Dynamics and Mission Design Near Libration Points, Vol. IV: Advanced Methods for Triangular Points

KA = M

0

which satisfies MT JM = J.

Lemma 3.7 H(X,Y,Q,Pe)

The complete change of variables transforms the initial Hamiltonian into H(x, y, 6,pe) = ojpe - -xTB21x

- xTB22y

+

-yTB12y.

Proof: Inserting the following expressions X

=

Pux + Puy,

Y

=

P2xx + P22y,

0

=

6,

P@

=

pe+xTF^x

+ yTF[2x

+ xTF^y

+

yTF^2y,

given by the change of variables, into H, and applying the definitions of Fij, i = 1,2, j — 1,2 given in Proposition 3.1, we obtain H(x,y,0,pe)

= u>pe - -xTB21x

- -xTB22y+

Since each term is an scalar, in particular yTB\\x of Lemma 3.3, the result is obtained.

3.4.2

Practical

Implementation

-yTBux+

— xTBf1y,

of Proposition

-yTBi2y.

and using B22 = —Bjx •

3.1

Our target is to take the current Hamiltonian up to the second order part H(X,

Y, 6 , Ye)

= UJSYQ + H2(X,

Y, 0 ) ,

and to perform a change of variables which leaves it as H(x,y,0,ye)

= usyg + uiixxy! + w2x2y2 +w 3 x 3 ?/3,

where u>i, u>2 and 0J3 are complex numbers.

74

Normal Form of the Bicircular

3.4.2.1

Model and Related

Topics

Case of the Hamiltonian Expanded in Complex Variables

First of all we identify the matrices Qi, Q2 and Q3 of Proposition 3.1 writing the initial Hamiltonian in the form

H(X, Y, 0 , P 0 ) = wP© + XTQ1X

+ XTQ2Y

+ YTQ3Y.

We consider the differential equations associated with the variables X and Y, noted by Z = QZ in Proposition 3.1. Then, we compute the monodromy matrix numerically, integrating the periodic linear system of equations P = QP, starting with P(0) = h, during a time span equal to the period of the perturbation which is^L. us

Once the monodromy matrix M is given, we compute its eigenvalues. These appear as pairs of complex numbers, u;, Vi, i = 1,3 satisfying m = v~l, i = 1,3. Anyone of each couple can be selected in order to be in the matrix Dm of Lemma 3.3. The remaining ones of each couple fit in D^1 in such a way that DmD^ = I3. Since in our problem the manipulator assumes that the out of plane components are uncoupled from the in plane ones, a little care must be taken when selecting the element (3,3) of Dm. Having DM, we can compute the matrix 5 of Lemma 3.3, and using the Lemma 3.1 its inverse 5 _ 1 . Then, we have M = SDMS~X, with S symplectic. From Dm we compute Df, of Lemma 3.3, taking the logarithms of the values of the diagonal divided by ^ZL. These complex eigenvalues are determined except in multiples of cosy/—!- Any determination can be chosen, and as we are going to show, these values will be the coi, ui2 and CJ3 of the new Hamiltonian. Taking —Df, we have DB of Lemma 3.3, then we can compute B according to

B=

SDBS-\

Since by Corollary 3.1a linear constant (with respect to time) change of variables in the components X, Y given by a symplectic matrix S, can be completed with 0 = 8 and PQ = pe, we compose the change Z = Pz, of Proposition 3.1 with z = Sz, obtaining Z = PSz (later the matrix PS will be called again P). If we introduce this change in the differential equations Z = QZ, we obtain PSz + PSz = QPSz, that is, i = S~1(P~1QP - P~1P)Sz. But P is computed in such a way that P = QP - BP, what implies z = S~1BSz, that is, z = DBZ, concluding that in the variables z = (xT, yT)T, the system of differential equations is autonomous and diagonal. The implementation of this fact is as follows. Starting with the matrix P(0) = IQ, the differential equation P = QP — PB is integrated numerically over a period (^~units of time). The values of P = PS, are stored equally spaced in time in a prefixed number of points in order to perform a Fourier analysis. At the same time, the values of Fu, P12, -F21 and F22 are computed taking into account that due to

Normal Form to Order Two

75

the change of variables given by S now their expressions are: F&

=

--(Pl1QiPii+PFiQ2P2i+PliQ3P2i), LJ

PL

=

^~Db - -(P&QiPn

+ PI2Q2P21 + PI2Q3P21),

LJ

ZLJ

Fl

=

^Db-^(P[1Q1Pl2+P^lQ2P22

F22

=

--(PUQ1P12

+

+

P^Q3P22),

P&Q2P22+PMQ3P22).

Then, performing the symplectic change X

=

Pnx + P12y,

Y

=

P21x + P22y,

0

=

0,

Pe

=

T

Pe+x

F^x

+ yTF^x

+ xTF2Tly + yTF2T2y,

according to Proposition 3.1, the new Hamiltonian is H(x,y,e,pg) 3.4.2.2

= u>spe + xTDby

:= cuspe + wi^u/i + uj2x2y2 + u>3x3y3.

Case of the Hamiltonian Expanded in Real Variables

Although this case is very similar to the previous one, some care must be taken in order to get the real normal form of the second order part of the Hamiltonian. As in the case of complex variables, we consider the differential equations associated with the variables X and Y, noted by Z = QZ in Proposition 3.1. Then we compute the monodromy matrix numerically, by integrating the periodic linear system of equations P = QP, starting with -P(O) = Ie, during a time span equal to the period of the perturbation which is —. We remember that P = QP is the system of variational equations associated to the equations of motion. Then the monodromy matrix M can be obtained directly by integrating the variational equations over the periodic solution of the initial (not necessarily expanded) Hamiltonian. Once the monodromy matrix M is given, we must compute the matrix B of Lemma 3.3 which can be considered as the logarithm of M divided by the period ^ZL. The point is that the matrix B must be real, since we want the real normal form. So, spite many of the following computations are performed in complex arithmetic the final result is real. First we compute the eigenvalues of M. These appear as pairs of complex numbers, u;, Uj, i = 1,3 satisfying Ui = v~l, i = 1,3. Anyone of each couple can be selected in order to be in the matrix Dm of Lemma 3.3. The remaining ones of each couple fit in D^1 in such a way that DmDmx = I3. Since in our problem the manipulator assumes that the out of plane components are uncoupled from the in plane ones, a little care must be taken when selecting the element (3,3) of Dm.

76

Normal Form of the Bicircular

0 0 0 -Wl

0 0

0

0 0 0 0 0

U>2

0 0 0 0

Model and Related

-ui3

0 0 0 0

Wl

0 0 0 0 0

-W2

Li

Wl

W2

L3 Li L5

-0.77780671E-01 0.30050393E+00 0.30050393E+00

0.17901829E+01 -0.13852201E-01 -0.13852201E-01

Table 3.2

0

Topics

0 0 UJ3

0 0

ot ^3

-0.84140544E-01 -0.78810538E-01 -0.78810538E-01

JB matrix for the real normal form around some equilibrium points.

Then we compute the (complex) symplectic matrix of the change of basis, S, of Lemma 3.3, having M = SDMS-1, being M real and S and DM complex symplectic matrices. From Dm we compute Df, of Lemma 3.3, taking the logarithms of the values of the diagonal divided by —. Then taking — Df, we have DB of Lemma 3.3, and we can compute B according to B = SDBS~ . This matrix B must be real. In order to obtain the real normal form of the second order part of the Hamiltonian we need the decomposition B = SDgS~l in real symplectic matrices B = RJBR"1 where now JB stands for the real Jordan form of B. This fact is accomplished getting the real symplectic matrix, R, from S by storing by columns the real and imaginary parts of the complex eigenvectors which we know they are complex conjugated. Once we have the real symplectic matrix R, JB is computed according to JB = R~lBR. The values of JB for the normal form around some different equilibrium points are given in Table 3.2. Since by corollary [3.1] a linear constant (with respect to time) change of variables in the components X, Y given by a symplectic matrix S, can be completed with 0 = 6 and P@ = pg, we compose the change Z = Pz, of Proposition 3.1 with z — Rz, obtaining Z = PRz (later the matrix PR will be called again P). If we introduce this change in the differential equations Z = QZ, we obtain PRz + PRz = QPRz, this is z = R~l(P-lQP - P~1P)Rz, but P is computed in such a way that P = QP - BP, what implies z — R~xBRz, this is z = JBZ, Concluding that in the variables z — (xT,yT)T, the system of differential equations is autonomous with its matrix in Jordan form. The implementation of this fact is as follows. Starting with the matrix P(0) = I6, the differential equation P = QP - PB is integrated numerically over a period ( ^ units of time). The values of P = PR, are stored equally spaced in time in a

77

Normal Form to Order Two

prefixed number of points in order to perform a Fourier analysis. At the same time the values of Fn, F 1 2 , F21 and F22 are computed taking into account that due to the change of variables given by R now their expressions are: FK

^JB21--(PKQlPn+PFlQ2P21+P2T1Q3P2l),

=

CO

ZOJ

F,12

—

7T~JB11

(PwQlPll

+ P12Q2P2I +

P22Q3P21),

CO

LLO

FlT

= ~

^JJB22--(PI1QYPI2+

*M

=

~JB12 LUJ

PKQ2P22 + P&Q3P22)

~ -(PT2Q1P12

+ PI2Q2P22 +

P&Q3P22).

CjJ

Then performing the symplectic change X

=

Pnx + P12y,

Y

=

P2ix + P22y,

0

=

9,

P& =

pe+xTF^x

+ yTF[2X + xTF2T1y +

yTFl2y,

according to Proposition 3.1, the Hamiltonian is H(x,y,9,pe)

= usPe - ^XTJB2\X

- xT JB22V +

^y1JBUV,

and taking into account that the matrix JB has the form:

I JB

0 0 0 W2 0 0 -Wl 0 0 0 0 0 \

0 wx 0 0 0 0 0 0 0 0 0 -0J3

0 \ 0 0 0 0 U>3 0 0 0 —W2 0 0/

that is, the hyperbolic part appears in the main diagonal and the central ones in the diagonals of JB\2 and JB21, we have finally the real normal form H(x,y,e,pe)

=u)Sps + ^ l f a i + Vi) + ^2^22/2 + ^z{x\

+j/|).

We note that introducing the canonical change of variables given by the usual complexification (x = (x + y/^ly)/y/2, y = (y/^lx + y)j\f% in the first and third couples of coordinates we get the complex normal form (we denote the new complex variables by the same name as the real old ones) H(x,y,9,pe)

= uspe + w i V - l z i j / i +Lo2x2y2

+U3V-lx3y3

78

Normal Form of the Bicircular

Model and Related

Topics

This complexification of the couples of variables, related with the central part, is necessary if we want to put in normal form the terms of the Hamiltonian of order greater than two using an efficient algorithm as we are going to see in Section 3.5. So, if the complexification is not performed at the beginning of the expansion, it must be performed (for the central variables) after doing the translation which cancels the linear part of the Hamiltonian and the Floquet change of coordinates which puts the second order part in the (real) normal form. 3.4.3

Applying

the Obtained

Change of

Variables

In the previous subsection we show a change of variables X

=

Pnx + P12y,

Y

=

P2ix + P22y,

0

=

9,

Y@

=

ye+P2{x,y,6),

where Pij, i = 1,2, j — 1,2 are 3 x 3 matrices whose elements are 27r-periodic functions in the variable 9 = wst, and p2(x,y,9) is an homogeneous polynomial of degree two in the variables Xi, yj, i = 1,2,3, j = 1,2,3, with 27r-periodic coefficients in the variable 9, followed by an eventually complexification of the couples of variables related to the central part, which diagonalizes the Hamiltonian H(X, Y, 0 , Ye) = oosYe + H2(X, Y, 0 ) , putting it into the form H(x,y,9,yg)

+oj2x2y2+^>3yr-lx3y3.

= ujsye + uiV^lxiyi

After performing the change of variables given by the generating function G\ to the initial Hamiltonian, we obtain the Hamiltonian oo

H(X,Y,e,Ye)

= iOsY8+ ^

HW(X,Y,Q),

1*1=2

where H\k\(X, Y, 0 ) are homogeneous polynomials of degree |fc| in the variables Xi, Yj, i = 1,2,3, j = 1,2,3 with 27r-periodic coefficients in the angular variable 0 . We note that if we insert the above changes of variables (Floquet + the eventually complexification) into this Hamiltonian, we obtain oo

H(x,y,9,ye)

= cosye +ui<s/^lxiyi

+ u2x2y2 + uzV^xzy?,

+ ^

H\k\(x,y,0),

|fc|=3

where H\k\(x,y,0) are homogeneous polynomials of degree \k\, like H\k\(X,Y,Q), which appear as the result of inserting the part of the change of variables associated with X and Y into H\k\(X,Y,0).

Normal Form to Order Two

79

In terms of algebraic manipulation this is an enormous task. So we have decided to expand again the Hamiltonian. In fact, we note that the translation which cancels the linear part of the Hamiltonian and the change of variables given by the Floquet theorem, are computed numerically using the equations of motion obtained from the Hamiltonian (3.3). This is essentially, by integrating the variational equations over the periodic solution and performing the corresponding Fourier analysis. Then the above expansion of the Hamiltonian must not be performed, the following one using the last variables will give us the Hamiltonian in normal form up to degree two. A crucial point, which must be taken into account, is that if we expand the initial Hamiltonian in complex variables, and we perform the normalization up to some degree, obtaining a complex Hamiltonian in normal form, it is not clear which is the change of variables that inserted in the normal form give us a real Hamiltonian. So, in our computations we followed the second way, computing first the translation and the Floquet change of coordinates in its real form, and after introducing the complexincation of coordinates to the couples associated with the central part of the Hamiltonian. In this way, introducing the inverse of the complexincation in the final complex normal form of the Hamiltonian it gives us a real normal form. 3.4.4

Second Expansion

of the

Hamiltonian

In this Section we want to obtain an expansion of the original Hamiltonian (3.3), which we recall is: H

=

^(Y2 + Y2 +

Y2)+X2Y1-X1Y2-aX1-^X2 1-/* + (X2 - X2E)2

y/{.Xi - X1E)2

V(Xi -

*IM) 2

+ X2

+ (x2 - x2My + x2

rris " y/{X! - X1S)2 + (X2 - X2S)2

+ X2

±({X2 + P) sin^ - (Xi + a) cos0),

a

s

such that it must be in the normal form: oo

H(x,y, 9,ye)= usye + uiV^xiyi

+ u2x2y2 + ui3y/^lx3y3

+ ^

H\k\(x,y,

6),

1*1=3

where H^\(x,y,0) are homogeneous polynomials of degree |fc| in the complex variables (xi,x2,x3,yi,y2,y3), with polynomials in the variables e9^^, e~9v^, (0 = cost), as coefficients. Here u>\, UJ2 and u>3 are real numbers. For this purpose we follow the same way as in Section 3.2, but in the original coordinates (real or complexified) we will introduce also the change of coordinates

80

Normal Form of the Bicircular Model and Related

Topics

given by the generating function Gi, which is supposed to be already computed, the Floquet change of coordinates given in the previous subsection and the complexification of the variables related to the central part, if we start with the original coordinates in real form. Suppose that x, y, refer to some old coordinates and x, y, to the new ones in each change of coordinates, f The change of coordinates given by any generating function G of the Lie method is explicitly u = u + {u,G} + \{{u, G}, G} + ! { { { « , G}, G}, G} + • • •, where u is any of the old variables and u is its corresponding new one. We remember that G\ = gu(8)xi

+gi2(0)yi

+ gn(0)x2

+

gu(0)y2,

where gu{6), g\2(0), gn(0) and gu(9) are trigonometric polynomials in the variable 0. Then, the change of coordinates given by G\ is: xi

=

xx+gu,

2/i

=

2/1-5H,

X2

=

X2+gi4,

272

=

2/2 - 9 i 3 ,

PB

=

i fgu

gi2

P6

-2\-Mm-lB9ll

, gi3 +

gu

\

dGx

lB9lA-^9lZ)-^^

whereas the other variables remain unchanged (^3 = x 3 , y3 =2/3, 0 = 9). The Floquet change of coordinates in the previous Section is explicit: x

=

Piix + Pi2y,

y

=

P2\x + P22y,

0

=

0,

PS =

Pe+p2{x,y,0),

where P ^ , i = 1,2, j = 1,2 are 3 x 3 matrices whose elements are trigonometric polynomials in the variable 0 = ust, and p2(x,y,0) is an homogeneous polynomial of degree two in the variables a; = (a;i,a;2,a;3) r , y = (2/i,2/2,2/3)T, with trigonometric polynomials in the variable 0 as coefficients. We will assume that this Floquet change of coordinates contains as well the complexification of the couples of variables related to the central part of the Hamiltonian, in case that the original coordinates are the real ones (this only implies that the matrix P is multiplied by some complex symplectic matrix). This is: the complexification is performed after the translation and the Floquet changes of coordinates instead before these changes.

Normal Form of Terms of Order Higher than Two

81

Inserting the Floquet change into the one given by G\ we obtain the global change of coordinates: z

=

Pz + V,

9

=

9, l/5ll

where z = (xT,yT)T,

5l2

V = (VxT,VyT)T,

Vj = (gi2,9u,0)T,

. #13


N

W = (-VyT,VxT)T and

117T0

,

/

Q\

with (-9ll,-g13,0)T

Vf =

and P has the 3 x 3 matrices Py, i = 1,2, j = 1,2 as components. These expressions are inserted in the original variables of all the terms of the initial Hamiltonian (3.3), and in the polynomials which appear in the recurrences which expand the potentials of the Earth, Moon and Sun of the Hamiltonian in Section 3.2. The expression of the recurrences are the same ones as before but now all the recurrences depend on 9 (in the first expansion the Earth and Moon potentials do not depend on 9 and the computation of the recurrences are much faster). Moreover, now the fixed polynomials which appear in those recurrences, for instance p2, are no longer homogeneous polynomials. In the case of p2 it is a complete polynomial of degree two with independent term included. Then the polynomials Gn are no longer homogeneous polynomials either, and to perform the computations of the Gn with a high degree of accuracy at each n-step, the product of a complete polynomial of degree n, by a complete polynomial of degree one (including independent terms), plus the product of a complete polynomial of degree n — 1, by a complete polynomial of degree two must be done. (Gn are not homogeneous polynomials but when n is big, the terms of low degree could have been dropped).

3.5

Normal Form of Terms of Order Higher than Two

In this Section we are going to see how to reduce the higher order terms of the Hamiltonian to autonomous form. The departing Hamiltonian is the one obtained in the preceding Sections, that is, a Hamiltonian of the form H(x,y,9,ye)

= wsye + H2(x,y)

+ '^2,Hr(x,y,9), r>2

with 3

H2{x,y)

= ^UiXiyi, i=l

Hr(x,y,9)

= ^ |A|=r

hkr{9)xk'y^,

(3.4)

82

Normal Form of the Bicircular

Model and Related

Topics

where A; = (fci,..., fce), k1 — (fci, Aft,^) and k2 = (&2, £4, A^). In the H2 part, of our concrete problem, we have now that wi and W3 are purely imaginary. The replace the previous values wi\/—T , W31/--I in this Section and we hope no confusion will be produced. The purpose is to obtain a (periodic) change of variables transforming this Hamiltonian in a simpler one. That is, we will try to eliminate the time dependence and, at the same time, to cancel the maximum possible number of monomials. The method we are going to use is, as before, based on Lie transforms. 3.5.1

The

Method

The main point of the method is contained in the following proposition, which is presented in a general setting. Proposition 3.2

Let us consider the Hamiltonian r-l

H = u>syo +H2(x,y)

+ ^2Hi(x,y)

+ Hr(x,y,6)

+ Hr+i(x,y,6)

+ •••,

»=3

where r > 2, H2(x,y) Y^ hkr{6)xklyk2

= Y^v&iVi fai e Q, Hi(x,y)

andhk{6)

= ^ ft*a;* yk , Hr(x,y,0)

=

= ^ft^expO^^l).

\k\=r

3

Let Gr(x,y,9)

x

= V_, 9r(@)

y

, be a generating function defined as follows:

\k\=r

(1) ifk1

^k2:

Jfc/m_

Ck

-

h

rfi

h

—

-

3^0

(2) ifk1 = k2: hk ffnJUJSV-l Then, the new Hamiltonian H' obtained from H by means of the change of variables given by the generating function Gr, H' = H + {H,Gr}

+ ^{{H,Gr},Gr}

+ ---,

satisfies that r-l

H1 = usyo + H2(x,y)

+ ^2Hi(x,y)

+ H'r(x,y) + H'r+1(x,y,6)

+ • • •,

83

Normal Form of Terms of Order Higher than Two

where H'r{x,y) = ]P(/i')~ar yk

and

(u>\k _ J ck [ ) r ~ \ hk

if k 2 i/^1 = k .

Remark: The value Ck that appears in the function Gr is arbitrary. Usually it will be selected equal to zero, unless the divisor < u>, k2 — k1 > were small, in which case it would be chosen equal to hk0. Later we shall come back to this point. In the proof of this proposition we shall use the following Lemma: L e m m a 3.8

If H2 and GT are defined as n

H2(x,y)

= Y^UjXjVj,

Gr(x,y,6)

= ^

j=l

{fix*yk\

9*

\k\=r

then we have {H2,Gr}

= ^

A

< u,,fc2 - fc1 > gkr{6)xkk\.k'y

\k\=r

Proof: {H2,Gr}=

J2iH2,9kr(0)xklyk2}=

Y,9kM{H2,xklyk2}.

\k\=r

\k\=r

The last step can be done because dH.2JdQ = dfo/dyg note that

{H2,xk\k2} = Y: J=I

= 0. To finish the proof we

d(xk yk ) 8(xk yk ) iVidyj -"**>—tei

= j;[ W j .(t j )// -u>;(*v*Va] j=l

=


> xkk'.k* y

Proof of Proposition 3.2: It is easy to see that H'r

= =

Hr + {ujsye,Gr} + dG

de

{H2,Gr} k\.kz

Hr-us^+Y,k2-kl>9kr(6)x

\k\=r

In the last step we have used Lemma 3.8. If we impose that H'r — J^ CkXk yk (where the c% are constants), it is obtained, from the last equation, that gk must

84

Normal Form of the Bicircular

Model and Related

Topics

satisfy ck = hkr(6) - UJS^- + < u,k2 - k1 > gk(0), dO that is, dak

US

<^k2~kl>

M~

1

9r(S) = K{9) - Ck.

2

Now, if k 7^ k it is not difficult to see that a solution of this differential equation is

C

Sr (*) =

\2 \

+E

< u, k2-k1> 1

r-

^

„

r - exP 0V=1).

^ J j w ^ v ^ T - < w, fe2 - ife1 >

'

2

The case k = fc is easier, and the corresponding periodic solution is k

hk

g {6) =• Y, ~^H

ex

P 0'*V=1).

Here, it has been necessary to select ck = hk0 in order to avoid secular terms. 3.5.1.1

•

Selecting ck

As it has been mentioned before, we have some freedom in the selection of a parameter of the generating function, that is, the value ck. We want the final Hamiltonian to be as simple as possible, and this implies that it would be desirable that all the ck were chosen equal to 0 (except, of course, the ones corresponding to the resonant terms k1 — k2). On the other hand, if all the ck are selected equal to zero, then the expressions of the generating functions contain the divisors < u,k2 — k1 >, that can be very small and produce convergence problems. For this reason we have implemented a threshold e in the program, in order to select Cfc = 0 only when the corresponding divisor is greater than e. Otherwise it is chosen to be equal to hk 0 to avoid the corresponding small denominator. With this, we expect to obtain a final Hamiltonian with only a few monomials, and valid over a relevant region of the phase space. 3.5.2

Implementation

The implementation of the algorithm described above relies on an algebraic manipulator, that performs all the basic operations needed during the process. 3.5.2.1

The Algebraic Manipulator

It contains the necessary routines to deal with power expansions (more exactly, with polynomials in 6 variables), where each coefficient is a Fourier series. We have developed, for Fourier series, the arithmetic operations (except division, that

Normal Form of Terms of Order Higher than Two

85

is not necessary in the program) plus differentiation. Using these operations, we have implemented the Poisson bracket between two homogeneous polynomials, in the following way:

{ k

I

J

k,l

\j=l

J

where the terms with \kl + lx\ = 0 and \k2 +12\ = 0 are missing. 3.5.2.2

The Main Algorithm

The process of putting the Hamiltonian in normal form is not very difficult, using the routines mentioned above. The method is as follows. For each degree r, the generating function Gr is computed according Proposition 3.2, selecting c^ in the way mentioned just above. Now, to explain how the corresponding change of variables is performed, let us consider the first of them, that is, r = 3. Let us write the Hamiltonian as H = cosVe + H2 + H3 + H4 + • • • + Hn-2 + Hn-\ + Hn, where, as usual, Hi stands for an homogeneous polynomial of degree i. The steps of the change are the following: (1)

{Hn-UG3}^P, P + Hn -> Hn, (2) {Hn_2,G3}^P, P + Hn-i —> Hn-\,

{P,G3}^P, P + Hn -> Hn, (3) {Hn^,G3}^P,

and so on. The same kind of scheme is applied for G4, G5,.... With this, we only need to store one Hamiltonian to do all the process (we also need some working space for P , but it only needs to be big enough to contain homogeneous polynomials of degree less or equal than n, that is less than the size of the whole Hamiltonian). Once one degree has been completed, the GT is stored in a (binary) disk file and the new degree is started. We note that all the Gr are stored in the same file, one after the other. Finally, when all the process has been completed, the Hamiltonian in normal form is written in a (ASCII) disk file. 3.5.3

Results

The algorithm described above has been implemented in FORTRAN 77, and it has been applied to the case of L4. Using a threshold e = 10~ 2 for the small divisors, the program has computed the normal form up to order 12. This expansion only

86

Normal Form of the Bicircular Model and Related

Topics

contains the exactly resonant terms (this is because there is not any small divisor less than 10~ 2 up to order 12), that implies that the Normal Form is integrable. 3.5.4

The Change of

Variables

It is also necessary to have the (periodic) change of variables that allows to go from the Hamiltonian in normal form to the initial one. This change will be used to translate results obtained with the normal form to the bicircular problem. Here we are going to describe the computation of the change of variables that goes from the final normal form to the Hamiltonian (3.5). Obviously, this change must be composed with the changes done in preceding Sections (Floquet change and translation of the origin) in order to reach the initial coordinates of the problem. The change of variables can easily be obtained from the generating functions G3, G4,..., Gn, applying the same scheme used in Section 3.5.2 to put the Hamiltonian in normal form: Let Xi, yi (1 < i < 3) be the variables of the normal form. Then we apply the following transformation: First step: x3i(x,y>t)

=

Xi + {xi,G3}+y{{xi,G3},G3}

y3(x,y,t)

=

yi + {yilG3}

+ -{{yuG3},G3}

xt(x,y,t)

=

x3 + {x3,G4}

+ ^{{x3,G4},G4}

yf(x,y,t)

= yf + {ylG4} + ^{{yf,GA},Gi}

+ ---, + ---,

Second step: + ---,

+ ---,

where x\ and y3 are the series computed above. This process is continued until x™ and yf are obtained. Then, the final change of variables will be

j/?

=

y?(x,y,t),

where x — (xi,X2,x3) and y = (j/i, y2, y3) are the variables of the normal form and ft y™ (1 < * < 3) corresponds to the variables of Hamiltonian (3.5). Obviously, the scheme used to do the Poisson brackets at each step is the same one used for the Hamiltonian (see Section 3.5.2). x

3.5.5

Going Back to Real

Coordinates

Finally, it is necessary to translate the final Hamiltonian, as well as the changes of variables, to real coordinates. The necessity of this step comes from the fact that we are only interested on the behavior of the system for real values of the coordinates.

Normal Form of Terms of Order Higher than Two

87

If this realification is not done, it is very difficult to study only the part of the final (complex) Hamiltonian that corresponds to the real part of the initial one. 3.5.5.1

The Hamiltonian

To realify the Hamiltonian, we have used the inverse of the change of variables used for the complexification Xi —

lPi

V2

—v/^lg, +pi Vi

V2

, i = 1,3,

and (as before) x2 and y2 have not been changed. As we need to insert the different powers of these expressions in the final Hamiltonian, the program computes once and stores the powers of the monomials x\y[. Then, it is not difficult to compute the different powers x1^y^x^3y%4[x\l'y^B (the powers x^y^ are the same ones as x\y{) as an homogeneous polynomial of degree |fc| in the new variables q and p. Then, the sum of these polynomials produces the realified Hamiltonian. We can write this Hamiltonian in an simpler form by defining the actions 7

1 = 2 ( X l + 2/l)'

/2 =

^ 2 '

J

3 = 2 ^ 3 + 2/3)•

We give that Hamiltonian in Table 3.3, where the three first columns contain the exponents of Ii, I2 and ^3, and the fourth one contains the corresponding coefficient. 1 0 0 2 1 0 1 0 0 3 2 1 0 2 1 0 1 0 0 4 3 2 1 0

0 1 0 0 1 2 0 1 0 0 1 2 3 0 1 2 0 1 0 0 1 2 3 4

0 0 1 0 0 0 1 1 2 0 0 0 0 1 1 1 2 2 3 0 0 0 0 0

-.3005039252506557D+00 -.1385220057080626D-01 . 1004006523604956D+01 .3270045540584760D+00 -.3324999040181953D+01 .8515077021376744D+00 .2307565086566667D+00 .2577837871812799D+00 -.4032569929542033D-02 .3096780901433248D+02 -.2146275456385827D+04 -.4565506222871429D+03 . 1037922842258645D+03 .2494045964589006D+01 -.4784863921619558D+02 .3321899325302790D+02 -.2270782098281886D+00 .2607311336225566D+01 .9052424722400662D-03 -.2709111320415782D+05 .1288209631845397D+07 -.1748183316981774D+07 -.1379292451759401D+06 .2499474130723898D+05

2 1 0 3 2 1 0 2 1 0 1 0 0 6 5 4 3 2 1 0 5 4 3 2

2 3 4 0 1 2 3 0 1 2 0 1 0 0 1 2 3 4 5 6 0 1 2 3

1 1 1 2 2 2 2 3 3 3 4 4 5 0 0 0 0 0 0 0 1 1 1 1

.1380185818563243D+08 -.1807321686550926D+08 .3996634109602074D+07 .1711773835404635D+06 .5520658628803922D+05 -.1864225143531067D+07 .7164000260814325D+06 .1632111440703841D+04 -.5699313229911387D+05 .5304991639454239D+05 -.9347880155609670D+02 .1299414585487088D+04 .1551444240916563D+01 .8912688288853617D+10 -.4715529712174978D+12 .1858745324134717D+13 -.1116963084432885D+12 -.8672410442035261D+12 -.2411838508681229D+11 .2987999377729337D+10 .5578588877780112D+09 -.2821114699272034D+11 .5250659284283364D+11 -.2505479615606837D+11

Normal Form of the Bicircular Model and Related Topics 3 2 1 0 2 1 0 1 0 0 5 4 3 2 1 0 4 3

0 1 2 3 0 1 2 0 1 0 0 1 2 3 4 5 0 1

1 1 1 1 2 2 2 3 3 4 0 0 0 0 0 0 1 1

-.3353580692587518D+04 .4931293498255790D+05 -.3120287556473054D+05 .1018100448948390D+05 .6040532947455749D+01 -.1616391528463035D+04 .1318002728888036D+04 -.1944360487197618D+01 .5049658419448990D+02 .4157193549100161D-01 -.2448924023843400D+07 -.4052415718452325D+08 .8247843021482749D+09 -.1253144279500538D+10 -.5407213341839783D+08 .7974901100840953D+07 .1501613346456005D+07 -.1360849493836872D+08

1 0 4 3 2 1 0 3 2 1 0 2 1 0 1 0 0

4 5 0 1 2 3 4 0 1 2 3 0 1 2 0 1 0

1 1 2 2 2 2 2 3 3 3 3 4 4 4 5 5 6

-.1052914435151395D+11 .1782373214959049D+10 -.1075748940803764D+09 .9001096107686855D+09 .4716462061283490D+09 -.1622067585088890D+10 .4078275712719006D+09 -.3296022936390645D+07 .8144952535614336D+07 -.1020454848295031D+09 .4405125443309778D+08 .8616212602968118D+05 -.2160288537221536D+07 .2189796196165091D+07 -.4191316945624839D+04 .3877318738835637D+05 .5876577678574162D+02

Table 3.3: Hamiltonian in normal form corresponding to L4.

3.5.5.2

The Change of Variables

The reahfication of the change of variables is split in two parts. The first one is very similar to the one of the final Hamiltonian: the realifying change of variables is substituted into the (complex) change of variables found before. The second part is to write this realified series into a binary file in a real format. This process depends strongly on the variables we are working with. Let us see this point with more detail. First, let us suppose that we want to realify a variable that has been complexified. As it has been mentioned above, the (still) complex change of variables is something like x = x' + ,

y=y +

02(x',y'), 02(x',y'),

where the "primed" variables correspond to the normal form and the "unprimed" ones correspond to the original Hamiltonian (3.5). Now, we insert the reahfication in both sides of the change to obtain Ip V2 lq + p V2

-lp> + 02(q',p'), y/2 -yf-iq'+p' + 02(q',p'). ^/2

The next step is to use either of these two expressions to isolate q =

q(q',p')

A Test Concerning

the Normal Form . . .

89

and p = p(q',p')\ q can be isolated from the first equation by taking real parts and multiplying by %/2, and p is obtained from the same equation by taking the imaginary part times —y/2 (a similar process could be applied to the second equation to obtain the same expressions). This means that it is enough to compute only one of these expressions in order to obtain the realified change of variables. This is what we have implemented, though that during the checks of the software, both were computed to check if the same expressions were obtained from any of them. We also note that, in our problem, the right-hand sides of the above equations depend on all the variables, and we must realify only with respect to first and third (positions and momenta). Now, let us suppose that we are realifying with respect to the second variable (both position and moment, X2 and 2/2)- This means that these variables (x2 and 1/2) are not directly changed. The only change performed is on the right-hand side of the expressions x = x' +

02(x',y'),

y = y' +

02{x',y'),

where the realification of xi, j/i, £3 and 2/3 is substituted. With this, we obtain a couple of expressions, that are already real series (imaginary parts must become zero). Note that here, unlike before, we need to compute both expressions. 3.5.6

Checks of the

Software

The final check, used to verify the correctness and accuracy of all the software implementing the above operations, is the following: take an initial condition of the Hamiltonian in normal form, and integrate it numerically to obtain an orbit of this system. Transform this orbit, using the change of variables, to the original coordinates of the bicircular problem (this means to use also the Floquet change of variables and the translation of origin). Now, take the first point of this orbit as initial condition for the bicircular problem, and integrate it numerically to obtain a new orbit, that must coincide with the previous one. This test has been done for different degrees of the change of variables, showing an agreement according to the degree used. 3.6

A Test Concerning the Normal Form Around the Small Periodic Orbit Near L5 in the Bicircular Problem

The Normal Form obtained so far shows that, around the equilibrium point which replaces of the periodic orbit having the synodic period of the Sun, the Hamiltonian behaves like a center x center x saddle. The action variables related to the centers are I\ and I3, while I? is related to the saddle. Along the local formal center manifold, defined by I2 = 0, we can compute the normal hyperbolicity. This is given

90

Normal Form of the Bicircular Model and Related

Topics

by some function A , the derivative of the Hamiltonian with respect to I2 evaluated at points of the form (Ii, 0,13). Of course, at the origin A is the coefficient of I-zFigures 3.6 and 3.7 show curves, in the (Ji, J3)-plane of constant value of A. This function is increasing when we go from the right lower corner to the left upper one. In Figure 3.6, A ranges from -0.018 to -0.002, while in Figure 3.7 ranges from -0.017 to -0.002. The curves are computed using terms of the Normal Form up to degree 5 (Figure 3.6) and 6 (Figure 3.7). The criterion used to stop the computations is that the contribution of the terms of the last degree must be less that 1 0 - 4 . This is a very mild requirement, but a stronger condition would be satisfied only in a very small neighborhood around the origin. The poor convergence conditions of the Hamiltonian are also seen in the fact that, for points in the /i-axis the contribution of the terms of order 6 is larger than the one of order 5.

o . 04

O . 035

O . 03

O . 025

O . 02

O . 015

O.Ol

O . 005

~ O

o.OOO]

0.OO0-4

O.OOOS

O.OOOS

o.u^i

~-~

.

Fig. 3.6 Lines of constant unstable eigenvalue from the Normal Form of the Bicircular model around the periodic solution replacing L5, using expansions to order 5 in the actions.

However, we can learn something from the previous plot. Using a higher order Normal Form (not available for memory requirements) it would be possible to compute lines with small values of A (say, A close to zero). For A = 0 we are on the boundary of the region where the behavior is of the type center x center x saddle and beyond the boundary a region where 3-dimensional tori appear is found. In this region many points should be stable. From both figures, it seems that this boundary reaches the /3-axis at some value close to 0.028. Taking into account that the ^3 action is essentially |((/f -\-p\), and (93,^3) is close to (z,z), we have that the boundary must cut the z-axis, assuming z = 0, near \/2 x 0.028. This gives a value of z close to 0.24, in good agreement with the values of z, in the numerical explorations of the bicircular problem, for which a significant fraction of points starting with zero synodic velocity subsists for a long time. See next Section to obtain a much more accurate determination of this bifurcation.

A Test Concerning

the Normal Form . . .

91

Fig. 3.7 Lines of constant unstable eigenvalue from the Normal Form of the Bicircular model around the periodic solution replacing L5, using expansions to order 6 in the actions.

For completeness we display the values of A, wi and u>3 (which are the frequencies again) for some values of I\ and I3 (I 2 being zero).

h

Wi

UJ3

A

order

0.0

0.0

0.000981

0.000004 0.017695

-0.3005039 -0.299886 -0.296519

1.0040065 1.004233 1.003867

-0.0138522 -0.018 -0.008

5 6

h 0.0

A value of wi = -0.290680 is attained at h = 0.000831, I3 = 0.039994 (see Figure 3.6). We can summarize the variations of the frequencies and A by saying that, always in absolute value, W3 and A decrease from the right lower corner to the left upper one in Figures 3.6 and 3.7, while UJ\ decreases when going from the origin to the right upper corner. A numerical test concerning the robustness of the stability of orbits in the bicircular problem has also been done to stress on the change of the stability properties at some level of z. We describe it next. In Table 1.2 some initial conditions for periodic orbits of the vertical family of the RTBP were displayed. Then, some of these initial conditions were taken as initial conditions for the bicircular problem. We have performed the following experiment. As initial conditions we have taken, on the plane z = 0, the values corresponding to the vertical periodic orbits of the RTBP (with z-amplitude roughly 0.1, 0.2,..., 0.9) but with displacements in each one of the positions and momenta (but z) having values -0.02, - 0 . 0 1 , 0, 0.01, 0.02. Also we have taken 9 equally spaced values for the initial phase of the Sun (equal to 0, 1/8,..., 7/8, revolutions). This gives, in all 3125 initial conditions on positions and momenta and 8 initial phases. This has been integrated for the 9 different levels of z-amplitude. A summary of results is displayed in the next Table

Normal

92

Phase Level 1 2 3 4 5 6 7 8 9

Form of the Bicircular

Model and Related

Topics

0

1

2

3

4

5

6

7

8

total

vmimi

vmima

2755 2738 1329 496 289 322 592 251 359

122 132 40 45 65 27 67 49 26

187 169 402 211 169 95 208 50 121

28 39 40 64 56 37 54 37 26

25 39 440 226 150 90 214 54 133

6 4 51 71 82 37 40 34 21

1 3 224 127 146 99 290 81 204

0 1 45 82 67 48 42 41 42

1 0 554 1803 2101 2370 1618 2528 2193

724 788 9070 17678 19734 20763 16679 21643 20045

0.0004 0.0002 0.0001 0.0002 0.0002 0.0005 0.0207 0.0979 0.2466

0.0370 0.0452 0.0442 0.0763 0.1378 0.2131 0.2900 0.4107 0.5596

Table 3.4 Numerical test concerning the robustness of the stability of the vertical orbits. See explanation in the text

Here level refers to the approximate ^-amplitudes of the orbits. Each initial condition and which each initial phase has been integrated till 1000 synodic revolutions of the Sun, unless a escape (y less than 0 at some moment) is produced. For a given amplitude and initial conditions, it can happen that the point subsists only for some of the phases. That number of phases ranges between 0 and 8. The number of points , 4>(k), which subsist on a level for a given number of phases , k, is displayed on Table 3.4. Also the cumulative sum J2 k * <j>(k) is given under the heading total of the above Table. Finally we show the minimum and maximum value, on a given level, of the minimum modulus of the velocity along the orbit, to see how far are those orbits from the ones used before (in Chapter 1) to do numerical simulations, which were started with zero synodic velocity. As a conclusion, we realize that a big part of the points subsist for all the phases for values of the ^-amplitude larger than 0.3 and even an important part for 0.3.

3.7 3.7.1

On the Computation of Unstable Two-dimensional Tori The Equations

and the

Algorithm

The vertical periodic orbits of the circular 3-dimensional RTBP give rise, in the case of the bicircular problem, to two-dimensional tori. One of the frequencies is the vertical one, the other being the solar frequency in the synodic system. Let £, 7/, C be the coordinates with respect to the L5 point in the synodic system (for the Li case is similar). Then, the equations of motion are ('= d/dt)

r," = C" =

-m-at+ln -C+—V

+ ^V,

On the Computation

of Unstable Two-dimensional

Tori

93

where a = (\/27/4)(l — 2/i) and V is the potential accounting for the higher order terms of the RTBP plus the solar effect:

v = (i-M)E^((1/2)g-(^/2),?)p"+^E^("(1/2)c"(V5/2),?)p" n>3

\

J - ^ V ^ P

(ti

+

P + P-

V2)

J C0S

n>3

\

P

/

( ^ s Q ~ fa + V 3 / 2 ) sin(a;sf) ^\ # n

»* B VH

«

Jar

In the last expression one has p2 = £2 + rf + (2, R2 = (£ + /i - 1/2)2 + fa + \/3/2) 2 + C2, and a s , m j , ws denote the radius of the orbit, the mass and frequency of the Sun. We write first f = £0 + £, 77 = rj0 + rj, C = Co + C, w h e r e £0, % (Co = 0) are the coordinates of the periodic solution, with frequency uis, which replaces L5 in the bicircular problem. Let ip = exp(iwgi). Then £0 = S * £00*^*', Vo = S * Tlook'4'k- The coefficients are known from the numerical solution and numerical Fourier analysis. We look for a solution of the form (with complex coefficients)

l,3,k

C = £&*aW> with I > 0, j,k £ Z, where 0 = exp(«a;i), a is some (vertical) amplitude, which is being considered as a parameter, and u is an amplitude depending (vertical) frequency, of the form u = OJQ + a^a 2 + W4Q;4 + .... As normalizing condition on a we choose Ci,i,o = 1/2, 0,i,o = 0, VZ > 1. As the solutions are real, we have £i,j,k = £; -j -k a n d similar relations hold for the 77 and C coefficients. Furthermore, the symmetries of the problem imply that in £, 77, the indices / and j are even, in the C they are odd, and in all the cases \j\ < I. Let us denote by & the sum J2j,k £iik$ipk- So £ = a£i + a 2 £ 2 + — Similar representations are used for rj, C- First we discuss how we have computed the terms in Ci (£1 and 771 being zero) and then how to proceed, in a different way, to compute ?n> I n i Cn> n > Z.

