /r S(r). From (2.4.2) we can show that the kernel k(r,u), as a function of r, satisfies the differential equation
1 L
dr
2
k -» - Q(r).
(2.4.4)
Q- 0,
(2.4.5)
4r
Let us now choose Q(r) to satisfy " 1 dr
cf
4r
where a is an unknown constant.
Then from (2.4.4) and (2.4.5) we see
that (u2 - a2)k(r,u) - Q(r).
(2.4.6)
Substituting this in the homogeneous equation (2.4.3) we have
= .o r"1, QCr) SM_ u —a2 du
Q(r)
2
which implies a = co
o
tanh
a.
(2.4.7)
Thus the most general solution of the homogeneous integral equation (2.4.3) which is finite at the origin is
Q(r) - A/r I (ar),
86 where a satisfies the transcendental equation (2.4.7).
Adding this
solution to any particular solution of equation (2.4.1) will yield the general solution to equation (2.4.1).
To discuss the transform method
for the above problem, let us now define the following Bessel integral transforms : CO
Q(s) = ( /r JQ(sr)Q(r)dr
CO
B(s) = f r J (sr)B(r)dr, J 0 ° CO
k(s,u) = [ A
J (sr)k(r,u)dr.
Then equation (2.4.1) gives
Q(s) = B(s) + u>
k(s,u)du.
(2.4.9)
o i1 But from (2.4.4) we see that Q(s) = u k(s,u) - P(s,u) where
J ^ v sr) 00
P(s,u) = lim e ->0
d k
— dr
+
k — 4r
dr
- s k (s,u),
using the fact that k(r,u) - 0(/r) for small values of r. 2 2Thus Q(s) = (u + s )k(s,u).
Substituting this in equation (2.4.9)
gives B(s)
Q(s) =
(2.4.10)
a
1 - -—s tan
(s)
87
Since B(r) = e
/r,
Q(s) =
1 _ — /-, 2,2 (1+s )
Inverting, we have
Q(r) = /i
1 i-
Li
i — —° » • - 1s 1 tan s
\ sQ(s) J (rs)ds. J0 o
By contour integration it can be shown [Hunt
2 *(r) = -ft Q(r) = 2 °
(1
~°
i
] that for large r
2 2 Ko(ar) + 0(e~ r //O
}
(a -1+0) )
o
1 2 2Q7 I exp(-ar)
2
2 2 2a (1-a )
(2.4.11)
(a -1+ai ) o (b,) Spherical
symmetry
:
Let us consider a homogeneous infinite spherical medium which has a point source at the centre.
The integral equation for the source
function 5(r) is given by [cf. Smith 0)
, equations (2.5), (2.6)]
««
r8f(r) = B(r) + ^
[E (|t-r|) - E (t+r) ] tSf (t)dt J
(2.4.12)
0
where 0)
S(r) = j
oo 2
j
—t
[E1(|t-r|) - E1(t+r)] ^ - dt.
o) (0 < o) < 1) is the albedo of single scattering. o o introducing the functions
l(r) =
B(r),
r > 0
-B(-r),
r > 0,
<
(2.4.13)
As before by
88 and setting Q ( r ) = r ? ( r ) , the Integral equation (2.4.12) c a n b e rewritten as
Q ( r ) = < K r ) + ■£■ J
E ] _(| t-r | )Q(t)dt.
(2.4.14)
—00
We n o w define the two sided Laplace transforms : CD
Q(s) = J
e_srQ(r)dr,
*(s) = f
e
Sr
<6(r)dr.
—sr If the equation (2.4.14) is n o w multiplied b y e and integrated and the resulting terms are rearranged, w e obtain [Smith
]
QO-gg.
(2.*.M>
w _. where M(s) = 1 — —tanh (s). s We also obtain from (2.4.13) that
?(s) =
(tanh(s)}
s
= - —
u> o
[M(s) - 1] .
(2.4.16)
Inverting w e have „c+i<» -j,
«W-H!
. sr
. *%r"-
By taking a suitable contour it can b e shown [Smith
Q(r)
. 2°W} u> (a -1+0) ) o o
e_ar
] that
- V + -2 f ^ r h ■
<2-^7>
u> J l (N(t)} +7T o
where a is the positive root (< 1) of the transcendal equation
89
a — o> tanh o M(t exp (± U))
and (N(t) ± i*} = —
(2.4.18)
(a) ,
.
From the above expression for r?(r), it is possible to obtain the asymptotic expressions for the intensity as r —> =° and for small r. We have [Smith^16^] 2 Kr,±/0
°
2
;e
K
-or (1 + 0 < ± ) } , 0 < ft < 1,
(2.4.19)
[a -l+o) ] [ l T a / i ] r for r »
1 and 1
Kr,/0
\_
~
•
r [ril - / j 2]. 2
ri
2
2
S
2
sin i
2
r[l-/i ]
f o r s m a l l v a l u e s of
n ,1
2
2
_ a q-cx ) l o g r + 0 ( 1 ) , 0 < n < 1 2 , j r - s i n {(1-/J ) } ' a —l+a>
1
1
Kr,->0 -
lf/1
{1—/* )
1
2 2 g (1-a ) log r + 0(1) , 2 a —1+0)
0 < /i < 1. (2.4.20)
r.
90 References
1.
Ambarzumian, V. A., Theoretical Astrophysics, Pergamon Press, London, 1958.
2.
Chandrasekhar, S.,
Radiative Transfer, Clarendon Press, Oxford,
1950; also Dover Publications Inc., N.Y., 1960. 3.
Cuperman, S., Engelman, F. and Oxenius, J., Radiation from a spherical Plasma.
Non Thermal Impurity
The Phys. of Fluids 5^, 108,.
1963. 4.
Davison, B., Sykes, J. B., Neutron Transport Theory, Clarendon Press, Oxford, 1958.
5.
Golberg, M. A., Solution Methods for Integral Equations.
Plenum.
New York, 1978. 6.
Heaslet, M. and Warming, R.,
J. Quant. Spectro. Radiative
7.
Hummer, D. G. and Rybicki, G. H., Mon. Not. Roy. Astro. Soc.
Transfer 5, 669, 1965.
152, 1, 1971. 8.
Hunt, G. E.,
Radiative Transfer in a Homogeneous cylindrical
9.
Inonu, E. and Zweifel, P. F.,.
atmosphere, Siam. J. Appl. Math., 1_6, 1255, 1968. Developments of Transport Theory.
Academic Press, London, 1967. 10.
Kho, T. H.,
The Combined-Operational Method in Radiative
Transfer, MSc. Thesis, Singapore University, 1971. 11.
Kho, T. H.,
Non-coherent Scattering in Spherical Media, Ph.D.
Thesis, Singapore University, 1974. 12.
Kourganoff, V. and Busbridge, I. W., Basic methods in transfer problems.
Oxford University Press, 1952; also Dover Publications
Inc., New York, 1960. 13.
Leong, T. K.,
Probabilistic Method in Radiative Transfer and
Neutron Transport Problems, MSc. Thesis, Singapore University, 1969. 14.
Leong, T. K.,
Radiative Transfer Problems in Scattering Media
with Curvature, Ph.D. Thesis, Singapore University, 1971. 15.
Miklin, S. G., 1957
Integral Equations, Pergamon Press, London,
91 16.
Smith, M. G.,
The Isotropic scattering of a concentrated
ray pencil from a point source, Proc. Cambridge Philos. S o c , 60, 105, 1964. 17.
Sobolev, V. V.,
A treatise on Radiative Transfer, Van Nostrand,
New York, 1963 18.
Unno, W. and Kondo, M., Publ. Astron, Soc. Japan, 2*5, 347, 1976.
19.
Viskanta, R.,
Radiative Heat Transfer.
Fortsehritte des
verfahrenstenchnik, 2_2, 51, 1984. 20.
Wilson, S. J., Wan, F. S., and Sen, K. K., Astrophysics and Space Sci 6_7, 99, 1980.
95 Chapter III SPHERICAL HARMONIC AND DISCRETE ORDINATE METHODS
3.1
Introduction In this chapter, we shall discuss and demonstrate a group of
approximate methods which had its beginning in the well known (12) (19 21) Milne—Eddington approximation and Schuster—Schwarzschild Milne—Eddington approximation and Schuster—Schwarzschild approximation. The origin of spherical harmonic method may be traced approximation. The origin of spherical harmonic method may be traced to the former and that of Wick— Chandrasekhar's discrete ordinate method to the latter.
In spherical harmonic method, the specific
intensity is expanded into a series of Legendre polynomials P.(^J) which form a complete orthogonal set within the range
(—1,1).
The
series is generally taken to be a truncated one, the point of truncation depending on the accuracy required.
On the other hand, in
the Wick—Chandrasekhar method of discrete ordinates, analytical representation of intensity is avoided.
The determination of specific
intensity I(T,|I) at any optical depth is limited to certain discrete ordinates fj.. (i = ±1, ±2 ... ±n) which are selected by Gaussian quadrature.
Chandrasekhar stated that the above two methods are
equivalent in all details.
However, both of these schemes face the
difficulty of analytical representation of boundary conditions at free surfaces, where the specific intensity is discontinuous.
In
plane—parallel problems and in some simple cases of spherical models, the situation is met by using different types of equivalent boundary conditions.
The single interval spherical harmonic method and the
method of discrete ordinates, however, succeeded in giving fairly reliable results in a wide variety of radiative transfer problems in (12) plane—parallel medium. Kourganoff drew attention to some of the serious limitations of the single interval spherical harmonic method. It is expected that in the extension of the method to radiative transfer problems in curved geometries, these shortcomings will be carried over.
96 In what follows we shall demonstrate the use of the single interval spherical harmonic method (§3.2), modified double-interval spherical harmonic methods (§3.3) and discrete ordinate method (§3.4) in solving radiative transfer and transport problems in curved geometry.
An
excellent analysis of the use of the single interval spherical (12) harmonic method in slab medium is given in Kourganoff and that of discrete ordinate method in slab and spherical geometry in Chandrasekhar
3.2
.
Spherical harmonic methods in plane—parallel medium The origin of the single interval spherical harmonic method can
be traced to the works of Eddington and Gratton . However, it (4) was left to Chandrasekhar to systematise the scheme and to suggest a general procedure for solving integro—differential equation of transfer by this method to any order of approximation.
This method
was extensively used to solve various radiative transfer problems in plane—parallel medium in stellar atmosphere and in neutron transport. Though our main interest is to discuss basic mathematical methods in spherical and cylindrical geometries, we shall reproduce here a short review of the development of the method in solving transfer problems in slab medium.
It is seen that in depth analysis of the use of the
method in plane—parallel medium suggests modifications for its application to other geometries. The essential feature of the method lies in the representation of specific intensity I(r,/j) in the form of a finite series in Legendre polynomials P.(/i), where /* = cos 6, $ being the angle between the intensity and the outward drawn normal to the plane surface at the depth T. For example,
t I(r,M) =
o
I I,(OP,00, i-0
(3.2.1)
where P^(j0 is orthogonal in the interval (-1,1) and this is the interval over which n varies.
97 The radiative transfer equation for diffuse radiation in the simple model of semi—infinite, plane-parallel, scattering medium for axially symmetric radiation field is given by [cf. equations (1.3.6), (1.3.7)]
M
dI
^ , / j ) - Kr.it) ~ | J
p C ° ; (M,/x') = |j- J
where
p(0)(M,/x')Kr^')dM',
(1.3.6)
p(p,v,/i' ,*' )dp' .
( 1 . 3(1.3.7) .7)
The source function,
*
, -+1
=i J
,.
P
w
(M,/.')i(fy)^ .
-1 The specific intensity I(r,ji) in this case is azimuth independent and other symbols have their usual meanings labelled in Chapter I. The boundary conditions under which the transfer equation (1.3.6) has to be solved is given by : (a)
absence of incident radiation from outside at the free surface
defined by r = 0,
i.e. 1(0,-;*) = 0
and
(b)
for
0 < fi < 1
^(r)e _T -> 0 as r -> <=.
(3.2.2)
(3.2.3)
The condition (b) implies absence of infinite source of radiation as T -> oo. In representing the specific intensity I(r,/i) by a finite series with Legendre polynomials as the basis functions, it is usually hoped that a fairly accurate solution of the problem can be attained retaining a reasonably small number of terms in the expansion. Substituting (3.2.1) in the transfer equation, using appropriate recurrence formulae for P.(ji)'s and equating the coefficients of
98 various Legendre polynomials, we get a set of linear differential equations in I . ( O . Chandrasekhar
(4)
assumed a solution of the form 1,(0 = A^e-1"" ,
where A and k are constants.
(3.2.4)
Substitution of (3.2.4) in the set of
linear differential equations in I„(0 yields a set of linear algebraic equations, the consistency of which requires a determinental equation A(k) = 0 to be satisfied.
This gives us the roots k. as
0, 0, ± k. (i = 2,3...n). '
'
l
Then the solution (3.2.4) reads as -k.r 1,(0 = I a i
Ai(ki)e
X
(3.2.5)
where the constants are to be found from the boundary conditions. While the exact boundary condition is realised at the lower boundary T -> °°, it cannot be done at the free surface, r = 0.
Instead,
equivalent boundary conditions are used. In place of 1(0,—/J) = 0 , 0 < ft < 1, the following boundary conditions were used. (4) (a)
Chandrasekhar
used the boundary condition
vm V 0 ) - 1 ° V 0 ) L p.^p/iodn. m—0
(3.2.6)
0
(13) (b) Mark in connection with the mathematically (b) Mark in connection with the mathematically case of neutron transport in plane parallel medium, used case of neutron transport in plane parallel medium, used
analogous analogous the boundary the boundary
condition which in present notation should read 1(0,^) = 0,
(3.2.7)
where fi'.s are some strategic values of n within the range and were taken as the roots of P
+1(M)
= 0, n being an odd integer.
99
(c) Marshak's r
(14)
equivalent boundary condition was
u
J
I ( 0 , / » ) P 2 M ( M ) ^ = 0,
Later Wang and Guth
(25)
I - l,2...n
(3.2.8)
used the same method to obtain an
approximate solution of transport equation for multiple scattering of neutrons and charged particles, the scattering being anisotropic. The comparative values of (I(0,/i)/F) obtained by Chandrasekhar are shown in the following table.
(4)
The results obtained by single
interval spherical harmonic method in different approximations are compared with exact solution.
Table
3.2.1
1(0,/*)/P p
Second approx.
Third approx.
Exact
0
0.4397
0.4440
0.43301
0.2
0.6168
0.6232
0.62803
0.4
0.7842
0.7884
0.79210
0.6
0.9463
0.9475
0.95009
0.8
1.1052
1.1035
1.10536
1.0
1.2620
1.2578
1.25912
Kourganoff
(12)
(p.90—101) made a thorough analysis of single
interval spherical harmonic method used in the case of Milne problem or its analogues.
Davison and Sykes
(p.124—129) discussed the
merits and demerits of the different types of equivalent boundary conditions and concluded that odd order approximations were destined to (12) yield superior results. Kourganoffs analysis leads one to the following conclusions.
100 (i)
The exact boundary conditions cannot be used in this method.
(ii) The discussions of Davison and Sykes
in support of the odd
order approximation over even order were of no consequence.
The salient points of Kourganoff's
(12)
discussion can be stated
as follows. We have seen that the determinantal equation A(k) = 0 has roots 0, 0, ±k. (i = 2,3...n). 2n.
so the number of roots of the equation are
As the boundary condition (3.2.3) implies —T
I(r ,fi)e
-» 0
only positive k'.s are possible.
as
r -> =>,
So (n-1) constants of c. are zero.
From the constancy of flux F (which is built—in, in the equation of transfer (1.3.6)), one more constant can be determined.
The remaining
n constants are to be found from the boundary condition at the free surface, which in the present representation implies that
1(0,-p) = 0 = l2£Q
I i (0)P i (-M) for 0 < p < 1.
This is satisfied for all values of fj. in (0,1).
(3.2.9)
This is equivalent to
an attempt at determining n constants from an infinite set of linear equations.
Hence, certain arbitrariness in the determination of
constants cannot be avoided.
Use of various equivalent boundary (13) For example, Mark met it
conditions is an attempt to by pass it.
by choosing some strategic values of p for which the condition (3.2.9) holds good. Kourganoff
(12)
tried to limit this degree of arbitrariness by
imposing a minimum condition on 1(0,—/j).
r1 a =
He suggested that
2 [1(0,-/0] dfi = mln,
J
0
which is equivalent to the condition
(3.2.10)
101 1(0,-/i) = 0 ,
0 < n < 1
(3.2.11)
in the least square sense. This, however, gives 2n+l equations to determine n constants (12) p.94-95]. Thus the arbitrariness in the
[cf. Kourganoff
determination of constants is minimised but not removed.
Kourganoff
traced this source of defect to the fact that the function I(r,/x) which is discontinuous at the free surface at /i = 0 was represented by a finite number of continuous terms.
His suggestion for the
improvement of the situation was to try a double—interval representation of specific intensity. This was, in fact, suggested by (12) ootnc Yvon [Footnote Kourganoff , plOl], but elaborately demonstrated by (16) Mertens Mertens
represented I(r,/i) as I (T ,fi) and I (r,/i) at the two
separate ranges (0,1) and (—1,0) of fi. He wrote
I+(r,fO = Z" = 0 (2i+l)I*(r)Pi(2M-l); 0 < p < 1,
(3.2.12)
I_(r,/i) - I" = 0 (2i+l)l](r)Pi(2/x+l); -1 < p < 0.
(3.2.13)
as P (2/z-l) is orthogonal in (0,1) and P (2/i+l) in (-1,0). The boundary conditions of the problem were taken as I~(0) - 0,
I+(r)e
T
I (r)e
T
i - 0,1,2...it.
(3.2.14)
• as T -> "o.
(3.2.15)
-> 0 ■* 0
With this representation, the main steps of the single interval spherical harmonic method were gone through with necessary adaptation.
102 The end results were encouraging.
As far as the transfer problems in
plane—parallel medium is concerned, this method was free from two main defects of single interval spherical harmonic method, namely (i) and
(ii)
the even
the exact boundary conditions could be used here, there was no preference for odd order approximations over order ones contrary to the suggestions of Davison and
Sykes (6) . However, some new difficulties cropped up. (i)
It was found to adversely affect the critical size
calculations of neutron transport. (ii)
The extrapolated end points calculated for neutron transport
in slab medium were found to be unreliable. (iii) Above all, it was found that the method was not adaptable to the solution of transfer problems in spherical geometry.
The
method did not give correct flux integral in the spherical case. Double—interval (or half—range as it is sometimes called) method (24) has been used with some modifications by Sykes , Gross and Ziering
, Max Krook
and others.
They are however essentially
equivalent to Mertens' method and share its limitations. On close examination of Mertens' representation of specific intensity, it was found that in an attempt to provide for discontinuity at the free surface, the discontinuity of representation (25) at fi = 0 had been carried to the interior. Wilson and Sen introduced at this stage a modified double-interval (or half—range) spherical harmonic method which retained the advantages of the double—interval representation of Mertens and at the same time removed its defects.
The aim was to seek a suitable spherical harmonic method
which would be equally effective for tackling transfer problems in plane—parallel and spherical medium. Before considering the applicability of single and modified double—interval methods for spherical geometry, we examine the
103 position of modified double—interval spherical harmonic method vis—a—vis other SHM's and exact solutions in the case of planeparallel medium. The representation of intensity in the modified double—interval method is taken as follows :
I+(r,M) = A(r) +
LI n XI (2i+l)I^(0^(2M-l); 0 < /i < 1
(3.2.16)
£=0
I_(r,,i) = A(r) +
n I (2^+l)I~(r)MP (2/x+l); -1 < y. < 0
(3.2.17)
£=Q
where for slab media A(r) was taken as A(r) = A T .
(3.2.18)
The boundary conditions of the problem were stated as I~(0) - 0,
I+(r)e
T
-> 0
I,(r)e
T
-» 0
i-0,l,2...n.
(3.2.19)
• as T -> co.
(3.2.20)
It is to be noted that in (3.2.16) and (3.2.17), I (T,H)
and I_(r,/x)
are continuous at ft — 0 at all T. With this changed representation of specific intensity the procedure outlined for single interval SHM is followed.
The
recurrence formula used is 1
£+1 „
^ (2 ^ ±:L) " 2i+I [— W
,„ . ^ ,- 2i+l P„ (,„ _,_,, £ P 2 2/i±1) +
^ *-2" i
2 M(^±1}. (3.2.21)
The comparative values of the law of darkening I (0,/x) obtained by different methods are compared in Table (3.2.2) and the percentage (27) error in each case is given [cf. Wilson and Sen p.364].
104 Table
3.2.2 I+(0,M)/F
/J
Exact
Single interval SHM Chandrasekhar (4) error 2nd approx.
A%
Iterated double interval (16) Mertens error 2nd approx. A%
0
0.4330
0.4397
1.55
0.4331
0.1
0.5401
0.02
0.5227
1.89
0.5339
1.20
0.2
0.6280
0.6168
1.78
0.6241
0.62
0.3
0.7112
0.7014
1.27
0.7092
0.28
0.4
0.7921
0.7842
1.00
0.7913
0.09
0.5
0.8716
0.8657
0.68
0.8717
0.01
0.6
0.9501
0.9463
0.40
0.9509
0.08
0.7
1.0280
1.0261
0.18
1.0292
0.12
0.8
1.1054
1.1052
0.02
1.1069
0.14
0.9
1.1824
1.1838
0.12
1.1841
0.15
1.0
1.2591
1.2620
0.23
1.2610
0.15
Modified double-interval SHM (27) Wilson and Sen
/*
error 1st approx.
A%
error 2nd approx.
A%
0
0.4346
0.36
0.4334
0.1
0.5298
1.91
0.5345
1.04
0.2
0.6196
1.34
0.6248
0.51
0.3
0.7059
0.75
0.7097
0.21
0.4
0.7898
0.29
0.7918
0.04
0.5
0.8720
0.04
0.8721
0.06
0.6
0.9530
0.31
0.9512
0.12
0.7
1.0330
0.49
1.0295
0.15
0.8
1.1123
0.62
1.1072
0.16
0.9
1.1909
0.72
1.1844
0.15
1.0
1.2671
0.79
1.2605
0.11
0.09
105 It is clear from the above table that the second approximate results obtained by different methods for the law of darkening in plane—parallel medium are comparable in accuracy and the values of I (0,^)/F obtained are close to those of exact solutions.
However,
the modified double-interval SHM has the advantage that it can be extended to the solution of transfer problem in spherical geometry without any major change.
It may be cautioned here that the emergent
intensity should not be calculated by direct substitution in the series representation of specific intensity.
The source function ?(r)
should be calculated first and the emergent intensity obtained from the general solution of the equation of transfer which in the semi—infinite grey case reads as CO
I (0,n)
= f Sf(t)e_t/'i jp
(3.2.22)
The reason for this has been explained in Wilson and Sen
(27)
[p.365
lines (11-18)]. It has already been mentioned that Mertens'
double—interval
SHM though succeeds in using exact boundary conditions at free surface fails to give correct flux integral in transfer problems in spherical geometry.
The reason for this is the discontinuous representation of
intensity at fi «■ 0 not only at the free surface but also, at all levels of the atmosphere. The formulation of the correct flux integral in the transfer problems in spherical geometry requires in the spherical harmonic methods the continuity of intensity representation at /i - 0 at the inner levels.
The single interval and the modified double—interval
SHM satisfy this condition, but Mertens' method does not.
Hence, they
can be used to solve transfer and transport problems in spherical geometry.
As an illustration of the use of the two methods, we
consider a simple problem of neutron transport in spherical geometry.
106 3.3
The spherical harmonic methods in spherical geometry : Single interval spherical harmonic method The single interval spherical harmonic method has been widely
utilised for solving radiative transfer, neutron transport and heat transfer problems in spherical medium, inspite of its limitation that exact boundary condition could not be used at the free surface.
To
illustrate the use of the method, we take the following simple model of neutron transport problem, which was proposed and solved by Marshak14). Model : An infinite, source free, non-capturing medium surrounds a black core of radius "a" absorbing completely all the neutron falling on it. A neutron current density exists in the direction —r, r being measured from the centre of the black core.
Fig
3.3.1
The medium is supposed to scatter neutrons isotropically without changing their velocity. Our aim is to obtain the neutron density at any point and calculate the values of "extrapolated end point". The equation of neutron transport appropriate to this problem in spherical geometry is given by
/• MlWil
+
1=1 ^ £ l
+
,M) _ 1 j + 1 t w w
,
(3.3.1)
107 where ^/>(r,/i) is the angular distribution of neutron at r in the direction cos
/j measured from the outward drawn normal to the surface
at that point. The neutron density is defined as „+l
i r +1 J(r)
= j I
-1
^(r.AOd/i.
The boundary conditions under which the equation (3.3.1) is to be solved are :
(1)
V(a,M) = 0 for 0 < /j < 1.
(3.3.2)
This implies the presence of black absorbing core at r = a. (2)
V>(r,/i) - h
°°a s
r
-» "•
(3.3.3)
This indicates the absence of infinite neutron density at the remotest point from the centre of the sphere. In the single interval spherical harmonic method, the representation of ij>{v,fi) is proposed as
i *(*.*») = l2l0
. 5(2i+l)^i(r)Pi(/i).
(3.3.4)
P.(/x), the Legendre polynomial, is orthogonal in the range (—1,1).
It
is to be noted that the series is truncated after a finite number of terms. Now multiplying the transport equation (3.3.1) by P „ ( M ) , integrating over /i and using the recurrence formula (24+D/iP^/i) = (i+l)Pi+1(M) + ^ P ^ _ 1 ( M ) ,
(3.3.5)
one obtains the following set of linear differential equations in ^.(r) and i>'D(r) where V'p(r) is the differential of i>f(r) to r.
with respect
108
H
{T)Hi+1)r
im{ *-i
(T) +
_l_ T
*+i }
+
1
r(2i+l
l ( J>+1 "1 ( 1+7} (M)(i+ZH
7mM.
S {X.-TL )
\X.-T£.}
^r)'m~1)1p^M} (3.3.6)
* i ( r) - *0(r)fioi.
where S „ is the Kronecker delta function. If we choose Z
= 1 (PT-approximation), the equations (3.3.6) o 1 become equivalent to the following two equations. ^ ( r ) + 2^(r)/r = 0
-
(3.3.7)
i V;(r) f ^ ( r ) = 0
This has solutions ^(r)
0o(r)
=
_ 1_ 2 r
(3.3.8)
■
r + B■
where B is a constant. The boundary condition (3.3.2) is not consistent with the representation of V(r,M) by a finite number of Legendre polynomials. (14) Marshak used an equivalent boundary condition appropriate to the approximation used.
In P.—approximation, the boundary condition used
was
i
ni>(a,p)dn = 0,
(3.3.9)
'0
which is tentamount to requiring that the total current of neutrons entering the medium from the black core is zero.
In higher
approximations, a set of boundary conditions is to be used whose number is determined by the approximation at which we are working on. Hence, in the P.-approximation the boundary condition becomes
109
(3.3.10)
2 * Q (a) + ^ ( a ) - 0. Hence from (3.3.8) and (3.3.10)
(3.3.11)
—)/&.
B - 3(1 +
The second of equation (3.3.8) can be written as rV> = [ r - a + r ] B where r
The quantity r
(3.3.12)
= a - 3/B.
is called the "extrapolated end point".
In P.-approximation r
o =
2
/E3<
In P.-approximation, we set I
1 +
(3.3.13)
!^
= 3 in (3.3.6) and we get
r^£(r) + 2V»1(r) = 0 rV>;(r) + 2rV>2(r) + 6V>2(r) + 3r^(r) - 0
•
(3.3.14)
2r^(r) + 3r^(r) - 2^(r) + 12^(r) + 5r^(r) = 0 3r^(r) - 602(r) + 7rV>3(r) = 0. These equations are solved under the following boundary conditions. In P_—approximation, the eqivalent boundary condition, read as
c
Pi(M)^(a,M)d/i = 0
(3.3.15)
(i - 1,3).
These may be written explicitly as 4V>o(a) + 8^(3) + 5V>2(a) = 0 8^1(a) + 25V>2(a) + 32^3(a) = 0 .
1
(3.3.16)
110 The solutions of equations (3.3.14) are
* (r) - - r1 + B *o ^ ( r ) = -1/r
^r !
,
(3.3.17)
-kr ^ 2 (r) = -6/5r3 + G(l + 3/kr + 3/k 2 r 2 )^ * 3
-kr + -|5_+ _ ^ ) e , 3 3' 3kr k r k r
where k = /35/3
B =
G -
c
1 +
2 + 3^ J a
1 -2
2a ■
7kC 2a
l-£-ka
5 .22 k a
ka 14kCe
(l/8k2a)[l + 15/4a + 72/7a2] ( 1 + 1Z5) + 6_ u 96k; ka
U S . ); + _15_ ( U1 192k ,22 k a
( u1 +
+
(3.3.18)
35_, ;+ _15_ 96k ,33 k a"
We can rewrite the first equation of (3.3.17)to highlight the expression for extrapolated end point. In P.—approximation, it is rib (r) = B[(r-a+r ) - -ye o o
-k(r-a)
(3.3.19)
where B = 3/(a-r ) , 7 = [2G/(kB)]. Hence r , the extrapolated end point, is given by r = a - 3/B. o '
(3.3.20)
In the following table, we record the values of the extrapolated end points for the changing values of the radius of the black core in different approximations.
111 Table (3.3.1)
r 0
a P.—approx.
P„-approx.
0.5
0.286
0.323
1.0
0.400
0.480
2.0
0.500
0.608
5.0
0.588
0.682
CO
0.667
0.705
The single—interval spherical harmonic method was also used by .(4) Sen and Davison and Sykes for solving Chandrasekhar (18) transfer and transport problems in spherical geometry. Poisy used a scheme for making iterative corrections for curvature.
All the
single—interval spherical harmonic methods suffer from the limitation that the exact boundary conditions cannot be used at the free surface.
Modified
double-interval
(or half-range)
spherical
harmonic method :
We demonstrate the method for a neutron transport problem in spherical geometry [cf. Wilson and Sen
] of the same model as has (14) been considered by the single-interval SHM [cf. Marshak ]. The representation of ij>(r,n)
in the modified double—interval SHM
is given by
,6+(r>M) = A(r) + j j _ 0 (2i+l)^(r)^Pi(2|i-l) , 0 < JI < 1
(3.3.21)
i> (r,/i) = A(r) + E"
(3.3.22)
(2i+l)^.(r)/*Pi(2|*+l), -1 < n < 0,
112 where A(r) is a function of r only, P.(2/J-1) and P.(2/x+l) are orthogonal in the ranges (0,1) and (-1,0) respectively.
The inclusion
of the same term A(r) both in (3.3.21) and (3.3.22) ensures the continuity of intensity at p, = 0. The neutron transport equation in the present representation can be written as 3V>+(r,M)
"
3r
1-M +
"r
di>+(r,n)
aJT~ + V r ' M ) "
A(r) +
K-^i+h-
f o r 0 < fi < 1, 3V> (r,M) 1-M W (r,M) -+ + V> ( r , / i ) = 3/i r
v- - 3r
A(r) +
(3.3.23)
^o-V^l
*
J
f o r - 1 < /i < 0 .
(3.3.24)
We multiply equation (3.3.23) by P (2/j-l) and (3.3.24) by P (2/i+l) and integrate over fi in their respective intervals and using the recurrence formula
^< 2 ^) - afei [¥ W2"*1' * ^ V2**1' + I p,_iC2^D (3.3.25) we are led in the first approximation (£
= 1) to the following set of
differential equations. For i = 0
(| **'+2yfy + | 4 *o+2*J> + ^o+V^rV (f +'o -2i>\) + §
^o + *o + *l" i *l )
= 2A
- '< r >>
= 2A
' (r) '
(3.3.26)
(3.3.27)
For i = 1
(
C + 3 *'i> + \ <-K+ ^ *i> + <*o+3*I>= _A'(r)
(3.3.28)
113 (_^
l(>_
+ " / x ) + i (^Q+ ^itf~)+ dAo-3^~) = -A'(r)
(3.3.29)
denote differentiation with respect to r.
The boundary condition (3.3.2), in terms of the present representation is
A(a) = 0
V>Q(a) = 0
>
(3.3.30)
*i+(r,/j)/* d/i + J V_(r,/i)M d/j L-'O
=
1 2
4 3 ^o
+
) • V +2«i-^ 1 )|.
(3.3.31)
We also observe that adding (3.3.26) and (3.3.27), we have dF dr
2F r
(3.3.32)
which yields the correct flux integral for spherical medium,
F = F /r . o'
(3.3.33)
F is the constant of integration. It is clear from this that the o flux integral is satisfied irrespective of the form of A(r). Hence the form of A(r) cannot be guessed from the flux integral. However, it is to be noted that A(r) is the y.—independent term in the representation of i/>(r,/x) adn V_(r,/i) and as such it could be viewed as the generalistion of the term ^ of Marshak
(14)
.
in the single-interval SHM
114 Let us assume for A(r) the form A(r) = a + | + 2_ r
£_ r
+
+
e~ k r A r
B C 3> + ^ S r r
+
+
D where k * 0. ^ r ■
(3.3.34)
The choice of A(r) is usually dictated by the nature of the physical problem at hand.
However, no uniqueness is claimed for this
representation.
A'(r)=-^-^-!{ + e- k r r
r
kA r
r
kB+A 2 r
kC+2B 3 r
kD+3C 4 r
4D 5 r ;3.3.35)
As trial solution of the homogeneous equations (3.3.26) - (3.3.29), we take 7.
K. h. -kr -+ 6 1 1 f. + — + -o + l r 2 r
S.
* ± Cr) = -j + -3 r r
76
i(r)
=
«.
~3 + "3 r r
-kr +
Si f +
i
e
h
r
+
Z. p. 1 *1 ^ + —r 3 4 r r
i
*i
-i+
-3
r
r
Pi +
-4
i = 0,1, (3.3.36) i = 0,1,
r
where k »* 0. Substituting these into the homogeneous equations (3.3.26)—(3.3.29) —kr and equating the coefficients of e , we have (1 - ^ ) f + + (l-2k)ft + f - f, - 0 J o 1 o 1 (2-2k)f+ + 6(1 - ^ ) f t = 0 O
D
1
+ + 4k — — fo + fx + (1 + ^ ) f o - (l+2k)fx = 0
(3.3.37)
4k -(2 +2k)f + 6(1 + =^)f, - 0. O
3
1
The set of equations (3.3.37) will have non-trivial solutions only if A(k) = 0, where
115 (1 -
3 '
(2 -- 2k)
(1 - 2k) 4k
6(1
1
-1
0
0
A(k)
(3.3.38) 1
1
^k. (1 + y)
0
0
-(2 + 2k)
-(1 + 2k) 6(1 + ^ )
This gives us k - 0, 0, ±1.8257.
(3.3.39)
The negative value of k is rejected to avoid singularity as r -> <*> [cf. boundary condition (3.3.3)] k
Let
o
= 1.8257 1
((3.3.40)
It may incidentally be mentioned that in all cases of conservative scattering, the characteristic equation (3.3.38) has roots of the form, 0, 0, ±k. (i = l,2,...n).
The choice of A(r) in (3.3.34)
reflects this feature of the problem, where we have separated the expression corresponding to the double—zero roots from that for non—zero roots.
A polynomial in (1/r) is taken both within and
without the exponential terms.
As many terms of the polynomial are
retained as could be managed in a. specific approximation subject to the boundary conditions.
However, one cannot claim any uniqueness for
the present representation especially about the starting point of the polynomial. Substituting (3.3.34) and (3.3.36) in the differential equations (3.3.26) — (3.3.29), equating the coefficients of similar terms and making use of (3.3.33), (3.3.38) and (3.3.39), we can express all the constants in terms of three independent constants, say a, h , h . These three constants can be evaluated using the three equations in (3.3.30).
Thus V"(r,/i) can be determined.
116 Now the neutron density, J ( r ) is given by , r-1
.0
J(r) - f J| i> (r,/i)dM +J f V_(r,/i)d^ J 0
-l
C ~ ^o+ *1 + *1
A(r)
—k r (a + \ F )/r + e ° [X(r)] 4 o'
(3.3.41)
(28) [cf. Wilson and Senv ' p.435-436] Hence rJ(r) = a r +
F /a 4 o
T
-k r + re ° [X(r)]
= a[r - a + r ] + re
-k r o
(3.3.42)
[X(r)].
Thus r
Marshak
calls r
o "
a +
(
the "extrapolated end point".
the boundary condition (3.3.30), r values of a.
(3.3.43)
4 Fo)/a-
As a is known from
can be evaluated for different
These have been compared with Markshak's values in the
following table.
Table (3.3.2)
r
o
Single—interval spherical harmonic method
P,—approx.
P,—approx.
modified double—interval SHM , a (28) Wilson and Sen First approx.
Marshak
(14)
a P 1 —approx. 0.5
0.286
0.323
0.334
0.288
1.0
0.400
0.480
0.494
0.429
2.0
0.500
0.608
0.620
0.587
117 The salient feature of the modified double-interval SHM is the continuity of the representation of i/>(r,n) (i - 0 at interior levels of the atmosphere.
in two half-intervals at This made the method
equally effective in the case of plane-parallel and spherical geometry.
For solving the transfer problem, there is no need to look
for equivalent boundary conditions of Marshak type.
The exact
boundary conditions could be used in this method. The modified double—interval or half—range (as it is sometimes called) spherical harmonic method has been successfully applied to solve transfer and transport problems in spherical geometry [cf Wilson (29) and Sen ] such as (a) diffusion problems, (b) problems in finite medium and (c) those in finite shells of planetary nebula. It was (30 31) also demonstrated that [Wilson, Tung and Sen ' ] the method gives the peaking effect of Eddington factor [f(r) = K(r)/J(r), the ratio of the second to zero order moment of I(r,/*)] near the outer surface of a spherical media.
3.4
Method of discrete ordinates in spherical geometry
The approximate method of discrete ordinates was developed by Chandrasekhar medium.
for solving transfer problems in plane-parallel
The principle of the method was inherent in the
Lck<2 investigations of diffusion problem by Wick ,(12) of the method was also made by Kourganoff
. A thorough analysis
Chandrasekhar's discrete ordinate method was a generalisation of the idea of Schuster (19) and Schwarzschild (21) developed for solving the idea of Schuster and Schwarzschild developed for solving Milne problem in plane—parallel medium. They represented the Milne problem in plane—parallel medium. They represented the radiation field by two streams of radiation of different intensities in opposite directions described as incoming and outgoing radiations. However, Chandrasekhar
noted that for describing complex
radiation fields, division into two parallel opposite intensities was
118 rather a gross approximation.
In Wick-Chandrasekhar method of discrete
ordinates, the radiation field was divided into 2n streams in the directions p.
(i = ±1, ±2 ... ±n) with p_.
zeros of Legendre polynomial of order 2n.
= -p.,
where p'.s were the
The integral term in the
integro—differential equation of transfer was approximately represented by the weighted average of intensities at suitably chosen points in the range of integration.
For exmaple,
-+1 n I I(r,/»)d/i - I a l{r,(t,), J -1 j=-n J
(3.4.1)
where a'.s are the weights appropriate to the Gaussian quadrature formula used. In what follows, we shall demonstrate the use of discrete ordinate method for solving transfer problems in spherical medium.
Model : We consider the problem of transfer of radiation in a spherically symmetric, conservative, isotropic scattering extended stellar atmosphere.
In most astrophysical problems the atmosphere may
be taken to extend to infinity. Usually the scattering coefficient is found to vary as inverse power of radius r, which is greater than unity. The problem of transfer of radiation in spherically symmetric extended atmospheres was first considered by Kosirev the year 1934.
and Chandrasekhar
However we shall demonstrate here the use of discrete
ordinate method in solving transfer problems of above description [cf. Chandrasekhar(5), p.364-370]. The equation of transfer for this problem can be written as
M
3I(r,/i) 3r
l-/i r
SI(r,/i) 3/i
-
I(r,/i')d/i',
(3.4.2)
in
119 where a is the mass scattering coefficient.
Other symbols have their
usual meanings. Now replacing the integrals by Gaussian sums the transfer equation (3.4.2) is reduced to an equivalent system of linear equations in a finite approximation
31. A». a-= +
1-M 2
„, ~ (^)
= ~o1-
+ 4 ff S, a. I.
(3.4.3)
with i = ±1, +2, ... ±n. As explained by Chandrasekhar
(p.365), one faces here the
problem of deciding how the quadrature formulae designed to evaluate integrals at a given number of division fi. are to be used to determine (■=—)
. Instead of going into this question, Chandrasekhar
eliminated the term Cg—)
from (3.4.3) in the following way. i
Let
Q-00
2i+l
vi0i)
-Pi+1(^)
irmypi^>-
< 3 - 4 - 4)
(,.) ^ ^ ,
(3.4.5)
Now because dP (2i+l)Pi(/.)
<M) ^
dP
we have dQjC**)
d*t
P,
and
Q,(±l) = 0,
(3.4.6)
Then integrating by parts, we have
f ^[SK-J^V"'1^
(3 4 7)
--
where i = 1,2 ... 2n.
Therefore by the method of d i s c r e t e ordinates ,31,
£n 'lV"i> V
(3 4 8)
'-
120 aT
at The equation (3 .4. 8) gives 2n equations for determining (—) The equation (3.4.8) gives 2n equations for determining (-5—) "i at Gaussian division points in terms of the functions at the same P'aints. Gaussian division points in terms of the functions at the same points. Now multiplying equation (3.4.3) by a.P'(/0 and summing over all i and using equations (3.4.4) and (3.4.8), we have d dr
h VW1! + - ~° ^
r
a
Zi iV"i . ^ i
a, P ^ ) ^ + \ o I.
a. I. Z i a. P ^ . ) ,
(3.4.9)
where £ = 1,2 ... 2n. For i = 1,2,3,4, using the known expressions of Legendre polynomials, equation (3.4.9) gives us [cf Chandrasekhar
£ - y. a.M.i. dr
L
-3.
1 1
, p.366]
+ - y. a.M.I. = 0
1
r ^1
1 1
1
^ - y. a./x 2 I. + - L a . ( 3 / i ? - l ) I . = - a Y. a . / i . I . dr **x 1 1 1 r ^i iv ' I ' 1 ^1 r i 1 ^ - X- a./*. ( 5 / ^ - 1 ) 1 . + - Y. a . / , . ( 5 / i 2 - 3 ) I . - - f a [ . a . ( 3 / * 2 - l ) I . dr
L
-i.
11
1
1
1/1x
r ^1
i
*i
3
^ 1 1 1 1
d_ X. a . / i 2 ( 7 / i 2 - 3 ) I . + - Y. a.(35/i 4 -30/i 2 +3)I. = - a Y . a . p . ( 7 ^ 2 - 3 ) I . . 11 1 1 r ^1 1 1 'I . 1 °l r i 1 1 dr ^1
(3.4.10) The first of the equation (3.4.10) integrates to give the flux F = FQ/r2.
This gives the correct flux integral for spherical problems.