The standing equation is C" = —C + (9/d()V, and this, to order 1 in a, reads C" = —Ci + JT7^(£O>77O,0)CI- Furthermore, we keep C110 = 1/2, and we have to determine UIQ, where (Ciik^^)" = Cijk^^V-Owo + ku>s)2]- By equating terms k in <jpijj in the previous equation for Ci, we get a linear system (but the problem is nonlinear because of wo) in the Cijfc coefficients | j | < j m a x , 0 < k < kmax. As the system is homogeneous, one has to take the right value of UJQ to make the solution possible and some normalization, as it has been said. There are several possible ways to solve the problem. We can get OJQ from the numerical integration of the

94

Normal Form of the Bicircular Model and Related Topics

variational equations along the periodic solution (£o,»7o>0). Another possibility is to use an iterative process. We write (C{' + (i){k+1) = ^ ^ ( £ o , % , 0 ) ( C i ) ( * ) and we start with the initial guess Ci = ( l / ^ ) ^ (notice that we had to add to the representation used, the complex conjugate and that if k = 0 then the range of j in the representation is 0 < j < jmax)- Let us denote by c[k>(, tp) the product ^•l / (^o,^o 1 0)(Ci) ( ' : ^. As we want to keep Ci,i,o = 1/2, this gives immediately the value of w0 by equating the O +1) 2 terms in 0V = [1 - ( 4 * ) ] ( l / 2 ) = c<*>i0. Then the terms c } * ^ , (j,k) ? (1,0) are given by c}**1* = 4*j i/fe /(l - (;'4* + 1 ) + kcjs)2). When C ( * +1) is available, we substitute to get the new q ' and the process is repeated. Unfortunately, that process, as it is described, fails to converge. However, it has been seen that it has just one unstable direction. This suggests to use an extrapolation method (Aitken) that then leads to convergence. The value obtained for w0 is 1.0040065236..., in perfect agreement with the monodromy matrix. When £i is available, we could continue the iterative-extrapolation process. However, the number of unstable directions increases when we look for (£„, rjn, £„) (remark that if n is even we had to solve only for (£n,?7n) and if n is odd, only for £„). Hence, we need a different method. We describe first the procedure for (fn,J?n). We write the equations as d2V

/3 C

-

2

Vn - i^n

~ O-Vn + -gTf (&, Vo, 0)f„ + QTQ-(to,

dV -

_

-

""^"(fn-l'^n-uCn-l)

( 2

d2V

9

-a —

Sn-1 +

d2V

Vo, 0)%

, 2r

ln-l

d2V

a

Vn + C ~ I Zn + ^Vn + ^ - ( f r , % , 0)£ n + -g^-(Co,VO,0)r]n

-l'Vn-l'Cn-l)

Vn-1

2

£n-l

where ^„_ 1 means £o + a£i + • • •&n~1£n-i (and similar for 77, £) and [ ] | n denotes the terms in an. Notice that in the [ ] | n terms, there is a contribution in an when £, 77, £ are known to order n - 1 in a because of the nonlinearities in the derivatives of V and because of the powers of a appearing in ui in the time derivatives. For C« (n odd) the procedure is similar, but keeping £n,i,o = 0 allows the determination of ujn-\ by equating the terms in a71^1^0. We remark that the systems of linear equations which appear to get (^ n ,i)„) for n = 2,4,... have the same matrix. The same thing happens for £„, n = 3, 5,... Furthermore, the systems for different values of j are uncoupled. Some caution has to be taken with the poorly conditioned character of the matrices, specially in the £n case, j = 1. It has been found very useful, to decrease the numerical

On the Computation

of Unstable Two-dimensional

Tori

95

errors, to work in quadruple precision complex variables (or simulate it, depending on the software). The computation of the derivatives of V is shortened by the use of recurrences derived from the ones of Legendre polynomials. 3.7.2

Truncated

Power

Series

Results

The previous algorithm has been implemented with lmax (and hence, jmax) = 41 and kmax = 18. Partial results are shown in Table 3.5. From the results of Section 3.6, we know that the previous 2-dimensional unstable tori should be convergent up to moderate values of a (at most of the order of 0.25). In fact, the coefficients of to seem to give a radius of convergence (limn_yoo>n even con ) close to 0.219. If, instead, we consider the norm of £ n as |£ n | = J3 • k |£n,j,*|> w e can also get an idea of the radius of convergence of the expansions with respect to a. The same is true for the \r}n\, \(n\. All of them give essentially the same radius of convergence as w. Notice that for n large the numerical errors propagated by the successive solutions at the previous orders, start to show up, giving a slightly irregular behavior of the components (see Table 3.6). U)

0 2 4 6 8 10

. 1004006523604968D+01 -.4044304089436558D-02 -.2769536717320718D-03 .2075410948439819D-01 .5039834259073994D+00 .1162380610009936D+02

0 1 2 3 4 5 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3

.3745042052603002D-04 -.3818924233569763D-02 .2835152244081680D-02 -.1165679992790928D-05 - . 1367834679315942D-05 .4671017897076164D-07 -.1121097273213035D-02 .6458349043578387D-02 -.1196396967312828D-03 -.3280604486365375D-03 .7255393204866987D-06 .5152236432631926D-03 -.6213438582006870D-01 -.4709623399906364D-02 -.6041180284133439D-04 .1375508737621625D-04 .6039559637006229D-06 .6005293360713431D-01 -.2904788151237182D-03 .8921151012132396D-04 .8732600053784776D-06

Z

0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 0 0 0 0 0 -2 -2 -2 -2 -2 0 0 0 0 0 0 2 2 2 2

.0000000000000000D+00 -.1022178410639085D--02 -.3737789804083809D--02 .9766764114222627D--05 -.2454625154183015D--05 .9199193030594531D--09 -.1983737551477677D--03 -.8073050962742528D--02 .1141441416031814D--03 .4217844175632335D--04 -.4656517805432561D--06 -.8171297260147378D--35 -.1653426485620139D--01 .9271387612106054D--02 .8075956728464732D--04 .1819979638414314D--06 .1698969150493526D--06 .1059718083851778D+00 -.1058803231885023D--05 -.1604322848074182D--03 .1437432627955780D--05

96

Normal Form of the Bicircular Model and Related Topics 2 2

2 2

4 5

-.6965192217117018D-06 .5320169403776209D-08

.2599830018924677D-07 .1010431908133230D-07

0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 0 0 0 0 0 -2 2 2 2 2 0 0 0 0 0 0 2 2 2 2 2 2

0 1 2 3 4 5 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5

-.5078650806217793D-03 -.9473721629589176D-03 .3708869713954297D-02 .1833297779271841D-05 -.2389239527149953D-05 -.1240971626967073D-09 -.6738672336045910D-03 -.2657270002350977D-02 -.7169285446551132D-04 -.9932115265641339D-04 .3937342413347967D-06 -.1445182264456548D+00 -.1581741200787926D-01 -.8644180980115640D-02 .9007778542241214D-04 .6995234956886275D-06 -.9228407560224826D-07 -.1073586877153194D+00 .7246278095873423D-05 .1683744726810605D-03 -.1563620247124998D-05 -.3048689818817711D-07 .1079865523155058D-07

.0000000000000000D+00 .2686112212988688D-02 .1364511726610659D-02 .5610521400905868D-05 .1531406766470087D-05 .4665636020878598D-07 .1004085216124557D-02 .1288830693098316D-01 .3861804727808925D-04 .2352675805429158D-03 .3539691071992446D-06 .1543525837671003D-35 .4407381887758294D-01 .3984072433665252D-02 .4674387150685164D-04 .9018363205145137D-05 .7307695174304373D-06 .5976304284877290D-01 .3023447879546918D-03 .9806234943642511D-04 .9444098545915824D-06 .7674256488120364D-06 .5640523145870803D-08

1 2 3 4 5 0 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 0

.1530079512981396D-02 .9429100726308741D-02 -.3544156931199900D-04 .1035349651777416D-04 .1082273874774495D-06 .5000000000000000D+00 -.5781827239267952D-03 -.3875168837666344D-03 .6780047298688830D-05 -.1340869632616841D-05 -.2344335473013481D-07 -.1746498964093720D-03 .5458391884454499D-02 -.2506356025949434D-04 .1657824491484841D-04 .2662041574561901D-07 .2588776193225446D-01 -.1474779561091857D-01 -.5774738694169348D-03 .5685267362298090D-05 .1274183890785477D-05 .0000000000000000D+00

-

c

.2674146952022885D-02 .1552010121200999D-01 .1490474369293529D-05 .2019224459838174D-04 .1657606004114359D-06 .0000000000000000D+00 .1011315317909751D-02 .6379357799823534D-03 .9270634384520497D-07 .2504881739399120D-05 .3992220258967982D-07 .1676083365013458D-03 .1206007489653678D-02 .8640418756855263D-04 .4820058937694783D-03 .1493676040438885D-05 .4354466199137452D-01 .5705840637101454D-02 .3807800738358692D-04 .6954376052873851D-04 .2216101565724158D-05 .8665447349837359D-02

On the Computation of Unstable Two-dimensional Tori 3 3 3 3 3 3 3 3 3 3 3

1 1 1 1 1 3 3 3 3 3 3

1 2 3 4 5 0 1 2 3 4 5

-.9624161105162387D- -02 .2991845381536182D- -03 .1036223977493041D- -03 -.1087830623482743D- -05 -.3215306137835351D- -06 -.5407270139673915D- -03 .2937996270110554D- -05 .4577383977958574D- -05 -.8282561199575211D- -07 -.1180247864726370D- -07 .8968872182157179D- -09

.1660254894885864D- -01 .3950313400907980D- -04 .1142442993724218D- -05 -.1345228926370532D- -05 -.5981322123219733D- -06 -.1985229093854305D- -04 -.7020550707989282D- -05 .4048628534045530D- -05 .6646427458455313D- -07 -.4801404116373339D- -07 -.8749519934790405D- -10

Table 3.5: A small part of the solution obtained for the 2-dimensional unstable tori whose basic frequencies are the solar and the vertical ones, is shown. The values of u; as function of a and £, r] and £ as functions of a and the exponentials related to the two angles are given.

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

w .100400D+01 .404430D-02 .276953D-03 .207541D-01 .503983D+00 .116238D+02 .261223D+03 .576511D+04 .125580D+06 .270934D+07 .580404D+08 .123691D+10 .262616D+11 .556124D+12 .117566D+14 .248292D+15 .524165D+16 .110661D+18 .233726D+19 .494006D+20 .104511D+22

0 2 4 6 8 10 12 14 16 18

.489778D-02 .196064D+00 .899886D+00 .172544D+02 .332035D+03 .640986D+04 .124186D+06 .241562D+07 .471914D+08 .926194D+09

i .157560D+02 .382136D+01 .115518D+00 .202928D+00 .208225D+00 .210944D+00 .212863D+00 .214260D+00 .215292D+00 .216056D+00 .216618D+00 .217024D+00 .217307D+00 .217492D+00 .217600D+00 .217644D+00 .217639D+00 .217592D+00 .217514D+00 .217412D+00

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

.613512D-02 .138518D+00 .123815D+01 .237356D+02 .456873D+03 .882262D+04 .170993D+06 .332743D+07 .650326D+08 .127693D+10 .251957D+11 .499663D+12 .996014D+13 .199577D+15 .401983D+16 .813808D+17 .165577D+19 .338509D+20 .695257D+21 .143426D+23 .297108D+24

.158052D+00 .466772D+00 .228372D+00 .227959D+00 .227597D+00 .227188D+00 .226737D+00 .226247D+00 .225725D+00

1 3 5 7 9 11 13 15 17 19

.500341D+00 .573760D-01 .104372D+01 .199921D+02 .384768D+03 .742906D+04 .143958D+06 .280077D+07 .547271D+08 .107433D+10

i\

.210454D+00 .334476D+00 .228395D+00 .227930D+00 .227561D+00 .227147D+00 .226691D+00 .226198D+00 .225673D+00 .225123D+00 .224556D+00 .223978D+00 .223396D+00 .222818D+00 .222250D+00 .221697D+00 .221164D+00 .220654D+00 .220170D+00 .219713D+00

c

.295302D+01 .234461D+00 .228488D+00 .227944D+00 .227579D+00 .227168D+00 .226714D+00 .226223D+00 .225699D+00

98

Normal Form of the Bicircular Model and Related Topics 20 22 24 26 28 30 32 34 36 38 40

.182662D+11 .362062D+12 .721361D+13 .144470D+15 .290843D+16 .588521D+17 .119684D+19 .244575D+20 .502114D+21 .103540D+23 .214405D+24

.225178D+00 .224612D+00 .224034D+00 .223453D+00 .222874D+00 .222304D+00 .221749D+00 .221213D+00 .220701D+00 .220214D+00 .219754D+00

21 23 25 27 29 31 33 35 37 39 41

.211934D+11 .420179D+12 .837397D+13 .167751D+15 .337852D+16 .683700D+17 .139109D+19 .284283D+20 .583760D+21 .120527D+23 .249632D+24

.225148D+00 .224586D+00 .224001D+00 .223425D+00 .222828D+00 .222295D+00 .221694D+00 .221208D+00 .220677D+00 .220077D+00 .219731D+00

Table 3.6: For the variables u>, f, r\ and £ the nonzero norms of the coefficients for orders of a up to 41 are given, and also the estimated radii of convergence. .01 .02 .03 .04 .05 .06 .07 .08 .09 .10 .11 .12 .13 .14 .15 .16 .17 .18 .19 .20 .21 .22 .23 .24 .25

.114145D--13 .114284D--13 .113139D--13 .114804D--13 .118083D--13 .115082D--13 .111439D--13 .113659D--13 .115845D--13 .115082D--13 .118048D--13 .152708D--13 .148928D--12 .337156D--11 .669266D--10 .112226D--08 .162949D--07 .210096D--06 .246850D--05 .272526D--04 .292881D--03 .319453D--02 .364627D--01 .481799D+00 .142723D+02

.117545D--13 .125351D--13 .119991D--13 .119106D--13 .121951D--13 .121604D--13 .121118D--13 .122923D--13 .121509D--13 .127572D--13 .124831D--13 .175892D--13 .204631D--12 .490744D--11 .957271D--10 .156734D--08 .221698D--07 .277842D--06 .319889D--05 .352209D--04 .380884D--03 .426555D--02 .514659D--01 .589514D+00 .204970D+02

.137043D--15 .279290D--15 .395517D--15 .555112D--15 .700828D--15 .915934D--15 .106859D--14 .130451D--14 .154043D--14 .163758D--14 .181799D--14 .262290D--14 .291295D--13 .710931D--12 .148328D--10 .259666D--09 .391360D--08 .532121D--07 .660951D--06 .769544D--05 .873664D--04 .102051D--02 .126245D--01 .159405D+00 .403816D+01

Table 3.7: Test of errors for the unstable 2-tori obtained by the Lindstedt-Poincare method. The first column gives the amplitude a. The other three columns give the residual accelerations m. x, y and z, respectively.

A first test has been passed as follows. Consider a value of a a n d look at t h e approximate quasi-periodic solution provided by t h e formulas above. For a given

On the Computation

of Unstable Two-dimensional

Tori

99

value of t one can compute £, rj, £, £', rj', (', £",?/', C"- Then, from the positions and velocities, one obtains the acceleration from the direct equations of motion for the bicircular problem. The difference with respect to the values computed from the series is the residual acceleration. Table 3.7 shows some results for a time span of 100 adimensional units with time step 0.1. There is a remarkable agreement till a = 0.16 and still acceptable till a = 0.2. Figures 3.8 and 3.9 show the (x,y) and (x, z) projections of the analytic orbits computed for a = 0.16 and a = 0.20.

3.7.3

Discussion

Invariant 2-dimensional unstable tori have been found by a symbolic manipulation implementation of a version of the Lindstedt-Poincare method. By a suitable modification and an additional computational effort, one can compute also 3-dimensional unstable tori. The frequency to be added is equivalent to the long period frequency of the RTBP. We also note that the action of the Sun produces an increase of the vertical frequency (whose limit is 1 in the RTBP when the vertical amplitudes goes to zero). However, the behavior of the "vertical" frequency when the amplitude increases is preserved. An increase of the amplitude produces a reduction of the frequency. The boundary of convergence is associated to a bifurcation (as was already outlined in the previous Sections): the tori lose its hyperbolic character, becoming stable (in some weak sense). They can be considered

Fig. 3.8 Projection {x,y) (left) and (x,z) (right) of the 2-torus of z amplitude equal to 0.16. The final time is 1000 units (recall 2n is equivalent to the lunar period). The time step is 0.1 time units. The distance unit is the Earth-Moon distance.

100

Normal Form of the Bicircular Model and Related

Topics

aira=0.2,x-y,tf-100OO

Fig. 3.9 Projection (x,y) (left) and (x,z) (right) of the 2-torus of z-amplitude equal to 0.2. The final time is 10000 units. All the units as in the Figure 3.8.

as "big" KAM tori of the bicircular problem. Hence, in the next Section we shall search for higher vertical amplitudes to look for these stable tori and the size of the domain of stability. As sketched in Chapter 1, the loss of stability, for ^-amplitudes around 0.22, is due to a resonance between the "short period frequency" and the solar one.

3.8 3.8.1

Big Tori and Stability Zones An Autonomous Periodic Orbits

Intermediate

Vector Field and its

Vertical

For larger amplitudes one can look also for 2-dimensional tori (and also for 3dimensional and 4-dimensional tori). Despite this is feasible and we have some results in this direction, finally we present here a different approach. We go back to the equations as presented in Section 3.7. As the terms in V coming from the Sun (to be denoted as Vs) have the factors 1/ag, only a few terms are significant. Let us expand the Legendre polynomials and denote by qj the numerical factors ms/aJs. Only q3 to q-j are requested, at most. The function Vs has an expression of the form Vs = go + 5i,c cos 0 + 0i )g sin 0H

\- g6tCcos69 +

g6,ssin69,

where 6 denotes wst. Let x = £ - 1/2 + //, y - n + y/3/2. Then g0 depends only on p = x2 + y2 and q = z2. For the equations of motion we need dg0/dp, dg0/dp,

Big Tori and Stability

Zones

101

which are given by dgo dp dg0 dp

I + 1(9p ~ 36q) + 2k{75p2 ~ 9°°Pq + 6°°q2)' "f + H ( - 3 6 P + 2iq) + 2?6 (_450p2 + U0°Pq ~ 240
From this it follows that the "autonomous" contribution of the Sun, to be added to the equations of the RTBP, is 2x(dgo/dp), 2y(dgo/dp) and 2z(dgo/dq). Similar expressions are derived for the contributions in c o s # , . . . , sin 60, but shall not be used here. Adding the contribution of go to the RTBP we have an autonomous skeleton of the stable tori of the bicircular problem for large enough amplitude. Table 3.9 gives some of these orbits and their stability characteristics. 3.8.2

Simulations

Around

the Periodic

Orbits.

Region of

Stability

Let A be the initial point of one of the orbits in Section 3.8.1. We have taken (arbitrarily) z = 0 for t = 0. Let us look for the "top point", T, of such orbit, for which z is a maximum (and hence z = 0). Of course, in general T will have values of ±T and yr different from zero. Let XT, VT be the x and y coordinates of T. Then we select initial conditions of the form {xT + nxSx,yT + ny6y,ZT,XT,yT,0)They differ from the ones considered in Chapter 1, which were taken with x = y = z = 0. In some sense, the new conditions are "more adapted" to the possible stable tori. The stepsizes 5X and Sy in the simulations are taken equal to 0.005 and nx, ny are integers ranging from —40 to 40. As stability criterion we have used the same one of Chapter 1: y > 0 for all time of integration until some final value £/, which was taken to be 1024 lunar revolutions. Guided by the results on unstable tori of Section 3.7, we have considered periodic solutions with values of ZA (very close to z?) ranging from 0.24 to 0.9. Larger values are associated to periodic orbits which are unstable. Figure 3.10 shows a sample of results for different values of ZA- This give us a good feeling about the size of the region where stable tori can be found. The Sections have a rather sharp character for small values of ZA (up to 0.50) and for other values like 0.84, 0.88, etc. For values like 0.62 they look much more irregular. Points close to the "fuzzy boundaries" of the sections will probably escape in a longer simulation. Some additional resonances seem to show up also.

3.8.3

Frequency

Analysis

For the subsisting orbits found in Section 3.8.2 we have carried out a determination of the basic frequencies (following the procedure given in Appendix B). To this end we have considered the harmonics in x, y and z giving a contribution at least equal

102

Normal Form of the Bicircular Model and Related

Topics

Fig. 3.10 x and y displacements with respect to the "top" point of the periodic orbits of the RTBP plus the autonomous part of the Sun perturbation, which give rise to nonescaping orbits. The z-amplitudes are equal to 0.4, 0.5, 0.6 and 0.8. The units in both axes are 0.005 adimensional units in the x and y directions.

to 0.01. We consider that the orbit is not lying on a torus if the frequency determination process is unable to obtain the frequencies or if it considers that to obtain a good approximation by a quasi-periodic function more than 4 basic frequencies are required. Figure 3.11 shows the subsisting points after this frequency based filter has been passed. It displays the values corresponding to z-amplitudes equal to 0.4, 0.5, 0.6, 0.8. In the last one there appear several holes inside the main region. They seem to be associated to resonances. However, it is possible to have confined motion for a very long time even in the presence of a resonance. Figures 3.12 to 3.14 show the variation of the basic frequencies letting aside the one of the Sun (that is, the vertical frequency and the ones corresponding to short and long periodic orbits

Big Tori and Stability

Zones

103

in the RTBP). In the same figures several combinations of frequencies which give values close to zero have been also displayed as a function of the displacements in x and y of the initial point.

Fig. 3.11

Points as in Figure 3.10 which allow for a good determination of the 4 basic frequencies.

The Tables 3.8 and 3.10 give additional information on these frequency analysis. The first one gives the range of variation of the basic frequencies (not the solar one, of course) along the explorations done for the different levels of the z-amplitude. The second one displays the strongest resonances up to order 10. For each one of the levels of the ^-amplitude we look for the linear combinations of the basic frequencies which are uniformly small across the full section. In particular one can see the resonances displayed in Figure 3.12.

104

Normal Form of the Bicircular Model and Related 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(1.00365670,1.00439125) (1.00335670,1.00491989) (1.00295777,1.00420427) (1.00243323,1.00346385) (1.00176334,1.00281765) (1.00088662,1.00201358) (0.99967926,1.00124390)

(.93083716,-94071455) (.93699504,-95018051) (.95033153,-95635591) (.95793837,-96338729) (.96784821,-97236054) (.97572750..98141234) (.98654848,-99371418)

Topics

(.29029414,.29899431) (.27898123..29451710) (.26940446,-27712484) (.25538684,-26281568) (.24488907,-24888734) (.22716986,-23357627) (.20490696,-21493934)

Table 3.8: Range of variation of the main frequencies for some levels of the zamplitude. The first column contains the z-amplitude. Columns 2 to 4 contain the range for the vertical, the short period and the long period frequencies, respectively. We recall that the solar frequency is 0.92519599. 6.25698107457975 -.487849418376448

-.487849418376686 1.0000444493E-06

.8649503549232797 1.88299798419136

-.864950354922858 -.61359126695675

6.2569811009829 -.487849299332599

-.487849537420395 1.0000000000E-03

.8649502812513738 1.88299823143853

-.864949859125107 -.613589903439733

6.256983714966060 -.487837512662962

-.487861322628206 9.9999999999E-03

.8649429877036223 1.88302270731691

-.864900774354472 -.613454915788536

6.25724581279924 -.486645623370183

-.489038519447405 .1000000000000255

.8642134281766597 1.88545416244488

-.859984822433008 -.599963007792335

6.25804860116908 -.482871169924057

-.492588903241109 .2000000000000089

.8620007956037174 1.89262927690402

-.84499611868331 -.559168124572937

6.25941589990124 -.476015104019146

-.498442997261571 .2999999999998991

.8583119236900273 1.90395899621289

-.819695773828183 -.491523341042486

6.26139451651519 -.465147585154271

-.506480287504519 .3999999999991264

.8531646049317756 1.9185546496548

-.783555854248697 -.397687245698371

6.26405678597042 -.448778366937917

-.516473036725464 .4999999999990261

.8466307077298531 1.93527741249927

-.735708872860995 -.278752185914777

6.26751144658584 -.424589889829194

-.528004194955053 .5999999999994341

.8389063357511194 1.95283663528541

-.674718506710829 -.136108692822348

6.27192961980021 -.388890994818804

-.540304203302417 .6999999999994738

.8304555098853972 1.96990909926993

-.59800951058024 2.9372080044E-02

6.27762087280927 -.335254539425624

-.551848695784324 .7999999999990916

.8223557883367272 1.98527441375032

-.500209717676416 .220071608359958

6.285342755324400 -.2496109946717781

-.5588291350564894 .8999999999990472

.8175144172680369 1.997958452484640

-.3665491685839273 .4517635019627926

Table 3.9: Selection of orbits for the RTBP plus the autonomous part of the solar perturbation in the bicircular problem. For each orbit two lines of data are given. The first line contains the period, the initial values of x, y and px, the value of z being equal to zero. The second one contains py, pz and the two traces associated to the 4 eigenvalues different from 1.

Big Tori and Stability

Zones

Fig. 3.12 A rough 3-dimensional representation of the variation of some frequencies, for a given value of the z-amplitude, as a function of the x and y initial displacements. The frequency is shown along the vertical direction. Here the ^-amplitude is 0.4. From the top left plot to the right bottom one the six following frequencies have been represented: ui\, C02, W4, wi — 2o;3 + 3u;4, 2u>i - 4u>2 +U13 + 3o)4 and u\ - 4w2 + 3^3, where UJ\ to UJ4 denote the vertical, short period, Sun and long period frequencies, respectively.

106

Normal Form of the Bicircular Model and Related

Topics

Fig. 3.13 Same as Figure 3.12 for z-amplitude 0.8. The frequencies represented are wi, ui, UJ4, 2a>i - 3^2 + W3, wi + 3oJ2 - 4^3 - 014 and 3wi - 4^3 + 3a;4.

Big Tori and Stability

Zones

107

II

'//, .: ',

'', ., , • LI 7 /,/////' i'

m>

. "il/iilhi,

.//"

'S

\

I1' I 1 ! U,!i' W/f-.

\ •

^

,

-

W\k\U\U\\V

v

/

\ Fig. 3.14 Same as Figure 3.12 for z-amplitude 0.5. The frequencies represented are uii, u2, u>i, ui - 3w2 + 3a»3, 2LJI - 2oJ2 - u>3 + 3UM and 3u>2 - 4w 3 + 3^4.

II

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C O O O l O i O l C O O l C O O H ^ ^ ^ H H

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CO

t 0 C 0 C 0 C 0

Big Tori and Sti.ability Zones 3 - 2 0 - 4 3 1 -4 -1 3 -1 -1 -4 2 1 - 2 - 4 1 2 -2 -4 1 1 0 - 7 1 0 1 -7 0 2 0 -7 0 1 1 -7 0 0 2 -7 3 - 4 0 3 2 - 5 3 0 1 2 - 4 3 0 3 - 4 3

( -0.00907, 0.03133) ( -0.01236, -0.00752) ( 0.02208, 0.05722) ( 0.00453, 0.03072) ( -0.04398, -0.02171) ( 0.01966, 0.06905) ( -0.01068, 0.04307) ( -0.02885, 0.01661) ( -0.05909, -0.00951) ( -0.08948, -0.03543) ( 0.01431, 0.02404) ( 0.00185, 0.03109) ( 0.01331, 0.04728) ( -0.03931, -0.00038)

2 2 - 4 - 1 2 - 2 1 - 4 1 2 - 2 - 4 0 3 - 2 - 4 0 0 2 - 7 3 - 3 - 1 3 3 - 2 - 2 3 2 - 5 3 0 2 1 - 4 3 2 0 0 - 8 1 2 - 4 3 0 3 - 4 3

109 (-0.03665,-0.03012) (-0.04619,-0.00863) ( 0.02510, 0.05028) (-0.01666, 0.00643) ( 0.01068, 0.06268) (-0.02746,-0.01324) ( 0.00679, 0.02249) (-0.03502,-0.00717) ( 0.03146, 0.05720) (-0.09620,-0.03646) (-0.01239, 0.01727) (-0.05625,-0.02254)

z = 0.7 1 - 1 0 0 ( 0.03042, 0.03453) 1 - 2 1 0 (-0.01673, -0.00812) 1 0 0 - 4 ( 0.00633, 0.02326) 0 1 0 - 4 (-0.02673, -0.00747) 2 - 3 1 0 ( 0.01369, 0.02640) 3 0 - 3 - 1 (-0.01908, -0.01202) 2 - 1 0 - 4 (' 0.03894, 0.05435) 1 - 2 0 4 ( 0.03822, 0.05988) 1 - 1 1 - 4 (-0.03794, -0.02254) 0 2 - 1 - 4 ( 0.01674, 0.03940) 2 - 4 2 0 (-0.03346, -0.01624) 2 0 - 3 3 (-0.03612, -0.02484) 4 -2 -2 -1 (-0.03109, -0.02200) 4 -1 -3 -1 ( 0.01382, 0.02077) 3 1 - 4 - 1 ( 0.02445, 0.03485) 2 2 - 4 - 1 (-0.00872, 0.00411) z = 0.7 2 - 2 1 - 4 (-0.00586, 0.01148) 1 3 - 4 - 1 (-0.04223, -0.02662) 1 - 3 1 4 (-0.00866, 0.01647) 3 - 5 2 0 (-0.00304, 0.01828) 3 - 2 - 2 3 (-0.05142, -0.03562) 3 - 1 - 3 3 (-0.00454, 0.00782) 2 1 - 4 3 ( 0.00817, 0.02001) 2 0 0 - 8 ( 0.01267, 0.04652) 1 2 - 4 3 (-0.02524, -0.01177) 1 1 0 - 8 (-0.02036, 0.01578) 1 0 1 - 8 (-0.06398, -0.03109) 0 2 0 - 8 (-0.05346, -0.01495)

z = 0.8 1 - 1 0 0 ( 0.02030, 0.02628) 1 - 2 1 0 ( -0.03591, -0.02424) 0 0 1 -4 ( -0.00910, 0.01651) 2 -3 1 0 ( -0.01561, 0.00204) 3 0 - 3 - 1 ( -0.00632, 0.00197) 2 1 -3 -1 ( -0.02996, -0.02096) 1 - 2 0 4 ( -0.04731, -0.01917) 1 - 1 1 - 4 ( 0.01394, 0.03918) 1 - 1 - 1 4 ( 0.00615, 0.03282) 0 1 - 2 4 ( 0.03695, 0.06222) 3 - 4 1 0 ( 0.00468, 0.02832) 4 -2 -2 -1 ( -0.03785, -0.02226) 4 -1 -3 -1 ( 0.01709, 0.02826) 2 -3 0 4 ( -0.02486, 0.00472) 2 2 - 4 - 1 ( 0.02200, 0.03423) 2 - 2 1 - 4 ( 0.03659, 0.06387) z = 0.8 2 - 2 - 1 4 ( 0.02882, 0.05665) 1 3 - 4 - 1 ( -0.00223, 0.01383) 1 - 2 2 -4 ( -0.03958, -0.01294) 0 4 -4 -1 ( -0.02762, -0.00649) 4 - 5 1 0 ( 0.02498, 0.05461) 3 - 5 2 0 ( -0.05153, -0.02220) 3 -1 -1 -5 ( -0.06751, -0.03568) 3 0 -4 3 ( -0.01525, 0.00374) 3 0 - 2 - 5 ( -0.01538, 0.01778) 2 1 - 4 3 ( -0.03792, -0.01889) 2 1 -2 -5 ( -0.03905, -0.00489) 1 2 - 2 - 5 ( -0.06279, -0.02756) 0 1 1 - 8 (0.03385,0.08650) 0 0 2 - 8 (-0.01821,0.03303)

z = 0.9 - 1 0 0 ( 0.00636, 0.01414) - 2 0 0 ( 0.01272, 0.02828) - 3 0 0 ( 0.01908, 0.04243) - 3 1 0 (-0.05579, -0.03306) 0 0 - 5 (-0.07470, -0.02447) 1 0 - 5 (-0.08666, -0.03185) 0 - 3 - 1 ( 0.00904, 0.01992)

z = 0.9 1 - 2 0 5 ( 0.03923, 0.09894) 1 1 - 1 - 5 (-0.01189, 0.04300) 0 2 - 1 - 5 (-0.02416, 0.03562) 4 -2 -2 -1 (-0.04899, -0.03124) 4 -1 -3 -1 ( 0.01929, 0.03074) 2 - 1 - 2 4 (-0.02332, 0.02131) 1 - 4 3 1 ( 0.00984, 0.04248)

1 2 3 2 1 0 3

110 2 1 1 0 0 4 3 2 2

Normal Form of the Bicircular 1 2 0 3 1 -4 -4 -1 0

-3 -3 -2 -3 -2 0 1 0 -1

-1 -1 4 -1 4 0 0 -5 -5

(-0.00293, 0.01229) (-0.01559, 0.00491) (-0.03070, 0.00935) (-0.02885, -0.00246) (-0.03808, -0.00149) ( 0.02545, 0.05657) (-0.04943, -0.01892) (-0.06345, -0.01709) ( 0.00005, 0.05038)

1 1 0 0 5 4 3 3 1

Model and Related -2 1 4 2 -5 -5 -2 -1 -3

2 -3 -4 -3 0 1 0 -1 1

-4 4 -1 4 0 0 -5 -5 5

Topics ( 0.01101, 0.04546) ( 0.03677, 0.07344) ( 0.03281, 0.06501) ( 0.02939, 0.06416) ( 0.03181, 0.07071) (-0.04306, -0.00477) (-0.05343, -0.00971) ( 0.01137, 0.05776) (-0.02824, 0.03664)

Table 3.10: For each one of the z-amplitudes from 0.3 to 0.9 the most relevant resonances to order 10 are given. In each block the multiples of the following frequencies: vertical, short period, Sun and long period are shown, and then the range of values along the section through the given amplitude.

Chapter 4

The Quasi-periodic Model

In this chapter the Hamiltonians corresponding to motion near £4 or L5 in the Earth-Moon system are written. The full solar system and the radiation pressure are considered. The terms of the Hamiltonian which involve Legendre polynomials are expanded as power series in x, y and z with coefficients which are quasi-periodic functions of time (with incommensurable frequencies). A Fourier analysis of such functions as well as of the other functions needed for the equations and Hamiltonian is performed. We consider an inertial frame of reference with the origin at the center of masses of the solar system and the axes parallel to the ecliptic ones. The equations of motion of a spacecraft in the solar system can be written as k =G

£

A(RA-R)^

Ae{S,E,M,Pu-,Pk} I

A

^ '

where G is the gravitational constant, and R and RA are the position vectors of the spacecraft and the celestial body of mass A respectively. The summation is taken for the Sun (S), Earth (E), Moon (M), and planets (Pj). However, the above system of reference is not convenient to study the motion of a spacecraft in the vicinity of the libration points £4 or L5 corresponding to the Earth-Moon system. As it is usual, the (geometrical) libration points are defined as the ones that form an equilateral triangle with the Earth and the Moon. These points are placed at the instantaneous plane of motion of the Moon around the Earth (see Figure 4.1) This is, the plane containing both bodies and the velocities of one with respect to the other. We define a normalized reference system centered at the libration points and given by the unitary vectors ei, e*2 a n d &z'-

ei

=

TEM T-Z TEM

111

r,

112

The Quasi-periodic

Model

ez

=

TEM A TEM I TEM A rEM

e*2

=

e3 A e i .

TEM

TEM Fig. 4.1

M

M

Reference systems at the triangular equilibrium points.

Notice that the a;-axis points from the Earth to the Moon. This differs from the sense used for the RTBP and the Bicircular model, but we hope that the correct interpretation will be easily obtained from the context in each case. We introduce a modified mass of the Earth, E, to satisfy Kepler's third law K = G(E +

M)=nMaM,

where UM and AM are the mean motion and the semimajor axis of the Moon around the Earth. The remaining mass E = E — E will be considered as a perturbation. Then, in this normalized system (x, y, z), we adopt the following units: • The unit of mass is chosen such that G(E + M) = 1, • The unit of time is defined to have TIM = 1, • The unit of distance is taken as TEMTherefore, in the L\ case, the coordinates of the Earth and the Moon are:

{xE,yE,ZE) = ( - | , - ^ , 0 ) , = (|,-^,0).

(XM,VM,ZM)

In a similar way, for L5 we have: • (XE,yE,ZE) • {XM,VM,ZM)

= (-|,^,0), = (|,

2 »°)-

For a precise definition of the adopted system see [8].

The Lagrangian and the

4.1

113

Hamiltonian

The Lagrangian and the Hamiltonian

In the normalized system of reference centered at the libration point Li, i = 4, 5, the Lagrangian (see [8]) can be expanded in Legendre polynomials, P„(x), as: L

-{k2{x2+y2+z2)

=

+ 2kk(xx + yy + zz)

+2k2 (E(xy - yx) + F(yz - zy)) + k2(x2 + y2 + z2) +k2(Ax2

+ By2 + Cz2 + 2Dxz)}

-k2(xEx

+ yEy) - kk(xEx

2

-k {-EyEx l

+Kk~ {l

+ VEV) ~ k2(AxEx

+ ByEy +

DxEz)

+ ExEy + FyEz) anPn(cosEi)

-HM + VE) Yl

n>l

AeiS.M.Pi

Pk}

\

EA

rA

n>lKTAj

J

where a = (x,y,z) is the position vector of the spacecraft, v stands for a vector v in the normalized system, k = rEM is a time-dependent scaling factor, cos^li = (f J 4,a)/(f J 4a) and c o s ^ = (rEA,a)/(rEA
=

-92smScosS(l

E

=

9 cos S + SR (1 + R2)1/2'

F

=

*sin*

+ r

+

^,

R2)1/2

OR cos S (l + i? 2 )V2'

114

The Quasi-periodic

Model

R

= -. , 6 cos 5 where 6 and 8 are the geocentric longitude and latitude of the Moon, respectively. The momenta px, py, pz, are introduced through Pa = f£, ify = § | , Pz = § p From these relations it is easy to express x, y, z as functions of the positions and momenta:

±

=

¥ ~ kX + Ey' + kXBEyE'

V

=

^-^y-(Ex-Fz)

i

=

%~kZ~Fy

+ -yE + ExE,

+ FVE

'

The Hamiltonian function becomes H

= xpx + ypy + zpz- L =

2k^(Pl+Pl+Pl) k --r(Pxx+Pyy+PzZ)

+E(pxy

-pyx) +

F(pyz-pzy)

+ I -xE - EyE I px + I -yE + ExE J py + FyEpz + ^(E2 -k2(D

- A)x2 + y (E2 +F2- B)y2 + ^-(F2 - C)z2 + EF)xz

2

+k (A - E2)xEx 2 2

2

(E k

+

k \

+

+ k2(B - E2 - F2)yEy 2 2

,

2 2

(E k

+

EF)xEz

2

Fk

x +

+ k2(D + k \

+

2

yE

{— Y) z {— — Y)

-^(l-/iM+^)Ea"P»(C0S£;i)-^E(f) n>l

1

v^

£

n(C0s5!)

n>l

/

a cos A2

+

l v ^ /

a

^ " ^ . /

^ (-^c— rA £ [ji) \ EA

A6{S,M,Pi,-,Pk}

P

, v I

Pn{cosAl)

•

n>l ' J Taking into account that E - A = E + F - B = F2 - C = D + EF = 0, and after removing all the terms which only depend on time we get: 2

H

2

2

1 yyx = 2k2 7^(PI+PI+PD ' yy k • friPxX + PyV + Pz*) + E(pxy -pyx) + F(pyz

-pzy)

Some Useful

+ I -xE -\{l

Expansions

- EyE j px + I TVE + ExE 1 py + FyEpz

~m

+ VE) E

^ ^ ( c o s E i ) ~l^J2

n>l

1

115

^

v-^

I

^{S.M.P!,..../',,}

^

acosA2 '

p

(^-) n>l ^

n(^sS1)

5

'

lv-^/^A™/

EA

. . i

n>l

/

As it has been said the unit of distance is taken as TEM and as we have the origin at the triangular equilibrium points we also have that TM = 1, SO the above expression, with some rearrangement, reduces to H

2k2(pl+P2y+Pl)

=

k

-riPxX+Pyy+Pzz)

+ E(pxy - pyx) + F(pyz - pzy)

+ I ^xE - EyE J px + I -yE + ExE J py + FyEpz - - ( 1 - \xM + nE)

--//M

1 ^

^2anPn(cosEi)

-a; + ^ a " P n ( c o s M i )

|

acosA 2

fiA+iAVA

S~^ {

a

\"

D

t

A A

l

L - ^ - ^ 2 — + — ^ L U : P„(COS^) . EA *Ae7>\ n>AVA^ J with P = {5, Pi,..., Pfc}, %A = 1 if A = S and i^ = 0 for any other body.

4.2

Some Useful Expansions

We are interested in writing the Hamiltonian function as a series including terms of the following types E 0,ijkrXly3ZkF{vTt

+ tpr),

or

where v denotes px, p y or pz and F stands for one of the trigonometric functions sine or cosine.

116

The Quasi-periodic

Model

A Fourier analysis of the functions that appear in the Hamiltonian will give the coefficients, a^r, ar, the frequencies, i/r, and the phases, <pr, involved in the dominant terms. We deal separately with the terms which involve the Legendre polynomials. Using the normalized system of coordinates we have xxE + yyE + zzE cos£i =

1 /= - — (x + sV3y), 2a

=

arE xxM + WM + zzM

1 , n: . = —(x - sV3j/), 2a

cos Mi = arM with s = 1 for Li and —1 for L 5 . Therefore, it is not difficult to see that

£ «"*.(«»*) = £ — £ n>l n 2k

(r\/Vim v v °l

~

^ m=0

n>l

m!(n-2fc-m)! v

;

fc=0

«-^ V^

^

[

2

1^W2 ^ 1 -1

ni!n 2 !n 3 !

H2(n - k))l

>'

2ni+n-2fc-m„,2n2+m_2n3

n i + n 2 + n 3 = it

where f 1 if A == E, \ 0 if A = M, and fl \ -1

if 'A = E, if A = M.

A routine gives the coefficients of each monomial xly3zk up to a given order n = i + j + k. This routine has been checked comparing the development with the generating function 1 y/1 — 2a cos Ai + a 2 where A = E or M. It has been seen that to get a difference of the order of 10~ 5 , terms up to order 20 are required if the magnitude of a is 0.5, but only order 8 is required if o = 0.15. We note that the coefficients in the development of ^2 o-npn (cos Ai), n>l

for A = E or M are constants and so no Fourier analysis is needed for those terms. For the planets and the Sun, similar expansions are obtained, the coefficients of the monomials being functions of time. We have:

117

Some Useful Expansions For the planets 1

l

k~ HA—

/

\

n

Y\[^-)

Pn^osAx).

For the Sun k^ifis

+ ns)— J2 [•?- ) P n (cos5i).

In general, we have the following expansion

— > ~ fAL^\fAj k

k — ni

P„(cOsAi) = > — > ( - 1 ) * ^ - T ^ Z ^ 2" ^-^V ; (n-k)\ n — 2k

1

n—2k—l

^ ^ n i M f c - m - n j ) !E ^ E £f0 /i!W(n-2ifc-Z1-Z2)! x

^ V

1

^MV2 ^ V

3

n y+1a.2„1+/l2/2n2+;202n3+/3)

where 1$ = n — 2k — li — hThe terms of the expansion can be collected in the form E

/(
9li92.93

where

/(*,*,©)=:£ w l h « , (g) 1 (g)2 (g)3 (£f '1 »'2i'3

are The coefficients, cqiq2q3i1i2i3 and the exponents, qi,q2,q3,li,h,h> computed up to a given order n = q\ + qi + qz in the routine. The routine has been checked using the generating function

1

1 2

yjr\ - 2afA cos A\ + a ?A A routine has been implemented to compute the time-dependent functions at a given day. This routine will be used by the program that performs the Fourier analysis.

118

4.3

The Quasi-periodic

Model

The Fourier Analysis

We shall present in this section the results obtained for the time-dependent functions that appear in the Hamiltonian. As functions involving the motion of the Moon around the Earth we have studied the following four ones (these are the only functions needed): • • • •

Function Function Function Function

number number number number

r

1 2 3 E, 4

Related to the last line of the Hamiltonian, including all the effects of the Sun and the planets, only the terms coming from the Sun, if we neglect all the functions with amplitude less than 1 x 10~ 7 , must be taken into account. Associated to the Sun, if we write its contribution as ^fijk(t)xiyjzk, we must consider only the 19 functions that appear multiplying powers of x, y and z up to degree 3. In the following Tables we give the results of the Fourier analysis of these functions. All the computations have been done using the JPL ephemeris for a time span which goes from the MJD 3631 up to the MJD 26280.6. This time interval corresponds to 829 revolutions of the Moon and is the maximum one covered by the numerical ephemeris. The number of points used for the determination of frequencies and amplitudes (see Appendix B) is 2 16 . As it can be seen the maximum number of frequencies that appear in the developments is of the order of 100. With these number of frequencies the FFT of the difference between the initial function and its trigonometric approximation, has a maximum amplitude less than 10~ 6 for all the functions. All the frequencies can be written as linear combinations of five basic ones, which are defined up to unitary transformations. This five basic frequencies can be taken as the natural ones: • • • •

The The The The

mean longitude of the Moon (ni), mean elongation of the Moon from the Sun (n2), mean longitude of the lunar perigee (713), longitude of the mean ascending node of the lunar orbit on the ecliptic

("4),

• The Sun's mean longitude of perigee (n 5 ), Taking the value of ni equal to 1, the other ones take the following values: n 2 = 0.925195997455093,

The Fourier

Analysis

119

n 3 = 0.845477852931292 x l 0 ~ 2 , n 4 = -0.401883841204748 xlO" 2 , n 5 = 0.357408131981537 xlO" 5 . The identification of the frequencies obtained in the Fourier analysis, as a linear combination of the five basic ones, has been done in such a way that the product of the difference between the true frequency and its linear approximation, by the amplitude related to the frequency, is always less than 0.5 x 1 0 - 9 . This guarantees that if in the Fourier expansions we use the linear approximation of the frequencies, the contribution of each individual harmonic, for a time span of 10 years, has an error less than 0.4 x 10~ 6 . 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

0.00000000 0.85884677 0.99154522 1.85039199 1.98309044 2.84193722 0.07480043 0.13269845 0.78404634 0.91674479 0.92519600 1.01649246 1.06634565 1.71769355 1.77559157 2.70923877 2.97463566 3.70078399 3.83348244 0.15764568 0.70924592 0.83389954 0.93364720 0.99998928 1.12424367 1.14919090 1.64289312 1.70079114 1.78404992 1.90828644 1.91675194 1.92519242 2.05789087 2.63443834 2.76713679 2.91673764 3.62598356 3.96618089 4.69232921 4.82502766 0.00845120 0.02494723 0.06634922 0.14960086 0.20749888 0.23245326 0.72614475 0.79252453 0.84194794 0.85044042 1.00844763 1.05788730

0 -1 1 0 2 1 1 2 -2 0 0 1 2 -2 -1 -1 3 0 2 2 -3 -1 0 1 3 3 -3 -2 -1 1 1 1 3 -2 0 2 -1 4 1 3 0 0 1 2 3 3 -3 -3 -1 -2 1 2

0 2 0 2 0 2 -1 -2 3 1 1 0 -1 4 3 4 0 4 2 -2 4 2 1 0 -2 -2 5 4 3 1 1 1 -1 5 3 1 5 0 4 2 0 0 -1 -2 -3 -3 4 4 2 3 0 -1

0 1 -1 0 -2 -1 0 -2 1 -1 0 1 -1 2 0 1 -3 0 -2 0 1 -1 1 0 -3 -1 2 0 1 -2 -1 0 -2 1 -1 -1 0 -4 -1 -3 1 2 -1 0 -2 0 3 8 -1 6 1 -2

0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 -2 0 2 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 -2 0 -6 0 -6 0 0

0 0 0 0 0 0 -1 0 1 1 0 0 -1 0 1 0 0 0 0 0 2 0 -1 -3 0 0 1 2 2 0 3 -1 -1 1 1 -1 1 0 0 0 -1 0 0 -2 -1 1 -1 -3 3 3 -2 -2

0.50096324E+00 -0.56284520E-02 -0.38318020E-01 -0.18045857E-02 -0.61945169E-04 0.96678173E-03 -0.10214781E-03 0.19266959E-03 -0.36265258E-03 -0.31143737E-03 0.18058342E-03 -0.41495825E-04 0.10103185E-03 -0.55066561E-04 0.11037737E-03 0.20705959E-03 0.19067075E-03 -0.93913819E-04 -0.11748816E-03 0.30524614E-04 -0.14419789E-04 -0.14429892E-04 0.11727454E-04 -0.11931308E-04 0.17785600E-04 -0.97739167E-05 -0.32713038E-07 0.15397954E-04 0.39850078E-05 0.17824708E-04 0.56171706E-05 0.53033635E-04 0.15501911E-04 0.17645442E-04 0.47801281E-04 -0.17659301E-04 -0.13564128E-04 -0.18835088E-04 0.66357116E-05 0.59926820E-05 0.98601233E-06 0.47449669E-05 -0.58841132E-05 -0.16426565E-05 0.51124223E-05 0.13560671E-05 -0.18651237E-05 0.12051113E-05 -0.55346919E-05 0.79053826E-06 -0.13963257E-05 -0.30245707E-05

0.00000000E+00 -0.85706473E-02 -0.38960959E-01 0.83583737E-02 0.37228275E-02 -0.61212021E-03 -0.43937569E-04 -0.38255300E-04 -0.23938029E-03 -0.12707179E-03 0.22360472E-03 -0.20313830E-03 0.25987084E-03 0.12714332E-03 0.56273453E-03 -0.79940747E-04 -0.18140168E-03 -0.42529773E-04 -0.27412088E-04 0.12182316E-04 -0.25642301E-05 -0.63317621E-05 0.66326451E-04 -0.41997505E-04 0.12090571E-04 -0.23296892E-04 0.11980814E-04 0.22464643E-04 -0.11995004E-04 0.42377206E-04 -0.39484342E-04 -0.74539582E-04 -0.34460221E-04 -0.17278950E-04 -0.69872322E-04 0.28186876E-05 -0.25622435E-06 -0.62743902E-06 0.18075615E-04 0.98205086E-05 0.56647049E-06 0.30787177E-05 0.60485608E-06 -0.17355423E-05 0.10907192E-05 0.14156474E-05 -0.44673626E-05 0.25242413E-05 0.79999078E-07 0.65172679E-06 0.59190596E-05 -0.24925634E-05

120

The Quasi-periodic 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79

1.08283096 1.14115680 1.62598356 1.79249755 1.82544119 1.85883605 1.87533923 1.99154165 2.00803410 2.11578889 2.14073612 2.55963791 2.57654032 2.69233636 2.78403920 2.86688088 2.89983166 2.90829001 3.04943609 3.55120296 3.56808554 3.75868201 3.77558442 3.90828287 4.06513562 4.33053251 4.61752163 4.92405917

2 3 -3 -1 0 0 0 2 2 4 4 -3 -3 -1 0 1 2 2 4 -3 -2 1 1 3 4 8 0 5

-1 -2 5 3 2 2 2 0 0 -2 -2 6 6 4 3 2 1 1 -1 7 6 3 3 1 0 -4 5 0

0 -1 0 2 -2 1 2 -1 0 -4 -2 1 3 -1 1 1 -3 -2 -3 6 2 -2 0 -2 0 0 -1 1

-2 0 0 0 2 0 -2 0 -2 0 -2 0 0 0 0 -2 0 0 0 -6 0 0 0 0 0 0 0 21

-3 1 1 0 -1 -3 0 -1 -1 0 0 2 0 2 -1 -1 0 1 -1 -3 0 1 -1 -1 0 0 -1 0

Model

0.69511265E-06 -0.19610705E-06 0.97027576E-06 0.12283961E-05 0.76471799E-06 -0.83250093E-06 -0.37186865E-05 -0.30833433E-05 -0.52408183E-05 -0.68759211E-06 -0.12680580E-05 0.64740047E-06 0.19588363E-05 0.82633765E-06 -0.37881425E-05 0.43707374E-05 0.18609248E-05 -0.33233275E-05 -0.37718016E-05 -0.10198396E-05 -0.37314628E-05 -0.95537873E-05 0.14887010E-05 0.20056548E-05 -0.26487107E-08 -0.14714916E-08 0.18971399E-05 -0.20210758E-08