(3.4.11)
F
is a
constant. Further, as by definition /i!s are the zeros of P. (/j) , we have
£. a. P„ (/i.)I. = 0. ^1
1
2n 'l
1
(3.4.12)
121 Solution
in the first
approximation :
In the first approximation, we shall deal with the first two equations of (3.4.10) with a
=a
= 1 and ^
/*_.. = 1//3. We
have
/3
^l" and
1
F
o (3
-!"^"! r
li<X+l+I-l> - "\ \
( 3 - 4 - 14)
r
3 - | (I^I^) - q f,
Hence
R
+
"413)
| FQ J R ^ r
.
(3.4.15)
r
In (3.4.15), r = R denotes the free surface of the atmosphere and we have used the boundary condition 1 = 0
at r = R,
(3.4.16)
implying the absence of external incident radiation at the outer surface. If the atmosphere is semi—infinite I . and I
will tend to zero
as r -» <», and J - l ^ f ^ . r r Solution
(3.4.17)
in the second approximation :
In the second approximation, we retain all the four equations of (3.4.10).
In the last equation, the second term on the left hand side
is zero as fi'.s in this case are zeros of
P,(fj,).
Now we may write down the equations (3.4.10) as moment equations H = (F /4)r -2 ,
35M - 30K + 3J = 0,
^ + - (3K-J) = - CTH dr r
122 %- (5L-H) + - (5L-3H) - - | CT(3K-J) dr r J and |- (7M-3K) - -
where
CT(7L-3H);
J -| j ^
I d^ - i I a.I.
H- I f
IM *• - § Z a.^I,
x
(3.4.18)
K - | f " l/d, - | £ a 1 4 l i i r+1 _ 3.
lv
3T
M - | f ] I M V - | I v*tl . Putting
X - 3K - J and Y - 5L - 3H,
(3.4.19)
the basic equation in the second approximation can be written as
-ff--Kfi?
K=
-|x-}-.
dX dr "
« + V3 dr + *Y--|aX r 3 r
(3.4.20)
and the flux integral as H = F o /(4r 2 ). Further, if a = cr
(3.4.21)
(n > 1), where c is a constant, the optical
thickness r measured from r = => inward (r = 0, r = °°) is given by
123 CO
|
r
a dr = c r~ n+1 /(n-l).
(3.2.22)
This implies that ax = (n-l)r
and
T - (R/r) n 1 ;
(3.2.23)
where r = R is supposed to denote r = 1. From the relations (3.4.21) - (3.4.23) and (3.4.20) measuring 2 intensity in units of F /R , equation (3.4.20) takes the form
1 f 5dr + (n-11 T(n+l)/(n-l)
K
n-1 J
dX r + dr
T
(3.4.24)
4(n+l)
2 7 / iv X = 4 Y(n-1)r 3
(3.4.25)
(3-n)/(n-l)
and dY 4 5 d7 " Tn^l)7 Y " 3 X
(3.4.26)
r (3-*)/(-D
n=I
'
(3
-4-26)
Eliminating Y between the equations (3.4.25) and (3.4.26), the following differential equation is obtained d2X _ 2 dX _ 2(n+3)X . 2 (n-l)r dr . ..2 2 dr (n—1) T
Chandrasekhar
7r(3~n)/(n~1) 3(n-1) '
^.*.-J/J
made the following substitutions, namely z = kr,
and X
35 9
k = /35/3
_ _ L _ k -M)/(n-l) z (n + l)/2(n-l) '
(3.4.28)
and converted equation (3.4.27) to the form of a Lommel's equation for a purely imaginary argument given by
z
2 d24> ^ d . 2, 2 , M+1 — £ + z -r^ - (z +1/ )0 ~ - z^ , . 2 dz dz
(3.4.29)
124
" = 2(5^7
where
and
"
=
2j^=T>
'
(3.4.30)
The solution of this equation is given by [cf. Chandrasekhar c 4, = I (z) |
, p.369]
pz z M K^(z)dz + K^(z) J zMIi/(z)dz, c2
(3.4.31)
where I (z) and K (z) are the Bessel functions for purely imaginary argument. The arbitrary limits c. and c. are determined from the following facts. (a)
The study of the asymptotic behaviour of I (z) and K (z) as
z -> => suggest that c. = =>, since none of the quantities tend to infinity exponentially as z -» °°. (b) All quantities vanish at z = 0 . z (n+l)/2(n-l)^ (z)
^
However, K (z) diverges at z = 0.
Q
as
z
Then ^
(3.4.32)
Q
Hence for (3.4.32) to be true
c 2 - 0. Hence CO
Z
c4 - I (z) I z p K (z)dz + K (z) I z^I (z)dz. z U
(3.4.33)
Thus knowing 0, from (3.4.28) we can calculate X and using the equation (3.4.24) we obtain K and remembering that X = 3K — J, we can calculate J. It is clear that in any finite approximation, the part played by discrete ordinate method is to convert integro—differential equation of transfer for diffuse radiation in a spherically symmetric medium to a set of linear differential equations.
This can be done effectively (5) (22) by the scheme suggested by Chandrasekhar . Subsequently Sen
125 extended the method to the solution of transfer equations in spherically symmetric, non—conservative scattering medium. Chapman
(2)
has demonstrated that in solving problems of spherical
geometry, the single interval spherical harmonic method and Chandrasekhar's discrete ordinate methods are equivalent.
He adopted
a procedure similar to that of Chandrasekhar and used the single interval spherical harmonic method to solve the transfer equation in the first and the second approximations for an infinite spherically symmetric scattering atmosphere.
Using numerical methods, he computed
the source function for different r and compared them with those obtained from formal integration of transfer equation.
In the formal
solution, he evaluated the source function using the classical Eddington approximation 3K = J where K and J are the second and the zeroth moments of intensity.
He then used the iterative process to
find the approximate source function.
He also predicted that the
radiation becomes very sharply peaked near the surface and concluded that the single—interval spherical harmonic method failed to show this "peaking effect".
Hummer and Rybicki
developed a numerical method
for the solution of radiative transfer problem in a spherically symmetric medium.
Their scheme is based on iterating on an assigned
space dependent Eddington factor f(r) = K(r)/J(r), which was presupposed to take care of the "peaking effect" suggested by (2) Chapman . The essential feature of the method lay in assuming a form of f(r) which monotonically increases from T to 1, and developing an iterative scheme for calculating successive values of J(r) and (2) f(r). However, their scheme and also that of Chapman failed to give correct flux integral for spherical medium. /no\
It was also seen [Wilson, Tung and Sen
] that the Eddington
factor f(r) calculated from the moment equation, with J(r) obtained from double—interval spherical harmonic method, shows all the essential features of the "peaking effect".
It is also found that
J(r) obtained by this method compares well with those of Hummer and Rybicki
. Furthermore, in this method, the flux integral for
spherical geometry is exactly satisfied.
126 3.5
Discrete ordinate matrix method In the discrete ordinate method described above and for that
matter in the spherical harmonic methods, the general solution of transfer equation contains terms which increases exponentially with increasing depth, in addition to those decreasing with depth.
While
the presence of such terms does not create any problem for solutions in semi—infinite medium, in finite medium of moderate or large optical depth, they give rise to numerical difficulty.
The analytical
approaches described in §3.2, §3.3 and §3.4 of the present Chapter are workable only when the convergence of the series represention of the specific intensity (in the spherical harmonic methods) is rapid or the number of divisions of the variable p. into ^.(i = ±1, ±2 ... ±n) are not too large (in discrete ordinate method).
However, if this is not
the case, the number of simultaneous equations one has to deal with becomes unmanageably large for analytical solution and one has to appeal to numerical methods.
In the transfer problems in finite
medium with moderate and large optical depths, the presence of exponentially increasing term in the solution poses a formidable problem for the computer on account of the unwieldy size of the word length.
Considering this Mihalas
concluded that discrete ordinate
method was not suitable for numerical calculations. Wehrse
Schmidt and
proposed a "Discrete ordinate matrix method" for
eliminating the increasing exponentials by a matrix formulation, so that the resulting equations can be amenable to numerical solutions. The transition and reflection operators as well as the source function were derived free from terms exponentially increasing with optical depth.
For this Schmidt and Wehrse
expressed the general solution
of transfer equation in terms of a transition matrix. developed by Schmidt and Wehrse
The method
is designed to bring the discrete
ordinate and spherical harmonic method within the orbit of numerical techniques particularly in the case of transfer problems in finite medium of moderate and large optical thickness. discussed in detail by Schmidt and Wehrse
The method is
(p.341-368).
127 References
Optics and spectroscopy 14, 285, 1963.
1.
Barkov V. I.,
2.
Chapman, R. D., Ap. J. 143, 61, 1966.
3.
Chandrasekhar, S., Ap. J. 94, 444, 1934.
4.
Chandrasekhar, S., Ap. J. 99, 180, 1943; Ap. J. 101, 95, 1945.
5.
Chandrasekhar, S.,
6.
Davison, B. and Sykes J. B.,
Radiative transfer, Clarendon Press, Oxford,
1950; also Dover Publications Inc., New York, 1960. Neutron Transport theory, Clarendon
Press, Oxford, 1958. 7.
Eddington, A. S.,
Internal Constitution fo Stars, Cambridge,
England, 1926. 8. 9. 10.
Gratton L.,
Soc. Astron., Italy, 10, 309, 1937.
Gross, E. P. and Zeiring, S.,
Ap. J. 123, 343, 1955.
Hummer, D. G. and Rybicki G. B., Mon Not. Roy. Astron . Soc. 152, 1971.
11.
Kosirev, N. A.,
Ap. J. 94, 430, 1934.
12.
Kourganoff, V.,
Basic methods in transfer problem, Oxford
University Press, 1952; also Dover Publications Inc. New York, 196 13.
Mark, C , Phys. Rev. 78, 558, 1947.
14.
Marshak, R. E.,
Phys. Rev. 71, 443, 1948.
15.
Max Krook, Ap. J., 122, 488, 1955; 130, 286, 1959.
16.
Mertens, R., Simon Steven Supplement 30, 1954.
17.
Mihalas, D.,
18.
Poisy, P., Ann.
19.
Schuster, A., Ap. J. 21, 1, 1905.
20.
Schmidt, M and Wehrse, R., Numerical Rad. Transf. Edited by
Stellar Atmospheres, Freeman, San Francisco, 1978. d'Astrophys. 21, 151, 1958.
Kalkofen, W. Cambridge Univ. Press, 1987. 21.
Schwarzschild, S., Gottinger Nachristen, 41, 1906.
22.
Sen, H. K.,
Ap. J. 110, 276, 1949.
23.
Sen, K. K.,
Proc. Nat. Inst. Sc. India, 20, 12, 1954.
24.
Sykes, J. B.,
25.
Wang,
Mon. Not. Roy. Astron. Soc. Ill, 377, 1951.
M. C. and Guth, E., Phys. Rev. 84, 1092, 1951.
128 26.
Wick, J. C , Zs. f. Phys. 120, 702, 1943.
27.
Wilson, S. J. and Sen, K. K.,
Publ. Astron. Soc. Japan 15, 351,
1963. 28.
Wilson S. J. and Sen, K. K.,
Canad. Jour. Phys. 43, 432, 1965.
29.
Wilson, S. J. and Sen, K. K.,
Ann. d'Astrophys. 27, 46, 1964; 27, 264, 1964; 28, 348, 1965; 28, 855, 1965.
30.
Wilson, S. J., Tung, C. T. and Sen, K. K.,
Mon. Not. Roy.
Astron. Soc. 160, 349, 1972. 31.
Wilson, S. J., Tung, C. T. and Sen, K. K., Japan 27, 611, 1975.
Publ. Astron. Soc.
129 Chapter IV MOMENT METHOD
4.1
INTRODUCTION In moment methods, instead of the specific intensity, the angular
moments of the specific intensity are used and the integro-differential equation of transfer in specific intensity is replaced by a set of ordinary differential equations of its moments. The methods of Schuster( Feautrier
',
Schwarzschild^
, Eddington^
,
are all examples of moment method for solving transfer
and transport equations.
In fact, the spherical harmonic and the
discrete ordinate methods serve the same purpose of converting the integro-differential equation of transfer to a. set of ordinary (12) differential equations. Max Krook asserted that the three methods are essentially equivalent.
However, in view of the wide variety of
uses of the moment method and the number of modifications that have been proposed for it, it is thought that the method deserves consideration in a separate chapter.
In this chapter, we shall mainly
be concerned with its utility in solving transfer and transport problems in curved geometry
We shall consider in particular transfer
problems in spherical medium. We recall that the relevant equation of transfer for diffuse radiation in an axially symmetric spherical grey medium is given by [cf equation (1.3.28)]
„ "g^
+
I=M! aigwo _ ^ ( r ) [ I ( r i M ) _ s ( r ) ] i
(4.1.1)
where the source function ?(r) is given by equation (1.3.29), the other symbols have their usual meanings. The transfer equations may have to be solved under diverse types of boundary conditions.
Some of the standard ones are detailed below.
130 For example, let us consider a spherical shell medium bounded by radiiL r
and R(0 < r < R).
We recall tl that the specific intensity is
I(r,/i), where /J = cos 0 [cf. Fig 4.1.1].
Fig
4.1.1
(a) The absence of inward incident radiation at the outer surface implies I(R,-/i) = 0
for 0 < /i < 1.
(4.1.2)
(b) At the inner bounding surface, either (i) the diffuse radiation field entering the medium may be speficied, i.e.
I(r. ,(i) given for 0 < (i < 1,
(4.1.3)
or (ii)
the surface at r.. may be a reflecting surface with reflectivity
7(0 < -y < 1), i.e.
I(r1,/j) = 7l(r
-A<),
0 < /» < 1.
(4.1.4)
More complex boundary conditions may occur depending on the problem at hand. In the moment method, the transfer equation (4.1.1) is multiplied by /J (n = 0,1,2...) and integrated over ^ within the range (—1,1). For each n, we shall have an ordinary differential equation connecting moments of intensity of different orders.
Thus we have a set of
ordinary differential equations in r whose number depends on the value of n at which we chose to terminate. The moments are usually labelled as follows :
131
i r J = j I
I(r,/i)d^, (mean intensity, zeroth moment),
(4.1.5)
r+l
F = 2
+1 I(r,^)n d/x, (Flux, first moment) or H = T | I(r,/j)/x d/x, J -1 - l (4.1.6)
i r +1 K = j
2
I I(r,/i)/j dfj., (K-integral, second moment), -1
(4.1.7)
and higher moments
r+1 L = 2 I
M
w
= -;
+1
-1
3 I(r,/i)/i
d^i,
(4.1.8)
4 I(r,/i)/i d/i,
(4.1.9)
and so on. Now if for example, we set up two ordinary differential equations in moments of specific intensity by successively multiplying the transfer equation (4.1.1) by unity (fj. ) and /* and integrating over the range —1 to 1, the number of unknown moments occuring in these equations is three.
This feature of the number of unknowns being one
more than the number of equations set up to determine them is basic to the moment method in transfer problems.
So to solve any problem by
this method one needs to invoke a relation between the moments.
In
plane—parallel medium, this difficulty was circumvented by appealing to what is known as Eddington relation 3K = J, a relation between the zeroth and the second moment of specific intensity.
In fact, this
followed directly from Milne-Eddington approximation for solving transfer equation in slab geometry. to hold throughout the atmosphere.
Eddington's relation was assumed While this approximation yielded
good results in some special cases as for example in the case of Sun, its validity is questionable in many other problems of slab geometry.
132 In spherical geometry, Chapman
(3)
showed that the assumption of
the validity of Eddington relation throughout the medium is in error. A "peaking effect" of Eddington factor f(r) = K(r)/J(r) towards the surface was observed by Chapman , Hummer and Rybicki and Wilson, (23) Tung and Sen . Subsequent works on transfer problems in spherical geometry took account of the variable nature of Eddington factor.
4.2
Methods of solution using variable Eddington factor, f(r) (3) Chapman's revealing result was first expressed in his Ph.D.
thesis on "Radiative transfer in extended stellar atmospheres" method he employed was boundary conditions.
The
spherical harmonic method with equivalent It is essentially analogous, in principle, to
the single-interval spherical harmonic method described in Chapter III.
He reiterated the equivalence of spherical harmonic, discrete
ordinate and moment methods in solving the transfer equation in the same order of approximation.
He established the inadequacy of
constant Eddington factor assumption throughout a spherical medium and predicted a peaking effect of the Eddington factor f(r) towards the surface.
The mathematical methodologies involved in his calculations
were a combination of single interval SHM, discrete ordinates, moment methods and a formal solution of spherically symmetric transfer equation. Hummer and Rybicki
used a variant of Eddington approximation
with the explicit intention of incorporating in the process of solving transfer equation, the notion of peaking effect of Eddington factor f(r).
They started from the moment equations, one of which contained
f(r).
Initially the Eddington factor f(r) was assumed throughout the
medium to be either 1/3 or l-(2/3)exp(-/3r) , where /3 is the inverse of the radius at which optical depth is unity.
A numerical iterative
procedure was developed to calculate the successive improved values of f(r) and the source function.
The method employed by them is strictly
a numerical method of very high precision. detail in Chapter VII.
It will be discussed in
133
Wilson, Tung and Sen
(23)
also started with the moment equations.
They introduced f(r) into one of the moment equations and solved the moment equations for obtaining f(r) and ?(r) by the modified double-interval spherical harmonic method described in Chapter III. 4.3
Variable Eddlngton factor methods The variable nature of Eddington factor to a certain extent
complicates matters in the solution of transfer problem in spherical media.
However, in the moment method the problem of discontinuous
specific intensity at the free surface is avoided by averaging over angles.
Even though Eddington approximation proved fairly successful
in many problems of slab geometry, it was recognised early that direct application of this approximation would not yield reliable results (13) (2) near the surface of spherical media. McCrea and Chandrasekhar near the surface of spherical media. McCrea and Chandrasekhar constructed an approximate scheme in which for a thin atmosphere constructed an approximate scheme in which for a thin atmosphere surrounding a central star of radius "a", the radiation field was averaged over two separate regions p. < /x < 1 and — 1 < fi < p. , where 2 r r p.
= [1 — (a/r) ]. However, it was found that for this method to be
of any success, judicious choice of p. was of extreme importance. Moreover, the method failed for atmospheres of dense or moderately (9) (2) dense optical depth.
The works of Kosirev
and Chandrasekhar
were the fore-runners of the methodologies for solving transfer We shall illustrate below the scheme of combining and analysing problems in spherical media. the asymptotic solutions of two regions for solving transfer problems We shall illustrate below the scheme of combining and analysing in a grey spherical atmosphere. This rough analysis throws ample the asymptotic solutions of two regions for solving transfer problems light on the effect of sphericity and to demonstrate this we shall in a grey spherical atmosphere. (14) This rough analysis throws ample follow the treatment of Mihalas" [p.245-246]. light on the effect of sphericity and to demonstrate this we shall / (14) follow the treatment of Mihalas" [p.245-246]. The transfer equation for a grey, axially symmetric, isotropically / scattering spherical medium is given by The transfer equation for a grey, axially symmetric, isotropically scattering spherical medium is given by
ajj^l ^
3r
+
i V ai^Bl r
dp
_ ,(r)[I(r,ri _ ,(r)]i
(4.3,!)
134 where a(r) anad ^(r) are the attenuation coefficient and source function respectively.
The other symbols have their usual meanings.
For the isotropically scattering medium, the source function S(r) is equal to mean intensity;
i.e.
i r 3( (r) = i j
+i
I(rjOd/i = J(r).
Multiplying the transfer equation (4.3.1) successively by n
and fi and
integrating over /i in the range (—1,1), we have the following moment equations. dF(r) dr
2F(r)
0,
^ g l _ | [j( r) _ 3K(r)] - - J a(r)F(r) .
(4.3.2)
(4.3.3)
Introducing the Eddington factor f(r) = K(r)/J(r),
(4.3.4)
the equation (4.3.3) becomes |p (fJ) + (3f-l) l - - J c(r)F(r).
(4.3.5)
It may be mentioned that in deducing equation (4.3.2) the condition of radiative equilibrium has been invoked. Equation (4.3.2) yields th flux integral F = FQ/r2.
(4.3.6)
We define the optical depth
r(r) = J
a(r')dr',
(4.3.7)
135 where R Is the radius of the spherical medium. r »
We assume R »
1, the optical depth being measured inwards from r = R.
r and In this,
the radiation field is assumed to be isotropic and Eddlngton approximation is supposed to hold good, i.e. , f -» -~ . Hence the equation (4.3.5) becomes _ 3 Q-F 4
dJ dr
3 2 aF /r . 4 o
(4.3.8)
- j
This implies
3 f dr' 4 L0 r2
J(r) = F
(4.3.9)
+ C
The Eddington type boundary condition in the Milne—Eddington approximation [cf. Kourganoff
Chapter 1, p.87, Equation (19.4B)] is
given by F
1 J(0) - i F(0) - 14 -f
(4.3.10)
R
Then J(r)
F o
3
4R
f
2 2 (R /r )dr' + 2
(4.3.11)
0
This result was derived by Chandrasekhar
(2)
In LTE, 4
co
J(r) = f B (T)cU- = J 0 V
(4.3.12)
—
This relation is valid only deep down the medium r » surface, f -> 1.
This gives us
Near the
Then equation (4.3.5) reads as dJ dr
"^
1.
2J r
_1 4
o(r)F
fe (r2j(r)) H
\ a(r)Fr2 =
" 4
Q(r
>Fo
(4.3.13)
136 J(r) = I (Fo/r2)(r+c).
(4.3.14)
The boundary condition at free surface when f -» 1 is given by
(4.3.15)
J(0) - \ F(0) = i ( F 0 A 2 ) In this case, I(r,^) - I(r)5(^-1) and this implies that J(r) = K(r) = i F(r) as r -> R.
Hence
J ( T ) = i (FQ/r2)(r+l)
a result valid for <■ « 1, r = R. Following Kosirev (9) and Chandrasekhar (2) , we Following Kosirev and Chandrasekhar , we of opacity and take a(r) = C r of opacity and take a(r) = C r Then r(r) from equation (4.3.7) is given Then r(r) from equation (4.3.7) is given r(r).= C
(4.3.16)
consider the power law consider the power law
by by (4.3.17)
n-1 n n—1
taking, for simplicity, R as very large (R -» ■»). form of J as r -» 0, r »
Then the limiting
1 is
F T 3(n-1) o J(r) -* 4(n+l) 2 r Larson
(4.3.18)
interpolated between the expressions of J(r) in
(4.3.18) and (4.3.16) and deduced an expression for J(r) in the intermediate region as r3F , J(r) = ■
2
Ur J
n-1 n+1
, 1 ,n+l. 3 <5=I>
(4.3.19)
T +
This expression gives fairly accurate values of J ( T ) for ordinary run of T even in the regions away from the two extremes.
Comparison of
the results obtained by approximate methods is usually made with those from the numerical method of Hummer and Rybicki
.
137 4.4
Generalised Eddington relations in radiative transfer in spherically symmetric medium We have mentioned that the idea of variable Eddington factor in
transfer problems in spherical medium has been effectively made use of in the precise numerical method of Hummer and Rybicki
and that it
is built in into the modified half—range spherical harmonic method of (21) However, Unno and Kondo made a generalisation of
Wilson and Sen.
Eddington approximation by introducing an angular variable p. which allows the radiation field to be divided into two distinct streams. This n
is then found from the solution of the equation of transfer.
The radiation field was described approximately in terms of the moments of specific intensity, J (the mean intensity) and F (the flux).
The second and the third moments were expressed in terms of J
and F through the generalised Eddington relations.
In a way, this
method may be said to have evolved from the classical works of (13) (2) McCrea and Chandrasekhar for McCrea and Chandrasekhar for solving solving transfer transfer problems problems in in spherical geometry. However, /* in Unno and Kondo's method is much spherical geometry. However, p. in Unno and Kondo's method is much more flexible than that used by earlier authors as it is determined from the equation of transfer. (21) We state below the basic features of Unno and Kondo's method with reference to a radiative transfer problem in a spherically symmetric, anisotropic scattering participating medium. The two stream model of the specific intensity is taken as I+(r)
for
1 > p. > fi (4.4.1)
I (r,/x) I (r) v cos
for
/x < u < -1, r
p being the angle between the direction of the radiation beam of
frequency v and the radius vector at r from the centre of the sphere.
138 u is a function of r and 1/ to be determined from the transfer r equation. Then 1 r+1 1 J„ = g j ^ I y ( r , M ) * i - 2
,+1 :-' MS ■= ■ |
„(r)
=
Iu(x,fi)n
dp
(I + +I ) - p ( I + - I )
(4.4.2)
a-ftK-v
(4.4.3)
i / i /
U , V r .M)M 2 dM - §
r
J/
1/
(I++I ) - /x3(I+-I ) 1/ 1/ r v v
(4.4.4)
a-^x^-V
(4.4.5)
and
L/r)
r+1 = 2 J lw{r,n)p
3
1 dp = ±
From these relations, it follows that
3K
- J
1/
LJ
i/
Unno and Kondo
(21)
= %p
F,
(4.4.6)
1 2 = i (1 + /i )F .
(4.4.7)
1/
2
2
r
1/
1/
r
described these conditions as generalised
Eddington approximation. The boundary condition of the absence of incident radiation at the outer surface at r = R is given by I (R) for 1 > u > u 1/
I (R.M) -
I~(R) v
0
r
for u > u > 0 r
for
(4.4.8)
-1 < p < 0.
This leads to the relations
MR JVW
~ 2 < 1+ V F ,/ (R) + 3K^(R) = 0
(4.4.9)
139 and MR(1+^R)J^(R) - \
(l+/iR+^)Fi/(R) + L^R) = 0.
(4.4.10)
The equations (4.4.9) and (4.4.10) in view of the generalised Eddington approximation expressions (4.4.6) and (4.4.7) reduce identically to F (R) = 2(l+,OJ (R).
(4.4.11)
In addition to this, we shall have to deal with a boundary condition at the inner surface. The equation of transfer for diffuse radiation in a spherically symmetric participating medium is written as
a
vr^
" — a i — = —a „
i V avr'"> +
~f
^
V r ^ } + \ \ + o\ J. 1 P„(<*,^)Itf(r,p')^',
(4.4.12)
-1 where k U
a
and a V
are the absorption, scattering and the V
attenuation coefficients respectively and a p (ju,/ir) is the phase function given by
+ a ,w
=k V
V
V
p^/x./i') = 1 + X ] = 1 a
tne
= a fa V
V
U
(4.4..13)
Legendre polynomials.
The following thr e moment equations follow from the transfer equation (4.4.12). dF -r-^ + - F = - k (J -B ), dr r i/ i/ ^ i/
-r-^ + i ( 3 K - J ) = - i a ( l - i w a (1) )F dr T v v 4 i/ 3 t i/ »/
(4.4. 14)
(4.4..15)
140 and
~
dr
+ - (4L -2F ) = - % a l [(3K -J )+(l-w ) (J -B >- r? w a(2)/j F ] .
r
v
3 v
v
v
v
v
v
v
10 v v
Xv
(4.4.16) Eliminating K
and L
from the equations (4.4.14) — (4.4.16) by
equations (4.4.6) and (4.4.7), we have
\%z
(r2
V =-4WV-
(4 4 17)
--
r -r-^ + - ^ ^- (r3/i F ) = - r o ( l - i « a (1) )F dr „ 3 dr r i/ 4 v 3 v v f 2r
(4.4.18)
and dM 1-M v^ = dr ru
1+3/i 1 (2) r d 2 u; - I [a (1 - i u a ) + -; ^- (in r F) ] . 3 L v b v v 4u dr 2
(4.4.19)
In the generalised Eddington approximation, the equations (4.4.17) — (4.4.19) are the basic equations of transfer in the moment form for spherically symmetric medium. that ft
The special point to note is
is to be calculated from the transfer equation.
In the case of an extended spherical shell medium (bounded by r = a and r = R ) , scattering isotropically and the medium being grey and in radiative equalibrium (J = B ) , the boundary condition at the surface of the inner shell r = a can be written as
and
F - F a
(4.4.20) '
M r = M a = 0.
(4.4.21)
(4.4.21) ensures adherence to the classical Eddington approximation at the inner boundary [cf. (4.4.6)].
141 Then from (4.4.17), (4.4.20) and the radiative equilibrium condition, we have 2
r
2
(4.4.22)
a
and
1
J - \ a2F 2 a
2+ (1+/OR
"R "r 3 fR , ~2 _ ~2 + 2 J (a + R
J
r
r
2
Vdr ~)-2
(4.4.23)
rJ
[From equations (4.4.6), (4.4.11), (4.4.18) and (4.4.22); The equation (4.4.19) reads as
da dr
i
1-M|_t
2 " 3
a
'
(4.4.24)
The attenuation coefficient is supposed to be a given function of r. Unno and Kondo
(21)
calculated p. by Runge—Kutta—Gill method for
different values of a and concluded that RKG method shows instability 2 near r = a unless the spatial step A satisfies A In r < 9/(2ar) . For 2 p < 0.1, they approximated p. by
2 i
r
2 2 (1-a/r ) 1 + (4/9)a2r2(l-a2/r2)
™ -
(4.4.25)
They found that in the optically thin limit of a -> 0, \i = [l-(a/r) ], r (2) a result agreeing with that of Chandrasekhar And in the optically thick limit of a •» «, p. Eddington approximation.
is zero in keeping with the classical They also predicted the existence of peaking
effect of the Eddington factor towards the outer surface of the spherical medium. Assuming that a = r , Unno and Kondo compared the values 2 of r J obtained by the generalised Eddington approximation method for varying R with those from the exact numerical solution of Hummer and Rybicki (8) .
For thick shell (n = | , R = 100; n = 2, R = 30 and
142 n - 3, R = 10) and for thin shell (R = 0.1, n = 3/2, 2, 3), there were fairly good agreement between the two sets of results.
They also
studied the variation of J, the mean intensity and f, the Eddington factor with r for n = 2 and R = 10, and compared the values with those of Hummer and ERybicki
and Wilson and Sen
The results were in
good agreement. (21) For practical applications, the method of Unno and Kondo seems to be farily versatile and accurate.
The main advantage of the
method is the simplicity of its execution as compared to the other purely numerical methods. The fact that the dividing /J between the two streams of radiation is found from the solution of the equation of transfer ensures that any error arising out of the two stream representation of intensity is greatly smoothed out. It will be shown that with proper adjustment, this method could be used to tackle transfer problems of much more complexity in spherical medium.
Radiative
transfer
in spherical
dust shells
with a core :
(24)
Wilson, Wan and Sen examined the possibility of extending (21) Unno and Kondo's scheme by dividing the radiation field into three streams rather than two.
This division was suggested to take care of
the shadowing effect of the central core [cf. Chou and Tien and Sen
; Leong
] so that the exact boundary conditions could be used.
The radiative transfer problem in a non-grey spherical dust shell (r
< r < R) surrounding a core star of radius a (a < r1 ) was studied. The equation of radiative transfer for the diffuse radiation at a
frequency v is given by equation (4.4.12), where the symbols have their usual meanings.
The phase function for scattering p (/J,/J') is
given by equation (4.4.13). We represent the radiation field by a three stream model where the specific intensity represented by each stream is the average over each region.
We write
143
Ix(r)
f o r / i r < /x < 1
I(r,/i) = { I2(r)
for
0 < /i < p.^
^ I,(r)
for
- 1 < /i < 0.
(4.4.26)
Using t h i s , we have ,+1
'.w-ij
Ii/(r,/i)d/i = ^
(I1+I3) -
Fi/(r) = 2 J
I^
(iri3) + M ^ W
,(r) = I J , V 1 -"^ 2 d^ - \ .+1
'^(r) = 2 J
I ^ r . / i ) / * 3 d/i = |
d1+i3)
^ ( ^ - I ^
+ /ir(i2-i1)
(4.4.27)
d1-I3) + Mrd2-I1)
and +1
,
( r )
~ \ \
, ^ < r ^ > M 4 d/i =
From ( 4 . 4 . 2 7 ) , i t f o l l o w s
^
(I1+I3) +
A»r(I2-I1)
that
2L - F = 2/i (3K - J ) . i/ i> r u v (4.4.28) 5M - 3 K = /i (3K - J ) i/ i/ r J/ f
(4.4.28) gives us the generalised Eddington approximation in this model. Then writing out the four moment equations from the transfer equation (4.4.12) and eliminating L and M by (4.4.28), we have [cf. v (24) " Wilson, Wan and Sen^ ' p.100-101].
144 dF -JL dr
+ - F = - 4k ( J - B ) . r v v v v
dK ^ dr
+ r
Q
i dF 1 ^ 2 dr~
=
+
(3K - J ) v v
^ d dr
V
2T
u (3K - J ) r v v
(4.4.29)
1 — ■=■ w a 3 i/f
(4.4.30)
4u (3K - J ) r v v
+ -
r
4a j ^ [(3K - J ) + ( 1 - u ) ( J - B ) - i w a ( 2 ) ( 3 K - J ) ' 4 i/ i/ i/ i^ i/ 5 1/ 1/ »/ i/ 4~~
(4.4.31)
and
3 i/ 5 dr~
Q
V
where a
u (3K - J
a dr
+
rF ~ 2
v
)
u (3K - J ) *r i/ i/
a a + ft (3K - J ) + a, < j - ^ — + - 4 ^ - | 5 ( 3 K - J ) r i/i/ i/1 5 35 ; i/i/
= k
i/
+ a
, u> i / i /
-
]}]■
(4.4.32)
— o /a , a. (i) (i = 1,2) are the coefficients of tr y v
Legendre polynomial expansion of p (ft,ft')
[cf. equation (4.4.13)].
The
other symbols have their usual meanings. The boundary conditions of the problem can be stated as
Krr/i)
I1(R)
for
ftR < ft < 1
= {■ Ii2, ( R )
for
0 < ft < ftR
-o
for
- 1 < u < 0,
(4.4.33)
145 and ' 0
Kr 1 ,M)
■
1I , - 1
for
fi
for
0 < n < a
< /i < 1
II, = I 3
for
r
(4.4.34) r
1
l
-1 < u < 0,
l
1 2 (1-a /r ) [cf. Unno and Kondo^21^ p. 696].
with fi
From (4.4.27), (4.4.33) and (4.4.34), we have 6Kj/(R) - Fi/(R)(1+^R) + 2»R J^R) = 0, 6K (r..)(l-Mr v
1
i
) + F (r.)(l+^r +» r "
i
1
) - 0,
(4.4.35)
(4.4.36)
-i
and 2 ( 3 ^ ) -yrL)) - ^
where
F,^) - 0 ,
2 = /i (1 - a //r2,1/2 ) ' .
A»
(4.4.37)
(4.4.38)
1 F
and K /J
for this spherical dust shell problem can be calculated
from the equations (4.4.29) - (4.4.32) and the boundary relations (4.4.35) - (4.4.38).
Takeuti
(20)
drew attention to the fact that the slope of the
dependent variable fi
at the
inner boundary is very steep thus
complicating the direct numerical integration of the equations (4.4.29) — (4.4.32).
This difficulty was taken care of by introducing (24) a two—shell method [cf. Wilson, Wan and Sen ]. In the region near the inner boundary, equations (4.4.29) — (4.4.31) were solved with assigned fi
given by
2
2
1/2
/• - (1 - aVr )
146 together with the boundary conditions (4.4.36) and (4.4.37).
This
scheme of calculation was confined to a narrow region near the inner boundary where the shadowing effect of the core star was relevant. Beyond this region equations (4.4.29) - (4.4.32) were solved treating p.
as an unknown dependent variable.
The boundary condition (4.4.35)
was also used.
J , F and u were taken to be continuous at the v v r interface between the two regions. The flux, F and the Eddington factor K /J were calculated by the present method for the two—shell v v and thin shell approximation for a purely absorbing medium of finite thickness.
The Eddington factor exhibited "peaking effect" as
expected. It was found that the accuracy of this method in (21) calculating J and K /J was comparable with that of Unno and Kondo v v v (p. 699, model Al). The two methods described above are very simple and flexible in execution and they prove very useful in tackling relatively complex transfer problems in spherical atmospheres.
We shall consider next
the problem of light scattering by an optically thin, inhomogeneous, spherically symmetric planetary atmosphere illuminated radiation.
by solar
Approximate solutions of this asymmetrically illuminated
problem have been obtained by Minin and Sobolev
, Smoktii
,
Germogenova et al
and Bellman et al . We shall first consider (19) the method proposed by Sobolev for solving such problems and then solve the problem by the method of generalised Eddington approximation (24) [cf. Wilson, Wan and Sen ]. 4.5
Light Scattering by Planetary Atmosphere We shall assume that the planet is of radius r.. and is
illuminated by solar radiation with flux TTF. through an area normal to the direction of the sun.
An arbitrary point in the atmosphere of the
planet is located by its distance from the centre of the planet and the angle its radius vector markes with the direction of the sun. angle is designated by ij>.
The
147
Fig 4.5.1 The Fig. 4.5.1 defines the coordinates for a spherical atmosphere of radius R surrounding a planet of radius r.. . Parallel arrows indicate the direction of solar radiation. The direction of radiation at the given point P is designated by two angles 6 and
Hence
the specific intensity will be a function of (v,ij);9 ,ip) I = I(r,iM,
(4.5.1)
The scattering is taken to be anisotropic.
Then the transfer
equation for diffuse radiation can be written as [cf. equation (1.3.35) - (1.3.43)]
„ 31 sin 6 cos cp dl cos B 7— + 77 3r r dtp
sin 9 51 77 r 38
cot V> sin 6 sin
where a(r) is the attenuation coefficient.
The equation (4.5.2) is
the basic equation of transfer for diffuse radiation in an anisotropically scattering, spherically symmetric planetary atmosphere illuminated by parallel solar rays. In this the source function ?(r, V>; 8 ,
148 -2TT
TT
f(r,V>;«,' J( I(r,iM,
p(7*) is the phase
and ($ ,
cos 7' = cos 6 cos 6' + sin 0 sin 6' cos(
(4.5.4)
B (r,V>;0, F . p( 7 )e" T ,
(4.5.5)
cos 7 = — cos 9 cos V + sin 0 sin ^ cos (p.
(4.5.6)
where
T is the optical depth from the outer surface of the atmosphere to the given point. „R, -2 „, 2. 2,1/2 T = I"1 (r" sin^V A+ O " ' " dz rcosV1
(4.5.7)
where R
2 2 2 1/2 = (R - r sin i>) ' .
(4.5.8)
If the optical depth is measured from the sun to the given point of the atmosphere,
T
! a(r2 sin2V> + z 2 ) 1 / 2 dz. rcosV1
(4.5.9)
From the equations (4.5.2) and (4.5.3) under appropriate boundary conditions, one could, in principle, derive an integral equation for ?.
However, the resulting integral equation is rather complicated.
149
Sobolev
(19)
suggested an approximate method for solving this
transfer problem.
This was basically a moment method, which we
outline below. To start with he approximated the phase function as p(7) = 1 + p
cos 7.
(4.5.10)
With the phase function of the type (4.5.10), equation (4.5.3) reads as 9 = uJ + op
H cos 6 + o>p1 G sin 6 cos
+ x F.(l — p, cos 8 cos V + Pi sin 8 sin V
cos
(4.5.11)
where we have taken
J = fl ^
,
H = f I cos 8 ~
,
G = f I sin 8 cos
(4.5.12)
Hence the source function £(r ,i/>;8 ,ip) and the specific intensity I(r ,ip;8 ,
4JT
4TT
directions and we have 3H 2 „ 1 9G cot i> „ .. .. au -T — + - H + --7-7 + 2- G = -a(l-u>)J t+ 7— F.e , 3r r r dip r 4 1
(3-«p1)H - - i ~
(3^pl)G
[cf. Sobolevv
' p.220].
- I F l P l cos * e"T,
_ - i_ " + I F.Pl sin * e T
,. _ ... (4.5.13)
(4.5.14)
(4.5.15)
150 Now we shall assume that a = a(r) and (3-opj) are constants in the atmosphere.
Then substituting H and G from (4.5.14), (4.5.15) into
(4.5.13), we have an equation in J(r,V0 given by V2J
_ ^M ¥- = ° 2 (k 2 j - f) a(r) 3r
(4.5.16)
,2 . where V' is the Laplacian operator
1
(4.5.17)
(3-«P1)(l-w),
t'(r) is the derivative of a with respect to r, and
f= T 4
F
-e
T
P
3 - wp1 +
l
Sl
or
l
" * dT dip
P
l
COS
a
* 8T dr
(4.5.18)
Remembering from (4.5.9) that CO
,2 . 2 , 2,1/2, a(r sinV' + z ) dz. 1
rcosV we have
f = I F. e
[3 + (l-w)Pl]
(4.5.19)
The boundary conditions for solving (4.5.16) are : (i)
absence of diffuse incident radiation (—1 < \J, < 0) averaged
over the angular distributions at the outer boundary, i.e.
j
= :2H,
(4.5.20)
(This is a consequence of classical Eddington approximation 3K = J, which holds strictly for plane parallel medium in the interior), (ii) the diffuse radiation reflected to the medium at the inner boundary r = r. is a fraction of the total radiation incident on it.
151 This implies -T(r V ) J + 2H - a[J - 2H + F. cos V e ],
where a is the albedo for the planetary atmosphere.
(4.5.21)
The approximated
equation (4.5.16) is solved to determine J under the boundary condition (4.5.20) and (4.5.21) of which (4.5.20) is relevant to the plane parallel medium.
Once J is known, we can obtain H and G from
equations (4.5.14) and (4.5.15).
Then ? can be found from (4.5.11),
where the last term is now replaced by j F.p(7)e"T. p(7) is the actual phase function. Once y is determined, I can be calculated by the direct integration of the equation of transfer (4.5.2).
Solution of the equation when the attenuation exponentially with altitude :
coefficient
Of several particular cases considered by Sobolev
(19)
decreases
one was
that of a(r) decreasing exponentially with altitude. -(r-r )/H a(r) = a(r1)e
.
(4.5.22)
TC. is the radius of the planet. H is the scale height giving the 1 * geometrical thickness of a homogeneous atmosphere with a = a(r.) and geometrical thickness of a homogeneous atmosphere with a = a(r.) and of the same optical thickness as the given atmosphere. of the same optical thickness as the given atmosphere. Then (4.5.16) takes the form 2
d J
7T + 9r
—
,2
(
r
+
1 . 3J
H->ar *
+
1
d
1-7-2T M r
s i n i>
where f i s g i v e n by ( 4 . 5 . 1 9 ) ,
, .
(sln
. 3J,
+ 3*> ="
2 . . 2 T *.. (k J
"f} '
., ,_
(4
nn.
-5'23)
152 / 2 2 2 - ( / r sin V + z -*)/H* e dz.
=o with
T - a(r)J
(4.5.24)
rcos^ Sobolev (19) wrote a simplified version of equation (4.5.23) assuming that the scale height H
is much less than the radius r.. of the planet.