0.20139840E-05 0.30905466E-05 O.58782843E-06 -0.11688616E-05 0.19383603E-05 0.96033987E-06 0.37273277E-05 0.52758532E-05 0.79315122E-05 -0.38373548E-05 0.30237398E-05 -0.15836313E-05 -0.34446990E-06 -0.42156746E-05 -0.14602140E-06 0.42774097E-08 -0.44617904E-05 0.25497216E-05 0.13994614E-05 0.35673859E-06 -0.26691539E-05 0.17087508E-05 0.16327518E-05 0.14742583E-05 0.79000000E-02 0.81000000E-02 0.19301582E-05 -0.25274982E-08

Table 4.1: Results of the Fourier analysis of the function l/(2k2). In the second column we give the value of the frequency. In the next five columns the integers that allow to write that frequency as a linear combination of the five basic ones. In the last two columns we give the coefficients of the cosine and sine terms respectively.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

0.00000000 0.99154522 1.85039199 0.85884677 1.98309044 2.84193722 0.78404634 0.91674122 0.92519957 1.01649246 1.06634565 1.71769355 1.77559157 1.92519242 2.70923877 2.76713679 2.97463566 3.70078399 3.83348244 0.13269845 0.93364720 0.99999643 1.12424367 1.14919090 1.64289312 1.70079114 1.78404992 1.90828644 1.91674479 2.05789087 2.63443834 2.91673764

0 1 0 -1 2 1 -2 0 0 1 2 -2 -1 1 -1 0 3 0 2 2 0 1 3 3 -3 -2 -1 1 1 3 -2 2

0 0 2 2 0 2 3 1 1 0 -1 4 3 1 4 3 0 4 2 -2 1 0 -2 -2 5 4 3 1 1 -1 5 1

0 -1 0 1 -2 -1 1 -1 0 1 -1 2 0 0 1 -1 -3 0 -2 -2 1 0 -3 -1 2 0 1 -2 -1 -2 1 -1

0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0 1 0 -1 0 1 -1 0 1 0 0 0 0 -1 -1 0 0 1 2 2 0 1 -1 1 -1

-0.98023206E-09 0.38471807E-01 -0.14403334E-01 0.70468979E-02 -0.44101026E-02 0.10163942E-02 0.17623938E-03 0.11395765E-03 -0.20586034E-03 0.20609372E-03 -0.27297568E-03 -0.12837375E-03 -0.94984421E-03 0.13145383E-03 0.12859253E-03 0.11413065E-03 0.23393363E-03 0.79474390E-04 0.44939729E-04 -0.94951636E-05 -0.59281224E-04 0.41951925E-04 -0.25265111E-04 0.24899599E-04 -0.11586384E-04 -0.36751972E-04 0.16549730E-04 -0.48112269E-04 0.45924283E-04 0.42268659E-04 0.27337614E-04 -0.47582034E-05

O.00000O00E+00 -0.37836977E-01 -0.31097009E-02 -0.46277959E-02 -0.73375403E-04 0.16053026E-02 -0.26699989E-03 -0.28676551E-03 0.16319052E-03 -0.42095935E-04 0.10614048E-03 -0.55619121E-04 0.18630835E-03 0.93575913E-04 0.33308088E-03 0.78083613E-04 0.24591397E-03 -0.17549049E-03 -0.19260122E-03 -0.47837303E-04 0.10594634E-04 -0.11059200E-04 0.37144271E-04 -0.10432752E-04 -0.24244069E-07 0.25164808E-04 0.54608728E-05 0.20245292E-04 0.56455491E-05 0.19015479E-04 0.27923610E-04 -0.29763519E-04

The Fourier 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 82 84 85 86 87

3.62598356 3.75868201 3.96618089 4.69232921 4.82502766 0.07480043 0.15764568 0.20749888 0.70924592 0.79252453 0.83389954 0.84194079 0.85044042 0.98735243 0.99593274 1.00844763 1.05788730 1.08286866 1.14115680 1.19904410 1.22400563 1.62599071 1.69274631 1.70926575 1.79249755 1.82544119 1.85883605 1.87533923 1.99154165 2.00000715 2.00803410 2.11578889 2.14073612 2.55963791 2.57653675 2.69233636 2.77558084 2.78403920 2.85038485 2.86688802 2.89983166 2.90829001 3.04943609 3.49328511 3.55118671 3.56808554 3.77558442 3.90828287 4.06513562 4.33053251 4.55963076 4.61752163 4.75022366 4.92405917

-1 1 4 1 3 1 2 3 -3 -3 -1 -1 -2 -11 2 1 2 1 3 4 4 -3 -2 -3 -1 0 0 0 2 2 2 4 4 -3 -3 -1 0 0 1 1 2 2 4 -3 -2 -2 1 3 4 8 -1 0 2 5

5 3 0 4 2 -1 -2 -3 4 4 2 2 3 13 -1 0 -1 0 -2 -3 -3 5 4 5 3 2 2 2 0 0 0 -2 -2 6 6 4 3 3 2 2 1 1 -1 7 6 6 3 1 0 -4 6 5 3 0

0 -2 -4 -1 -3 0 0 -2 1 8 -1 -1 6 0 -6 1 -2 6 -1 -3 -1 0 0 7 2 -2 1 2 -1 0 0 -4 -2 1 3 -1 0 1 0 1 -3 -2 -3 2 0 2 0 -2 0 0 1 -1 -3 1

0 0 0 0 0 0 -2 0 0 -6 2 0 -6 10 7 0 0 -8 0 0 -2 0 2 -6 0 2 0 -2 0 0 -2 0 -2 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 21

Analysis 1 1 0 0 0 -1 0 -1 2 -3 0 1 3 -2 -3 -2 -2 -3 1 -1 3 3 0 -3 0 -1 -3 0 -1 2 -1 0 0 2 -1 2 -2 -1 -2 1 0 1 -1 1 3 0 -1 -1 0 0 0 -1 0 0

0.47576913E-06 -0.27662061E-05 0.85247315E-06 -0.33409456E-04 -0.16050152E-04 0.36049847E-05 -0.16783972E-05 0.70035920E-06 0.16680516E-05 -0.16617654E-05 0.41530778E-05 -0.13983479E-06 -0.63678667E-06 -0.54576958E-06 -0.16979487E-05 -0.59118882E-05 0.30939496E-05 -0.21439271E-05 -0.34961320E-05 -0.14516796E-05 0.10695974E-05 -0.95356646E-06 0.18936652E-05 0.12041155E-05 0.11867239E-05 -0.18796374E-05 -0.10965552E-05 -0.40479964E-05 -0.62740337E-05 0.52650386E-06 0.39306622E-05 0.64051497E-05 -0.37172335E-05 0.24565570E-05 0.39984746E-06 0.67363310E-05 0.28639232E-05 0.23662223E-06 0.76647982E-07 -0.13728701E-06 0.55658538E-05 -0.32519268E-05 -0.18475126E-05 0.23234846E-06 -0.73951512E-06 0.41546623E-05 -0.30752651E-05 -0.24469432E-05 0.33966252E-08 0.10157632E-08 -0.67660962E-05 -0.35448362E-05 -0.92492023E-06 0.37294127E-08

Table 4.2: Results of the Fourier analysis of the function

0 1 2 3 4 5 6 7 8

0.00000000 0.99154522 0.85884677 1.85039199 1.77559157 1.98309044 2.84193722 0.07480043 0.13269845

0 1 -1 0 -1 2 1 1 2

0 0 2 2 3 0 2 -1 -2

0 -1 1 0 0 -2 -1 0 -2

0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 -1 0

0.10000129E+01 -0.76309221E-01 -0.10475587E-01 -0.45311995E-02 0.27471305E-03 -0.12317341E-03 0.22655388E-02 -0.22135919E-03 -0.13350654E-03

121 -0.25214604E-04 -0.15451377E-04 -0.25577357E-04 0.12266818E-04 0.97936484E-05 -0.83756253E-05 0.42093096E-05 -0.32523621E-05 -0.94152474E-05 0.79387343E-06 -0.94082627E-05 -0.46504736E-05 0.80719426E-06 0.18191633E-05 0.57140465E-06 -0.13912129E-05 -0.37584576E-05 0.52321133E-06 -0.21119247E-06 0.93222099E-06 0.57579129E-07 0.15178037E-05 -0.32286060E-06 0.39716298E-07 0.12184116E-05 0.75229229E-06 -0.96294411E-06 -0.40432252E-05 -0.36413023E-05 0.97342395E-06 0.25833413E-05 -0.11466918E-05 -0.15529083E-05 0.10084953E-05 0.21685978E-05 0.13216809E-05 0.54819152E-05 -0.61807804E-05 0.14285537E-05 0.69201067E-05 0.23300746E-05 -0.42219538E-05 -0.49994131E-05 -0.10686598E-05 -0.18533402E-05 -0.58044653E-05 0.28023850E-05 0.33276781E-05 0.81000000E-02 0.83000000E-02 0.10663897E-05 0.34857710E-05 0.13243568E-05 -0.28017368E-08 k/k.

0.00000000E+00 -0.77589580E-01 -0.15951499E-01 0.20987373E-01 0.14004684E-02 0.74026186E-02 -0.14344231E-02 -0.95218002E-04 0.26505349E-04

The Quasi-periodic

122

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76

0.78404634 0.91674122 0.92519957 0.93364005 1.01649246 1.06634565 1.71769355 1.92519242 2.70923877 2.76713679 2.97463566 3.70078399 3.83348244 0.15764568 0.70924592 0.83389954 0.84194079 0.99999643 1.00846550 1.12424367 1.14919090 1.64289312 1.70079114 1.78404992 1.90828644 1.91674479 1.99154165 2.05789087 2.63443834 2.91673764 3.56808554 3.62598356 3.75868201 3.96618089 4.69232921 4.82502766 0.06634565 0.14960086 0.20749888 0.23245326 0.72614475 0.75910269 0.76736044 0.79252453 0.85044042 0.92629512 0.94171543 0.98308687 0.98735243 0.99593274 1.05788730 1.08283096 1.09134488 1.14115680 1.19904410 1.22399848 1.56809269 1.62599071 1.69274631 1.70926575 1.75679353 1.78176416 1.79249755 1.82544119 1.83352371 1.84198564 1.85883605 1.87533923

-2 0 0 0 1 2 -2 1 -1 0 3 0 2 2 -3 -1 -1 1 1 3 3 -3 -2 -1 1 1 2 3 -2 2 -2 -1 1 4 1 3 1 2 3 3 -3 -2 -1 -3 -2 0 -1 1 -11 2 2 2 1 3 4 4 -4 -3 -2 -3 -2 -3 -1 0 -1 -1 0 0

3 1 1 1 0 -1 4 1 4 3 0 4 2 -2 4 2 2 0 0 -2 -2 5 4 3 1 1 0 -1 5 1 6 5 3 0 4 2 -1 -2 -3 -3 4 3 2 4 3 1 2 0 13 -1 -1 -1 0 -2 -3 -3 6 5 4 5 4 5 3 2 3 3 2 2

1 -1 0 1 1 -1 2 0 1 -1 -3 0 -2 0 1 -1 -1 0 1 -3 -1 2 0 1 -2 -1 -1 -2 1 -1 2 0 -2 -4 -1 -3 -1 0 -2 0 3 -1 -16 8 6 -7 7 -2 0 -6 -2 0 7 -1 -3 -1 2 0 0 7 9 17 2 -2 4 5 1 2

0 0 0 0 -2 0 0 0 0 0 0 0 0 -2 0 2 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 2 -13 -6 -6 -15 -8 0 10 7 0 -2 -8 0 0 -2 0 0 2 -6 5 -3 0 2 -6 -6 0 -2

1 0 1 -3 0 -1 0 -1 0 1 0 0 0 0 2 0 1 -1 3 0 0 1 2 2 0 1 -1 -1 1 -1 0 1 1 0 0 0 -1 -2 -1 1 -1 2 0 -3 3 0 -3 -1 -2 -3 -2 -3 3 1 -1 1 2 3 0 -3 3 -1 0 -1 1 3 -3 0

Model - 0 65150846E-03 - 0 60878417E-03 0 34839173E-03 0 24952126E-04 - 0 . 83431719E-04 0 .20505760E-03 - 0 10181910E-03 0. 13466369E-03 0 .47710053E-03 0 11111245E-03 0 37863865E-03 - 0 .23259439E-03 - 0 .26479172E-03 0 57857980E-04 - 0 25014557E-04 - 0 23868884E-04 - 0 10514757E-04 - 0 22174992E-04 - 0 . 33214394E-05 0. 59584568E-04 - 0 . 19676161E-04 - 0 . 51352011E-07 0. 37696120E-04 0. 87662688E-05 0. 34805136E-04 0. 96448790E-05 - 0 . 61191352E-05 0. 31303207E-04 0. 40337141E-04 - 0 . 41683365E-04 - 0 . 81666559E-05 - 0 . 33534962E-04 - 0 . 21377019E-04 - 0 37352643E-04 0. 15964706E-04 0 13175673E-04 0 59303399E-05 - 0 37135097E-05 - 0 . 84767102E-05 0 26244526E-05 - 0 16211940E-05 - 0 14751707E-05 - 0 10124765E-05 0 19919857E-05 18200113E-05 0 16737190E-05 0, 75471876E-06 0 17618941E-05 0 36037961E-05 0 11034954E-05 0 69152384E-05 -0 14335383E-05 0, 55223783E-06 0. 40187308E-06 -0. 15121830E-05 0. 63060998E-07 0. 47948821E-06 0. 23146962E-05 0 43553084E-06 - 0 . 49023048E-07 0. 82298620E-06 0. 24426450E-07 - 0 . 22519342E-05 0. 13822760E-05 0. 11769670E-05 0. 21004269E-06 - 0 . 16350607E-05 - 0 70894767E-05 -0

-0.43000638E-03 -0.24184559E-03 0.43960563E-03 0.12625247E-03 -0.40848201E-03 0.52733935E-03 0.23501290E-03 -O.18913508E-O3 -0.18419503E-03 -0.16241520E-03 -0.36020721E-03 -O.1053382OE-O3 -0.61782747E-04 0.23092325E-04 -0.44482722E-05 -0.10512205E-04 0.33832812E-06 -0.84179142E-04 0.11638127E-04 0.40532532E-04 -0.46928134E-04 0.21762071E-04 0.55065333E-04 -0.26348199E-04 0.82647367E-04 -0.78159192E-04 0.10514026E-04 -0.69560126E-04 -0.39494670E-04 0.66597342E-05 -0.58431170E-05 -0.63170456E-06 0.38256858E-05 -0.12444971E-05 0.43483030E-04 0.21595775E-04 -0.58398992E-06 -0.39239658E-05 -0.18109650E-05 0.27417094E-05 -0.38669651E-05 -0.44453178E-07 -0.15571104E-06 0.42305349E-05 0.14624003E-05 0.11626250E-05 0.98898327E-06 0.78276184E-06 0.11074849E-05 0.33896593E-05 -0.57142571E-05 0.40008889E-05 -0.15998651E-05 0.64536690E-05 0.23592753E-05 -0.19692583E-05 0.11092389E-05 0.14574199E-05 -0.25284166E-05 -0.18872852E-05 0.76341898E-06 0.14080582E-05 -0.21506689E-05 0.34664692E-05 0.12620764E-05 -0.11217473E-05 0.18746381E-05 0.70833349E-05

The Fourier 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109

1.93324440 2.00000000 2.00803410 2.04943967 2.08289010 2.11578889 2.14073612 2.55963791 2.57654032 2.69233636 2.77558084 2.78403920 2.85039557 2.86688088 2.89983166 2.90829001 2.98309044 3.04943609 3.10734483 3.49328511 3.55118671 3.68388158 3.77558442 3.90828287 4.06513562 4.19791085 4.33053251 4.48482676 4.55963076 4.61752163 4.75022366 4.76713321 4.92405917

1 2 2 3 2 4 4 -3 -3 -1 0 0 1 1 2 2 3 4 5 -3 -2 0 1 3 4 8 8 -2 -1 0 2 2 5

1 0 0 -1 0 -2 -2 6 6 4 3 3 2 2 1 1 0 -1 -2 7 6 4 3 1 0 -4 -4 7 6 5 3 3 0

0 0 0 -3 6 -4 -2 1 3 -1 0 1 0 1 -3 -2 -2 -3 -5 2 0 -2 0 -2 2 -2 -2 1 1 -1 -3 -1 1

-2 0 -2 0 -8 0 -2 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 -12 21 -12 0 0 0 0 0 21

Analysis 3 0 -1 0 3 0 0 2 0 2 -2 -1 1 -1 0 1 0 -1 3 1 3 2 -1 -1 0 0 0 0 0 -1 0 0 0

-0.11665142E-05 -0.93650665E-06 0.32019975E-05 0.12679379E-05 -0.49738314E-06 -0.96081882E-05 0.60790524E-05 -0.35833892E-05 -0.63021890E-06 -0.96865090E-05 -0.23497126E-05 -0.33827259E-06 -0.53906582E-07 0.13242974E-07 -0.87077285E-05 0.50414595E-05 -0.32708062E-06 0.28118228E-05 0.82488682E-06 -0.32829277E-06 0.98837876E-06 0.64345334E-06 0.40589740E-05 0.33446301E-05 -0.42181043E-08 -0.73993094E-09 -0.13624081E-08 0.13399280E-05 0.88660249E-05 0.46282705E-05 0.12463995E-05 -0.12196674E-05 -0.51003638E-08

0.18304172E-06 0.16134256E-05 -0.21195234E-05 0.77319458E-07 0.10080120E-05 -0.17334559E-05 -0.25462259E-05 0.14666904E-05 0.35992142E-05 0.18978816E-05 0.45221137E-05 -0.87821896E-05 0.21127480E-05 0.99664046E-05 0.36377777E-05 -0.65578352E-05 0.12857963E-05 -0.75981894E-05 -0.62806530E-06 -0.15148864E-05 -0.24774514E-05 -0.96693048E-06 0.36994125E-05 0.45482257E-05 0.36002941E-08 0.69894002E-09 0.20709377E-09 0.86185512E-06 0.13961657E-05 0.455O0859E-05 0.17894698E-05 0.75588432E-07 -0.51738913E-08

Table 4.3: Results of the Fourier analysis of the function E.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

0.00000000 0.84637316 1.00401884 0.01247362 0.14517206 0.77157273 0.92921841 1.07881927 0.06232681 0.07881927 0.08727762 0.21997249 0.69676515 0.92117358 1.13671729 1.16166452 1.70521993 1.77156558 1.83791838 1.86286561 1.92921484 1.99556406 2.69676515 2.85441083 0.71367113 0.78003465 2.62196472

0 -1 1 0 2 -2 0 2 1 1 1 3 -3 0 3 3 -2 -1 0 0 1 2 -1 1 -3 -2 -2

Table 4.4

0 2 0 0 -2 3

—1 -1

—1 -1 -3 4 1 -2 -2 4 3 2 2 1 0 4 2 4 3 5

0 0 0 1 -1 0 0 0 -1 0 1 -1 0 0 -2 0 1 0 -1 1 0 -1 0 0 2 1 0

0 1 -1 -1 -1 1 -1 -1 1 -1 -1 -1 1 1 -1 -3 1 1 1 -1 -1 -1 1 -1 1 1 1

0 0 0 0 0 1 1 -1 -1 -1 0 -1 0 -1 0 0 0 -1 0 0 0 0 0 0 -1 3 1

0.57132158E-09 0.55183661E-03 0.33968131E-03 0.90778293E-04 0.94089327E-04 0.42093116E-04 0.13864503E-04 0.14614183E-05 0.20469237E-05 -0.34374476E-05 0.25171493E-05 0.48146101E-05 0.18313480E-05 -0.38752400E-05 0.10066046E-05 -0.15310581E-06 0.32169012E-06 0.21602084E-07 -0.86216297E-07 0.12538121E-05 0.86403975E-06 0.33530983E-06 0.37805574E-05 0.44680551E-05 0.48715890E-06 -0.35471568E-06 0.29353702E-06

0.000O0000E+00 0.46667175E-03 0.63787109E-03 0.26867188E-04 0.86550505E-05 0.12810985E-04 0.11088573E-04 0.17554999E-04 0.24951390E-06 -0.63189590E-06 0.21299913E-05 0.26169837E-05 -0.23639226E-06 -0.77370166E-05 0.12494085E-05 -0.13108168E-05 -0.27063921E-05 -0.22148609E-05 -0.11295565E-05 -0.22873999E-05 -0.20201281E-05 -0.10612121E-05 -0.29132165E-05 -0.12643416E-05 0.60519439E-06 -0.37201507E-06 -0.52762637E-06

Results of the Fourier analysis of the function F.

124

The Quasi-periodic

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

.00000000 .07480043 .13269845 .15764568 .78404634 .85884677 .91674479 .92519600 .99154522 1.06634565 1.70079114 1.77559157 1.85039199 1.92519242 1.98309044 2.00803768 2.70923877 2.76713679 2.77558799 2.84193722 .06635995 .20749888 .70924592 .83389954 .93364720 1.01649246 1.78407690 2.63443834 2.70083241 2.91673764 3.83348244 .14960086 .23244611 .72618602 .79254597 .84192649 1.12424367 1.14919090 1.62599071 1.69274631 1.70923519 1.71768282 1.90829716 1.91678964 1.93323725 2.08283811 2.69233636 2.78403920 2.86688445 2.99958290 3.70078399 3.75868201 3.76713321 4.69232921

0 1 2 2 -2 -1 0 0 1 2 -2 -1 0 1 2 2 -1 0 0 1 1 3 -3 -1 0 1 -2 -2 -2 2 2 2 3 -4 -3 -1 3 3 -3 -2 -2 -2 1 0 1 3 -1 0 1 3 0 1 1 1

0 -1 -2 -2 3 2 1 1 0 -1 4 3 2 1 0 0 4 3 3 2 -1 -3 4 2 1 0 4 5 5 1 2 -2 -3 5 4 2 -2 -2 5 4 4 4 1 2 1 -1 4 3 2 0 4 3 3 4

0 0 -2 0 1 1 -1 0 -1 -1 0 0 0 0 -2 0 1 -1 0 -1 -1 -2 1 -1 1 1 7 1 6 -1 -2 0 0 9 8 -1 -3 -1 0 0 1 2 -2 5 0 0 -1 1 1 -1 0 -2 -1 -1

0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 2 0 -2 -6 0 -6 0 0 0 -2 -6 -6 0 0 -2 0 2 0 0 0 -6 -2 -2 0 0 -2 -2 0 0 0 0

0 -1 0 0 1 0 1 0 0 -1 2 1 0 -1 0 0 0 1 0 0 3 -1 2 0 -1 0 -1 1 3 -1 0 -2 -1 0 3 -3 0 0 3 0 -1 -3 3 3 1 -1 2 -1 0 0 0 1 0 0

Model

1.32667303D-03 .56728618D-04 .67947147D-04 -.14384840D-04 .83797010D-04 .16212940D-02 .20130816D-04 -.17516145D-04 .46635815D-03 .12227145D-04 .21633247D-04 .48663822D-03 .61712003D-02 -.39305887D-04 -.10645641D-04 -.13728215D-04 .97273481D-05 -.70308398D-05 -.22248841D-04 -.96545130D-05 -.23798783D-05 .20028452D-05 .24450369D-05 .36810703D-05 -.72498711D-05 -.50143235D-05 -.75657290D-05 -.17817181D-05 -.22026487D-05 -.13080744D-05 -.45367313D-05 .89710147D-06 -.66268029D-06 .76383110D-06 -.50845724D-06 .39408611D-06 -.29813311D-06 -.16391205D-05 .69079416D-06 -.18520921D-05 -.49156842D-06 .72222949D-06 -.87281816D-06 -.90007476D-06 -.59140892D-06 -.49030328D-07 -.55712808D-06 -.57261546D-06 .30084838D-06 -.25976175D-06 .83818090D-06 -.17429337D-06 .37958998D-06 -.50380392D-O6

.000OO00OD+O0 .26192552D-04 .66315274D-04 -.51186577D-05 -.39814600D-04 -.62790122D-04 -.91935986D-05 -.29044002D-05 .65141871D-04 .51339366D-05 -.18087278D-05 .16412684D-03 .55587990D-02 -.86916653D-04 -.67066050D-05 -.27438264D-04 -.62463060D-04 -.14632488D-04 -.12253072D-04 -.23382408D-03 -.11293285D-05 .45221295D-05 -.27714429D-05 -.39733035D-05 -.21166470D-05 -.11065429D-05 .62695302D-06 -.69156609D-05 -.50344802D-06 .34164477D-05 .48300352D-05 .11118461D-05 -.62869364D-06 .21809449D-06 .55989322D-06 -.52943272D-06 .14083515D-05 -.11993520D-05 -.37625259D-06 -.45996082D-06 .31620788D-06 .84763948D-06 -.15241159D-06 .62086669D-06 -.47698853D-06 -.73961946D-06 -.49430426D-06 .90532506D-06 -.50763584D-06 .73850940D-06 -.55314126D-06 .48060404D-06 .13660784D-05 -.89505202D-07

Table 4.5 Results of the Fourier analysis of the function number 1 related to the Sun. In the second column we give the value of the frequency. In the next five columns the integers that allow to write that frequency as a linear combination of the five basic ones are given. In the last two columns we give the coefficients of the cosine and sine terms respectively.

The Fourier Analysis

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

.00000000 .07480043 .13269845 .15764568 .78404634 .85884677 .92519600 .99154522 1.70079114 1.77559157 1.85039199 1.92519242 1.98309044 2.00803768 2.70923877 2.76713679 2.77558799 2.84193722 .06632225 .14960086 .20749888 .70924592 .83389954 .91674122 .93364720 1.01649246 1.06634565 1.69274631 1.78409119 2.63443834 2.70081811 2.91673764 3.83348244 .23244611 .70119752 .72619317 .75909911 .79254597 .85038127 1.12424367 1.14919090 1.62599071 1.70923519 1.71768282 1.90829716 1.91669280 1.93323725 2.08283811 2.69233636 2.78403920 2.86688445 2.99958290 3.70078399 3.75868201 3.76713321 4.69232921

Table 4.6

0 1 2 2 -2 -1 0 1 -2 -1 0 1 2 2 -1 0 0 1 2 2 3 -3 -1 0 0 1 2 -2 -2 -2 -2 2 2 3 -3 -4 -2 -3 -1 3 3 -3 -2 -2 1 2 1 3 -1 0 1 3 0 1 1 1

0 -1 -2 -2 3 2 1 0 4 3 2 1 0 0 4 3 3 2 -2 -2 -3 4 2 1 1 0 -1 4 4 5 5 1 2 -3 4 5 3 4 2 -2 -2 5 4 4 1 0 1 -1 4 3 2 0 4 3 3 4

0 0 -2 0 1 1 0 -1 0 0 0 0 -2 0 1 -1 0 -1 -7 0 -2 1 -1 -1 1 1 -1 0 7 1 6 -1 -2 0 1 9 -1 8 0 -3 -1 0 1 2 -2 -7 0 0 -1 1 1 -1 0 -2 -1 -1

0 0 0 -2 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 6 0 0 0 2 0 0 -2 0 2 -6 0 -6 0 0 -2 2 -6 2 -6 0 0 -2 0 0 0 0 6 -2 -2 0 0 -2 -2 0 0 0 0

0 -1 0 0 1 0 0 0 2 1 0 -1 0 0 0 1 0 0 3 -2 -1 2 0 0 -1 0 -1 0 3 1 -1 -1 0 -1 -1 2 1 3 -3 0 0 3 -1 -3 3 -3 1 -1 2 -1 0 0 0 1 0 0

2.53651289D-03 .12472804D-03 -.65814788D-04 -.27993973D-04 -.36853088D-04 -.52240759D-04 -.13948990D-04 .21309258D-03 -.18544593D-05 .16580471D-03 .55980890D-02 -.86112571D-04 -.63942684D-05 -.28023100D-04 -.62576048D-04 -.14647879D-04 -.12359680D-04 -.23396990D-03 -.73765081D-06 .25005152D-05 -.43356790D-05 -.25607864D-05 -.56583965D-05 .18025616D-05 -.24089915D-05 -.14636136D-05 .50585830D-05 -.62879670D-06 .29063705D-06 -.69337790D-05 -.43036014D-06 .34154960D-05 .48106294D-05 -.13313671D-05 -.43167582D-06 .20615173D-06 -.49279695D-06 .53669192D-06 -.10471755D-05 -.13249967D-05 -.10241364D-05 -.38784602D-06 .30838762D-06 .92834095D-06 -.11205213D-06 .47822709D-06 -.50380353D-06 -.77159159D-06 -.49571288D-06 .90437492D-06 -.50634940D-06 .73528930D-06 -.58516080D-06 .47796263D-06 .13701333D-05 -.90354717D-07

125

.00O000O0D+00 .54701102D-04 .59143800D-04 -.10808332D-04 -.79239281D-04 -.15411644D-02 .11670515D-04 -.19488444D-04 -.21990616D-04 -.49223077D-03 -.62169839D-02 .38855389D-04 .11092170D-04 .14036282D-04 -.97556882D-05 .70313390D-05 .22335823D-04 .96148001D-05 -.21983691D-05 .27443101D-05 .14409453D-05 -.23112565D-05 -.57137570D-05 -.92987811D-05 .88343432D-05 .63718387D-05 -.21835943D-05 .23380518D-05 .75381463D-05 .17863385D-05 .22360370D-05 .13065738D-05 .45218369D-05 -.13124775D-05 -.47081789D-06 -.72533840D-06 -.19991069D-06 .47136776D-06 -.72882301D-07 .30779589D-07 -.87411594D-06 -.70721295D-06 .49147410D-06 -.76570857D-06 .90461464D-06 .22750174D-06 .62348820D-06 .53406631D-07 .56176447D-06 .57212674D-06 -.29997367D-06 .25938338D-06 -.87794209D-06 .17418011D-06 -.37873419D-06 .51012014D-06

Results of the Fourier analysis of the function number 2 related to the Sun.

The Quasi-periodic

126

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

.00000000 .01247362 .06232681 .07037164 .07882284 .08727405 .14517206 .21997249 .69677230 .71367471 .77157273 .78007235 .84637316 .92117358 .92921841 .97907160 1.00401884 1.07881927 1.13671729 1.16166452 1.70521993 1.76311795 1.77156915 1.83791838 1.86286561 1.92076363 1.92921484 1.99556406 2.62196472 2.69676515 2.77961040 2.82946360 2.85441083 2.98710928

T a b l e 4.7

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

0 0 1 1 1 1 2 3 -3 -3 -2 -3 -1 0 0 1 1 2 3 3 -2 -1 -1 0 0 1 1 2 -2 -1 0 1 1 3

0 0 -1 -1 -1 -1 -2 -3 4 4 3 4 2 1 1 0 0 -1 -2 -2 4 3 3 2 2 1 1 0 5 4 3 2 2 0

0 1 -1 -1 0 1 -1 -1 0 2 0 7 0 0 0 -2 0 0 -2 0 1 -1 0 -1 1 -1 0 -1 0 0 0 -2 0 -2

0 0 -1 1 0 -1 0 -1 2 0 1 3 0 -1 1 0 0 -1 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0

Model

-5.00215708D-09 .76905197D-04 .11358634D-05 -.31671718D-06 -.33304235D-05 .27480070D-05 .52393303D-04 .11733233D-05 .72252697D-06 .11796366D-05 .32813995D-04 -.27466123D-06 .69513344D-03 -.83302524D-05 .17377971D-04 -.37729228D-06 .74210793D-03 .17050579D-04 -.34524101D-06 -.13277898D-05 .11150853D-O5 .13198814D-05 -.20975327D-05 .18006920D-04 .86279827D-06 .58646251D-06 -.23715489D-05 .13801753D-04 -.41281150D-06 -.86245578D-06 -.97691150D-07 -.26686711D-06 .14044104D-05 .32295217D-07

.OOOOOOOOD+00 -.50993570D-04 .14059284D-05 -.67684324D-06 -.63922477D-06 -.80187428D-07 .76268701D-04 .50357239D-05 -.18098003D-05 -.10594003D-06 -.32084497D-04 .43066587D-06 -.26120431D-03 -.66623738D-06 -.74440285D-05 .32665733D-05 .15014593D-04 .74622544D-05 .26961817D-05 -.50640346D-06 .71218407D-06 -.11697123D-07 .11708154D-05 .79151242D-05 .12021364D-05 .20690046D-06 .30776235D-06 .14043906D-04 -.67053416D-06 -.60304660D-05 -.54595721D-06 -.75244049D-06 -.57955360D-05 -.80878532D-06

Results of t h e Fourier analysis of t h e function n u m b e r 3 related t o t h e Sun.

.00000000 .06634565 .07480043 .13269845 .14960086 .15764568 .20749888 .23244611 .70924592 .78404634 .79252810 .83389954 .85884677 .91674479 .92519600 .93364720 .99154522 1.01649246 1.06634565 1.12424367 1.14919090 1.69274631 1.70079114 1.70923519 1.71806223 1.77559157 1.78406975 1.85039199

0 1 1 2 2 2 3 3 -3 -2 -3 -1 -1 0 0 0 1 1 2 3 3 -2 -2 -2 -1 -1 -2 0

0 -1 -1 -2 -2 -2 -3 -3 4 3 4 2 2 1 1 1 0 0 -1 -2 -2 4 4 4 3 3 4 2

0 -1 0 -2 0 0 -2 0 1 1 8 -1 1 -1 0 1 -1 1 -1 -3 -1 0 0 1 -3 0 7 0

0 0 0 0 0 -2 0 -2 0 0 -6 2 0 0 0 0 0 -2 0 0 -2 2 0 0 8 0 -6 0

0 -1 -1 0 -2 0 -1 -1 2 1 -2 0 0 1 0 -1 0 0 -1 0 0 0 2 -1 -3 1 -3 0

1.32667106D-03 -.21812668D-05 .57382946D-04 .52703810D-04 .97747951D-06 -.14171081D-04 .28388011D-05 -.64441349D-06 .18875593D-05 .40061650D-04 -.76234649D-06 .16207640D-05 .45973410D-03 .14040438D-04 -.11345868D-04 -.58350443D-06 .26186917D-03 -.41810261D-06 .33387407D-05 .45115599D-06 -.67307510D-06 -.12011239D-06 .60672396D-05 -.53551953D-06 -.56632402D-06 .49046509D-04 -.45095546D-05 -.87180623D-03

.00000000D+00 -.30454688D-06 .24684871D-04 -.10463761D-04 .10367424D-05 -.56532368D-05 .60675353D-06 -.64595816D-06 .33475879D-06 .26443324D-04 -.11417151D-06 .71030468D-06 .70003113D-03 .57295110D-05 -.72564291D-05 -.32965752D-05 .26625939D-03 -.20454470D-05 .85871380D-05 .30785703D-06 -.16043101D-05 -.75321021D-06 .88436514D-05 -.27492638D-06 .11693813D-06 .24998546D-03 -.59628701D-05 .40382531D-02

The Fourier 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

1.91674479 1.92519242 1.98309044 2.00803768 2.63443834 2.70078756 2.70923877 2.76713679 2.77558799 2.78403920 2.84193722 2.85038842 2.91673764 3.70078399 3.76713321 3.83348244

1 1 2 2 -2 -1 -1 0 0 0 1 1 2 0 1 2

1 1 0 0 5 4 4 3 3 3 2 2 1 4 3 2

-1 0 -2 0 1 0 1 -1 0 1 -1 0 -1 0 -1 -2

0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0

Analysis 1 -1 0 0 1 1 0 1 0 -1 0 -1 -1 0 0 0

-.14336296D-05 .27829068D-04 .10901108D-06 .84291188D-05 .25494193D-05 -.85412459D-06 .29469382D-04 .45788956D-05 .48299052D-06 -.53557363D-06 .98794383D-04 -.14751763D-06 -.18037898D-05 .44141993D-06 -.10495164D-05 -.32350299D-05

-.83285556D-06 -.39091010D-04 -.65302367D-05 -.12533828D-04 -.24918905D-05 -.21280624D-05 -.11338556D-04 -.66916500D-05 -.25776912D-04 -.21989997D-07 -.62551902D-04 .49531958D-06 .29211726D-06 .30090513D-06 .98988370D-06 -.75457832D-06

Table 4.8: Results of t h e Fourier analysis of t h e function n u m b e r 4 related t o t h e Sun.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

.00000000 .06634922 .07480043 .13269845 .15764568 .20749888 .70924592 .72619317 .78404634 .79252810 .83389954 .84196419 .85884677 .91674479 .92519600 .93364720 .99154522 1.01649246 1.06634565 1.12424367 1.14919090 1.62599071 1.69274631 1.70079114 1.70923519 1.71806223 1.77559157 1.78406975 1.85039199 1.90829359 1.91675194 1.92519242 1.93323725 1.98309044 2.00803768 2.08283453 2.63443834 2.69233636 2.70078756 2.70923877 2.76713679 2.77558799 2.78403920 2.84193722 2.85038842 2.86688445

0 1 1 2 2 3 -3 -4 -2 -3 -1 -2 -1 0 0 0 1 1 2 3 3 -3 -2 -2 -2 -1 -1 -2 0 1 1 1 1 2 2 3 -2 -1 -1 -1 0 0 0 1 1 1

0 -1 -1 -2 -2 -3 4 5 3 4 2 3 2 1 1 1 0 0 -1 -2 -2 5 4 4 4 3 3 4 2 1 1 1 1 0 0 -1 5 4 4 4 3 3 3 2 2 2

0 -1 0 -2 0 -2 1 9 1 8 -1 5 1 -1 0 1 -1 1 -1 -3 -1 0 0 0 1 -3 0 7 0 -2 -1 0 0 -2 0 0 1 -1 0 1 -1 0 1 -1 0 1

0 0 0 0 -2 0 0 -6 0 -6 2 -6 0 0 0 0 0 -2 0 0 -2 0 2 0 0 8 0 -6 0 0 0 0 -2 0 -2 -2 0 0 0 0 0 0 0 0 0 -2

0 0 -1 0 0 -1 2 2 1 -2 0 -3 0 1 0 -1 0 0 -1 0 0 3 0 2 -1 -3 1 -3 0 2 3 -1 1 0 0 -2 1 2 1 0 1 0 -1 0 -1 0

3.31197758D-08 -.21887869D-05 -.75042369D-06 .17602208D-04 -.24770742D-06 -.96501122D-06 .64341972D-06 .74743514D-06 .50501099D-04 -.20713463D-06 .23741700D-05 .14942806D-07 .13412749D-02 .70323680D-05 -.16855899D-04 -.76982159D-05 .23613149D-03 -.53088909D-05 .10263538D-04 -.86611086D-06 -.11162755D-05 .41564180D-06 -.19941107D-05 .17955287D-04 -.55744731D-06 .26029879D-06 .50531393D-03 -.11866036D-04 .81326657D-02 -.81134535D-06 -.11072762D-05 -.77535850D-04 -.75789453D-06 -.12415960D-04 -.25585847D-04 -.41121641D-06 -.49986051D-05 -.74097733D-06 -.42769972D-05 -.22720013D-04 -.13402382D-04 -.51773622D-04 -.42256182D-07 -.12521846D-03 .99924413D-06 .55087737D-08

.00000000D+00 -.12690002D-05 .17426495D-05 .88658815D-04 .61652823D-06 .45210091D-05 -.35870595D-05 -.20215040D-06 -.76507575D-04 .14631342D-05 -.54097664D-05 -.57558880D-06 -.88084330D-03 -.17232283D-04 .81347873D-06 .13619302D-05 -.23222917D-03 .10841801D-05 -.39901846D-05 .12711639D-05 .46801992D-06 -.67298662D-06 .31822385D-06 -.12321514D-04 .10576742D-05 .12625164D-05 -.99136905D-04 .89046843D-05 .17557973D-02 .33153731D-06 .19192600D-05 -.55220592D-04 -.13380140D-06 -.20397447D-06 -.17211717D-04 -.62786711D-06 -.51121500D-05 -.14452919D-06 .17156310D-05 -.59056939D-04 -.91708060D-05 -.10162082D-05 .10704039D-05 -.19777918D-03 .29041505D-06 -.58956065D-06

The Quasi-periodic

2.91673764 2.99958290 3.63443477 3.70078399 3.75868201 3.76713321 3.83348244 4.69232921

46 47 48 49 50 51 52 53

2 3 -1 0 1 1 2 1

1 0 5 4 3 3 2 4

-1 -1 1 0 -2 -1 -2 -1

0 -2 0 0 0 0 0 0

-1 0 0 0 1 0 0 0

Model

.57767889D-06 .14671155D-06 .36726985D-06 .62148866D-06 .90306006D-07 .19829806D-05 -.15043676D-05 -.49330771D-06

.36085779D-05 .76876983D-06 .59102976D-06 -.92477012D-06 .50301797D-06 .21016265D-05 .64493468D-05 .16546736D-06

Table 4.9: Results of t h e Fourier analysis of t h e function n u m b e r 5 related t o t h e Sun.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

.00000000 .01247362 .06232681 .07882284 .08727405 .14517206 .21997249 .69677230 .71367113 .77157273 .78005091 .84637316 .92117358 .92921841 .93767676 .97907160 1.00401884 1.07881927 1.13671729 1.16166452 1.70521993 1.76311795 1.77156915 1.83791838 1.86286561 1.92076363 1.92921484 1.99556406 2.62196472 2.69676515 2.77961040 2.82946360 2.85441083 2.98710928

0 0 1 1 1 2 3 -3 -3 -2 -3 -1 0 0 0 1 1 2 3 3 -2 -1 -1 0 0 1 1 2 -2 -1 0 1 1 3

0 0 -1 -1 -1 -2 -3 4 4 3 4 2 1 1 1 0 0 -1 -2 -2 4 3 3 2 2 1 1 0 5 4 3 2 2 0

0 1 -1 0 1 -1 -1 0 2 0 7 0 0 0 1 -2 0 0 -2 0 1 -1 0 -1 1 -1 0 -1 0 0 0 -2 0 -2

0 -1 1 -1 -1 -1 -1 1 1 1 -5 1 1 -1 -1 1 -1 -1 -1 -3 1 1 1 1 -1 -1 -1 -1 1 1 -1 1 -1 -1

0 0 -1 0 -1 0 -1 2 -1 1 -3 0 -1 1 2 0 0 -1 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0

1.10535890D-09 .11366872D-03 .25727497D-05 -.36622375D-05 .31727745D-05 .11783295D-03 .60405846D-05 .18297404D-05 .82213572D-06 .42200036D-04 -.10527955D-05 .55303184D-03 -.38689175D-05 .13907204D-04 -.10466778D-05 .21971809D-05 .34O39O97D-03 .14782340D-05 .23483297D-05 -.15259897D-06 -.15093479D-06 .55237001D-06 -.40180294D-05 .13836532D-05 -.60687707D-06 .64838647D-07 -.28312308D-05 -.55492794D-05 .37582543D-06 .47563971D-05 .41928040D-06 .51449318D-06 .56471854D-05 .70711986D-06

.O000O00OD+0O .33641718D-04 .30810400D-06 -.70834746D-06 .26361121D-05 .10842500D-04 .32879945D-05 -.20702368D-06 .10156499D-05 .12836329D-04 .10685893D-06 .46765226D-03 -.77226807D-05 .11132735D-04 -.20168044D-06 .12225240D-05 .63916396D-03 .17623396D-04 .28852137D-05 -.13094798D-05 .11211375D-05 .10652080D-05 -.24013188D-05 .18433984D-04 .11323287D-05 .54114712D-06 -.37122958D-05 .17672709D-04 -.67310739D-06 -.36419702D-05 -.34305270D-06 -.59055834D-06 -.15860178D-05 -.36220944D-06

Table 4.10: Results of t h e Fourier analysis of t h e function n u m b e r 6 related t o t h e Sun.