This gives 2 d J -2
+
2 1 dJ . 1 d J HT ai + "2 ^ 2 -
3r
*
r
a
2/r-2 T x-. ( k J f)
(4
" -
,, c oc * -5-25)
3TJ>
It should be noted that the steps taken to go from the equation (4.5.23) to (4.5.25) are akin to replacing a spherical medium to plane—parallel medium.
The equation (4.5.25) was solved by
/TON
Smoktii
to obtain the numerical values of J.
Then introducing T = a(r)H^,
t « a(r1)r1V,
(4.5.26)
where r is the optical depth and t is the optical distance in a horizontal plane, the equation (4.5.25) can be rewritten as
4 + r£i24 = k - 2 J-f, ar2
where r
is the total depth of the atmosphere.
Minin and Sobolev
gave a method of approximate solution of
the equation (4.5.27) which runs as follows. (H^/r..) «
(4.5.27)
LT J a t 2
1 and initially assumed that tj) < -^.
They considered With this limitation,
they calculated T ( T , ^ ) , the optical path of direct solar radiation between the upper boundary of the atmosphere and the point of coordinate (r ,V>) • Through a series of approximations, they established that for angles i/> not close to JT/2 , one could take for (19) many applications [cf. Sobolev p.224 - p.227]
153 T(r,tf) - rb<*) where
(4.5.28)
b(i/>) = sec V-
The relation (4.5.28) was derived from the relation (4.5.24)
T(r,V>) = a(r) J
exp - [/ r 2 sin2^ + z 2 - rl/H^ dz rcosV> ^ *■ • CO
= a(r) j
r'dr'
exp[-(r'-r)/HJ
(4.5.29)
/~72 2 r 2 " / r' —r s m v Setting r' = r + H x. (r + H^x)dx
CO
T = a(r)H
[ e~X J 0
(4.5.30)
A
2 2 2 2, r cos V + 2rH x + H x ) * *
Then using the first of the relations (4.5.26), we can write T(r,V) = Tb(u.V),
(4.5.31)
where H. (1 + — x)dx r
00
b(u,V)
-I •"" / 0
/
E_ H
where
(4.5.32)
H H " * * (1 —sin V" + — x)(l + sin ij) + — x)
*
!i
H
*
in u,
u = T/T
(4.5.33)
(4.5.34)
From (4.5.19) and (4.5.28) we have f - ():
(4.5.35)
154 To get the solution of (4.5.27), we recall that it was obtained on the assumption that the atmosphere is practically stratified in plane—parallel layers.
It was also assumed that in (4.5.35),
b = sec T/>. This along with the numerical analysis made by Sobolev suggested that b is constant through out the atomosphere.
This
assumption was fairly accurate for i/> sufficiently away from TT/2 . However, if it was taken to be true for the whole atmosphere, the derivative with respect to t in (4.5.27) could be neglected and equation (4.5.27) could be approximated as
d*J
k2J
f.
(4.5.36)
d,2 In (4.5.36), f is given by (4.5.35) and in it h(ij>) is given by (4.5.32).
The equation (4.5.36) could at best be called the first
approximation.
However error due to this approximation goes on
mounting as we approach ij> = ?r/2. The solution of the equation (4.5.36) can be written as „ kT _, —kr J = Ce + De
- i~_ r'r J
2k 0
k(r-r') -k(r-r') — e dr ' , for u> < 1 < '> e
f T
(4.5.37) and J = CC +Dr -
[
f(r')(T-T')dr', for w - 1.
(4.5.38)
C and D are constants to be determined from (4.5.20) and (4.5.21) Once J(T,I/>)
is found, the specific intensity at any point of
the medium can be known by direct integration of the equation of transfer. Sobolev
(19)
demonstrated the method with reference to two
problems of practical importance, namely
155 (i)
the brightness of a planet close to the terminator (V> = T / 2 ) ,
(ii) the brightness of the zenith. We shall record here the application of the method in the second case, namely calculation of the brightness of the zenith. In this model a purely scattering (w =■ 1) atmosphere of finite thickness r is considered which surrounds the surface of a planet o with scattering albedo a. We also take p. = 0 which ensures isotropic or Rayleigh scattering. J is calculated from (4.5.38), C and D being found from (4.5.20) and (4.5.21). At r - 0,
J(0) = 2H(0).
(4.5.39)
Furthermore, from (4.5.14), we find that 3H = |^ .
(4.5.40)
These two relations substituted in (4.5.38) imply that
J(0) = C,
(|i) OT
-D-3H(0).
(4.5.41)
T=U
Now combining (4.5.39) and (4.5.41) J(0) = C = 2H(0) = | D.
(4.5.42)
The boundary condition at the lower boundary is given by (4.5.21). For example J(r ) + 2H(r ) = a[J(r ) - 2H(r ) + F. cos V e~T] o o o o i
or
(l-a)[C + Dr
- [ f(r)(r -r)dr]
+ 2H(TQ)(l+a) = aF i e~
cos T/>. (4.5.43)
aT
This along with the fact that C = 2D/3, (•=-) °
OT
T"T
= 3H(r ) , gives us O
O
156 T
T
[^ +(l-a)r ]D = (l-a)f ° f(r)(r -r)dr+ \ J j o 0 °
(1+a) [ ° f (r)dr+aF.e_Tcos i>. J 1 0 (4.5.44)
The last term vanishes when ip > »■ . With C and D known, we can find J and hence the specific intensity at any point of the atmosphere. The brightness at the zenith as seen from the surface of the planet is found as follows. Let I(V0 be the brightness at the zenith, i/) being the zenith distance of the sun. T
I(V0 = I ° J(r,V)exp[-(r -r)]dr + I (V>) , J 0
(4.5.45)
where I.. (ip) is the zenith brightness produced by first order scattering. Substituting (4.5.38) in (4.5.45), we have
I(V-) = D U
- 3 + 3 e
°J + J ° f(r)[l-ro+T-e
°
]dr + I1(^.).
(4.5.46) From (4.5.35), since w = 1
f(r) - 3
p
_e-rbW
(4
5 4 7 )
For V > TT. (4.5.47) is not valid in the region in the shadow area. (19) took the relation to be valid over the entire
However, Sobolev atmosphere.
He argued that the error would not be large as f in the
unilluminated region would be very small. Substituting (4.5.47) in (4.5.46), we have
157
IW) = D
T
1 1 o - 3+ 3
3F.r
'o
l
e
4b
, r —T -br -, b o o b=l | e -e
ixW
where
+ „ F.
=
P W
^b->K b r °l-'o (4.5.48)
I1W, —br
—r o
o
(4.5.49)
b-1
4
From (4.5.44), D is given by '4 3
3F. +
(1
-
a)T
o3
D
i
4b"
(
l-a)|r<>_| 4 + ■» ae
Sobolev
(19)
o
+
| e '"I + | (l+a)
r
-br
i
1—e
b cos ij>
(4.5.50)
calculated the values of zenith brightness from the
relations (4.5.48) - (4.5.50) for r
0.1 and 0.3, a = 0.2 and 0.8
for scattering according to Rayleigh's phase function. It was found that there was fairly good agreement between the computed and observed results. Thus we see that Sobolev's method succeeds in solving radiative transfer problems in spherically symmetric planetary atmospheres illuminated by solar radiation. asymmetric.
The radiation field in this case is
However, the number of approximations introduced in
converting equation (4.5.16) to (4.5.27), in stating the boundary condition (4.5.20) and expressing the optical depth as in (4.5.28) suggest that we are dealing with atmospheres not far away from plane—parallel model.
So the agreement between the observed and
calculated results simply indicate that in the problem considered, the effect of sphericity is not very much pronounced.
We shall demonstrate
below the use of the generalised Eddington approximation method for tackling such problems [cf. Wilson et al
(25) ].
158 4.6
The generalised Eddington approximation method in solving problems of
light scattering by an optically thin, inhomogeneous
spherically
symmetric planetary atmosphere
We take the same model of a planetary atmosphere as in the previous section 4.5. The transfer equation for diffuse radiation, the source function, the optical depth from the outer surface of the planet to a point of the atmosphere, the phase function etc are given by equations (4.5.1) to (4.5.11) in terms of the moments of specific intensity. symbols have their usual meanings.
The
[cf. Fig 4.5.1].
Thus the transfer equation for diffuse radiation is . 91 cos 9 cos * 3i +
sin 8 cos
sin 8 31
H - ~T~
r
cot ib sin 8 sin cp 31
38
r
r„ T .
3^ -
Qt?
-I] ■
(4.5.2) where where
I = I(r ,i>;8 ,
*(r ,*;#,,») = ^ T
J
dxp'J I(r,<M',V>')p(-y')sin 8' ; 8 ,
F.p(7)e"T,
(4.5.5)
with
"I1 . i:
, 2 . 2 . 2.1/2 . a(r s m y> + z ) dz, (4.5.7)
r cosyi R x - (R2 - r 2 sin2i/01/2,
and
p(7) - 1 + p
cos 7
[cf. equation (4.5.10)].
9 = uJ +
+ -r F.(l
_
PT C O S 9
cos
^ + P-. sin 8 sin V cos
(4.6.1)
159 where the moments introduced are
J
- J1 £ • » - ; . « . . g .
0l
- ji
s i n 0 cos
dn
(4.6.2)
47T
The other symbols have the meanings characterised in §4.5. (25) Wilson and Sen further introduced the following moments.
K - J I cos2* g ,
L - J: L - I cos
3
a d" 9 -r47T
G
1 I sin d cos 0 cos cp •
I sin 6 cos 6 cos
G_ -
s
rT . 2, 2 dfi -;— 4 = J I sin 9 cos cp 4w
T
(4.6.3)
fT . 2 . . 2 dfi I sin 6 Sin
Using the equations (4.5.2), (4.6.1), (4.6.2) and (4.6.3), the following, differential equations are obtained [cf. Wilson and (25) Sen1- ; , p.109]
3H 3r
2H r
1 1 v dip
cot V „ r 1
,
,v
au „ -T 4 l
T
3K 1 ,_„ „ 1 dG2 cot K 3 - + - (3K-J) + - V T - + *- G„ = -aH + a 3r r r 3V> r 2
(4.6.4)
u p . cos i/1 F. e
-up,H
-T
12 (4.6.5)
|i + I 3r r
(4L-2H) + i ^ r 3i/>
c o ^ r
^ 3
R
uF.
a>J
T~
+
l
e
12
-T (4.6.6)
-
and |-wp..G.. 3G 3G 3G v - = + — - + - ^-r- + c o t YVCG.-G,.) = -aG n + a - s 3r r r B\j> 4 5' 1
wp. s i n V F . H
e
-T
12 (4.6.7)
We t a k e t h e r a d i a t i o n f i e l d t o be a p p r o x i m a t e d by a t h r e e s t r e a m (25) model [ c f . Wilson and Sen ] , where t h e i n t e n s i t y i n each r e g i o n i s
160 represented by an average value.
This division is suggested by the
shadowing effect of the planet and is described by
I =
I..(r,V>) + f(r,V>)cos cp, for
/i < /* < 1
I„(r,V0 + f(r,i>)cos
0 < \i < y,
ip,
for
I (r,V) + f(r,V)cos
where
/x = cos 8
and
M
=
r
11
(4.6.8)
-1 < /i < 0,
t o 2-, 1/2
ir
(4.6.9)
r1 being the radius of the planet. For an optically thin atmosphere (4.6.9) can be used through out the whole atmosphere. From equations (4.6.8), (4.6.2), (4.6.3), we have J = 2 [( I 1 + I 3 ) + ' 1 r
(4.6.10)
H - i [(I r I 3 ) + ^ ( I g - I ^ ]
(4.6.11)
=i
+4(.i1-i1r.
(4.6.12)
L = | [(^-Ij) + ^ ( I g - ^ ) ]
(4.6.13)
K
G
G
[(I1+I3)
l - -8 f
(4.6.14)
= 0
(4.6.15)
2
G
3 = (*/32)f
(4.6.16)
G
4 = j (J-K)
(4.6.17)
G
5 = j (J-K).
(4.6.18)
161 From equations (4.6.10) to (4.6.13), we get (4.6.19)
2(2L-H) - /.r(3K-J) After Unno and Kondo
(21)
, we may call this relation a generalised
Eddington approximation. Then combining (4.6.4) - (4.6.7), (4.6.14) - (4.6.18) and (4.6.19), we have 3H ___ 2H
ai
+
A
n
,3f
_
,.
F" + si {M + f cot * >
, . .T
= a( 1)J
+
^
-wp1H
+
(4.6.20)
—*-
(4.6.21)
12
2H] + - ^ (3K-J) + ^
/ „ + 4a = -4aK y- uJ +
-T "Fie
-T up 1 cos TJ) F.e
|^ + - (3K-J) = -aH + a 3r r
f j [M r (3K-J)
a
[|| + f cot *
T
wF.e l
(4.6.22)
and 1 8_ .J-K. r 3i/> L 2 J
7raf
r HWT>.
u>p. sin ib F.e 1
+ a
24
f + -
-T
1
(4.6.23)
12
These equation are solved under the following boundary conditions. (i)
The diffuse incident radiation (—1 < /J < 0) at any point on
the outer surface, averaged over angular distribution, is put equal to zero.
That is at r = R Z-K
I.
(I, + f cos (p)dip = 0
i.e., 1 , - 0 .
>
(4.6.24)
162 From e q u a t i o n s
(4.6.10)
to
(4.6.13),
we h a v e b y
(4.6.24)
6K(R) - 4 H ( R ) ( 1 + p R ) + 2/i R J ( R ) = 0 .
(4.6.25)
(ii) The diffuse radiation reflected to the medium at the inner boundary r = r.. is a fraction of the total radiation incident on it. Averaged over angular distribution at any point in the inner boundary, it gives -Z7T
j
J
2W
( I . . + f c o s
i
J ,-.
__
( I , + f c o s (p + F . c o s V e ~>
1
)d*>,
where a is the albedo for the planetary surface, i.e.,
I
= a(I
+F. cos V e ~ T ) .
(4.6.26)
From equations (4.6.10) - (4.6.13), it follows that [J(ri) + 2H(r;L)] = a[J(ri) - 2H(r;L) + F t cos V e _T ] .
(4.6.27)
It may incidentally be mentioned that on using the classical Eddington approximation 3K = J in the moment equations (4.6.20), (4.6.21) and (4.6.23) and in the boundary conditions (4.6.5) and (4.6.27) we recover Sobolev's moment equations (4.5.16) — (4.5.18) and the boundary conditions (4.5.20) and (4.5.21). Method of solution
:
To solve the moment equations (4.6.20) -(4.6.23) under the boundary conditions (4.6.25) and (4.6.27), the following forms of J, H and K are assumed.
The choice of these forms is usually suggested
by the physical nature of the problem and the boundary conditions.
163 -T J - J cos \b e +
-T cos ^ e + ^„(r)
K
cos
- K
o
i> e
-T
+
(4.6.28)
4>~(T)
3
Using equation (4.6.28) and the boundary conditions (4.6.26) and (4.6.27), we then have aF. aF. aF. ^ , H -± and K = - ^ o 2 o 4 o 6 Next we use equation (4.6.23) to obtain f. This gives J
(4.6.29)
-T a
™ o-wp1)f
"Pl
sin
^
F
ie
aF
— 2 —
i
r
_T e [sln
*+
cos
vr is determined in the approximation of Sobolev. dtp
that the scale height H
«
AT
* W'
<4-6-30>
That is we assume
r, .
o(r) - o(r1)exp[-(r-r1)/H#],
(4.6.31)
T = a(r)HA,
(4.6.32)
T = T sec ij>.
(4.6.33)
These three equations are the same as (4.5.22), (4.5.26) and (4.5.28) The approximation (4.6.33) is accurate for angles V n °t close to
n/2. Then f is given by
™
(3-wp1)f
, _ —T sec ib awp, sin y> F.e g
_ — T sec ib aF.e " r [sln V> + T tan V] (4.6.34)
164 f obtained from (4.6.34) is substituted in the moment equations (4.6.20) - (4.6.22) and they are solved numerically under the boundary conditions (4.6.25) and (4.6.27) to obtain the mean intensity J. (25) Wilson and Sen calculated J for different (r-r..) , for different zenith angles i/> and different albedo "a" of planetary surface, setting F. = 1, r T
o
= 6400 km, R = 6500 km, H^ = 8 km, optical thickness
= 0.1, oi = I, rp. - 0 L [cf. Wilson and Sen( l
5)
p.112].
Once J is known, intensity at any point can be easily calculated.
Brightness
at the zenith
the terminator
as seen from the surface
of the planet
near
:
At the terminator, i/> = TT/2 . As the expression (4.6.33) is not valid close to \j> — ff/2, instead of this expression, we shall use for T, the following general expression
X dr T(r,*) - a(r)f exp[-(r'-r)/H.] ' ' - ,* S f J 2 r * / ,2 2 . 2. / r' —r sin y>
T(r,tf) = a(r)J
(4.6.35)
r'dr'
exp[-(r'-r)/H^J
/~2
2 sin7l.V
/ r' —r
+ 2a(r sin ^)
r'dr' exp[-(r'- r sin VO/H^.] r sin ij) / ~2 2 . 2 , / r' —r sin y> f > |,
a(r) and T are given by (4.6.31) and (4.6.32).
(4.6.36) We use the moment
equations (4.6.20) to (4.5.23), and the three-stream approximation (4.6.8). The boundary condition at the upper surface at r = R is given by [cf. (4.6.25)] 6K(R) - 4H(R)(l+/iR) + 2 M R J(R) = 0.
(4.6.25)
165 The boundary condition at the lower surface at r ■ r., assuming that the surface reflects according to Lambert's law, can be stated as
f a[J(r ) - 2H(r ) + F
cos ^ e~T] , i/> < ^
J(r t ) + 2H( ri ) =
(4.6.37)
$> |
L a[J(rx) - 2H( ri )], (251 [cf. Wilson and Sen^ , equation (2.19)].
When using the differential equations (4.6.20) to (4.6.23) in the —T shadow region, we set the term F.e to be zero, as in this region there is no direct illumination by solar radiation. For the solution of the equations (4.6.20) to (4.6.23) we use the following form in the first approximation. 3F
— J
- (
i
, -T cos V e
V ^ 2
0
-T—
*>i cos V e
,
V ^ ~-
H = i
(4.6.38)
i>>\
^ 0
raFi
/ "T
K = I -g— cos y e , (
■
0
,
y> < ^
*>i
These forms are suggested by the physical condition of the problem and satisfy the boundary conditions (4.6.25) and (4.6.37). We then use equation (4.6.23), (4.6.35) and (4.6.36) to determine f.
T and its derivative with respect to i/> are computed from (4.6.35)
and (4.6.36) and not from (4.6.33), as we are dealing with the terminator ij> = -~.
Once f is known the numerical integration of the
166 differential equations (4.6.20) — (4.6.22) is done to determine J. The boundary conditions (4.6.25) and(4.6.37) are used. The brightness at the zenith near the terminator is computed from the relation
I(V0 = J ° jj(r,V) + 2 ^ - Fie"T|exp[-(ro-r)]d7-,
3 2 where p(V0 = 7- (1 + cos \j>)
(Rayleigh scattering).
(4.6.39)
(4.6.40)
We take to = 1 indicating that the atmosphere is purely scattering. The total optical depth is taken to be 0.1 and 0.3, p. = 0 , r
= 6400 km, R = 6500 km, H^ = 8 km, F. = 1000. We compare the results with those obtained by Sobolev (19) (p.231), under classical Eddington approximation. 4.6.1
The comparative results are given in Table
167 Table 4.6.1 r
= 0.1
0
a - 0.8
a = 0.2
* Wilson and
(19) Sobolev^ ;
Wilson and
(19) Sobolev1* ;
25
^J ^> ben
ben 84
16.3
14.4
29.3
16.7
86
15.6
12.2
22.1
13.7
88
11.3
9.3
14.2
10.2
90
6.3
92
2.15
94
0.41
96
0.03
0.03
5.56
6.6
5.95
2.04
2.25
2.19
0.40
0.43
0.43
0.04
0.04
r
= 0.3
0
a = 0.8
a = 0.2
* Wilson and
Sobolev (19)
Wilson and
Sobolev (19)
Sen
ben
84
29.6
25.3
40.2
30.6
86
21.3
18.9
27.7
22.4
88
13.1
12.1
15.8
14.3
90
6.3
6.26
7.02
7.39
92
2.11
2.2
2.37
2.61
94
0.40
0.43
0.45
0.51
96
0.03
0.04
0.04
0.04
For V < ^. there is measurable difference in the value of the (19) brightness at the zenith calculated by Sobolev's method and that (25) by the three—stream moment method of Wilson and Sen Sobolev's calculations were done on the assumption of adherence of classical Eddington approximation (3K = J ) .
168 References
1.
Bellman R. E., Kagiwada, H. H., Kalaba, R. E. andUeno, S., Icarus 11, 417, 1969.
2.
Chandrasekhar, S., Mon. Not. Roy. Astro. Soc. 94, 443, 1934.
3.
Chapman, R. D., Ap. J., 143, 61, 1968.
4.
Chou Y. S. and Tien, C. L., J. Quant. Spectro. Rad. Transfer, 6, 919, 1963.
5.
Eddington, A., The internal constitution of stars, Dover, 1926.
6.
Feautrier, P. C., R. Acad. Sc. Paris, 258, 3189, 1964; Ann. d'Astrophys, 30, 125, 1967; 31, 257, 1968.
7.
Germogenova, T. A., Koporova, L. I. and Sushkevich, Atmos. and Oceanic Phys. 5, 731, 1969.
8.
Hummer, D. G. and Rybicki, 'G. H., Mon. Not. Roy. Astro. Soc. 152,
9.
Kosirev, N., Mon. Not. Roy. Astro. Soc. 94, 430, 1934.
10.
Larson, R., Mon. Not. Roy. Astro. Soc, 145, 297, 1969.
11.
Leong, T. K. and Sen, K. K. , Publ. Astron. Soc. Japan 23_, 99,
12.
Max Krook, Ap. J, 1 ^ , 488, 1955.
13.
McCrea, W., Mon. Not. Roy. Astron. Soc. 8ji, 729, 1928.
14.
Mihalas, D., Stellar atmospheres, W. H. Freeman and Co., San
15.
Minin I. N. and Sobolev, V. V., Kosmichesk Issled
1,
1971.
1971.
Francisco, 1978.
1, 287, 1963; 2, 610, 1964. 16.
Schuster, A., Ap. J. 21, 1, 1905.
17.
Schwarzschild, K., Gottinger Nachr. 41, 1906.
18.
Smoktii, 0. I., Atmos and Oceanic Phys. 2. 281, 1967.
19.
Sobolev V. V., Light scattering in planetary atmospheres, Pergamon Press, Oxford, 1975.
20.
Takeuti, M., Publ. Astron. Soc. Japan, 31, 199, 1979.
21.
Unno, W. and Kondo, M., Publ. Astron. Soc. Japan, 28, 347, 1976 29, 673, 1977.
169 22.
Wilson, S. J., and Sen, K. K., Ann d'Astrophys. 28, 348, 1965.
23.
Wilson, S. J., Tung, C. T and Sen, K. K., Mon. Not. Roy. Astro.
24.
Wilson, S. J., Wan F. S. and Sen, K. K., Astrophys and Sp. Sc.,
Soc. 160, 349, 1972., Publ. Astro. S o c , Japan, 27, 611, 1975.
67, 99, 1980. 25.
Wilson, S. J. and Sen, K. K., Astrophys and Sp. Sc. 67, 107, 1980. 71, 405, 1980.
170 Chapter V AMBARZUMIAN'S PHYSICAL METHOD : PRINCIPLE OF INVARINACE
5.1
INTRODUCTION The physical and mathematical methods of Ambarzumian were
primarily designed to take care of transfer problems in slab geometry. Ambarzumian's physical method was refined considerably by Chandrasekhar (13) . The basic features of Ambarzumian's physical and mathematical techniques are outlined in §2.3, Chapter II.
The aim of
the methods was to know the law of darkening without going through the process of determining of the source function first.
In slab
geometry, the method has been extensively used by Chandrasekhar (13), (11) (23) (12) (11) (23) (12) v v v . For a Busbridgev ' Stibbsv ' and Busbridge and Stibbsv homogeneous slab of finite optical thickness, it was shown that the homogeneous slab of finite optical thickness, it was shown that the radiation field in the medium could be uniquely specified in terms of the scattering and transmission functions. Chandrasekhar (13) (p.162—164) stated four principles for obtaining intensities in the outward and inward directions and also the diffuse reflection and transmission functions.
He obtained the integral equation for
H—function in semi—infinite and X- and Y-functions in finite plane—parallel medium by direct analysis of diffuse reflection and transmission of light.
He also recovered the well—known
Hopf—Bernstein relation for Milne problem.
Ambarzumian's physical
method led to two very powerful methods for tackling complex transfer problems in plane—parallel semi—infinite and finite medium, namely (a) Invariant imbedding (particle counting) and (b) Probabilistic methods. In what follows, we shall describe the use of the invariant imbedding and probabilistic methods in solving transfer problems in spherical and cylindrical media.
171 5.2
INVARIANT IMBEDDING (PARTICLE COUNTING) METHOD Though our main interest is to consider the basic methods for
solving transfer problems in spherical and cylindrical geometry, we shall for completeness recount the fundamental features of the use of invariant imbedding method in slab medium.
Apart from any other
reasons, this study appears to highlight the criteria which distinguishes the transfer problems in curved geometry from those in plane parallel medium. We shall demonstrate the scheme with reference to the problem of multiple scattering by an inhomogeneous plane layer of finite thickness.
The thickness of the slab is taken to be the parameter
describing the system. A narrow layer of thickness Ax is added to one end of a layer of inhomogeneous medium whose bounding surfaces are x—distance apart. the surface a flux TTF. is assumed to fall at cos~ u to the inward l o normal to the surface at A.
Fig
Let cos
5.2.1
n denote the angle between the outward drawn normal and the
emergent intensity.
At
172 By Ambarzumian's principle, the intensity of light reflected from \i at a point 0 of the surface A
the whole medium in a direction cos hole me is given by
1(0,M) = F. r(/i,/io;x).
(5.2.1)
The reflection function r(/i,/i ;x) gives the measure of the reflected energy from the whole medium appearing at the bounding surface A per unit area per unit time along cos
n given that the energy incident on
the surface A per unit area per unit time along cos
\x
to the normal
is one unit. Let u(x) denote the albedo for single scattering.
Following the (4) procedure of Ambarzumian technique, Bellman and Kalaba obtained for r(/i,/j ;x) the following imbedding relation. . . . . ... r(/i,/i ;x+Ax) = r(/i,/» ;x)(l o
o
Ax /i
Ax. , w(x)Ax ) + —-.
fi
4/i
1 1 u(x) Ax f . , . . , w(x) f . . dp" r(M',u ;x)d/i' + -±f^^+ - V ' T, I r(/i',u Ax | r(/i,/i";x) - ^ '0 0 „1 A + w(x)Ax f r(/i',u ;x)d/i' I r ( M , M " ; x ) ^ . J0 J0 ft
(5.2.2)
In equation (5.2.2), only the first order terms in Ax have been retained. The terms in (5.2.2) can be identified as follows.
The
first term accounts for the Lagrangian residue of intensity due to absorption losses in passing through Ax in the way—in and on the way-out.
The second term is the contribution from the direct
scattering from the layer Ax of radiation incident on it in a given direction cos
/i. The third term is the contribution from the light
that is scattered in the layer of thickness Ax and reflected from the slab extending from 0 to x.
The fourth term is the contribution from
the light reflected from the layer 0 to x and scattered in the layer Ax.
The fifth term is the contribution from reflection in (0,x)
followed by scattering in (x,x+Ax) and then reflection in (0,x) again.
173 We introduce the scattering function S(/J,JJ ; X ) defined by S(/J,U
r(
"-V
x)
;x)
(5.2.3)
w
Then from (5.2.1), F i S( M ,M o ;x)
KO, n)
(5.2.4)
n
4
and letting Ax -> 0 in (5.2.2), we have
aax
S(M.VX) + u>(x) 1 +
S(M,M0;X)
ljS(^» ; x )^][l + ^\(,-„ o ;x)^
(5.2.5)
In homogeneous, Isotropic scattering, semi-infinite medium, S(/i,/i ;x) is replaced by S(/i,/j ) , co(x) = ta , a constant and we have
1 —
1 H
S(|.,MO)
" -0[l + | J
B<|,.M-)^] [l+ |
j' SO.' ,„.)<£ (5.2.6)
[cf. equation
(2.3.27)].
Now writing
i r1
*0*,x) - 1 + I J S(/i,/.»;x)
(5.2.7)
and solving (5.2.5), we have
S(/i,/io;x)
jw-&-y
(x-y) <»>(y)'MMo-y)'0(M,y)dy-
(5.2.8)
Assuming that at the upper bounding surface x = 0, the scattering function is zero, i.e., S0i,/io;0) - 0
(5.2.9)
174 and substituting (5.2.8) in (5.2.7), we have
o
| expi- U + j - (x-y)L(y)^(/io,y)^(M,y)dy
4K/i,x) = 1 + ^ J
(5.2.10) This has a feasible computational solution giving us <^(/i,x).
And
knowing $(/i,x), we can determine S(/j,/i ;x) from equation (5.2.8) and hence 1(0,^i) from equation (5.2.4). Incidentally it may be mentioned that for homogeneous, isotropic scattering in semi-infinite medium, S(/J,/* ) satisfies the integral equation (5.2.6). Remembering that S(/J,/J ) is symmetrical, we may define
H(/i) = 1 + \
f
S(M,M') ^ T - 1 + \
\
S(/i',M) ^ r
,
(5.2. 11)
and the integral equation (5.2.6) can be written as
f
+
fyl»>»o>
=" 0 H(M)H(M 0 ).
(5.2.12)
Substituting (5.2.12) in (5.2.11), we have
H
(5.2.13)
This is the familiar non—linear integral equation for Chandrasekhar's H—function. We have described above the essential elements of invariant imbedding technique of Bellman and Kalaba (4) for solving transfer: and transport problem in plane-parallel medium. Wing
Bellman, Kalaba and
utilised the scheme for solving various complex problems of
time-dependent and time-independent neutron transport.
Wing
175 treated the diffuse reflection problems in plane geometry by a sort of unified approach in conjunction with the linear transport equation. (4) Bellman and Kalaba extended the method to non-linear transfer equations.
Bellman, Kalaba and Ueno
derived from the transfer
equation an integral equation for scattering function for slab geometry and applied the principle of invariant imbedding to various problems of geophysics and astrophysics. contributions of Preisendorfer
To this we may also add the
and Matsumoto
, who used the
principle of invariant imbedding in a slightly modified form.
5.3
INVARIANT IMBEDDING (PARTICLE COUNTING) METHOD IN SPHERICAL GEOMETRY One of the earliest examples of the application of the invariant
imbedding method to transfer problems in spherical geometry was that (2) of Bailey He obtained the equation for reflection function for spherical shell media considering the effect on the transfer equation of the differential change in the shell radius.
He studied the
corresponding changes in the angle of incidence and emergence. Bailey's calculations brought to light an omission in the particle counting method of Bellman, Kalaba and Wing
. Bailey and Wing
attributed this to the fact that they did not take into account the effects of "geometrical convergence" and "oblique incidence".
These
two features in fact differentiate the study of transfer problems in (2) spherical geometry from that in plane geometry. Bailey and Bailey (3) and Wing's discussions referred mainly to the problems of neutron transport in spherical medium. Taking into consideration the effects of geometrical convergence and oblique incidence which arise out of the sphericity of the medium, Bellman, Kagiwada, Kalaba and Uneo
treated the problem of diffuse
transmission of light from a central point source through an inhomogeneous, isotropically scattering, spherical shell medium. they put the method of particle counting on a firmer footing. (25) used the corresponding method for solving transfei
Tsujita
problems in an infinite cylindrical shell medium.
Thus
176 It may be noted that the difference between the particle counting and invariant imbedding method as they are used in radiative transfer and transport theory is rather cosmetic.
It may be argued that the
particle counting method supplies some basic details of invariant imbedding method.
For example, the particle counting method utilises
the equation of transfer to derive the reflection and transmission functions.
Bellman, Kagiwada, Kalaba and Ueno
derived the
integro-differential equations for the scattering and transmission functions S(R,r. ;jt,/i ) and T(R,r -/i,/z ) respectively and I(R,^) in inhomogeneous anisotropically scattering spherical shell medium (R > r > r. ) utilising the equation of transfer in conjunction with the assumed law of diffuse reflections
i
r1
Kr.M) - JJ; J I.
S(r,r i;M , M ')I inc (r,- M ')d M ',
(5.3.1)
(r,—/i') defining the incident intensity at any depth r. This led to the relation F. I(R,/i) = ji S(R,r i ; M l M o ).
(5.3.2)
(5.3.2) represents the angular distribution of intensity emerging from the outer surface r = R.
This relation, in fact forms the basis of
the method of invariant imbedding. In the "particle counting" method, the reflection coefficient p(n,H
;x)dv [cf. Bellman, Kagiwada, Kalaba and Ueno
] was defined
as the total reflected energy per unit area per unit time from the outer surface r = R. In the expression for p, the range (cos
fi,cos
(/i+d/i)) denotes the angular interval of emergent
intensity and (cos
/j. ,cos
incident radiation, cos
(/a +d/i )) denotes the angular range of the
p is measured from the positive and cos" y.
from the negative r-direction, x the total depth of the medium is a parameter.
The integro-differential equations for reflection and/or
transmission coefficients p , t were built up by physical reasoning,
177 considering the change in the coefficients p, t due to the addition or subtraction of a layer Ax to the total depth x of the medium. feature is the characteristic of invariant imbedding method. intensity of radiation is considered azimuth independent.
This The
No direct
reference is made to the integro-differential equation of transfer in setting up the integro-differential equations for reflection and/or transmission coefficients. /o\
Bellman, Kagiwada, Kalaba and ueno
used the particle counting
method for tackling the problem of diffuse reflection of solar rays by spherical shell atmosphere. This is an example of azimuth dependent (29) radiation field. Uesugi and Tsujita utilised the technique of invariant imbedding for solving azimuth dependent transfer problems in spherical and infinite cylindrical shell media. From the point of view of mathematical methodology, the difference between the particle counting and the invariant imbedding method is very thin. We shall avoid treating them separately.
An externally inner surface
illuminated :
spherical
shell
medium with a
reflecting
We shall outline below the features of the invariant imbedding technique [cf. Bellman, Kagiwada, Kalaba and Ueno
] to derive the
integro-differential equations for the scattering functions, transmission functions and intensities in an inhomogeneous, anisotropically scattering, source free spherical shell medium with inner and outer radii r.. and R (0 < r. < R) respectively.
The inner
surface at r - r. reflects radiation isotropically according to Lambert's law with a constant surface albedo, A.
The conical flux of
radiation TTF. per unit area is incident uniformly on the surface r = R at an anele cos u 0 < u. < 1 to the inward normal to the surface. ° o o is assumed to be axially symmetric, and Further, the radiation field Further, the radiation field is assumed to be axially symmetric, and hence the source function and intensity are azimuth independent. For hence the source function and intensity are azimuth independent. For simplicity, we have assumed that the medium is grey.
178 The transfer equation for diffuse radiation in this case is given by [cf. equations (1.3.28) and (1.3.29)] ai(r,/i) r
dx
,
+
1
p. ~ ^
rr
2
:
dl(r,ii)
dn
- - a(r) [I(r,/i) - *(r)]
(5.3.3)
The source function S(r) is given by
c
. . +1 »(r) - ^ - J p(/i,/i')I(r,A«')dAi' + B]_(r) + B Q (r). u(r) = <7(r)/a(r) is the albedo for single scattering.
(5.3.4) a(r) is the
scattering coefficient and a(r) is the attenuation coefficient. p(/u,jiz') is the phase function for scattering.
For simplicity.
We
shall take the scattering to be conservative and isotropic implying p(/x,/i') = 1.
B (r) is the contribution to the source function from
internal sources other than scattering.
For source free medium,
B., (r) = 0 . B (r) is the contribution to ^(r) from reduced incident 1 o radiation resulting from the incident flux at the outer surface. B (r) in this case is given by [cf. equation (1.5.30)].
Bo(r) = U (r)
F. ^
IfJ
exp
{-
Fig 5.3.1
I o(r')ds'
'}
(5.3.5)
179
-n
r > RA
Now assuming that
+1 ?(r) = 5*|Hl | I(r,^')p(^,M')dju'
F. l + w(r) 4
r
p
o * 1 A,
:X
][l]
.- fRAt
P"H
^ a(r' )ds' }■
(5.3.6)
r
At
The equations (5.3.3) — (5.3.6) are to be solved under the boundary conditions I(R,-AO = 0
for
(5.3.7)
0 < A* ^ 1,
and I(r 1 , M ) = 2A J" I(r1,-At')At'dAt' + F J ^
'Us. |f L.] 2 exp - u
"o
H r iV.
r R/t
.(r')ds't r
l"
for 0 < At < 1,
(5.3.8)
where A is a constant albedo for reflection at the lower surface r — r. , reflecting according to Lambert's law.
/i is the value of \i
at r = r. One realises that the term
—r—
I
| in the expression for
M
o the source function spells the basic difference between the expression given here and that in the case of plane—parallel medium illuminated on one free surface by a uniform flux.
This term arises out of the
combined effects of geometric convergence and oblique incidence consequent upon the curvature of the layers in spherical media.
The
integro—differential equations for scattering and transmission functions in spherical medium differ from those in slab medium on this account. The law of diffuse reflection of radiation by a. spherical shell is given by I(r,A
°
=
i
Ta J
r1
s r
( .>V /i '' i ' )I inc (r, ~ M ' )d/ ''-
(5.3.9)
180 S(r,r •/!,/*') is the scattering function and I.
(r,-/j') (0 < fi' < 1)
is the intensity of the inward directed radiation in the azimuth independent case.
l
o
R
Wr'-"'> -7- \~*
°a(r')ds' 5(M'-A<0) + Kr,-M').
exp r/i
o
(5.3.10) From equations (5.3.9) and (5.3.10), we have F
I(r,AO
i
0
r rRA
,2
r Mo i
f R-r 1 €exp L JL J
.R/i
a(r')ds' S ( r , r ; ^ , M *) 1 0
r n/*
r1
i
+ ±- J
(5.3.11)
S(r,r1;^,^')I(r,-/J')d/i'.
At r = R, remembering the boundary condition (5.3.7), we have for emergent intensity
(5.3.12)
I(R,M) = ^ S ( R , r i ; / ^ o ) . If the angle made by the path of a ray be cos at r' and cos
fi
H
/i at r, cos /x'
at R, then
2 2 2 '2 2 2 r (1-/) = r'Z(l-M <■) = R Z ( l - ^ ) ;
(5.3.13)
2 2 r (l-/i ) = constant.
i.e.
This implies that SJJ.
_
dv
1 — (i
(5.3.14)
/xr
Further, we know that 2 rZ(l -
or
*2 2 Z ) = R^(l -
Mo
R2(l - j ) l
""o
=
2
2 /o)
181 This implies
* ^o
R
ar
2 2 < X ~ "o> * 3
(5.3.15)
Now letting r -» R and taking note of the boundary condition (5.3.7), after differentiating (5.3.11) with respect to r, we have d_
ar
Kr./i)
2 F.1 1+ , 2 F. 1-u A* c i o «j LL 4 " R, 2 " 4 " R, 2
°"
r=R
F.
q(R)s
l
^
+
1M
as 3R
^ |
o as
R/x 3/i J 'o o
S
(5.3.16)
where S = S(R,r^;u,u ) . 1 o In view of the transfer equation
a3r and
_ 1
I(r,At)
F. ,,?;;. ^
S(R,r 1 ;^,M 0 ) - *(R
r=R o(R)
-5- I(r,-/i) or r=R
where ?(R,/i) = u(R) 1 +
f(R,-(i) ,
ir 1
4
0 < fi < 1,
S(R,ri;M',Mo)
^
(5.3.17)
(5.3.18)
(5.3.19)
2 J„ From (5.3.16), (5.3.17), (5.3.18) and (5.3.19), we have the following partial integro—differential equation for scattering function
as + i=e! as 3R
^ o as_
R/i 3A»
= a(R)
RA»
5M0
i
i_
V
MQ
l+ i J o S ( R , r 1 ; M " > , 0 ) ^ +
+
2 o. 2 ^ o S n 2 2
RM M 0
iJoS(R,r1;,,,')^;
U0I0S(R'rl;"^)S(R'rl;""'"o)T?. ■
(5.3.20)
182 where S = S(R,r •/*,/* ). The relevant boundary conditions under which (5.3.20) is to be solved are
S ( R , r 1 ; / i , / i ) = ttAfifi
for r
= R
f o r /i > 0 (5.3.21)
S(R,rn;u,u ) = 0 i o
f o r - 1 < ^ < 0.
The equation (5.3.20) is the requisite invariant imbedding equation for the scattering function to be solved under the conditions (5.3.21).
Then from (5.3.12), one can determine the emergent
intensity.
The integro—differential
equation
for transmission
function
:
In the absence of reflecting inner surface at t - r., if one is interested in finding the emergent intensity at r = r. in a direction cos
(—/i) (0 < it < 1) , one can write I(r
-n)
= I(r,- M )(£-)(^) 2 expT- f M
+
i
r1
27;L
r
T(r
l
L
a(r')d s '
J ^
-ri;^')Iinc(r--'i')d''
where T(r,r,;u,u') is the transmission function and I.
(5.3.22)
(r,—u') is
given by (5.3.10),
with
* I ix = \l -
and
s = XIX
2 r .2-.1/2 (1-/)
I '
*
(5.3.23)
(5.3.24)
In (5.3.22), the first term on the right hand side accounts for the direct transmission of intensity I(r,—n) of diffuse radiation in the direction —/*.
The second term accounts for the diffuse transmission
of the inwardly directed incident radiation I.
(r,— ti) by the
spherical shell layer of thickness (r—r.) below the surface at r.
183 By an approach, similar to that in the case of scattering function, and remembering that the initial condition of the problem will read as I(R,-ji) - 0
for
(5.3.25)
0 < n < 1,
we obtain the partial integro—differential for transmission function, T(R,r1;M,/io) as 2-L
§1 + L-^2 §1 . 1 ^o ST_ q(R) R/i
3R
dfi
o = a(R)
„ 2 2 RM M Q
a
du.
Ru
o
2
o
( T ] III)2 - (- J^HI1
T
+
JJ>-'i=<"-.> ^ }
♦ | J I».r i;i .,,') SKI + 1 J J I(R,rlil.,,')s(R,rli»-,1.0
d/i' d/i"
(5.3.26) T - KR.r^p,/^) and
where
r, .2 1/2 and
"o
=
(5.3.27)
1 - (1-V)
The boundary conditions for the problem are
T(R,ri;M,/i0) = 0
for
R = r±
> T(R,r ;/j,/i ) - 0
for
(5.3.28)
0 < p. < 1.
The diffusely transmitted intensity is derived as F.
R,
I(rlf-/i) = Y1 "o
rR^
F.
Jv°.(r)
ds 5<"-"o>d"o
+
1
5= T(R,ri;/z,M0) (5.3.29)
184 with
2
"o
1/2
(5.3.30)
1 - (1-^) 1
<* -f
c
0,
2 2 2 if r, < R (l-/x )
l
otherwise;
*■ 1, otherwise; and S (n—fi ) is the delta function. and S (n—fi ) is the delta function. Thus the method of invariant imbedding utilised the global Thusof the method of invariant the global attitude Ambarzumian's physical imbedding principleutilised of invariance for solving transfer problems in spherical geometry. Bellman, Kagiwada, Kalaba attitude of Ambarzumian's physical principle of invariance for solving and Ueno problems also solved the radiative transfer problems in (a) a transfer in spherical geometry. Bellman, Kagiwada, Kalaba and Ueno
also solved the radiative transfer problems in (a) a i and
spherical shell medium surrounding a black core emitting radiation and that in (b) a spherical shell medium surrounding a vacuum core with a central point source.
The mathematical approach for all these (9) problems were identical. Bellman, Kagiwada and Kalaba and equations (21) Rybicki developed algorithms for numerically solving the equations (5.3.20) with (5.3.21) and (5.3.26) with (5.3.28). 5.4
INVARIANT IMBEDDING IN CYLINDRICAL MEDIUM Bellman and Kalaba
and Tsujita
used the invariant
imbedding method to obtain the functional equations governing the (25) Tsujita also used
diffuse reflection from a cylindrical medium.
the method for obtaining the dissipation function for radiation in (29) such medium. Uesugi and Tsujita calculated the functional relations for the reflection function of an incident searchlight beam in slab, spherical and cylindrical media. They use the principle of (13) invariance of Chandrasekhar and showed that in the case of constant conical flux of radiation uniformly incident on the outer (2) boundary, their results reduced to those of Bailey , Bailey and Wing
and Ueno, Kagiwada and Kalaba
. Heaslet and Warming
,
on the other hand, solved the auxiliary equation of source function in a homogeneous, infinite cylindrical medium with internal emitting source and uniform boundary condition.
185
In what follows, we outline the method as used by Tsujita
(25)
for obtaining the functional equation for dissipation function for a transfer problem in an infinite, inhomogeneous cylindrical shell medium.
Model
:
We consider a cylindrical shell region bounded by radii r..
and R(r1 < r < R), the axis of the cylinder extending to infinity. The material of the shell is supposed to be inhomogeneous, absorbing and re—emitting radiation isotropically with axial symmetry.
The
shell medium is assumed to surround an absorbing core of radius r.. . It is assumed that when photons pass through a distance A at a distance x from the central axis, each one has a probability of a(x)A + 0(A ) of suffering collision. The fraction of colliding photons being re—emitted isotropically is u(x).
Fig 5.4.1
The direction of photons incident uniformly on the outer surface of the cylindrical shell is specified by (r)',/i') where r\' = cos $' and fj,' = cos tp' [cf. Fig 5.4.1].
186 The reflection function p(R;r; ,/xjr;' ,/j' ) and the dissipation function L(R,r1 ;r?' ,fi')
p(R;T7,^;r7' ,/i' )
are defined as follows.
Expected number of photons emerging per unit
/T7
area per unit time on the outer surface x = R in a direction (r/./x) when a photon is incident at
(TJ'.A4') on the outer surface.
L(R,r1;t)',M') = Expected number of photons being absorbed in a region (R,r ), when a photon is incident from the direction (rj' ,n')
onto the surface x = R.
The projection of the ray of incident photons on a plane perpendicular to the axis makes an angle <j>' with the normal at x = R.
Now adding a
shell of infinitesimal thickness A to the medium under consideration, the projection of same ray makes at x = R + A an angle 4>' - 8' with the normal at that point.
Fig 5.4.2
Then from Fig 5.4.2 R sin <j>' = b,
d^' = -
tan
-6' = | t a n <j>' + 0 ( A 2 )
.^'dR,
6<j>'
- tan 4>' + 0(A )
(5.4.1)
187 Then expanding cos('-S') around <j>' , we have 2 cos(^'-S<^') - cos <j>' + | 1 ~ M ' R ,z
+ 0(A 2 ).
(5.4.2)
Let the photons be incident at R + A in a direction (r/'
,ti'+Sti'),
where fi' + Sn' = cos <j> (say) . Each one of these photons has a probability of suffering collision in the region R and R + A of magnitude a(R)A(r;' ,n'+Sn')
.
The fraction of colliding particles absorbed in (R.R+A) is a(R)A(r;',/i'+5/i') {l-u(R)}.
(5.4.3)
The fraction of colliding particles which reach R from the direction (.V' >M' ) is w(R) 4TT
a(R)A(»j' tfi+Sfi'),
(5.4.4)
where from (5.4.2) ^ — ^ — (R+A)cos - R cos '
A(r)' ,/i'+d^' )
l-r,'1'
/
A
- + 0(A 2 ).
(5.4.5)
u'/l-n'' Then the fraction of the incident particles absorbed in (r.. ,R) after a collision in (R.R+A) is
4 ^ " a(R)A(fj' ,n'+6y) 4
"
J
I
J
f L(R,r, ;>?",/i") d, ?" d ^"
-i o
x
/
.
(5.4.6)
rr~2 1-/1"
Further, the fraction of incident particles directly transmitted to R without any collisions is given by 1 - a(R)A(rj' ,fi'+Sfi'). The fraction directly transmitted and absorbed in (r1,R) is
(5.4.7)
188 [1 - o(R)A(fj',/i'+«^')]L(R,r1;fj',/*').
(5.4.8)
The fraction reflected by the layer (R.r.,) and absorbed in (R,R+A) is
l-a(R)A(ij',/i'+Sji')||
I ^^-p(R;r?",M";r?',/i')A(r?",M")a(R)[l-1o(R)]. J 10
-
Ay
(5.4.9) The fraction back scattered into R = r after a collision in (R,R+A) and absorbed in (R,r.) is
[l-a(R)A(ir,*••+*/*')] I
-
1i0 ^ / ! ( E ; l ) ' , ^ ; l | ' ^ ' ) M i I
A7
-+1 r + 1 .1 r1
l,
^")«Wl7
v dt?"'du"'
(5.4.10) M L(R,r .;ij"',/!"') °" J J -l 0 / 2 1 U / l-/i"' The p r o c e s s e s i n v o l v i n g two or more s c a t t e r i n g c o n t r i b u t e s o n l y terms x
9
of t h e o r d e r 0(A ) . Then L(R+A,r.;r)'
,n'+6y.')
= a(R)A(ii'^'+J(i')(l-»W) .+1 .1 + a(R)A(f,',p'+fiM')fJ | L(R,r-,;7",M") -1J0
d>? d/1
"
/iV
"
+ [ l - a ( R ) A ( . j ' , / i ' + ^ ' ) ] L ( R , r 1 ; r ? ' ,/*')
+ [l-a(R)A(!,',/*'+«/»')][
f
1
df? dM
"
"
- ° A7 x p(R;i 7 ",/i";»,' I /i')a(R)A(»? , , ,A« , , )(l-w(R)) x
/3(R;^'',^
,
';^'J/i')a(R)A(r;",/j")(l-w(R))
+ [l-a(R)A(lj',/i'+fi^')]f
I
gg^jgl-
p(R;i>",M-;^-,M')B(R)A(t>"./i'')
+ [l-a(R)A(lj',/i'+fi^')]f
j
gg^jgl-
p(R;q",^";v-,/i')q(R)A(r?",Ai")
lJ
- ° A 7*3^SZx^g> J l J1 L (R,,;,.-,,..') x -gl
J
f L(R,r.;,-,M-) ^
lJ
- °
^
A^
+0(A
1
2
).
+ 0(A2).
(5.4.11) (5.4.11)
189 Substituting
f o r A from
(5.4.5),
we h a v e from
(5.4.11)
L ( R + A , r 1 ; i j ' , / i ' + 5 M ' M ^ R . r ^ r j ' ,/*') A q(R)(l-<j(R))
4TT
i-n'2
iti'+sn')/
1
a(R)u>(R)
(/i'+5/i')/l-r,':
r+1 r1 x -1
J
dq"d/t" L(R,r
,;r,",n»)
Ay
0
a(R)L(R,r1;T/' ,M')
)AV
(M'+5M'3
+ a(R)(l-u(R))
a(R)u>(R)
^ > J
.+1 -1
,+1
x f
j
.1
J
"0
.1
Now l e t t i n g A -» 0 i n
3r
+
lz^l^_ r/i'
(5.4.12), o(R)
+
3/i'
dr;"d/i" M
f L(R,r,;,-',M-')
(1-I»"2)(1-/I"2)
MV
P(R;r,»,»>■;,■,»■
-1 " 0
3_
dr?"d/t n
I p(R;r,",/i";r?' , M ' ) - 1 J0
"/(l-»?"2)(l-/i"2)
-^^11+0(A2). / 1-^"' Z
( 5 . 44 . 1 2 )
we h a v e
UR.r^TT./O
H'/M' q(R)tj(R)
«*
[l-w(R)]
M'/l-r)'2
+ a(R)
x
-+1 .1
LI -1
"0
47T/J
P
j
L(R,r1;»?",/i") /
1-M"
dr?"d/j' i
P
dr;"d^"
r i 1 0
•AV" - '
p(R »J" ,M" ;»?' ,P )
1-»(R)J + ^ g ± j
r !
"/(l-r?"2)(l-M"2)
L(R,r i ; J 7 "',M"':
dr;"'d^"'
JV^
1
J
(5.4.13)
190 The integro-differential equation (5.4.13) is the functional relation for the dissipation function which is to be solved under the initial condition L(r1,r1;r?',M') = 0,
(5.4.14)
implying the absorbing nature of the core. The equation (5.4.13) contains the reflection function p(R;»? ,H; IJ' ,/i') . This was obtained by a similar application of the method of invariant imbedding with appropriate initial condition [cf. (25) Tsujita^ , p.473], We have
(a [H -
L +
8 ^ 1-M'2 a i v h ._ ~W afi + -&- W ~ -^zl'tov-™ 1-M2
a(R)
+
f\
———-1 p ( R ; » ? . / ' ; » ? ' , / * ' )
r+1 f ,_ J _ x J Q p(.R;v,p;i"
„.
.M") /
x\——i—A.H 1
W
/
,. •» >
J* i-n'--2
/7~2 u(R)a(R) 4*
,
-+1
2 71
1-/1
-1
j P{K;n'.M';nu^u) -- i 1 o 0
+ [
dr^dju" >—-2 V
J
l-V'
-
dr)"dfiu /
2
\ 2~'■>
/iV (1-rT )(!-/*" ) (5.4.15)
As the core of radius r. is purely absorbing, for ir the initial condition for solving the problem, we can write P<.*l_;v,l*iri' ./*') = 0.
(5.4.16)
5.5. PROBABILISTIC METHOD IN SPHERICAL AND CYLINDRICAL GEOMETRY The probabilistic method was first proposed by Sobolev solving transfer problems in plane-parallel medium.
( 22)
for
He defined a
function measuring the probability of a photon absorbed at a particular optical depth to reappear at the surface of the plane-parallel medium in a particular direction.
In terms of that, he
expressed the scattering function, the transmission function and the
191 specific intensity.
He, then, appealed to Ambarzumian's principle of
invariance to obtain the law of darkening. Ueno
( 9fi *)
On the other hand,
built up a stochastic model of multiple scattering of photons
and applied this statistical theory of radiative transfer to solve a wide variety of transfer problems in finite and semi-infinite plane—parallel media.
His approach was based on the Markovian
properties of the photon diffusion process.
He demonstrated that from
Chapman—Kolmogoroff equation, one could obtain the stochastic integro—differential equation for photon diffusion process, assuming a resonable nature of conditional probability. Ueno's method is indeed (281 mathematically elegant. Uesugi relied on an axiomatic approach of defining the scattering and transmission functions in terms of four suitably chosen probability functions for forward and backward scattering of photons.
Apart from other advantages, probabilistic
method served to supply some basic ingradients of Ambarzumian's physical technique by looking into probable behaviours of photons in radiative transport. We demonstrate below the use of the probabilistic method for solving transfer problems in spherical geometry [Leong and Sen
'
]
An example of the application of the method to transfer
problems in cylindrical geometry will also be given. Model
: An inhomogeneous, anisotropically scattering, source—free
spherical shell medium is bounded by surfaces of radii r.. and R (0 < r. < R).
The inner surface is taken to be a perfect absorber.
A conical flux wF. per unit area is incident on the outer surface at x _1 an angle cos fi to inward normal. The radiation field is supposed to be axially symmetric so that the source function and Intensity are azimuth independent.
I(r,^) and I(r,—fj,) denote the outward and inward
intensities respectively (0 < \L < 1) .
192
Fig 5.5.1
The equation of radiative transfer for diffuse radiation in this case is given by [cf. equation (1.3.28)]
au^X
+
i^_ aii^o . _^(r)I1(ril0 _f(riM)]i
(5.5.1)
where the source function, ?(r,jj) is given by .1
*(r,/i) - <S$p- J
I(t1^)it(t,|i1(i,)d.
(5.5.2)
-1 with F. r^o , . l a>(r) 5 - f?(r, -M*0 ,M) *
r
a' 2 r
exp - f ° Q(r')ds' , L
J
rit
> R/I^7 Bo(r,/0 - -j
(5.5.3) 0,
r < R / 1-M 2 .
The first term of the right hand side of (5.5.2) takes care of the diffuse radiation part of the source function and the second term gives the part arising out of the reduced incident flux.
a(r) is the
attenuation coefficient, w(r) = a(r)/a(r) is the albedo for single scattering, o(r)
being the scattering coefficient.
phase function given by
r/(r,^,/i') is the
193
r2n
l nir.n.p')
- i-
r,(r,/i',(p',/i>
(5.5.4)
We have labelled the phase function here as r) instead of p to avoid —1 * confusion with the notation for probability functions, cos ft is the angle made by the direction of the incident flux 7rF. with the inward normal to the surface at r, so that 1/2 r
o
L J
(1 - M Q )
* s = ru . o
and
(5.5.5)
(5.5.6)
The equation (5.5.1) is to be solved under the boundary conditions I(r
/i) = 0 ,
0 < fi < 1 (5.5.7)
I(R,-M) = 0,
0 <
/J
< 1.
The first of this takes account of the perfectly absorbing nature of the inner surface and the second suggests that there is no inward diffuse radiation at the free surface R.
Probability
of emergence of photon from the medium :
* Let p(r,— fi ,fi;R)dfi
be the probability of a photon moving at angle
—1 * cos fi to the inward normal at r, emerging from the medium at r = R within fi and fi + dfi.
We define q(r,/i' ,fi;R)dfi
to be the probability of
a photon emitted at r in a direction fi' to the outward normal emerging at R within (fi,fi+dft)
to the outward normal.
The phase function,
rf(r,—fi
,fi') may be viewed as the probability of a photon at r being ° -1 * -1 scattered from cos (—fi ) to cos fi' direction. It may be mentioned that rf{x,fi,n')
Then
L
p(r,-/i*,/i;R) = ^ p - J
= »?(r,/i' ,fi).
ij(r,-jT,/i')q(r,/i',/*;R)d/i',
where u>(r) is the albedo for single scattering.
(5.5.8)
194 Probabilistic
integro—differential integro-differential
equation
:
A small layer of geometrical thickness AR is added to the sphere of radius R.
Then the angle cos y. subtended at R with the outward
drawn normal changes to cos /i at R + AR along the line of observation, where /* = cos $ and /i - cos(0-5 ) [Fig (5.5.2)]
Fig 5.5.2
Then
fi = n + ^ -
and
As - —
AR
[cf. Leong and Sen
AR +0(AR)
+ Q(AR)
(5.5.9)
(5.5.10)
, p.467, equation B3]
Hence retaining upto the first order terms in AR * — — p(r,-p ,/i;R+AR) dfi R+AR]' * ! a(R)AR dp = p|l - + |i J* p ( r , - / i * ( p ' ; R ) ^ ^ R p(Rl/i',M;R)dM'|dM.
(5.5.11)
195
The term
[
R+ARl — —
on the left hand side of (5.5.11) provides for
geometric convergence.
The first term on the right hand side of
(5.5.11) accounts for the probability of the passage of photon through AR without absorption.
The second term takes care of the contribution
from the multiple scattering within the layer AR. The equation (5.5.11) can be rewritten as
* 1-u p(r,-/j ,/i + _/ o R/J
fR+AR"
AR; R+AR) 1 - ^ A R R/i
= p(r,-/io,M;R)|l -
IR
a(R)AR
*
AR
2 Jn p(r,-u ,/i';R)a(R) ^- p(R,/i',^;R)dM', (5.5.12)
where d/i has been written as
An = d/i
1 - ^ A R
+ 0(AR).
(5.5.13)
RM
[cf. Leong and Sen
p.467 equation B4] .
Rearranging the terms and making AR -¥ 0, we have d_
3R
1-fi2
1+M
d_
R/i 8fi
a(R)
,2 2
R,
5
oOO
p(r,-/xo,M;R)
"
p(r,-/i*,M';R)p(R,/z',/i;R)
^
(5.5.14)
The equation (5.5.14) is the probabilistic equation for photon emergence from the spherical medium.
Emergent intensity
:
To find the expression for emergent intensity, a scheme is (19) followed which resembles that of Minin for plane-parallel medium. Remembering that q(r',/i',^;R)d/i is the probability of a photon emitted at r in a direction cos
\L' with the outward drawn normal emerging at
196 r = R in the interval (/j,/i+d/i) , the emergent intensity at r = R in the \i can be written as
direction cos
KR.AO - i J
!
g(r,M')q(r./»',A»; R)
\
a(r)dr M
d>'
(5.5.15)
Here we have used (5.5.10). g(r,n')a(r)drdQ' is the measure of the quantity of radiant energy emitted per sec per unit area by an elementary volume and optical thickness dr [dr = a(r)dr] contained between r and r + dr in the p'
direction cos
to the outward normal at r within the solid angle
dQ' . In the present model the outer boundary at r = R is illuminated by a conical flux TTF.
per unit area incident at cos u to the inward J° - . —1 * normal and the amount of energy absorbed at an angle cos ^i at r
within the optical depth dr = a(r)dr is given by
*F.[
c-sKim-o •<*•"■■
a(r)dr, for r. > R 1 Q
XVo .
(5.5.16)
Then g(r,/j') is given by
g(r.M')
u(r)F. , n i o = —
sHi]'
rR^
|
exp
°a(r')ds' '/(r,-M0,M') • (5.5.17)
Hence, from (5.5.8), (5.5.15) and (5.5.17) the emergent intensity I(R,JJ) is given by
F. -R r /*
-i ] ( 1 ]
«*->-^ r
l
2
rR^
exp
-I : a ( r r
) ds' p(r,-^o,/i;R)a(r)dr.
^„
*o (5.5.18) *
Hence if p(r,-u ,/i;R) is known, we can obtain the emergent intensity.
197
Scattering
function
:
Multiplying the integro-differential equation (5.5.14) by
( and r e a r r a n g i n g ,
If
<3R
+ Q(R)M
exp
- J ° o(r')ds''
o a(r)dr
we h a v e
m"
a_
R
o •* "o '
^ a ( r ' ) d s ' k>(r,-/io,/i;R) a ( r ) d r
exp-{- I
.[[■" o l f R ^
,. R M
2
exp-UJ
r
o
°
a(r')ds'lp(r,-po,/i;R) a(r)dr
M„
mm-
+ f k * L .
f r R/i o
exp-^-l ^ •>■ r/x ' o
o
1
*
a(r')ds'^p(r,-/io,/j;R)
x a(r)dr + Qa(r)dr
Mtt?) I r r M„
9
0LL
f R ] 22
/i
r
r -Ru 1 * d/i' exp-J-! ^ a ( r ' ) d s ' W>(r , - ^ , / i ' ; R ) p ( R , / i ' , / i ; R ) r
^. x
(5.5.19)
a(r)dr,
where
ra(r)
Q =
1_ - j
-
1—U -, r U f"0 I
*
R/i
.R/i
-, 2
r
exp<j-,
°
a(r')ds'j.p(r,-Mo,M;R)
Now s i n c e 2 r1"/* oi a Ru 3u 'o-' o
IR/i
[(
[( A*
I
* [r
] [ l ]
ex
"
f r R/J o
P
exp-M
{-J^a(r')ds'}p(r,-M^:R)
^
1
*
a(r')ds'>p(r,-/io,/i;R)
o o
)[
R
1 a(R)R e x p | - |
° a ( r ' ) d s ' | p ( r ,-/<*,/»;R)
+ Q,
(5.5.20)
198 we have
UQ
- * ] ( ! ] «p{-J^«(r')ds'}p(r,-^,p:R)
a(R)
a(R) u o
, ^ o Ru
9 da o
[[$)[ i ] 2 > 4 f : . « , , -
^ o Ru o
o
a(r)dr
x p(r,—/x , £i;R) a ( r ) d r -
Qa(r)dr.
(5.5.21)
We define the scattering function as
S(ri:/*o,M;R) = |
[ - | ] [ f ] exp'j-J" ° a(r')dS'}p(r,-u*,u;R)a(r)dr.
V "o
(5.5.22)
Now substituting (5.5.21) in (5.5.19) and using (5.5.22), we have
a_ 3R
+
i-u2 d Ru
3M
+
^ _ 3 _ M°+ ^ RMoaMo-
a(R) P(R,-/I O ,M;R)
R
m
Q
^ 2
fi
< >£
i_ MQ.
S(r 1>Mo ,M;R)
+ ^ J s(r1,^o,/x';R)p(R,M',M;R)
The term of the type p(R,-/z , u;R)
^
(5.5.23)
in (5.5.23) gives the
probability of the photon at the outer surface r = R being scattered from the direction cos
(—u ) to cos
p. after absorption at r = R.
It
can be written as P(R,-MO>M;R) = o(R)
(R,-/io>/i) + i j
J
|j expi-J
x IJ
^a(r")ds"|p(r,-u'*,M;R)
/«•) 3£. a(r)dr K
w(R) is the albedo for single scattering at r = R.
(5.5.24)
199 The first term on the right hand side of (5.5.24) takes care of the deflection of the photons impinging on the outer surface r = R in the direction cos
n.
(—n ) to the direction cos
not penetrate the medium.
These photons do
The second term gives the contribution of
photons which penetrated the medium and then were scattered and ft at r => R.
finally emerged in the direction cos
Then the equation (5.5.23) reduces to
, ^
a 3R
a
Rp
, ^ o R/J
dfi
,
a dfi
i r
a(R)
(R,-/io,M) + 2 J
...fi
1
[/i
/i
W
2 . 2 2 x> Ru
2
S(r1,/io,/i;R)
u o
R) r?(R,-M0>-^')S(r1,M' ,A»I
d^'
1 + ^
i,(R,^',/i)S(r
^ JQ
+ Jf
« ,/i';R) - ^ J-
O
Ji
f S(r « ,M«;R)7(R,M',-M")S(r ^",M;R) ^ 1
^
(5.5.25)
This is the integro—differential equation for scattering function S(r1 ,ji ,/J;R).
This equation is found to be identical with that
obtained from the invariant imbedding method by Bellman, Kagiwada, Kalaba and Ueno model.
for the scattering function of the same spherical
The equation (5.5.25) is to be solved under the conditions
for
S(r1,M0,^;R)
-1 < /j < 0 (5.5.26)
S(r1>Aio,/i;R)
0
for
r,
R.
It is to be noted that over here a two point boundary value problem has been converted into an initial value problem. indeed the basic feature of Ambarzumian's physical method.
This is The
equation is solved numerically by an algorithm developed by Bellman, (9) (21) Kagiwada and Kalaba and also by Rybicki
200 In this method it has been possible to give a probabilistic interpretation of the scattering function. From (5.5.18) and (5.5.22), the emergent intensity is given by F. I(R,M) = ^ S(r 1 ,^ o ,M;R). The method has also been used by Leong and Sen
(5.5.27) for solving
transfer problems in spherical shell medium with (a) an emitting source surrounding a black core and (b) a central point source.
These
problems involve the simultaneous use of the scattering and transmission functions.
Uesugi's
scheme of deriving a family of
fundamental equations governing the diffusion of photon in slab geometry has also been extended to transfer problems in curved geometry [Leong and Sen
] by suitable description of forward
and back scattering with reference to a transfer problem in cylindrical media. Wang
constructed a general linear operator equation governing
the specular and diffuse radiation for radiative transfer in a spherical medium.
In some special cases, his equations reduced to the
functional >nal equ equations obtained by Bellman, Ueno et al .(15,16) and Sen
(ft ft 9 7 ^
and Leong
201
Probabilistic model for radiative transfer infinite, cylindrical shell medium :
problems in
inhomogeneous,
(17) In solving this problem, Leong and Sen
adapted for curved
geometry, a scheme of solution, which was originally suggested by Uesugi
for solving transfer problems in slab medium.
Uesugi's
(28 "i
technique for plane-parallel medium was built on the
foundations of the probabilistic methods of Sobolev and Ueno.
He
suggested that the photon diffusion process in an inhomogeneous, finite plane—parallel slab bounded by r = r
and r =• r- (0 < r
< r«)
can be described in terms of four probability functions p(/i,z.r;r , r . ) , q(/J,z, r ; r , f.,) , p (/*, z , r fr Q , T ^ ) ,q (/z,z, T ; r Q , r^) .
Fig
5.5.3
Of these, pdji denotes the probability that a photon emitted at T will reappear at z (< r) in the direction (/i,/j+d/j) ^ > 0.
qd/i denotes the
probability that a photon emitted at r re-emerges at z (z > r) in the direction^,/i+dji), ^ > 0. p d/i is the probability that a photon emitted at f reappear at z (> r) in the direction (^,/i+d/i) with /J < 0 and q d/i that of photon emitted at r reappearing at z (< T) (/i,/j+d/i), /i < 0.
within
A set of integro-differential equations for the
scattering, transmission, back scattering and back—transmission functions were obtained, the scattering and transmission functions
202 being expressed in terms of the four probability functions.
The
results obtained by this method were found to agree well with those of other methods. /no\
Uesugi's
approximate method for solving photon diffusion
problems in plane—parallel medium is based on the following two postulates : (i) A photon will travel in a direction \i through an optical thickness T without absorption and then will subsequently be absorbed in the medium within the interval r and r + dr with probability exp(-r//j) -^ .
(ii) A photon absorbed at large r will be re-emitted in the direction between \i and \i + d/j with probability d/i
2
U(T)
where to(r) is the albedo for single scattering. In what follows, we demonstrate, how this scheme of describing the radiation field in terms of four probability functions can be used to solve transfer problems in curved media.
Here, again, the
inclusion of the idea of "oblique incidence" and "geometric convergence", which played a major part in the extension of methods of plane geometry to those in curved geometry, comes to the rescue.
We
develop here a probabilistic model for tackling transfer problems in an inhomogeneous, infinite cylindrical shell medium surrounding (a) a perfectly absorbing cylindrical core with external illumination and (b) an emitting source in the form of a cylindrical black core.
The probability
of photon emission
:
An inhomogeneous, isotropically scattering, source free, infinite, cylindrical shell medium bounded by surfaces of radii rn and R (0 < r
< R) is considered.
The free boundary surfaces are such
that a photon emerging from there never returns to the medium again.
203 The photons wandering in the medium will either be scattered or absorbed.
The direction is specified by r/ and /J, where r\ = cos 9,
li = cos tp as shown in Fig 5.5.4
Fig 5.5. 4 8 is measured from the upward vertical and
d
AV
(r),/i) p e r unit are;i of surface sing emitted at r (< z) for ! after b< /i > 0.
W e also introduce '-^L- as the probabil ity of photon emergence at r = z
q(r;z,r/,^;r n ,R)
1
AV
within the solid angle
dr/d/j !
A-„
2
along the direction (T?, M) per unit area
of surface after being emitted at r (> z) for /x > 0; * P (r;z,T7,-/j ;r r R;i
AT
as the pro!lability of a photon emerging at
2
r = z within the solid angle
drjd/z
A-/
along (*?,—M ) per unit area oi
surface aft er being emitted at r (> z) for ^ > 0, and lastly
204 q (r;z,r/ ,-^;r.,R)
^ ^
as the probability of a photon emerging at
AV
along (r) ,-p)
r = z within the solid angle —
per unit area of
AV surface after being emitted at r (< z) for /z > 0. As a first step, we set out to establish the integro—differential equation for p(r;z, 17 ,ft; r.. ,R) . The function p(r; z , r; ,/u;r1 ,R) consists of four parts. (1) A photon emitted at r will emerge at r = z along (77,^) within dQ Z/i
(=
) without absorption through
r\
drjdfj, iL "
r
" AV
AV p
a(r')dr ._, , ,„ . — — - with probability
1
ZM
4TT
r
dr) dfi
A7 -G3-AV 2
—
t/j
^
exp<
Lr
"A7
q(r')dl' _
"AV
HI(MU z
J
or
p, dn =
N
*|
da 4JT
L Jr
" AV
J L
-^?)
(5.5.28)
/x, /J denote the cosines of the angles made by the horizontal projection of any line of observation with the outward normal at r = z and r = r respectively. H, fi
are related by
z(l-/i)=r(l-/i
)•
(5.5.29)
a(r) is the attenuation coefficient and & = Vfi , and the Heaviside step function
205 " r
P
0,
p -
2 ,
/
H *■
/ 2 -/ 1 - ^ - ° z
1,
/ i - 4
;
z
z"
(2) A photon emitted at r (< z) will first be absorbed within the depth interval t and t + dt where z < t < R, will be re-emitted there in some direction and then suffer multiple scattering between the layer z to R before emerging at r = z along the direction (r?,/j).
The
probability is given by
P
2dn
=
S
f
z
q( t ; z .'?.A<;r 1 ,R)w(t) 4f J
diV*
X
a(t)dt
dr;'d/i'
f
,2
^
M'/l-W A1~
0
i
0
rtM'
exp
a(r')d^'
■ J»'*A-v2
J
Ml,
/ T
t*
(5.5.30)
;
9
assuming di = —j- + 0(dt ). fi',n'
*
are the cosines of the angles subtended by the horizontal
projection of any ray with the outward normals at r = t and r = r respectively. u(t) is the albedo for single scattering at r = t. (3) Within the layer r to z, a similar process as in (2) with q(t;z,rj,/i;r1 ,R) replaced by p(t;z ,rj ,/i;r..,R) will yield
3*° - S fr p ( t ; z ' w i r r E W t ) 4J0 J0 exp
r
X
V
diV* V
UJ *J4
a(t)dt
drj'dn'
,7iV/
,
,2
„
'
ftM'
a(r')dr
" '"'Vi-v 2 -
V^ -/'' i - 4t" l>.
(5.5.31)
206
W
s imi l a i ■ly w i t h i n t h e Is
V"
=
dn r r
AT
4
4w J
X
p(t;z,ij,ji;r,,R)«(t)
4
x
r;L
r^ dfl'
■k
a(t)dt
J
r1 r1
T rr/i'
*
o o
(r')di
n
exp -
J
L Jt/i.
'
I-,'2
df?'d/i'
/i'/l-r,'
2
/l-^'
(5.5.32)
2
p ( r ;z ,r? ,/ijr.. ,R) i s made up of t h e c o n t r i b u t i o n s from P. ( i = 1 , 2 , 3 , 4 ) . Then p(r;z,t?,/i;r1,R.) _ 1
'**' > J
47T
+ -
|
* Jz
6XP
T f2^
a(r')di'~
J * J
2 ,
«[/^* - / I - rz 2
J
z
q(t;z,ij,Ai;r-,R)a(t)| x
J
o Jo
exp- |
L Jr/
' *'
a(r')di'
+ -T l J
r
p(t;z,i,,(.;r
X
R)o(t)
J
22
0 0
r
/ l - , '
/ ( l - / i ' 2 ) ( l - ^ ' ; -)
r]dfl'
2 J
/
HL-
2 ,
"t2 J
*
a(r')di''
dtdri' d/j'
x —
L
)
exp - j LJr/i
J
*
'' A V
dtdr)' d/j' / 22 M7 (l-ix' )(l-r,
fr d n '
2 ,
-- / i - rr "t2 J
* + - [ p(t;z,f/,/i;r 'rJr1
X
R)a(t)f J
x —
[ 0 0 J
a(r')di'l
exp - f LJt/J
d t d t j ' d/j'
2
/
— .
/ a-/*- )a-,,- '")
2 -1
tjdn-
(5.5.33)
207 By e q u a t i o n
(5.5.29),
we h a v e
*
2
z r
(5.5.34)
*
Since dfi'
dp'
an'
we h a v e ,
inserting
an' art'
(5.5.34)
in
2
!/ I - M '
dri'
n
-
ii'*
2
(5.5.33)
p(r;z,»7>/i;r1,R)
(r >di =L-4-r^ ' i [vW, - / -4] " /iV L r
J
v
/i
v
'
R + - fJ
*
1 q(t;z^
Z
) M
; r x R ) a ( t ) Jf
y
Z
1
Jj
o o
x
r t/i' [r')&Z exp - f LJr/i-*y 2
dtd„'dM' M'/
-[''-I V J
i-,'
H
(l-M'2)(l-r?'2)
["'-Z7?
+ i f p ( t ; M l ? ; r 1 B M t ) f f exp[-f ^r ' ) d i ' " x J J "Jr o o LJr/i.*^r dtd,'d„' 2
H
M' -
2
/x'/(l-M' )(l->?' )
y
N
i »J
/*-£
* r + i |J
*
ri
.1 .1 p ( t ; z , r / , / i ; r 11 , R ) a ( t ) J
X
—
J
o o
r .r/i' expL JtM-
. :')&l'~ -^
^r
dtd»?'d/i'
M'/ (1-^
2
[»
when r < z . 2
(5.5.35)
208
F o l l o w i n g t h e same scheme of d e r i v i n g ( 5 . 5 . 3 5 ) , t h e e x p r e r s s i o n f o r t h e o t h e r t h r e e p r o b a b i l i t y f u n c t i o n s c a n be o b t a i n e d . q(r;z,//,/i;r1,R)
i rz
: ,r? , ^ ; r . ,R)cr
Kj
*xr i (V )
o ( r )cU'"
J
° o
L J ^ //l -^ . z J
2
r
i[ r q ( t; 2 l ,,, : r 1 ,R)c r (t)f 1 f1 «pf-f",,*-<*,)«U'" 51 J
X
z
[>' )
v
/
"
J
0 J0
L Jt/1.
/
y 1—fj
2-
dtdr?'dM'
V*J +
J
dtdr/'d^'
L-*J / +
(t)JJ 0JJ 0
if/
2
T
(1-M' )(!-»?' )
I fR q ( t ; z , , , ( ; ; r . , R W t ) [ 1 f' expf f ^ w J J r 0J0 L V
dtd d
*£J
;' -'
2
Q
-i
H [ , . - / T ^ ] , * , , . > „ (5.5.36,
* p (riz.r/.-^jr.^R)
1 - 4.
eX
?
[
J
J- A
<J0oJo0 JJ
x f
r
JJ
r
V-
r r/J
* ,
a ( r ' ) d i ' 1 f/x I] * 5
exp -
r
+
')d&'~
,
1 TrR * , P (tjz.v.-^r
HT/I' L Jr M - / .22 -I - L r u ' / 1-*?' J
, _,.-. ^rCtjf1
J J
-
/
f1 c-rpf
° o
■
L
r
2.
t: -1
y
f^'
Q
(r')drl
V/M.2J
R)a(t)
209
.z
*
+J
q
( t i z . r j , - / ! ; ^ ,R)cr(t
L
>j l r 1 «pf-r' B ( t ' ) d i ']l J
f/i'
]
0J0
I V
dtd/u'dr/'
*
- V V / ,,
/ 1 - ) ? , 2 JJ
for r > z ,
,2, .. ,2.-
(5.5.37)
and q
(r;z,r;,-Ai;r1,R)
■K
rJr pR * (tj,s,»7,
+
fV(t;Z,,,-,;r1,R).(t)r1
_ 1 1
L L
zz
-/i;r1,R)a
X
r
JJ
'
J
^ M *'"/-$] -;>]
I - V **/7 / . ,2 i I
0 JJ 00
J
['expf
0 0
r
f / (
I - V /, /
' ^ ' U
,2 J 1—t?
-
/
■
hA 7 ?)
f q*(t ! «., l - t ,;r..R)a(t>r 1 f' expf f ' •<''>«"'] X J r.. 0J0 L J t / i ' / , ,2' J
+
IT*' 1
dcd/x'dr?'
1
2
2
(l-Al' )(l-r7' )
In p a r t i c u l a r
r
(5 5 38)
J
p ( r ; R , ? 7 , / i ; r . . ,R)
1
H
+
eX
i 7T
P1- J *
a(r )di : 2 / l - -n
} v
J
rr R p(t; R ,,,,;r 1 ,R).(t)f 1 J
-r
X
J
,(,. - A 7 ?) J
f" expf
0 0
J
f ^ / ^ ' ^ ' } *
I r/x' A
rrp(t:R,,,M;r1,R>(t)f1J1exp{-rr'i' 1 J r0 J0 I "V
J
1
nt +
Lv
dtdfj,'
,2 i
7
h Z ?)
-
drj'
^'/(1-M'2)(1-^'2)J
(3.3.iy;
210
q(r ; r r M;: r i ,R)
■ i Ur
+ -r
l
q(t r
r?,^;r
1
1
0
| 0
R)a(t)[ J
expj- r ^ ' a C f X U ^
V/^^J
J
1
q ( t ;: r r
»7 ,M;: r l
,R)<
J
o)
J
'|V ] U
"'
A-
0
+
■
r
2-,
~t2J
,2 J-l
dtd/x'dr;'
J
/
)di'Tl
a(r
expJ- 1
M'~ *•
(5. 5 .40)
2
"'/fl-*» Hl-.'V
P ( r ; r 1 , 7 , -M;r 1 ; R) _ 1_ 4TT
a(r
expj- |
-17
-i r «K
+ l
r *( t n[\ P L r
r
r ,R)o l ' " ' ~M 1
* P ( t ; Trn.
Jl
~n;r i
,R)cKt)fJ
i01 -j-r^'^X'-/^ -] I. V ' / , „,2 J ^ '
0 J
tfcJ
exp I H o J
0
"|V ] *
a(r')di'V
^Jt^'
7i-u-2lJ
dtd/i'dr/'
(5 .5 .41)
^•V/a-^xi-vV
and q (r;R,r? ; - p ; r 1 ( R )
- i [!>*■•—!■•>•<< £ -4 £ . ^ > f 7 ^ -3 +
rr * r
i
q (t;R,i7,
-M;r 1 ,R)o
(t)f
J
0
f exp(-f M
J
*
« ( r )di'Vl
L Jt/i'
0
-n
,2 JJ
dtd^'dri'
( 5 . .5 • 42)
V - V (i-^' )(i-v' ) M
2
2 J
211 Since we have
R 2 (l- M 2 ) - r 2 (l-/ 2 )
(5.5.43)
** dn W 3R
(5.5.44)
2 _ R(l-/i ) * 2 Mr
* 3M = [ r J | * J an
(5.5.45)
Ue now derive the integro—differential equation for p(r ,R,i; ,/i;r1 ,R) . Differentiating equation (5.5.39) with respect to R, using equation (5.5.44) and after some rearrangement, we have 3P p < r ; R , ^ , p ; r 1 , R )
"fc exp [~
TR/J
J
g(r')dl'
* /
T_ £^_| a(R)M
[ft | a ( r ) r R ( l - f ) , _^_ R ( l - / ) *r 2 p *2M r* 2 V J x-n1 M /J r /J /J r
i
+ *
H
hA-^) R
J
7 [--A ?] IV*/—71 J
r1
J a(R)p(R;R,r7,/ 1 ;r 1 ,R) 1 JQ
>o
a 1 J. ♦J rJ p(t;r,,,|,;r k -<- ,RMt)dtJ •,•«»< J HV C/ ^T^fH'-/ ?] hA?) -2+
S
f r^
1
L
M
1
a(r')di')
J
/ l-TJ
f1 exp{- f "<''>«"')
f |= + | ^ Pp((t ;r,,,,;r t ; r , r ) , / i1;,R)a(t)dtf r 1 , R ) a ( t ) d t J1 f expf f ^ - f c ^ l ) + J J J rr , d R L Jt/i' / T T J <0 0 J 1 / 1-r?'
x x "Kl
; ^'^==1. [N"' " 7 ( i - , ' ) ( i V )
Ll
JM
V
22
2
J
(5.5.46) (5.5.46)
(l-M' )(l-r?' )
Differentiating (5.5.39) with respect to n and then multiplying by „ , we have Ru
212
3g-l I P
. 2_ i V e R/i p..
4TT
tp
r_r^ «(r:)di;
] j
*
- AV~ '* :R
-l 1-M2
f 1
fR/i
f
t? "RT" e x p f 1r/»
+
-r , J
? L
o ( r ' ) d i L'
AV
AV A 7 \L_
ry
a(r)r
+
M
•kJ
H J
.-/ - i' ?
i
a_ IR
}[;*
2
*2
r
/j
J
*■
R
'
1 .1
;t ^ fc ,Wi..,.„,1.w«,j j o . 4r"'j rA-,' = ^ > f - H3 A
+ — 7T
-R ,
Ra(R)
*
Jl
.—
rx
*
2
^ ^"( t ; M , , r i , R )0 .0t ) If ft 4' r/ 7 ^ }] I72 R/i
a
1
X
J J
h t ' "I
J
M
d t d ^ ' dr)'
LL'*-L- /
2
(5.5.47)
2~J
F u r t h e r , we may w r i t e a(R)
MAV
-1
+
1-n R
2n
p(r;R,r;,/i;r1,R)
"
rR^
exp{_ r \ *nn**L\
^*>- KI• rl/x/7 Vl-,2 J
_
*
a(R)
"A7
1-M
R/i
2n
2 ,
Hhi - /
1
1 1 t i ^)p(t;R,,,,;rX1 ,R) f f (J t )[ f exp{-f ' ',^^j J J
^R/J
00
x H[M'-/T4)
l r^'/^7^2-'
213
+
l,-1
j J^L.kL\p(t;M,M;
r jn'
R)ff(t)rYexjr
E ^ > ^L
« ^f 'S l"7
J
From ( 5 . 5 . 3 9 ) ,
2
J|
J
0 0
^/(l-M'^d-r,'2) r r R i"'
R)J J e x p j J
0
d/i' dr;'
p(r;R,r/' ,/x' ; r rR)
1.1
- — — p(R;R,,,M:r
J
^Jru'
0
* ■
V ^
"2 J V
/~
M
J
r . /7~?-\
d^i' dr?'