0 1 2 3 4 5 6 7 8 9 10 11 12

.00000000 .06633850 .07480043 .13269845 .14960086 .15764568 .20749888 .23244611 .70924592 .78404634 .79252810 .83389954 .85042969

0 1 1 2 2 2 3 3 -3 -2 -3 -1 -2

0 -1 -1 -2 -2 -2 -3 -3 4 3 4 2 3

0 -1 0 -2 0 0 -2 0 1 1 8 -1 6

0 0 0 0 0 -2 0 -2 0 0 -6 2 -6

0 -3 -1 0 -2 0 -1 -1 2 1 -2 0 0

1.46447062D-03 -.45083306D-06 .72228940D-04 -.43080619D-04 .14701833D-05 -.16092203D-04 -.22249156D-05 -.76313953D-06 -.16523019D-05 -.35850759D-04 .70449953D-06 -.39488572D-05 -.89224062D-06

.0000O00OD+O0 -.19096494D-05 .31076689D-04 .85534223D-05 .15590481D-05 -.64182522D-05 -.47341610D-06 -.76287644D-06 -.30146047D-06 -.23664922D-04 .88394054D-07 -.17337813D-05 .35323248D-06

The Fourier

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

.85884677 .91674122 .92519600 .93364720 .99154522 1.01649246 1.12424367 1.14919090 1.69274631 1.70079114 1.70923519 1.71774197 1.77559157 1.78406975 1.85039199 1.91678607 1.92519242 1.98309044 2.00803768 2.63443834 2.70078756 2.70923877 2.76713679 2.77558799 2.78403920 2.84193722 2.85038842 2.91673764 3.70078399 3.76713321 3.83348244

-1 0 0 0 1 1 3 3 -2 -2 -2 -3 -1 -2 0 0 1 2 2 -2 -1 -1 0 0 0 1 1 2 0 1 2

2 1 1 1 0 0 -2 -2 4 4 4 5 3 4 2 2 1 0 0 5 4 4 3 3 3 2 2 1 4 3 2

1 -1 0 1 -1 1 -3 -1 0 0 1 8 0 7 0 5 0 -2 0 1 0 1 -1 0 1 -1 0 -1 0 -1 ~2

0 0 0 0 0 -2 0 -2 2 0 0 -6 0 -6 0 -6 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0

Analysis

0 0 0 -1 0 0 0 0 0 2 -1 3 1 -3 0 2 -1 0 0 1 1 0 1 0 -1 0 -1 -1 0 0 0

129

-.41735717D-03 -.94282020D-06 -.11207476D-04 .83234102D-06 .54867599D-04 .68690915D-06 -.51427462D-06 -.26886276D-06 .21026378D-06 -.62609901D-05 .53031704D-06 .31004250D-06 -.50139291D-04 .43924086D-05 .88439769D-03 .47522024D-06 -.27313352D-04 -.10802180D-06 -.87917094D-05 -.25628008D-05 .85894473D-06 -.29581358D-04 -.45886553D-05 -.53755400D-06 .53433724D-06 -.98937375D-04 .14677751D-06 .18014043D-05 -.48659966D-06 .10557580D-05 .32122249D-05

-.63552644D-03 -.37477048D-06 .92797904D-05 .47060286D-05 .55788399D-04 .33653740D-05 -.34875798D-06 -.64034467D-06 .12533128D-05 -.91406184D-05 .25991813D-06 -.52386504D-06 -.25557580D-03 .58980484D-05 -.40962795D-02 .23959586D-06 .38372214D-04 .64650069D-05 .13073160D-04 .25056204D-05 .21561441D-05 .11382004D-04 .67066077D-05 .26002679D-04 .22361583D-07 .62644323D-04 -.49396118D-06 -.29117651D-06 -.32175638D-06 -.99351358D-06 .74941582D-06

Table 4.11: Results of t h e Fourier analysis of t h e function n u m b e r 7 related to t h e Sun.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

.00000000 .01247362 .06232681 .07037164 .07881927 .08727405 .14517206 .17011930 .21997249 .69677230 .71367113 .77157273 .78005091 .84637316 .92117358 .92921841 .93767319 .97907160 1.00401884 1.07881927 1.13671729 1.16166452 1.70521993 1.76311795 1.77156915 1.83791838 1.86286561 1.92076363 1.92921484 1.99556406 2.62196472

0 0 1 1 1 1 2 2 3 -3 -3 -2 -3 -1 0 0 0 1 1 2 3 3 -2 -1 -1 0 0 1 1 2 -2

0 0 -1 -1 -1 -1 -2 -2 -3 4 4 3 4 2 1 1 1 0 0 -1 -2 -2 4 3 3 2 2 1 1 0 5

0 1 -1 -1 0 1 -1 1 -1 0 2 0 7 0 0 0 1 -2 0 0 -2 0 1 -1 0 -1 1 -1 0 -1 0

0 -1 1 -1 -1 -1 -1 -3 -1 1 1 1 -5 1 1 -1 -1 1 -1 -1 -1 -3 1 1 1 1 -1 -1 -1 -1 1

0 0 -1 1 -1 -1 0 0 -1 2 -1 1 -3 0 -1 1 1 0 0 -1 0 0 0 1 0 0 0 1 0 0 1

-8.47103940D-09 .23181220D-04 -.17037129D-06 -.26853326D-06 -.50002332D-05 .13471390D-05 -.75345962D-05 -.33442206D-06 -.21322106D-05 -.22232423D-06 .89097157D-06 .13526984D-04 .79469697D-07 .48339868D-03 -.74129745D-05 .12059379D-04 -.17904936D-06 -.17006060D-05 .66040867D-03 .18848034D-04 -.17534396D-05 -.14435603D-05 .13681923D-05 .12046785D-05 -.24835651D-05 .19994515D-04 .13540902D-05 .64030216D-06 -.38148400D-05 .19140686D-04 -.69364421D-06

.00000000D+00 -.78310854D-04 .14491283D-05 -.81575282D-06 -.91345697D-06 -.16164599D-05 .81805443D-04 .42831962D-06 .39182373D-05 -.19727042D-05 -.71970223D-06 -.44459605D-04 .96599775D-06 -.57161997D-03 .36833157D-05 -.15047658D-04 .94391742D-06 .30685224D-05 -.35168443D-03 -.15475311D-05 .14488014D-05 .17109436D-06 .18009854D-06 -.62703190D-06 .40949152D-05 -.15007899D-05 .72756829D-06 -.73847508D-07 .28742286D-05 .60128287D-05 -.38779333D-06

130

The Quasi-periodic

31 32 33 34 35

2.69676515 2.77961040 2.82946360 2.85441083 2.98710928

-1 0 1 1 3

4 3 2 2 0

0 0 -2 0 -2

0 1 0 0 0

1 -1 1 -1 -1

Model

-.37296644D-05 -.35358089D-06 -.60534764D-06 -.16234223D-05 -.37092587D-06

-.48711325D-05 -.43285084D-06 -.52779877D-06 -.57815154D-05 -.72485479D-06

Table 4.12: Results of t h e Fourier analysis of t h e function n u m b e r 8 related t o t h e Sun. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

.00000000 .06632225 .07480043 .13269845 .14960086 .15764568 .20749888 .23244611 .78404634 .83389954 .85039557 .85884677 .91674479 .92519600 .93364720 .99154522 1.01649246 1.06634565 1.14919090 1.69274631 1.77559157 1.85039199 1.91674122 1.92519242 2.00803768

0 2 1 2 2 2 3 3 -2 -1 -1 -1 0 0 0 1 1 2 3 -2 -1 0 1 1 2

0 -2 -1 -2 -2 -2 -3 -3 3 2 2 2 1 1 1 0 0 -1 -2 4 3 2 1 1 0

0 -7 0 -2 0 0 -2 0 1 -1 0 1 -1 0 1 -1 1 -1 -1 0 0 0 -1 0 0

0 6 0 0 0 -2 0 -2 0 2 0 0 0 0 0 0 -2 0 -2 2 0 0 0 0 -2

0 3 -1 0 -2 0 -1 -1 1 0 1 0 1 0 -1 0 0 -1 0 0 1 0 0 -1 0

-2.79114073D-03 .27291675D-05 -.129618O1D-03 -.96235980D-05 -.24478976D-05 .30262543D-04 -.61389440D-06 .14071931D-05 -.42104529D-05 .23306060D-05 .98673296D-06 -.42380858D-04 -.13101784D-04 .22553341D-04 -.24898975D-06 -.31673689D-03 -.26900219D-06 -.32961451D-05 .94172147D-06 -.81884248D-07 .10915700D-05 -.12592171D-04 .94120473D-06 -.51400061D-06 .36328990D-06

.0O000O00D+00 .20607619D-05 -.55766539D-04 .19095328D-05 -.25964945D-05 . 12071294D-04 -.13378086D-06 .14085564D-05 -.27789025D-05 .10245115D-05 -.59568979D-06 -.64502772D-04 -.53460095D-05 -.20231866D-05 -.14093915D-05 -.32204789D-03 -.13199117D-05 -.84765327D-05 .22445013D-05 -.50372932D-06 .55930932D-05 .58027542D-04 .63782361D-06 .72025000D-06 -.53882013D-06

Table 4.13: Results of t h e Fourier analysis of t h e function n u m b e r 9 related to t h e Sun.

0 1 2 3 4 5 6

.00000000 .06634922 .92519600 1.78404277 1.91674122 2.70078756 2.77558799

0 1 0 -1 1 -1 0

0 -1 1 3 1 4 3

0 -1 0 1 -1 0 0

0 0 0 0 0 0 0

0 0 0 0 0 1 0

1.37213339D-08 -.97009348D-06 -.31969629D-05 .75557742D-06 .86877960D-07 .49735621D-06 .78112742D-05

.00O00000D+00 .95284132D-07 -.39608427D-05 -.23534812D-05 -.75298671D-06 -.59286183D-06 -.39330553D-05

Table 4.14: Results of t h e Fourier analysis of t h e function n u m b e r 10 related t o t h e Sun. 0 1 2 3 4 5 6 7 8 9 10

.00000000 .06634922 .79249755 .92519600 1.70924234 1.78404277 1.91674122 2.70078756 2.77558799 2.85038842 3.76713321

0 1 -2 0 -2 -1 1 -1 0 1 1

0 -1 3 1 4 3 1 4 3 2 3

0 -1 2 0 1 1 -1 0 0 0 -1

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 1 0 -1 0

1.76674076D-08 -.59208664D-07 -.66024454D-06 -.38339464D-05 -.61294842D-06 -.69941155D-05 -.17267555D-05 -.17876764D-05 -.11843428D-04 .34699852D-07 -.47266828D-06

.000O000OD+O0 -.60051615D-06 .34688889D-06 .30942811D-05 .58916203D-07 -.22456589D-05 -.19932770D-06 -.14995168D-05 -.23521983D-04 .56350664D-06 .13929763D-05

Table 4.15: Results of t h e Fourier analysis of t h e function n u m b e r 11 r e l a t e d t o t h e Sun.

The Fourier Analysis

0 1 2 3 4 5

.00000000 .07882284 .78002393 .93766961 1.77156915 1.92921484

Table 4.16

0 1 2 3 4

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4

0 1 2 3 4 5

0 1 2 3

0 -2 0 -1 1

0 3 1 3 1

0 1 -1 1 -1

0 1 1 0 0

-7.54320595D-11 -.73953440D-06 -.95441158D-06 -.46998950D-05 -.43195787D-05

0 0 0 0 0

.O0000O00D+0O .71885097D-06 .39016495D-06 -.10566354D-06 -.18394507D-05

0 1 -2 0 -2 -1 1 -1 0 1 1

0 -1 3 1 4 3 1 4 3 2 3

0 -1 2 0 1 1 -1 0 0 0 -1

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 1 0 -1 0

1.59209190D-08 -.11793841D-05 .33280737D-06 -.36123348D-05 .58238536D-07 -.22232029D-05 -.13485130D-06 -.15074576D-05 -.23612398D-04 .56296170D-06 .13960437D-05

.000O0000D+00 .11581141D-06 .63590291D-06 -.44749626D-05 .60690442D-06 .69234411D-05 .11659209D-05 .17970950D-05 .11888691D-04 -.34613793D-07 .47377179D-06

0 1 -1 0 1

0 -1 2 1 1

0 -1 0 0 -1

0 0 0 0 0

0 0 1 0 0

-5.70280716D-08 .40897787D-05 .95562324D-06 .13202893D-04 -.12576400D-06

.00000000D+00 -.40165495D-06 .50372321D-06 .16357195D-04 .10929100D-05

0 1 0 -1 -1 0

0 -1 1 3 4 3

0 -1 0 1 0 0

0 0 0 0 0 0

0 0 0 0 1 0

1.84446849D-08 -.73183746D-07 -.43585959D-05 .22823203D-05 .60224502D-06 .39783075D-05

.00000000D+00 -.74183136D-06 .35180175D-05 .73295141D-06 .50527479D-06 .79017258D-05

Results of the Fourier analysis of the function number 16 related to the Sun.

.00000000 .07882284 1.77156915 1.92921484

Table 4.21

.OOOOOOOOD+00 -.52068215D-06 -.40137506D-06 -.52314853D-06 -.23056524D-05 -.21195287D-05

Results of the Fourier analysis of the function number 15 related to the Sun.

.00000000 .06634922 .92519600 1.78404277 2.70078756 2.77558799

Table 4.20

-1.2521617D-09 -.27014188D-05 -.39088138D-06 -.20861977D-06 .51465910D-07 .90274901D-06

Results of the Fourier analysis of the function number 13 related to the Sun.

.00000000 .06634922 .85039557 .92519600 1.91674122

Table 4.19

0 0 0 0 0 0

Results of the Fourier analysis of the function number 14 related to the Sun.

.00000000 .06634922 .79249755 .92519600 1.70924234 1.78404277 1.91674122 2.70078756 2.77558799 2.85038842 3.76713321

Table 4.18

0 0 1 1 0 0

0 -1 3 1 3 1

Results of the Fourier analysis of the function number 12 related to the Sun.

.00000000 .78002393 .93766961 1.77156915 1.92921484

Table 4.17

0 1 -2 0 -1 1

131

0 1 -1 1

0 -1 3 1

0 0 0 0

0 -1 1 -1

0 0 0 0

-1.51727087D-09 -.28920361D-05 -.53875695D-07 -.93769788D-06

.00000000D+00 -.55540986D-06 .23963468D-05 .22020369D-05

Results of the Fourier analysis of the function number 17 related to the Sun.

132

The Quasi-periodic

.00000000 .06634922 .85039557 .92519600 1.91674122

0 1 2 3 4

Table 4.22

0 1 Table 4.23

0 1 -1 0 1

0 -1 2 1 1

0 -1 0 0 -1

0 0 0 0 0

0 0 1 0 0

Model

-7.30413683D-08 .27867998D-06 .53360653D-06 .16909554D-04 .11570260D-05

.00000000D+00 .28260071D-05 -.10121870D-05 -.13648010D-04 .13308174D-06

Results of the Fourier analysis of the function number 18 related to the Sun.

.00000000 .07882284

0 1

0 -1

0 0

0 -1

0 0

1.01030772D-09 .18646025D-05

.00000000D+00 .35888072D-06

Results of the Fourier analysis of the function number 19 related to the Sun.

Chapter 5

Nominal Paths and Stability Properties

5.1

Introduction

This chapter deals with the computation of some quasi-periodic orbits near the Lagrangian points L4 and L5 of the real Earth-Moon system. The approach followed here is similar to the one used in [9]. For this reason, some technical parts already done in this previous work have been cited instead of including them here. The procedure used can be summarized as follows. First, we will study the model built in Chapter 4. Then, the solution found for that model will be refined numerically for the JPL model. Finally, the stability properties of this orbit will be computed by means of a variational study. As we have seen in Chapter 4, the equations of motion for a particle close to the equilateral points can be approximated by x-xE

x = P(7)

15

PE

x + xE (1 - flM)

3 ' PM

/1 MM - XE(1

o ~

^ 2(1M)

+P(1) + P(2)x + P(S)y + P(4)z + P(5)x + P(6)y 2

2

+P(21)a; + P(22)xy + P(23)xz + P(24)y V-VE

V = P(7)

(1 -

HM)

PE

+P(10)y + P(ll)z 2

y

-VE -fJ-M -

'

+ P(25)yz + P(26)z

VE

-(1 ' PE

(5.2)

+ P(12)x + P(13)y + P(U)z 2

+ P(28)xy + P(29)xz + P(30)y

z = P(7)

+ P(8) + P(9)x

PM

2

+P(27)x

(5.1) 2

HM)

Z -3 — MM ' PM

+ P(31)yz + P(32)z

+ P(15) + P(16)x (5.3)

+P(17)y + P(18)z + P(19)y + P(20)i 2

2

+P(33)x + P{M)xy + P(35)xz + P(36)y + P(37)yz + P(38)z

2

where rps, rpM denote the distances from the particle to the Earth and Moon, respectively, given by r%E - (x - xE)2 + (y - yE)2 + z2, r2PM - (x + xE)2 + (y ~ VE)2 + z2. We recall that xE = - 1 / 2 , yE = -y/3/2 for L4 and xE = - 1 / 2 , 133

134

Nominal Paths and Stability

Properties

yB = V3/2 for L5. The functions P(i) that appear in the equations are of the form: m

m

ij cos dj + ^ 3=1

Bij sin 8j,

3= 1

with 9j = Ujt + (fij, where t denotes the normalized time. Values for all of the coefficients, frequencies and phases can be found in Chapter 4. The first step is to expand (in power series of x, y and z) the nonlinear terms of the equations of motion. The program used has been the program EXPAN.F described in [9].

5.2

Idea of the Resolution Method

To introduce the method used let us start with a short description of the method used in [9]. First, let us write the equations of motion as x

=

f!{t,x,y,z,x,y,z),

V

=

f2(t,x,y,z,x,y,z),

z

=

f3(t,x,y,z,x,y,z).

Define G — (Gi, G2, G3), where Gi — fi - x, G? = f2are looking for quasi-periodic solutions of the form x

=

V

=

y ^ xkexp((k,

y, and G3 = fo - z. We

u)ty/^l),

^Vkexp^k^^y/^l),

where w = (uii, u>2,..., wr)* is a known set of basic frequencies the ones that appear as basic frequencies, in the developments of the functions P(i), and (k,u>) denotes the inner product of k = (fci, k2,..., fcr)' and u. We consider x, y and z truncated up to some order. After substitution, G may be considered as a function of x^, yu and Zk, which means that looking for quasi-periodic solutions of this problem is more or less equivalent (this depends on the order of approximation taken for the trigonometric expansions of a;, y, and z) to look for a zero of the function G : M™ —> Rm, where m denotes the total number of coefficients. To solve this equation we used a Newton method. Due to the fact that the perturbation is too big to take the zero solution as initial condition, we used a Newton continuation method: for this reason we added a continuation parameter (h) multiplying the nonconstant terms of the perturbations. Thus, when h is equal to 0 we have an autonomous differential system with an equilibrium point near

Idea of the Resolution

Method

135

the origin (the equilibrium point is not the origin itself because the constant terms of the perturbations are not multiplied by h). This point is close enough to the origin to be found by the Newton process starting from the zero solution. Then we increased h and this point became a small quasi-periodic orbit (see [11]) that was found by the Newton method (otherwise the value of h must be reduced to an intermediate value). Now the value of h is increased again and the Newton process is started from the solution found for the last value of the parameter. If no problems appear (that is, bifurcations and/or turning points), this process can continue until h reaches the value 1. Unfortunately, in this problem those difficulties are found, not allowing h to reach the final value 1 (to obtain more details, see [9]). In the present work we have started running the software mentioned just above, with the new model obtained in Chapter 4, to find that h can not reach (again) the value 1. The maximum value we can reach is some value close to h = 0.5. This fact seems to be related to some low order resonance. To try to solve this problem, we have proceed in the following way: first, we have identified the main frequencies that seem to be relevant in the solution we are looking for (this has been done by means of several runs using different thresholds). Then, the coefficients taking part in the Newton method are fixed: we select the ones corresponding to these relevant frequencies (that is, the continuation method is done with a fixed number of arguments). This allows to consider the parameter h as one of the variables that take part in the Newton method. With this, the method we have implemented works as follows: let G be the function defined above, going from an open domain of E" + 1 to E n (n being the number of coefficients related to the selected relevant frequencies). The exponent n + 1 indicates that the parameter h is considered as an unknown coefficient. Note that the equation G(p) = 0 defines a curve of E " + 1 , and we are interested in the intersection of this curve with the hyperplane h = 1. Let po = (^c^o) be a point of that curve (at the beginning this point is taken equal to 0) such that G(po) = 0. Prom the Jacobian of G it is possible to compute the tangent direction vo to the curve, and to define p\ as XQ + Svo • Then this point is refined by means of a Newton method to obtain the next point of the curve, p\. Note that, during each step of Newton method, the linear system we have to solve has n equations but n + 1 unknowns. We solve it by taking the solution of the system of minimum Euclidean norm (this is more or less equivalent to look for the point of the curve closest to pi). Note that this method can overcome some of the difficulties mentioned above (like turning points), but it can not solve completely some of the others (like bifurcation points: sometimes we can "cross" the bifurcation and continue the curve in the other side, but we can not detect the number of branches and their directions). To have a good analysis of a bifurcation it is necessary to look at the Hessian matrix on the bifurcation point (this is true for a "generic" bifurcation, in some (degenerated) cases being necessary to compute higher order derivatives), but for this problem that computation overcomes by far our actual computing capacities.

136

Nominal Paths and Stability

5.3

Properties

The Algebraic Manipulator

We are going to describe the algebraic manipulator used in this problem. This manipulator is based in the one used in [9]. All the basic operations are the same, and their description can be found there. Here we will only comment the differences in the high level routines. 5.3.1

High Level

Routines

To simplify the reading, we have grouped the routines in common blocks, according to their functions. Let us comment each group separately. 5.3.1.1

Input/output

Routines

The only differences here come from the fact that the formats of the different data files have been changed. 5.3.1.2

Evaluation of the Function

The routines that evaluate function G (see Section 5.2) have been modified in order to include the new perturbing functions P(i), i = 2 1 , . . . , 38. 5.3.1.3

Evaluation of the Jacobian Matrix

As before, we have done corrections to take into account the effect of the new perturbing functions. Moreover, we have written the routines that compute the derivatives with respect to h. The algorithm used is very similar to the one used to compute the other parts of the Jacobian matrix. 5.3.1.4

The Newton Method

This is the most different part of the program. We have deleted the routines that take care of adding new frequencies, and we have rewritten the Newton routine itself. As mentioned above, at each step we have a point (let us call it XQ) of the curve defined by G(x) = ( d (x),..., Gn(x)) = ( 0 , . . . , 0) (x € Rn+1), and we want to compute the next one (from now on, x\). To do it, the program goes through the following steps: (1) Computation of the tangent direction to the curve at XQ\ to do this, we note that this curve can be seen as the intersection of n surfaces (defined by the equations Gi(x) = 0, i = 1 , . . . , n) inside Rn+1. Then, the Jacobian of G can be seen as n vectors (of n + 1 components), each one orthogonal to the corresponding surface in XQ . This implies that the tangent direction can be obtained by finding an orthogonal direction to these vectors. This direction is the one of the kernel of the Jacobian matrix, that can be found easily

Results with the Algebraic

Manipulator

137

using Gaussian elimination. In fact, the result of this part of the program is a vector of n + 1 components (let us call it vi), orthogonal to all the rows of the Jacobian matrix of G, positively oriented with respect to them (v\ is taken as the last element of the basis) and satisfying ||t>i|| = 1. (2) Choosing orientation along this direction: as the direction pointed by v\ above could not be the right one (we want to move forward on the curve, even through turning points and bifurcations) we need to check that orientation. At the beginning of the run (h = 0), the orientation is chosen such that h increases (or, equivalently, such that the component of vi corresponding to h be positive). During the run, if the scalar product between vi and VQ (here VQ is the tangent vector obtained in the previous point of the curve, pointing to the right direction) is positive, v\ will be well oriented. Otherwise, we will change its orientation: vi = —v\. (3) Computing a new point: the new point of the curve is approximated by xi = x0 + 5v\, where 5 is a small value (we will come back to this value later on). As we want to achieve h = 1 exactly (note that h is one of the components of xi), we check that component to see if it is bigger than 1. In that case, the value of 5 is decreased in order to obtain h = 1. Note that we are looking at the curve as the solution of a certain differential equation, and we are integrating that equation using the Euler's method. (4) Refining the new approximation: here starts the Newton method. As it has been explained before, we try to solve G(x) = 0 using as initial guess the value x\. As the linear system obtained (let us call it Au — f) has n + 1 unknowns and n equations, we select the solution closest to the origin (in the sense of the Euclidean distance). This is done by a Gaussian elimination-based algorithm, to avoid the normal equations AtAu = Alf that are usually ill-conditioned. If more than a certain number of iterations are needed, the value of 5 (see above) is reduced (typically, 5 = (5/2), x\ is recomputed and this step is started again. If the value of S is too small, the program stops issuing the corresponding error message. Moreover, when the scalar product < VQ, V\ > is computed, we take the opportunity of computing the angle a between the two vectors. If a is lower than a certain value, S is increased (5 = 5x2).

5.4

Results with the Algebraic Manipulator

We have done several runs with that program. To explain them, let us introduce some control parameters of the program: t l is a threshold used to read the perturbing functions (when a coefficient has amplitude lower than t l is neglected), eps is the threshold used in the operations of the algebraic manipulator (all the coefficients that appear during the operations whose amplitude is lower than eps

138

Nominal Paths and Stability

Properties

are dropped, for more details see [9]) and zer is the threshold to decide if the value of the function is 0 or not. The degree of the expansion of the nonlinear part of the equations can be set to any value between 1 and 9. In all the runs we are going to comment, the degree has been fixed to 6. Initially, we have selected t l = 10~ 4 , eps = 10~ 8 and zer = 1 0 - 5 . With this, we have found a turning point close to h = 0.5. The continuation of the curve backwards in h seems to go out of the domain where the expansions are valid. Then, a way to try to avoid those difficulties can be to increase the threshold t l : we will read less perturbing frequencies and then we will have less resonances. We have done that using t l = 10~ 3 and we have still found the turning point, but using t l = 1.5 x 1 0 - 3 we do not have any turning point before reaching h = l. This solution is the one used below as a solution obtained with the algebraic manipulator. The next step has been to try to improve this solution, decreasing the value of t l and applying Newton method using it as initial guess (here we keep h = 1). In this case, the method has been found divergent. To explain this phenomenon, we note that the solution with t l = 1.5 x 10~ 3 contain the "biggest" effects of the perturbation. This implies that it is more or less good for short time intervals, as it is ensured by Gronwall's Lemma. Unfortunately, in this case the effect of the resonances dropped when selecting t l = 1.5 x 10~ 3 becomes relevant if the time interval is longer. This implies that the coefficients of the Fourier expansion found by the algebraic manipulator for this case, are very far from the real ones. Thus, they are not suitable as a starting point of a Newton procedure.

5.5

Numerical Refinement

In order to obtain a good nominal orbit we have implemented a modification of the classical parallel shooting method (see [18] or [9]) to get a solution of the JPL model very similar to the one found with the algebraic manipulator. To explain the modifications introduced in the parallel shooting (that can also be found in [10]), let us include here a short description of the classical method. First of all, we split the time span [a, b] in which we want to find the nominal orbit in several pieces [ti,ti+i], i G {0, . . . , n - 1}, verifying that t0 = a, tn = b and h — ti+i — ti = (b — a)/n (different time intervals can be used but we have chosen here all of them equal). Now let i be a value between 1 and n — 1 and let T be the function defined as follows: if xt is a point of the phase space, then Ti(xi) — yt+i, where yi+\ is the value of the solution defined by the initial conditions (U,Xi) at time ij+i, under the flow of the JPL model. Now we consider the vector (x0,... ,xn), where Xi are coordinates in the phase space corresponding to time ti. We can define T = (To,..., Tn~i) hi the usual way: (x0,...,xn)

>—>

(yi,...,yn),

Numerical

Refinement

139

and note that, if the values Xi are all from the same solution of the JPL model, then we should have that yt = Xi, 1 < i < n. Then we impose these conditions and this leads us to solve a nonlinear system of Qn equations and 6n + 6 unknowns. To do this, we have used again a Newton method and we have taken as initial guess the values provided by the solution obtained using the algebraic manipulator. As we have more unknowns than equations, the classical procedure adds six more conditions: the initial position (the three first components of XQ) and the final one (the three first components of kept fixed. With this, the system we deal with has 6n equations and the same number of unknowns. The problems we can find with this version of the method are, basically, that the (extra) boundary conditions can force the solution in a nonnatural way, giving convergence problems when we try to refine long time spans. In order to avoid this, we have used the following modification: instead of adding extra conditions, we apply Newton's method directly. This leads to a linear system with more unknowns than equations, having (in the general case) an hyperplane of solutions. Then, we select one of these solutions to proceed. Now, the criterion to select this solution has to be fixed. It is natural to look for the solution nearest (with respect to some norm) to the initial condition of the Newton's method, and this leads us to choose the point of the hyperplane nearest to the origin. We have used the Euclidean distance to select that point, but it is possible to use other distances. This is more or less equivalent to search for the orbit of the real system closest (in the sense of the norm used) to the one found with the algebraic manipulator. 5.5.1

The

Program

This is essentially the program described in [9], but with some modifications inside its routines. The biggest one is inside routine SISBAN, because it is the one that looks for the solution of minimum norm. There are several ways of computing that solution, but due to the fact that the linear system we deal with has a band matrix and the dimension is very large, we have avoided the normal equations (that are usually ill-conditioned), and we have looked at that point as the orthogonal projection of the origin on the hyperplane of solutions (as the distance is the Euclidean one, the scalar product is the Euclidean one). With this, is easy to see that the point we are looking for is s

x = a + y^^CjVj, i=0

where a is a point of the hyperplane, s is the dimension of that hyperplane, Vi are their director vectors and c* are the components of the array c defined as Mc — b, where M = (m^), b = (bi), m^- = < V{,Vj >, bi = - < a, Vi >, i £ { 1 , . . . , s} and j £ { 1 , . . . , s}. In the actual case we have that s is equal to 6. Finally, in order

140

Nominal Paths and Stability

Properties

to compute the point a and the vectors vi, the program uses Gaussian elimination with partial pivoting. 5.5.2

Nominal

Paths

Now, using the program described above, it is possible to find orbits for the JPL model with a behavior similar to the one of the planar solution obtained by the manipulator. We have refined three orbits: all of them start at the year 2000.0 (MJD 18262.0), the orbit 1 lasts for 146 revolutions of the Moon and orbit 2 lasts for 200 revolutions of the Moon. For these two orbits we have used as initial condition for the parallel shooting the quasi-periodic orbit provided by the algebraic manipulator. Orbit 3 has been obtained using as initial condition the L4 point, and it has been refined for 146 revolutions of the Moon. It is not possible to refine this orbit for a time span of 200 revolutions of the Moon due to convergence problems in the parallel shooting. These orbits are plotted in the next figures. Figure 5.1 shows the (x, y) projection of orbit 1. We have used a dotted line, plotting a dot each 0.25 days. Figures 5.2 and 5.3 show the same as Figure 5.1 but for orbits 2 and 3. Figures 5.4 to 5.9 show the same orbits, but using the xz and (y, z) projections. To compare these orbits, we have drawn each coordinate against time. Figures 5.10 to 5.15 show how the x coordinate depends on time, for different time intervals. Note that orbits 1 and 2 seem to be the same, but orbit 3 is slightly different, specially at the boundaries. Similar plots (shown in Figures 5.16 to 5.21) are obtained for the y coordinate. The z coordinate (shown in Figures 5.22 to 5.27) is worth a more detailed remark: Note that the three orbits are completely different in size. This seems to be related to the length of the time interval used to refine the orbit: the vertical amplitude increases with the length of the time interval. Moreover, the kind of motion looks like an harmonic oscillator (related to the vertical mode of the RTBP) plus perturbations. Indeed, the vertical behavior has a frequency which has very small changes when the z-amplitude is changed. Compare to the results given in Chapter 2 for the RTBP and in Chapter 3 for the Bicircular model. To obtain a quantitative comparison of these orbits we have performed a Fourier analysis of the three coordinates, to identify the main frequencies of the motion. This analysis has been done by the method described in Appendix B, and the results are shown in Tables 5.1 to 5.9. In all these Tables we have used the same notation: w denote the frequencies found by the Fourier analysis, and cu, sw denotes the coefficients of the corresponding sine and cosine.

Numerical

Refinement

141

.#K,

m "'^tSgjgl^. J l i l f i

- O . I . —O . 0 8 - 0 . 0 6 - 0 . 0 4 - 0 . 0 2

Fig. 5.1

O

0.02

0.04

0.06

O.OS

O.i

{x,y) projection of orbit 1.

* to2.xyz•

—O . i

-O.Q8 - 0 . 0 6 - 0 . 0 4 - 0 . 0 2

Fig. 5.2

O

0.02

0.04

(x,y) projection of orbit 2.

0.06

O.OS

O.I

142

Nominal Paths and Stability

Properties

.mmmmrn^, mmmm. H»sit« —O . 0 6

-O . 1 - O . 0 8 - 0 . 0 6 - 0 . 0 4 - O . Q 2

Fig. 5.3

O

O.Q2

0.04

0.06

0.08

O.l

(x,y) projection of orbit 3.

'tol.xyz•

-O.004

-

-O- 0 0 6

_J

i~

-O . X - 0 . 0 8 - 0 . 0 6 - 0 . 0 4 - 0 . 0 2

Fig. 5.4

O

0.02

0.04

(x, z) projection of orbit 1.

0.06

0.08

O.l

Numerical

143

Refinement

»i

- v . . . . . . . r :>,TV • .

m

- O . 0 0 6

-

—O . X

—O.OS - 0 . 0 6 - O . 0 4

Fig. 5.5

- 0 . 0 2

0.02

0.04

0.06

O.OS

O-l

(x,z) projection of orbit 2.

1 s.** ^ -~ ,r-F'^5F^S%C» .> - ^ J

-O.006

L _J

-O . X

1_

-O.OS - 0 . 0 6 - 0 . 0 4

Fig. 5.6

- 0 . 0 2

(x,z)

O

0.02

0.04

projection of orbit 3.

0.06

O.OS

O.X

144

Nominal Paths and Stability

-O . DQ6

Properties

-

-0.08

- 0 . 0 6

- 0 . 0 4

Fig. 5.7

- 0 . 0 2

Q

O - OZ

0.04

O.OS

O . OS

0 . 0 6

0 . 0 8

(y,z) projection of orbit 1.

O . 0 0 6

$&.m

-O.OS

— O . 0«5

- 0 . 0 4

Fig. 5.8

- O . Q 2

0 . 0 2

0 . 0 4

(y, z) projection of orbit 2.

Numerical

145

Refinement

* to3.xyz•

—O.003

- 0 . 0 8

—O.OS

- 0 . 0 4

Fig. 5.9

—O.D2

0.02

0.04

0.06

0.08

{y,z) projection of orbit 3.

— O . 06

18250 18300 18350 18400 18450 18500 185SO 18600 1B65Q 18700 1B75Q

Fig. 5.10

Orbits 1 and 2. Modified Julian days against x coordinate.

146

Nominal Paths and Stability

Properties

2 0 0 0 0 2 0 0 5 0 2 0 X 0 0 2 0 X 5 0 2 0 2 0 0 2 0 2 5 0 2 0 3 0 0 2 03 5 0 2 04 0 0 2 0 4 5 0 2 0 5 0 0

Fig. 5.11

Like Figure 5.10, but for a different time interval.

- o . 04

2 1 7 5 0 218QO 2XS50 21900 2 1 9 5 0 2 2 0 0 0 22050 22100 2 2 1 5 0 22 2 00 2 2 2 5 0

Fig. 5.12

Like Figure 5.10, but for a different time interval.

Numerical

Refinement

X 8 2 S O 1 8 3 0 0 1 8 3 5 0 X 8 4 0 0 1 8 4 5 0 X S S O O 1 8 5 5 0 XSfiOO 1 8 6 5 0 1 8 7 0 0 1 S 7 S O

Fig. 5.13

Orbits 2 and 3. Modified Julian days against x coordinate.

2 000 0 20 0SO 20X00 20X50 202 00 20250 20300 203 50 20400 2 04S0 2 05 00

Fig. 5.14

Like Figure 5.13, but for a different time interval.

147

Nominal Paths and Stability

148

Properties

2 1 7 5 0 2 1 8 0 0 2 1 8 5 0 2 3.900 2 1 9 S O 2 2 0 0 0 2 2 0 5 0 2 2 1 0 0 2 2 1 5 0 2 2 2 0 0 2 2 2 5 0

Fig. 5.15

Like Figure 5.13, but for a different time interval.

O . 04 ^

1S250 XS300 1S3SO 1 S 4 0 0 1S45Q 18500 18550 13600 18650 18700 187SO

Fig. 5.16

Orbits 1 and 2. Modified Julian days against y coordinate.

Numerical

Refinement

2 0 0 0 0 2 0 05 0 2 0 1 0 0 2 O 1 5 O 202 00 2 0 2 5 0 203 0 0 2 0 3 5 0 2 04 0 0 2 0 4 5 0 2 0 5 0 0

Fig. 5.17

Like Figure 5.16, but for a different time interval.

2 1 7 5 0 2 1 B O O 2 18 5 0 2 1 9 0 0 2 1 9 S O 2 2 0 0 0 2 2 0 5 0 2 2 1 0 0 2 2 1 5 0 2 2 2 Q O 2 2 2 5 0

Fig. 5.18

Like Figure 5.16, but for a different time interval.

149

Nominal Paths and Stability

150

Properties

XS2SO 1 S 3 0 0 1 S 3 5 0 1 B 4 0 0 1 8 4 5 0 1 8 5 0 0 1 8 5 5 0 1 8 6 0 0 X 8 6 5 0 1 8 7 0 0 1875C

Fig. 5.19

Orbits 2 and 3. Modified Julian days against y coordinate.

20000 2 0050 20100 20150 20200 2025Q 20300 20350 20400 204S0 20500

Fig. 5.20

Like Figure 5.19, but for a different time interval.

Numerical

Refinement

2 17 5 0 2 1SOO 2 1 8 5 0 2 1 9 0 0 2 1 9 5 Q 2 2 0 0 0 2 2 0 5 0 2 2 1 0 0 22 1 5 0 2 2 2 O O 2 2 2 S O

Fig. 5.21

Like Figure 5.19, but for a different time interval.

«i|J|jj|j|'|jriT|]. O.002

—O.002

-

'iffl™™^ j

—O.004

-o.ooe

II llll 111111 . •

18250 18300 18350 18400 18450 18500 18550 18600 18650 18700 18750

Fig. 5.22

Orbits 1 and 2. Modified Julian days against z coordinate.

151

152

Nominal Paths and Stability

Properties

-0.006 2 0 0 0 0 2 0 0 5 0 20X0 0 Z0150 20200 20250 20300 2 0 3 5 0 20400 2 0 4 5 0 2 05 00

Fig. 5.23

Like Figure 5.22, but for a different time interval.

-O.002 -

2 1 7 5 0 2 X B O O 2 1 8 5 0 2 1 9 0 0 2 X 9 SO 2 2 0 0 0 2 2 0 5 0 2 2 1 0 0 2 2 1 5 0 2 2 2 0 0 2 2 2 5 0

Fig. 5.24

Like Figure 5.22, but for a different time interval.

Numerical

Refinement

—O.004

—O.006 1S250 18300 18350 18400 18450 18500 18550 1S6O0 18650 18700 18750

Fig. 5.25

Orbits 2 and 3. Modified Julian days against z coordinate.

O . 002

—O.002

2 0 0 0 0 2 0 0 5 0 2 0 1 0 0 2 0 1 S O 2 0 2 0 0 2 0 2 S 0 2 0 3 0 0 2 03 5 0 2 0 4 0 0 2 0 4 5 0 2 0 5 0 0

Fig. 5.26

Like Figure 5.25, but for a different time interval.

153

I

c

poi P I

oi p

*

to ^1 o to to

cn o o PI PI

CO CO CO

cn -j

to ^1

1-^ to CO CO O M to O PI

1-*

I-*

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to

00

^1

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PI

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Pi

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CO ^ 1 -~i CO M cn 0 0 ro H* ^ J 0 0 cn ^ 1 I-* c> o cn ^ 1 •r* ^^ 11 ^ 1 CO ^ i cn o ro ro £ > CO cn cn tOo cn cn I O ^roi itti- * o CO ^ 1 O cn o o O o o o O O o Poi Pi PI PI PI Pi

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*> *> *»

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cn

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cn cn ^j cn ro cn to O 0 0 cn o CO O to r> cn cn O i it- t o cn o CO CO o cn cn cn ^ 1 CO cn ro cn CO tCO CO ^ i ^ i ro cn cn cn o o o o o r> O O o o P I P I P i P i PJ P i P I P I P i P I

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co to o o

p-*

t

o+ o

o c-> C3 r> o o PI

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CO

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IO CO rf> to o to 0>

o o o o o o o o o o o o o o o o o o

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0+3

E+0

Numerical Refinement

cu

U)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

OOOOOOOOE+OO 65544630E-01 13273590E+00 20761740E+00 72515420E+00 78410200E+00 85887270E+00 92770310E+00 99177890E+00 11246450E+01 17175990E+01 17754060E+01 18504490E+01 19234170E+01 19830260E+01 27091720E+01 27670700E+01 28420310E+01 37008880E+01 38336060E+01

Table 5.2

-0 0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 0 0 0 0 -0

0O0OO0OOE+O0 78822460E-01 15411520E+00 78039310E+00 84637600E+00 91295580E+00 97908530E+00 10056130E+01 10705190E+01 55957810E+02 56032580E+02 56190230E+02 56264980E+02 11214390E+03

Table 5.3

{ci + styi-

Su

0 -0 -0 -0 0 -0 -0 -0 -0 0 -0 0 0 -0 0 0 0 0 -0 -0

10000000E+01 29807140E-03 12033290E-02 12209710E-03 37157620E-04 13056180E-02 35538390E-01 25338880E-02 59724100E-02 80644030E-04 15285290E-03 94939940E-03 13089100E-01 38681260E-03 68343880E-03 49904010E-03 78696030E-05 93576640E-04 22448860E-03 10042150E-03

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

OOOOOOOOE+OO 30001090E-03 27242650E-02 17202470E-03 12943820E-03 15175370E-02 39605690E-01 41164980E-02 91991130E-02 11553280E-03 73653930E-03 11050090E-02 16731600E-01 40348590E-03 68348330E-03 68627570E-03 11815770E-03 14617820E-02 22744800E-03 11857170E-03

Fourier analysis for the y coordinate of orbit 1.

cu

OJ

0 0 0 0 0 0 0 0 0 0 0 0 0 0

23916130E-03 34057680E-04 24440980E-02 12118080E-03 12399010E-03 77348460E-03 17482390E-01 32442210E-02 69967130E-02 82730740E-04 72050410E-03 56540760E-03 10422190E-01 11479040E-03 77923190E-05 47109790E-03 11789530E-03 14587840E-02 36571680E-04 63045790E-04

155

-0 0 0 0 -0 -0 0 0 0 0 0 -0 -0 0

17828620E-03 29771510E-02 15098830E-03 98529270E-04 26586880E-03 11144810E-03 18911910E-03 34662690E-03 15241780E-03 92522900E-04 26468240E-02 25352670E-02 83512850E-04 17972860E-02

(4+4)1/2

su 0 -0 -0 -0 -0 0 0 -0 0 -0 -0 -0 -0 -0

10000000E+01 62701730E-02 16642980E-03 87478900E-04 27535260E-03 11344170E-03 33712460E-04 25211940E-03 12881810E-04 43900870E-04 11203610E-02 13624870E-02 48493250E-04 75883770E-03

0 0 0 0 0 0 0 0 0 0 0 0 0 0

OOOOOOOOE+OO 69410740E-02 22471390E-03 13175950E-03 38276O0OE-O3 15902f30E-03 19210040E-03 42861920E-03 15296120E-03 10240980E-03 28741750E-02 28781850E-02 96571170E-04 19509150E-02

Fourier analysis for the z coordinate of orbit 1.

156

Nominal Paths and Stability

OOOOOOOOE+OO 66095090E-01 13128870E+00 20744760E+00 30263390E+00 68893070E+00 78426890E+00 85887410E+00 92824420E+00 99158020E+00 10163570E+01 11245250E+01 17177320E+01 17757670E+01 18504480E+01 19238720E+01 19829910E+01 27093270E+01 27676600E+01 28420100E+01 37008970E+01 38335850E+01

Table 5.4

-0 -0 0 0 -0 0 0 0 0 0 -0 -0 -0 -0 -0 0 -0 -0 -0 -0 0 0

0 0 -0 0 -0 -0 0 0 0 -0 -0 -0 -0 -0 -0 0 -0 0 0 0 0 -0

cw

00000000E+00 65998010E-01 13289840E+00 20732490E+00 30263580E+00 72610950E+00 78401940E+00 85887410E+00 92835310E+00 99158010E+00 10163290E+01 11242620E+01 17177120E+01 17757760E+01 18504480E+01 19239720E+01 19830000E+01 27093250E+01 27677160E+01 28420100E+01 37008970E+01 38335840E+01

Table 5.5

10000000E+01 56155640E-03 79095180E-03 18014270E-03 10636400E-02 32596090E-04 61759590E-03 10966270E-01 69773450E-03 47414120E-02 21072120E-03 18076140E-03 83016600E-03 11624000E-02 16881660E-01 99768270E-04 29552160E-03 63793560E-03 13481120E-03 16802440E-02 18809170E-04 51564840E-04

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

OOOOOOOOE+OO 59284350E-03 40629800E-02 29629020E-03 14315150E-02 13837370E-03 21814430E-02 56257460E-01 48957000E-02 13095260E-01 25140740E-03 18305300E-03 87607720E-03 13825140E-02 20239360E-01 47179680E-03 87523990E-03 80792410E-03 14957650E-03 17549740E-02 25674050E-03 13835290E-03

Fourier analysis for the x coordinate of orbit 2.

u 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

17792260E-02 19004680E-03 39852480E-02 23523710E-03 95807300E-03 13447960E-03 20921920E-02 55178280E-01 48457250E-02 12206750E-01 13712120E-03 28873780E-04 27988500E-03 74844490E-03 11164280E-01 46112750E-03 82383970E-03 49576150E-03 64800250E-04 50666970E-03 25605050E-03 12838460E-03

(4+4)1/2

su

cu

U!

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Properties

-0 0 -0 -0 0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 0 0 0 0 0 -0

14596610E-03 50496340E-04 22377780E-02 13394630E-03 24099890E-03 12637250E-03 83121260E-03 17276290E-01 13154740E-02 75693540E-02 75771550E-04 66925500E-04 70858430E-03 75914590E-03 10529700E-01 17770120E-04 88378630E-05 42353030E-03 11207130E-03 14709830E-02 41417450E-04 62142940E-04

(cl+sl)^

sw 0 -0 -0 -0 0 -0 -0 -0 -0 -0 0 0 -0 0 0 -0 0 0 0 0 -0 -0

10000000E+01 25028320E-03 14226640E-02 10018460E-03 82106800E-03 18322730E-04 12641620E-02 35486690E-01 32174370E-02 54265890E-02 16634490E-03 92087550E-04 19082760E-03 84832440E-03 12993910E-01 37418910E-03 68444820E-03 53233050E-03 54032770E-04 98832690E-04 22341720E-03 10069020E-03

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

00000000E+00 25532640E-03 26517210E-02 16726800E-03 85570620E-03 12769390E-03 15129500E-02 39468660E-01 34759710E-02 93135920E-02 18278940E-03 11383820E-03 73383030E-03 11384010E-02 16724720E-01 37461080E-03 68450530E-03 68026000E-03 12441670E-03 14742990E-02 22722380E-03 11832270E-03

Fourier analysis for the y coordinate of orbit 2.

157

cw

U)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

OOOOOOOOE+OO 12289640E-01 78841430E-01 14412830E+00 78021370E+00 84622190E+00 91268900E+00 93820240E+00 97911940E+00 10039190E+01 10696640E+01 19955230E+01 39890840E+02 40807790E+02 40882410E+02 41040090E+02 41114920E+02 81768710E+02 81843780E+02

Table 5.6

-0 0 0 0 0 -0 -0 -0 0 0 0 -0 -0 0 0 -0 -0 0 0

00000000E+00 63911230E-01 13269580E+00 20758840E+00 30530050E+00 78401840E+00 85887070E+00 93078550E+00 99157500E+00 11242660E+01 17175960E+01 17752970E+01 18504470E+01 19239380E+01 19831620E+01 27092770E+01 27675380E+01 28420260E+01 37008860E+01 38336010E+01

Table 5.7

0 0 -0 0 -0 -0 0 -0 0 0 -0 -0 -0 -0 -0 -0 -0 -0 -0

Sw 10000000E+01 16097450E-04 85250310E-02 18937560E-03 13286610E-03 11075440E-03 16968000E-03 53400190E-04 31710170E-04 31970240E-02 78764670E-04 26184260E-03 11155740E-03 48529740E-04 22314730E-02 18983220E-02 74198640E-04 46041240E-04 83144120E-03

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

i/2 (
Fourier analysis for the z coordinate of orbit 2.

cu

LJ

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11412920E-03 31004400E-03 41703410E-02 53888960E-05 12984070E-03 44961750E-03 13662810E-03 16639830E-03 18321330E-03 41943300E-02 19559550E-03 20827480E-04 12324860E-04 11427510E-03 45218930E-02 38640680E-02 12611020E-03 65937280E-04 20584610E-02

-0 0 0 0 -0 0 0 -0 0 -0 -0 -0 -0 0 -0 -0 -0 -0 0 0

17592650E-02 47694200E-03 36372780E-02 22603370E-03 22453500E-04 21742370E-02 55422730E-01 10455290E-03 12106450E-01 59556370E-04 33791290E-03 98048840E-03 11333240E-01 50961280E-03 78159860E-03 47219870E-03 61902410E-04 49391210E-03 25676900E-03 12807460E-03

(ciJ+'ii)1'2

sw 0 0 0 0 -0 0 0 0 -0 -0 -0 -0 -0 0 -0 0 0 0 0 -0

10000000E+01 32529730E-03 29872230E-02 19473750E-03 94968190E-04 29741210E-03 10727580E-01 51661780E-02 48356930E-02 17537160E-03 81509400E-03 95105330E-03 16777060E-01 10170800E-03 36389410E-03 65541820E-03 13901870E-03 16681680E-02 12947510E-04 52981300E-04

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

00000000E+00 57731450E-03 47067280E-02 29835200E-03 97586450E-04 21944840E-02 56451400E-01 51672360E-02 13036490E-01 18520840E-03 88236240E-03 13659650E-02 20246290E-01 51966320E-03 86215740E-03 80780240E-03 15217780E-03 17397510E-02 25709520E-03 13860060E-03

Fourier analysis for the x coordinate of orbit 3.

158

Nominal Paths and Stability

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

UJ OOOOOOOOE+00 62474140E-01 13269970E+00 20760510E+00 72517170E+00 78408730E+00 85886950E+00 93077770E+00 99157510E+00 11243040E+01 17174220E+01 17745460E+01 18504490E+01 19240310E+01 19831570E+01 27092490E+01 27669870E+01 28420300E+01 37008860E+01 38336060E+01

Table 5.8

00000000E+00 78823690E-01 15441820E+00 78040540E+00 84641680E+00 91309020E+00 97918410E+00 10048510E+01 10703280E+01 55957820E+02 56032580E+02 56190230E+02 56264970E+02 11214390E+03

Table 5.9

0 0 -0 -0 0 -0 -0 -0 -0 0 -0 0 0 -0 0 0 0 0 -0 -0

10000000E+01 57082490E-04 11578190E-02 12108190E-03 35950390E-04 13021040E-02 35508510E-01 20519810E-02 53313630E-02 91599700E-04 95512820E-04 10611440E-02 13089700E-01 40977440E-03 67362920E-03 51186030E-03 69195910E-05 93454940E-04 22448720E-03 10041410E-03

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

OOOOOOOOE+00 24883150E-03 27161510E-02 17191480E-03 12929040E-03 15193870E-02 39601090E-01 37183710E-02 92718980E-02 11429840E-03 74215410E-03 10734210E-02 16731190E-01 41094730E-03 67530900E-03 68130790E-03 11762200E-03 14617420E-02 22741950E-03 11856670E-03

Fourier analysis for the y coordinate of orbit 3.

cu

U)

0 0 0 0 0 0 0 0 0 0 0 0 0 0

18801570E-03 24219560E-03 24570170E-02 12204050E-03 12419170E-03 78298260E-03 17532600E-01 31009120E-02 75858210E-02 68363810E-04 73598230E-03 16188460E-03 10420780E-01 31026320E-04 47602410E-04 44964370E-03 11741830E-03 14587520E-02 36402090E-04 63048270E-04

(4+4)1/2

su

cw -0 -0 -0 -0 -0 -0 -0 0 -0 -0 -0 -0 -0 -0 -0 0 0 0 0 -0

Properties

-0 0 0 0 -0 -0 0 -0 0 0 0 -0 -0 0

17504580E-03 29821410E-02 16644610E-03 98501260E-04 22854060E-03 11863890E-03 18654350E-03 26332500E-03 15229580E-03 92710030E-04 26468200E-02 25352240E-02 83845390E-04 17974440E-02

(cl+sW

su 0 -0 -0 -0 -0 0 0 -0 -0 -0 -0 -0 -0 -0

10000000E+01 62669030E-02 13893240E-03 87156250E-04 28611750E-03 10635950E-03 41725840E-04 94290440E-03 75164910E-06 43484800E-04 11203440E-02 13625440E-02 47961630E-04 75845090E-03

0 0 0 0 0 0 0 0 0 0 0 0 0 0

00000000E+00 69402620E-02 21680990E-03 13152460E-03 36618860E-03 15933470E-03 19115320E-03 97898360E-03 15229760E-03 10240160E-03 28741660E-02 28781750E-02 96593820E-04 19509110E-02

Fourier analysis for the z coordinate of orbit 3.

The Neighbourhood of the Almost Planar Nominal

5.6

Paths

159

The Neighbourhood of the Almost Planar Nominal Paths

To study both the stability properties of the computed solutions and the relative motion of particles running in orbits close to these ones, we have performed a numerical integration of the variational equations on the JPL model. Then, computing the eigenvalues of the corresponding matrices it is possible to obtain approximations to the three Lyapunov largest exponents (let us call them A?, A° and A3) related to these orbits, by means of

A° = lim \j(t) = lim J

t—>oo

ln

IM*>",

t—>oo

1 < j < 3,

t

where <Jj(t) stands for the j t h eigenvalue (we have sorted them using the size of their moduli) of the variational matrix at time t. Usually we have that |Ai(J)| > |A2(*)| > IA3 (*) I = 0, and occasionally, we have |A 2 (t)| = 0 or |A 3 (t)| > 0 but not too far from 0. In Figure 5.28 we have displayed the behavior of Ai(t), corresponding to the orbit 2. Note that it seems to have a limit value close t o 3 . 5 x l 0 - 3 . This is a very mild instability, offering no problems concerning station keeping. Figure 5.29 shows the behavior of X2(t) (we remark that it is less than Ai), and Figure 5.30 contains a plot of the sum Ai(£) + A2(i) + A 3 (i). This sum gives an idea of the total amount of instability. A similar study has been also carried out for the other almost planar solutions. The results are very close to the ones presented here. Finally, to have an idea of the motion close to that orbit, we have performed the same kind of study that in [9]. That is, we have computed the eigenvectors of the variational matrix at the end of the orbit, that is, after 146 lunar revolutions for orbits 1 and 3 and after 200 lunar revolutions for orbit 2. As an example we have included such values for the orbit 2 (see Table 5.10). We see clearly two stable and two unstable directions, and a central one. As the central eigenvectors are complex, we have taken their real and imaginary parts as a basis of the real part of the eigenspace spanned by them. Then, we have multiplied these six vectors against the variational matrix at each day of the orbit, starting from MJD 18262.0. The plots obtained shows the same behavior as the ones obtained in [9], and for this reason they have not been included here. 0"l 02 03 04 05 06

Table 5.10

= = = = = =

0 -0 0 0 -0 0

510754408925D+06+0 372533317267D+02+0 280407706561D+00+0 280407706561D+00-0 272022258649D-01+0 341075278953D-03+0

OOOOOOOOOOOOD+OOv^ OOOOOOOOOOOOD+OOi/17! 959882512228D+00V/::T 959882512228D+00V-1

OOOOOOOOOOOOD+OOV^7! OOOOOOOOOOOOD+OOV^T

Eigenvalues of the variational matrix of orbit 2 after 200 revolutions of the Moon.

160

Nominal Paths and Stability

Properties

Fig. 5.28 Maximal Lyapunov exponent of orbit 2. The horizontal axis are days, with origin at the beginning of the orbit (MJD 18262.0).

Fig. 5.29

Like Figure 5.28, but for the second Lyapunov exponent.

The Neighbourhood of the Nominal

Paths

j^Wyw^M*^M*y»wyw»*w^wv**'

Fig. 5.30

Like Figure 5.28, but for the sum of the three Lyapunov exponents.

161

Chapter 6

Transfer to Orbits in a Vicinity of the Lagrangian Points

The purpose of this chapter is to study the transfer of the satellite from a GTO in the vicinity of the Earth to the quasi-periodic orbits, both to the ones obtained in Section 5.5.2 which can be considered as the substitute of the equilibrium point of the RTBP, and to the ones obtained in section 1.4, which are stable orbits, of rather large ^-amplitude, and sitting in a rather big vicinity of the Lagrangian points. The main ideas of the transfer follow [2]. This is, in order to save fuel consumption in terms of delta-v (An) the satellite will have several gravity assists of the Moon (swingbys). In spite of the fact that each swingby increases the transfer time in approximately one month, at least for a moderate number of swingbys (one or two), the An saving will be more than 100 m/s for each swingby. Taking into account both the cost of the transfer and the transfer time, although other strategies with one or three swingbys have been also studied, the following one with two swingbys and of a time span of about two months, has been selected in order to find the best ones. TRANSFER STRATEGY SELECTED • Departure from the G T O . A tangent impulse ranging approximately from 650 m/s to 700 m/s when the satellite is at the periapsis of the GTO raises the apoapsis up to more than 300000 km. This manoeuvre can be split in two, by means of a phasing loop, in order to satisfy the launch window requirements. (See Section 6.1.3 for the specifications concerning the GTO orbit). • Change of Inclination. Approximately between 4 and 5 days after the departure, when the satellite is at the apoapsis relative to the Earth, it is inserted in the apoapsis of an orbit having period of about 9 days. The change in inclination as well as the magnitude of the manoeuvre, depend both strongly on the final target orbit and on the type of gravity assist that the satellite will receive from the Moon. • Periapsis Passage 1. About 4.5 days after the former manoeuvre the satellite passes approximately to 12 000 km from the center of the Earth. If 163

164

Transfer to Orbits in a Vicinity of the Lagrangian

•

•

•

• •

•

•

•

Points

desired transfer strategies with more than one periapsis passage before the first swingby can be studied as well. First Swingby. About 9 days after the Change of Inclination,! and near the apoapsis relative to the Earth, the satellite passes between 10 000 and 40 000 km from the center of the Moon. The satellite goes into an orbit of a period approximately half the period of the Moon's orbit. Apoapsis Passage 2. About 2 days after the swingby the satellite reaches an apoapsis relative to the Earth. The distance from the center of the Earth is of about 425 000 km. Periapsis Passage 2. About 7 days after the Apoapsis Passage 2 the satellite is at a periapsis relative to the Earth, roughly 40000-50000 km from the center of the Earth. Apoapsis Passage 3. About half the period of the Moon after Apoapsis Passage 2 the satellite is again in an apoapsis similar to the number 2. Periapsis Passage 3. Manoeuvre. About half the period of the Moon after Periapsis Passage 2 the satellite is again in a similar periapsis. At this moment a manoeuvre varying from 15 m/s to 60 m/s depending on the final target orbit is performed in order to obtain another swingby with the Moon. Second Swingby. About 27 days after the First Swingby when the satellite is near the apoapsis, it passes again near the Moon at a distance which could go from approximately 10 000 to 50 000 km. The satellite goes into an orbit of eccentricity and period with roughly resemblance to the Moon's orbit. Apoapsis Passage 4. Periapsis Passage 4. Several days after the Second Swingby the satellite is in the apoapsis relative to the Earth at a distance from the Earth little more than the Earth-Moon mean distance. Eventually about 15 days after the apoapsis the satellite is in a periapsis relative to the Earth at a distance from the Earth little less than the EarthMoon mean distance. Insertion QPO Between 19 and 25 days after the Second Swingby for the target orbits of section 5.5.2 and much less precise time span for the target orbits of section 1.4,the satellite reaches its destination. An insertion manoeuvre must be performed. For the orbits of section 5.5.2 it is near 120 m/s. For the orbits of section 1.4 the size of the manoeuvre depends more both on the precise orbit and epoch.

One has to take into account that the previous figures are only approximations and the real situation for each case could be slightly different. Although this strategy came out as the best in terms of Av-transfer time, may be some other strategies shorter (res., longer) in transfer time but more expensive (res., cheaper) in Av, can be interesting. As we will see in the next section, the computations are done in such a way that it is very easy to modify the strategy and to look for optimal transfer trajectories within the new strategy. We will give also some results in the

Computations

of the Transfer

Orbits

165

corresponding section.

6.1

Computations of the Transfer Orbits

Although the sketch of the transfer trajectories goes from the GTO around the Earth to the target QPO orbit near the Lagrangian point, the computations follow just the inverse way. That means that the insertion epoch to the QPO will be fixed and we will obtain the departure conditions from the GTO. If the departure conditions are not satisfactory it could be changed by means of phasing loops, changing the insertion date or changing the target orbit in case of the orbits of section 1.4. Before proceeding to explain the methodology, let us comment how we characterize a manoeuvre through all this work. Let us suppose that the satellite at some epoch has position r\ and velocity v\, usually relative to the Earth. A manoeuvre will be a change of the former velocity to a new one, v{, defined by a magnitude Av and two angles, a and /3, in the following way: v

l = vl + &vRvd,

where R is a 3 x 3 orthogonal matrix whose respective columns are unitary vectors obtained from v\, h\ Avls, hls where h\ = r\ f\v\ and Vd = (cos (i cos a, cos (3 sin a, sin j3)'. We have adopted the convention that the magnitude Av can be positive or negative and the angles a and /? must be in the interval [ ^ , ^]. In this way "to brake" becomes clear with negative Av values. Manoeuvres will be denoted by M(a, /?, Av) and it is clear that a measures an in plane direction, positive "out" of the orbit, and /3 an out of plane direction, positive towards the sense of the angular momentum of the orbit. A routine, CVANGL, computes v{ given r\, v\, a, j3 and Av. Another routine, CVRNG does the inverse function, given r\, vls and v{ it computes a, )3 and An. 6.1.1

Looking for Arrival

Conditions

at the

QPO

Firstly we are going to look for the arrival conditions at the QPO which come from a passage near the Moon. Since the target orbits have a very mild instability or they are stable, we can not expect to arrive at it asymptotically in a reasonable time span in the cases where it is theoretically possible. Some manoeuvre at the arrival point must be performed in order to have the insertion. So the way of computing these insertion manoeuvres will be essentially to select the arrival epoch and an insertion manoeuvre. Then, to subtract the insertion manoeuvre from the target conditions at the QPO and to integrate backwards in time the obtained conditions, till a encounter with the Moon produces a significant decrease of the energy of the satellite's orbit with respect to the Earth. This simplified description of the method differs depending on the type of the target QPO.

166

6.1.1.1

Transfer to Orbits in a Vicinity of the Lagrangian

Points

Case of the Target Orbits of Section 5.5.2

The QPO orbits of section 5.5.2 are the orbits which substitute the Lagrangian equilibrium point. Seen in the synodic system their ^-amplitude is very small and it is quite close to the geometrically denned equilibrium point. This means that seen from the Earth these orbits are close to the Moon's orbit, they lie almost in the same orbital plane and the satellite is advanced or late with respect to the Moon, depending whether we consider the one which substitutes L\ or the one which substitutes £5 respectively. As a consequence of this fact many manoeuvres (of moderate magnitude) which keep the satellite in the same orbital plane will have a passage near the Moon soon or later. So, for this case, a rough sweep of values and the integration backwards produces a good (and big) set of insertion values. Starting with a swept of values with manoeuvres of the type M(Q, 0, Av) varying Av from —200 to 200 m/s, a big set of orbits "falling" to the Earth can be obtained integrating backwards the initial condition r\, v{. For the numerically refined QPO orbits around Li the suitable values of Av are in agreement with the negative values given in [2], ranging from —140 to —90 m/s. However, very occasionally, for Av values outside the former interval, for instance ~ —45 m/s, some orbit appears that has a close encounter with the Moon (may be passing at less than 3 000 km). A short time later another encounter with the Moon, this time at a greater distance, produces an orbit with periapsis relative to the Earth of a few km. In spite of the fact that these transfer orbits could appear as less expensive, we have discarded most of them because of their forced appearance. A final swept is done for the selected insertion epochs, moving now moreover the Av in a larger interval containing the suitable one, the angles a and j3 in the respective intervals [—20°, 20°], [—10°, 10°] in steps of 5 degrees. A big amount of insertion conditions are obtained. Selecting only the ones which have a periapsis relative to the Earth of less than 80 000 km, we found that we are dealing with orbits whose final eccentricity is between 0.65 and 0.77 and which passed between 10000 and 70000 km from the Moon. The explorations reveal, as it can be seen in Table 6.1, that the suitable Av range had no variations and the suitable ranges for a and /? are fair big. Another interesting fact is that the swingby will happen between 15 and 22 days prior the insertion manoeuvre, being most of the results between 20 and 22.

6.1.1.2

Case of the Target Orbits of Section 1.4

Orbits of section 1.4 are no longer contained in the Moon's orbital plane. Their ecliptic inclination is about 30 degrees for the orbit of z-amplitude 0.5, being the inclination of the Moon near 5 degrees. In spite of this fact, we can advance that these target orbits have their inclination close to the inclination selected for the GTO orbit wnich is 7 degrees with respect to the equatorial plane, this is about 30 degrees with respect to the ecliptic plane, and so, in principle, the total Av for

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168

Transfer to Orbits in a Vicinity of the Lagrangian

Points

we save the part of change of inclination when doing the insertion manoeuvre at the target point. For this purpose we select an insertion epoch and then, using a two-body problem approximation, routine CRNDIL computes the epochs when the Moon crossed the orbital plane of the target orbit. These epochs appear in pairs differing approximately in half the period of the Moon. We can select for instance the ones which are within one month before the insertion manoeuvre, and then another routine, RDVZIL, using again a two-body problem approximation, computes the insertion manoeuvre at the target point from an orbit which passes through the target point at the insertion epoch and it is on the Moon* at the time given by routine RDVZIL. In case that more than one elliptic orbit is possible (transfer time greater than the period of the smallest possible one) the cheapest one is chosen. The orbits which imply a big amount of Av, say greater than 200 m/s, for the insertion manoeuvre are discarded. Using each of the former approximations and the routine CVRNG, one can compute the approximations for the insertion manoeuvre which are of type M(an,0, Ai>0)Then, using the JPL model, a similar swept process as for the case of target orbits which substitute the equilibrium point can be done but only around the values an for a, 0 for /? and AVQ for Av. In the same way we select as the suitable transfer orbits the ones which have the periapsis near the Earth. Table 6.2 shows part of a typical output. 6.1.2

Computing

the Successive

Swingbys

At the end of the former procedure we know the values of the insertion manoeuvres which come from a near encounter with the Moon for a selected insertion epoch. As we said previously integrating backwards the arrival condition, prior the insertion manoeuvre, we have a gravity assist from the Moon which results in an orbit having a periapsis close to the Earth. Next step will be the computation of the manoeuvre which must be done in this periapsis, such that integrating backwards again, has a passage near the Moon obtaining an orbit with less energy with respect to the Earth and with a lower periapsis than the previous one. In principle this manoeuvre is requested to be of the type M(0,0, Av), that is, tangent to the orbit in the direction of the velocity of the satellite relative to the Earth. The computations are done in routine PSWP and can be outlined in the following way. Let us suppose that we have the satellite in the previously mentioned periapsis at the epoch t. As we need the Moon near the apoapsis of the new orbit, firstly we compute the true anomaly, vQ, of the Moon when it is in the apoapsis line of the satellite's orbit. This value can be approximated easily by the difference in true *For practical purposes, since the best swingbys always take place when the Moon and the satellite are in conjunction, being the Moon between the Earth and the satellite, it is better to look for these initial approximations choosing a point in the intersection line and away from the Moon the usual distance of the swingbys. This is, for instance, 20 000-40 000 km instead of "on the Moon".

Computations

Way 18262.0 18262.0 18262.0 18262.0 18262.0 18262.0 18262.0 18262.0 18262.0 18262.0 18265.0 18265.0 18265.0 18265.0 18265.0 18265.0 18265.0 18265.0 18265.0 18265.0 18265.0 18265.0 18265.0 18265.0 18265.0

Q

P

-4.0 1.0 6.0 -9.0 -4.0 1.0 6.0 11.0 16.0 21.0 -45.0 -50.0 -50.0 -55.0 -55.0 -60.0 -60.0 -53.0 -53.0 -58.0 -63.0 -58.0 -58.0 -63.0 -68.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

At; 53.0 53.0 53.0 56.0 56.0 56.0 56.0 56.0 56.0 59.0 76.0 79.0 82.0 85.0 88.0 94.0 97.0 83.0 86.0 89.0 92.0 92.0 95.0 104.0 113.0

of the Transfer Orbits

Ai M 23.6 23.6 23.5 23.6 23.6 23.5 23.5 23.5 23.4 23.4 26.4 26.5 26.5 26.7 26.6 26.8 26.7 26.6 26.5 53.3 26.8 26.7 26.6 26.8 27.1

Moon-D 26973.5 25092.5 26259.0 13503.7 8911.8 6825.1 7332.6 10832.8 17577.1 10211.0 6448.5 17376.0 4911.4 23055.6 11047.3 26106.4 16351.7 17946.6 6234.6 22673.2 59215.7 17601.5 8765.8 20275.0 41589.3

169

Earth-D 71779.4 67275.1 73635.8 40781.4 40573.2 45036.2 38490.8 35682.1 55937.1 60374.2 56800.2 44561.1 60363.5 59190.7 34174.1 71821.3 54861.0 40414.6 40313.1 34288.1 70747.5 36753.2 68939.7 79206.2 43774.8

Ecc 0.699 0.713 0.688 0.823 0.822 0.801 0.829 0.841 0.757 0.759 0.772 0.799 0.744 0.743 0.846 0.708 0.776 0.813 0.809 0.833 0.695 0.828 0.731 0.703 0.803

Table 6.2 Insertion manoeuvres selected from the sweeping process with insertion days 1.0-JAN2000 (18262.0 MJD 1950.0) and 3.0-JAN-2000 (18265.0 MJD 1950.0), for the stable target orbit of z-amplitude 0.5 near the equilibrium point L 5 . In addition to the a, 0 and Av values of the insertion manoeuvre, AtM indicates the time span in days since the encounter with the Moon, Moon-D the approximate Moon distance in km when the swingby, Earth-D the approximate periapsis distance in km from the Earth of the obtained orbit integrated backwards, and Ecc, the approximate eccentricity of the obtained orbit seen from the Earth.

longitudes: VQ

~ fts + uis +

•K

- ftm - u)m,

where ft and ui denote respectively the longitude of the ascending node and the argument of periapsis of the orbits of the satellite (subindex s) and the Moon (subindex m) computed at the epoch t. Then, using Kepler's equation, we compute the time span, At, since the Moon had true anomaly v = v0. If we want the satellite at the apoapsis at the time t — At, the time span At must be an odd multiple of the semiperiod of the orbit of the satellite. At this time we have two options. The first one is to impose the odd number of semiperiods before the encounter. The second one is to select the number of semiperiods in such a way that the magnitude of the

170

Transfer to Orbits in a Vicinity of the Lagrangian

Points

manoeuvre for accomplishing this fact be the smallest one. In the later case the number of semiperiods is chosen as the odd number nearest to At HT asir V as' where as is the semimajor axis of the satellite's orbit (with respect to the Earth) and fj, is the gravitational parameter of the Earth. Once the odd number of semiperiods, n, is chosen, one can compute the energy of the new orbit by means of

1 t

(imn\\

~ 2 I At )

'

and from it the velocity at the periapsis

where r p is the periapsis distance from the Earth. Of course the manoeuvre will be M(0,0, Av) where Av = vp - vs and vs is the modulus of the velocity of the satellite when it is at the previous periapsis obtained integrating backwards. We note that this procedure can be repeated integrating the orbit backwards after the swingby till another periapsis with respect to the Earth is found. Then another manoeuvre for producing another swingby can be computed and so on. When the approximation to the real value of Av is found, a small swept of values can be performed in order to take the most advantage of the swingby. But in this case instead of doing the swept of values of Av whose step should vary according to the magnitude of the initial Av, it is better to vary the encounter point with the Moon. Then instead of having true anomaly v = v0 for the Moon when the satellite is at the swingby periapsis, we require that the Moon have true anomaly v — i/0 + Av. It has been observed that the most advantageous result is obtained for values of Av between 20 and 25 degrees, and it seems independent of the swingby. Another interesting fact is the chosen value n. For the case of the transfer orbit from the GTO to the target point using two swingbys, the local optimization does not imply global optimization. That is, usually the smallest periapsis manoeuvre which produces the desired encounter with the Moon is obtained with n = 3 and it is done nearer than the Moon's orbit. But using n = 1, implying that the period of the new orbit will be about half the one of the Moon, the swingby produced behind the Moon's orbit is much more effective. In spite of the fact that the periapsis manoeuvre is quite more expensive than for n = 3, the save in global consumption can be greater than 100 m/s.

Computations

6.1.3

Departure

of the Transfer

from the GTO.

Orbits

Change of

171

Inclination

The final procedure is to connect the obtained orbit coming from the previous sections with the GTO around the Earth. The selected GTO has 300 km of periapsis height^, 35 786 km of apoapsis height and equatorial inclination of 7 degrees. When possible, the optimum transfer is obtained raising the apoapsis when the satellite is at the periapsis of the GTO by means of a manoeuvre M(0,0,Av), and then to perform a manoeuvre of insertion with an eventual change of inclination when the satellite reaches the apoapsis, that is, a manoeuvre of the type M(0,/3,Av). So we have, firstly, to look for the insertion point where the (eventual) change of inclination must be performed. This is accomplished integrating backwards the last obtained periapsis after the desired number of swingbys. The selected apoapsis is usually the first one we found since it minimizes the transfer time, but any other one obtained integrating backwards an additional fixed number of revolutions can be considered. Let (a 2 ,e2,i2,^2,W2,7r) be the orbital elements of the satellite at the selected apoapsis, and ai, e\ and i\ the fixed elements of the GTO orbit. We look for a transfer orbit from the periapsis of the GTO to the apoapsis where the insertion point is. The orbital elements of the transfer orbit will be (at, et, it, fit, wt, u), where the first three ones are easily determined since we know the periapsis and apoapsis distance for the transfer orbit and moreover we impose it = iiThe routine CONEA computes the approximation of the real velocity of the transfer orbit in the insertion point in the following way. The vector velocity at the apoapsis has the expression

y at{l + et) where Q is an unitary vector which can be written in terms of the orbital elements as Q = (cos flt

sm w

* +sin flt cos ut cos it, sin flt sin u>t — cos flt cos ut cos it, cos uit sin itY.

So we have only to compute the ascending node, fit, and the argument of periapsis, u)t, of the transfer orbit, which will be the same ones for the GTO since we require their coplanarity and the alignment of their periapsis lines. An unitary vector pointing towards the apoapsis in terms of the orbital elements can be written down as A = (— cos fl cos u> + sin fl sin ui cos i, — sin fl cos u) — cos fi sin a; cos i, — sin w sin i)'. Since we know the unitary apoapsis vector for the transfer orbit, let us denote it by +The radius of the Earth has been taken equal to 6378.145 km.

172

Transfer to Orbits in a Vicinity of the Lagrangian

Points

R = (ri,r2,r3) t , imposing that A = R one gets the following equalities: sin ut

=

sin flt

—

-r3 sin it' r\ sin uit cos it ± T2 cos ut

If the right-hand side of the first equality has modulus greater than 1, the proposed transfer from the GTO to the insertion point can not be accomplished. If this happens, the inclination of the GTO should be changed or another optimal strategy, probably with three manoeuvres, should be designed. However, in our case this value has been at most 0.35 in absolute value for all the transfers studied, and so not any other strategy has been implemented. In our case the former equalities have two possible pairs of solutions for the angular unknowns fit and uit- This can be seen in the following way. Consider the orbital plane of the orbit that contains the insertion point. Since the transfer orbit we are looking for must be in the orbital plane of the GTO connecting its periapsis with the apoapsis of the former one, the required condition when the orbits have different inclination, is that the intersection line of both planes must contain the periapsis-apoapsis direction of the orbit which contains the insertion point. When both orbits have the same inclination the required condition is the coplanarity. In both cases two solutions for this required condition can be found rotating the GTO orbit around its angular vector direction, that is, changing the argument of the ascending node. When the argument of the ascending node is fixed one rotates the GTO orbit in its plane, this is to change the argument of periapsis, till the periapsis direction of the GTO and the apoapsis direction of the orbit which contains the insertion point are opposite aligned and, because of the choice of the argument of the ascending node, contained in the intersection set (a line or a plane) of the orbital plane of both orbits. From the two solutions one chooses the pair Q, w, that implies less change of inclination in the apoapsis manoeuvre from the transfer orbit to the insertion point. Since the periapsis manoeuvre at the GTO for raising the apoapsis, are in both solutions of the same magnitude, and the modulus of the velocity at the apoapsis of the transfer orbit are also the same for both solutions, the selected one with the former criterion will be of course the best in terms of Av. From the orbital elements it is easy to find the right one since (sin i sin ft, — sini cosfl, cos i)*, is a unitary vector perpendicular to the orbital plane and in the sense of the angular momentum. As a result of the previous calculations in routine CONEA, we know an approximation of the arrival velocity v at the insertion point. Since all the above computations are done using a two-body problem model we must improve it to the real one. This

Computations

of the Transfer

Orbits

173

is done in routine COJPL where a Newton procedure is implemented in the modulus of v, in order that the integration backwards of the insertion position with velocity v, gave a periapsis near the Earth equal to the periapsis of the GTO orbit. Finally a periapsis manoeuvre of type M(0,0, Av) is done in order to obtain the GTO orbit of the desired size. Of course, because of the perturbations, for keeping the initial direction of v found in the approximation procedure, or for performing the periapsis manoeuvre at the GTO of type M(0,0, Av), some slight modifications in the orbital elements i, fi and UJ of the GTO from the given, or obtained ones, can be expected. But their change is so small that even it is not worth to modify slightly the procedure. However if some requirements should be satisfied, one could add them in the last part of the global transfer procedure which is the optimization.

6.1.4

Global Optimization

of the

Transfer

Using the algorithm of the previous sections, one is able to found transfer orbits from the periapsis of a GTO till a target point lying in a selected quasi-periodic orbit at a given epoch. The transfer orbits depend on the manoeuvres performed during the way and one should do them in such a way of taking the maximum profit of the swingbys, minimizing the Av consumption. So starting from a set of manoeuvres found in the previous sections, we look for a local minimum of the Av consumption using a conventional routine of minimization. In our case the variables for the optimization are the manoeuvre at the insertion point of the QPO and the manoeuvres at the periapsis in order to get the swingbys. Since each manoeuvre is of type M(a,/3, Av) depending on two angles, a and f3 and one magnitude Aw, the total number of variables is three when we look for optimal transfers with one swingby or it is six when we look for optimal transfers with two swingbys. The manoeuvres done at the periapsis of the GTO and the one of change of inclination seen in the above section, are not included in the set of variables since they are performed in the optimal way and are determined by the previous manoeuvres (integrating backwards). However they could be included, for instance if we require that the slight variation in inclination, mentioned at the end of the previous section, is not desired. Seeing how the computations are performed, one can include easily other variables eventually restricted in a range, such as the insertion epoch at the quasiperiodic orbit or the inclination of the GTO. Analogously one can include the restrictions which must be satisfied by the transfer orbit. In our case the only ones included have been the distances from the transfer orbit to the Moon for each swingby. The routine FTRTL does the transfer from the QPO orbit to the GTO, backwards in time, once a set of manoeuvres is given. Full information about the transfer is contained in its output values. From it, it is easy to implement an objective function and the desired constraints.

174

6.2

Transfer to Orbits in a Vicinity of the Lagrangian

Points

Summary of the Results

Using the algorithm of the previous section, we have searched the transfer possibilities for a rather big set of arrival conditions, and different nominal target orbits. Most of the given results have the insertion epoch during the first month of the year 2000. Nevertheless, sometimes other insertion epochs have been explored as well. 6.2.1

On the

Magnitudes

In our computations we look only for transfer trajectories lasting for a time span, counted from the GTO departure till the QPO insertion, not much larger than two months. This restricts the number of swingbys at a maximum of two for what we could call an ordinary transfer. However, as we will see, it is possible to have transfer trajectories with two swingbys lasting for about one month, that extended to three swingbys last for about two months. The explorations are done according to the following sketch. From a selected set a, P, Av, of arrival conditions which followed backwards in time give an approach to the Earth after having a swingby with the Moon (INI1), firstly we compute the departure from the GTO and we optimize the obtained trajectory from the GTO to the QPO. We have a local optimum transfer trajectory with one swingby. Now we have two sets of arrival conditions at the QPO, the selected first one, INI1, and the optimum one (OPTl). Then for each one we compute the periapsis manoeuvre which produces a second swingby with the Moon when integrating backwards, let us call them INK and INI12 respectively. These new transfer trajectories are extended till the GTO and then used as starting point of the optimization procedure. We get two local minima, OPT2 from INK and OPT12 from INI12. The results of these computations for different insertion epochs and several target orbits are summarized in Tables 6.3, 6.4, 6.5, 6.6 and 6.7. One could expect that the values of the optimized variables in OPT2 and OPT12 were very close. This is not always true specially for the angular variables a and (3 of the insertion manoeuvre. However most of the times the Av values of the manoeuvres are very similar and the total amount of Av is of the same order of magnitude. Moreover the plots of the transfer trajectories have slight differences as we can see in Figure 6.1 in page 189. Roughly we could say that the percentage of saving on the total amount of Av is as follows INI1 ^ 5 INK ^

OPT2, or

INI1 ^

OPTl ^

INI12 ^

OPT12,

what implies that the addition of the swingby is the really important thing in the whole procedure. An inspection of the Tables shows that many times the way INI1 -> OPT2 or OPT12 gives a Av saving of 200 m/s so the second swingby usually implies a saving of more than 100 m/s.

May n 18262.0 1 18262.0 2 18262.0 3 18262.0 4 18262.0 5 18262.0 6 18262.0 7 18262.0 8 18262.0 9 18262.0 10 18262.0 11 18262.0 12 18262.0 13 18262.0 14 18262.0 15 18262.0 16 18262.0 17 18262.0 18 18262.0 19

a -20.0 .0 -20.0 .0 -20.0 .0 -15.0 .0 -15.0 .0 -15.0 .0 -15.0 .0 -15.0 .0 -10.0 .0 -10.0 .0 10.0 .0 10.0 .0 15.0 .0 15.0 .0 15.0 .0 15.0 .0 15.0 .0 20.0 .0 20.0 .0

INI1-INI2 Av 0 -5.0 .0 .0 .0 5.0 .0 -10.0 .0 -5.0 .0 .0 .0 5.0 .0 10.0 .0 -10.0 .0 10.0 .0 -10.0 .0 10.0 .0 -10.0 .0 -5.0 .0 .0 .0 5.0 .0 10.0 .0 -10.0 .0 -5.0 .0

-140.0 36.8 -140.0 47.8 -140.0 36.9 -140.0 54.9 -140.0 51.7 -140.0 49.3 -140.0 51.7 -140.0 54.9 -140.0 47.8 -140.0 47.8 -140.0 44.1 -140.0 44.0 -140.0 53.1 -140.0 47.2 -140.0 43.9 -140.0 47.2 -140.0 53.0 -140.0 45.3 -140.0 56.2

AVT

1236.7 1060.8 1197.6 1028.6 1238.6 1066.7 1187.5 1005.4 1227.3 1041.1 1241.0 1060.1 1223.7 1063.5 1182.6 1031.6 1258.5 1058.7 1250.1 1085.7 1289.4 1086.9 1279.5 1092.5 1231.9 1054.3 1275.1 1077.4 1286.6 1091.0 1269.7 1080.5 1219.0 1049.1 1224.4 1035.4 1192.4 1015.6

a .0 .0 .7 .0 .5 .0 -1.2 .0 .9 .0 1.4 .0 1.4 .0 1.5 .0 -.7 .0 1.2 .0 3.0 .0 3.3 .0 4.0 .0 4.0 .0 3.6 .0 3.6 .0 3.6 .0 5.2 .0 4.5 .0

OPT1-INI12 AD 0 7.2 .0 8.0 .0 7.7 .0 6.6 .0 8.4 .0 8.2 .0 8.1 .0 8.2 .0 6.4 .0 7.8 .0 8.3 .0 8.6 .0 8.9 .0 9.1 .0 8.9 .0 8.4 .0 8.6 .0 8.9 .0 9.3 .0

-132.9 55.7 -133.2 55.9 -133.0 55.8 -132.8 55.7 -133.4 53.7 -133.3 53.8 -133.2 53.8 -133.2 53.8 -132.7 55.7 -133.1 53.8 -133.4 54.0 -133.5 54.0 -133.7 54.0 -133.8 54.0 -133.6 54.0 -133.4 54.0 -133.5 54.0 -133.8 54.1 -133.9 54.0

OPT AVT

1172.7 1031.3 1172.6 1028.7 1172.6 1029.7 1172.8 1033.5 1172.6 1030.4 1172.5 1029.7 1172.5 1029.4 1172.5 1029.5 1172.8 1034.2 1172.6 1029.2 1172.5 1028.6 1172.5 1028.8 1172.5 1028.7 1172.5 1029.1 1172.5 1029.0 1172.5 1028.3 1172.5 1028.7 1172.5 1028.0 1172.5 1029.1

a

0

-20.0 .0 8.8 -.4 -20.0 .0 14.6 .3 13.3 -.3 11.8 1.4 11.4 1.3 13.1 -1.7 15.6 -.1 14.5 .6 23.5 .6 24.2 .9 24.0 -.4 23.1 .6 23.5 .1 24.1 1.2 23.6 -.5 20.6 .0 24.0 .1

-5.0 .0 -9.2 -1.2 5.0 .0 -8.7 .2 -8.8 .5 -8.9 .3 -9.1 .1 14.6 .4 -8.5 -1.5 -8.7 -.4 -7.5 -1.2 -7.4 -.9 -7.5 -.6 -7.5 -2.0 -7.5 -1.8 -7.5 -1.6 -7.5 -.4 -10.3 .0 -7.5 -.2

-

18262.0 20 18262.0 21 18262.0 22 18262.0 23 18262.0 24* 18262.0 25* 18262.0 26* 18262.0 27* 18262.0 28*

20.0 .0 20.0 .0 20.0 .0 .0 .0 10.0 .0 10.0 .0 15.0 .0 15.0 .0 15.0 .0

.0 .0 5.0 .0 10.0 .0 .0 .0 -10.0 .0 10.0 .0 .0 .0 -10.0 .0 10.0 .0

-140.0 55.8 -140.0 56.2 -140.0 45.5 -130.0 24.0 -110.0 36.8 -110.0 38.1 -110.0 39.9 -90.0 48.9 -90.0 49.5

1199.9 1027.0 1184.3 1020.9 1210.1 1051.3 1275.3 1085.0 1245.3 1002.9 1250.3 1025.9 1235.5 1001.9 1158.9 922.5 1162.8 947.1

3.6 .0 3.7 .0 4.0 .0 3.4 .0 8.1 .0 10.5 .0 11.0 .0 18.1 .0 19.1 .0

8.8 .0 8.4 .0 8.8 .0 8.4 .0 -9.0 .0 9.2 .0 -7.0 .0 -7.6 .0 7.2 .0

-133.6 54.0 -133.5 54.0 -133.6 54.0 -133.4 54.0 -109.6 51.5 -110.3 51.3 -109.8 51.1 -90.6 51.1 -91.0 48.3

1172.5 1029.0 1172.5 1028.3 1172.5 1028.6 1172.5 1028.4 1150.3 971.3 1158.9 970.8 1150.9 972.5 1138.5 952.1 1147.7 960.2

24.3 .0 23.5 .7 24.2 .3 .0 .0 10.0 .0 10.0 .0 15.0 .0 15.0 .0 15.0 .0

-7.5 -.6 -7.6 -.5 12.2 .0 .0 .0 -10.0 .0 10.0 .0 .0 .0 -10.0 .0 10.0 .0

Table 6.3: T h i s t a b l e jointly with t h e table 6.8 on page 184, s u m m a r i z e s t h e c o m p u t a t i o n s t h e equilibrium point a n d was improved by t h e parallel shooting procedure for 146 revolu a "*" are t h e ones with an "extra" swingby. See t h e explanations in t h e captions of table

Iday n 18262.0 1 18262.0 2 18262.0 3 18262.0 4 18262.0 5 18262.0 6 18262.0 7*

a -20.0 .0 -15.0 .0 15.0 .0 20.0 .0 .0 .0 10.0 .0 10.0 .0

INI1-INI2 AD 0 .0 .0 -5.0 .0 -10.0 .0 -5.0 .0 .0 .0 -10.0 .0 10.0 .0

-140.0 48.7 -140.0 51.4 -140.0 55.0 -140.0 56.2 -130.0 28.3 -110.0 44.7 -110.0 40.4

AVT

1197.1 1020.9 1231.6 1043.4 1237.5 1036.7 1198.0 1020.7 1266.9 1071.7 1212.4 978.9 1242.0 1013.1

a .8 .0 -.2 .0 5.0 .0 5.4 .0 4.0 .0 7.1 .0 11.6 .0

OPT1-INI12 Av 0 10.3 .0 8.6 .0 10.6 .0 11.2 .0 10.5 .0 -6.5 .0 12.0 .0

-134.1 55.8 -133.3 55.6 -134.5 54.1 -134.9 54.2 -134.3 54.1 -108.6 51.6 -111.7 51.2

OP AUT

a

0

1175.2 1029.0 1175.4 1034.5 1175.0 1031.2 1175.0 1032.1 1175.0 1031.6 1149.0 967.7 1160.4 970.5

11.9 .2 12.8 .9 24.0 .0 23.9 .0 .0 .0 10.1 .0 10.0 .0

-7.2 -.7 -7.2 -.4 -5.8 -.2 -5.8 -.4 .0 .0 -10.4 .0 10.0 .0

CN O in

o o TH

CN T H

957.3

o oi o

896.3 O CN

o oo o m o

886.2

Ol r-i

-10.6 .1

960.3

m

-11.1 -2.4

917.2 923.8 913.6 914.2 913.3 913.7 887.5 942.5 929.3 875.5 877.6 874.8 876.5 877.6 918.8 908.9 956.7 886.6 981.2 939.5

177

935.1

19.2 -3.7

to o

977.0

in o 00 O to O CN O 00 ^ CO CO tCN t> Ol

949.0

-10.4 .0 CN

881.1

m o

886.4

-20.2 -3.2

878.5

Ol rH t- CN

CO O CO t~ TP Ol 00 t00 00 00 in ^-i CN '

CO O CN O in o to O in o t- O CN CN to to 00

o oo o m in

to ^-t co in CO i to

to O CN rH to o in Ol

rH O TH O in o CO 00

877.6

878.9

CN

o o

in

883.5

942.5

of the Results

Ol Tf Ol '

o o m

o

O CN CN O 00

rH T* CO

o oo o o

945.8

1010.3

-109.1 51.2 -90.1 56.3 -91.0 50.5 -137.9 61.7 -136.4 61.6 -138.6 61.7 -137.6 61.7 -121.3 61.5 -107.2 48.6 -108.2 47.7 -125.1 62.8 -125.4 63.3 -125.1 64.2 -125.4 63.2 -125.7 63.6 -98.0 45.5 -98.4 45.8 -114.1 67.8 -116.5 77.2 -107.5 80.9 -125.8 88.0

Summary

o CN rH tO rH O O

896.8

10.4 -.3

935.6 CN

O CN

~& O )

915.8

20.7 -.1 00 rH

tT Ol TT rH

o o o

CN

o

-10.0 .0 -10.0 .0 19.4 -8.2 -20.4 -4.0 -24.4 -2.8 15.6 -.3

937.1

Ol O Ol

t> Ol

'

CO lO

00

00 O rH O CO O to CN Ol

to

t- o rH O

rH

o o m

-11.0 .0

o o o

to C N

00 CN 00 rH

00 O CO O to

-20.0 .2 -15.9 .0

986.4

rH O

CO CN to i

t~ rH

CO

CO O CO O CN CN CO

o oo oo o m o

CN

o o

18.7 1.5 -15.0 .0

973.8 CN

o oo o -10.0 .0

-110.0 44.7 -90.0 44.1 -90.0 45.9 -132.8 60.4 -135.3 60.4 -130.4 61.4 -130.3 61.5 -119.6 91.8 -110.0 32.5 -110.3 43.9 -137.2 63.7 -133.2 64.4 -129.1 65.5 -128.2 65.9 -130.1 66.1 -100.0 40.3 -100.0 37.4 -120.4 71.2 -116.5 77.5 -116.0 82.0 -120.8 88.3

o

-20.0 .0

in oi

1- O Oi

to o

o o o

CN

-10.2 .0

1149.4 969.1 1144.5 942.5 1139.6 957.4 1169.0 993.8 1168.4 988.5 1167.9 988.2 1167.9 988.1 1150.0 966.1 1144.5 967.0 1141.0 945.5 1163.2 969.3 1161.9 945.8 1169.8 899.6 1161.8 943.1 1161.8 941.8 1122.6 946.5 1123.1 921.4 1167.1 989.1 1178.9 960.4 1196.3 991.2 1168.9 1001.3

o o m

in

o o ^r o

-tf O Ol O CO O rH O in o to O in o CO to to CN CN

CN

o oo oo o in in o

-18.7 .0

-10.5 .0

-109.0 51.2 -90.0 56.2 -90.9 50.6 -131.3 59.1 -132.4 59.1 -133.2 59.7 -133.3 59.7 -120.7 59.9 -107.2 48.6 -108.2 47.7 -123.8 63.3 -125.2 63.7 -125.2 64.0 -125.5 63.8 -125.7 63.8 -97.9 45.5 -98.4 45.7 -114.4 67.6 -114.5 75.2 -106.5 79.7 -122.9 87.5

o o

CN O

m

rH

o o m

o o o

CN

-10.0 .0 -15.0 .0

CN O rH O CO O CN 00 Ol

in

-10.0 .0 -10.0 .0

-20.0 .0

o o m

in

-20.0 .0 -20.0 .0

o oo oo oo oo oo oo oo oo oo oo oo oo oo oo oo oo o o in in m in o o o o o

in

o oo oo o

-15.0 .0

1213.3 979.3 1192.1 986.5 1171.3 943.8 1208.4 1000.6 1199.8 1031.0 1209.6 973.1 1204.4 989.8 1386.7 1954.5 1243.3 1022.9 1177.0 985.6 1243.1 936.5 1237.3 955.5 1225.2 926.9 1215.5 928.7 1204.7 990.8 1146.5 967.1 1160.9 966.8 1177.8 998.3 1212.3 977.1 1215.2 1006.1 1171.8 947.6

00 O Tf O Ol CN

o o -15.0 .0 -15.0 .0

-10.0 .0 -10.0 .0

-110.0 44.7 -90.0 44.1 -90.0 45.9 -140.0 58.5 -140.0 58.3 -130.0 49.7 -130.0 50.0 -130.0 63.7 -110.0 32.5 -110.0 43.9 -140.0 59.6 -140.0 59.1 -130.0 63.6 -130.0 63.4 -130.0 65.6 -100.0 40.3 -100.0 37.4 -120.0 70.5 -110.0 75.8 -110.0 80.5 -120.0 87.8

rH O O

-20.0 .0

o oo o

in

18268.0 19 18268.0 20 18268.0 21 18268.0 22 18268.0 23* 18268.0 24* 18271.0 25 18274.0 26 18277.0 27 18280.0 28

o o

18262.0 8* 18262.0 9* 18262.0 10* 18265.0 11 18265.0 12 18265.0 13 18265.0 14 18265.0 15 18265.0 16* 18265.0 17* 1 18268.0

15.0 .0 .0 .0 .0 .0 10.0 .0

18283.0 29 18286.0 30 18289.0 31 18289.0 32

.0 .0 .0 .0 .0 .0 .0 .0

-130.0 44.9 -130.0 46.2 -130.0 50.6 -130.0 51.3

1199.7 1000.0 1183.5 987.5 1175.9 1012.1 1162.9 1000.7

12.2 .0 2.3 .0 -6.0 .0 -4.9 .0

10.0 .0 7.8 .0 4.6 .0 5.0 .0

-128.5 50.0 -129.2 48.7^ -130.2 49.7 -130.0 49.8

1145.6 952.2 1141.3 970.7 1156.9 1010.7 1156.9 1011.0

6.2 -2.3 -11.2 -3.5 -15.8 -2.8 -9.0 -3.2

-14.3 -1.4 -12.7 -1.4 -11.0 -1.1 -10.3 -.4

Table 6.4: This t a b l e jointly with t h e table 6.9 on page 186, summarizes t h e c o m p u t a t i o n s d t h e equilibrium point a n d was improved by t h e parallel shooting procedure for 200 revolut a "*" are t h e ones with an "extra" swingby. See t h e explanations in t h e captions of t a b l e

Iday n 18262.0 1 18262.0 2 18283.0 3 18286.0 4 18286.0 5 18289.0 6 18289.0 7 18295.0 8 18298.0 9 18307.0 10 18310.0 11 18313.0 12

a -31.0 .0 -44.0 .0 72.0 .0 26.0 .0 36.0 .0 -11.0 .0 -31.0 .0 81.0 .0 55.0 .0 83.0 .0 70.0 .0 33.0 .0

INI1-INI2 An 0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0

63.0 22.9 70.0 16.5 141.0 22.8 68.0 14.2 77.0 19.5 66.0 22.6 72.0 20.4 139.0 44.2 73.0 38.6 188.0 19.8 137.0 20.0 76.0 27.9

Al/T 1154.6 1054.4 1170.9 1053.4 1197.2 1005.4 1163.4 994.1 1147.4 955.4 1158.0 1008.7 1145.8 984.2 1218.8 1059.9 1193.0 1010.4 1261.1 1087.4 1195.0 1023.2 1150.9 992.4

a -39.4 .0 -39.5 .0 74.0 .0 32.8 .0 39.5 .0 -20.5 .0 -20.5 .0 83.8 .0 67.2 .0 87.6 .0 66.0 .0 21.0 .0

OPT1-INI12 Av 0 -1.5 .0 -2.0 .0 -1.3 .0 -9.7 .0 -5.6 .0 -1.5 .0 -1.6 .0 -17.7 .0 3.4 .0 11.7 .0 .3 .0 .7 .0

68.4 23.6 68.5 23.6 147.6 27.6 74.7 23.1 79.4 26.6 68.7 25.7 68.7 25.7 163.1 45.4 96.6 51.8 225.1 29.4 122.1 30.3 69.1 29.6

OPT ADT

a

0

A

1133.7 1025.5 1133.7 1025.3 1183.4 1029.2 1142.3 953.3 1126.7 975.8 1117.2 974.1 1117.2 974.1 1194.8 953.2 1149.4 996.4 1224.8 1064.6 1157.4 994.4 1111.4 953.7

-37.8 1.7 -44.9 -2.6 72.0 -.1 32.9 -2.4 33.5 -3.0 -18.5 1.1 -27.1 -2.0 81.2 -.1 61.2 -.8 85.6 -.3 64.1 -.8 14.7 -.3

-6.7 1.5 -6.5 -1.2 -1.8 .0 -9.5 -.5 -7.7 -.7 -7.0 2.6 -4.1 -.7 -.2 .1 5.6 .6 -3.1 -.4 2.8 -.3 2.7 -.3

6 2 7 1 14 1 7 1 7 1 6 2 7 1 14 4 8 4 21 2 11 2 6 2

18319.0 13 18319.0 14

87.0 .0 -51.0 .0

.0 .0 .0 .0

152.0 47.6 98.0 -1.9

1198.5 1025.7 1321.8 1141.2

86.9 .0 -64.7 .0

-19.0 .0 -4.0 .0

159.3 75.7 115.9 29.3

1184.0 1072.9 1154.6 999.8

86.4 -.1 -62.8 2.2

.7 .0 -.2 .5

151 44 111 28

Table 6.5: This table jointly with t h e table 6.10 on page 186, summarizes t h e c o m p u t a t i o n s 0.3. See t h e explanations in t h e captions of t a b l e 6.6 on page 180.