X
^
(5.5.48)
2
we may w r i t e
-4ff(R)p(R;R,T/,/i;r 1 ,R)J
- 4j
}j
'■
-
/
x
"
*
'/(1-M'2)d--?'2)
i p(R;R,f ? ,M;r 1 ,R)J
[ p(t;R,r,' tff jx^,,R)
dr;' d ^ '
"7(i-p' 2 )a-,' 2 )
l.i r .t/i" a(r")di" x a ( t ) f [ exp-j-f J 0 J0 *- J r
""AV
x 1^1
J
J..„ A ?l '
dtdr,"d/J"
HL" " / 1
**" ^ " / ^ W J '+J - 4
^p(R;R,?,n;rrR)(
/ xf r
x a(t)
J
J
|
'o o
drj' dfj,'
f»( t.;i:, ■■,' ,/,.': r , ,R) /i
/ r
i exp-M 0 J0 LJ tu"
tt(r")di-g 1 /T,72
^"/w-
I
J
7(l-/i'2)(l-r;'2) dtdr?"d/i"
*
1
7d-M"2)(l-'/"2) (5.5.49)
214 Adding equations (5.5.46), (5.5.47), (5.5.48) and (5.5.49), we have
*(r,R,fj,/i) = - J
(5.5.50)
$(t;R,*j,/i)k(r,t)dt,
where $(r;R,i7,^) =
r i . . k^l. §- + dR dfi an "*" Rfi D.. a..
/
T
y.-/
°( R ) _ iz^_ip(r;R,rj,/i;r ,R) r x
1
„ 2or
1-ri
4t7(R)p(R;R,r?,M;r1,R)|
j p(r;R,r?' ,/i' ; r l tR)
'oJ o
d»j' d/z'
X M
(5.5.51)
2
2
' / (!-/:' )(l-r?' )
and .1 .1 J
J
0 0
rtJ*' J
L r^' A
T2
J
y*
V
H|M'-/ 1 - V
drj' d/j' M
'/(l-/.'2)(l-r/'2)
when r < t,
k(r,t) = -
a(t)
r1
L'o Lo P1
J
exp
#.*/*' q(r')dl') f/i'
H^A7 . . J v J "7(iV)dV) dr/' d^j'
when r a t .
(5.5.52)
The equation (5.5.50) is the homogeneous equation corresponding to the inhomogeneous integral equation (5.5.39) of Fredholm type with the same kernel.
Therefore, for the solution of the non—homogeneous
equation (5.5.39) to be unique, the corresponding homogeneous equation (24) (5.5.50) must have a trivial solution [cf. Tricomi p.64]. Hence *(r;R,»;,/i) - 0.
(5.5.53)
215 That is 1 - M 2 §_
8_
aa
4. a„
a(R) _ 1-/J2' /
_j
2
4a(R)p(R;R,t;,/i;r1,R)J
p(r;R,i7,^;r1,R)
J
j p(r ;R,r;' ,/J' ; rR) ^ 0 o J
d>7' d/i' M
'/(1-M'2)(l-'?'2) (5.5.54)
Similarly differentiating (5.5.39) with respect to r1 and taking account of equation (5.5.41), we obtain j ^ r - p(r;R,r;,yu;r1)R)
A A
J 'o o
-4<7(r1)p(r1;R,Tj,/i;r1,R)j
p (r ;r1> rf ,-fi> ; r^R) —
J
dr)'
dfi'
/(l-,i'2)(l-r,'2) (5.5.55)
The integro—differential equations for p (r ;r. ,r/' ,—fi; r. ,R) can likewise be obtained as
a5 r
i
+
r
HL
i"
°
M/L^
2
1-/
p (rir^.-zij^.R)
r./x J
A A
J
- ^ ( r ^ p (r1;r1,»7,-/i;r1,R)j j p (rjr^t?' .->*' jr^R) 0 J0 dr;' dfj,' M
2
(5.5.56) 2
'/(l-^' )(l-»?' )
and |g p*(r;r 1> ^- M ;r 1 ,R)
= Uo(.R)p (R; r i, - p ; ; r r i
,R)J J
P ( r ; R , * ' ,M' ; r r ,R>-
dr;' d|i'
? "'/o^'* i' ) ( H ' ) (5.5.57)
216 The Scattering
and
TransmissionI functions
:
The f o l l o w i n g four f u n c t i o n s a r e d e f i n e d : S(z,»?,M;R,-»? 0 ,-/i 0 ;r 1 > R)
= |
p ( r , z , T 7 , ^ ; r 1 ,R)
r
i
R ■4-
expj
_ ^r(r-)drja(r)dr J
V
r / / . 2 ° 7 ^"o
J
LL
R
+ f q(r.z,,,M;r z
R ) [ - | ] [ | ' exp-{Mi -1
o
L
|
„. - i
'-
o /
l-r, Q
'
j-CT(r)dr,
(5.5.58)
* T (z,r),/i;r1,r)1,M1;r1,R)
p(r;z^,/i;r1>R)[4][^]exp.-
- J r
M
l
+ J
- Wi-,?
l
r"i-i rrii
rR
z
q(r;z,»j>^;r1,R) —
T 1 ^ r '> di 'V(r)dr J
f
vj[r-l exi 1
yL
* rr"i
r
-
l"l/l-4
{t)dr>
;
(5.5.59)
T(z)t/,-M;R1>-»?o,-/io;r1>R)
rR *
p (r;z,r7,-/i;r
+
and
r
f«
q ( r ; z , r ; ( - / i ; r 1 ,R)
i
f rR/io exp-j- | „. 1 r Mo
,R) — o
a
" R j ex
V '
f P1
a ( r ' ) d r ) , VJ - * '^a(r)dr // , 1 - ,2 ' Q
rR/io a ( r ' ) d i ' \ . . , * , Mr)dr,
o / l-„
(5.5.60)
217 S (z,rj,-/i;r1.-r/l,-/i1;r1,R)
- f
P*(r : Z ,, 3 ,-, : r i > R )ftl&l e xpf
f
1
-^i^}.(r)dr
¥1/1-,
J
r
( l] f I" 1 a(r')di'\ . . . ' |-a(r)dr. i- I q (r;z,rj,-M;r1,R) r"n — expj- I r. "i V l" / l-r;1
,. , ,,, (5.5.61)
(—TJ ,—/i ) occuring in (5.5.58) and (5.5.60) specifies the direction of the incident flux at r - R, u denotes the direction cosine of the o angle made by the horizontal projection of the reduced incident flux angle made by the horizontal projection of the reduced incident flux with the inward normal to the surface at r
>:-{>
1 -R *2
1/2
i - n:
The horizontally projected line element in these equations is given by * In the equations (5.5.59) and (5.5.61), (i;1,/i.) denotes the direction of the incident flux at r = r., n,
is the direction cosine
of the angle made by the horizontal projection of the reduced incident flux with the outward normal to the surface at r
r-
2
•;-{'
1
- r" 2 l-I X ~ "1
and r/i- denotes the horizontally projected line element.
Multiplying
the integro—differential equation (5.5.54) by
V v
fR ]
v Uj
e x pf
R rr^o ^.
q(r')di'1 , .
rJ r /-F^r r ) ' 'o / 1-n
re—arranging, simplifying and integrating the terms of the resulting integro—differential equation over r between the limits r.. and R and using the equation (5.5.44), we have, [cf. Leong and Sen
(17)
p.67]
218
fl. , kJL. L. . 3R
R/i
*I
R r
a(R)u
*U
1_ * 1 o
s(R,»?,A»;R,-»?0,-/i0;r1,R)
/ \ 2
iir
v
R 1
' exp
M„
rrR^ %
r
r
1
a ( r ) r /
< )
RM
r /*
i1 /A 1-n2
tt R
JL +
3,u
o aa((rr '' )) dd ii '' \
r
"o / l - r ,
I-K
** i
Ru
*
l-»)
2
ee xx pp
I)J
a(R)
a(r)dr
o
f}
a(r')di'
' o / 1-n
p(r;R,r?1/i;r1,R)a(r)dr
CT(R)p(R;R,fj,/i;r ; r 1 ( 1R,R) )
1 1 1+41 1+4 | S< ( RR,,rrjj'',, // i ' ,; R , - » ) J 0 0
x
„
: ; R , r j , M ; r 1 , R)
J
flr l
x =
2
,
dr;' d/J'
, - M -;.rr1^F. R )M
(5.5.62)
' / ( l - ^ ' ) ( l - r ? ' )i 2
2
Since 1_/i
2 o
S
f o f l ]
-RTT^T LL~* l r °°
° °
* *oo
^ R,u
^_2 ** «
eXP
L
„
I' R5 r
K.
*
M
^^o o
f
a(r')di'
*/
■p(r;R,»7,,u,r1,R)
/l-r,
o a(r')di'1 , D ->p(r;R,v,,u;r
o
a(r')dl'
pCr.R.r/./iir^R)
o /l—f;
a(r)r / s~
l - ^ r ^ 2 R/i
@W-{-C I rJ * T ^ F ! :
J
exp
rlvro o ( r ' ) d i ' l r /* 2 ' o /A l-ri
x p(r;R,r?,,tt;r1.,R). T h e n we h a v e f r o m
R) ,R)
1-r?
o R/i
exp
F
/ l - r , 22
'o /
A-? A 2
r
rR-M
i
a(R)R
4-v-
V
/
— I-
rR/io
f
(5.5.62)
(5.5.63)
219 L
a
3R
i-„2 a
^o a
R„ o
3MO
fl,i
RM
1
+ a(R)^
^2 + j
!
+
• \ - 2^ * ^ = '/1-,
=
o
i
2 2 o
F)]
R
S(r,fj,/i;-jj o ,-^ o ;r 1 ,R)
o
J
O' a(R)p(R;R,r,,/i;r
R) 1 + 4 [ | S(R, JJ ' ,/i' ,R,-»? -M ; r 0 J0 x
dr?'dM'
— M
R)
(5.5 (5.5.64)
2
2
'/(l-M' )(l-»,' )
The term p(R;R,T7 ,^;r1,R) can be obtained from equation (5.5.39).
In
(5.5.39), 2 2 2 *2 t^(l-/x'Z) = r (I-/*' ),
.5.65) (5.5
from which, we obtain
From ( 5 . 5 . 6 5 ) ,
a^_ _ r 1 1
r^
flM' " I r J
^.*
(5.5 !5.5.66)
( 5 . 5 . 6 6 ) and ( 5 . 5 . 3 9 ) , p u t t i n g r = R, we o b t a i n
p(R;R,fZ,M;rrR) =
^
11 1 + 4 f | S(R,//,M;R,-r?' - / i ' ; r J J 0 0 x —
R)
dn'du' 'o ' o 2
( 55.5.67) .5 2
"V(i-/i; )(i-fj; ) Putting (5.5.67) in equation (5.5.64), we obtain
\a_ + iVa_ ^ ^ _ [3R
RM
a^
RM O a %
;
+ o(R)i
'
V
l-»7
+
» + £ +i ^ 2
' u / !-»?_
R
\ S(R,ri,v;R,-r,
-n
;r ,R)
220 a(R)
1 + 4
„1 ,1 S(R,r;,/i;R,-r;',-/;';rn ,R)dr7'd/i'1 <• <• "^' 'o o 1 o o
J0J"0
A.'/(l-^'2)(l-»7:2)
rl rl
LI. 0 "0
1 + 4
S(R,r/' ,u' ;R>-r;o,-/io;r1,R)dt7'd/i' (5.5.68) 2
M'/OVKH' )
This is the integro-differential equation for scattering function which is to be solved under an initial condition appropriate to the specific problem. The integro-differential equations for the transmission and scattering functions T(r. , r) ,-/j;R,— rj ,—n ;r.,R), T (R,rj ,u;r.. ,rj u ;r.. ,R) , and S (r. ,q,-/i;r. ,ij. .—fi. ;r, ,R) can likewise be obtained starting from the relevant integro- differential equation for probability function.
For example, the integro-differential
equation for transmission function T is given by [cf. Leong and Sen<17>
p 69,70]
"«>
a L 9R
1-M 2
Ru ' o ^Ano
ff(R)
<*W
° .
a
R ^9
my cxp(L T
L
/ .
1" R T(r1,r?,-/j;R,-r?o>-^o;r1,R)
2
«<*'>«"']. 2 j
V/i-„
.1 ,1 T(r, , r?,-Ai;R, —rj' ,-u' ;rn ,R)drj'du'
+ 4
J0I"0 — c
c
I
^'
2
u
rl
O
O
2
S(R,r/',u';R,-r; ,-u ;r.,,R)drj'&\i'
oJ o
where u is u
1
u ' / (1- M ' )(1-^' )
1
1 + 4
O
'o
as r -» R.
Similarly for T , we have
(5.5.69)
"'/(l-u'W)
221 8_ _,_ l-M2 3R
R,
g(R)
3
1-/
aM
/
r
T (R,r?,/i;r1,r/1,^1;r1,R) T
1 ,1 S(R1r?,M;R,-r;' o -M';r o 1i ,R)dr,'dM' o o
1 + 4
4TT
g(R)
+
2
J
°J° 5
"o/(i-,: 2 )(i-„; 2 )
rM-Ci^iA^f
M-p rV
M.
)_di'\ +
4
U °°
f1 f1T (R."'.M';r1,^1,/x1;r1,R)dv'd/i' J
"'/(i-^xi-v2) (5.5.70)
and for S
S (r 1 ^,- M ;r 1 ,, 1 , Ml ;r 1 ,R)
M
a(R) 4w
K x P f rM ° (r '> dr " ♦ 4/- f
.1 pl T(r1,i?,-M;R,-t?^,-M;;r1,R)d»j^d^
L
"
L
Wi-,
2
■^i r r i
+4
1
1 T (R.r,' ,/i';r
JJ, - 'O'O
^
0 -'0
r
M'/:
o/ (-u
w
, ,2
-I Vl/^[ .RM
exp
2
o x1-"; >
1
g(r')dl'
»j j. ;r R)d»?'d/J' (5.5.71)
ii- f~ ,2. ... X / (1-M' )(!-»?' )
The integro—differential equations (5.5.68) -(5.5.71) are to be solved under initial conditions specific to the particular problem on hand.
The emergent intensities
:
Let g(r)g(r)dr denote the emission distribution function for photon per unit area per unit time.
Then at any level z, the
intensity in two directions is given by the following expressions.
222 / i / 1-n
I(z,Tj,/i) = I
p(r;z,r;,M;r1 ,R)g(r)a(r)dr
+ J
q(r;z,r7lM;r1,R)g(r)a(r)dr,
/ 2 r u/ 1-r; I ( z , r j , - / i ) = I
* (r;z,rj,-/j;r1,R)g(r)a(r)dr p
r
+ |
In p a r t i c u l a r ,
\J
(5.5.72)
q
(r;z,r?,-/j;r1,R)g(r)a(r)dr.
(5.5.73)
at z = R
1-M2 I(R,r?,M) = J
p(r;R,r?,/i;r1,R)g(r)a(r)dr,
(5.5.74)
and a t z = r ,
u / 1-f?2 I(r1,r,,-(i)
= J
p*(r;r1,^,-/i;r1,R)g(r)a(r)dr.
(5.5.75)
If 7rF and TTF, denote the incident fluxes at r = R and r = r, rper o 1 1 unit area normal to the directions (—rj ,—/i ) and (IJ.,/1..) respectively, then
, x, ^
F
° f«>l f R 1
g(r)a(r)dr = ^
+ ^
—
a(r)
-
/ expj-
fR/1° " ( r ^ d i ' l , "o^ —/ l_- A„ _ l d r
r
£M- r « - } - <....w
•"n <*i
L"l
Hence from ( 5 . 5 . 7 2 ) ,
/ l ^
( 5 . 5 . 7 3 ) and ( 5 . 5 . 7 6 ) , we have
223
2
ft
-r>I(z,r/,^) - ^- S(z,Tj,/i;Rl-r/o,-/Jo;r1,R) F
l
*.
+ 2p Tm (z,t;>/i;r1,»/1)/i1;r1,R),
(5.5.77)
and i
r
2
HY 1-rj
I(z,?7,-/i) = ^
T(z,r?,-/i;R,-r/o,-Mo;r1,R)
F l * + 2^ S (z,ij,->i;r1,»j1>/.1;r1,R).
Thus if the scattering and transmission functions S, T, S
(5.5.78)
and T
are known, the intensity at any layer of the cylindrical shell medium can be determined. We consider the following two special cases. (A) The cylindrical shell medium surrounds a perfectly absorbing cylindrical core with external illumination; (B) The shell medium surrounds an emitting cylindrical black core. An inhomogeneous, isotropically scattering, source free infinite shell medium bounded by surfaces of radii r.. and R(0 < r.. < R) is considered. absorber.
In the case (A) , the inner surface r = r.. is a perfect
A uniform incident flux wF
per unit area normal to the o
direction of incidence (—rj ,—p) r = R.
r
is falling on the outer surface
In case (B) , the cylindrical core emits uniformly TTF.. per unit
area normal to the direction of incidence (r;.. ,Mi) at r = r. . The initial conditions at the free surfaces for diffuse radiation are given by I(r1,r7,^) = 0 ,
0 < ji < 1,
-1 < 17 < l
I(R,fj,-/i) = 0 ,
0S(.<1,
-1
(5.5.79)
224 The emergent intensities are given by equations (5.5.74) and (5.5.75), where g(r) is given by F
w(r)
4
o *
<
fR r
e x p
f.r:^
r R ^,
for
case (A) (5.5.80)
S(r) = / x
Fi
r
1 \
»,l
H«pf- rr^^ ( r ' ) ^ ' \
forcase (B).
In other words, 2"- S(R,»7,/J;R,-TJ ,-/i ;r
R)
for case (A)
(5.5.81)
I(R,t7,/i)/i/ 1-rj = • L
1 * ^- T (R,»j,M;r1,r;1,/J1;r1,R)
for case (B) ,
and 2|- T(r1,»j,-/i;R,-r;o,-Mo;r1,R)
for case (A) (5.5.82)
K r 1 > '7>-M)M/ l-,2= 1 * 2|- S (r1,*7,-M;r1,tj1>/i1;r1,R)
It may be recalled that S and T (5.5.70) and T and S
for case (B)
are given by (5.5.68) and
by (5.5.69) and (5.5.71).
These equations are
solved under the initial conditions S(R,n,u;R,-n ,-u, ;r-,R) = 0 for -1 < /* < 0; -1 < n < 1 'o o 1
(5.5.83)
S(R,r/,/i;R,-r/ ,-u ;r..,R) = 0 for r.. = R o o 1 1 T (R,rj,^;r
rj M -r
R) = 0
T(r- ,n,-/*;R,-n ,-u jr., ,R) 0 1'' ^' 'a o 1
when r
when rn 1
= R
(5.5.84)
(5.5.85)
225 and S (r1>rj,-/i;r1>T?1,/i1;r1,R) - 0 for -1 < n < 0,
-1 < r, < 1 (5.5.86)
S (r1,»),-/j;r1>»j11p1;r1(R) = 0 for ^
Rybicki
(21)
= R.
suggests that for securing uniqueness of solution
it is advisable to break up the scattering and transmission functions into diffuse and reduced incident radiation parts, to solve for the diffuse part numerically and then construct the total scattering and transmission functions.
However, even if the work is done with the
total functions, the passage from the total to the diffuse scattering function is not difficult [cf. Bellman, Kagiwada, Kalaba and Ueno (7) p.8-10].
226 References
1.
Ambarzumian, V. A., Astron. Zu, 19, 1, 1942.
2.
Bailey, P. B., J. Math. Analy and Appl. 8, 144. 1964.
3.
Bailey, P. B., and Wing, G. M. , J. Math. Analy. and Appl. £, 170, 1964.
4.
Bellman, R. andKalaba, R., Proc. Nat. Acad. Sc. 42, 629, 1956.
5.
Bellman, R., Kalaba, R. and Wing, G. M., Proc. Nat. Acad, Sc. 43, 517, 1957; J. Maths. Mech. 7, 149, 1958; 7, 749, 1958; 8, 249, 1959; 1, 280, 1960.
6.
Bellman, R., Kagiwada, H., Kalaba, R. and Ueno, S., J. Math.
7.
Bellman, R., Kagiwada, H., Kalaba, R. and Ueno, S.,
Phys. 9, 909, 1968.
RM-5402-PR-Rand external publications, 1966. 8.
Bellman, R. , Kagiwada, H. , Kalaba, R. and Ueno, S., ICARUS, 1_,
9.
Bellman, R., Kagiwada, H., Kalaba, R., Jour. Comput. Phys. 1,
365, 1969; 11, 417. 1969.
no. 2, 245, 1966. 10.
Bellman, R., Kalaba, R. and Ueno, S., The Rand Corporation
11.
Busbridge, I. W. , Ap. J. 133_, 198, 1961.
12.
Busbridge, I. W., and Stibbs, D. W. N. Mon. Not. Roy. Astron Soc.
report. 1, no. 3, 1962.
London 114, 2, 1954. 13.
Chandrasekhar, S., Radiative Transfer, Clarendon Press, Oxford, 1950; also Dover Publications., New York 1960.
14.
Heaslet, M. A., and Warming, R. F., J. Quant. Spectrosc. Rad. Transfer 5, 669, 1965.
15.
Leong, T. K. and Sen, K. K., Ann. d'Astrophys. 31, 467, 1968.
16.
Leong, T. K. and Sen, K. K., Publ. Astron. Soc. Japan 21, 167, 1969; Jour. Sing. Nat. Acad. Sc. 3, 1973 (supplement).
17.
Leong, T. K. and Sen, K. K., Publ. Astron. Soc. Japan 22, 57, 1970.
18.
Matsumoto, M. Publ. Astron. Soc. Japan, 18, 456, 1966.
19.
Minin, I. N. Soviet. Astron. 8, 528, 1964.
227 20.
Preisandorfer, R. , Proc. Nat. Acad. Sc. 44, 320, 1958. 44, 323, 1958.
21.
Rybicki, G. B., Jour. Coraput. Physics 6, 131, 1970.
22.
Sobolev, V. V., A treatise on radiative transfer, Van Nostrand, New York 1962.
23.
Stibbs, D. W. N. Mon. Not. Roy. Astron. Soc. London, l_n, 493,
24.
Tricomi, R. G., Integral Equations, Interscience Publishers Inc.
1953.
New York, 1957. 25.
Tsujita, J. Publ. Astron. Soc. Japan 19, 468, 1967, 21, 15, 1969.
26.
Ueno, S., Ap. J., 126^, 413, 1957; J. Math, and Mech. ]_, 629, 1958.
27.
Ueno, S., Kagiwada, H., Kalaba, R., The Rand corporation research, memo. RM-6061-PR; J. Math. Phys. 14, 855, 1973.
28 29.
Uesugi, A., Ann. d'Astrophys. 2j>, 263, 1963. Uesugi, A. and Tsujita, J., Publ. Astron. Soc. Japan, 21, 370, 1969.
30.
Wang A. P., J. Math. Phys. 14, 855, 1973.
31.
Wing G. M., J. Math, and Mech. 7, 757. 1958.
228 Chapter VI Solution of integral equation of transfer in curved geometry.
We have seen in Chapter I that the equations of transfer in spherical and cylindrical geometry and their boundary conditions are transformable into integral equations of Fredholm type.
In the
present chapter, we discuss some of the basic methods for obtaining approximate or exact solutions of them by analytical techniques.
We
have considered in particular Neumann series method, the method of integral transforms, F —method,, Pincherle—Goursat kernel method and combined operations method as applied to transfer and transport problems in spherical and cylindrical media.
6.1
(a)
Neumann series method for homogeneous medium in curved geometry
Spherical
Geometry :
For the simple model of a homogeneous spherical medium scattering isotropically with spherically symmetric internal source B(r), the integral equation for the source function ?(r) is given by [cf. equation (2.1.1)] w K r?(r) = rB(r) + -^- \ k(r,r'.)r'?(r' )dr' ,
(6.1.1)
where the kernel
k(r,r') = (E1(|r-r'|) - E^r+r'))
(6.1.2)
with the exponential integral E.. (x) given by
Ex(x) - J e Xt p . Case (i)
(6.1.3)
Let us first consider the simple case where the given
229 i.e.
B(r) - B . o
From Chapter II, Section 1, equation (2.1.4) we note that the Neumann series solution of equation (6.1.1), for this case, can be written as riO -,m
«>
r»(r) = rB Q + B Q J U £ Am(r) m=l
where
A (r) -
(6.1.4)
*■ '
r k (r,r')r'dr Jn m
and k is the iterated kernel defined in Chapter II §(la). Since the m kernel k(r,r') is non—negative in the region 0 < r, r' < °°, the iterated kernel k is also non-negative and hence each term in the
m
°
infinite series (6.1.4) is non—negative.
However each term in
equation (6.1.4) is smaller than the corresponding term when R is replaced by «° in the upper limit of the integral.
Thus the
convergence of the series can be established if the convergence of the majorant (when R is replaced by ») can be shown. We will first show that A {r} = 2r. We have
AV)
J0 k.. (r, r ' ) r ' dr' - I '■ J J
-
0
t dt
f
f
- ( r + r ' ) t dt d r '
1
fr r'dr'fV< r - r '>^ C J0 J1 CO
=
LJ
-|r-r'
r
+
f r ' d r - f e ^ r ' " r ) t ^ - f r 'dr' f e ~ ^ ' >fc * J J J r l 0 l
J
00
CO
00
- r t dt r , r ' , f r t dt f , - r ' t , , P - r t dt e -— r ' e dr'+ e — r ' e dr'e — J c J J J1 t JQ l r l
on
r'e
dr'
230 CO
r
V
-rt dt
"
00
J
J
l
e
, r't r' e
-r't r e
r
t2
_ r t
dt
j * .
, -r't -r'e
e
-r't
J
t
fc
rt oo e-rt —rt dt ■ rt — re _ e l * ■' " St 2
e
'
J
l
e
-r't t2
J
r
00
'0
■a rt r e dt
1 1 t'-l
-,00
, -r't r'e
<» rtdt
u
re rt , ertn L
c
t"J
J
f
JL 4:2
00
i
2r
Jr s ^ 1 t
= 2r. Using the definition of the operator A we have CO
CD
A2{r} - [ k(r,t) I k(t,r')r'dr' J J 0 0 CO
= [ k(r,t)A1(t}dt J 0 CD
- [ k(r,t)2tdt = 22r. Continuing this process we see that the series
I
F
1=1 ^ which converges for u> < 1.
J
A»(r) = r E (. )' m-1 Thus the Neumann series solution (6.1.4)
for £(r) is valid in the whole physical space for w
< 1.
231 Case
(ii)
Let us now consider the case of a spherical enveloping
atmosphere with a point source at the centre.
In this case the
source function S(r) satisfies the integral equation [cf. Germogenova ( 1 1 ) equation (1.8)] B e -r o + A (r?(r); r
r?(r)
(6.1.5)
where
A(r?(r)} -
u> „R ^ ' a
JE1(|r-/,|)-E1f/r2-a2
+/P2-a2}\
+
7
k*(r,p)
[
(p)dp (6.1.6)
with d/i
k (r,p) = r J 2
i >.
r
r>
2
P -a
2M 2. (l-/i )
x exp ^ / a 2 - r 2 ( l - / i 2 )
}■
- /p2-a2(l-/i2)
- /i(r-a) k
(G. I . / )
The c o n s t a n t 7 i s a non—negative f a c t o r
denoting the f r a c t i o n
of
e n e r g y r e f l e c t e d by t h e c o r e a t r -= a.
We w i l l now show t h a t when
R = «, 0 < All)
< w
l - j l j
(r+a)
(6.1.8)
and hence the Neumann series solution
B e o
00
m
•B e- r o
-}
(6.1.9)
of (6.1.5) with R = <» will be convergent if a
< 1 since we know that
r?(r)
+
Z A
m-1
E (x) < 1.
(1)
[Abramowitz and Stegun v ' ] .
°
To establish (6.1.8), 1 1 let us consider the different contributions to A (1) from (6.1.6).
From (6.1.7) we have
232 00
j
1
k (r,p)dp = r J
exp /l-(a/r)
-/ir + / a - r (l-/i )
2
2 exp(^a - A //J - a '-■ ( 1 - / J' )
"ff
d?W.
(6.1.10)
2
2
i 2 ,) // p - a /(1-/J
Using the s u b s t i t u t i o n / 2 T7. 2, z = / p -a (1-^ ) for the inner integral we obtain
0 < Jr"
aM
Also
exp(Ma-z)dz
_ 1_ M
/ 2+ 2Z 2, / z +a (l-/i )
I/ l - ( a / r )
a
a
2
f1
L a3
J
1< I a
o < M 5 1.
exp Uur + /a - r (1-/U )4i 2
with the substitution ■f) i/ = pr - /a - r (1-
reduces to / 2
2 —1/
e
2r
-
2 2 r —a 2
- 1
di/.
r—a Since the exponential
integral
E n (x) -
e J
l
- x t at n—l J — = x x tn
n y
satisfies the recurrence relation [Abramowitz and Stegun
p.229]
233
W
x )
= 5e
- xE (x) n
we have 00
2E,(r-a) - E,(/r2-a2)
k*(r,p)dp = E (r-a) - \
+ 0
a
I
i
T" '
< E2(r-a) We also have CO
E1(|r-p|)dp = 2 - E2(r-a) and E 1 f/r 2 -a 2 + /p2-a2
using the relation
dE (x) n dx
)dp > J" E^r+^dp =• E2(r+a)
-En_l(x)
Hence
f}{l)
< ^ 2 - E2(r-a) - E2(r+a) + 7E2(r-a)
< w
1 - i E2(r+a)
thus establishing the inequality in (6.1.8).
Let us now demonstrate
the convergence of th the Neumann series for the case u> = 1. o (6.1.8) we note that 0 < Am{l) = A™ 1{A1{1))
< A
"H1 - s E 2 (r+a) }
From
234 < Am
2
| A ( 1 ) - I A (E2(r+a))
< A m 2 |l - | E2(r+a) - | A (E2(r+a)}
}
Continuing this process we obtain the inequality . m—1 0 < Am{l) < 1 - | ^ A n (E2(r+a)). n=0
(6.1.11)
Hence the Neumann series I A n (E (r+a)J n=0 converges for all r > a and its sum is not greater than 2. But —
since
and
r
< 2 I — ] E . (r+a)B e a a 2 o
En(r+a) > 2v '
(6.1.12)
-(r+a) =r+a+2
2(a+l) 1 a(r+a+2) " r
this last inequality follows from the inequality r(a+2) > a(a+2). Hence the Neumann series solution (6.1.9) of the integral equation (6.1.5) is convergent and its sum is bounded by
235 I
m=0
2
a+1
4(a+l) a
B Q e a Am(E2(r+a))
o
6.2; Integral transform methods in curved geometry In Chapter II, §4 we noted that certain simple problems of transfer in curved geometry can be solved analytically using integral transform methods.
In this section we shall obtain in detail the
results quoted in Chapter II.
(a)
Cylindrical
Geometry (infinite
medium) :
The integral equation for the source function ?(r) in a homogeneous, infinite, non—conservative cylindrical medium which has a —r line source distribution B(r) = e /r is given by [cf. equation (2.4.1)] CO
/r S(r) = /r B(r) + u> k(r,u)du, o J,
0 < u> < 1, o
(6.2.1)
where
k(r,u) = /? K (ur) J /t I (ut)/t S(t)dt ° 0 1
CO
+ I (ur) I /t K (ut)/t ?(t)dt . o J o K (t) I (t) are the Bessel functions of purely imaginary arguments o o and a) is the albedo for single scattering.
°
Following Hunt
(14), we let
Q(r) = /r £(r) and define the integral transforms
236 Q(s) =
/r J (sr)Q(r)dr,
-1
B(s) -
r J (sr)B(r)dr
and CO
k(s,u) - f /r J (sr)k(r,u)dr,
where J
is Bessel's function of zero order, o Taking the transform of equation (6.2.1), we obtain
« r
a
Q(s) = B(s) + w ! k(s,u)du 0 Ji We have shown in Chapter II (equation (2.4.10)) that B(s)
Q(s)
w 1 -
Since B(r) = e
/r, B(s)
o
tan
Sn
(l-2)1/2 '
We also noted in Chapter II that the general solution of the homogeneous equation CO
Q(r) = to I k(r.u)du, o J„ which is finite at the origin is given by
Q(r) = A/? I (ar) o where a satisfies the relation a = u
o
tanh
a.
(6.2.2)
237 To invert the equation (6.2.2) we write 00
Q(r) - / t J
sQ(s) J (sr)ds.
Using Bessel's identity [cf. Watsonv 2J where H
,H
o
o
o
p.201]
+
H<2> o
are the Hankel functions of order zero, we have
00
CO
2 f sQ(s)J (rs)ds •= {
J0
- H ^ o
(6.2.3)
o
J0
CO
sQ(s)H(1)(rs)ds + f
o
JQ
= Lx(r) + L 2 (r),
sQ(s)H(2)(rs)ds
o
say.
(6.2.4)
We will first consider the integral j sQ(s) H(1)(rs)ds,
\ where P.. is the contour in the s—plane consisting of the real axis 0 < K(s) < R, the first quadrant of the circle |s| = R and the imaginary axis between 0 and iR, indented to the right of the branch at s = i and the simple pole at s = ia. H
o
- (rs)
e
On the large circular arc
and hence the contribution to L. from this 1
portion is arbitrarily small.
As there are no singularities within
€ , we have L x - - | P . V. |
pQ(ip) H(ol)
aPr)dp\
00
- | pQ(ip) H(ol) (ipr)dp
+ i7rRa,
(6.2.5)
238 where R is the residue of the integrand at s = ia and P. V. denotes a the principal value. Defining 2
+1
\P l , N(p)s = -P - 1log l^-jl M/
o
we note that uoN(p)Q(ip) = 2p(.l-p2)~1/2,
0 < p < 1,
and
I 2
u> [N(p)+iw]Q(ip) = - 2ip(p -l)
_1/2
,
(6.2.6)
p > 1-1 .
Substituting (6.2.6) and using the relation H
[cf. Watson
Wr) 1
(37)
o1)(i'r)
=-^Ko("r>'
p-78] in equation (6.2.5), we have
= ^ { P .
V.
V l
p1 K ( p r ) d p
1 J
^
1
0 ( l - p 2 ) 1 / 2 N(p)J 2
4
+ "o~ u i
2 2 1/2 K Q (pr)dp 2a ( 1 - a V ' K (or) 1/2 " 1 ( ps— 1 )=r-^ [N(p)+iir] + (a2-l+u ) J. o -
p
To evaluate L„ we consider the integral sQ(s) H (2) (rs)ds,
where G
is the mirror image in the real axis of the contour C .
This gives
L2(r) = -
2
«0* I
rl
2 P K^(pr)dp
J0 (1-p")-'" (1-2)1/2
i
N(p)-J1
/, r"
2 P K (pr)dp
J (p2-l)1/2[N(p)-ijr] w" V* "1 2a 2 (l-a 2 ) 1/2 K (or)
+—
°
(a -1+w ) o
239 Combining these two, we have from (6.2.3) and (6.2.4) 2a 2 (l-a 2 ) 1/2 K ( « )
1 1/2 /9
*(r) - r- Q(r) - —
,
°
(a -1+u ) o
p2N(p)K (pr)dp
,
f-
o
. 2 n.l/2r.,2. . , 2 , '1 (p -1) [N (p)+?r ]
+a)A. J w J-
(6.2.7) (b)
Spherical
Geometry
(infinite
medium)
Let us now consider a homogeneous infinite spherical medium which has a point source at the centre. source function 9(v)
The integral equation for the
is given by [cf. Chapter II, (2.4.14)]
Q(r) = <6(r) + ^
J
E ^ | t-r| )Q(t)dt
(6.2.8)
—CO
where Q(r) = r?(r) and ^(r) = B(r), r > 0. (33) Following Smith , we define the two sided Laplace transform CO
Q(s)
■L e
Q(r)dr,
CO
*<s) = J
■L
e s r *(r)dr.
Transforming (6.2.8) and then inverting we have [cf. equation after (2.4.16)] Q(r)
T/ .. sr 1 .1°° f (s)e ds 27Ti J M(S)
(6.2.9)
where 1
M(s) = 1
tanh o. log 2s
= = 1I
(s)
1+s 1-s
and *(■)
2s
->2 M(s)-1
240 To evaluate the integral (6.2.9) we consider the following contour integral (r > 0) :
f
« - si L W1 -■
(6.2.10)
where C is the contour shown in the Figure 6.2.1 with p large and 8 small compared to unity.
Fig. 6.2.1 Let Q ( < 1) be a positive root of M(s) = 0.
Then the contour E? will
contain a simple pole at M(s) = —a and hence
f(r) =
If \ ~ar ^(-a)e y(-a)e
M'(-a)
where the derivative o
M' (s)
log 1
2s' =
1 s
1+s 1+s
1 - M(s) -
o s(l-s') o 1-s2^
Thus 2 2 2a (1-a )
f(r) u
o
—ar e
(l—o) —a )
o
But from (6.2.10), using the contour P, we have
(6.2.11)
241 2 „ i f(r)
_ f1" ] -iP
**><*>
ds
+
f K/2
lp
M(S)
?(pe")exp(r,e"+lg)dg VUpe19) — ifl i.9 -*• <£(-l+5e ) e x p ( r « e +19) r^ dfl J TT M(-1+Se )
7 , _ 0 -sr yv + '^ ds + i « e J l + « M(se ) rP
_ rP J
*(-s)e
Sr
ds
+
M(se 1 , r )
l+S
r~lr/2
^(pelg)exp(rpeil9+ig)dg
•"-*■
M(pel5)
= 2»rl [X 1 + I 2 + I 3 + I 4 + I 5 + I g ] ,
say.
(6.2.12)
In the cut plane M(s) is 0(1) as s -> =° while <£(s) is 0(-i—r) . Hence s
I I i
the contribution to the integrals I„ and I, is at most 0((-j—r) by Jordon's lemma.
Similarly on the small circle around s = -1, the
integrand is of order 5 log 5.
In the limit when p -» <*> and 5 -> 0, I
tends to Q(r) and hence we have
f(r)
- Q(r) + o>oii J. [{ 2 - M ( s e w > 1
- J2-M(se 1,r ) -
w
M(se
x
M(se
.
)J
Sr 1 se _.-s ds.
i i r , JI
M(se )
For s > 1,
-
r
1
) — 1 + ~—
"
= ~ {N(s) + in) zs Similarly M^e" 1 *) = ^ (N(s) - iir]. /s
(6.2.13)
242 Thus oo
!'(,')
r
0(r) ; i -jl--^ ^ -7T } e co (N (s) + n ) > o
sr
ds.
(6.2.14)
From (6.2.11) and (6.2.14), we have o 2., 2. _. . 2a (1-a ) -ar Q(r) = - 5 — 5 — e co (a -l+co„) o U
(cj Spherical
geometry
-ar . » 2 -sr. e 4 f se ds — + -s—7) 5- ■ J co l {N (s)-hr } o
(finite
/,- o icv (6.2.15)
medium) :
In contrast to the infinite medium the boundary effects have to be considered in a finite medium.
To determine these effects we consider
a spherically symmetric medium of finite radius R with a point source at the centre, the medium scattering isotropically.
The corresponding
integral equation for the source function ?(r) can be written as in the previous case as co -K Q(r) =
E1(|t-r|)Q(t)dt
(6.2.16)
where co „R *
exp
<
t}
dt
(6.2.17)
and Q(r) = r9t(r) . Cassell
suggested the following modification of Smith's
, integral
transform method to solve the above spherical problem in finite medium. Let
Q*(r) = Q(r) + e _r /r,
then (6.2.16) and (6.2.17) can be written as -r c o „R —r CO K. r * * Q (r) = ^ - + Y J {^Clt-rD - E 1 (t+r)| Q (t)dt. '0
(6.2.18)
243 Defining the integral transform
r
c Q(s) - I Q (r)sinh(sr)dr
(6.2.19)
0
and taking the transform of (6.2.18), we have [cf. equation (2.4) Cassell(8)] M(s
«<•> -1 . * £ ) -1 f [ a -R(t-s)) ^ p _ exp(-R(t+s)) t+s x
1 +
a>o Q(t)
^ J
(6.2.20)
dt.
If we now let w o Q(s) q(s) = | exp(-sR) -j 1 + — g
'}
then (6.2.20) can be rewritten as
Q(s)
M(s)
I i„E(^, ♦ r {=&« - =¥?u>*
(6.2.21)
Let us now define Q (r) to be zero for r > R, then from (6.2.19) CO
Q(s) = f Q*(r)sinh(rs)dr J 0 and the inversion formula becomes 1
Q*(r) = K
IX
lira [
x-*° - I X
_
_
e S r Q( s)ds.
Let us first consider the non—conservative case 0 < o
(6.2.22)
< 1. The first o term of (6.2.21) tends to a constant as s H> <• and so its inversion term of (6.2.21) tends to a constant as s H> <• and so its inversion
must be taken as a first Cesaro limit.
244 Then
X Urn f'
e
x-*» —ix
Sr
log[(l+s)/(l-s)] M(s)
] + 1
s
2n
ds
X
= — residue at s = a of the expression
e S r log[(l+s)/(l-s) M(s)
(1 }
V
2 2 -ar 2a (1-a )e + 4
2
co (a-l+u
o
2jri J
<» 2 f _ p exp(-pr)dp
"u2 J J l
)
o
e S r log[(l-s)/(l+s)] ds M(s)
2
T~
[N (P)+TT ]
o
■
If we denote the solution (6.2.15) for the infinite spherical medium (r) by Q , then the above quantity can be written as Q_
-r e r
The contribution from the next term in (6.2.21) is 1°°
1_
exp(-s(R+r)} M(s)
?ri
i2a(1-a
J
)exp(-a(R+r))P
u> -1+a o
2
t+s
q(t)
^1
4_ r p exp(-p(R+r)) J
"o l
2
P
2
J
[N (p)-Br ]
l
q(t) dt. t+p
For r < R, the inversion contour for the remaining term must be distorted into the left-hand half of the cut on the s—plane.
The
contribution from this term is then -2a(1-a )exp{-<«(R-r)) id — 1 + a
o
Thus
«
r agi dt _ *_ r
J 1 t+a
p exp(-p(R-r)) 2 2 a)o J"1 [N (p)+*r ]
P
f J
]
q(t) dt. t+p
245 Q(r) - Q*(r) - e r /r = Q (r) -
q(t)dt t+a
4a(1
~ a } exp(-aR)sinh(ar) I J u -1+a 1 o 00
_ 8_ |*°° g exp(-pR)sinh(pr)d/9 f™ q(t)
I,
[N2(p)+^2]
o Jl
2.23) (6.2
dt.
Thus we will know Q(r) provided q(t) is computable.
We will now
derive a series type representation for q(t) whose convergence increases as R increases.
£
We have from (6.2.21)
M(s)q(s) - J- exp(-sR) - f
Let
1KB)
and if 1 < a < oo, let V (°0 =
fe
- 2?ri ~ J
- ^i^)j
q ( t ) d t
.