Iday n 18262.0 1 18262.0 2 18262.0 3 18265.0 4 18265.0 5 18265.0 6 18265.0 7 18271.0 8 18274.0 9 18274.0 10 18274.0 11 18277.0 12 18277.0 13 18277.0 14

a 6.0 .0 1.0 .0 16.0 .0 -55.0 .0 -53.0 .0 -58.0 .0 -58.0 .0 68.0 .0 36.0 .0 41.0 .0 46.0 .0 -10.0 .0 -5.0 .0 5.0 .0

INI1-INI2 Av 0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0

53.0 8.7 56.0 13.5 56.0 7.9 88.0 14.9 83.0 16.4 89.0 -4.4 92.0 7.8 101.0 17.8 59.0 27.4 62.0 39.8 65.0 43.2 49.0 38.8 49.0 26.7 49.0 27.0

AUT

a

1130.8 985.1 1022.2 886.0 1069.0 979.7 1023.9 880.5 1046.6 972.3 1177.3 1008.5 1075.1 978.7 1191.8 1081.0 1092.3 980.8 1039.3 952.0 1095.9 1036.7 1005.1 906.4 1040.0 806.2 1124.5 1009.9

2.3 .0 3.3 .0 3.5 .0 -51.9 .0 -51.3 .0 -51.8 .0 -51.3 .0 76.8 .0 47.3 .0 47.5 .0 48.6 .0 -13.1 .0 -14.3 .0 -13.0 .0

OPT1-INI12 At) 0 3.9 .0 5.2 .0 6.7 .0 22.6 .0 25.8 .0 21.2 .0 24.9 .0 25.2 .0 12.1 .0 13.8 .0 17.8 .0 -9.8 .0 -2.7 .0 -6.1 .0

55.6 14.6 55.1 15.2 55.3 15.1 89.6 15.9 89.8 36.1 88.7 15.9 89.2 36.2 138.1 39.1 69.4 41.1 70.1 41.0 71.8 41.8 49.9 41.6 50.0 40.9 49.5 41.7

OPT2 AVT

973.8 836.8 982.7 894.8 982.7 895.6 985.7 833.3 992.5 887.2 986.3 835.1 992.8 890.5 1026.2 989.8 994.8 865.1 994.4 867.0 1007.2 944.6 993.8 931.8 982.2 854.9 994.0 931.9

a 2.3 .0 2.3 -.3 2.2 .0 -51.8 .0 -51.8 -.1 -52.0 .0 -51.8 -.2 71.3 .2 39.0 -.2 42.1 -.1 44.3 .0 -10.0 .0 -6.3 .0 3.5 .5

0

Av

.1 -.6 1.4 .1 1.3 -.2 10.3 .0 14.0 .4 7.1 .1 11.0 -.1 .5 .0 -.5 -.1 -.5 .0 .1 .0 .0 .0 -.1 .0 -1.0 .3

55 15 55 15 55 15 84 15 85 15 84 15 84 15 113 24 61 27 63 40 65 42 49 38 49 31 48 42

18280.0 15 18280.0 16 18286.0 17 18289.0 18 18289.0 19 18289.0 20 18289.0 21 18292.0 22 18292.0 23 18292.0 24 18295.0 25 18295.0 26

-58.0 .0 -63.0 .0 59.0 .0 15.0 .0 25.0 .0 30.0 .0 30.0 .0 -22.0 .0 -32.0 .0 -42.0 .0 94.0 .0 94.0 .0

.0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0

77.0 43.3 86.0 43.2 82.0 -16.5 51.0 3.5 54.0 35.7 54.0 4.2 57.0 14.5 56.0 -8.7 59.0 35.6 65.0 7.2 210.0 6.7 213.0 .7

1132.6 1072.0 1091.6 1046.7 1128.4 1012.0 1041.2 903.1 991.6 902.6 1123.8 962.0 973.2 808.5 1041.5 893.9 984.0 877.0 1042.5 919.7 1286.1 1158.5 1288.9 1126.7

-60.5 .0 -60.4 .0 70.4 .0 29.9 .0 30.6 .0 30.6 .0 29.9 .0 -32.5 .0 -32.7 .0 -32.7 .0 97.5 .0 97.5 .0

-26.6 .0 -26.3 .0 -21.3 .0 -.8 .0 -.9 .0 -.5 .0 -1.5 .0 19.8 .0 18.6 .0 17.3 .0 17.5 .0 17.7 .0

90.6 39.6 90.2 40.1 113.6 37.1 57.1 14.2 56.7 12.8 56.7 12.8 57.1 14.2 63.8 13.6 63.5 13.7 63.0 13.8 250.0 35.0 250.0 35.4

1004.6 951.9 1005.5 969.4 1011.1 902.7 970.9 790.4 981.5 896.1 981.5 895.5 970.9 790.4 967.5 785.8 967.6 786.7 967.9 786.2 1089.4 1055.9 1089.8 1061.6

-59.8 .0 -62.1 .0 68.9 -.3 29.6 -.2 29.1 -.4 29.6 .0 29.6 .0 -32.8 .0 -33.5 -.5 -32.7 -.4 94.3 .0 94.0 .0

-.5 .0 .0 .0 -7.5 -.1 1.1 -.1 .8 -.1 .8 -.1 .0 5.1 .1 9.4 -.2 8.1 .0 .5 .0 .0 .0

1

2 2

Table 6.6: This table jointly with the table 6.11 on page 187, summarizes the computatio 0.5. In the first column we have the insertion day (Iday) and an index (n) which relates o table. Then in each row of the remaining four columns, two manoeuvres (a, /3 and Au) are correspond to the insertion manoeuvre and the second one (same row as n) to the periaps The columns of A ^ T contain the total amount of Av needed for the transfer. When two as Iday) corresponds to the transfer with one swingby (only the insertion manoeuvre is th as n), to the transfer with two swingbys (the insertion manoeuvre and the periapsis mano initial transfer with one swingby (AX>T of the same row as Iday is its cost). INI2 to the in INIl the periapsis manoeuvre (AUT of the same row as n is its cost). OPTl refers to the o with two swingbys adding to OPTl a periapsis manoeuvre. Finally OPT2 and OPT12 a INI2 with respect to the insertion and periapsis manoeuvre.

INI1-INI2

CN

820.9

00 CO ID

o

767.0 CO xF

800.7

CD XT'

801.4

45.6 -.1

CD

801.0

CO

788.7

-16.1 .0 -15.3 .0 -15.2 .0

o o CN O CO O o o

797.2

O r-i ID

•H/

ID xf

800.8

830.2 829.9 831.6 784.6 784.6 784.6 784.8 777.5 840.5 842.5

of the Results

CN C~

CD CO

791.8

TH O CO O CN rH rH O rH O ID CO CN rH

CN

860.0

r~ o 00 O ID rH CD O

CN C~

o o CD O o o Oi O OI O CO rH b- rH

COCD

1040.0

C-

836.6

-21.5 .1

OI

O O rH O O O) CO CO

CN rH

769.9

809.3

829.7

Summary

767.7

t- o t- o rH O CO O O rH

00 CO

co to> ID

792.7

118.9 11.4 119.3 11.2 119.7 11.4 122.1 10.2 66.1 14.4 65.9 14.7 66.6 14.2 66.4 14.2 56.6 13.9 107.6 24.4 108.6 22.3 59.0 25.7 59.1 25.9 60.4 21.3 59.1 26.5 58.5 26.5 45.9 21.0 45.4 25.7 45.7 25.5

h

814.3

-11.4 .0 -12.7 .0 -11.8 -.1 -16.9 .0


CN

CN '

rH O CN O rH CD CD rH CN O CN CD CN xf *C* ••H CO CN

o o CN CD o o CO O r- O r- O CN O iO CN o o CO O o o o o O rH CO rH ID CN CO O l> O o o

799.2

<1

CQ.

799.0

o o o o rH O CN O t- o

<

00 CD

o o

-17.2 .6 -19.8 .4 -20.5 .0

59.4 7.0 66.4 5.2 70.5 4.7 57.0 -53.5 95.0 -84.9 105.7 29.6 53.0 -.7 55.9 16.8 58.5 26.1 59.8 26.0 59.2 26.3 43.7 12.9 46.0 26.0 47.0 26.7

a

OS

o o

CD

-22.1 -.3

101.7 .5 106.0 1.1 109.2 8.8 118.7 6.6

OPT12

<

<

<X1

ID CO

CD ID

CO O ID O r~ O r~ O ID O c- O b- O CD CD CD CD CD CN CN

ID

ID O 00 O CN O CO O CO O rH O rH O CN O 00 O CN O ID O CO CD CD CN

OI CO

CD

850.2 852.8 849.1 850.8 849.5 851.9 845.3 850.2 814.7 808.2 814.7 808.0 814.5 808.2 814.7 808.2 807.1 800.0 829.2 853.1 827.0 852.6 796.8 815.5 796.8 815.4 777.5 793.1 796.8 815.5 796.8 815.4 761.1 777.9 780.1 804.0 779.2 803.0

a

00

OI CO

CD CO

118.3 10.0 118.8 10.1 118.7 10.0 121.6 9.3 66.6 10.6 66.6 10.6 66.9 10.5 66.5 10.6 56.9 10.4 106.2 21.0 107.7 20.3 60.3 22.0 60.3 22.0 60.4 20.2 60.3 22.0 60.3 22.0 45.9 20.6 45.6 22.4 45.8 22.3

h

Oi CO

CO

r-

o o o o o o o o o o o o o o

CO CD

-16.2 .0 -15.4 .0 -15.3 .0

CO O ID O

0> CO

~H

t- o t^ o o> o o o o o CN O o o

CO

-21.7 .0

-11.5 .0 -12.8 .0 -12.3 .0 -17.2 .0

0PT2

<

<


CO O

CO

1096.2 861.5 1007.7 844.6 1134.9 933.8 913.3 915.8 1142.6 799.4 1075.5 853.8 1115.6 866.7 896.8 870.9 90.7 1001.0 128.9 1069.9 934.3 881.5 95.7 843.2 867.5 780.7 956.5 870.4 853.7 855.4 1172.2 1049.2 1177.2 970.2 1052.6 900.8 942.3 818.1

a

b-

103.0 10.4 106.0 1.1 115.0 19.8 118.0 6.6 59.0 2.4 62.0 16.5 65.0 15.7 71.0 5.2 68.0 -51.7 95.0 -84.9 107.0 31.4 53.0 -.7 56.0 16.7 59.0 34.8 65.0 24.8 71.0 34.1 44.0 27.4 47.0 31.3 47.0 27.2



r-

-16.0 .0 -26.0 .0 -21.0 .0

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o

co

o o o o o o o o o o o o o o o o bCN CN o ID ID o

co

-39.0 .0


a

c

18262.0 1 18262.0 2 18262.0 3 18262.0 4 18265.0 5 18265.0 6 18265.0 7 18265.0 8 18268.0 9 18277.0 10 18277.0 11 18280.0 12 18280.0 13 18280.0 14 18280.0 15 18280.0 16 18283.0 17 18283.0 18 18283.0 19

0PT1-INI12

800.8 800.8 780.7 800.9 800.8 767.1 786.6 786.2

181

18283.0 20 18283.0 21 18289.0 22 18292.0 23 18292.0 24 18292.0 25 18295.0 26 18295.0 27 18295.0 28

-31.0 .0 -41.0 .0 70.0 .0 29.0 .0 34.0 .0 44.0 .0 -26.0 .0 -31.0 .0 -36.0 .0

.0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0

50.0 14.5 56.0 -44.4 115.0 18.4 62.0 9.2 65.0 14.7 71.0 19.3 60.0 18.2 63.0 11.8 69.0 -52.5

893.9 834.4 98.7 1027.9 1063.0 919.6 1049.5 846.3 876.7 837.3 1056.1 910.9 1062.3 874.3 1070.7 845.0 80.5 1042.9

-16.2 .0 -15.4 .0 72.7 .0 39.9 .0 39.9 .0 39.9 .0 -21.6 .0 -22.0 .0 -22.0 .0

-4.2 .0 -4.6 .0 -12.2 .0 -7.6 .0 -7.5 .0 -6.7 .0 5.1 .0 3.1 .0 3.3 .0

45.8 20.8 45.4 22.5 126.1 8.5 69.3 10.3 69.3 10.3 69.1 10.4 59.4 10.1 59.8 9.2 59.8 9.2

761.9 778.2 780.8 805.0 842.7 859.1 803.6 812.4 803.6 812.3 803.8 812.1 795.5 803.5 778.6 779.4 778.6 779.7

-33.0 .0 -36.2 .2 72.2 .8 26.6 .2 39.2 .3 39.2 .7 -21.4 .2 -32.9 .8 ^22.4 -.5

-.3 .0 .3 .0 -2.8 -.2 -1.1 .0 -1.7 .1 -3.8 .2 .4 -.1 .0 -.2 .5 .2

1

Table 6.7: This table jointly with the table 6.12 on page 188, summarizes the computatio 0.7. See the explanations in the captions of table 6.6 on page 180.

Summary

of the Results

183

As we stated in the previous section, the transfer with one swingby requires three manoeuvres and the transfer with two swingbys four. The values of these manoeuvres can be seen in Tables 6.8 to 6.12 which are associated, respectively, with Tables 6.3 to 6.7. For a selected target orbit, the magnitudes and epochs of the manoeuvres are essentially independent of the insertion day. Moreover one can see that the addition of the second swingby implies an important reduction of the second manoeuvre, which is usually performed between 4 and 5 days after the departure from the GTO. Almost all the saving due to the addition of the second swingby reverts to this manoeuvre. These Tables also contain the Moon distance at the swingbys which usually is between 15000 and 40000 km. A description of a trajectory from the GTO to the QPO with two swingbys can be seen in Table 6.15 on page 192. Another interesting thing are the orbits labeled with a "*" in the Tables 6.3 and 6.4. Some of them are represented in Figures 6.3, 6.9 and 6.10 on pages 199, 205 and 206 respectively. These transfer orbits have two encounters with the Moon separated, approximately, half the period of the Moon, this is about 15 days, just before the insertion manoeuvre. These type of orbits can only be found when targeting to an orbit which lies almost in the same place as the Moon's orbit. The simulations revealed that using a transfer orbit like this, sometimes the insertion manoeuvre could be lowered to approximately 50 m/s, from the 100-120 m/s which is the usual range when targeting to the orbits which lie in the Moon's orbital plane. The bad fact is that with an insertion manoeuvre of about 50 m/s the distance to the Moon in the swingby prior to the insertion can be of only about 3000 km. A description of a trajectory from the GTO to the QPO with two plus an extra swingby can be seen in Table 6.16 on page 193.

184

Transfer to Orbits in a Vicinity

of the Lagrangian

ONE SWINGBY Way n 18262 1 18262 2 18262 3 18262 4 18262' 5 18262 6 18262 7 18262 8 18262 9 18262 10 18262 11 18262 12 18262 13 18262 14 18262 15 18262 16 18262 17 18262 18 18262 19 18262 20 18262 21 18262 22 18262 23 18262 24* 18262 25* 18262 26* 18262 27* 18262 28*

Ai>i ti 679.5 .0 679.4 .0 679.4 .0679.5 .0 679.4 .0 679.4 .0 679.4 .0 679.4 .0 679.5 .0 679.4 .0 679.4 .0 679.4 .0 679.4 .0 679.4 .0 679.4 .0 679.4 .0 679.4 .0 679.4 .0 679.4 .0 679.4 .0 679.4 .0 679.4 .0 679.4 .0 679.1 .0 679.2 .0 679.3 .0 679.7 .0 679.8 .0

Av2

A«3

*2

*3

360.3 4.2 360.0 4.2 360.1 4.2 360.5 4.2 359.8 4.2 359.8 4.2 359.9 4.2 359.9 4.2 360.6 4.2 360.0 4.2 359.7 4.2 359.6 4.2 359.4 4.2 359.3 4.2 359.5 4.2 359.7 4.2 359.6 4.2 359.4 4.2 359.3 4.2 359.5 4.2 359.6 4.2 359.5 4.2 359.7 4.2 361.5 4.2 369.4 4.2 361.9 4.2 368.1 4.2 376.8 4.2

132.9 34.5 133.2 34.4 133.0 34.5 132.8 34.5 133.4 34.4 133.3 34.4 133.2 34.4 133.2 34.4 132.7 34.5 133.1 34.4 133.4 34.3 133.5 34.3 133.7 34.3 133.8 34.3 133.6 34.3 133.4 34.3 133.5 34.3 133.8 34.2 133.9 34.3 133.6 34.3 133.5 34.3 133.6 34.3 133.4 34.3 109.6 43.0 110.3 42.9 109.8 42.9 90.6 43.0 91.0 43.0

MDi

Points

TWO SWINGBYS AKT

Aui

Av2 *2

42582.8

1172.7

42776.1

1172.6

42573.6

1172.6

42961.4

1172.8

42729.0

1172.6

42716.6

1172.5

42669.8

1172.5

42669.0

1172.5

42847.3

1172.8

42759.7

1172.6

42685.9

1172.5

42654.2

1172.5

42686.0

1172.5

42700.6

1172.5

42695.5

1172.5

42676.2

1172.5

42640.5

1172.5

42599.1

1172.5

42573.5

1172.5

42649.6

1172.5

42655.9

1172.5

42678.0

1172.5

42653.8

1172.5

12401.8

1150.3

12465.5

1158.9

12370.8

1150.9

3727.6

1138.5

3753.5

1147.7

687.0 85.2 .0 4.6 98.1 686.9 .0 4.6 687.0 86.5 .0 4.6 85.4 687.0 .0 4.6 687.0 84.9 .0 4.6 687.0 84.6 4.6 .0 687.0 84.5 4.6 .0 86.0 687.0 4.6 .0 86.1 687.0 4.6 .0 85.1 687.0 4.6 .0 84.0 687.0 4.6 .0 83.6 687.0 4.6 .0 83.9 687.0 4.6 .0 84.7 687.0 4.6 .0 84.2 687.0 4.6 .0 83.6 687.0 4.6 .0 84.1 687.0 4.6 .0 84.8 687.0 4.6 .0 83.8 687.0 4.6 .0 83.5 687.0 4.6 .0 84.1 687.0 4.6 .0 687.2 110.9 4.6 .0 84.4 687.0 4.6 .0 93.6 685.8 4.6 .0 685.7 74.7 4.6 .0 99.1 685.9 4.6 .0 684.8 88.7 4.5 .0 685.6 104.1 4.6 .0

Al>3

Auj U 56.7 142.4 36.8 61.1 55.9 134.8 36.8 61.7 56.7 141.3 36.8 61.1 56.7 142.3 36.8 61.1 56.8 142.6 36.8 61.1 56.7 143.0 36.8 61.0 56.7 143.1 36.7 61.0 56.7 141.8 36.8 61.1 56.6 141.7 61.1 36.8 56.6 142.6 61.1 36.8 56.8 143.5 36.8 61.0 56.8 143.9 36.8 61.0 56.7 143.7 36.8 61.0 56.7 142.8 36.8 61.1 56.8 143.2 36.8 61.0 56.8 143.8 36.8 61.0 56.7 143.4 36.8 61.0 56.8 142.8 36.8 61.1 56.8 143.7 36.8 61.0 56.8 143.9 61.0 36.8 56.8 143.4 61.0 36.8 56.6 146.2 61.2 36.9 56.8 143.1 36.8 61.1 51.6 109.6 70.6 36.6 51.4 110.3 36.5 70.4 51.1 109.8 36.7 70.5 48.9 90.0 36.5 70.1 48.3 91.0 70.2 36.4 *3

AtJT MD2 41778.8 25223.7 42612.6 27006.7 41886.0 25241.9 41599.5 25003.1 41872.0 25165.8 41623.8 25242.8 41486.9 25122.7 41843.5 25270.2 41672.2 25266.8 41417.6 25072.0 41644.0 24905.3 41514.4 24837.8 41478.0 24917.6 41627.5 24986.8 41735.1 24867.5 41630.0 24831.1 41557.5 24869.1 41812.1 25174.8 41603.1 24880.9 41567.5 24828.8 41609.9 24909.1 41148.5 23739.6 41631.0 24858.8 12398.1 24005.5 12453.0 25644.2 12372.2 23789.5 3849.8 31160.7 3780.4 27819.9

971.3 975.8 971.5 971.3 971.3 971.3 971.3 971.4 971.4 971.3 971.2 971.2 971.2 971.3 971.2 971.2 971.2 971.3 971.2 971.2 971.2 1000.9 971.3 940.7 922.1 945.9 912.4 929.0

Table 6.8 Results for the orbit which substitutes the equilibrium point and improved by parallel shooting for 146 revolutions of the Moon. Associated with Table 6.3 on page 176 trough index n, this Table contains information on the manoeuvres (Av in m / s ) and its corresponding time span elapsed since the departure (t in days). The first manoeuvre (index 1) corresponds to the departure from the G T O and the last one (index 3 or 4) to the insertion in the Q P O . It contains also the Moon distance (MD in km) at the swingby. T h e values for O N E SWINGBY correspond to O P T 1 of Table 6.3 and the values for T W O SWINGBYS to the best result in terms of total Av (AvT in m / s ) between OPT12 and O P T 2 .

Summary

of the Results

TWO SWINGBYS

ONE SWINGBY Iday n 18262 1 18262 2 18262 3 18262 4 18262 5 18262 6 18262 7* 18262 8* 18262 9* 18262 10* 18265 11 18265 12 18265 13 18265 14 18265 15 18265 16* 18265 17* 18268 18 18268 19 18268 20 18268 21 18268 22 18268 23* 18268 24* 18271 25 18274 26 18277 27

ti

679.4 .0 679.5 .0 679.4 .0 679.4 .0 679.4 .0 679.1 .0 679.3 .0 679.2 .0 677.4 .0 679.8 .0 677.7 .0 677.7 .0 677.6 .0 677.5 .0 677.3 .0 680.5 .0 681.0 .0 675.8 .0 675.6 .0 675.5 .0 675.5 .0 675.5 .0 681.8 .0 681.8 .0 673.0 .0 670.2 .0 669.1 .0

Au2 ti 361.7 4.2 362.6 4.2 361.1 4.2 360.7 4.2 361.3 4.2 361.3 4.2 369.4 4.2 361.2 4.2 377.0 4.0 368.8 4.2 360.0 4.0 358.3 4.0 357.2 4.0 357.0 4.0 352.0 4.0 356.8 4.3 351.9 4.3 363.6 3.9 361.1 3.9 369.2 3.9 360.7 3.9 360.6 3.9 342.9 4.3 342.9 4.3 379.7 3.8 394.2 3.7 420.7 3.6

Au3

MDi

AUT

Aui

All 2

687.0 .0 687.0 .0 687.0 .0 687.0 .0 687.2 .0 685.8 .0 685.7 .0 685.9 .0 684.2 .0 685.5 .0 686.4 .0 686.4 .0 686.4 .0 686.5 .0 686.2 .0 686.3 .0 686.2 .0 685.1 .0 684.2 .0 685.2 .0 684.2 .0 684.3 .0 686.5 .0 686.3 .0 681.3 .0 680.1 .0 678.0 .0

ti 93.3 4.6 94.1 4.6 84.8 4.6 84.9 4.6 110.5 4.6 93.4 4.6 74.2 4.6 96.4 4.6 86.6 4.5 96.8 4.6 27.6 4.6 29.8 4.6 18.1 4.6 18.5 4.6 18.6 4.6 100.4 4.7 87.2 4.7 2.5 4.5 4.6 4.5 .3 4.5 3.7 4.4 4.1 4.5 88.8 4.7 78.4 4.7 4.7 4.3 12.2 4.2 101.0 4.1

t3 134.1 34.4 133.3 34.5 134.5 34.2 134.9 34.2 134.3 34.3 108.6 43.0 111.7 42.9 109.0 42.9 90.0 44.5 90.9 43.0 131.3 33.8 132.4 33.7 133.2 33.6 133.3 33.5 120.7 33.0 107.2 44.7 108.2 44.6 123.8 34.3 125.2 34.1 125.2 34.0 125.5 34.0 125.7 34.0 97.9 47.8 98.4 47.8 114.4 34.9 114.5 35.7 106.5 34.3

185

42755.3

1175.2

42460.6

1175.4

42746.9

1175.0

42807.1

1175.0

42805.4

1175.0

12393.6

1149.0

12459.3

1160.4

12386.7

1149.4

3910.1

1144.5

3687.8

1139.6

38527.2

1169.0

38302.3

1168.4

38958.0

1167.9

38857.4

1167.9

37040.1

1150.0

13005.5

1144.5

14242.6

1141.0

34683.2

1163.2

34547.5

1161.9

34057.3

1169.8

34567.5

1161.8

34550.7

1161.8

16568.6

1122.6

16681.2

1123.1

31694.9

1167.1

31378.6

1178.9

37538.3

1196.3

A«3

MDi MD2 136.6 42166.2 61.4 25970.5 136.2 42023.6 61.5 25989.0 143.1 41521.6 61.0 24805.0 142.9 41451.7 61.0 24790.9 146.5 41422.9 61.2 23955.7 108.7 12391.0 70.6 24091.8 111.7 12455.7 70.3 25780.1 109.1 12388.7 70.6 23876.3 90.1 3845.8 71.9 25525.4 91.0 3658.7 70.3 25192.7 137.9 38560.3 61.2 16510.8 136.4 38589.8 61.3 16692.9 130.4 39548.5 61.5 17989.0 130.3 39840.7 61.6 17776.0 121.3 37265.4 60.9 15931.7 107.2 13010.5 72.3 25030.4 108.2 14244.2 72.0 26808.3 125.1 33063.3 61.8 21461.0 125.4 32492.3 61.9 16380.3 125.1 33989.6 61.5 21658.9 125.4 32228.1 61.8 16494.4 125.7 32709.0 61.8 16928.0 98.0 16580.5 75.2 25616.9 98.4 16689.0 75.2 25976.2 120.4 29681.3 62.2 15233.7 116.5 35066.8 64.5 11919.3 116.0 35960.3 64.0 16918.5 Av4

AUT

U

56.3 36.8 56.3 36.8 56.8 36.8 56.7 36.8 56.7 36.9 51.7 36.6 51.3 36.5 51.2 36.7 56.3 36.4 50.5 36.8 61.7 36.9 61.6 36.9 61.4 36.8 61.5 36.8 61.5 36.9 48.6 36.9 47.7 36.8 62.8 36.0 63.3 36.3 64.2 36.1 63.2 36.3 63.6 36.3 45.5 36.8 45.8 36.7 71.2 36.2 77.5 36.9 82.0 36.4

973.2 973.5 971.6 971.6 1000.8 939.6 922.8 942.5 917.2 923.8 913.6 914.2 896.3 896.8 887.5 942.5 929.3 875.5 877.6 874.8 876.5 877.6 918.8 908.9 877.6 886.4 977.0

186

Transfer to Orbits in a Vicinity

18280 28 18283 29 18286 30 18289 31 18289 32

676.9 .0 680.5 .0 681.3 .0 680.5 .0 680.5 .0

369.0 4.0 336.5 4.2 330.8 4.3 346.2 4.3 346.4 4.3

122.9 33.7 128.5 34.2 129.2 35.0 130.2 35.5 130.0 35.5

43703.7

1168.9

43723.5

1145.6

42523.2

1141.3

41493.9

1156.9

41497.2

1156.9

of the Lagrangian

699.0 .0 687.9 .0 687.9 .0 687.3 .0 687.3 .0

27.0 5.8 44.2 4.8 54.9 4.8 88.4 4.7 89.9 4.7

88.3 37.3 51.6 37.2 48.8 37.3 50.2 37.2 50.3 37.2

Points

120.8 62.1 129.0 62.3 134.8 63.4 139.1 64.0 137.7 63.9

44802.3 8529.1 45385.3 24590.7 44141.6 26356.9 42590.8 28315.4 42725.8 28266.4

935.1 912.6 926.4 965.0 965.2

Table 6.9 Results for the orbit which substitutes the equilibrium point and was improved by the parallel shooting algorithm for 200 revolutions of the Moon. Associated with Table 6.4 on page 178 through index n. The comments can be seen in the caption of Table 6.8.

ONE SWINGBY Iday n 18262 1 18262 2 18283 3 18286 4 18286 5 18289 6 18289 7 18295 8 18298 9 18307 10 18310 11 18313 12 18319 13 18319 14

Av2 ti

687.2 .0 687.2 .0 687.7 .0 688.2 .0 687.6 .0 687.6 .0 687.6 .0 681.6 .0 679.1 .0 688.4 .0 687.1 .0 687.0 .0 681.9 .0 687.1 .0

At>3

MDi

TWO SWINGBYS Aux

A«i

*3

378.2 4.7 378.1 4.7 348.2 4.7 379.4 4.8 359.8 4.7 360.8 4.7 360.8 4.7 350.1 4.3 373.7 4.1 311.3 4.8 348.2 4.7 355.3 4.7 342.7 4.3 351.5 4.7

68.4 42.1 68.5 42.1 147.6 35.6 74.7 39.2 79.4 38.8 68.7 41.7 68.7 41.7 163.1 32.1 96.6 34.8 225.1 32.3 122.1 35.2 69.1 38.2 159.3 29.4 115.9 44.3

30687.4

1133.7

30701.6

1133.7

30510.1

1183.4

27459.0

1142.3

31347.2

1126.7

30833.6

1117.2

30826.8

1117.2

15697.7

1194.8

22319.3

1149.4

26346.9

1224.8

30413.5

1157.4

30735.4

1111.4

20099.3

1184.0

29016.0

1154.6

682.3 .0 682.4 .0 684.2 .0 683.8 .0 683.8 .0 683.8 .0 683.8 .0 681.1 .0 680.7 .0 685.9 .0 684.9 .0 684.8 .0 681.9 .0 685.0 .0

AU2 *2 208.9 4.4 208.0 4.4 152.8 4.5 168.6 4.5 169.0 4.5 172.0 4.5 171.9 4.5 2.1 4.3 48.5 4.2 102.4 4.6 144.4 4.5 154.7 4.5 61.6 4.3 145.4 4.5

Av3 t3 17.3 35.2 17.4 35.2 20.2 35.4 18.9 35.3 18.8 35.3 17.9 35.2 17.8 35.2 45.9 35.5 48.3 35.6 22.5 35.4 22.6 35.2 21.8 35.2 44.8 35.3 22.3 35.2

ti 74.9 67.0 75.4 67.1 141.5 60.9 75.8 64.0 75.6 64.0 72.3 66.8 72.4 66.8 163.2 58.3 84.7 61.6 223.4 57.7 117.3 60.5 67.6 63.5 151.9 55.2 122.1 69.5

MDi MD2 22885.3 48667.9 22971.0 48739.1 22738.5 43630.4 23599.2 46181.0 23610.1 46440.4 23019.9 47321.8 22952.9 47318.8 14920.9 15366.0 18356.9 16622.9 18725.4 40257.0 22369.7 44918.6 23107.1 47095.5 16516.1 19282.4 21789.4 45999.7

AVT

983.4 983.2 998.7 947.0 947.2 946.0 945.9 892.3 862.2 1034.2 969.1 928.9 940.2 974.8

Table 6.10 Results for the orbit of z-amplitude 0.3. Associated with Table 6.5 on page 179 through index n. The comments can be seen in the caption of Table 6.8.

Summary

187

of the Results

TWO SWINGBYS

ONE SWINGBY Av2

Iday n

ti

ti

A«3 *3

18262 1 18262 2 18262 3 18265 4 18265 5 18265 6 18265 7 18271 8 18274 9 18274 10 18274 11 18277 12 18277 13 18277 14 18280 15 18280 16 18286 17 18289 18 18289 19 18289 20 18289 21 18292 22 18292 23 18292 24 18295 25 18295 26

694.2 .0 693.9 .0 693.9 .0 694.4 .0 694.2 .0 694.4 .0 694.2 .0 682.4 .0 682.5 .0 682.5 .0 682.2 .0 682.2 .0 682.5 .0 682.2 .0 682.5 .0 682.5 .0 694.4 .0 694.7 .0 694.5 .0 694.5 .0 694.7 .0 694.9 .0 694.9 .0 694.9 .0 683.7 .0 683.7 .0

223.9 5.3 233.7 5.3 233.5 5.3 201.7 5.3 208.5 5.3 203.2 5.3 209.5 5.3 205.7 4.4 242.9 4.4 241.8 4.4 253.2 4.4 261.7 4.4 249.7 4.4 262.3 4.4 231.6 4.4 232.9 4.4 203.1 5.3 219.2 5.4 230.3 5.3 230.3 5.3 219.1 5.4 208.8 5.4 209.2 5.4 210.1 5.4 155.7 4.4 156.2 4.4

55.6 40.1 55.1 40.2 55.3 40.2 89.6 43.3 89.8 43.4 88.7 43.3 89.2 43.4 138.1 32.4 69.4 35.5 70.1 35.5 71.8 35.6 49.9 38.9 50.0 38.7 49.5 38.8 90.6 42.0 90.2 42.0 113.6 36.6 57.1 39.9 56.7 40.0 56.7 40.0 57.1 39.9 63.8 43.1 63.5 43.1 63.0 43.1 250.0 29.4 250.0 29.4

MDi

AUT

9327.7

973.8

12994.5

982.7

12945.5

982.7

8684.8

985.7

11955.1

992.5

8722.6

986.3

12025.0

992.8

9127.6

1026.2

7290.3

994.8

7236.9

994.4

10117.5

1007.2

10227.3

993.8

7261.1

982.2

10263.3

994.0

9246.7

1004.6

9677.3

1005.5

12737.2

1011.1

9457.1

970.9

13456.9

981.5

13417.9

981.5

9452.8

970.9

9041.1

967.5

8967.2

967.6

9095.8

967.9

9466.0

1089.4

9689.9

1089.8

Aui 687.4 .0 687.4 .0 687.4 .0 687.9 .0 687.7 .0 687.9 .0 687.6 .0 686.3 .0 681.7 .0 678.9 .0 680.6 .0 679.9 .0 681.1 .0 681.0 .0 679.8 .0 681.2 .0 688.0 .0 687.7 .0 687.7 .0 687.7 .0 687.7 .0 687.9 .0 687.9 .0 687.9 .0 696.3 .0 697.2 .0

Av 2

At> 3

Al>4

ti

t3

U

15.7 35.7 15.7 35.7 15.7 35.7 15.7 35.9 15.7 35.8 15.7 35.8 15.7 35.7 38.5 35.7 41.7 35.4 40.7 35.9 42.3 35.6 41.5 36.0 41.7 35.3 41.1 35.2 39.6 35.0 42.4 35.5 15.0 36.0 14.3 35.9 14.3 35.9 14.3 35.9 14.3 35.9 14.0 35.8 14.0 35.8 14.0 35.8 35.0 37.1 35.4 37.3

55.7 64.7 55.8 64.7 55.9 64.7 91.1 68.1 85.6 67.9 90.6 68.1 84.7 67.9 138.1 58.9 69.5 61.7 63.6 61.9 65.3 61.7 50.4 65.7 49.8 64.8 49.9 64.8 82.0 67.9 86.2 67.7 105.3 61.5 57.0 64.5 57.0 64.5 57.0 64.5 57.0 64.5 64.0 67.7 63.7 67.7 63.5 67.7 250.0 57.3 250.0 57.5

54.3 4.7 54.1 4.7 54.0 4.7 30.0 4.7 40.7 4.7 30.8 4.7 43.2 4.7 .1 4.6 .0 4.3 6.9 4.1 .0 4.2 .6 4.2 .0 4.3 .0 4.3 7.6 4.2 .0 4.3 17.9 4.8 30.2 4.8 30.2 4.8 30.3 4.8 30.2 4.8 18.4 4.8 19.0 4.8 19.2 4.8 9.3 5.5 18.1 5.6

AVT MD2 9066.5 30985.3 9067.5 30992.2 9067.5 30983.0 8538.1 29065.0 8727.0 30106.6 8549.9 29115.7 8783.1 30517.7 7205.5 27825.3 7652.3 20997.4 8069.7 14024.5 8000.5 18981.5 7006.2 10353.8 7640.9 20223.3 7510.6 20140.4 7136.8 17555.5 7893.4 20062.7 9270.7 28574.9 9329.0 29941.6 9321.8 29946.3 9333.0 29956.5 9325.2 29947.9 8990.6 29120.9 8996.4 29184.2 9002.4 29136.6 9051.0 38162.8 9254.6 39144.9

813.1 813.0 813.0 824.8 829.6 825.1 831.3 862.9 792.9 790.1 788.2 772.3 772.7 772.1 809.1 809.8 826.3 789.2 789.2 789.2 789.2 784.3 784.5 784.6 990.7 1000.8

Table 6.11 Results for the orbit of z-amplitude 0.5. Associated with Table 6.6 on page 180 through index n. The comments can be seen in the caption of Table 6.8.

188

Transfer to Orbits in a Vicinity of the Lagrangian

ONE SWINGBY Iday n 18262 1 18262 2 18262 3 18262 4 18265 5 18265 6 18265 7 18265 8 18268 9 18277 10 18277 11 18280 12 18280 13 18280 14 18280 15 18280 16 18283 17 18283 18 18283 19 18283 20 18283 21 18289 22 18292 23 18292 24 18292 25 18295 26 18295 27 18295 28

ti 694.1 .0 694.1 .0 694.1 .0 694.1 .0 694.1 .0 694.1 .0 694.1 .0 694.1 .0 694.2 .0 686.3 .0 686.3 .0 686.3 .0 686.3 .0 686.5 .0 686.3 .0 686.3 .0 686.4 .0 686.2 .0 686.1 .0 686.4 .0 686.2 .0 693.8 .0 693.8 .0 693.8 .0 693.8 .0 693.9 .0 694.0 .0 694.0 .0

Av2 ti 37.8 5.2 36.2 5.2 36.8 5.2 29.6 5.2 54.0 5.2 54.0 5.2 53.5 5.2 54.0 5.2 56.0 5.2 36.7 4.6 33.0 4.6 50.2 4.6 50.2 4.6 30.6 4.6 50.2 4.6 50.2 4.6 28.7 4.6 48.3 4.6 47.3 4.6 29.6 4.6 49.2 4.6 22.9 5.2 40.5 5.2 40.5 5.2 40.9 5.2 42.2 5.3 24.8 5.3 24.8 5.3

AJ> 3

MDi

TWO SWINGBYS AUT

t3 118.3 34.8 118.8 34.8 118.7 34.8 121.6 34.8 66.6 37.9 66.6 37.9 66.9 37.9 66.5 37.9 56.9 41.1 106.2 34.9 107.7 34.9 60.3 38.0 60.3 38.0 60.4 38.0 60.3 38.0 60.3 38.0 45.9 41.0 45.6 41.1 45.8 41.1 45.8 41.0 45.4 41.1 126.1 34.7 69.3 37.8 69.3 37.8 69.1 37.8 59.4 41.0 59.8 40.8 59.8 40.8

Points

7773.9

850.2

7821.2

849.1

7787.2

849.5

7703.2

845.3

7897.8

814.7

7896.6

814.7

7869.6

814.5

7893.4

814.7

7795.2

807.1

6738.2

829.2

6666.3

827.0

6780.3

796.8

6781.2

796.8

4745.1

777.5

6779.6

796.8

6780.7

796.8

4678.4

761.1

6684.7

780.1

6660.1

779.2

4677.9

761.9

6685.8

780.8

7443.5

842.7

7528.9

803.6

7529.5

803.6

7547.3

803.8

7484.0

795.5

5251.6

778.6

5255.7

778.6

A-ui ti 691.7 .0 690.7 .0 694.9 .0 695.4 .0 698.4 .0 693.4 .0 698.4 .0 693.7 .0 698.2 .0 699.1 .0 699.4 .0 699.7 .0 693.9 .0 698.9 .0 699.0 .0 699.0 .0 699.0 .0 699.0 .0 699.0 .0 698.8 .0 699.0 .0 697.6 .0 691.0 .0 697.9 .0 697.9 .0 697.6 .0 697.3 .0 697.3 .0

Av2 *2

5.1 5.0 1.3 4.9 1.5 5.2 .1 5.3 5.6 5.5 8.1 5.1 5.4 5.5 1.0 5.1 8.8 5.5 9.4 5.6 12.2 5.7 38.4 5.8 .5 5.2 .1 5.6 16.3 5.6 16.3 5.6 1.2 5.6 16.5 5.6 16.1 5.6 .4 5.6 17.1 5.6 4.6 5.5 32.4 4.9 14.9 5.5 15.2 5.5 17.6 5.5 1.4 5.5 2.0 5.5

Au 3 t3

At>4

U

.5 101.7 60.4 36.4 1.1 106.0 59.5 36.3 8.8 109.2 59.7 36.4 6.6 118.7 59.5 34.0 66.1 14.4 63.4 35.9 59.4 7.0 63.4 36.7 66.6 14.2 63.4 35.9 70.5 4.7 63.7 34.3 56.6 13.9 66.5 35.8 24.4 107.6 62.5 36.5 22.3 108.6 62.2 36.3 53.0 .7 63.8 36.2 55.9 16.8 64.7 36.8 60.4 21.3 65.0 36.3 59.1 26.5 65.7 37.0 59.2 26.3 65.6 37.0 45.9 21.0 68.0 36.2 45.4 25.7 68.6 36.8 45.7 25.5 68.6 36.7 45.8 21.2 68.0 36.2 45.3 25.6 68.6 36.7 11.2 127.6 60.1 35.4 61.7 .4 63.4 36.8 68.8 13.5 63.2 36.0 68.7 13.5 63.2 36.0 59.1 12.5 66.3 35.7 59.7 9.5 65.4 35.0 59.8 9.5 65.4 35.0

MDi MD2

AUT

14927.7 18480.8 7426.1 31389.5 6125.8 49890.5 4873.4 87734.0 9190.2 80731.0 11081.8 25355.7 9128.0 81640.6 6627.9 45031.3 9057.6 81772.0 7207.4 85791.9 6903.1 98647.0 5598.0 5241.1 5049.2 46888.6 4615.5 93489.7 7863.5 73706.2 7797.5 75506.6 4554.3 97075.9 7449.2 79585.6 7400.1 81003.1 4535.4 93562.8 7431.5 80039.0 8112.2 88786.9 12398.0 19553.7 8630.9 82189.7 8626.2 82397.6 8434.4 84147.3 5066.6 98019.1 5066.1 99309.0

799.0 799.2 814.3 820.9 784.6 767.7 784.6 769.9 777.5 840.5 842.5 791.8 767.0 780.7 800.9 800.8 767.1 786.6 786.2 766.2 787.0 841.0 785.6 795.2 795.3 786.8 767.9 768.5

Table 6.12 Results for the orbit of z-amplitude 0.7. Associated with Table 6.7 on page 182 through index n. The comments can be seen in the caption of Table 6.8.

Summary

of the Results

189

Fig. 6.1 Two local minima for the same insertion epoch. The target orbit is the one with zamplitude 0.7 with insertion epoch 18262.0 MJD (l-JAN-2000). The orbits correspond to number 2 in Table 6.7 on page 182 sections OPT2 (top one) and OPT12 (bottom one). Continuous line: spacecraft. Dashed line: Moon's orbit.

IN EACH LINE: INDEX,P,E,I,0,W,V,ORBIT'S ENERGY, PERIOD (DAYS), PERIAPSIS OF THE ORB EPOCH (WHEN INDEX=0) OR TIME SPAN PREVIOUS TO EPOCH OF INDEX-1 (WHEN INDEXOO). 0 397448.4 .026 28.99 245.01 138.75 289.17 -.37E+10 28 .92378 387212 .86434 1 433557.8 .128 29.30 246.51 105.00 321.60 -.34E+10 33 .74515 384358 .43383 9 .97596 31197..79280 91.24 174.68 360.00 -.76E+10 9.23 2 57419.8 .841 31197,.79280 91.24 174.68 360.00 -.61E+10 13..92979 9.23 3 57412.6 .872 77.29 176.34 360.00 -.74E+10 10..31594 4 13504.3 .966 30.44 6870,.14430 8..61601 77.27 176.31 180.00 -.84E+10 6849..96154 5 13435.4 .961 30.50 8..61608 77.27 176.31 180.00 -.84E+10 6851..89244 6 13439.1 .961 30.50 77.26 176.14 360.00 -.75E+10 10..24930 7 13132.4 .966 30.55 6678.,14286 .44008 77.26 176.14 360.00 -.61E+11 6678 .14286 8 11530.1 .727 30.55 12-•NOV-1999 7h:39m: 3s. (MJD 18212.3259212) DEPARTURE: Oh: Om: Os. (MJD 18274.0000000) INSERTION QP : 13-•JAN-2000 TOTAL TRANSFER TIME (DAYS): 61 .6740788 TOTAL DELTA-V (M/'S): 792 .897

**************************************** IN EACH LINE: I N D E X , P , E , I , 0 , W , V , O R B I T ' S ENERGY, PERIOD (DAYS), PERIAPSIS OF THE ORB EPOCH (WHEN INDEX=0) OR TIME SPAN PREVIOUS TO EPOCH OF INDEX-1 (WHEN INDEXOO).

5.29 126.44 355.20 0 389493.0 .076 5.34 126.25 154.22 1 316810.7 .128 2 91930.2 .707 3.88 125.58 179.81 3 91357.9 .705 3.85 125.46 180.07 4 13127.1 .958 30.50 125.52 180.01 5 13101.9 .962 30.31 125.58 179.88 6 11530.1 .727 30.31 125.58 179.88 DEPARTURE: 12-•DEC-1999 16h:45m:30s. INSERTION QP : 16-•JAN-2000 Oh: Om: Os. TOTAL TRANSFER TIME (DAYS): 34 .2935102 TOTAL DELTA-V (M/S): 1196.285

345.29 -.38E+10 28 .27238 186.46 -.46E+10 21 .07802 9 .09981 .00 -.81E+10 180.00 -.82E+10 8 .91262 180.00 -.94E+10 7 .24234 .00 -.85E+10 8 .46394 .00 -.61E+11 .44008 (MJD 18242.7064898) (MJD 18277.0000000)

362115 .28621 280875 .52761 53844..27191 53593..90138 6705,.84429 6678,.14321 6678,.14321

-

Table 6.13 Information output produced by the programs in terms of the orbital elements. Th number 9 (two swingbys) and to Table 6.9 on page 186 number 27 (one swingby). (Distances the ecliptic one.)