.2 ( 66.2.24;
q(t) »dt t-s
li-m V'(s) s->cr±iO
Then q(a) = ^ (a) - ^ (a). If we let s -» cr ± iO in (6.2.24), we obtain {N(o)
+ i7r)V> (cr) - IN(o) - iw}^ O ) CO
I
—
exp(-aR) + exp(-2crR) f 3 I H
dt
where N(a) _ 22 - log O
C7+1
(7-1
We look for a series representation of q(t) of the form
q(t) =
E q, (t)exp(-2kR) k=0
(62 (6.2.25)
246 where q u (t) satisfies the relation i r" q k ( t ) 0 (s) = - i ^ — dt. *V ' litx J 1 t-s The Vv.(s) satisfy the following iterative relations : (N(
with
(6.2.26)
G (a) = — exp(-crR) o w o
and for k > 1,
G (a) = exp( 2(li-a)R} tC
q r" — k -—l ( t ) dt
J -.
exp(2(l-a)R) ^
L.+CJ
C<_1 ~ V l
( t )
> tfe
Let Z(CT) be the modulus of N(CT) + irc, so that N(CT) + iw = z(ff)exp(i/3(a) ) where f}{o) decreases from w to 0 as a increases from 1 to •.
Let
r(s) -I f i^l dt, T J n t-S
so that its values above and below the real axis, 1 < a < <*>, are 00
r^a) = - p.v. f §iSi dt + i/3(a) T
J 1 t—S
rQ(a) ± i^(a), say.
If we let X(s) = S s -> =°. Also
ex
P ^ r ( s ) ) , then X(s) is 0(1) as s -> 1 and as
247 - a exp {-r (CT) T
X~(a)
i0(o))/(a-l)
and hence (6.2.26) Is equivalent to a )
*k(a)
X (a)
X (a)
V
Hence A
/ ->
Gk(«')(«r-l)exp
Gk(t)(t-l)exp(ro(t))
X(s)
z(t)t(t-s)
dt +
— s
where J, is a constant, k Thus qk(c) = $k<<*) -
\ W a exp(-r (a))
- _ i rN(a) O v) Z(a) z(a)
^_2
k
a-1
- G (t)(t-l)exp(r x (t))
k p.y.f v f JS
•VJ1
" s°°'
z(t)t(t-a)
dl
exp(-ro(a)) ^-1
"Jk "
exp(-ro(a)) If we let v-(a) - — 7 — 7 7 — r r — , then z(cr)(<7-l)
N(a)Gk(a) qk(f) = — 2 — ~ z (a)
a
x("
Gk(t)dt
) P.V.J
1 z"(t)x(t)t(t-a)
- xO>J k .
Let us now define the integral operators A.. , A_ as follows:
A1(f(<7)} =
N<„)f<_o) _
ax{a)
p v
j"
z (a)
and
1 z (t)x(t)t(t-a)
a
A2(f(a)) = exp{2(l-a)R} J
f(t)dt
f(t)dt t+<7
(6.2.27)
248 Then we have qk(<7) = A1(Gk(<7)) - Jkx(a)
A
! A 2 l q k-l ( C T ) ) " V ( C T ) -
(6.2.28)
Performing this reduction k times
qk(a) = (A j A ^
(q
(a))
-
J.CAJAJ)'
I
j-1
(A A ) * A1(Go(a)) -
I
JAAA)
ix(°))
k
- j f J(x(°))
Thus 1 q(a> = i- I exp(-2kR) (A^A ) A (a exp(-aR) o k=0
+ A(R)
I exp(-2kR)(A1A„)K{X(a): L k=0 *
where A(R) = -
I exp(-2kR)J K k=0
The constant A(R) in (6.2.29) can be obtained from (6.2.21).
(6.2.29)
Since
Q(s) is regular at s = a, we have U>
1 +
a
a
J.
exp(-aR) t+a
exp(qR) t-a
q(t)dt = 0.
(6.2.30)
Substituting for q(t) in the above equation, we can determine the constant A(R). Noting that CO
" x(t)dt t-a
* 0
249 as the integrand is everywhere positive, we see from (6.2.29) and (6.2.30) that A(R) is of order exp(-aR) for large R.
Hence the only
part of q(t) which can be of larger order than exp (-R) on 1 < t < <° is A(R)x(t).
Thus from (6.2.23) and (6.2.29)
2 oo Q(r) - Q (r) - 4B(R) a ( 1 ~ q \ exp(-aR)sinh(aR) I ^ - dt u) —1+a J T t+a o i i _. 4_ B ( R s f P exp(-p(R-r))dp r y(t) w Ji 2. . J. t+p r o 1 z (a) 1 + 0 (exp(-R))
(6.2.31)
where B(R) = —
o
exp(-aR)
Al^
exp(-2aR) t+a
X(t)dt
-1
B(R) is in fact the part of A(R) which is of greater order than exp(—R) Let us now consider the conservative case where u> = 1 . In this case o M(s) has a double zero at the origin s = 0 and hence the condition for M(s) has a double zero at the origin s = 0 and hence the condition for Q(s) to be regular there is CO
1 - 2 f J
q(t)
(1 + Rt) ^
i
= 0,
(6.2.32)
tr
and so A(R)
{2 h ( t ) CO
X
(1+Rt) ~
|
+ 0 (exp(-R)).
Taking the contour for the inversion to the right of the origin, the inversion of the first two terms in (6.2.21) will differ from the nonconservative case only in that the contribution from the pole at a will be absent while in the remaining term, the contribution from the pole at —a will be replaced by that from the double pole at s = 0.
250 The r e s i d u e a t s «= 0 of 2 expts(R-r)} f M(s) J.
is
"
6
K q(t) (
J
q(t)dt t-s
> ( P . . •; j - ^ |
and this, from (6.2.32), is equal to (■ q(t)dt _ 3 J
C
l
Thus f o r u> = 1, we have o * Q (r)
.°°
3 + 4 f
•'l 00
exp(-pr)dp _
6r
z2(p)
p exp(-pR)sinh(pr)dp
-.J £ Hence
Q(r) = Q^r) - 6r J
CO
2
E
2
z (/>)
T q(t) dt ■'l
C
f
q(t)dt
J,
t+p
q(t)dt ^ 1
p exp(-pR)sinh(pr)dp z2(p)
(*
~J1
q(t)dt t+
"
X(t)dt
= (Q (r) - 6B(R)r f ^ i ^ =° J, t
_4B(R) |
p£ eexp(-p(R-r))dp x p ( - p ( R - r ) ) d p Jf 2
z (p) + 0 (exp(-R)).
*(t)dt t+p
(6.2.33)
251 6.3
The F„ Method N / o c 01 \ The F to The F method method was was used used by by Siewert Siewert et et al al to solve solve some some basic basic problems problems of of neutron-transport neutron-transport and and radiative radiative heat heat transfer transfer in in spherical and cylindrical media.
In the initial stage, the method was
employed primarily to calculate the surface characteristics through the (29) albedo or the transmission factor. Siewert and Maiorino utilised the method to obtain the mean intensity as a function of optical variables for a finite sphere with a point central source. Siewert (31) and Thomas used an integral transformation technique and the F —method for solving the albedo and internal source problems in neutron transport for an isotropically scattering, bare cylinder of infinite length. distribution.
They also computed neutron flux and current
They further employed the scheme for solving radiative
transfer problems in isotropic scattering spherical and cylindrical media.
We outline below some of the salient steps of the application
of the F
(a)
method to curved geometry.
Cylindrical
geometry
(infinite
medium) :
Consider a homogeneous, isotropically scattering, non—conservative cylindrical medium of infinite length and radius R.
We assume that no
external radiation is incident at the free surface.
Let Q(r) denote
the internal source distribution and let the boundary condition be taken
I(R,/ i = 0 ,
0 < fi < 1.
For the above model, we define I(r) +1 I(r) = 2J(r) = I I(r,/j)dM
"I.
J(r) being the mean intensity I(r) satisfies the integral equation [cf. Siewert and Thomas equation (8) and equation (6.2.1) of this chapter with u = -] I(r) = cooK{T(r)) + (l-c4>o)K(Q(r)}
(6.3.1)
252 where K is the integral operator defind by
X{f(r)} = [
K (r//i) [ I (t//i)f(t)t dt
+ I (r//i)
and I , K
K (t//i)f(t)t dt
dfj,
~2
(6.3.2)
are the Bessel functions of purely imaginary arguments,
u
is the albedo for single scattering. Following Siewert and Thomas
, we let
W(r) = WQ I(r),+ (l-*>o)Q(r),
(6.3.3)
and r *(r,/x) = KQ(r//x) j
IQ(t//i)W(t)t dt
r I0(X/M) J
Ko(t/Ai)W(t)t dt.
(6.3.4)
Then (6.3.1) becomes
I(r) = f «(r,M) % 0
(6.3.5)
ft
and hence the mean intensity can be calculated if $ (r,/j) can be obtained.
Following Mitsis
[cf. Siewert and Thomas
, equation
(15)] equation (6.3.4) is differentiated to obtain, for ft e (0,1) and r 6 (0,R), r
d2 . 2 3r
Id r 3r
1_ 2 • (r,M) - -»a n
I *(r,A») %
-
By differentiating (6.3.4) and using the relation K^(z) = - K (»),
(6.3.6)
253 we also have at r - R,
K1(V*i)*(R.M) + MKo(R/M) J J *(r,M)
= 0,
r=R
M 6
[0,1]-
(6.3.7)
This Is the boundary condition. Viewing the present cylindrical problem as a pseudo—plane problem Siewert and Thomas
attempted a solution of (6.3.6) in the form
*
+ *(-i/o,/i)]Io(r/i/o)
+ [ *.(»)[ + (*,fi)
+ *(-»>,AO]I ,
(6.3.8)
where tf(±i/,/i)
and
= -^ ^
u P.V.
(-=-)
+ (1-w
\J+\I
o
v tanh~ i/) 5 (i/T/i ) .
«*,,,.) - £ -0(^-) • u
(£,fi) are the generalized eigenfunctions appropriate to the plane geometry [cf. Case and Zweifel
Chapter 4].
v
is the positive zero
of w
A(z) = 1 + ^
z
(■
d£_
and $ (r,/i) denotes a particular solution of (6.3.6) corresponding to the non—homogeneous source term.
For the present analysis we assume
$ (r,u) can be found exactly for simple source terms Q(r). For P instance when Q(r) is a constant we have $ (r,/j) = n 2 Q. Using the instance when Q(r) is a constant we have $ (r,/j) = n Q. Using the full—range orthogonality condition of the eigenfunctions [cf. Case and Zweifel (7 \p71]
ce-£')
J
- l
/* *(e./0rf(e\/0d/* - o, i, v e i/ u [o,i]
(6.3.9)
254 we have from (6.3.8) d/x [*«./*> -
i
0,
(6.3.10)
where 7 (x)
= Io(x)/I1(x).
Setting r = R in (6.3.10), using (6.3.7) and putting 0(x) = K.(x)/K (x), we have
[*«,*«) --
1
*<-?,*o][p + ?7(ve)«VM)]*(R.M) ^ = w o .
(6.3.11)
where £R(£)
a
-J.
[ * « M)
H-<
f- •p (r,/i) ft)] •P (R,M)-?7(R/0 dr
^.
(6.3.12)
r-R-
Equation (6.3.11) is taken as the constraint relation to determine (261 In the F method proposed by Siewert and his collaborators,
$(R,JJ).
they take an approximate representation of $(R,/j) in the form
$(R,/I)
- n
Y. aj*a:>
(6.3.13)
a=0
the degree of approximation being determined by N. Substituting (6.3.13) in (6.3.11) they obtain for ? e v
U [0,1] the relation
N I aa[EQ(£) + 7(R/OD Q (?)] = R ( 0 , a=0 where
E(0
- i [ MQ+1 E^.M) --
0(-£,/i)]dM,
and Da(«
l r1
i4ii.it)
-
^(-e,M)]^(R//i)d/i.
(6.3.14)
255 Equation (6.3.14) is used to determine the expansion coefficients a . By taking a set of collocation points (£.) in 1/ U [0,1], equation (6.3.14) reduces to a system of linear algebraic equations for the unknown coefficients a . Knowing these expansion coefficients we can now substitute (6.3.13) into (6.3.8) and use the full-range orthogonality relation to obtain the unknown function A ( 0 . [cf. Siewert and Thomas^
'(1984) p.109 equation (33)]
-If A ( 0 = [N(OI (R/O] -U
N
L
a EJO -
I
a=0
f1 J
I ^ ^ H H , / . ) ]V
0
We have
R,^}, (6.3.15)
where the normalization factors [cf. Case and Zweifel
N(„ o )
u> 0 3 ™ 2 ~ V0
rwo _ 1
L02 -i
u
and N(v) = v
-1 (1 — 00 v tanh o
] are
2 ■
0
2 V v) +
2 2 2 * 4;
Knowing A(£), we can now use (6.3.8) and (6.3.5) to obtain I(r) in the form.
[cf. Siewert and Thomas
(1984) p.109, equation (36) with
F = 0]
J fr ; = A(:/_ yiji 0
0
fvj
0
: I A(.)I o (r/i/)di/ + I
$ p (r,/i)
,,
2
(6.3.16)
Hence we can find the mean intensity J(r).
(b)
Spherical
geometry
(finite
medium) :
We consider a homogeneous, isotropically scattering, spherically symmetric, non—consevative medium with internal source distribution Q(r). Thomas
The equation of heat transfer can be written as [cf. Siewert and (31) , equation (2), p.59]
256 w r i (r./O-^J I(r,M')dM' + f(l-wo>Q 1
" a? + r (1 -" >a£ + *
(6.3.17)
where the symbols have their usual meanings. We assume that the temperature at the free surface r = R is zero and so also the coefficient of diffuse reflection.
Let the boundary
condition be taken as I(R,-M) = 0 ,
ft e [0,1].
Viewing this problem as a pseudo—plane problem and using the F approximation N I(R,/i) =
I
<±afia,
ft > 0,
a=0 we obtain the constraint relations determining the expansion coefficients in the manner described in the previous section [cf. (28);
(6.3.14), Siewert and Grandjean^
\
a=l
V W
p.96].
-exp(-2R/„ )Ca(„ )]
[ B 0 ( O - exp(-2R/i/ )C o (i/ ) ]
(6.3.18a)
and
E V
W
~
*=0
ex
P(-2RA0)CQ(i/o)] = 0 ,
aQ = 1
where v , v„ are defined as before with
o
0
B
o « > - S~ - 1 - « log (1 + |) o *
V «
- « B a-1<^ " S I
C Q ( 0 = 1 - f log (1 + |)
C
a ( ^ " " « Ca-1«)
+
Si
(6.3.18b)
257 As in the previous section, we can show that the mean intensity J(r) ( 311 [cf. Siewert & Thomasv '(1985) with F - 0 and 0 = 0] is given by 1 I(r) - 2J(r) = J" I(r,M)«fc -1 I(r) can be expressed in the form
Kr)
exp r {W<"o> ['
. R-r. . R+r. (- - — ) - exp(- - — ) o o
.
f W(y) exp(
R-r. . R+r. — ) - exp( — ) di/
where m i
r
N
W(?) - 2N(?) R B Q ( 0 + B 1 ( 0 + I L
aQCa(^)
(6.3.17)
Q=0
The constants v
and the normalization functions N(£) are as described
in the previous section. The examples given above are mainly concerned with homogeneous (32) media. Attention may be drawn to the works of Thynell and Ozisik developed for solving heat transfer problems in spherical and cylindrical media.
Integral form of transfer equations in
isotropically and anisotropically scattering inhomogeneous spherical and cylindrical media are established.
In the general case, the
phase function for anisotropic scattering is expressed as a truncated series in Legendre polynomials. incident radiation is defined.
A quantity called generalised The integral form of transfer
equations containing in them the boundary conditions of the problem are of Fredholm type or coupled equations of Fredholm type involving generalised incident radiation. numerical methods.
These equations are solved by
258 Their method and the F —method are in a sense generalisation of (24) transformation idea of Mitsis used for the analysis of the critical—size problem of neutron transport. The generalised incident (32) radiation of Thynell and Ozisik is the moment or the combination of moments of specific intensity defined for the radiative transfer of stellar atmosphere.
6.4
Pincherle—Goursat Kernel Method For Solving Integral Equation of Transfer in Curved Geometry While many examples exist of the use of the integral equation
method for solving transfer problems in homogeneous spherical and cylindrical media, very little has comparatively been done with respect to inhomogeneous medium.
In what follows, we recount such a method
applied to transfer problems in shell media of spherical and cylindrical geometries. For this, advantage is taken of the
regional division of
intensities in the construction of integral equations suggested by Germogenova
. The resulting Fredholm type of integral equations
for inhomogeneous, isotropically scattering spherical and cylindrical atmospheres are solved by the Fredholm theorem for general L„—kernel. It is shown that the procedure of expressing the L„—kernel as a sum of a Pincherle—Goursat kernel and another L„—kernel proved very effective in solving the integral equations of transfer in specific cases [cf. (21 22) Leong and Sen ]. Some problems of spherical shell media and infinite cylindrical shell media are considered and exact solution obtained when the attenuation coefficient vary as inverse power of radial distance. Spherical
shell
Media :
An inhomogeneous isotropically scattering, spherical shell medium bounded by surfaces of radii r.. and R {0 < r.. < R} is considered. radiation field is supposed to be axially symmetric, so that the intensity and source function are azimuth independent.
The
259 The equation of transfer for diffuse radiation field is given by (21 22) [cf. Leong and Senv ' , equation (2.1)].
M
aiir^x) + l V aiir^) . _ ^ ( r ) I ( r > M ) 3r r 9M
+f
.(r>i
(6.4.1)
where +1 f'(r)-2jEij
I(r,/i')d/x' + B;
I(r,/i) is the intensity at r along the direction cos the outward radius r.
(6.4.2)
\i measured from
o> (r) is the albedo for single scattering
u (r) = (7(r)/a(r) , (0 < w £ 1), where tr(r) and a(r) are scattering and attenution coefficients respectively.
y(r)is the source function.
B'(r) is the contribution to the source function from reduced incident o radiation at the bounding surface or surfaces. B'(r) is the contribution from internal sources other than scattering. Equations (6.4.1) and (6.4.2) are to be solved subject to the boundary conditions
where
I(R,-/J) = 0 ,
0 < /* < 1,
I(r r jO - yl(.xv-ti),
0 < y. £ 1,
0 < 7 < 1.
(6.4.3)
(6.4.4)
260 We write the transfer equation (6.4.1) as dl(r,/i) ds
o(r) I(r,/i) + ?'(r).
(6.4.5)
s is the distance measured along a pencil of radiation, s = 0 being taken as the point on r = R on the pencil. Formal integration of (6.4.5) using the boundary condition (6.4.3) yields s s I(r,/i) = [ 9'(r')exp[- [ a(r")ds" ]ds' .
(6.4.6)
I(r,/i) is divided into three parts I., I„, I, in respective range of /i [cf. Fig 6.4.1].
Fig 6.4.1
Division of I(r,/i) into I. (r,/i) , I„(r,/i) and I_(r,/i) (according to the ranges of /i) is shown in Fig 6.4.1.
When the pencil of
radiation is confined to region (1) we designate I(r,/i) as I1(r,/i) with - 1 < y. < 0; in region (2) as I (r,/i) with 0 < n < /l-(r / r ) ' and in region (3) I„(r,^) with /l-(r 1 /r)
< p. < 1.
When -1 < fj. < 0,
ds'
-r' dr' / ,2 2. n 2, / r' - r (1-p )
(6.4.7)
261
Fig 6.4.2 Diagrammatic relation between s, s', and r, r', n in region (1) of Fig 6.4.1.
Now inserting equation (6.4.7) in (6.4.6) and rearranging, we obtain
Ir")r"dr" dr' l1(r,^) - J »'(r')r'exp -J — ^ r/..„2 r2 (1-/*V ) // r',2 - r2,, (l-^i2,) / r'
(6.4.8)
/ 2 For 0 < /i < /l—(r1/r) , the line of integration is from (0,s ) and from (s ,s) and in this range I„(r,/u) is given by
I2
sP
„s + J W'(r')exp-J 0 "s
a(r"')ds' ds'.
(6.4.9)
Writing this as sum of two integrals we have r
s
I2(r,/i) = j
P
r .s
Sf'(r')exp - jj
9'(r')exp
- [
.& \ P
+ J
|a(r")ds" ds
a(r")ds" ds' .
(6.4.10)
262
s=0
Fig 6.4.3
Diagrammatic relation between s, s' and r, r', /i in regions (2) and (3) in Fig 6.4.1.
For 0 < s' < s P
we have -r'dr'
ds' =
/T2
2TT
(6.4.11)
27
/r' -r (1-// ) and for s < s' < s P ds'
r' dr'
(6.4.12)
2
/P -r-(l-/i^) IT'
Then from (6.4.10), (6.4.11) and (6.4.12), we have
r fr r '
R I2(r,/i)
f'(r')r'
:XV
e x p - i\
/ r/l-/x
dr'
A
2 -
2... 2 , r (l-/i ) dr'
/ ,2 2.n 2, / r' - r (l-/i )
J
/ 2 r/l-/i
fr JJ + 2 //
a ( r " ) r "d r "
\
9 527; 2oJ ■>/ r~^2 r/l-/i / r" - r (l-/i
( r ' ) r ' exp
Lr r
Q r
( " )r"dr"
'/~7/2
/
r"
27; 27J
- r (l-/i )
(6.4.13)
263 The region (3) in Fig 6.4.1 is divided into two parts (3a) and (3b). In the region (3b) the pencil of radiation proceeds from s = 0 to the surface of the core r.. , where this point on the line element is designated as s = s.. and here -r' dr'
ds'
(6.4.14)
/T2 2~; 27 /r' -r (l-/i ) Hence from (6.4.6)
"(3b)
R r r' (r1,-/i1) = J *'(r')r' exp - J
q(r")r"dr"
dr'
.2 2, l/r»' - r^(l-/ip
/ ,2 2,, 2T /r' -r1(l-/i1) (6.4.15)
From (6.4.5), we have in the region (3a) S
I(r,/i)exp [
1
a(r')ds'
S S
r S'
Ids' 5(r')ds' exp j a(r»)
- Kr1,M1) " J
S
l
l (6.4.16)
where s = s. denotes the point of the line element at r - r. in the region (3a). From (6.4.16) and (6.4.4) we have
I
(3a)(r'")
= 7l
ds'
(3b)(rl--"l)exp
1
exp — s1
L
a(r")ds
9'(r')ds',
(6.4.17)
s
where /1, fi^ a r e r e l a t e d by
r /l-/i
and
ds'
= r.
/I-/1,
r'dr'
/ 7 2 27, 27 / r ' - r (l-/i )
(6.4.18)
(6.4.19)
264
Then from ( 6 . 4 . 1 5 ) , (6.4.17), (6.4.18) and ( 6 . 4 . 1 9 ) , we have
rR I,(r,/i) = 7
r'
S
+
exp :-{/:r.
r.
+
1
exp
„r J
a(r")r"dr"
"(r')r'dr' r . J / „2 2 / n 2.-I / 7 2 27: 2 1 / r" - r (1-/J ) / r ' - r (l-/i
■c
9> ( r ' ) r ' d r ' a(r")r"dr" ' AT2 2 , , 2, J / 7 2 27; 2 / r" - r (l-/i ) /r* - r (l-/i
(6.4.20)
The i n t g r a t e d i n t e n s i t y over n i s given by 0 / 2 I(r l A l )d/i - [ I (r,/i)d/i + f / 1 _ ( r l / r ) I (r,/i)d/i J -1 -1 0
♦t/ l - (
I 3 (r,/x)d/2. ri
/r)
(6.4.21)
2
Substituting I (r,/j), I (r,/i)and I (r,/i) in (6.4.21), from (6.4.18) ; (6.4.13), and (6.4.20), choosing only those a(r") which leave the integrands positive and integrable, and interchanging the order of integrations over \i and r' which are permissible, we obtain
r:
I(r,M)d/i
a r - [ S' r " d r " \ LJ l J r -1 rA^2 2,. 2.J / r" - r (1-u )
♦ pk- exp - f 0
I t
J
r /
+r ^77
J
r
a r r dr
d
( ") " " I
2
2
2
^
2
ry^^J/r" -r (l-, )JVr' -r2(l-,2)
265
g(r")r"dr" f expj- J J 7. , , , ,2 <- r ' / " 7 2 2 . , 2., /l-(r'/r) / r " - r (1-/* ) J
J" S ' ( r ' ) r ' d r '
+
/l-Cr^r)2
J/
X (r'/r)
+7
r
^ r L r/(l-
f *'(x')r.drY r
l
g(r")r"dr" '
exp
/l-^./r)Z
M
2
)
2T; 27J O r/(l-^ 2 )J /Ai2 r " - r (l-/z ) J / r ' exp-{f+j L
r
l
r
r
}
d/i
2727
- r (1-/J )
g(r")r"dr"
d/x
/u2 2 7 27 /,2 2 7 27 l ./r" -r (1-jx ) /r' -r (1-^x ) (6.4.22)
In the right—hand side of equation (6.4.22), the terms are now expressed in terms of r, r', cp' [cf. Fig 6.4.1] using the following relations. We use in the first term of the first bracket
/ ,2 2.. 27 Ar2 +r' - 2 —2rr'cos
(6.4.23)
and in the first term of the second bracket
T/T +r'2-2rr'cos cp' = r/z - A ' 2-r2(l-/i2)
(6.4.24)
and substituting for all other terms
/r2+r'2-2rr'cos
(6.4.25)
266
j<(r) = f i l l
s i n tp' dtp' f 5 ' ( r ' ) r ' 2 d r ' f 1 exp|-f a(r")ds" ~~2 2 J L J r, 0 s' r + r ' — 2 r r ' c o s
J
*
+ 7
.s,
ff(r)
(r')r'
dr'
exp
„s
(r )ds
rtLM. > " "
s i n ip'dip' 2 ,2 0 , r +r - 2 r r ' c o s (p (6.4.26)
+ B'(r) where B ' ( r ) = B n '(r) + B ' ( r ) I o tp, satisfies the relation fl 2 /r +r' —2rr'cos cp..
/2 2 /T2 2 /r -r- + /r' -r
(6.4.27)
That is cp. represents the angle tp' when the points r and r' lie on the opposite sides of the point of contact on the tangent line dividing region (2) from (3). The equation (6.4.26) leads to the same result obtained by Cuperman, Engelmann and Oxenius (9) [equation (3.9)] when r.. -> 0, i.e. tp
= 7r.
Equation (6.4.26) can be written as
r ?'(r) = A [r ?'(r)] + rB'(r),
(6.4.28)
where the operator A [r 9'(r)] is given by
Ar[r Sf'(r)] = ?lp-
jk(r.r') + 7k*(r ,r') } r '? ' (r' )dr'
| r
(6.4.29)
l
with the kernels 1*1 k(r,r' ) = rr'
J
0
f ■N(r,r' ,
(6.4.30)
267
Cr.r') - r r ' f exp{-N*(r,r' ,v>)] g ^JE^El ip.. *• -* r +r' -2rr'cos
where
(6.4.31)
(6.4.32)
N(r,r' ,
1
and
(CO
a(r")ds",
N (r.r'.v*) =
(6.4.33)
where s.. denotes the point on the line elements at r = r.. in the region * (3a) and s.. denotes the point on the line element at r — r.. in the region (3b). Determination of the kernels, inversely
as integral
when the attenuation
coefficient
varies
power of r :
Let the attenuation coefficient be given by a(r) = a r o where a
,
m>0
(6.4.34)
is a constant and m is an integer.
Hence writting -(2n+l) a r o
when
m =
2n+l,
n - 0,1,2...
when
m =
2n,
n = 0,1,2...
«(r) = i
(6.4.35) -2n a r o
and the corresponding N (r r'.ip') and N (r,r',
N„
n(r,r',
when m = 2n+l (6.4.36)
N (r,r','), zn
when m = 2n
N. ,(r, r ',<»'), zn+l
when m = 2n+l
and
N (r,r' ,
(6.4.37) N. ( r, r' ,
when m - 2n.
268 "ft
The explicit forms of N (r.r'.cp') and N (r,r ,
s=0
Fig 6.4.4
Connection between s, s', s" and r, r', r" through
From Fig 6.4.4, we have 2 r"
2 2 = (s-s") - 2r/i(s-s") + r ,
(6.4.38)
when m = 2n+l, in view of equations (6.4.32) and (6.4.33)
N
2n+l
ds"
° J s ' ((
„s2 o , „„ 2,(2n+l)/2 s-s") - 2r/i(s-s") + r } "
(6.4.39)
I n ( 6 . 4 . 3 9 ) , we w r i t e s — s" = y ,
and
r Si/l-/i y-r/j
2
= s i n h x.
The sine rule in the triangle OAA' [Fig 6.4.4] gives
(6.4.40)
269
XV r'
sin ip'
(6.4.42)
t
2
2
/ rr +r' - - — 2rr'cos cp'
Integrating (6.4.39) we have [cf. Leong and Sen^ 9
*?
M
/
,
,\
N 2n+1 (r,r' l(p ') °
'p.106]
n
a (r +r' —2rr'cos cp') n-1 . ,.k r , ° v (-1) fn-1
2n 2n
.
r
sin
r'
E ks" [k
2n cp'
v
k=0 n>l
,,2k+l , ,,2k+l (r'-r cos
x
(6.4.43)
7~2 ^ ^ cos
It is to be noted that the above expression is true only for n > 1. For n =0, using (6.4.40), (6.4.41) and (6.4.42), and integrating (6.4.39), we have „ . , ,. . ,-1 (r'-r cos cp'\ N- (r ,r' ,
a
o
. ,-lfr' cos ycp'-r sinh <—; . I r' sin cp (6.4.44)
For all n > 0, equations (6.4.44) and (6.4.43) give the complete expressions for N„
.(r.r'.cp').
For m = 2n, following similar steps, we get n-1 , k V 1 n 2n-2t\ 2n-2k-l Jl 2n-2t / r 2 . .2 — k=l t-1 'r +r' —2rr'cos cp' n>l a
N2n(r,r',1p')
2 2 (r +r' -2rr'cos cp')k 2k ,2k . 2k r r sin
X -
+
r(2n-l)
r
r'—r cos cp' ,2n-2k-l
r—r' cos cp' 2n-2k-l
a rr' sin a?' o
( 2 n ^ ( n ) } 2 / 2 ^ ,2 : ; , v " /r +r' -2rr'cos cp' X
/ 2 ,2 „ , ,. n (r —r' —2rr'cos cp') 2n ,2n . 2n , * r r sin cp'
(6.4.45)
270 Above equation is valid also for n - 1, in which case the summation term on the right hand side in zero. For n = 0 /2 2 /r +r' —2rr' cos
N (r,r' ,
(6.4.46)
We now denote the integral in (6.4.30) with N replaced by N ^
as
exp[-N (r,rr ,
k (r,r*) - TT' m Jn
(6.4.47)
2 2 r + r' — 2rr' cos
This integral will be evaluated for some particular values of m. When m = 0, from (6.4.27), (6.4.46) and (6.4.47), we have
nr~i
^ i ,2 2
„/r —r,
+ /r' -r,
z (r,r') = o JI ,i r—r' r °°
Q
ol r - r 'l
exp(-a x) o
, — x
OO
a
exp(-rj) €
,2
-^
(6.4.48)
J\/^A - ^ -*l}
(21) This leads to [cf. Leong and Senv , p.106-107] kQ(r,r') -= E1(ao|r-r'|) - E1
[..{^
♦ y^{ }j
<s.4
49)
where °0
L < X ) - | exp(-r,) ^ J l ■»
(6.4.50)
As R ■) », the expression for k (r,r') is the same as that derived by Davison and Sykes^
As r, -> 0, k (r,r') 1 o leads to the equation derived by Cuperman, Engelmann and Oxenius (9) (equation (3.13)].
[p.211, equation (15.49)].
271 When m = 1 and a
o
- 1.
Putting m = 1 in equation (6.4.47) and using equation (6.4.44)
k^r.r*) = log
i2
[2 2 ^ /72 2 /r -r^ + /r' -r
r + r' +
fl
|r-r'|
2
r-r'
/~2
2 2 '
(6.4.51)
+ r' + /r -r 1 + /r' -r
When m = 2
k2(r,r')
r 2 ^ ,2 2 —11r +r —x -2a x cos o 2rr'
r/2 2 A A72 2 /r -r + /r' -r exp r-r'
/ 2 2 2 2 /{x (r-r') } {(r+r') -x )
(6.4.52)
when m = 3
k3(r,r') =
dx x
1f PT^L /r -r
^ / ,2 2]
+ /r' -r^ /(r+r') r -2a ° _J_1 dl exp " l- ? 2 / « ' (r-r')/(r+r')j
(6.4.53)
From (6.4.30) and (6.4.31), we find that k(r,r') and k (r,r') are formally similar, except for the limits of integrations.
Therefore,
following a similar procedure we can find the explicit forms of * —m k (r,r') for a(r) = a r , where o m = 0,1,2,3. Solution
of
the Integral
Equation
Of Transfer
:
The integral equation of transfer is given by (6.4.28) as r.%' (r) = A [r?'(r)] + rB' (r) where
i rR
Ar[rJ'(r)] - i
f)(r,r')[r'?'(r')]dr'
(6.4.54)
272 O(r.r') - a(r)[k(r,r') + 7 k (r,r')].
with
(6.4.55)
k(r,r') and k (r,r') are given by (6.4.30) and (6.4.31).
We shall
show now that Q(r,r') is an L„-kernel i.e., ,R „R
I lJ l CT(r,r')drdr'< M, r
(6.4.56)
r
where M is some finite quantity.
In other words what is wanted is
that the integral (6.4.56) is bounded. From (6.4.30) and (6.4.31)
*J?
s k(r,r') = rr' i ' exp{-N(r ,r',
j
-f
i
. »,*/
,
s i n
, s i
k (r,r') - rr' I exp{-N (r,r',
(6.4.30)
(6.4.31)
From (6.4.30)
k(r,r') < rr
■f01 r2+r'2-2rr' cos
» n
< rr
'J 0 r2+r'2—2rr' cos
(6.4.57) which implies that k(r,r') < log
r+r' r—r'
(6.4.58)
In exactly the same way, we obtain that k (r,r') < log ' Then from (6.4.55), (6.4.58) and (6.4.59), we have
(6.4.59)
273
n(r.r') < <7(r)(l+7) log
r+r' r-r'
(6.4.60)
i.e.
J n2(r,r')dr' < a 2 (r)(l+ 7 ) 2 |
(log
r+r' r-r'
dr.
(6.4.61)
It can be shown that
A
[log
~ r
| dr' < 2 J
[log(r+r')j dr' + J
[log(r-r')j dr'
(6.4.62) The first term on the right—hand side of (6.4.62) is bounded. Therefore j [log (r+r')j dr' < Mj
(6.4.63)
Mn is finite. The second term on the right hand side of inequality (6.4.62) can be written as
|
[log |r-r'|j dr' = J
[log(r-r')| dr' + J
[log(r'-r)| dr'
as € -» 0.
(6.4.64)
Integrating the terms on the right hand side of (6.4.64), we have
j [log |r-r'| j dr' - (R-r)log(R-r)[log(R-r)-2l + (r-r^logCr-r^ r
l x [log(r-r1)-2| + 2(8.-^).
This implies that there exists some finite M„ such that
(6.4.64)
274 & r
.2
|log |r-r'|ldr' < M 2 . r
(6.4.65)
l
Then from (6.4.61), (6.4.62), (6.4.63) and (6.4.65), we have D
I n2(r,r')dr' < 2a2(r)(l+7)2(M1+M2). r
(6.4.66)
l
Now assuming that cr(r) = a r
where n is zero or an integer, positive
or negative, we obtain
|R
|R
r
r
l
fi2(r,r')dr
M 1 +M 2
dr' < 2 ( l + 7 ) 2 ^ a o { R 2 n + 1 - r 2 n + 1 } < M , 2n+l '
(6.4.67)
l
which is bounded. Hence fi(r.r') is an L„—kernel. Then using the method of E. Schmidt which has been generalized by Picone, the kernel fi(r,r') may be decomposed (in an infinite number of ways) into the sum of a suitable degenerate kernel S(r,r') and f 35) another L -kernel T(r,r') [cf. Tricomiv ,p.64 Sec. 2.4]. We can write n(r.r') = S(r,r') + T(r,r'),
(6.4.68)
where the degenerate Pincherle—Goursat kernel
S(r.r') =
s I Xi/(r)Yi/(r'), i/=0
(6.4.69)
and the L.-kemel T(r,r') is such that its norm ||T(r,r')|| can be made as small as possible.
275
T(r.r')
12
rR j
rR j
r
r
l
m 22
T (r,r')dr dr' < 4,
(6.4.70)
l
The condition (6.4.70) ensures that the Newmann series for T(r,r') is convergent. The decomposition offl(r,r')into the sum of two kernels as in (6.4.68) can be done by expanding fl(r,r') as a double cosine series [cf. Mikhlin (23) , p.22, §5], i.e.,
«..-»- i ^ H£%MEih
(6.4.71)
where
A pq
45 o (p)6 (p)5o (q) (q) „R 45 R r
(R-^r
„R R
!
r
s r r O ( r , r ' ) c o s { ^ - j cos<3
(6.4.72)
R r
]
l - lJ
r -~ 5Q(p) = \ *■ 1
and
,\ 1 drdr
when p = 0
(6.4.73)
when p * 0
If we now take S(r.r') =
l A c o s j ^ l cos(§^l , p,q=0 P q lR~rii lR-riJ
(6.4.74)
we would g e t
T(r,r
> -1 - f e ~ f~im ■>? qi r.. , s,*„„«-{&■} e} p=0 p=0 q = s + l
r*1
l-
1J
I.
]J
Jp7rr 1
JqT'
l
w
+ p=s+l 1 1q=0. K„ « « &V »- & 1 rH
J
(6.4.75
s i s chosen l a r g e enough f o r t h e e q u a t i o n ( 6 . 4 . 7 0 ) t o h o l d good.
276 From e q u a t i o n ( 6 . 4 . 6 9 ) , we have s I
S(r.r') =
Xi/(r)Yi/(r')
i/=0
where Xy(r) and
= cos
i^-).
<J. <<.!<>)
r,) =
C0£
vV '> - £L \% { f^e}} •• r
q
q=0
cos
"-
1J
((66 44 777)7)
-- --
S(r,r') is then a degenerate kernel of (r,r'), since the number of terms in T(r,r') is infinite, this part of the kernel is non— (23) degenerate [cf. Mikhlin P-19, §4] Let H„(r,r') be the resolvent kernel corresponding to T(r,r'). Now from (6.4.54) and (6.4.68), we may write R
i t*'(r)r] = =:
where
r r
l
T(r,r')[5'(r')r']dr' + A(r),
i rR
A(r) = [rB'(r)] + |
r
l
S(r,r')[*«(r')r']dr'.
(6.4.78)
(6.4.79)
From equation (6.4.78)
i rK [*'(r)r] = A(r) + ~\
r
l
H (r,r')A(r')dr',
(6.4.80)
where H (r,r') is the resolvent kernel corresponding to T(r,r') given by CO
HT(r,r') =
I ( i ) n _ 1 T (n) (r,r'). n=l
(6.4.81)
T (r,r') denotes the nth iterated kernel of T(r,r'). In veiw of (6.4.75), equation (6.4.81) takes the form
277 ,;
: n 1 U^ "
HT(r,r<)= l
h
«* [*
™l\*2
£ p,q1,q2...qn
n=l
"h%-lqn
5 ( q , ) 5 V( M q „ ) . . . 5 (q -) o^V o 2' ov^n-l q wr'^
*"={!%} °°i-hr)
r
where
)
(6 4 2
-"'
is defined as follows.
e P,qr..qn
If
) denotes the summations over p,q.,...q P,q r q 2 .-.q n
ranges 0 to s or (s+1) to »,
within the
V*
) denotes the sum of all 6 P.q r q 2 -..q n
possible permutations of p,q1 q....q
in
) with the (P.q r -.q n )
exclusion of the permutations of the type where two of its summation signs in
) occur side by side in the form P,q r ..q n
Iq . q I. , -q .L=0 q.I -„=0Wlth q i " P ' q l' q 2"-- q nx Ml+1
H
M
l
^l+l
since the summations are being taken care of in the expression for S(r,r'). From (6.4.78) and (6.4.79), we may write 1 "R [?'(r)r] = [rB'(r) +^\ S(r,r')r'?'(r')dr'] r
i r
+i
r
l
1r
l
H (r,r')[r'B'(r') + |
r
l
S(r',r")f(r")r"dr"]dr'. (6.4.80)
278 After rearranging, we have
[It'(r)r]-|J
[S(r,r") + | J
HT(r,r')S(r' ,r")dr' }*' (r")r"dr"
R + [rB'(r) + i1 f I HT(r,r')r'B'(r')dr'].
(6.4.81)
Hence
[r»'(r>] - | J
R
s \ I
r, <-v=o
X*(r)Y
J
(6.4.82)
where Xj/(r) = X^(r) + i j
= -{&%}
+
H^r.r')X^(r')dr'
? j" H T< r - r '> C ° S {^} dr '
(6 4 83)
"-
and * i r A (r) = rB'(r) + | I H (r,r')[r'B'(r')]dr*.
(6.4.84)
The solution of equation (6.4.82) is given by [cf. Tricomi
(35)
p.58 ] [tf'] - A*(r) + i ^ J
[J
JDoi/Yo(r') + B ^ Y ^ r ' ) + D ^ Y ^ r ' ) + ..
:')}x*<
+ D Y(r') si/ s where A is non—zero and is given by
X ( r ) A (r')dr',
(6.4.85)
279 , _ 1 1 2aoo
1 2 a ol
1 2 a 10
1 l
1 2 a 20
A =
1 2 a ll '•' 1 2 a 21
1 2 a so
1 2 a os
•••
1 2 a ls 1 2 a 2s
'"
(6.4.86)
.. 1 - i «
2 a sl
2
ss
D, , denotes the cofactor of the (h,k) element of the determinant A and hie
-hk^J
<(r')Yk(r')dr'.
(6.4.87)
Using equations (6.4.77), (6.4.83) and (6.4.87), we have
■*<■ j 0 % f - { ^ } H ^ } H
+
r,
HT(r-,r")coS{g: ^
1
(6.4.88)
dr" dr'
which leads to
A
^ kC
=
kh(R_rl)
1
v
lb
I
" a=0
(h) ov '
A
fR fR u /
icq J J T ^ r"1 . 'I r.
n
/q«'\ LK-*iI c 1J
/tar"'!, „. . L
LK_riJ 1J
(6.4.89)
In particular, when the attenuation coefficient a(r) =a and 7 ^ 0 , r o we replace H (r,r') by H_, (r,r') in the relevant equation and it is o g i v e n by
280
HT ( r , r ' ) = I o
n=l
A n* P q i \ q 2 A q 2 q 3 " " "Aqn-lqn L 6 ( q n ) 5 ( q 0 ) . . .5 (q -,) 6 p , q 1 , q 2 . . .qn oVHlyoVH2' ovvln-l
rR-r.. > n - 1 4
. x coss?
<-M
/-q irr' >■ cos- R-r.
(6.4.90)
where A is given by pq A Pq pq
J "'{i^}•
46 (p)5 (q) _R „R r or on r 2 (R-r^ '^ r. ,
fp7rr \
x ^(r) E (a r-r' ) - E i o' ' 1 + 7r
cos
fe-l <
■ - « & $ *
■
U^A ♦ ^M}]
*o|r/*+/r'2-r2(l-Ai2))- + 2a o /r 2 -r 2 (l- M 2 )
exp
/l-
(6.4.91)
/T2 27; 27 /r' -r (l-/ii )
Further if a and cr are constants, it can be shown that the resolvent of the kernel Q (r,r') of the integral equation (6.4.54) exists [cf. Leong and Sen
(21)
, 1971, p.114].
We can then set
0 (r,r') = T (r,r') o o
(6.4.92)
Hence from (6.4.85), "(r) = A (r)
1 rR
i.e.,
[r?'(r)] = rB'(r) + i J
r
H T (r,r')[r'B'(r')]dr],
l
°
(6.4.93)
(6.4.94)
(r,r') is the resolvent kernel of T (r,r f ). This result is o 0 the same as that derived by Germoeenova ( I D , [Theorem 1, p.1271. the same as that derived by Germogenova , [Theorem 1, p.127].
where H
1
281 It Is also clear that for the homogeneous case o(r) — a
and
a(r) = a , are constants. Ci (r.r') is an L„—kernel even when r. -» 0. o
o
Z
Hence the above result holds in this case.
I
Letting r- -» 0 and putting
7 = 0 , we find that To(r,r') -> ao[E1(ao|r-r'|) - E^aJr+r'|)].
(6.4.95)
Hence from (6.4.94), we have
i rK [rS'(r)] - rB'(r) + i
where
r
L (r,r*)[r'B'(r')]dr',
l °
LQ(r,r') - I ( \ ) n=l
with T o
n1
T
(6.4.96)
(6.4.97)
(r,r') as the nth iterated kernel of T (r,r'). o
This result is the same as that of Cuperman, Engelmann and (9) Oxenius^ ' [p.113, Equation [5.1]]. Case II
-3 Let a(r) = a r and 7 = 0 . o
We replace H (r,r') by H H
3
(r,r') is given by
-
(r,r') in relevant equations and
Pql\V"\-lqn L S ( q . ) S ( q 0 ) . . - 5 (q . ) e FP , q l ' q 2 - - - q n ° o^n-r
,R-r,,n~l
H (r,r<) - I U —
3
A
x cos{^-} c o s { ^ } with
(6.4.98)
282 46 o (p)6 o (q) A Pq
"
(R-r,)
2
rR
LII
:os-
2E£_1
{^}
cos
/ r 2 - r 2 + /r' 2 -r 2 |/(r+r') x cr(r)
exp (r-r')/(r+r' ) |
I
-2«o rr'
i
drdr'.
2f C
l-£
(6.4.99) The value of rSJ'(r) can be obtained by substituting (6.4.98) in (6.4.82). Thus we see that the boundary value problems defined by Fredholm type of integral equations (6.4.28) characterise transfer problems in an inhomogeneous, isotropically scattering spherical shell medium with axially symmetric radiation field.
They are amenable to exact
solutions as long as the attenuation coefficient varies as some inverse power of the radial distance from the centre of symmetry. This has been achieved by using the Fredholm's theorem of general kernels.
Its main feature is the decomposition of the L„ kernel
n(r,r') into a sum of a suitable Pincherle—Goursat kernel S(r,r') and another L„ kernel T(r,r').
It has also been shown that in the
particular cases of homogeneous medium, the results lead to those of -,(9) j r, (H) Cuperman et al and Germogenova The same method has also been shown to be equally effective in solving transfer problems in inhomogeneous, isotropically scattering infinite cylindrical shell media with the attenuation coefficient (22) varying as some inverse power of radial distance [Leong and Sen ].
6.5
Ambarzumian's Mathematical Method Ambarzumian's mathematical method was first developed as a tool
for solving Milne's integral equation. of Busbridge
The combined operations method
was a direct consequence of this technique.
As
pointed out in Chapter II of this treatise, the essential features of the method may be enumerated as follows.
In this an auxiliary
283 integral equation corresponding to the integral equation of transfer is developed and its solution is sought in the form of Neumann series whose existence and uniqueness are established.
The scattering and
transmission functions are defined in terms of N-solutlon.
The
integro—differential equations for scattering and transmission functions are obtained from auxiliary equation in a form suitable for numerical calculation as an initial value problem.
The original two
point boundary value problem is converted into an initial value problem.
The emergent intensity is calculated in terms of the
scattering and or transmission functions.
In short, this is a method
for obtaining the law of darkening without going through the knowledge of source function as is normally done. method.
This is thus a global
The method has been successfullly utilised by Busbridge
and Uesugi
for solving transfer problems in homogeneous and
inhomogeneous, coherent and non—coherent scattering, finite and semi—infinite plane parallel media. The combined operations method has also been used to solve diffuse reflection and transmission problems in spherical and cylindrical media [Kho and Sen
"
]. We demonstrate some of these
attempts. Combined
Operations
Method In Spherical
Medium :
We consider a homogeneous, isotropically and coherently scattering spherical medium of radi^
R.
The diffuse radiation field is supposed
to be axially symmmetric so that the source function and intensity are azimuth independent. The equation of transfer in this case for the diffuse radiation field is given by [cf. Chapter I, equations (1.3.28) - (1.3.29).]
p. ^ where
2 I(r,M) + ±qK- J- I(r,M) = -<*(r)[I(r,/x) - *(r)]
(6.5.1)
284 <*» < r >
r1
»(r ) - - ^
|
Kr,/i')p(M',M)d^' + BQ(r) + B]_(r).
(6.5.2)
Here u (r), the albedo for single scattering and a(r), the attenuation coefficient are taken constant throughout the medium. is isotropic p(/u' ,/z) = 1 .
As the scattering
B (r) is the contribution from the reduced
incident radiation and B.. (r) gives the distribution of the internal source of radiation if any. If B (r) arises from a net flux of radiation TTF. incident at the 0 * angle cos -1fi to boundary surface at r = R and in a direction making an boundary surface at r = R and in a direction making an angle cos
n
to
the inward normal, it is given by [cf. Chapter I equations (1.5.32) (1.5.33)]. „ , -
r)
td F. o i
V - — [?]'
_o *
2 exp (-aRji ) cosh(ar/i*) H(r-R/l-/i ),
(6.5.3)
where 0 < u , a < 1. o o -1 * cos ii is the angle between the direction of incident flux and inward drawn normal at r. 2 2 2 ( W *2Z: 0o ) R (!-//> = r ■(l-^
Also
(6.5.4)
and the Heaviside unit step function H is given by
2
H(r - R / \1-M N) = \
/ 2 1 when r > R/l-u o
(6.5.5)
0 when r < R/l-u Hence the source function for this model is o
+l
*
+
I(r,M)d/i + B]_(r)
F.
o l( R l
2
r V.
J
22 o exp(-aR/i )cosh(aru )Hi .r - I R/ 1-u, o * o o o
(6.5.6)
285 Boundary Condition
:
The transfer equation (6.5.1) - (6.5.6) is to be solved under the boundary condition l(R,-/i) - 0,
(6.5.7)
0 < (i < 1
That is the diffuse radiation in the inward direction is zero at the boundary.
Integral
Equation For The Source Function
:
Following a scheme first suggested by Cuperman, Engelman and (9) Oxenius Lus , the integral equation for the source source function is given by [cf. Chapter I equations (1.5.24) and (1.5.25)] a) R r?(r) = ^- | t?(t)k(r,t)dt + rB(r) J 0
(6.5.8)
k(r,t) = a[E.(a|r-t|) - E (a(r+t))]
(6.5.9)
where
and (6.5.10)
B(r) = BQ(r) + B]_(r)
B (r) in the homogeneous case from (6.5.3) is written as
Bo(r) .
co F. f R | 2 O 1 2 r
*
r^oi —
r
/ n
exp(-aR/i ) cosh(ar/i ) H r—R/l—n
and the exponential integral E.. (x) , by [cf. equation (1.5.11)]
_ .. E (x)
i
r
- I
e
-xu du
-n-
r
-u du
e
(6.5.11)
Since E.. (x) is a decreasing function for x > 0, the kernel k(r,t) is non-negative.
It is also symmetric, i.e. k(r,t) = k(t,r)
(6.5.12)
286
Auxiliary
Equation
:
For simplicity, we assume that there is no internal source of radiation, i.e. Ik (r) = 0.
Now replacing ^(r) by f(r) such that
u F. *(r) = - ^ f(r;R,M 0 ),
(6.5.13)
from the integral equation (6.5.8) we have w R rf(r;R,M o ) - TT~ \ 2 J„ t£(t;R,/io )k(r,t)dt R2 "o + 2 — —v exp(—aR/x ) cosh(ar/x ) H r - R y W
This is the auxiliary equation.
(6.5.14)
The solution of the auxiliary
equation (6.5.14) can be expressed as a Neumann series provided it is convergent. We rewrite the auxiliary equation (6.5.14) as
:(r;R,M ) - D M t ?(t;R,/i ) 'o R,u r| o o R2 ''o + 2 — —x exp(—aRfi
* ) cosh(ar/j ) H
M3)
(6.5.15)
where the linear operator „ L is defined by R,u r ' o w
_R
lt^{«t:a,i.0)}-^J^f(t;»,i,o)k(r.t)dt.
0 < t < R
and
(6.5.16)
0 < w < 1. o
Defining r?(r,R,/i ) as the N-solution of equation (6.5.15), we have
287
xf(x;R.,0> - I m=0
1" {g(t;R,M0)} . o
"-
(6.5.17)
'
where R2"o g ( t ; R , ^ o ) = 2 — — exp(-o:R/i o ) c o s h ( a t / i o ) H t - R / l - M 2
R,H
L i s the mth i t e r a t e of the operator _ r o
r
L and
R,u r "o
L°{g(t;R,Mo)!
R
Existence
(6.5.18)
gC.P..V
And Uniqueness Of The N—Solution Of The Auxilary
Cuperman et al
and Germogenova
of the series (6.5.17).
(6.5.19)
Equation
:
have proved the convergence
However, we shall present below a. proof which
closely resembles that of Busbridge Theorem : Let g(t;R,j* ) be a bounded function for 0 < t < R, such that |g(t;R,/* )| < C (a finite non—negative constant).
(6.5.20)
Then
lR, M L r^(t;R," o
S\\*
r
m
(6.5.21)
where
*-\*.,y»\-
(6.5.22)
Therefore, the series
Z R,, m=0
W*W0»
(6.5.23)
'o
is uniformly convergent, if 0 < x < 1-
(6.5.24)
288 From (6.5.16) and (6.5.20) |
L r (g(t;R )Mo ))|
(6.5.25)
o
where
* " l R ,MoL r ( 1 ) lSince k(r,t) is non—negative a;
R
u
0 < x = ^ J k(r,t)dt < -^ j k(r,t)dt. CO
Writing
k(r,t)dt in the explicit form by (6.5.9) and (6.5.11) and 0 interchanging the order of integration, we have J
CO
k(r,t)dt = 2[1 - E2(ar)]
(6.5.27)
Then from (6.5.26), we have 0 <
X
(6.5.28)
^ " Q [1 - E2(aR)] < [1 - E 2 ( Q R ) ] .
But 0 < E2(x) < 1.
(6.5.29)
From (6.5.28), if follows that 0 < X < 1. This is what is given in (6.5.24). By iteration, the relation (6.5.25) can be generalized to (6.5.21) CO
CO
L £ R,M r m=0 r o
c I m=0
}l
;(t;R,u )} \«**>»J
Lr m=0
xm = £■ , if o < x < i. *
II
g(t;R,M0)
(6.5.30)
289
That is
I m=0
R,M
o
L jE(t;E,» ) r
'
<6"5-31>
* j ~ -
This proves that the series (6.5.23) is uniformly convergent. Therefore N—solution of auxiliary equation exists. Now if w R *
(6.5.32)
is a homogeneous integral equation corresponding to the non-homogeneous auxiliary equation (6.5.14), in order that the non—homogeneous equation has unique solution, the homogeneous Fredholm integral equation (6.5.32) must have the trivial solution [cf. (35) Tricomi p.64]. This result is used to obtain the integro— differential equation for ?(r;R,/j ) [cf. (6.5.56)].
Emergent Intensity
:
The outward and inward intensities of the internal diffuse radiation field are given by
Kr,/x ) =
f
/
5
Sf(t)exp(-a(r/-t/J')) H _ ^
* a dt f(t)exp{-a(r/i +t/j')}——
+
(6.5.33)
R/I-/ and * r * a dt I(r,-/i ) = S?(t)exp(a(r/i -t/j')} — ~
(6.5.34)
where 0 ■£ ft , ft'
S 1
and r2(l-/i*2) - t2(l-M'2) - R2(l-/i2).
[cf. Cuperman et al
, Equation (3.4)].
(6.5.35)
290
Fig 6.5.1 The inward intensity I(r,-/iz ) and outward intensity I(r,/j ) at r. Hence the expression for emergent intensity assumes the form
I(R,/i) = [
?(r)exp(-aR/i) 2 cosh( Q r/) 2 _ |
(6.5.36)
R/lV
or
I(R,/x) = f £(r)exp(-aR/i) 2 cosh(ar/x*) 2 _ ^ H(r-R/1-^2) ,
(6.5.37)
where ?(r) is given by (6.5.8), H is the Heaviside unit step function [cf. (6.5.5)], 0 < fi, (i*Sl, and 2
*2
2
2
r (1-M ') = RZ(1-/)
(6.5.38)
Now"remembering that for B.. (r) = 0, a; F.
*
= a/a,
we h a v e f o r B 1 ( r ) = 0
CTF.
I(R,/x) = T-^1 [ J
R
0
'(r.R./j
°
)exp(-aR/i)
* * I 3 2 cosh(arM ) (dr//j*)H(r-R/l-M ) (6.5.39)
291 where £(r;R,/i ) is given by (6.5.14), and a is the scattering coefficient.
Scattering
Function
And Its
Reciprocity
:
We define the scattering function as r — * * / 2 ?(r;R,/j )exp(-aR/j) 2 cosh(ar/j ) (/J/M )dr H(r-R/l-/i ) ,
S(JI,/J ;R) = a
(6.5.40) where the symbols have the meanings given earlier. Hence for B. (r) = 0, the emergent intensity I(R,/i) is given by F. I(R,M> = ^ S( M ,M o ;R).
(6.5.41)
The reciprocity of the scattering function S(/J,/J ;R) can be established as follows. From (6.5.17), (6.5.18) and (6.5.40), we may write
S(fi,fi
;R) -
Y
m=0 x
a \ [ - }\^r
R,M
J
oLrJV^
L
r I I
exp(-aRAi) 2 cosh(aR/i*)H(r-R/l-/j2)dr
^
{(
^(""^o'
2 cosh Qt
( Mo)H(r-R/l-/22 (6.5.42)
Let f(t;R,y. ) and g(r;R,/j) be two non—negative functions for 0 < t, r < R. Then from (6.5.16) and (6.5.12), R g(r) L J 0 R,/i
w -R -R {f(t)}dr=/ g(r)dr f(t)k(r,t)dt J J 0 0 ^ z
- ■I* JJi n
J
f(t)dt 0
J
g(r)k(t,r)dt 0
f(t)_ L (g(r))dt. K,M t
(6.5.43)
292 By induction the result can be generalised to mth iterate as
' g(r) L" 0 R,M r o
J
{f(t))dr-| J 0
f(t)
L" lg(r))dt.
(6.5.44)
R>/i C
Hence from (6.5.42) and (6.5.44), we have S(u,u ;R) = S(u ,«;R).
(6.5.45)
This establishes the reciprocity of scattering function.
It may be
mentioned that the inversion of the order of integration and summation involved in securing the step (6.5.45) is justified since f(t) and g(r) are non—negative functions.
The Integro—differential
Equation For The Scattering
Function
:
A homogeneous integral equation corresponding to the non—homogeneous auxiliary equation (6.5.14) is derived. 2 2 2 *2 R (l-/i ) = r (1-u ) o o
Since we have
3
* %
-R(l-^>
3R
and flu
r
(6.5.46)
2 * " o
*
o flu ^ o
=
'Rl rr
2
"o * *
(6.5.47)
Differentiating the auxiliary equation (6.5.14) with respect to R and using (6.5.46) and remembering that
k(r,R) = 2^ J
- ^ ] .] '
exp(^aR0l) 2 cosh(aru*) — i H M ^ / l - ^
[cf. Kho and Sen
(6-5.48)
Appendix A (equation Al) p.239], we obtain
293
a_
ry(r;R,/i )
3R w
rR a
■£■ J
| g [ t S ( t ; R , p o ) ] k ( r , t ) d t - Q(R,r,po)
j
+ -z — ?(R;R,/j ) zr o j
2
+
R
R(1 +
~V
2 *2
-oR(l-On
+
-
exp(^aRM,)2 c o s h ( a r / i 1 ) — i H 1 1 *
Q
-
R2
r
v:/T4) r
2 , l
R2 r^oi * r-R/l-/x2 % — - j exp(-aR/i Q )2 c o s h ( a r / i o ) H
x exp(-aR/z )2 sinh(ar/x )H r - R / W >,
(6.5.49)
where
Q(R,r,,o) - 2 f^ff
,/W;
5|r-R/l^7
exp(—aR/i ) cosh ( a r ^ )
"o as -=— = 5 ( x ) , a d e l t a f u n c t i o n , ax Now d i f f e r e n t i a t i n g e q u a t i o n ( 6 . 5 . 1 4 ) w i t h r e s p e c t t o /i , u s i n g 2 ° ( 6 . 5 . 4 7 ) and m u l t i p l y i n g i t by ( l - / i )/RA» , we o b t a i n 5
[ r ^ ( r ;R "o»
Io5 L - R ^ 2 a f [ ^ ( t ; R , M o ) ] k ( r , t ) d t 0 o o l-/x 2 +
0 2 LR/i
o
R(l~//) 2 *2 r /x o
p
o
r^ ol
X
Q(R,r,^)
* H r - R / l - u 2> > exp(-aR/i )2 c o s h ( a r / i )
n
aR(l-/j ) o " R2. oo-1 exp(-aRj* )2 sinh(aR/x )H r-R/l-yu r ~* J r u M, o u
+
a(l-M2)
+
(6.5.50)
294 From ( 6 . 5 . 1 4 ) ,
r e p l a c i n g fi dfi
J
u>
r?(r,R,/i ) - i = / f J-
0
Mx
^
b y /;..
R
Aft. k(r,t)dt
|
tf(t;R,/i L
0
JQ
+ — j
'o
exp(-aR/i1)
) M
l
* «^1 2 c o s M a r / ^ ) — j H/a-j-y 1
2> r R
(6.5.51)
In (6.5.51), the order of Integration has been interchanged. permissible as the integrands are non—negative.
This is
Further the validity
of the steps taken in (6.5.49), (6.5.50) and (6.5.51) can be justified from the theory of N-solutions [cf. Kho and Sen Furthermore 2 1+1* tr ? ( r ; R , / i
R/x
_R f
u>
rU"
2
+
J
0
-
1 <-
)>
a + —
a
ttSS ( t ; R , i j )
2 R/i o
° + SL 'a
( 6 . 5 . 1 4 ) , we may w r i t e
r?(r;R,/i
l1+fi + 2
1+M Ru 'o
from e q u a t i o n
£
Appendix B p.240].
R2 rMoi
o
)
t?(t;R,^
o
)
]} k ( r , t ) d t *
— - * exp(-aR/iQ)
2 cosh(ar/Jo)H
[T-R/1-P 21
(6.5.52)
Equations (6.5.49), (6.5.50), (6.5.51) and (6.5.52) lead us to the following homogeneous integral equation having the same kernel as the inhomogeneous auxiliary integral equation (6.5.14).
^1(r;R,/io) where
- ^ J
r J
o
^1(t;R,/iQ)k(r, t)dt,
(6.5.53)
295
fc
i<*'^V - k
+
r?(r;R,M )
I+P: ^
H*
^r£-
i*»t«;^0)i
[tf(r;R,uo)] + =- [rSf(r;R,M0) ]
R/I
.1
* f(R;R,/i ) **
M
— l
(6.5.54)
The homogeneous integral equation (6.5.53) corresponds to the nonhomogeneous auxiliary equation.
And for the auxiliary equation
(6.5.14) to have an unique solution 61(r;R,/io) = 0.
(6.5.55)
That is
a fg [rf(r.R.M0)] +
2 2 1+ "n a ^o R ^ ^ [^(r:R.%>l ~ ~ T trS(r:R,P0)] o o R/i o
1_
dp + — [r?(r;R,u )] - | ?(R;R,u ) r»(r;R,/i ) — - . **<> o 2 o JQ 1 n±
(6.5.56)
Multiplying (6.5.56) by
°fa. *
exp(-aR^) 2 cosh(ar/J ) H C-R/I-M 2 1
and integrating over the range (0,R) and inverting the order of integration on the right-hand side (which is permitted due to non—negative nature of the integrands). we have
296
r a
u
*
Cj exp(-aR/i) J 0 / l-/i
+
R,
o
S(M>Mo,R)
S(MlMo,R) + —
r
o
dr H
fr-R/CT
S(/I,AIO,R)
R/J
r1
= 5 P(R;R,/0 «£
fl^^'V
l+M
a
o
2 cosh(ar/i )
^i
(6.5.57)
Sdi.^.W - r J0
1
M-L
w h e r e S(/u,/i ,R) i s g i v e n b y ( 6 . 5 . 4 0 ) . Since 9
9
^o
o
IT(l-/i ) - r (l-/i
z
),
we h a v e *
9 _ R(l-M ) 2 * r /i
3M_
aR and
,
2 Rs M r * " ^ J u
3n
an Differentiating
(6.5.58)
(6.5.59)
( 6 . 5 . 4 0 ) w i t h r e s p e c t t o R and u s i n g
( 6 . 5 . 5 8 ) , we
obtain
3R S ( * . . V R >
= ca ? ( R ; R , M o )
-
CT/1-/I
[ 1 + exp(-2c*Rfi)]
S R/l-At
,R,A»
I + CT [ + «T J
^ 0
ft
^
J
0L
/j*aR/l-/i
L
2 cosh(ar/i*) | ^ ? ( r ; R , / i o ) dr H r - R / l - / i 2
exp(-aR/j)
2 cosh(ar/i*) | ^ ?(r;R,/io)
2 *2 r 2*2
r
2 cosh
exp(-aR/j)
r R f R ( i - /J 2 )
"J oJL
^ f exp(-aR/i)
°J LM J
<-
'IN' 1
- ^j
exp(-aR/i)
i.„ j
J
dr H | r - R , / l - / i 2
2 cosh(ar/i )5(r;R,/i
)dr H o
r-RA^Jl
297
QR(1
+
L
)
~ ^
[^r|exp(-aRM) J
r/i
^ft
2 s i n h ( a r / O S ( r ;R,/x ) d r H | r - R / l - j i 2 ] .
J
'
*■
(6.5.60) Differenting (6.5.40) with respect to /i, using (6.5.59) and 2 multiplying by (1-y* )/(R/i) , we have
^fcs<-VR> = a / l - / i 2 ? R / 1 - / I 2 R,M o
f R ri- M 2 J 0JL R/i „ 2
x
H^
*
R(l-/i 2 ) 2 *2 r fi
exp(—aR/i)
a(l- M 2 )l *i
r * / 2 ' 2 c o s h \i aR/1—p.
KUxp(-aR/i)
2 cosh(ar/i*)?(r;R,/i
)dr
H|r-R/l-//
**
*
*
x p ( - -aR^i) i> s i n h ( a r / i , f ( r ; R , / i
)dr H r-R/l-/*2
(6.5.61) And f u r t h e r from
(6.5.40)
- M £ - S(/i,M o > R) + J S ( ^ , M O , R )
-.n-***!i °f [- H J
0
L
*
exp(-aR/i) 2 cosh(arji ) y ( r ; R , / i ) d r H
R/i
Adding ( 6 . 5 . 5 7 ) ,
r-RAVJ. (6.5.62)
(6.5.60),
( 6 . 5 . 6 1 ) and ( 6 . 5 . 6 2 ) , we o b t a i n
298
_a_ so ,R) •"o
a
,R) + 3R S(M **o
2
<7^(R;R,/i
Now from
S(R;R,IJ
,R) 9/i S(M,■ " o
2
p 2 u2 R/x
-
a_
+
o
Hf*•"o ,R)
)
+
■K- +
KM," o ,R)
d/i
i r1
r
n
(6.5.63)
1 + exp(-2aR^)+ S ( / i , / j R) — L L ^ ->Q Pil
(6.5.14)
and ( 6 . 5 . 4 8 ) ,
we h a v e
)
= 1 l+exp(-2aR*z
ar )+=■ ? ( r , R , / z
r )dr
exp(-aR^i ) 2 c o s h a r p
* **il —- H
r.. /T?i ^V 1 -?]
1 d/z = l1-+ e x p ( - 2 a R / i Q )
R /z ^ _ — exp(-aR/j ) 2 c o s h ( a r / i ) ? ( r ; R , / x ^ 1 J 0 ^.
+ ^ '0
x H r-R/l-M,
l+exp(-2aR/j
r1
i ) + ,
S(/i 0
)dr
(6.5.64)
dM i u ,R) — 1
(6.5.64)
Therefore from (6.5.63) and (6.5.64), the integro—differential equation for S(fi,fi
d
3R
+
r
I=M!^ R/J
3/i
+
^
,R) can be written as
R^
^ o
3/i
i r
= a a l+exp(-2aRu )+ T L ° ^
J
o
_ o
^
■'
+
„ 2 2 Ru u o
^
d/J S(/i
-
M M 'o
ii r
S(M,/io,R)
i r1
d
^i"
u , R ) - — l+exp(-2aRii)+ ^ S ( t i . p . ,R) — z ^1-1 L J0 **!-
(6.5.65)
299 This integro—differential equation is to be solved under the initial condition S(n,fi
,0) = 0
for
0 < /x, ft
< 1.
(6.5.66)
The problem is suitable for numerical integration [cf. Bellman et (2) (251 alv ; and Rybicki'' '] . Once S(/i,/j ,R) is known, the emergent intensity I(R,/x) can be calculated form equation (6.5.41) for the case B.(r) = 0. When B1(r) * 0, an integro—differential equation for emergent intensity I(R,j*,/i ) can be derived involving S(fi,fi
,R) and under
appropriate initial condition it can be solved numerically.
[cf.
Kho adn Sen ( 1 5 ) p.236]. The extension of the method to the solution of diffuse reflection problem by an isotropic, non—coherent scattering, homogeneous spherica (19) medium can be done [Kho and Sen ] without much difficulty. It may (13) be mentioned here that Gruschinske and Ueno used a technique suggested by Bellman to obtain the numerical solution of Fredholm type of auxiliary integral equation for radiative transfer in spherical medium. Combined Operations
Method In Cylindrical
Medium :
For solving transfer problems in cylindrical medium, we follow the same basic procedures
We set up a non—homogeneous auxiliary
integral equation of transfer, seeking the solution of it in the form of Neumann series.
We define scattering and/or transmission functions
and derive the integro-differential equation for scattering and/or transmission functions in a form solvable numerically as initial value problems.
We outline below an application of the method to solve
transfer problems in cylindrical medium [Kho and Sen
].
For simplicity we consider a homogeneous, isotropically scattering, infinite cylindrical medium of sectional radius R.
300 The equation of transfer for diffuse radiation is given by [Chapter I equations(1.3.49) - (1.3.50)]
ai(r,r?,/i) ^ 1-AT 31. 3r r 3^
.
A1-IJ2
= -a[I(r,ij,/i)-»(r)],
(6.5.67)
where
*(r>
=
w
+1
Tk J
dr?
+1
J
.
Kr.ij./i) — 5 ^ - + B(r)
vC?
and
B(r) = B (r) '+ B, (r). o i
(6.5.68) (6.5.69)
I(r,r/,/i) is the specific intensity and ?(r) is the source function 77 and fj. axe the cosines of the angles ij> and 6 respectively [cf. Fig. 6.5.2] . a) — a /a o
and
0 < u> < 1. o
(6.5.70)
c*> is the albedo for single scattering, a and a are the scattering and attenuation coefficients, B (r) is the contribution to £(r) due to o reduced incident flux and B., (r) the cylindrical distribution of reduced incident flux and B.. (r) the cylindrical distribution of internal sources, if any.
Fig 6.5.2
301 Let TTF. be the net flux of radiation incident at the boundary surface at r - R and in the direction (-n v
by [cf.
equation
BQ(r)
(1.5.45)]
^[1]
r^o T*
*l
ex
p
—aRfi
,-u ) .
'o
Then B (r) is given
o
j/U
o
2 cosh
arjj
*>
L/VJ
r-R/l-/j2
,
oJ (6.5.71)
where H is the Heaviside unit step function.
„2,, 2, 2 , , *2 V R
Also
0<^,/i < 1, 'o 'o
(6.5.72)
-1 < r; < 1, o
Hence the source function is
»(r) = ^ J
di, J
I(r,r,,M)
a) F. +-
d/i
^7
5
+ B1(r)
exp^-aR/i o //l^ J22
aru coshl
o
^
o
"1
H r-R/l-/i2
(6.5.73)
The transfer equation (6.5.67) with (6.5.73) is to be solved under the boundary condition that the diffuse radiation in the inward direction at the outer boundary is zero, i.e.,
I(R,??,/*) = 0 ,
-1 < M < 0,
-1 < r? < 1.
(6.5.74)
The integral equation for transfer for this cylindrical system can be written as [cf. equation (1.5.41)]
302
•rf(r)
= u> / t S f ( t ) k ( r , t ) d t + /r"[B (r)+B ( r ) ] o J„ o 1
where B ( r ) i s g i v e n by ( 6 . 5 . 7 1 ) , w i t h [cf.
(6.5.75)
Chapter I , e q u a t i o n 1 . 5 . 4 ]
00
a2(rt)1/2 J
Ko(ary)Io(aty)dy,
t < r, (6.5.76)
k(r,t) = • CO
- a2(rt)1//2
f
K (aty)I (ary)dy,
t > r,
where I (x) and K (x) are the modified Bessel funct ions of the first and second kinds of order m. k(r,t) is non—negative and symmetric. (6.5.77)
k(r,t) - k(t,r). Auxiliary
Equation
:
Taking B ^ r ) = 0 and replacing ?(r) by f(r), such that
t(r) - ! 5 _ i J ( r ; R , v / i o ) ,
(6.5.78)
We get the auxiliary equation as
/ r ? ( r ; R , r ) o > / , o ) = WQ J
/t
_Rf^o"
/-[ *
f(t;R,r?o,^o)k(r,t)dt
exp -OTRM
/yd?
2 cosh
ar/i
H|r-R/l-^2
UvJ
(6.5.79) The s o l u t i o n of t h e a u x i l i a r y e q u a t i o n ( 6 . 5 . 7 9 ) can be w r i t t e n i n t h e form of a Neumann s e r i e s p r o v i d e d i t i s
convergent.
303 We w r i t e *(r;r,f,o,/io) =
R
'V\>
L r {*(t;R.»? 0 .M ))
2
R 1 fMo -aRfi //1-f) r lhf|eXp
A
| 2 cosh
arji
*^ c-R/1-^2
lyCT (6.5.80)
Here the operator R
-Vo
L ( f ( t ; R , r ? ,u )} d e f i n e d between 0 < t < R i s r o 'o
given by 1/2
r oL
[f t ; R , i , ,/i r o o
o
) -co
f(t;R,f, °J
n
k(r,t)dt, u ) o o 0 < u> < 1. (6.5.81) o
We write the N—solution of the auxiliary equation as
f(r;R,i7o,po) -
£ L™{g(t;R,7/i )}, (m - 0,1,2,...) m=0 R, n , u. o o
(6.5.82)
where
Rf"o r(t:R,r,qlMo) = t [ ^ l e : < p
atfi' -aRfi //l-ri''
\ / 'Nil
r-R/11-/*21
UvJ
(6.5.83) L is the mth iterate of the operator r
L and r 1
R.V .M ' 'o o
R.»/ .A o o (6.5.84)
L r (g V „ 0 ) , R,»?o>juo
Existence
And Uniqueness
Of N-Solution
Of The Auxiliary
Equation
The existence of N—solution of the auxiliary equation is essential.
We prove the following theorem.
304 Theorem : Let g(t;R,r; ,ft ) be a function such that
|g(t;R,tj ,/i ) | < C (finite non—negative constant)
(6.5.85)
for 0 < t < R, then Lr{g(t;R, V /i o ))| <
cxm
(6.5.86)
R.f o ,P o
where m = 0,1,2... and X
(6.5.87)
Lrl(l)}|
=
R
'Vo
Then the series Lr(g(t;R,r?o,/io)) m=0 R, n , u, o o
(6.5.88)
is uniformly convergent, if (6.5.89)
0 < X < 1. From (6.5.81) and (6.5.85)
L {g(t;R,i7 /* )] < CX R,i7 ,n r o o
(6.5.90)
where x i s g i v e n by ( 6 . 5 . 8 7 ) . Again from ( 6 . 5 . 8 7 ) , we have
L {1}
o
As k ( r , t )
o
■
R
r
t
^ / 2
° J,
i s non—negative and 0 < u
< 1,
k(r,t)dt.
(6.5.91)
305
rR f t 1 1 / 2
°*"oJ0U]
rR f t l
1/2
k(r,t)dt S J o [ I
Hence
k(r,t)dt.
(6.5.92)
1/2
r1
k(r,t)dt.
(6.5.93)
Now writing k(r,t) explicitly, interchanging the order of integration and remembering that
f x I (x)dx - x I (x)
I x K (x)dx = - x K (x)
(6.5.94)
KQ(x)I1(x) + K1(x)IQ(x) = ^
and we get
R f >l/2 » J l-| I k(r,t)dt = 1 - aR J K1(aRy)Io(ary) ^
.
(6.5.95)
Then from (6.3.93) we have CO
0 < X =S 1 - aR J
K 1 (aRy)I o ( Q ry) ^
(6.5.96)
However I (x) and K (x) are non—negative and I (x)is an increasing m m m function for any order of m, so the second term of the RHS of (6.5.96) is positive.
Then from (6.5.94), we have
00
QR
CO
CO
[ K 1 (aRy)I o (ary)^ < aR J K 1 (aRy)I o (aRy)^ < J
^
= 1.
(6.5.97)
Thus 00
0 < aR [ K (aRy)IQ(ary) 2Z < 1.
(6.5.98)
306 Hence the relation (6.5.89) follows, 0 < x < 1-
i.e.
By iteration, the result (6.5.90) can be generalised to (6.5.86) and we have
* m=0IR, n
L r {g(t;R,f,o,/io)) ,u o o
L r (g(t;R, V M 0 ) m=0 R,n , u o o
%c I xm -£-,
if o < x < l.
X
m=0
(6.5.99)
L r (g(t;R,,o,/,o) m=0 R,n ,u o o
(6.5.100)
1-X
This proves that the series is uniformly convergent and that the N—solution of the auxiliary equation exists.
Emergent Intensity
:
From direct integration of transfer equation, the outward and inward intensities of the diffuse radiation are given by
*
r
I(r,i?,M ) =
a ?(t)exp
+ [
*
/l-i?
^'/1-r?
a
?(t)exp
a dt
a dt
* * (r/j +t/i')
/1-r?
(6.5.101)
t* Wl-ri
and I(r,r7,-^ ) =
»(t)exp
/v
(-r/i +t/i')
a dt
'/£?
(6.5.102)
307 where rj is the cosine of the angle that the radiation pencil makes with the vertical (cf. Fig. 6.5.2) and n, n
and /i' are the cosine of
the angles made by the projection of the pencil on the horizontal plane with the outward drawn normal at R, r and t respectively.
Then
2 2 2 2 2 *2 R^(l-/) = tZ(l-M' ) = r (1-M *),
(6.5.103)
0 < H, H , H' < 1.
(6.5.104)
Hence the emergent intensity is given by [cf. Kho and Sen
r I(R,rj,/i) -
-o&p/A-ri1
2 cosh
r
ar/i "I
Vi
?(r)exp
V
adr */i
p.157]
H
2 (6.5.104)
where ?(r) is the source function and H is the Heaviside unit step function. In particular, when B.. (r) = 0,
K.
at.
I(R,rj,M) = —
J
2 Sf(r;R,r?o,^o)exp - a R / i / / l - f j
2 cosh
ar/x I adr W.
V
x H r-R/l-^2
*/
2
(6.5.105)
where ?(r;R,r7 ,/u ) is given by (6.5.79)
The Scattering Scattering
Function
Function
And The Integro-differential
Equation
:
We define the scattering function S(/i,r?;/i ,r) ;R) as
For The
308
S(,n,ri;no,rio;K)
-
=a
rR~ f -aRn/A-r,2 ? ( r ; R , n ,/J. )exp J0 ° ° I X
2 cosh
ar/i
*^
Vl-r, 2J
2 ** H r-n/l-/* d r ,
*-u ■> ■fi ■■
r
(6.5.106)
I
so that
L
K
■ ^ -.v>/*^
F. r. l
=
"KM**!*!*_i5
o*
(6.5.107)
'
4/i/l-r; when B 1 (r) = 0. The reciprocity relation
S(/io>r,o;M,»?;R) i,n ,R) = S(Ai,»);Aio,r?o;R)
(6.5.108)
follows from (6.5.77) [cf. Kho and S e n ( 1 5 ) ] . The next step is to build a homogeneous Fredholm integral equation corresponding to the non—homogeneous auxiliary integral equation (6.5.79) and having the same kernel.
Then in order that the
non—homogeneous integral equation has unique solution, the homogeneous (35) integral equation must have a trivial solution [Tricomi , p.64].
Let *(r R
"o "o>
- u> o
r
$(t R .M Q )k(r t ) d t "o
be such a homogeneous Fredholm integral equation.
(6.5.109)
Then $ will be
constructed in a manner similar to that in the case of spherical media (cf. Kho and Sen
) making use of the superposition principle.
the auxiliary equation (6.5.79), we obtain
From
369
M ^ :o a
l
a_
a: 3R
RM
1
3M
o
D
^o
R/i
r1
-*(R;R,i,o,Po) J
g
+
2
i A
/r?(r;R,r;o,/jo)
=■ 2
M O /I-H 0 dr?
i
rL
d/J
— - ^ J /^(r;R^1)Ml) 2 J0
'°/
[cf.
7^2
( 6 . 5 . 1 1 0 ) , we h a v e u s e d t h e e x p r e s s i o n f o r k ( r , R )
Kho a n d S e n
(16)
=
-
7T
given by,
,p.164-166)].
1/2 L/2 k(r,R)
(6.5.110)
j .
l-i|
In deriving
i
(fj
1
r i
drj,
1
'i r
——— j
'°/iV°
exp
/—2
f
-a&4i1/Yl~y1
r
2 cosh
°rMl /1-'71
1 "i " ("l"!-/ -^
1
d/i
l
H-i/l-ii^
(6.5.111)
j
w h e r e 0 < /*i , /J-I ^ 1> —1 ^ »J, < 1 . Multiplying equation
(6.5.110)
^* expf-aR^/A-f?2
i n t e g r a t i n g over
by
2 cosh
°r/i
XV
1
H|r-R/l-/i2
(0,R) and i n v e r t i n g t h e o r d e r of i n t e g r a t i o n on t h e
RHS, we g e t
of J
H; e x p f - ^ R M / / ^ 0 /i
+ -
o
a a« o *"\>
RM„
l . 2
RM
2 coshf-^- |g *(riR.r^./^dr
X
o_ / 2
2
S(M,'?;MO1'?0,R)
H ( r - R / C• ? )
310
5*
O
where S(p,f),/J A g a i n from
(6.5.106),
'
a
I
R/i
/
RM
=
CT?(R;R,n
o
-
- s s(^,»7;/i o ,»? 0 ;R)
£7 }]
(6.5.113)
and ( 6 . 5 . 1 1 3 )
R/i 3 M o o
8U
I X
=■
4-
.r J>- . _ i ^ r
+ a
a
,/i ) l1 + e x p I —2aR^/
Adding ( 6 . 5 . 1 1 2 )
3R
M
(6.5.112)
r
(6.5.106),
2 cosh _ ? ^ J
- 5 e x p [-oR^/yO-^ 0 /*
+ a?(R;R,r/
l /—
1 1
we h a v e
2 1—p, D 2
3 3iu
1—/J
d M
, T7 , R ) i s g i v e n b y
2
"a3 a 3R
-
r 1 dr>\ r1 O J. / =• J „ '0 / 2 J0
+
2
_
Z~~22
,u ) 1 1+e o
„ 2 2 R/i /J
R
c
S(M.'/.M0.'/0;R)
-/ J
M/T
+ *
J
0
Z
-
?
J
S(/i,ij,/* 0
1
ly ;R) L
/—2
(6.5.114) From (6.5.79) and (6.5.111), we have
? ( R ; R , » 7 o , M o ) = 1+exp
r / 2 1 I f, 11 -2aRu//lV + i L o o J
d a ,f?i
1 l rr iS(/* , « , , ;R) 1 1 o o ffJ()^-^J0
a/1 * * 1
l 1-. / — j
(6.5.115)
311 Hence from (6.5.114), we have the i n t e g r o - d i f f e r e n t i a l the s c a t t e r i n g
function 2
i
3
+ IZEL
SR '
RM ^
equation for
+
^oJ_
•
RMQ 3M 0 "
2
2
^ o
1
R+
R/i2M2
a
■'—L- + L
/i/l-r/
i 1 J r\
n vl-y
x S(/i,T7,/io,*?o;R)
p[-
= a1 1 + e x p -2QR/I
x
o
v£7 I*
/
1 + e x p — 2aRiz i/VlV
d>7.
K°/
1-f
J:
d/x s
(/*]_. I 1 » M „ . » 7 „ ; R )
°
°
7^2
i r1
d d " i r1 "i — i S ( / i r ) , JJ xM l > f ? 1 ; R ) 1 0 / ^ 0 * f—2>
*lri
r
l
(6.5.116) This i s to be solved numerically under the i n i t i a l
S(/i,tj;/i
o
,r/ ;0) = 0, o
0 < u, u, < 1, o
condition
- 1 < IJ , r? < 1. o
(6.5.117)
Once the scattering function is known, the emergent intensity I(R,r7,it) can be calculated from (6.5.107) for the case B.. (r) = 0 .
As in the
case of spherical medium, the integro-differential equation for emergent intensity can be derived for the case B.. (r) i* 0 and can be solved under appropriate initial condition [Kho and Sen
,
p.161-163]. Though the examples given in the two earlier articles concern the use of combined operations method for solving diffuse reflection problems in homogeneous, coherently scattering, spherical and cylindrical media, the method can be aptly utilised to solve diffuse reflection and transmission problems in inhomogeneous, non—coherent
312 scattering spherical and cylindrical media with frequency dependent and frequency independent source functions [Kho and Sen
Sen
].
We quote below some of the basic results obtained [Kho and (19) , p.237-255] for the problem of diffuse reflection and
transmission for an inhomogeneous, isotropically and non—coherently scattering, spherical shell medium with frequency—dependent source function.