Summary

191

of the Results

** BASIC INFORMATION OF TRANSFER TO THE EARTH ** THE LINES CONTAIN: TOTAL NUMBER OF MANOEUVRES (SWING-BYES) INDEX, EPOCH (MJD), ALF, BET (DEG), DV (M/S) ECLIPTIC BARYCENTRIC POSITION (KM) AT THE ABOVE EPOCH (JUST BEFORE THE MANOEUVRE) IDEM AS 3: BUT VELOCITY (KM/DAY) SUCCESSIVE LINES REPEAT FROM 2: TO 4:

12.099612 18274.0000000000 47.417417 •.5452771979918038E+08 .1362986262787502E+09 •.2378542596046598E+07 .8878209421638642E+06 .002628 .005330 18274.7052542506 .1461411458645808E+09 .1273591425123313E+08 .2006191645677591E+06 •.2191212886047768E+07

69.513945 .1929735673214534E+06 . 1508664577159104E+05 41.699330 .1497123583544888E+05 •.6680634068624242E+05

***********************************

** BASIC INFORMATION OF TRANSFER TO THE EARTH ** THE LINES CONTAIN: TOTAL NUMBER OF MANOEUVRES (SWING-BYES) INDEX, EPOCH (MJD), ALF, BET (DEG), DV (M/S) ECLIPTIC BARYCENTRIC POSITION (KM) AT THE ABOVE EPOCH (JUST BEFORE THE MANOEUVRE) IDEM AS 3: BUT VELOCITY (KM/DAY) SUCCESSIVE LINES REPEAT FROM 2: TO 4:

1

18274.0000000000 .734195 -.485037 -106.452406 -.6209334713401063E+08 .1338734486413875E+09 .5246832320647672E+05 -.2472070339441753E+07 -.1105199586248708E+07 .8302385005186017E+04

Table 6.14 Output of the programs in what we call basicfilewhich can be used by any program. These examples correspond to Table 6.11 on 187 number 9 (two swingbys) and to Table 6.9 on page 186 number 27 (one swingby) and are related with Table 6.13.

192

Transfer to Orbits in a Vicinity of the Lagrangian

6-N0V-1999

4h (MJD 18206.21)

10-N0V-1999 15h (MJD 18210.66)

PERI DEPART APO MANOVR

15-N0V-1999 2h (MJD 18215.12) 19-N0V-1999 lh (MJD 18219.08) 21-N0V-1999 15h (MJD 18221.63)

PERI SBY-1 APO

28-N0V-1999 17h (MJD 18228.71) 5-DEC-1999 3h (MJD 18235.15) 12-DEC-1999 llh (MJD 18242.48)

PERI APO PERI MANOVR

17-DEC-1999 llh (MJD 18242.48) 19-DEC-1999 22h (MJD 18249.93)

SBY-2 APO

27-DEC-1999 21h (MJD 18257.89) 6-JAN-2000 16h (MJD 18267.70)

PERI APO

7-JAN-2000

Oh (MJD 18268.00)

TOTAL DELTA-V: TRANSFER TIME:

INS QP

ED= DV= CI= ED= MD= DV= CI= ED= MD= ED= MD= ED= ED= ED= DV= CI= MD= MD= ED= MD= ED= ED= MD= ED= DV= CT=

Points

6678.KM 684.27 M/S .00 DEG 355874.KM 608006.KM 4.14 M/S .15 DEG 7557.KM 16929.KM 431743.KM 195385.KM 42217.KM 438444.KM 48236.KM 63.57 M/S .00 DEG 805906.KM 32674.KM 382848.KM 99192.KM 244751.KM 397720.KM 394800.KM 397853.KM 125.66 M/S 1.81 DEG

( 1.06RE)

(56.49RE)

( 1.20RE) (68.53RE) ( 6.70RE) (69.59RE) ( 7.66RE)

(60.77RE) (38.85RE) (63.13RE) (63.15RE)

8 7 7 . 6 5 M/S 6 1 . 7 9 DAYS

Table 6.15 Example of a trajectory description from the GTO to the Q P O . The transfer orbit has two swingbys with the Moon. ED stands for Earth Distance and MD for Moon distance. It corresponds to Table 6.9 on page 186 number 22.

Summary

23-0CT-1999 19h (MJD 18192.81)

PERI DEPART

28-0CT-1999 llh (MJD 18197.49)

APO MANOVR

2-N0V-1999 6h (MJD 18202.27) 6-N0V-1999 16h (MJD 18206.70) 9-N0V-1999 6h (MJD 18209.25)

PERI SBY-1

15-N0V-1999 19h (MJD 18215.83) 22-N0V-1999 8h (MJD 18222.36)

PERI

29-N0V-1999 14h (MJD 18229.60)

PERI MANOVR

4-DEC-1999 16h (MJD 18234.70) 9-DEC-1999 Oh (MJD 18239.00) 19-DEC-1999 22-DEC-1999 22-DEC-1999

2h (MJD 18249.12) 3h (MJD 18252.14) 7h (MJD 18252.32)

26-DEC-1999 15h (MJD 18256.65) 6-JAN-2000 4h (MJD 18267.20) 7-JAN-2000

Oh (MJD 18268.00)

TOTAL DELTA-V: TRANSFER TIME:

193

of the Results

APO

APO

SBY-2

APO PERI SBY-3

APO PERI

APO INS QP

ED= DV= CI= ED= MD= DV= CI= ED= MD= ED= MD= ED= ED= MD= ED= DV= CI= MD= ED= MD= ED= MD= ED= MD= ED= ED= MD= ED= DV= CI=

6678.KM 686.47 M/S .00 DEG 366842.KM 645373.KM 88.79 M/S 15.88 DEG 12863.KM 25622.KM 427352.KM 160938.KM 45336.KM 433661.KM 789300.KM 49820.KM 45.52 M/S .00 DEG 33662.KM 444574.KM 140179.KM 296210.KM 16567.KM 374197.KM 18093.KM 267460.KM 398045.KM 397460.KM 397853.KM 97.98 M/S .86 DEG

( 1.06RE)

(58.23RE)

( 2.04RE) (67.83RE) ( 7.20RE) (68.84RE) ( 7.91RE)

(70.57RE) (47.02RE) (59.40RE) (42.45RE) (63.18RE) (63.15RE)

9 1 8 . 7 6 M/S 7 5 . 1 9 DAYS

Table 6.16 Example of a trajectory description from the GTO to the Q P O . The transfer orbit has two plus an "extra" one swingbys with the Moon. ED stands for Earth Distance and MD for Moon distance. It corresponds to Table 6.9 on page 186 number 23.

194

Transfer to Orbits in a Vicinity of the Lagrangian

6.2.2

On the Shapes of the Transfer

6.2.2.1

Looking at the Orbits in the Sidereal Frame

Points

Orbits

Using a reference frame centered on the Earth, the transfer orbits are seen basically as an elliptical orbit with eccentricity between 0.7 and 0.9 that due to encounters with the Moon becomes more circular until it reaches essentially the Moon's orbit shape, eventually with a different inclination (see Table 6.13 on page 190 for an example of the evolution of the orbital elements). As we mentioned previously when explaining the methodology of the computations, the transfer possibilities are very different according to the target orbit. Targeting to orbits of section 5.5.2, that is, to orbits which essentially substitute the equilibrium point and so they lie close to the Moon's orbital plane, one can say that the transfer orbit "rotates" according to the Moon, in the sense that the longitude of the periapsis of the GTO varies depending on the insertion epoch (see Figure 6.2, ending on page 198). The only exception happens with the orbits mentioned at the end of the previous subsection and which can be seen in Figure 6.3. When targeting to orbits of section 1.4, this is to orbits lying far from the Moon's orbital plane, the swingby must happen in the intersection line of both planes. So, we have essentially only two possibilities for the value of the longitude of the periapsis, when the target orbit is selected (see Figures 6.4, 6.5 and 6.6). Using the figures and its related Tables (see the captions in the previous figures) when enough data is available (for instance using the Tables associated with the target orbits with z-amplitude 0.5 and 0.7), it seems that one could say that the optimal transfers have a "periodicity" in the insertion epoch. According to this, during a time span it is better to perform the swingby in one side of the intersection line. Then we have another time span for which the transfer possibilities, at least the ones that need a low value of total Av, seem to disappear. It follows another time span where again appear good possibilities, but in the other side of the intersection line. Later they seem to disappear again, and so on. When more swingbys are included the longitude of the periapsis of the GTO is not strongly changed for the orbits lying near the Moon's orbital plane nor for the ones lying far from it, as one can see in Figures 6.7 to 6.16 which are shown from page 203 on. 6.2.2.2

Looking at the Orbits in the Synodic Frame

The behavior of the transfer trajectories in the synodic frame is very different from the behavior in the previous reference frame, and in some way seems much more clear. The rotation of the longitude of the periapsis for the orbits lying near the orbital plane, and the two possibilities for the orbits which lie far from the orbital plane are absorbed by the change of coordinates. One could say that we have only one type of transfer orbit, even when comparing transfer trajectories to different target orbits.

Summary

of the Results

195

This becomes clear seeing the figures which contain only one swingby (Figures 6.2, 6.4, 6.5 and 6.6). One could say that the most important difference comes from the fact that the selected orbits which lie close to the orbital plane of the Moon are near L4 and the ones with big ^-amplitude are near L 5 . Followed from the insertion point backwards in time, all the transfer orbits seem to be guided by the unstable manifold of some object in the central part of Li and then ejected towards the Earth, following the stable manifold of the same object. When we add more swingbys in spite of the fact that the figures seem to load themselves with other (complicated) loops, a little further inspection reveals that the behavior is always the same and the objective is to repeat the former procedure again and again. Even the orbits with an "extra" swingby could be explained in the same way. In that case the transfer orbits, when followed backwards in time, initially follow the unstable manifold of some object in the central part of Li they escape again by the stable manifold, but now towards the Moon, and return near to Li following again the same initial procedure (see Figures 6.3, 6.9 and 6.10). However in this case the invariant manifolds associated with the objects in the central part of L2 could also play an important role. Taking all this facts into consideration, seems clear that the transfer trajectories to orbits near Lagrangian points using gravity assists from the Moon, and so being much less expensive, are in fact the natural way of reaching the selected target orbits. Further computations should be done using the restricted three-body problem as starting point, in order to study the intersections of the previously mentioned manifolds, to obtain in this way the transfer orbits in a more systematic way and just using the optimization procedure to modify slightly the previous orbits when the full solar system model is used.

196

Transfer to Orbits in a Vicinity of the Lagrangian

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197

198

Transfer to Orbits in a Vicinity of the Lagrangian

["

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Fig. 6.2 Sidereal and synodic representation of the transfer orbit with target orbit the one improved for 200 revolutions of the Moon. The dashed lines in the sidereal representation correspond to the Moon's orbit. The insertion epochs are selected from 18262 MJD (l.O-JAN-2000) to 18286 MJD in steps of 3 days. The represented orbits come from Table 6.8 on page 184 section O P T 1 . Their respective indices are 1, 15, 19, 25, 26, 27, 28, 29 and 30. The values of the manoeuvres and Moon distances can be seen in Table 6.9.

Summary

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199

of the Results

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Fig. 6.3 Sidereal and synodic representation of the transfer orbit with an "extra swingby" (first two rows) compared with the ordinary one (last row). All the orbits correspond to Table 6.3 on page 176 section O P T 1 numbers 24, 27 and 1. The insertion date for all of them is 18262 MJD, (1.0-JAN-2000). Seen in the synodic coordinates, orbits 24, 25 and 26 of Table 6.3, and orbits 7, 8, 16, 21 and 22 of Table 6.4 are as the first one. Orbits 27 and 28 of Table 6.3 and orbits 9 and 10 of Table 6.4 are as the second one. All the remaining orbits of these Tables are as the last one. Dashed lines in the sidereal representation correspond to the Moon's orbit.

200

Transfer to Orbits in a Vicinity of the Lagrangian

i

1

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!

1

1

1

r-i

i—i

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r

r

Fig. 6.4 Sidereal and synodic representation of the transfer orbit with target orbit the one of z-amplitude 0.3. The two possibilities on the longitude of the periapsis are shown. No substantial difference can be seen between the 3 rows in the synodic representation apart from the twist in the GTO orbit. Rows 1 and 2 show the difference in the sidereal representation for different insertion epochs. The represented orbits correspond to Table 6.10 on page 186 orbits 3, 6 and 9. In the same Table orbits from 1 to 7, and 10, 11 and 12 have the longitude of the periapsis equal to the orbits of the first rows, while orbits 8, 9 and 13 have it equal to the orbit of the last row. Dashed lines in the sidereal representation correspond to the Moon's orbit.

Summary

1

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201

of the Results

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'

r

,

r

Fig. 6.5 Sidereal and synodic representation of the transfer orbit with target orbit the one of zamplitude 0.5. The same comments as in Figure 6.4 apply here. The represented orbits correspond to Table 6.11 on page 187 orbits 1, 6 and 12. In the same Table orbits from 1 to 7 and from 17 to 24 have the longitude of the periapsis equal to the orbits of the first rows, while orbits from 8 to 16 and 25, 26 have it equal to the orbit of the last row.

202

Transfer to Orbits in a Vicinity of the Lagrangian

-i

1

l « m

-T

1

1

Points

1—-r

JQtttt JWXWJ -"MOW

1—~r

i

n

_,

r

Fig. 6.6 Sidereal and synodic representation of the transfer orbit with target orbit the one of zamplitude 0.7. The same comments as in Figure 6.4 apply here. The represented orbits correspond to Table 6.12 on page 188 orbits 1, 6 and 12. In the same Table orbits from 1 to 9 and from 22 to 28 have the longitude of the periapsis equal to the orbits of the first rows, while orbits from 10 to 21 have it equal to the orbit of the last row.

Summary

T

_J

;

1

1

t__j

1

•40KBC -3UO00 -?HOOO lOOOM

1

1

I

I

0

«HO»

of the Results

203

1

1

r—I

(—i

—r

1

1

r

i

_i

i_J

l,„, t

J

i

i

i

£OMOft 30OOOO
-|

4«

9

9*

\

Fig. 6.7 Sidereal and synodic representation of a transfer orbit with 2 swingbys targeting to the nominal orbit which substitutes the equilibrium point and was improved using parallel shooting for 146 revolutions of the Moon. Projections (x,y) and (x,z) are represented. The orbit corresponds to Table 6.8 on page 184 number 1. Dashed lines in the sidereal representation correspond to the Moon's orbit.

204

Transfer to Orbits in a Vicinity of the Lagrangian

,

.„-,....

J

T

3M0M

s~

/ - / 700009

\

Jtmoo

\ \

^

-WQ00O

A -... [

.
~i

*e»oo

i"

*" -

-

•/

\

MM0«

J

T

{ S-—«^-~^

0

1*1000

r\ £^ 1 n- -r

*0Q#u>

; ^_.-' J '

J , ., mm_i loooao

_„i

B

„„i

...J_

iogco6 Tcgooo J O * »

— i — i ——,.,—

r

,.,,.

,„„ •««»

i—

-

VKiaaa

-

SWQCO

1«0CP

Points

-

-

0

-

IOOWW

awofc

-

-

-

3WW» -^em

-

J—i

i ,., i .

~0OOM> Doddoa -fooem -irtooa

,

a

...J

...I. - i -

ioooo» IttaOQ

KUOOO

' «ow»

Fig. 6.8 Sidereal and synodic representation of a transfer orbit with 2 swingbys targeting to the nominal orbit which substitutes the equilibrium point and was improved using parallel shooting for 200 revolutions of the Moon. Projections (x,y) and (x, z) are represented. The orbit corresponds to Table 6.9 on page 186 number 19. Dashed lines in the sidereal representation correspond to the Moon's orbit.

Summary

of the Results

205

jfloooo 44QO0Q i<xaat> •icoooq

~1

L

1

i"''

1

r-

!.

««K»C

..... , -

JOQOD?

-

?00000

-

ICOOOC

" = = ^ = = 5 ,

0 '

•OOOOO

a*-—^^,^

' ' • *

^

20DOOO

-

306060

<<e«rao ogwco

;ooooa

IOOWO

3

mMoa

^aaooa

raooo

•«wao

Fig. 6.9 Sidereal and synodic representation of a transfer orbit with 2 plus an "extra" one swingbys targeting to the nominal orbit which substitutes the equilibrium point and was improved using parallel shooting for 200 revolutions of the Moon. Projections (x, y) and (x, z) are represented. The orbit corresponds to Table 6.9 on page 186 number 9. Dashed lines in the sidereal representation correspond to the Moon's orbit.

206

4430PM

TYansfer to Orbits in a Vicinity of the Lagrangian

71

>

-

1

T

T• -

1

o

..m>

-

/

-

/ ^ -

:

v

xi \ ;, -

/

%^\*yi Si

.300003 ™

" -

\ \ \ \

/

... [•••.....L^^^Q.X...

-26000b

•400006

1

\X

/

*•

•Dana

I

*^

HXKWO

mow

7

}

Is _^-/s

\v^^ '--i-.

" -

—

*

«oooofl

i > ^ — < .1&W0O iooeae; -ictwoo

t

t

i

o

"

i i»0Wo

T "

I

< Moaa)

1

i SOOox

T

r tooooa

"•

-

JOOOQO

-

300000

,

'•00000

i»—«=|—....

-taoooe

MOOOO

,

, <

0

...

-

-

-

IPOOM

-*CODDI>

Points

t

<

-«H»oo j«xao ;ooooe -lOOooo

i »

i

—

tooott ?3aa» 3ac

Fig. 6.10 Sidereal and synodic representation of a transfer orbit with 2 plus an "extra" one swingbys targeting to the nominal orbit which substitutes the equilibrium point and was improved using parallel shooting for 200 revolutions of the Moon. Projections {x,y) and (x,z) are represented. The orbit corresponds t o Table 6.9 on page 186 number 23. Dashed lines in the sidereal representation correspond to the Moon's orbit.

Summary

/ /.

L-t

-l

-Mown -3M<M0 Jooooa

loaooo

IWUU

207

of the Results

*-*

3OOO0Q 4Q3OQ0 *0WO0

.

;

• — "

•

T

•

1

1

••—

—r,,",m

-

/<

1^.../.>^W

-

^ J

^

•

^

"

-

^

^—rrfr^T.

„

..,

^

1

1

1

<

[

Fig. 6.11 Sidereal and synodic representation of a transfer orbit with 2 swingbys targeting to the orbit of ^-amplitude 0.3. Projections (x,y) and (x, z) are represented. The orbit corresponds to Table 6.10 on page 186 number 6. Dashed lines in the sidereal representation correspond to the Moon's orbit.

208

Transfer to Orbits in a Vicinity of the Lagrangian

1

1

r

J

1

1

i

i

n

1

1

i

1

_J

i

i

i

-MXKXXI -3OM0O -JtOOOti loQPOb

i, 3

Points

L.

r

i

u.—i-—i

1P00OQ fOWDO J M l A M

•OOOXi

Fig. 6.12 Same as Figure 6.11 but with the longitude of the periapsis in the other possibility. The orbit corresponds to Table 6.10 on page 186 number 9.

Summary

J

-*6&Q

1

1_

;0MQO -IMar

.>qsooo

'

ft

'

imM

'

&1W

1

of the Results

209

L

ISOQCO *WWO

Fig. 6.13 Sidereal and synodic representation of a transfer orbit with 2 swingbys targeting to the orbit of z-amplitude 0.5. Projections {x,y) and (x,z) are represented. The orbit corresponds to Table 6.11 on page 187 number 12. Dashed lines in the sidereal representation correspond to the Moon's orbit.

210

Transfer to Orbits in a Vicinity of the Lagrangian

-MXXKM .3O0MO .7UDOO . l o g m

«

<MM

Points

HJOOCD 3QOM0 «TOW

Fig. 6.14 Same as Figure 6.13 but with the longitude of the periapsis in the other possibility. The orbit corresponds to Table 6.11 on page 187 number 1.

Summary

T

i

!

1

"" i

'" i

1

1

of the Results

211

r

Fig. 6.15 Sidereal and synodic representation of a transfer orbit with 2 swingbys targeting to the orbit of z-amplitude 0.7. Projections (x,y) and (x,z) are represented. The orbit corresponds to Table 6.12 on page 188 number 9. Dashed lines in the sidereal representation correspond to the Moon's orbit.

212

Transfer to Orbits in a Vicinity of the Lagrangian

i

1

1

i

|

1

i

i'

!"-]

i—i

1

Points

1

i

r

Fig. 6.16 Same as Figure 6.15 but with the longitude of the periapsis in the other possibility. The orbit corresponds to Table 6.12 on page 188 number 17.

Appendix A

Global Stability Zones Around the Triangular Libration Points in the Elliptic RTBP

In the elliptic problem the libration point is still present and it is linearly stable for H = py[00n, e = ey[oon. We have performed simulations similar to the ones we did for the circular problem (see Chapter 1). Now the system is not autonomous. So we have to take into account the initial phase of the Moon. We have selected for this phase the values j / 8 , j — 0 , . . . , 7. The results depend strongly on the value of that phase. In Figures A.l to A.9 we display the "stable" region (starting always with zero synodic velocity) for values of z equal to 0, 0.10,..., 0.80. For comparison purposes we have also included the corresponding data for the circular problem. The horizontal variable is the angle a and the vertical one is p. For a we have taken in all the figures the range from a = 0.4305 (to the left) till a = 0.2005 (to the right). The window for p depends on z according to Table A.l. z 0.00 0.10 0.20 0.30 0.40

p (-0.040, (-0.040, (-0.050, (-0.060, (-0.080,

0.040) 0.040) 0.030) 0.020) 0.000)

z 0.40 0.50 0.60 0.70 0.80

p (-0.080,0.000) (-0.100, -0.020) (-0.120, -0.040) (-0.148, -0.068) (-0.180, -0.100)

Table A.l: Windows in p for different values of z in Figures A.l to A.9 In general, there is a reduction on the size of the stable zone with respect to the circular case. This is specially true for small values of z. Furthermore (see Figure A.2, corresponding to z = 0.10) one can see the effect of the resonances, which seem to split the stable region in several components. Table A.2 shows statistics of the points that subsist up to the final time tf = 1000 x 27r. As it is seen in Chapter 1, the effect of the Sun is more dramatic than the one of the eccentricity. So, the elliptic RTBP does not seem to be a model very close to the real one for this kind of study.

213

214

Global Stability Zones Around the Triangular Libration Points in the Elliptic

z=0.0 851 838 263 243 363 363 -18 -20 11 9 global limits input limits z= 0.05 860 850 256 251 360 364 -18 -21 10 12 global limits input limits

z=0.1 850 248 369 -20 8

841 258 369 -22 8

global limits input limits z= 0.15 1187 1158 275 271 367 369 -21 -24 5 5 global limits input limits z=0.2 1038 1095 252 246 377 379 -28 -20 -1 0 global limits input limits

total 1188

237 363 -10 17

total 1217

237 366 -17 11

total 1312

242 368 -23 12

total 1357

265 370 -25 6

total 1205

240 391 -23 4

9099 1389 1277

237 362 -13 14 235 230

235 366 -27 17 366 400

9365 1427 1322

235 368 -13 13 235 230

1199

1235

1122

245 363 -21 18 -27

275 359 -17 10

-180

244 363 -19 15 18 20

1268

1270

1151

245 373 -24 17 -26

269 368 -17 12

239 365 -26 16 373 400

-180

260 369 -22 13 17 20

10009 1557 1493

1336

1420

1200

240 369 -25 15 -28

256 370 -18 10

234 370 -26 12 233 230

233 368 -28 15 374 400

-180

258 374 -22 13 15 20

11533 1602 1713

1643

1533

1340

238 370 -28 11 -32

245 367 -21 7

250 371 -32 8 238 230

251 371 -30 16 371 400

-180

248 368 -25 9 16 20

9387 1277 1276

1254

1238

1004

230 363 -21 4 -29

258 363 -29 0 5 20

262 374 -26 -2

232 365 -21 5 230 230

238 364 -22 5 391 400

-180

RTBP

Appendix A

z= 0.25 1994 1911 262 263 376 376 -30 -33 -1 -3 global limits input limits z= 0.3 2262 2241 237 234 397 383 -43 -40

-8

-7

global limits input limits z= 0.35 2097 1955 250 242 378 382 -48 -44 -10 -12 global limits input limits z=0.4 2658 2562 259 248 382 390 -61 -56 -22 -10 global limits input limits z= 0.45 2537 2368 250 244 382 385 -67 -67 -29 -19 global limits input limits

total 1895

255 379 -28 -1

total 2345

238 381 -45 -5

total 2104

243 397 -53 -11

total 2610

251 391 -61 -15

total 2269

258 386 -63 -25

16501 2026 2104

230 374 -30 2 230 230

245 373 -31 1 395 400

19753 2527 2701

215

2246

2183

2142

231 395 -38 0 -38

234 375 -34 -3

-180

247 377 -33 0 2 20

2664

2605

2408

237 377 -52 -4 -52

237 377 -40 7

230 380 -47 -2 400 400

-180

244 377 -41 -7 7 20

17908 2196 2342

2487

2429

2298

230 380 -53 -10 -53

243 382 -47 -14

234 400 -44 -4 230 230

234 383 -53 4 397 400

-180

235 379 -48 -12 4 20

22630 2710 2972

3140

3114

2864

231 380 -66 -20 -66

255 389 -63 -21

233 384 -50 -10 230 230

232 382 -64 -18 397 400

-180

232 397 -56 -21 -10 20

20544 2474 2479

2805

2932

2680

235 380 -83 -28 -83

239 380 -66 -23 -19 20

240 380 -68 -29

250 394 -62 -17 231 230

232 385 -66 -27 232 230

241 383 -71 -28 386 400

-180

216

Global Stability Zones Around the Triangular Libration Points in the Elliptic

z=0.5 2673 2495 240 253 380 383 -82 -80 -37 -37 global limits input limits z= 0.55 2178 2084 266 255 377 378 -92 -105 -41 -49 global limits input limits z= 0.6 1315 1424 241 285 371 372 -97 -96 -67 -61 global limits input limits z= 0.65 948 1020 274 293 364 366 -109 -119 -75 -76 global limits input limits z= 0.7 577 642 270 287 378 387 -126 -130 -94 -94 global limits input limits

total 2481

263 400 -86 -37

total 2201

283 395 -92 -49

total 1451

22374 2699 2846

231 386 -87 -36 231 230

248 380 -85 -38 400 400

18870 2307 2363

3066

3151

2963

231 378 -84 -40 -89

243 385 -84 -30

-180

240 383 -89 -29 -29 20

2526

2650

2561

260 376 -95 -53

242 376

262 399 -95 -50 399 400

-105 -180

250 375 -92 -54 -41 20

11979 1417 1457

282 376 -95 -48 242 230

-101

-45

1634

1718

1563

283 398

265 381

265 381

272 399

277 374

-100

-103

-103

-102

-62

-65 241 230

-44 399 400

-63 -103 -180

276 378 -99 -53 -44 20

8658 1154 1132

total 1138

-103

-68

1131

1128

1007

265 397

245 365

247 384

230 363

271 364

271 381

-121

-123

-129

-138

-126

-118

-81

-81 230 230

-88 397 400

-84

-87

-138 -180

-69 -69 20

total

6018

822 278 386

927 236 380

846 252 396

703 231 396

763 242 374

738 235 397

-134

-146

-142

-142

-136

-132

-95

-97 231 230

-77 -87 397 -146 400 -180

-89 -77 20

-92

RTBP

Appendix

z= 0.75 892 944 284 287 373 394 -146 -144 -108 -98 global limits input limits z= 0.8 438 459 295 303 365 365 -157 -158 -135 -137 global limits input limits z= 0.85 284 264 306 317 356 370 -166 -167 -153 -154 global limits input limits

total

927 282 372 -149 -115

total

A

217

7558

967 1019 1031 288 281 261 366 362 378

927 261 362

851 293 368

-156 -122 -161 -180

-151 -103

-149 -120

-160 -118

-161 -116

261 230

394 400

-98 20

4237

557 291 364

603 295 387

612 282 393

602 255 372

494 237 358

472 293 355

-161 -132

-167 -111

-169 -115

393 400

-163 -136 -111

-157 -137

237 230

-167 -144 -169 -180

total

20

1770

248 322 359

206 322 358

171 318 345

163 313 342

195 284 341

239 310 350

-171 -154

-174 -128

-175 -160

370 400

-171 -156 -128

-168 -154

284 230

-174 -159 -175 -180

20

Table A.2: Statistics of subsisting points in the elliptic problem after a time of integration equal to 1000 times the period of the Moon. For each value of z, as shown, we display 8 columns. For the j t h column the value of the true anomaly of the Moon at t = 0 is equal to (j-l)/8 revolutions. Initial points are taken for p and a with step equal to 0.001. The ranges of variation of a and p are given as input limits. For each value of z the second row displays how many points subsist. The third and fourth ones, the minimum and maximum observed values of a in the points that subsists. The fifth and sixth rows give the observed extrema for p. Finally, next to z, we give the total number of points that subsist adding the corresponding to all the initial phases and, at the bottom, the global extrema of a and p for the same phases.

218

Global Stability Zones Around the Triangular Libration Points in the Elliptic

RTBP

Fig. A.l Representation of the stable region for z — 0.0. The horizontal variable is the angle a and ranges in (0.4305, 0.2005). The vertical variable is p and ranges in (-0.040,0.040).

Appendix

m±--- •

"^•^•n

A

219

\

-#jimm&^

<-1.vj5gMfi^)^j' "

Fig. A.2 Representation of the stable region for z = 0.1. The horizontal variable is the angle a and ranges in (0.4305, 0.2005). The vertical variable is p and ranges in (-0.040,0.040).

Global Stability Zones Around the Triangular Libration Points in the Elliptic

'^^^^m*-

RTBP

i

"n

.^

EST

i-

Fig. A.3 Representation of the stable region for z = 0.2. The horizontal variable is the angle a and ranges in (0.4305, 0.2005). The vertical variable is p and ranges in (-0.050,-0.030).

Appendix

^^m^^^BP""11^^

I0HP ipfiiWr. 'fnti*-1*'

A

221

•n^^^^^^^^^^^^mm^:,

^^^^^BHUPPW^*'*

Fig. A.4 Representation of the stable region for z — 0.3. The horizontal variable is the angle a and ranges in (0.4305, 0.2005). The vertical variable is p and ranges in (-0.060,-0.020).

Global Stability Zones Around the Triangular Libration Points in the Elliptic

RTBP

Fig. A.5 Representation of the stable region for z = 0.4. The horizontal variable is the angle a and ranges in (0.4305, 0.2005). The vertical variable is p and ranges in (-0.080,0.000).

Appendix

A

223

•j

5^

...l^P^JH^w*:.-:..:

^^H^^^::

H p p n

Fig. A.6 Representation of the stable region for z = 0.5. The horizontal variable is the angle a and ranges in (0.4305, 0.2005). The vertical variable is p and ranges in (-0.100,-0.020).

224

Global Stability Zones Around the Triangular Libration Points in the Elliptic

RTBP

Ifeizziz zzisfc !••'•

i

1-

£flfct

: • • • ; —

••

i^pito^

^K

Fig. A.7 Representation of the stable region for z — 0.6. The horizontal variable is the angle a and ranges in (0.4305, 0.2005). The vertical variable is p and ranges in (-0.120,-0.040).

Appendix

K- •

*

A

225

*E

5+

*-Sggpa^

^iijb.._,

r^.

• *?;>-

4*

*&•,

!

*

a *r *

'

&,-..

»

-

•

•

.

NB £E

Fig. A.8 Representation of the stable region for z = 0.7. The horizontal variable is the angle a and ranges in (0.4305, 0.2005). The vertical variable is p and ranges in (-0.148,-0.068).

226

Global Stability Zones Around the Triangular Libration Points in the Elliptic

- ;"""

RTBP

™*13Bi^5>r-

pr-:1!

*..I-V^

^IF'

g^^*

^IP**

I

LI

! :%flk,

Fig. A.9 Representation of the stable region for z = 0.8. The horizontal variable is the angle a and ranges in (0.4305, 0.2005). The vertical variable is p and ranges in (-0.180,-0.100).

Appendix B

Fourier Analysis

B.l

Introduction

This Appendix is devoted to Fourier Analysis, despite this is a well-known topic, because of the specific requirements we have. Indeed Fourier Analysis is used, in most of the applications to the analysis of time series, to extract the relevant signal hidden in a relatively big amount of noise. Now, the level of noise will be very small but, on the other hand, we want to know the dominant frequencies up to a high level of accuracy and to determine with small error the related coefficients. Two applications have been mainly done in this work. The first one is to obtain the dominant terms of the Hamiltonian around the geometrical triangular libration points of the Earth-Moon system when the full solar system is taken into account (Chapter 4). It is well-known that the real solar system is not quasi-periodic. However, for moderate time intervals we can assume that it is quasi-periodic with small error. The second one is to recognize if a solution of the equations of motion (either in the case of an autonomous Hamiltonian or a nonautonomous one depending on time in a periodic or quasi-periodic way) obtained by numerical integration, seems to be on a torus (Chapters 1 and 3). If this is true, the Fourier Analysis should display frequencies having as basic set the basic set of the exciting terms appearing in the Hamiltonian plus a set whose cardinality is the dimension of the torus. In both cases it is clearly seen the need of a very accurate determination of the frequencies. Indeed, the model has been used to derive, by symbolic manipulation, approximate quasi-periodic solutions. The main difficulties are due to resonances, and to detect them properly we must know quite well the frequencies. Also to obtain the basic set, within a reasonable tolerance, the frequencies of the numerical solution should be well-known.

227

228

Fourier

B.2

Analysis

The Method

The proposed method consists of several steps. Assume that we have some time interval [0,T] and we discretize it by introducing equispaced values of sampling, tj, by means of tj = j x At, where At — T/N. A function to be analyzed, / , is known at the N values of t: to, ti,...,tpr-i. The size of the sample, N, is assumed to be even. To complete a Discrete Fourier Transform (DFT) we introduce the functions: ck (t)

-

cos

sk(t)

=

sin

(2-rrk \

,

[•jrt),

n

N

„

JV

* = 0,..., y ,

[2-Kk \

,

( ^ J , k = 0,...,--l,

and the discrete scalar product N-l

j=0

Then one has ~-

0 if

>

==

y

< Co, Co >

==

< CN/2,CN/2

^ $kt Sm ^

~-

< sk,sk>

=

0 if k ^ m, N_

^ Ck , Sm ^

~=

^ Ck ) Cm *> < ck,ck

k ^ m,

if

k^0,k^N/2, > =

N,

~~ T ' o.

Hence we can introduce N/2

N/2-1

F = ~^2akck + ^2 k=0

bksk,

k=l

where < f,ck> ak = < ck,ck >

, < / , sk > Dk — < sk,sk

>

and the functions / and F coincide at the sampling points. The function F is a trigonometric polynomial with all the frequencies multiple of 1/T. B.2.1

The DFT of Simple

Inputs

Now we compute the transform for simple functions that are to be known for all t and sampled at tj.

The Method

229

If /(f) = 1 we get do = 1, all the other coefficients being zero. If f(t) = cos f

—t

we obtain 6 *

~

4 N {

1 **

-

(

2N\

2 +

s m j ^ i u + k)) s i n ( £ ( w + fc))

fcos(^ +

+

sin(^7r(c-fc))\ sin ( $ ( " - * ) ) ) '

fc))-cos(^7r(u;

N "'''"' 2 '

+ fc)) (R1)

sin ( * ( « + *)) COS ( $ ( - 0 , + fc)) - COS ( ^ 7 T ( - U > + fc)) \ sin(£(_w +

fc))

)>

jV i

'---2

X

'

where 8 = 1 for' fc = 0 and A; = N/2, and <5 = 2 otherwise. In a similar way, for

f(t) = sin (^—ij , we obtain *

"

6 f cos (%(h, + fc)) - cos ( ^ 7 r ( a ; + fc)) 4iV^ sin^ioj + k))

{ii 2)

-

cos (ft (a; - k)) - cos ( ^ 7 r ( c ^ - k)) \

sin(£(u,-*)) . 'sin(^^-fc)) ""' sin(£(W-fc))

) '

N U

"-"2'

sin(M^7r(o; + fc))\ JV _ l s i n ( £ ( W + fc)) ) ' * '---'2 ''

The formulas above give some insight on the problems that can occur in performing a DFT. First, if u> < N/2 we have ft (a; + k) < TT. Otherwise, the terms having sin(ft(w + k)) in the denominator can give rise to some problems. This is quite natural, because to approximate cos(2^pt), for instance, by the c^, Sfc functions, we should have UJ well inside the interval [0,N/2]. Otherwise we have the phenomenon called aliasing. If we assume u in [0, N/2] and bounded away from 0 and N/2, then sin(ft(w + k)) is bounded away from zero and, when N is large, the terms having sin(ft(w + k)) in the denominator are comparatively small. Now consider the terms with sin(ft(w — k)) in the denominator. Introduce e = u> — k. For e finite and N going to infinity, the formulas (B.l) give ak

*

sin 2-nt "2^-'

230

Fourier Analysis 1 — cos 27re bk

sin 2ire

while (B.2) give 1 — cos 2TT6

a-k

2-KC sin 2TTC

bk

2ire

If we let e tend to zero, we have a^ = 1, bk = 0 in the first case, and a^ = 0, bk = 1 in the second one, as it should be. It is better to look at the modulus of the harmonics of index k, i.e.,

Pk =

(al+bl)^.

Using the approximations above sin7re Pk «

, ire

in both cases ((B.l) and (B.2)). Given w we can find k* such that | u — k* |< 1/2. Let e = u — k*. If we consider either of the functions /2TTW \

COsl-jT-H,

f2irw

Sill I y l

we have sin7re Pk' «

• ire This is 1 for e = 0 but for | e |= 1/2 we obtain 2/ir. Hence, if the frequency LO/T is not a multiple of 1/T we do not determine the coefficient correctly. Furthermore, if we look for the contribution of the nearby harmonics, we obtain pk*+m = 0 for m — ± 1 , ± 2 , . . . if e = 0, but for e ^ 0 the coefficients are different from zero. For instance, for e = 1/2 then Pk>+i = Pk' and 1 Pk*+m ~ | m - 1 / 2 | TT'

This fact could be expected due to the following reason. The approximating function F is, by definition, T-periodic, but the initial function, / , can have a different period or even be not periodic at all. In fact F gives a good approximation of the T-periodic function / * defined as follows: /*(*) = /(*) for f*(t) = f{t-mT)

te[0,T\,

for te[mT,(m

+

l)T],meZ.

The Method

231

In what follows we shall assume that the function / is analytic. But the function /* can even be discontinuous at mT. We claim that the jth Fourier coefficient of a T-periodic Cr function g, such that g^ is Lipschitz, decreases as | j |~( r + 1 ). Indeed, let us compute the exact Fourier coefficient f7

2

9(t) cos I ~jt

1 dt.

9(t) cos ( -^jt

I dt

By integration by parts 2 f1

2

f

•

—

)

(*) sin ( ~jt)

dt.

JO

21, j J Repeating the integration by parts r times we obtain

j

I

T

27TJ

loT9ir)(t)CS(^jtyt

where cs denotes either cos or sin. If we only know g to be Cr we only can bound the integral by TM, where M = sup te r 0 T i | g^T\t) \ and, therefore, | a,- |= 0(j~r). However, if g^ is Lipschitz with constant L we can write jTTflW(*)CB(yjt)dt

g(r){n_)+g(r){t)

n=0

J

—

,(->(»?)) c a f e J A.

v

As g(r)(n j) is a constant, the related integral is zero. On the other hand | g^(t) — g^(nj) \< Lj for t € [ n ? , (n + l ) y ] , and each integral can be bounded by L j x j . The total integral can be bounded by ^—, proving the claim. In our case, in general, /* is just Lipschitz in [0, T) with jump discontinuities at nT Vn 6 Z. To cope with this difficulty we can use different filters. The goal of filtering is to produce a function /*, related to / , and such that /* has a better degree of differentiability.

232

Fourier Analysis

B.2.2

Several

Filterings

Instead of / * we introduce the T-periodic function fHl /*=/(*) (l-cos ( ^ ) )

defined as follows

for

te[0,T).

This is the so called Hanning filter. We see that no matter which values / has at 0 and T, fHl has a double zero at them. Then fHl is C2 and (fHl)^ is Lipschitz in [0, T) and has a possible discontinuity at mT. Hence, the (exact) Fourier coefficients behave like \ j \~3. We shall check, later on, that this also happens for the related DFT. Hanning filter can be iterated. So we define fHm as a T-periodic function by fHm = f(t)qm

( l - cos ( ^ ) )

for

te[0,T),

where qm is a numerical coefficient such that the average of

is one, i.e., 1m =

(2m-1)!!'

The function fH™ is C2m and (/ f f »)( 2 m ) is Lipschitz in [0,T). Therefore the (continuous) Fourier coefficient a, behaves like | j |-( 2 m +!). Let us recompute the DFT after using Hanning filter. Let a • m, b • m, be the TT

discrete Fourier coefficients of / by means of afm

=

m

J

J

. They are related to the coefficients a,j, bj of /

p [ a m ) ^ a r o - i ) ( o i + i + a i - i ) + am-2)(«i+2+a>-2)-...

(-ir{im)(aj+m+aj-m)},

+

m and a similar imar formula ioimuia noma holds for tux b• b " with the same numerical coefficients. instance

1

TT

H,

o-j

2

=

a0-.- -aj+i

For

1 2

- -aj-x

1

+ -aj+2

1

+ g«j-2-

Hence we obtain formulas similar to (B.l), (B.2) by weighted summation. If, as we did before without filtering, we assume ui bounded away from 0 and N/2 but in [0, N/2] and N large enough, introducing again e = u — k, we have the

233

The Method

limit behavior: sin27re(m!)2

-Hr, bHm k

(l-cos27re)(m!) 2 27rV'ro(e)

„ ~

if /(«)

=

cos(^—*

and (l-cos27re)(m!) 2 2iripm(e)

^#m

sin27re(m!)2

&f"

27rV>m(e)

if /(t)

=

.

/2TTCJ

sinf—i

where

t(«) = n (e+*)i=—m

In particular if we select k* such that e* = cuk* satisfies | e* |< 1/2 and we put e — e* + q, we see that

vHkr+q = [{aHkr+q? + (b«™+qn^ = od q r 2 ^ 1 ), as expected. The quick decrease of the coefficients has the following advantage. Assume /(*) = c o s (wi—

t) +cos i^Y*)

>

and Wi (resp. W2) has fcx (resp. fc2) as closest integer. Then the contribution of the frequency u>2 to the fcith harmonic, p^m, is of the type | W2W1 |-( 2 m + 1 ) and the harmonics are better separated. However it is not advisable to take m too large. Indeed, filtering produces a local broadening, because if e* = 0 then the coefficients p^m are zero only for | k — k* |> m. In the applications we used mainly H\ and Hi-

Fourier

234

B.2.3

Determination

Given f(tj),j

Analysis

of the Frequencies:

= 0,...,N

First

Approximation

— l first we multiply by the filtering factor 2irtj

1 - cos ——Then we recompute the DFT obtaining the coefficients of cos and sin and we get the amplitude of the fcth harmonic p^m. We know that if / is the superposition of several harmonic terms: f(t) = ^ a r c o s ( w r — t ) + /?rsiri ( w r — i j and the uir are not too close, then the power spectrum {pkm}k=o,...,N will display peaks for the values of k which are closer to the true frequencies uir. Now we assume we are able to determine p^m for noninteger values of k. This is not possible by using the DFT, but we can compute the (continuous) Fourier spectrum {p^m} where s ranges in [0, N/2] in a continuous way, by means of a m

"

= ff

fHm(t)

cos

(sytjdt,

and a similar formula for b^m. Of course, the numerical integration should use only the values of / at the points ts. To this end we have used the 5 points rule (Bode's rule)

/,

f(t)dt = ^

(7/(to) + 32/(tx) + 12/(t 2 ) + 32/(t s ) + 7f(U)) - ^g^1^)

>

and then ftN

/

N i 1

/-

ru„+4

f(t)dt= f(t)dt= £Y, //

/(*)*•

This requires TV to be a multiple of 4. The global error is 0(N~6). We know that the maxima of p^m should occur for s very close to the theoretical values of u/r. Therefore, starting at the value of k for which a peak is found we look for local maxima by computing pfm numerically with the rule given above. This also produces some values of the a^m and b^m coefficients. B.2.4

An Alternative

Method

to Compute

the

Coefficients

The method of the preceding section suffers from the fact that /(£) is made of several (possible infinite) harmonics. Therefore all the harmonics contribute to the determination of p^™. Filtering makes this problem not too severe but usually the maxima found using the method described before are slightly away from the

235

The Method

true frequencies. Also the coefficients have some noise coming from the remaining frequencies. Assume, for the moment, that the frequencies are known. Let us denote by ak,w, bk u the expressions of (B.l) and by ak,u , bk,w the ones of (B.2). If some filter is used we have, in a similar way, expressions like ak ™, etc. A function

/(*) = E Q r cos (WrY"0 +@rS[n i^Y1 will produce, if we use an iterate filter Hm, values of DFT like /J OLrak%ur + Prak,wr,

r

r

If / contains only q harmonics we can take q values of k and write down this linear system a

K (/)

=

h„(f)

=

? z l ar®kr ,u>r + Praic,.,wr , r=l ^2arhr,Ur+Prhr,u,r, V= r=l

l,...,q

of dimension 2q, to determine ar,/3r. If we select kr to be the closest integer to uir and the CJT frequencies are not too close, the system is close to 2 x 2 block diagonal and iterative methods such as block Jordan or block Gauss-Seidel are convergent. Of course, if / has a constant component, ao, one has to use an additional equation (just selectfco= 0). This determination of coefficients is better because at each frequency we take into account the tails of the remaining ones on the DFT. B.2.5

The Effect of Errors

on the

Frequencies

For the moment we assume again that / has just one harmonic: r/,\ /27ru; \ „ . (2-KUJ f(t) = a cos I —t) + p sin ( —t Let k be the closest integer to u> and e = w — k. Now suppose that we have determined an approximation of w, a/, and let e' = w' - k. Using the method of the preceding section we determine the coefficients a' and /?'. We have (

a m

" (f) \ „ !EZ[£Mlo

(a)

„ sinTre' (m!) 2

/,x

236

Fourier

Analysis

where Rc stands for the matrix cos 7re sin ne sin ire cos ire With the approximate frequency, u>', and the coefficients a', /?', we have an approximate function /'(*) = a' cos ( -^-t)

+ /?' sin | ^T ^ t

\ T "J ' '

V

The error we have is / ' — / . By construction the harmonic p^m (/' - / ) is zero. Let us look at the nearby harmonics of the residue / ' — / : o-k+jW - / ) } _ sinTre' (ml)2 / ,\ sinTre 1 & ? + , • ( / ' - / ) ) ~ TT il>m(e'+j)ttt'\<}') rr V™(e + i) R*

(?) •

Using the expression of i?c< given above we have for the modulus of the harmonics of the residue

^ ( / ' - / ) « ( « ' 2 + ^ ' 2 ) 1 / 2 sin ire 7T

(ml)U2 1pm(e'+j)

V'm(e') V'm(e) •0m(e'+ j) ipm(e + j)

If, furthermore, we assume 6 — e' — e to be small then we can approximate the last term in the previous expression by d ( i>m(e') de \ipm(e' +j) The derivative is computed by means of v+e v + e+ j B.2.6

Determination

= >E

II

v= — n \u= — n,u^v

of the Frequencies:

u+ t u + e + j J (v + e + j)2 Improvement

From the last section it is possible to obtain an estimate of 5 and therefore, if we know e', try to obtain a better value of e. For this it is better to use j ^ 0 small, for instance j = 1, j = - 1 or just the sum of both: g = p^x(f' — f)+p^1(f —/)• The last function should be zero (in the case of / possessing just one harmonic) if u/ = LJ. Then we can try to minimize g as a function of the frequency close to k. To improve the frequencies for a general (quasi-periodic) function / one can use an iterative process. As soon as we know some frequencies and the related coefficients in an approximate way, we can subtract the contribution of these frequencies from the initial function. When a given frequency has been (partially) improved we can subtract its contribution from / to improve in turn the remaining ones. This is time consuming and should be done only if there are frequencies which are rather close and have important amplitudes. But this is usually the case if we have many

An Application

to the Analysis

of Orbits of the Restricted

Problem

237

basic frequencies and to have a good approximation of / several hundreds of terms are required. B.3

A n Application to the Analysis of Orbits of the Restricted Problem

To offer just an illustration we shall show here the results of the Fourier Analysis of x(t), y(t), z(t) for orbits of the circular restricted 3-dimensional three-body problem. The starting parameters for the numerical integration of the solution are: p = —0.04, a = 0.32, z = 0.4. The total time span covered by the integration has been 4096 lunar revolutions and 16 points per revolution have been used. The first terms of the analysis are shown for the 3 coordinates. Keeping just terms with amplitude larger than 6 x 10~ 7 , the respective number of harmonics (n x , ny,nz) are (102, 96, 85). The basic frequencies are • 0.99982230481, " vertical" frequency, • 0.28236325432, " short period" frequency, • 0.96077727857, " long period" frequency. The fact that the frequencies are known accurately and no problem occurs in the identification of a basic set, seems to indicate that really the motion takes place on a 3-dimensional torus. Tables B.l, B.2, B.3 show part of the results of the Fourier analysis for the example. In Table B.4 we display part of the output when determining the basic frequencies and identifying the linear combinations.