For the details of the calculation, the readers may refer to (19) Kho and Sen^ ' . An inhomogeneous spherical shell medium of inner and outer radii r and R is considered (0 < r. < R) . The equation of radiative transfer for diffuse radiation for inhomogeneous, isotropically and non—coherently scattering spherical medium with frequency—dependent source function is given by [Chapter I equation (1.3.30)-(l.3.32)]. 2 P- 9 I ( g ; " ' M ) + -=jr~ + §£ Kr,*,fO - -o(r)v(i/)[I) = a(r)
(6.5.119)
tp{v) being the absorption profile. The other symbols have their usual meanings. The source function ?(r,i/) is given by [cf. equation (1.3.33)]
»(r,v) - 2 p . J
R (
^^
)
dv' J
Kr.i/'.jOd/t. + g(r,„);
(6.5.120)
/1 O \
where R(y',i/) is the redistribution function [cf Kho and Sen 0 < co(r) < 1, and g(r,i/) = B (r,i/) + B (r) .
] (6.5.121)
313 B (r,i/) accounts for the contribution from the reduced incident ° . radiation I (i) arising from an external source and given by
«>(r)
Bo(r,v)
C^-j"1'1''—»*
(6.5.122)
B 1 (r) is the contribution from internal sources, if any. One of the standard boundary conditions under which equation (6.5.118) is solved is given by [cf. Chapter I, equation (1.4.10)].
(6.5.123)
I(R,^,- M ) = 0 and I(r1,v,+fi)
(6.5.124)
= 7l(r1,i/,-/i)
for 0 £ v S «,
0 < /i < 1.
7 gives the measure of reflectivity of the inner shell (0 < 7 < 1 ) . In particular, if the incident radiation is in the form of a conical monochromatic flux JTF.(I/ ) incident at r - R at an angle cos °
1 0
I (R,i/. ,/i-),
u , o
intensity of the radiation incident at the outer surface
r = R, is given by I^R.i/^) -
F,(f ) 1 «(i/o-H/1)«(po-fi1) 2°
(6.5.125)
where S is the Dirac delta function. Hence [cf. Kho adn S e n ^ 1 9 ) p.240, equations (2.14) - (2.16)]
B o (r,,)
u(r)
x U
,2 - R(v
f
-
v)
1
n,
— A — d"i i I (R,»/n ,*»-,) -1
(R^.r/i^H
^-A?
+ 7 ^ (R/^.rj^) dji
(6.5.126)
314 where /i. , /J1 , M-I are the cosines of the angles made by the incident ray at R, r and r. respectively;
0 < MXi /*___. M L ^ 1-
Also
R 2 (l-^) = r2(l-M*2) = r 2 (l-M 2 ),
(6.5.127)
N. (P1,P{) - exp{^p(i/1)Z(p1,pp) + exp{-
M^ (Pj_.Pi) = exp[^p(1y1)(Z(p1,r1Ai1) + Z(p^ . r ^ ) ) ]Hr . - R / W (6.5.129) a(r)ds,
(6.5.130)
Z(p.p') = Z(-p'.-p),
(6.5.131)
Z(p,p') = and
V
Fig 6.5.3
ds is an element of distance in the direction of transfer, and p, p' are the distances measured from the normal to this direction.
315 Integral equation for the source function ?(r,i/) is given by [cf Kho and Sen ( 1 9 ) equations (2.23)-(2.25)]
r?(r,i/)
w(r) J
0
di>' I t?(t,i/')k(r,t,i/,i/')a(t)dt + rg(r,i/), J r1 (6.5.132)
where the kernel k ( r , t , v , v ' ) is given by .1 k
|
N^,(rM ,t/i')H(r-t) 0
+ N , (tp'.r/i )H(t-r)H
1 -
d/j'
+ 7M . (r/i ,t/i')
*'
(6.5.133) where M
and N
are given by (6.5.128) and (6.5.129).
The kernel
k ( r , t , v , v ' ) is non—negative and symmetric, i.e. (6.5.134)
k(r,t,i/,*v') = k(t,r,i/,i/').
The auxiliary equation was found out for the case B.. (r) = 0, with I (R,j/_ ,u.) taken as o 1 1
V ^ i ' V "2 sovV^v-V'
(6.5.135)
(where 5 is the Dirac's 5-function) and ?(r,i/) and 9(r,i/) are related as 9(r,i/) = y ^
f(r,i/;R,i; ,/i ).
The auxiliary equation reads as [cf. Kho and Sen - (2.30)]
(6.5.136) (19)
equations (2.28)
316
r ? ( ir , i / , R , i /
,/i ) - i
o o
r°° [ dv,
2 J„
R (
r ;
t£(t,i/';R,i/
1 J_
y>
R2 r
M.
,/x ) k ( r , t , i / , i / ' ) a ( t ) d t
o o
N (Ryu , r / i * ) H i/ ' ' o ' ' o o
r-R/l-/i2
+ 7M (R/i , r u ) 1/ o o o
(6.5.137)
where * 0 < u , /1 < 1 'o o
and
2 2 2 *2 R (l-/i ) = r (l-/i ) o o
<7(t) = u ( t ) a ( t ) .
(6.5.138)
(6.5.139)
It was shown that N-solution of the auxiliary equation (6.5.137) exists and it is unique when the homogeneous integral equation corresponding to the non—homogeneous auxiliary integral equation (6.5.137) has only trivial solution. From the auxiliary integral equation (6.5.137), the following homogeneous integral equation corresponding to it is obtained namely [cf. Kho and Sen*' ^ equation (3.10) p.244]. a 1 ,r>°° -R r (r,i/,R,i/ ,/i ) = i[ di/'| * (t,i/';R,i/ ,/i )k(r, t ,v ,v' )a(t)dt
(6.5.140)
and for the uniqueness of N—solution of auxiliary equation (6.5.137). (6.5.140) will have only trivial solution i.e,
$„(r,iv;R,i/ ,u ) = 0. z 0 0
(6.5.141)
If further, we consider the core of the shell to be perfectly absorbing ( 7 = 0 ) , the equation (6.5.141) implies [cf. Kho and Sen (19) p.243-246.] that
317
\a
[an
+
iV
r) R/i 3/^ +
= er(R) R(i/
h
1 2
K R
i/) + i
2
r)
"o
3u
2
5-4 + a(R)
o
.1
f R(i/' ,i^)d^'
S(i/',t>
o'j
,u.,u
;R)
s(v,v
,Mi^ 0 ;R)
d/J 1 d„
j^.^.^R) — ~ 1
CO
Jo
R(i/
♦ 1 j *.
o
i»
)di/'
i;
S(i/
1/'
,At " l R)
[ R(i/» ,i/')di/' f S O "
au p/ J
o
11
o
lnl rr M n JQ
F
l
(6 5 142) where S (v ,v
,y. ,fi
R ;R) ==
i/;R,
o■ * . > ^
t*
N (R/j,r/i*)H r-R/l-/i2 ]a(r)dr i. J
(6 5 143) and 2,, * 2 , 2,, 2, R ( 1 - M ) •= r (1-ji )
(6 5 144)
The recip rocity relation (6 5 145)
S(i/,f ,u,i» ;R) == S(i/ ,v,/i ,u;R) o o
for the scattering function holds only when the recip rocity relation (6 5 146)
R(v" ,») = R(i/,i/') for the redistribution function holds. Given the redistribution function R(i/'
•0 and
the abso rption profile
¥>(") . the equation (6.5 .142) is solved numerically for the scattering funct Lon s o . f
,A*.At ;R) under the initial condition
0
for
0 < /i,
(l S i , 0
0
1
(6 5 147)
= 0
0 < i/,
<
J/
O
00
318 Once the scattering function S(i/,i/ ,/i,/i ;R) is known, the emergent intensity I(R,i>,/i) can be intensity I(R,i/,/i) can be equation equation (3.5) (3.5) p.243] p.243] the the °° I(R,i/,/0 = j - J" iu1
° from ° [cf. Kho and Sen (19), calculated calculated from [cf. Kho and Sen , relation relation
1
j
Io(R,v,^1)S(i/,i/1,/i,/x1;R)d^1, 0 < /* £ 1. (6.5.148)
We have demonstrated above that the combined operations method is adaptable for the solution of transfer problems in curved geometry when the source function is frequency dependent. It has also been used to ' (20) tackle problems of moving medium in curved geometry [Kho and Sen ].
319 References 1.
Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions (Ninth Printing), 1970.
2.
Bellman, R. , Kagiwada, H, Kalaba, R. and Ueno, S., RM-5402-PRRand External Publ. 1966.
3.
Busbridge, I. W., Quart. Jour. Math (Oxford), 6, 218, 1955.
4.
Busbridge, I. W., Mon. Not. Roy Astron. Soc. London, 115, 521, 1955, 116, 304, 1956, 117, 516, 1957.
5.
Busbridge, I. W., The Mathematics of Radiative Transfer, Cambridge University Press, 1960.
6.
Busbridge, I. W., Ap. J. 133, 198, 1961.
7.
Case, K. M. and Zweifel, P. F., Linear Transport Theory, Addison, Wesley, London, 1967.
8.
Cassell, J. S., Proc. Camb. Phil. Soc. 64, 711, 1968.
9.
Cuperman, S., Engelmann, F. and Oxenius, J. Physics of Fluids, £>, 108, 1963.
10.
Davison, B. and Sykes, J. B., Neutron Transport Theory, Clarendon Press, Oxford 1958.
11.
Germogenova, T. A., Astrofizica, 2, 251, 1966.
12.
Grandjean, P. and Siewert, C.E., Nucl. Sci. Engy 69_, 156, 1979.
13.
Gruschinske J. and Ueno, S., J. Quant. Spectrosc. Rad. Transfer 11
14.
Hunt, G. E., Siam. J. Appl. Math. 16, 1255, 1968.
15.
Kho, T. H. and Sen, K. K., Astrophys. Sp. Sc. 14, 223, 1971.
16.
Kho, T. H. and Sen, K. K., Astrophys. Sp. Sc. 16, 151, 1972.
17.
Kho, T. H. and Sen, K. K., Astrophys. Sp. Sc. 18, 363, 1972.
18..
Kho, T. H. and Sen, K. K. , Astrophys. Sp. Sc. 21,
19.
Kho, T. H. and Sen, K. K., Astrophys. Sp. Sc. 21, 237. 1973.
20.
Kho, T. H. and Sen, K. K., Astrophys. Sp. Sc. 2j), 247, 1973.
21.
Leong, T. K. and Sen, K. K., Publ. Astron. Soc. Japan 23, 99,
641, 1971.
Publ. Astro. Soc. Japan 22, 365, 1971.
39, 1973.
1971. 22.
Leong, T. K. and Sen, K. K., Publ. Astron. Soc. Japan 2J3, 247, 1971.
320 23.
Mikhlin, S. G., Integral Equations, International Series of monographs in Pure and Applied Mathematics, Pergemon Press, London, 1954.
24.
Mitsis, G. J., "Transport solutions to the monoenergetic critical problem", ANL-6787, Argonne National Lab. 1963.
25.
Rybicki, G. B., J. Comput. Phys. 6, 131, 1970.
26.
Siewert, C. E., Astrophysics. Sp. Sc. 58, 131, 1978.
J. Quant.
Spectrosc. Rad. Transfer. 21, 35, 1979. 27.
Siewert, C. E. and Benoist, P., Nucl. Sci. Engng. S9_, 161, 1979.
28.
Siewert, C. E. and Grandjean, P., Nucl. Sci. Engng. 70, 96, 1977
29.
Siewert, C. E. and Maiorino, J. R., J. Quant. Spectrosc. Rad. Transfer 12, 435, 1979.
30.
Siewert, C. E. and Thomas Jr. J. R., Nucl. Sci. Engng. 87, 107, 1984.
31.
Siewert, C. E. and Thomas, Jr. J. R., J. Quant. Spectrosc. Rad. Transfer 34, 59, 1985. Nucl. Sci. Engng £7, 107, 1984. IMA. J. Appl. Math. 34, 323, 1985.
32.
Thynell, S. T. and Ozisik, M. N., J. Quant. Spectrosc. Rad. Transfer
35, 349, 1986; 36, 497, 1986.
33.
Smith, M. G., Proc. Camb. Phil. Soc. 60, 105, 1964.
34.
Smith, M. G., Proc. Camb. Phil. Soc. 6JL, 923, 1965.
35.
Tricomi, F. G., Integral Equations, Interscience Publishers. Inc.
36.
Uesugi, A., Publ. Astron. Soc. Japan, 14, 102, 1962.
37.
Watson, G. N., A treatise on the theory fo Bessel functions (2nd-
New York, 1957.
edition)
1980.
321 Chapter VII Numerical Methods for Transfer Problems in Spherical Geometry
7.1
Introduction Effective numerical methods to solve transfer problems have been
recently documented in the two books edited by Kalkofen
. As
the authors described the methods in great detail, we will only attempt to give an overview of the methods here.
The interested
reader is well advised to consult these treatises edited by Kalkofen for further details.
We shall take the model atmosphere to be having
spherical symmetry with the star at the centre.
The atmosphere is
assumed to be grey and scattering anisotropically with a phase function p(r,/j,/j').
The relevant equation of transfer for diffuse
radiation is [cf. equation (1.3.28)]
" ^F^
+
^
^ I j 1 ^ - - W I K r , . ) - »(r,,)],
(7.1.1)
where the source function +1 *(r,ji) - 2^2- [ p(r,fi./i')I(r,|i')«^' + [l-«
(7.1.2)
a is the attenuation coefficient and w(r) is the albedo for single scattering.
The transfer equation (7.1.1) is to be solved under
appropriate boundary conditions. 7.2
Direct method using impact parameter Let the spherically symmetric shell atmosphere be bounded between
r the core radius, and R the outer radius. We define the impact c parameter p as the perpendicular distance of a ray from a parallel ray passing through the centre of the atmosphere.
These rays would be the
directions of a distant observer of the star.
The other coordinate z
322 is defined as the distance along the ray from the centre of the atmosphere [see Fig 7.1.1).
The coordinates (p,z) are related to the
polar coordinates (r,0) as follows z = r cos
p = r s in 8 ,
, 2 2,1/2 r = (p +z ) ' .
(7.2.1)
Fig 7.1.1
Instead of writing the transfer equation in polar coordinates as in (7.1.1) we can also write it in (p,z) coordinates.
If the impact
parameter p is fixed then the transfer equation along the fixed direction of the ray can be written as +
ai-(T.p) 8T
f(T,p) - Kr.p)
(7.2.2)
where the + and — sign refer, respectively to radiation toward and away from the external observer.
The optical depth r along the ray is
defined by dr(r,p) = —a dz. As the equation (7.2.2) is like the plane—parallel situation, we introduce the mean intensity—like and flux—like variables u(r,p), v(r,p) as follows :
323 u(r,p) = i [I+(r,p) + I (r,p)]
(7.2.3a)
v(r,p) - i[I+(r,p) - I~(r,p)].
(7.2.3b)
Forming the sum and difference of equations (7.2.2) we obtain
3r 0T 3u 37 = VThe above two equations can be combined to obtain the second order differential equation ,2 2-S - u - *. 3r
(7.2.4)
We solve the above equation subject to the following boundary conditions : At the upper boundary r - R, I 3u -3 3T
= 0 , which can be written as ,
U(T
max
. ,p)
r(R
where
T =— max J
2
-P2)1/2
a dz.
The inner boundary condition depends on whether the ray intersects the core shell r = r or misses it and intersects the central plane z = 0. c In the latter case we expect from symmetry considerations that In the latter case we expect from symmetry considerations that v = 0
v= 0
—»?-0.
In the former case we set u = v taking that the core is empty.
i.e.
In the former case we set u = v taking that the core is empty. du
i.e.
■=- -
u.
324 The second order form of the transfer equation (7.2.4), first derived (2) by Feautrier , is very suitable for numerical computation. A difference equation is obtained by discretization of all the (2) Feautrier suggested that the second order differential
variables.
d2u. operator =■ be replaced by using the relation dr d2u.
—2 -
(a u
i i-i
+
Vi+ v w
(7 2 5)
-°«
- -
dr where the coefficients a, b, c can be selected so that the equation * (7.2.5) is exact for any quadratic variation of u. with r As the (7.2.5) is exact for any quadratic variation of u. with r As the source term 5 usually contains the scattering integral, which involves an integration over the angles, we need to replace this integral by an angular—quadrature formula.
If m is the number of rays that hit a
typical shell of radius r, we define the m—vector U. = (u. ■ u. .;...; u. ) ~i i,l i,2 l,m containing all the angle components of the intensity at a single depth i, and the m—vector B. = (B. ■ ...; B. ) ~i 1,1 i,m containing the angle components of the thermal source term at any single depth i.
With the above discretization, the transfer equation
(7.2.4) together with the boundary conditions can be written as a block tridiagonal system of the form
- AHi_l + BHi - C H i+ i - Sf where the matrices A, B, C are all of dimension (mxm).
(7.2.6) In this
Feautrier scheme, the matrix B is full but the matrices A and C are diagonal. Later Auer [Kalkofen (4) ,p.237] suggested a fourth order d2u accurate scheme to replace the second order derivative —=■. AT
He
325 proposed that the coefficients a, b, c, d, e, f, g be obtained from the relation
Wi
+
Vi
+ c u
i i+i
instead of (7.2.5).
+ e
2 d u._
i ^¥
+ f
2 d u.
i —5
AT
+
AT
2 d u
.
*i — V - °-
(72 7)
-
AT
This scheme still leads to a block tridiagonal
system but the matrices A and C are now full. solved by a Gaussian elimination scheme.
The system (7.2.6) is
It may be remarked that the
above scheme can be extended to non—grey atmospheres by replacing the vectors U., B., to represent angle— frequency components of the radiation at a single point instead of just the angle components. 7.3
Method using moments and variable Eddlngton factors for isotroplc scattering Direct methods described in the previous article can become
unwieldy due to the large order of the matrices arising in the discretization processes.
Taking angular moments of the transfer
equation will reduce the number of independent variables and discretization may become possible.
Following the success of using
Eddington—type relations to reduce the number of dependent moment variables the following method has been proposed by Hummer and (3) Rybicki for the sphei spherical medium. Introducing the radial optical depth T by the relation AT
= —a dr,
equation (7.1.1) can be rewritten in the form
3i + iVai = dr
an
I
_?
d/x
(7.3.1)
The first two angular moments of equation (7.3.1), in the standard notation, are jjj: (r2H) « r2(J-Sf)
(7.3.2)
^ - — (3K-J) = H. dr ar
(7.3.3) '
326 If we set K/J = f, the Eddington factor, then equation (7.3.3) becomes d dr
(fJ)
_ (3f-DJ _ ar
(7.3.4)
H
To mimic the plane-parallel case, a sphericity factor q is now introduced by the relation
1* q dr i.e.
(fqJ) v u
in q - J
_ j . (fj) dr
gtm
(7.3.5)
ar
[(3f-l)/r'f]dr'.
(7.3.6)
r c Then equation (7.3.4) reduces to *- (fqj) - qH. Substituting this in (7.3.2) we have 1 d_ q AT q df lq
i f -dra ? ^ " n
-qf- <J-?>
Defining a new variable X by the relation dX = ^_ dr, we obtain r 2 A dX
4 [fqJ] - jr [J-^]q
(7.3.7)
For the isotropic case 9 — wJ + (l-a>)B. The equation (7.3.7) being a second order differential equation in J we can discretize the spherical coordinate X and solve it either by the Feautrier or modified Auer scheme.
But to carry out the scheme we must know
apriori the Eddington factor f, and to know this factor we must know the moments J and K. iterative scheme.
To overcome this dilemma, one adapts an
First an initial guess for f is made and equation
(7.3.7) is solved to obtain J and hence the source function 9. Knowing the source function 5 there are fast methods to obtain the
327 radiation intensity as the system is now uncoupled.
One method Is to
use the direct impact parameter scheme described in §7.1. radiation intensity Is computed, this knowledge can update the initially guessed Eddington factor f.
Once this
now be used to
The process is
repeated till a solution of desired accuracy in obtained.
As the
updating procedure requires the solution of the transfer equation for a known source function 9, one can develop a finite difference scheme of the transfer equation (7.1.1) in the original polar coordinates (r,/j) itself.
Equation (7.1.1) is rewritten in the form
2 3 M i£*-|! + | |- f(l-/x2)I] - -«[I-»]. 3(r )
(7.3.8)
Writing (7.3.8) for ±n and taking the sum and difference of these equations, we have
j ^ ^3 i rH dfil V w - ^ M I
(7.3.9)
3(r )
3 / i
^ l
5(r3)
+
i§- [
(7.3.10)
"
where u, v are as defined in equations (7.2.3) The angular derivatives, like all derivatives, are unstable computationally.
To overcome this difficulty equations (7.3.9) and
(7.3.10) are integrated over the cell [/*•_■>/9, 'i>j_-i/o]> where
M
i+l/2
= M
i-l/2
+ W
-j '
j = 1,2
m
(7.3.11) fi.,
and W. is the quadrature weight associated with the angular point m being the number of quadrature points. 0.
We also set /i_1 .. — p
These yield the following equations [cf. Mihalas and Mihalas
p.383 equations (83.100) and (83.104)]
.
=
328 9(r2v) ^ - + ±
3W.u. 1 3
3
r
a(r )
(I1
,
1
2 V " M J + V 2 j+l/2
-b - v
,}
J-l/2 VJ-l/2
(7.3.12)
W.a (£-u.). =w 3
]
and S(ru.)
)-
2
3W.ju 33
L
3(r' )
W 0. _ x > ! + — — J — u. + r j r
r
h+1/2
, 1
2 U " "1+1/2 i + l/2
, 2 - /•"j-l/2 ^ J - l / 2
U
j-l/2
= -a[W.u.v.1 3 J 3
(7.3.13)
The variables u j ± 1 / 2 , v j ± 1 / 2 are replaced in terms of their nodal values by a linear spline approximation.
The resulting differential
equation in the space coordinate r can be discretized to obtain a tridiagonal system [see Mihalas and Mihalas
p.385].
The present
discrete space scheme is economical compared to the impact parameter scheme but unlike the latter it does not use a true formal solution of the transfer equation. 7.A
The cell method using reflection and transmission functions Reflection and Transmission functions have been used to study
transfer problems as they represent physical processes that are natural in transfer theory.
In the cell method by Peraiah
and
his co-workers these functions are first computed across suitably defined cells and later computed over the whole medium using star products.
To describe the basic theory, we consider a cell with
boundaries n and n+1 (cf. Fig 7.4.1). boundaries are I
and I
1.
The specific intensities at the
We distinguish between the incoming and
outgoing intensities by the usual notation 1 , 1 .
In the spherical
case we introduce the variable U defined by U(r,ji) = 4?rr I(r,/j) to replace the intensity variable. defined in (7.2.3).
We note that U is not the same as u
329
Fig.
7.4.1
If /i.(l < ] < m) are the angular quadrature points we define the m—vectors.
Hn=
(U
n <*1>
"
H n = < U n
U
n
^
nV >
at each cell boundary n. We assume that there are certain linear operators which represent reflection and transmission of the incident radiation.
Let t(n+l,n)
represent the transmission at the boundary n+1 after incidence at the boundary n and interaction with the cell.
Let r (n,n+l) represent the
reflection at the boundary n+1 after incidence at the boundary n+1 and interaction within the cell.
Then the emerging radiation can be
obtained from the following relations : — U + .. = t(n+l,n)U+ + r(n,n+l)U~ , + E + (n+l,n) ~n+l — ~n — ~n+l — U~~ = r(n+l,n)U+ + t(n,n+l)U~ , + S~ (n+l,n) —n ~ ~n — —n+1 — where 2 , 2
are the internal source terms within the cell.
relations can be written in a compact form as follows :
These
330
u+
~n+l
= S(n,n+1) )
~n
+ 2 (n,n+l) ,
(7.4.1)
~n
where
S(n,n+1)
t(n+l,n)
r(n,n+l)
r(n+l,n)
t(n,n+l)
(7.4.2)
The relation in equation (7.4.1) is called the interaction principle. To facilitate the computation of emerging intensities between two adjacent cells, the star product is introduced.
The interaction
principle in the adjacent cell with boundaries (n+l,n+2) will be given by
c2 Eliminating U
S(n+l,n+2)
. ,U
~n+l
y; + 2
+ 2(n+l,n+2).
(7.4.3)
.. from equations (7.4.1) and (7.4.3) we can
obtain an interaction principle for the cell (n,n+2).
Writing this
interaction principle in the form
' C2' u ~n
= S(n,n+2)
U -n
+ S(n,n+2)
(7.4.4)
-n+2
we can obtain the elements of S(n,n+2) in terms of the elements of S(n,n+1) and S(n+l,n+2).
The new matrix S(n,n+2) is called the star
product of the matrices S(n,n+1) and S(n+l,n+2) and written S(n,n+2) = S(n,n+1) * S(n+l,n+2). To calculate the radiation field in the medium, the transfer equation is used to define the reflection and transmission operators in each cell.
Then the star product and boundary conditions are used to sweep
the computation across the whole medium and obtain the emerging intensities.
We will now illustrate the method of obtaining these
reflection and transmission operators for the spherically symmetric transfer problem.
331 The transfer equation (7.1.1) can be written as
^
fj {r2I(r,/i)J + i |- ( ( l V ) K r , ^ ) } = -«*fl(r,/0 - *
Using the variable U and setting b(r,/x) = 4wr B, where B takes care of the sources other than scattering , this equation can be written as
" a ? < u < r -"» a(r)
w(r)
+
i a
f
2
r
)U(r,/i) + a(r)U(r,/i)
„+l ! p(r,M,M')U(r,/i')d/i- + [l-a>(r)]b(r,/i)
(7.4.5)
and /TT/
^
\ \
/•aj (U(r,-M)) a(r)
»(r)
1 3
- - ^
(lV)U(r.-^)
+ o(r)U(r,-A.)
»+l | p(r,-M,M')U(r,M')dM' +
(7.4.6)
[l-u(r)]b(r,-fi)
where we have differentiated the forward and backward flow of radiation.
These equations are first integrated over the angular cell
coordinates (u. .. ,„,u. , .„) where u. . ._ are defined in Mequation "j-1/2 "j+1/2 ^j+1/2 (7.3.11). After this, equations (7.4.5) and (7.4.6) are integrated over the space cell coordinates (r ,r
1
) . In the above integrations,
variables subscripted by n+1/2 or j+1/2 are assigned suitable neighbouring nodal values by interpolation.
Comparing the resulting
equations with the interaction principle relation (7.4.2) one can obtain the required reflection and transmission operators.
Explicit
expressions for these operators can be found in Peraiah [Kalkofen (4) ,p.281]. This method has been found to be stable and maintained the flux conservation for the medium in radiative equilibrium (cf. 7.5.2)).
However it suffered from one drawback. As a
the discretization of the curvature term ^— was distributed unevenly, one obtained unphysical negative intensities in the vacuum (T — 0). Recently Peraiah [Kalkofen
,p.350] had proposed an integral operator
technique to overcome this difficulty while preserving the desirable
332 feautures of the cell method.
Consider a typical cell
with angular-space grid (fi. ,a. -) X (r ,r .. " j j —1 n n—1
Fig 7.4.2 The major idea is to first express the intensity in terms of certain interpolating coefficients as follows : U - U 0 Q
+
U01*
+
U10, + UL1«,
(7.4.7)
where r—r
5 - Ar/2 ' r
r =
+ r
n
1 =
M~M AM/2
M■+M• i
.1 n+1
H
-
and Ar =
(r
n-rn-l) '
A
" - "j - "j-r
The quantities U-_, U 0 1 etc are estimated in terms of the neighbouring nodal values (cf. Fig 7.4.2).
The main modification introduced in
this integral operator technique is to substitute this interpolating polynomial (7.4.7) in equations (7.4.5) and (7.4.6) and perform the integration by the operators X and Y given by
333
X
-H • **
A
. .
Y - | [ ... 47rr2dr,
d/ii
„
r
4 3 3 where V = •=■ 7r(r -r , ) . 3 n n-1 Finally as in the cell method, the resulting equations are rearranged in the canonical form of the interaction principle and the reflection and transmission operators obtained. be found in Peraiah [Kalkofen
Details of this modification can
,p.305], where various tests for
accuracy of the method have also been established. 7.5
Direct method using polar coordinates As the name suggests, in this method the transfer equation (7.1.1)
is discretizied to obtain a finite-difference scheme for the intensity at the space—angular points (r ,fi.).
The main concern is to develop
stable finite difference scheme for the angular derivative term which arises in the spherical case.
We first rewrite equation (7.1.1) in
the operator form (D + a - fi')I = a(l-w)B
(7.5.1)
where D is the differential operator given by 31
nT
DI s
" li
+
1 31 ... 2,
r at (1-"}
and Q' is the integral operator given by
r
and Q' is the integral operator given by
cr i
- a(r)w(r) 2
P ( r ,1* ,/*' ) I ( r ./*'
)d/j'.
O'l ^ a(r) ^ (r) f - i p(r,^,M')I(r,/,')d/J'. When we discretize the space and angle variables the operators D, Q' When we discretize the space and angle variables the operators D, Q' will be matrices. As the stability and accuracy of the algorithm depends very much on the matrix operator D, care must be exercised in (9) the discretization process. Peraiah and Grant observed that in the pure isotropic scattering case (u - 1, p - 1) integration of (7.1.1) yields
334 dF dr
+
2F r
+
r r J,
3/i
[(l-/i )I]d^ = 0.
(7.5.2)
As the third term is clearly zero, it is evident that the flux will be conserved to machine accuracy provided we use a discretization which is such that "the differentiation exactly cancels the integration".
This is guaranteed when we use for both processes the
same /i-grid, the same order of discretization and the values p = ±1 are included in the sets.
Using Gaussian quadrature, the angle
derivative in the operator D is written as d_ 3n
2
+
,
_ 1^
2
{( - ' j 1
W. J
i
+l/2j
I
+ j+l/2 "
1
2 ± -' J j-l/2j I J-l/2.
with I
+
J+l/2
(
"i +1 -"1+X/2>IT
+ (
".i +1/2"
" i + 1 " ""1
vv
x
i/2 - i al± V
and /*...._ as defined in (7.3.11). To discretize the radial coordinate, a half—implicit scheme was used. With this discretization a finite difference scheme of linear equations were derived from (7.5.1) for the components of the + intensities I (r,/i). Details of this scheme can be found in Kalkofen(4) (p.335). 7.6
Operator-perturbation methods The typical transfer equation couples the radiation field at one
point with all the other points within a certain distance called the thermalization length.
Hence an accurate numerical scheme must
include a large number of angle-space grid points.
The operator
perturbation technique is to solve the difficult coupled equations with fewer angular grid points and then correct the error by solving
335 the easier uncoupled transfer equation with a known source function for the much larger angular grid points. proposed by Cannon
This scheme was first
. Although there were certain deficiencies in
the original proposal, this method has recently been revived by Auer and Kalkofen [Kalkofen( \ transfer problems.
p.237 and Kalkofen (5 \p. 191] to solve
We will now describe Kalkofen's technique of
obtaining the correct perturbed equation to solve the transfer equation. The transfer equation (7.1.1) can be written in the operator form (cf (7.5.1)) with a = 1 and (l-w)B replaced by B) - DI = I - 9
(7.6.1)
9 = Cl'I + B.
(7.6.2)
Where D is the differential operator and Q' in the integral operator defined earlier in (7.5.1).
The formal solution of (7.6.1) is given
by I - (1+D)~V
(7.6.3)
In order to develop the perturbed equations of transfer, we write initially the formal integral equation for the source function in the form 9 = Q'(1+D)-1£ + B Z9 = B
i.e.
(7.6.4) fl'(1+D)_1.
9 - 1 -
where
(7.6.5)
Following Cannon's prescription to obtain the perturbed equations we introduce the approximate operator L for t and write t = L + (JE-L).
(7.6.6)
Next the source function is expressed as an infinite series as follows ,(n)
_
n
z
k=0
f
(k)>
* -
llmSf(n), n-»°°
336 The approximate integral equation is obtained from (7.6.4) and (7.6.6) as follows. L
n+1 ... n ... I f ( k ) - B - (JC-L) I f(k) k=0 k=0
i.e.
L f(n+1) = ,n
(7.6.7)
where
en = B - £ ? ( n )
(7.6.8)
Equation (7.6.5) is used to define the approximate integral operator L.
We define L - 1 - w' (l+d) _1
where a>' is the integral operator with fewer angle-quadrature points and d is the corresponding differential operator representing D.
To
compute the r.h.s. of (7.6.8) we use (7.6.5) to obtain £(SJn) - ? ( n ) - fi'(1+D)_1(3n). Although this involves the full integral operator fl', fortunately it operates on a known source function and hence can be evaluated very fast even with large angular grid points.
In practice we don't use
the perturbed integral equations (7.6.7),- (7.6.8) as the operators (1+D)
may not be easy to evaluate.
We return back to the
differential form and obtain the perturbed differential equations from the perturbed integral equations.
The integral equations had been
introduced only to facilitate the correct formulation of the differential perturbed equations. The coupled system (7.6.1), (7.6.2) can be written as
(1+D)I - 9 - O'l + B, i.e.
and
[(l+D)-n']I = B
9 = O'l + B.
(7.6.9)
(7.6.10)
337 From the above equations and equations (7.6.7), (7.6.8) we write the perturbed differential equations as follows : [ ( 1 + d W ] i(n+1) - ,(n) f (n + l)
_ a) ,.(n + l)
£(n+l)
_ S (n+1) _ , 0 0
+ £ (n)
(7.6.11b)
(7.6.11c)
? ( n + 1 ) = n < I ( n + 1 ) + B,
where
(7.6.11a)
(l + D)I ( n + 1 ) = S ( n ) .
These equations are solved with the starting solution e
(7.6.lid)
= B,
f<°> = 0 ( — *<°> = 0 ) . We will now check that the perturbed equations (7.6.7), (7.6.8) do infact yield the exact solution given by (7.6.5). Since e
= B, we have from (7.6.7) and (7.6.8)
f(1> = I T V
(1) £
-B-HTV
Substituting again in (7.6.7), (7.6.8) we have f ( 2 ) = L _1 B - L _1 £L _1 B,
i.e. f ( 2 ) = (1 - L h)
£ ( 2 ) - B - Z [L_1B + L _1 B - L _1 £L _1 B]
L X B.
Continuing this process we check easily that p(n+D _
£ (1-L _1 £) k L _1 B. k-0
Since the sum of this infinite series when n -> » is t converges to £
L and hence we obtain : 9 - (£ _1 L)L _1 B
i.e.
9 = £ _1 B,
(7.6.12)
L, the series
338
provided the eigenvalues of the operator (1—L unity in absolute value. Let us now describe Cannon's [Kalkofen
(4)
Z) are smaller than
,p.l57] scheme to adapt the
above perturbation operator technique for the spherical problem.
The
transfer equation (7.1.1) is cast in the form
5
1A
-a [I,
*Tidr
where
r
* J -= ! | J
(7.6.13)
p(r,/i,M , )l i d/i' + e^
£ = 0
(l-w)B, with
'-i ' '
I-, 2 " i - 1 dn
£ > 1.
The zeroth order equation is essentially the plane problem with the higher order solutions yielding the spherical dependence.
The
equation (7.6.13) is similar to the form (7.6.1) with a simpler differential operator and hence the operator perturbation technique can be used to solve this problem. representing the error term e
Care has to be exercised in
(£ > 1) as the angle derivative is
involved. The above operator perturbation technique has been described for the case when the integral form had a known source function term B. Modifications had to be incorporated if boundary conditions are given without any known internal source term.
Kalkofen
(p.191) has
described in detail the modification necessary for the radiative equilibrium problem with given boundary conditions.
339 References 1.
Cannon, C. J., J. Quant. Spectrosc. Rad. Transfer 1^, 627, 1973.
2.
Feautrier, P., C. R. Acad. Sc. Paris, 258, 3189, 1964.
3.
Hummer, D. G. and Rybicki, G.H., Mon. Not. Roy. Astro. Soc. 152,
4.
Kalkofen, W. (Editor), Methods in Radiative Transfer, Cambridge
1, 1971. University Press, Cambridge 1984. 5.
Kalkofen, W. (Editor), Numerical Radiative Transfer, Cambridge University Press, Cambridge 1987.
6.
Mihalas, D. and Mihalas, B. W., Foundations of Radiation Hydrodynamics,
Oxford University Press, New York, 1984.k, 1984.
7.
Mihalas, D., Stellar Atmospheres, W. H. Freeman and Co., San
8.
Peraiah, A., Mon. Not. R. Astro. Soc. 162_, 321, 1973.
9.
Peraiah, A. and Grant, I. P., J. Inst. Maths. Applies.12_, 75,
Francisco, 1978.
1973.
341 Glossary of symbols
a : Albedo for scattering by the planetary atmosphere a.,a. : Weights appropriate to Gaussian quadrature A : Surface albedo for scattering B (r) : Contribution to source function from reduced incident o — radiation B.. (r) : Contribution to source function from internal sources other than scattering B(r) : B (r) + B (r) ~ o — 1— B (T) : Planck function c : Velocity of light E : Radiant energy E.(x) : Exponential integral f(r) : Eddington factor f(r,V0 : An additive term in the three stream representation of intensity wF. : Incident flux l
?rF
: Flux
*(r) 8f(£,i/)
Source function
?(r,0 ,)
?'(r) G.,. . „ . : Moments of intensity i(i=l,2...n) H : First moment of intensity H(x) : Heavis:Lde unit step function
342 H(/J) : Chandrasekhar's H—function I(r,s,i/) : Specific intensity at r along s of frequency v I(r,s) : Frequency integrated specific intensisty I(z,8,
: Specific intensity in plane medium i
Specific intensities in the upward and downward direction
I (r,/j) '
in a plane medium
Id+(r,M,• Diffuse specific intensity Id_('",/i,
'
I. : Intensities at quadrature points I(TC,H,V)
: Specific intensity at frequency v
I(r,»7,—/x) : Specific intensity in cylindrical medium J(r,i>) : The mean intensity (zero moment of intensity) J(r)
: Frequency independent or frequency integrated mean intensity
K : K-integral (second moment of intensity) L : Third moment of intensity L(R,r 1 ;r}' ,/M' )
: Dissipation function
L„ : L„—kernel M : Fourth moment of intensity n : Refractive index of the medium n : Unit normal p(s,s') Phase function P(M,
p (fi,fi') : Phase function in the isotropic case * p(r,—p. ,/i,R) : Probability function for photon emergence p(r;z,rj,M;r1,R) -, ^ \ p (r;z,r;,-^;r1,R) >
Probability for photon emergence
q(r ,/J' ,/J;R) : Probability function for photon emergence
343 q(r; Z,t) .A»;r1>R)
> > Probability for photon emergence
*
q (r ; z, n,-n;rl,R)
Sphericity factor
q(r)
rO*,".> r(M> r
o
>
V
> Reflection function
x) i
: Extrapolated end point
R(
,v) 1
R(i/'
A
: Redistribution function
: Angle—averaged redistribution function
S(r,r»}i
: Pincherle—Goursat Kernel
S
S(n,
V
> Scattering function
'
x)
) : Azimuth averaged scattering function
S°(fi,ti' S(i/,
o
S(R, r
/i,/i ;R) : Scattering function in non—coherent spherical meidia
l ; M.M') I
>■ Scattering function in spherical medium
S
;^,^R) J
S(z,
"^;R>-"o'-%;rrR) \
function in cyl indrical geometry * f Scattering S (i:,»7i.-^;r 1 ,r/ 1 ,^ 1 ;r 1 ,R) '
T(r,VO
: Optical distance along a ray from the sun tci a point in the atmosphere
T(r,r«:l
: A L_—kernel
T ( r i ; / i :,
T(R,r
l;
,fi,fi')
o
,a> ) : Transmission function in plane geometry o : Transmission function in spherical geometry
T(z,> ? , - - M ;R,-r ?o ,-M o ;r 1> R) * T (2: , » ? :
W. :
, > Transmission function in cylindric :al
-":R'-V-Vri'R)
Quadrature weights
1
a(r,»0
1
o(r)
J
>■ Attenuation or extinction co«ifficient
344 7
: Reflectivity of the surface
7,7' : Angle between two directions of ray at the same point between which scattering occurs S(u,—u')
: Dirac's delta function
S.. : Kronecker 5—symbol ij e(r,v) : Emission coefficient or emissivity e (r,i/) : True emission coefficient g
e (r,s,i/) : Emission coefficient for scattering t)(,X,fi,ii')
: Phase function
ri : cos 9 (cosine of angle between the pencil of radiation and the direction of the axis of the cylinder) 9 : Polar angle between the direction of pencil of radiation and normal to the atmospheric layer 9 : Angle between the pencil of radiation and the direction of the axis of the cylinder k (r) : Volume absorption coefficient A
: Hopf's operator
fi,fi' ,H
: cos 9 (cosine of the angle between the direction of pencil of radiation and normal to the atmospheric layer) : ^ = cos
fj.. ,fi.
: Angle—point in discrete ordinate method
H
: Angular division in moment method to be determined from the solution of transfer equation
v ,i>' ,1/
: Frequency variables
p(R;rj ,ft;ri',n')
: Reflection function
<7(r,i/)
i
a(r)
> Scattering coefficient i
345 a (r,i/') : Total scattering coefficient a : Stefan constant r : Optical depth cp : Azimuthal angle of a point on a spherical surface $("') : Absorption profile V>(r,ji) : Angular distribution of neutron density V>+(r,/j) v > Angular distribution of neutron in two half-ranges V>_(r,M) i/i(r,i/) : Angle—averaged emission profile i/i(r)
: Angle—averaged emission profile for frequency independent intensity
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347 General References
Busbridge, I. W.
The Mathematics of Radiative Transfer, Cambridge
University Press, 1960. Case, K. M. and Zweifel, P. F.
Linear Transport Theory, Addison Wesley
1967. Chandrasekhar, S.
Radiative Transfer, Clarendon Press, Oxford, 1950,
also Dover Publications Inc. New York, 1960. Davison, B. and Sykes, J. B.
Neutron Transport Theory, Clarendon
Press, Oxford, 1958. Kalkofen, W.
Methods in Radiative Transfer, 1984; Numerical Radiative
Transfer, 1987, Cambridge University Press. Kourganoff, V.
Basic methods in transfer problems, Oxford University
Press, 1952, also Dover Publications Inc. New York, 1963. Mihalas, D.
Stellar Atmospheres, Freeman and Company, San Francisco,
1977. Ozisik, N.
Radiative Transfer, Wiley—Interscience Publ., 1973; Heat
Transfer, McGraw Hill Book Co. New York, 1985. Sobolev, V. V.
A treatise on Radiative Transfer.
Van Nostrand, New
York, 1962. Viskanta, R.
Heat Transfer in thermal radiation absorbing and
scattering media ANL—6170, Argonne National Laboratory, Argonne ILL, 1960. Viskanta, R.
Radiation Transfer and interaction of convection with
Radiation Heat Transfer. Advance of Heat Transfer Vol. 3 p.175-251, Academic Press, New York, 1966.