238

.28236325432E+0 .96077727857E+0 .19996446095E+1 .67841402441E+0 .10388673167E+1 .12431405333E+1 .39605077007E+0 .56472650864E+0 .75650409856E+0 .17172813552E+1 .22820078639E+1 .29604218882E+1 .78090038140E-1 .11368751571E+0 .20427320001E+0 .36045330718E+0 .47414082093E+0 .84708976287E+0 .16391912997E+1 .19215545561E+1 .19177756805E+0

Fourier

.89181897E--1 .10596182E--1 -.16570924E--1 -.29127325E--2 -.48547982E--2 -.53049813E--4 .51927500E--3 -.19112621E--2 -.39552707E--3 .22078443E--2 .14328186E--2 .69332264E--3 .32151203E--3 .87827238E--4 -.27898199E--3 .24141449E--3 .12318795E--3 .31863236E--3 .13051112E--3 -.62407193E--3 .58299172E--4

Analysis

-.13006751E--1 .78273477E--1 -.34050893E--1 -.23541247E--2 -.37821354E--2 .28533780E--2 .14070839E--2 -.23488309E--3 -.70933623E--3 -.74566731E--3 -.11785921E--2 .58031637E--3 -.19767708E--3 .50561222E--3 -.10440763E--3 -.17476396E--3 .18188448E--3 .65488119E--4 -.13144583E--3 .95670455E--4 .12582472E--3

.9013E--1 .7899E--1 .3787E--1 .3745E--2 .6154E--2 .2854E--2 .1500E--2 .1926E--2 .8122E--3 .2330E--2 .1855E--2 .9041E--3 .3774E--3 .5132E--3 .2979E--3 .2980E--3 .2197E--3 .3253E--3 .1852E--3 .6314E--3 .1387E--3

1158 3936 8192 2780 4256 5093 1623 2314 3100 7035 9348 12127 321 467 838 1477 1943 3471 6715 7872 787

.406E--11 .107E--09 .195E--10 .558E--10 .142E--07 .306E--09 .444E--10 .936E--11 .220E--07 .430E--12 .388E--11 .656E--10 .141E--07 .136E--10 .211E--08 .647E--09 .130E--08 .108E--09 .336E--08 .126E--08 .140E--09

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Table B.l Output of the Fourier analysis for the x component of the orbit of the 3-dimensional circular restricted three-body problem. Mass parameter equal to the one of the Moon (fj. = 0.012150582). Initial conditions: x = (l + p) cos(2ira)+fj,, y = (l + p) sin(27ra), z and zero synodic velocity, where p = —0.04, a = 0.32, z = 0.4 . Time interval: 4096 revolutions of the Moon. Only the harmonics with an amplitude larger than l.E—4 have been displayed. T h e columns display the following: First column: frequency (we set the frequency of the Moon equal to 1); second and third columns: cosine and sine components; fourth column: modulus of the harmonic; fifth column: the index of the related peak in the D F T (frequency 1 has an index equal to 4096+1); sixth column: an estimate of the error in the determination of the frequency; seventh column: number of order of the harmonic corresponding to the order used in the determination of the frequencies. For the numerical integration a constant time step equal to 1/128 of revolution was used, but only one point every 8 points was stored.

An Application

.28236325433E+0 .96077727873E+0 .19996446095E+1 .56472650864E+0 .67841402454E+0 .10388673331E+1 .39605077002E+0 .12431405342E+1 .17172813552E+1 .19215545562E+1 .22820078639E+1 .29604218882E+1 .78090069472E-1 .11368751582E+0 .20427319713E+0 .47414081637E+0 .75650402850E+0 .84708976298E+0 .13212306504E+1

to the Analysis of Orbits of the Restricted

.46672354E- - 1 .52820425E- - 1 - . 3 4 0 4 9 1 7 2 E - -1 - . 3 3 9 8 3 3 6 0 E - -2 - . 4 1 2 0 0 7 5 0 E - -2 - . 5 1 9 7 1 3 4 6 E - -2 .10014948E- -2 - . 1 4 7 2 2 8 0 2 E - -2 - . 7 2 9 6 2 7 8 6 E - -3 .33617623E- -3 - . 1 2 1 7 4 2 4 7 E - -2 .59051356E- - 3 .18461042E- - 3 .64819891E- -4 - . 1 8 8 0 0 1 6 2 E --3 .14274696E- -3 - . 3 6 8 5 8 4 3 5 E --3 .22623991E- - 3 .28938216E- -3

Table B.2

.99982230481E+0 .39045026065E-1 .71745905040E+0 .12821855591E+1 .19605995951E+1 .32140828021E+0 .43509579589E+0 .15645488138E+1 .16782363214E+1 .22429628200E+1 .24331822789E+0 .60377153463E+0 .92173225190E+0 .13958730725E+1 .20386896356E+1 .29213768857E+1 .15273254155E+0

.5529E- - 1 .5854E- - 1 .3802E- - 1 .4373E- - 2 .5146E- - 2 .5846E- -2 .1310E- -2 .1810E- - 2 .2380E- -2 .8972E- - 3 .1898E- - 2 .9321E- - 3 .2778E- - 3 .1738E- - 3 .1883E- - 3 .1607E- - 3 .3836E- - 3 .3957E- - 3 .3519E- -3

1158 3936 8192 2314 2780 4256 1623 5093 7035 7872 9348 12127 321 467 838 1943 3100 3471 5413

239

.465E- - 1 1 .571E- - 1 0 .184E- -10 .869E- - 1 1 .192E- -09 .222E- -08 .261E- -11 .121E- -08 .390E- -12 .119E- -08 .396E- -11 .434E- -12 .173E- -07 .119E- -09 .499E- -08 .587E- -08 .481E- -07 .590E- -11 .652E- -07

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

.300E- -10 .392E- -10 .520E- -10 .770E- -11 .116E- -07 .218E- -09 .240E- -09 .331E- -09 .769E- -08 .178E- - 0 7 .334E- -09 .125E- -09 .669E- -09 .228E- -08 .574E- -10 .236E- -07 .250E- -09

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Same as Table B.l but for the y variable.

.33815652E- -0 .25164558E- - 1 .60199549E- - 2 .11703173E- - 1 - . 5 5 4 7 4 5 4 8 E - -2 - . 1 1 6 3 8 8 3 8 E --2 - . 6 8 6 2 1 8 1 7 E --3 - . 6 8 7 9 0 4 0 5 E --3 - . 2 4 1 0 0 4 8 6 E --3 - . 8 5 9 5 3 3 1 8 E -- 3 .18255035E- - 3 .44419152E- - 3 .35450029E- - 3 - . 1 2 3 0 4 1 9 7 E --3 - . 2 6 6 2 7 1 2 6 E -- 3 .68032421E- -4 .48148912E- -4

Table B.3

- . 2 9 6 4 2 3 8 0 E -- 1 .25226614E- - 1 .16907789E- -1 .27527763E- - 2 - . 3 0 8 2 9 5 7 4 E --2 .26776937E- -2 .84510796E- - 3 - . 1 0 5 2 9 2 7 2 E --2 - . 2 2 6 5 3 9 8 8 E - -2 - . 8 3 1 8 9 0 6 8 E --3 - . 1 4 5 6 3 4 7 1 E - -2 - . 7 2 1 1 3 5 7 1 E -- 3 - . 2 0 7 5 7 1 0 9 E --3 .16121558E- - 3 - . 1 0 0 2 1 3 2 2 E - -4 .73701753E- -4 - . 1 0 6 1 7 7 7 3 E --3 - . 3 2 4 6 4 6 5 2 E --3 - . 2 0 0 2 3 2 7 9 E - -3

Problem

- . 1 7 9 8 2 3 7 0 E - -0 - . 5 9 1 4 9 2 0 3 E -- 3 .22836974E- - 1 .13312132E- - 1 .70502982E- -2 .98268431E- - 4 - . 8 4 9 5 8 7 0 8 E --3 .10846410E- - 3 - . 1 2 0 3 5 3 9 3 E --2 - . 2 9 4 5 0 1 1 3 E -- 3 - . 3 1 7 6 2 7 3 2 E -- 3 - . 7 4 4 5 3 1 7 8 E - -4 .29126562E- - 3 .26916844E- - 3 - . 4 4 1 2 3 7 1 4 E -- 3 - . 2 9 0 5 3 4 9 1 E -- 3 .11218689E- - 3

.3830E- -0 .2517E- - 1 .2362E- - 1 .1773E- - 1 .8971E- -2 .1168E- - 2 .1092E- -2 .6964E- - 3 .1227E- -2 .9086E- - 3 .3663E- - 3 .4504E- - 3 .4588E- - 3 .2960E- - 3 .5154E- - 3 .2984E- - 3 .1221E- - 3

4096 161 2940 5253 8032 1317 1783 6409 6875 9188 998 2474 3776 5718 8351 11967 627

Same as Table B.l but for the z variable.

240

Fourier

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

.9998223048 .2823632543 .9607772786 .9607772787 .2823632543 1.9996446095 1.9996446095 .0390450261 .7174590504 1.2821855591 1.9605995951 1.0388673167 1.0388673331 .6784140245 .5647265086 .6784140244 1.2431405333 1.7172813552 1.7172813552 .5647265086 2.2820078639 2.2820078639 1.2431405342 .3960507701 .3960507700 1.6782363214 .3214082802 .4350957959 2.9604218882 2.2429628200

.0000000E+00 •0000000E+00 .0000000E+00 -.1600000E-09 -.1000000E-10 .1200000E-09 .1200000E-09 .1750001E-09 .9000001E-10 .3000000E-10 -.1172000E-07 .1435000E-07 -.2050000E-08 -.2900000E-09 .0000000E+00 -.1600000E-09 -.4100000E-09 .1000000E-09 .1000000E-09 .0000000E+00 .4000000E-10 .4000000E-10 -.1310000E-08 -.1400000E-09 -.9000001E-10 .7660000E-08 .3500000E-09 .2800000E-09 -.1000000E-10 .1770000E-07

Analysis

.3829966E+00 .9012539E-01 .7898744E-01 .5853528E-01 .5528996E-01 .3801604E-01 .3786897E-01 .2517151E-01 .2361710E-01 .1772504E-01 .8971118E-02 .6154154E-02 .5846388E-02 .5145838E-02 .4373381E-02 .3745119E-02 .2853871E-02 .2379998E-02 .2330364E-02 .1925641E-02 .1898175E-02 • 1855276E-02 .1810045E-02 .1499844E-02 .1310420E-02 .1227432E-02 .1168025E-02 .1092105E-02 .9320638E-03 .9085858E-03

1 1 1 1 1 2 2 2 2 2 2 3 3 2 2 2 2 3 3 2 3 3 2 3 3 3 3 3 3 3

1 0 0 0 0 2 2 1 1 1 1 2 2 0 0 0 0 2 2 0 2 2 0 0 0 1 1 1 2 1

0 1 0 0 1 0 0 0 -1 1 0 0 0 -1 2 -1 1 -1 -1 2 1 1 1 -2 -2 -1 1 -2 0 1

0 0 1 1 0 0 0 -1 0 0 1 -1 -1 1 0 1 1 0 0 0 0 0 1 1 1 1 -1 0 1 1

Table B.4 Determination of the basic frequencies for the d a t a of Tables B . l , B.2, B.3. The three variables x, y and z are considered together (a given harmonic appears as repeated with different amplitudes if it appears in 2 variables). In the columns we show a number of order (according to the modulus of the harmonic), the frequency, the error in the identification, the modulus, the total order and the coefficients as a combination of the 3 basic frequencies.

Appendix C

Geometrical Bounds for the Dynamics: Codimension 1 Manifolds

In Chapter 1 we have mentioned the existence of manifolds of codimension 1 which can act as barriers for the motion. For instance they can confine the motion near a totally elliptic fixed point becoming (either in a purely mathematical sense or from a practical point of view) the boundaries of the stability regions.

C.l

The Center, Center-stable and Center-unstable Manifolds

For simplicity consider a fixed point at the origin of a system of ordinary differential equations. Assume that none of the eigenvalues of the differential of the vector field at the origin is equal to zero. We can classify the eigenvalues according to the real part: the "stable" eigenvalues, which have negative real part, the "central" or "neutral" eigenvalues, with zero real part and, finally, the "unstable" eigenvalues with positive real part. There are three linear vector spaces associated to each one of the groups of eigenvalues. In the stable subspace the motion tends to the origin for increasing time in an exponential way. In the unstable one the motion escapes from the origin exponentially fast. In the central subspace either the motion remains bounded from above and below or it escapes, at most, in a polynomial way. The stable, center and unstable invariant manifolds are the generalizations of the previous concept when the full nonlinear system is considered. They are invariant in the sense that a point which starts in one of these manifolds remains on it forever. This gives a way to compute, formally, the manifolds. Indeed, we should ask for a manifold such that the vector field restricted to it is tangent to the manifold. We can also define center-stable and center-unstable manifolds by requiring invariance and the condition that the tangent space at the fixed point is the sum of the corresponding vector spaces. If a fixed point has just one stable and one unstable directions, then the center-stable and center-unstable manifolds have codimension 1. These manifolds separate (at least locally) the space. Therefore they are good candidates to become boundaries of open sets. Notice, however, that we can have difficulties if the manifolds become extremely folded, because then it can be difficult 241

242

Geometrical Bounds for the Dynamics:

Codimension

1 Manifolds

to decide to which part of the space (say, inside or outside) belongs a given point. In this appendix we shall give the methods to compute the above mentioned manifolds and we shall do several applications. In all cases we consider analytic systems and, from now on, we restrict ourselves to the Hamiltonian case. For the stable and unstable manifolds general results assure the uniqueness and analyticity of the corresponding manifolds. The central manifold is neither unique nor analytic in general. However, it is formally unique. This means that the coefficients of the Taylor expansions can be obtained to all orders but the manifold is not unique because we can add to the formally obtained function any C°°-flat function.

C.2

On the Analytic Computation of Invariant Manifolds

Let i = F(z) be a differential equation having the origin as a fixed point. Assume that the system can be written as x

=

Ax +

f(x,y),

y

=

By + g(x,y),

where A is a matrix with all the eigenvalues having positive real part, while in the matrix B the eigenvalues have real part less than or equal to zero. Moreover, the functions / and g contain the nonlinear part of the equations, that is all the terms starting with degree larger than 1. Let us use x as independent variable for the parameterization of the unstable manifold. Then we should have y = h(x) for some unknown function h, which has no linear terms. The in variance condition is expressed as Dh(x) {Ax + f(x, h(x))) = Bh(x) + g(x, h(x)). By equating the coefficients in powers of the x variables in both sides of the preceding equation, all the coefficients of h can be determined uniquely. It is easy to check that no problem of small divisors appears. This simple formulation requires a big computational effort if h is desired to a high order, specially if the dimension of the x variable is high. This has been done for general systems of differential equations, and also (with the obvious modifications) for mappings instead of vector fields. If the center, stable, center-stable, etc. invariant manifolds are desired, instead of the unstable manifold, the procedure is similar. The programs performing these computations have been for general cases. However we shall not use them for the applications to the actual problem. Instead we have obtained some center-stable and center-unstable manifolds in a simple case by purely numerical methods that we describe in the next section.

The Center-stable

C.3

and Center-unstable

Manifolds for L3 in the RTBP

243

The Center-stable and Center-unstable Manifolds for L3 in the R T B P

As an illustration we consider the planar RTBP near L3. This libration point is of center-saddle type. Hence the center-stable and center-unstable manifolds have codimension 1. In fact L3 is the limit of the so called Lyapunov family of planar periodic orbits. Each one of them (at least locally) is hyperbolic. They have stable and unstable manifolds that one can imagine as tubes of dimension 2. The union of them, when we move along the Lyapunov family, gives the desired 3-dimensional manifolds. To visualize (and apply to our problem) these manifolds, we have cut them by a 2-dimensional manifold. We have selected as cutting manifold simply the points which satisfy the conditions x = 0, y = 0. Then the intersections are curves that we can plot easily. The procedure to obtain these curves on the plane of zero velocity is: a) Take a point on the ar-axis near L3 with x = 0. Look for a value of y such that one obtains a periodic orbit. b) Compute the variational matrix along the periodic orbit. Obtain the stable or unstable direction (as desired) on some Poincare section. c) Take initial points on the obtained direction near the periodic orbit (on the Poincare section the periodic orbit is seen as a fixed point) and transport them by the flow (backwards in the case of the stable manifold) until they reach a point with zero velocity. Probably one never obtains such a point if one starts the computations with a given point on the invariant direction. So one must look for the suitable initial point (that is, one has to solve some equation). d) When an initial point is obtained in the desired direction such that it goes to a point on the zero velocity plane, one uses a continuation method to proceed along the curve to be obtained. To this end one should move the periodic orbit along the Lyapunov family. The motion is not uniform, that is, sometimes we are approaching the libration point and sometimes we are escaping from it. When the curve progresses the required periodic orbits are confined to some domain which is relatively far from L3. This procedure has been applied to the planar RTBP for the mass ratio corresponding to the Earth-Moon problem, (1 = 0.12150582. Figure C.l shows the zero velocity curve on the (x, y)-plane, for the level of the Jacobi constant corresponding to L3. The window for the figure is (—1.05,1.15) x (—0.05,1.15). Inside that curve there are two curves which are the intersections of the center-stable and center-unstable manifolds of L3 with the plane of zero velocity. They are, roughly, at one half of the distance between the zero velocity curve and the circle of radius 1 around the Earth. Later on the curves start to become folded. However they leave

244

Geometrical Bounds for the Dynamics:

Codimension

1 Manifolds

free some domain. The "stable" region is contained in that domain, and it seems that when the curves progress they should define the boundary of the stable region. Figure C.2 shows the same picture but in (a, ^-variables region. In these variables the window is (0.5,0.0) x (-0.1,0.1). The stable manifold starts near L3 on the upper part, and the unstable one on the lower part, of course. Furthermore we have included in the picture the stable points obtained in Chapter 1. To see better the boundary, we have used a very small step size for points near the boundary of the stable region (that is, a step of 10 - 4 ), and we have continued that computations up to 10000 revolutions. Figure C.3 shows a magnification of the previous one. We have used the window (0.3,0.1) x (-0.02,0.03). Furthermore the size of the dots differs according to the subsistence time. The smaller ones escape after a time comprised between 100 and 1000 revolutions. The medium size ones escape after a number of revolutions ranging on the interval (1000,10000). Finally, the larger ones subsist for more than 10000 revolutions. One expects that, using the analytic method described above, it will be possible to compute easily a longer part of the manifolds to see the approach of the tongues to the stable region.

The Center-stable

and Center-unstable

Manifolds for L3 in the RTBP

245

Fig. C.l This figure shows the zero velocity curve on the (x,y)-pla,ne, for the level of the Jacobi constant corresponding to L3. The window for the figure is ( — 1.05,1.15) x (—0.05,1.15). Inside that curve there are two curves which are the intersections of the center-stable and center-unstable manifolds of L3 with the plane of zero velocity. The figure has been rotated 90° clockwise.

246

Geometrical Bounds for the Dynamics:

Codimension

1 Manifolds

Fig. C.2 The same as in Figure C.l but in ( a , p ) variables region. In these variables the window is (0.5,0.0) x (—0.1,0.1). The stable manifold starts near L3 on the upper part, and the unstable one on the lower. The dark region of the center of the figure corresponds to the stable points computed in Chapter 1. The figure has been rotated 90° clockwise.

The Center-stable

and Center-unstable

Manifolds for L3 in the RTBP

247

Fig. C.3 Magnification of Figure C.2. The size of the window is (0.3,0.1) x (-0.02,0.03). The figure has been rotated 90° clockwise.

Appendix D

Conclusions

To summarize the achievements of the present work we list some conclusions. We classify them in four parts. The first one presents the concrete results that have been obtained till now. In the second we describe the new achievements concerning methodology. The third part is devoted to possible uses of the concrete results for future space missions. Finally we close with some outlook and several proposals.

D.l

Summary of Achievements

The main goal has been the study of orbits in an extended vicinity of the triangular libration points of the Earth-Moon system. This depends, of course, on the model chosen for the description of the problem, that, in any case, has to take into account the three spatial dimensions. A reasonable list of models includes the circular restricted three-body problem, the elliptic one, the bicircular model, some intermediate analytic model and the real model, this one being understood as the one given by the JPL ephemeris. (1) For the RTBP it has been found that there are regions of stability that extend up to big distances from the libration point. The stability criterion asks, in all cases, for the motion to be constrained, in the space of positions, to a half space with constant sign of the y variable. That is, the sign of the y variable must be, all the time, the same one of the chosen libration point, either L4 or L5, in the synodic reference frame. If the initial conditions are selected as one point in the configuration space and zero synodic velocity, then the region obtained is a thin "shell". The shell is close to a revolution paraboloid with axis the vertical one passing through the Earth. The equatorial mean radius is close to the Earth-Moon distance. The amplitude of the shell, in the horizontal plane, is of the order of 1.2 times the Earth-Moon distance (see p.260). In the vertical direction one can reach distances from the equatorial plane up to 5/6 of the Earth-Moon distance. 249

250

Conclusions

The width of the shell changes along it. A significant value can be 1/25 of the Earth-Moon distance. It is larger for inclinations between 15 and 40 degrees, as seen from the Earth. An important feature is that the vertical mode is rather close to a harmonic oscillator, so that the vertical frequency changes very little with the amplitude, as revealed by the study of the corresponding family of periodic orbits and by an analysis of the Normal Form. Another typical feature is the "rough" character of the boundary of the shell. Indeed, we can not speak about a true boundary because even orbits starting close to the libration point can escape in very long time intervals by means of the mechanism known as Arnold diffusion. Furthermore, some of the resonances leading directly to instability have been detected. They change very much according to the part of the boundary we are looking for. Other initial conditions, without the restriction of taking the initial synodic velocity equal to zero, give an additional increase in the set of points leading to stable motion. Furthermore, there are minor stable domains in the vicinity of linearly stable periodic orbits, but they play a secondary role. (2) The elliptic RTBP has been only partially considered, because it has been realized soon that the main perturbation is due to the presence of the Sun instead of the lunar eccentricity. However, the main characteristics found for the circular model, are preserved for the elliptic one, at least for the mass ratio and the eccentricity of the Earth-Moon system. An important difference is due to the addition of an external frequency: the one of the lunar motion. This frequency is in exact resonance with the vertical frequency at the libration point. This prevents the stability of some points close to the libration points, but does not alter very much the global picture. (3) The main differences concerning the restricted problem are found when the influence of the Sun is considered. First, the libration points do not longer subsist (as it happens in the elliptic case), but they are replaced by periodic solutions, with period equal to the solar synodic period. In the case of the L 4 and L5 points, the corresponding periodic orbits are mildly unstable. Furthermore the instability is produced just in one direction. Nearby orbits have, beyond the period of the Sun, two additional periodic modes: One of them plays the role of the long period around the triangular points in the RTBP. The other one is a vertical mode. The short period mode appearing in the RTBP has disappeared due to resonance with the solar frequency and becomes a couple of unstable/stable directions. The vertical mode inherits, essentially, the character of the corresponding mode in the RTBP: it has a weak dependence of the frequency with respect to the amplitude. Furthermore, the frequency is slightly shifted (increasing by an amount of 0.4 per cent with respect to the vertical RTBP problem). This is due to the contribution of the autonomous part of the solar perturbation. The vertical family of periodic orbits subsists when we replace the RTBP model

Summary

of

Achievements

251

by RTBP plus the autonomous contribution of the Sun. It has stability until ^-amplitude near 0.9 Earth-Moon distance. When the full bicircular model is considered, the weak instability found for the periodic orbit that replaces the libration point, subsists till a zamplitude near 0.22 times the Earth-Moon distance. As a main conclusion, there are not regions of stable motion in the bicircular model for small zamplitudes in a vicinity of the small periodic orbit replacing L\ or L$. It is certainly true that the bicircular problem exhibits planar stable periodic solutions, but they are far away from the triangular points, and they seem to become unstable when the full set of perturbations is considered. A big set of stable orbits, lying on 2-dimensional, 3-dimensional and i 4dimensional tori, has been found for ^-amplitudes ranging from 0.22 to 0.9 in Earth-Moon units. At the bottom they lose stability by a resonance between the solar and the short period frequencies, while at the top the stability is destroyed by a resonance between the vertical frequency and the short period one. In the "lateral" parts different frequencies play a significant role to destroy the stability, depending on the location of the orbit. (4) A good analytic model has been derived for the equations of motion, using a "natural" reference frame, already introduced by the authors and coworkers in earlier work. It reproduces the true (time-dependent) vector field with an accuracy of 10~ 6 adimensional units of acceleration (1 adimensional unit is close to 2.6 mm/s 2 ) on a domain of radius of the order of 0.25 around the (geometrically defined) libration point. The model depends quasi-periodically on time, with 5 basic frequencies: The ones of the mean motion, perigee and node of the Moon around the Earth, and the ones of the mean motion and perihelion of the Earth around the Sun. With this model some quasi-periodic solutions (to be seen as a dynamical substitute of the triangular Earth-Moon libration points) have been found. The next step has been to refine these orbits to true solutions (numerical, of course) of the full JPL model. This has been done by using a parallel shooting technique, during time intervals reaching 200 lunar revolutions. They have a very mild instability. The dominant Lyapunov exponent is below 0.004 per day. This means that the separation from these orbits increases by a factor around 3.5 per year (on average). Hence, the requirements of station keeping are close to zero for them. These orbits are lying, essentially, in the Earth-Moon plane. (5) For the full JPL model, and following what we learned from the bicircular model, we have done many simulations starting, either at zero initial velocity in the chosen reference frame, or close to the stable "vertical" tori of the bicircular model. For time intervals up to 60 years, some of them show a remarkable stable character. This has been confirmed by Fourier

252

Conclusions

analysis, showing that they can be considered as quasi-periodic. The behavior of the frequencies when we range on the stable region, is very close to the one found for the bicircular problem. The main resonances leading to instability are the same ones. The effect of the eccentricity of the lunar orbit is seen to have a minor relevance, as well as the frequencies coming from the combined solar and lunar effects. (6) We have considered as nominal path for a space mission two classes of orbits. One of them is made by the orbits described in the previous item. The other one, by orbits with inclinations (with respect to the Earth-Moon plane) ranging from 20 to 40 degrees, and well located in the stable domain. The transfer to these orbits, starting from a GTO orbit (with 300 km of periapsis height, 35 786 km of apoapsis height and equatorial inclination equal to 7 degrees), requires moderate amounts of fuel. For the almost planar orbits the total Av required is less than 900 m/s, and for the ones lying in the stable zone is less than 800 m/s, if a good epoch is selected. The successive use of lunar swingbys (say, two is a good choice) is the main responsible for these moderate amounts.

D.2

O n the Methodology

We want just comment that the methodology used for the full work introduces much new ideas in the current development of the design of spacecraft missions. A systematic use has been made of the following tools: (1) A suitable reference frame such that it allows to pass from the circular restricted three-body problem to the elliptic one, to the bicircular, to several possible analytic intermediate models and to the full JPL (or whatever numerical model) taking into account all the main bodies of the solar system in a very clear, simple and efficient way. (2) The Normal Form theory, which allows to extract the dominant features of a vector field around a fixed point (or, eventually, around a periodic orbit, a torus, etc.). Several extensions had to be done with respect to the usual way, mainly to look for the normal form around objects different from a fixed point. This has been done by a previous step making use of the Floquet theory in the case of a periodic reference orbit. In any case, the manipulation of long d'Alembert series has been done by ad hoc symbolic systems developed by the authors. (3) The computer implementation of the Lindstedt-Poincare methods (and similar methods deviating slightly from it) to obtain analytic representations of approximated periodic and quasi-periodic approximated solutions of the equations of motion. (4) The theory of center, stable and unstable manifolds to look for the geomet-

On the Application

of the Results to the Design of Spacecraft Missions

253

rical explanations of the facts observed numerically. (5) Refined Fourier analysis procedures that allow very accurate determination of the frequencies and the amplitudes, when the analyzed function is quasiperiodic (or rather close to it). (6) As a rule, the point of view used to face to the many problems we have encountered, sits in the framework of the theory of Dynamical Systems.

D.3

On the Application of the Results to the Design of Spacecraft Missions

Concerning the applications of the present work, we would like to stress on the fact that we have fully demonstrated the existence of stable regions in an extended vicinity of the Earth-Moon triangular libration points (stable, at least for the usual time span encountered in space missions till now) and of orbits close to quasi-periodic ones near the geometrical libration point, which display a very small instability. The feasibility of the transfer orbits and the easy or not necessary at all station keeping, has been shown. The work has displayed several nominal paths that can be suitable for a spacecraft relatively close to the Earth and not suffering from too strong radioelectric noise. This can be suitable for scientific missions to be devoted to interferometry, or as parking orbits of groups of micro-spacecrafts ready to be used (with a very small initial manoeuvre and assisted by the lunar gravity) for small asteroids (Earth orbit crossing objects) and/or comets.

D.4

Outlook

What has been learned with this work suggests several extensions in Astrodynamics and other sciences. We list a few of the many items that can be developed from this starting point, by a combination and extension of the methods used here. (1) From a fundamental research point of view, the analysis carried out for the regions of stability around the triangular libration points, suggests that a similar study can be done for any system having fixed points (or periodic orbits or tori) with linear stability. At a first glance it seems that several different objects play different roles concerning nonlinear stability. Very close to the reference orbit Nekhorosev estimates can be rigorously derived, showing practical stability for very long time intervals. Probably they can be only applied in a somewhat small domain. Rough boundaries have been shown to appear. They are codimension 1 manifolds. Usually they are stable and unstable manifolds of some codimension 2 center manifolds. Points outside the domain sketched by these manifolds escape very

254

Conclusions

quickly. A third category of objects is related to resonances of higher and higher order. They connect the regions of slow diffusion with the ones of fast diffusion. Going "inside" through the chain of heteroclinic connections they develop, we have more and more stability. This one may be not perpetual, but several levels of "practical" stability can be denned. This can be enough for many applications. (2) As far as Astrodynamics is concerned, the manifolds discussed in the previous item, give regions of stability around a planet or a satellite. They also provide paths to be used at zero cost as part of transfer orbits. Related to this topic, we would like to make a reasonable proposal, that is seems to us that it would be a major step towards the analysis of spacecraft missions in the solar system. It is possible to compute (with a long and difficult combination of analytic, numerical, mass memory and graphical effort) all the stable/unstable manifolds of the center manifolds around the librations points (or their dynamical equivalents) of all the two-body systems in the solar system (say, Sun-Planet or Planet-Satellite). By using a regularization of the binary collisions, also the collision manifolds associated to the Solar system bodies can be studied, to look for the passages very close to all these bodies. Using both types of manifolds it is possible to obtain "all" the unpowered trajectories in the Solar System. This would allow to reduce the cost of design of the planetary or Earth-Moon missions to look at the catalogue and to search just for the required manoeuvres from a natural path to another one. In some sense, the problem of navigation in the solar system would be solved for ever.

Bibliography

[1] Arnol'd V.I. and Avez A.: Problemes ergodiques de la mecanique classique. Gauthier-Villars, Paris, 1967. [2] Companys V.: Transfer Orbits to the Equilateral Points of the Earth-Moon system. MASS Working Paper number 334. ESOC. [3] Diez C , Jorba A. and Simo C : A Dynamical Equivalent to the Equilateral Libration Points of the Earth-Moon Real System, Celestial Mechanics 50, 1329 (1991). [4] Dvorak R. and Lohinger E.: Stability zones around the triangular Lagrangian points, in Predictability, stability and chaos in N-body dynamical systems, Ed. by A.E. Roy, Plenum Press, 439-446 (1991). [5] Fontich E. and Simo C : The splitting of separatrices for analytic diffeomorphisms", Ergodic Theory Dynamical Systems 10, 295-318 (1990). [6] Giorgilli A., Delshams A., Fontich E., Galgani L. and Simo C : Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three-body problem, J. Differential Equations 77, 167-198 (1989). [7] Gomez G., Llibre J., Martinez R. and Simo C.: Station keeping of libration point orbits, ESOC contract 5648/83/D/JS(SC), Final Report (1985). Reprinted as "Dynamics and Mission Design Near Libration Points", Vol. I, World Scientific Pub. Co., Singapore, 2000. [8] Gomez G., Llibre J., Martinez R. and Simo C : Study on orbits near the triangular libration points in the perturbed restricted three-body problem, ESOC contract 6139/84/D/JS(SC), Final Report (1987). Reprinted as "Dynamics and Mission Design Near Libration Points", Vol. II, World Scientific Pub. Co., Singapore, 2000. [9] Gomez G., Jorba A., Masdemont J. and Simo C : Study refinement of semianalytic halo orbit theory, ESOC contract 8625/89/D/MD(SC), Final Report (1991). Reprinted as "Dynamics and Mission Design Near Libration Points", Vol. Ill, World Scientific Pub. Co., Singapore, 2000. [10] Jorba A.: On quasi-periodic perturbations of ordinary differential equations,

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Bibliography

Ph.D. thesis, Universitat de Barcelona (1991). [11] Jorba A. and Simo C : On quasi-periodic perturbations of elliptic equilibrium points, SIAM J. of Math. Anal. 27, 1704-1737(1996). [12] Laskar J.: A numerical experiment on the chaotic behavior of the solar system, Nature 338, 237-238 (1989). [13] Lazutkin V.F., Schachmanski I.G. and Tabanov M.B.: Splitting of separatrices for standard and semistandard mappings, Physica D40, 235-248 (1989). [14] Markeev A.P.: Stability of the triangular Lagrangian solutions of the restricted three-body problem in the three-dimensional circular case, Soviet Astronomy 15, 682-686 (1972). [15] McKenzie R. and Szebehely V.: Non-linear stability motion around the triangular libration points, Celestial Mechanics 23, 223-229 (1981). [16] Nekhorosev N.N.: An exponential estimate of the time of stability of nearlyintegrable Hamiltonian systems, Russ. Math. Surveys 32, No 6, 1-65 (1977). [17] Simo C.: Estabilitat de sistemes hamiltonians, Memorias de la Real Academia de Ciencias y Artes de Barcelona 48, 303-348 (1989). [18] Stoer J. and Bulirsch R.: Introduction to Numerical Analysis, Springer-Verlag, 1983 (second printing). [19] Szebehely V.: Theory of orbits, Academic Press, 1967. [20] Szebehely V. and Premkumar R.: Global sensitivity to velocity errors at the libration points, Celestial Mechanics 28, 195-200 (1982).

Updates with Respect to the Work Done for the European Space Agency

These last pages are devoted to update the bibliographical references given along the four Volumes on "Dynamics and Mission Design Near Libration Points". We list below contributions made by the authors and collaborators to these topics. A large part of them are an outgrowth of what is presented in the present Volumes. Many other contributions can be found in the literature and they can be traced back from the ones listed here. The reference [U45] contains, beyond some papers related to space missions and the celestial mechanics background, a large number of papers devoted to related areas. Essentially all the topics listed below are strongly related. They display a general philosophy consisting in the study of a concrete problem, or family of problems, in the context of dynamical systems (see, e.g., [U16]). These provide the geometric ideas about how the orbits fit in the phase space. Then, a combination of numeric and symbolic tools is developed to identify the key objects in the phase space and how are they related. Usually it is good to formulate a problem, first, in the more general context, and then to proceed to analyze several subproblems or simplified models. When these ones are sufficiently known, one goes back to the initial problem. For convenience we describe several main topics and the related papers in each one of them, despite the strong relation between topics allows to do this in different ways. • Simplified models. Some useful orbits based on the RTBP can be found in [U36]. For Hill's problem we refer to [U47], where a large chaotic region appears before reaching the level of the collinear points. The source is the nontwist phenomenon studied in [U44]. Tools like the ones displayed in [U04] allow the study in chaotic regions. A global study around collinear points in the RTBP is given in [U28]. Another interesting model is the bicircular (and quasibicircular) problem. It is studied in [U46] and [U01]. • Effective stability. Totally elliptic object can be nonlinearly unstable due to the presence of Arnold's diffusion. But it proceeds slowly and one can 257

258

Updates with Respect to the Work Done for the European Space

•

•

•

•

•

•

Agency

still have "practical stability". A seminal theoretical study was given in [U08] and also in [U37]. It was somewhat extended in [U38]. Applications can be found in [U31], for the elliptic RTBP and in [U20] under the general perturbations. The slow effect of the diffusion is bounded by the Nekhorosev estimates, in turn related to the exponentially small splitting of separatrices. For one of the simplest cases of this problem we refer to [Ull]. These smallness can be detected by allowing time (and phase space variables) to become complex. See [U41] for a general discussion. Quasi-periodic perturbations. When they are added to the equations of motion require, first, an analysis of its effect on fixed points. This was carried out in [U09], [U15] and [U17] near different libration points, mainly the Earth-Moon triangular points. A general study was undertaken in [U26]. The reducibility to constant coefficients in the linear case was studied in [U30]. In this context, and also related to a linear version of the breakdown of tori, see [U03]. For the theoretical study in the nonlinear case see [U32]. Effective implementations of the reducibility schemes are given in [U29]. Invariant tori. The classical approach to many studies in celestial mechanics based on the computation of families of periodic solutions, has to be completed with computation of invariant tori and their local properties. This is the contents of papers [U33], [U34] and [U35], both from a theoretical and applied point of view. See also [U42]. Effective computations of invariant tori near collinear libration points can be found in [U22] and [U24], where tori around halo orbits (or quasi-halo orbits) are computed. They belong to the corresponding center manifold. Methodological aspects. Methods to proceed to the effective computation of different objects appear in most of the papers. General descriptions of these methods, with examples, can be found in [U39], [U27] and also in [U42] and [U43]. Symbolic descriptions using numerical coefficients allow to obtain good analytic representations, useful not only for a qualitative study but also for quantitative purposes. Use of orbits near triangular points. This is the topic of several papers [U05], [U06], [U07], [U10] and [U21], where the orbits are described, the transfer to them is studied and a full mission analysis is carried out. The relative motion of two nearby spacecrafts is also analyzed. Transfer problems. The transfer to a halo orbit is studied in [U18] and the effect of the Moon in this transfer is the objective of [U14]. Transfers between different halo orbits are relevant. Indeed, one can proceed to different missions by changing the nominal orbit, or one can go to a target orbit passing through an intermediate one which could be reached with a lower cost. They are studied in [U19] and [U23] Station keeping. This is nicely studied when the local information around the nominal orbit is available. In the case of the Earth-Sun collinear L\

Updates with Respect to the Work Done for the European Space Agency

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point this is studied in [U12] and [U13]. For the much more difficult case of the translunar collinear point (the Li point of the Earth-Moon system) a comparison of different strategies is done in [U25]. For an sketch of the way to obtain the nominal orbit, we refer also to [U01] and [U02].

List of additional references: [U01] Andreu, M.A. and Simo, C : Translunar Halo Orbits in the Quasibicircular Model, in Proceed. NATO ASI on The Dynamics of Small Bodies in the Solar System, Ed. B. Steves and A. Roy, 309-314, Kluwer, 1998. [U02] Andreu, M.A.: The Quasi-Bicirculdr Problem, Ph.D. thesis, Universitat de Barcelona, 1999. [U03] Broer, H. and Simo, C : Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena. Bui Soc. Bras. Mat. 29:253293, 1998. [U04] Cincotta, P. and Simo, C : Conditional entropy: A Tool to Explore the Phase Space. Celestial Mechanics & Dynamical Astronomy 73:195-209, 1999. [U05] Companys, V.: Transfer Orbits to the Equilateral Points of the Earth-Moon system. MASS Working Paper number 334. ESOC. [U06] Companys, V., Gomez, G., Jorba, A., Masdemont, J., Rodriguez, J. and Simo, C : Orbits Near the triangular Libration Points in the Earth-Moon System, en Proceedings of the 44th Congress of the International Astronautical Federation, 1993, Graz, Austria, IAF 1993, Paris. [U07] Companys, V., Gomez, G., Jorba, A., Masdemont, J., Rodriguez, J. and Simo, C : Use of Earth-Moon libration points for future missions. Astrodynamics. Adv. Astronautic. Sci., 90:1655-1666, 1995. [U08] Delshams, A., Fontich, E., Galgani, L., Giorgilli, A. and Simo, C : Effective stability for a Hamiltonian system near an equilibrium point, with an application to the RTBP diffusion rate, Journal of Differential Equations 77:167-198, 1989. [U09] Dfez C , Jorba A. and Simo C : A Dynamical Equivalent to the Equilateral Libration Points of the Earth-Moon Real System, Celestial Mechanics 50:13-29, 1991. [U10] Flury, W., Gomez, G., Llibre, J., Martinez, R., Rodriguez, J. and Simo, C : Relative motion near the triangular libration points in the Earth-Moon system. In Proceed, of the Optical Interferometry Workshop, Granada 1987, ESA-SP 273. [Ull] Fontich E. and Simo C : The splitting of separatrices for analytic diffeomorphisms", Ergodic Theory Dynamical Systems 10:295-318, 1990. [U12] Gomez, G., Llibre, J., Martinez, R. and Simo, C : Station keeping of a quasiperiodic halo orbit using invariant manifolds. Proceed. 2nd Internat. Symp. on spacecraft flight dynamics, Darmstadt, October 86, ESA SP 225:65-70, 1986.

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[U13] Gomez, G., Llibre, J., Martinez, R., Rodriguez, J. and Simo, C : On the optimal station keeping control of halo orbits. Acta Astronautica, 15:391-397,1987. [U14] Gomez, G., Jorba, A., Masdemont, J. and Simo, C.: Moon's Influence on the Transfer from the Earth to a Halo Orbit around L\. In A.E. Roy, editor, Predictability, Stability and Chaos in the N-Body Dynamical Systems, 283-290, Plenum Press, 1991. [U15] Gomez, G., Jorba, A., Masdemont, J. and Simo, C.: Quasi-periodic Orbits as a Substitute of Libration Points in the Solar System. In A.E. Roy, editor, Predictability, Stability and Chaos in the N-Body Dynamical Systems, 433-438, Plenum Press, 1991. [U16] Gomez, G., Jorba, A., Masdemont, J. and Simo, C.: A Dynamical Systems Approach for the Analysis of the SOHO Mission. In Proceedings of the Third International Symposium on Spacecraft Flight Dynamics, 449-456, ESA SP 326, Darmstadt, 1992. [U17] Gomez, G., Jorba, A., Masdemont, J. and Simo, C.: A Quasi-periodic Solution as a Substitute of L4 in the Earth-Moon System. In Proceedings of the Third International Symposium on Spacecraft Flight Dynamics, 35-42, ESA SP 326, Darmstadt, 1992. [U18] Gomez, G., Jorba, A., Masdemont, J. and Simo, C.: Study of the transfer from the Earth to a Halo orbit around the equilibrium point L\. Cel. Mechanics, 55:1-22, 1993. [U19] Gomez, G., Jorba, A., Masdemont, J. and Simo, C.: Study of the transfer between halo orbits in the solar system, Advances in the Astronautical Sciences, 84:623-638, 1994. [U20] Gomez, G., Jorba, A., Masdemont, J. and Simo, C.: Practical stability regions related to the Earth-Moon triangular libration points. Space Flight Dynamics (1994) 11-17. [U21] Gomez, G., Jorba, A., Masdemont, J. and Simo, C : Mission analysis for orbits in a vicinity of the Earth-Moon triangular libration points. Space Flight Dynamics (1994) 18-23. [U22] Gomez, G., A., Masdemont, J. and Simo, C.: Lissajous orbits around halo orbits. Spaceflight Mech. Adv. Astronautic. Sci., 95:117-134, 1997. [U23] Gomez, G., Jorba, A., Masdemont, J. and Simo, C.: Study of the transfer between halo orbits. Acta Astronautica, 43:493-520, 1998. [U24] Gomez, G., Masdemont, J. and Simo, C.: Quasi-halo orbits associated with libration points. J. Astronaut. Sci., 46(2):135-176, 1998. [U25] Gomez, G., Howell, K., Masdemont, J. and Simo, C.: Station-keeping strategies for translunar libration point orbits. Spaceflight Mech. Adv. Astronautic. Sci., 99:949-967, 1998.

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[U26] Jorba, A.: On quasi-periodic perturbations of ordinary differential equations, Ph.D. thesis, Universitat de Barcelona, 1991. [U27] Jorba, A.: A Methodology for the Numerical Computation of Normal Forms, Centre Manifolds and First Integrals of Hamiltonian Systems. Experimental Mathematics 8:155-195, 1999. [U28] Jorba, A and Masdemont, J.: Dynamics in the center manifold of the collinear points of the restricted three-body problem. Phys. D, 132(1-2):189-213, 1999. [U29] Jorba, A, Ramfrez-Ros, R. and Villanueva, J.: Effective reducibility of quasiperiodic linear equations close to constant coefficients. SI AM J. Math. Anal., 28(1):178-188, 1997. [U30] Jorba A. and Simo C : On the Reducibility of Linear Differential Equations with Quasi-periodic Coefficients. J. Diff. Eq., 98:111-124, 1992. [U31] Jorba A. and Simo C : Effective stability for periodically perturbed Hamiltonian systems. In I. Seimenis, editor, Proceedings of the NATO-ARW Integrability and chaos in Hamiltonian Systems, Torun, Poland, 1993, 245-252, Plenum Pub. Co., New York, 1994. [U32] Jorba A. and Simo C : On quasi-periodic perturbations of elliptic equilibrium points, SI AM J. of Math. Anal. 27:1704-1737(1996). [U33] Jorba, A and Villanueva, J.: On the Normal Behavior of Partially-Elliptic Lower Dimensional Tori of Hamiltonian Systems. Nonlinearity, 10:783-822, 1997. [U34] Jorba, A and Villanueva, J.: On the Persistence of Lower Dimensional Invariant Tori under Quasi-periodic Perturbations. J. Nonlinear Sci., 7:427-473, 1997. [U35] Jorba, A and Villanueva, J.: Numerical Computation of Normal Forms around Some Periodic Orbits of the Restricted Three-Body Problem. Physica D 114:197229, 1998. [U36] Llibre, J., Martinez, R. and Simo, C : Transversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the Restricted ThreeBody Problem. J. Diff. Eq., 58:104-156, 1985. [U37] Simo, C : Estimates of the error in normal forms of Hamiltonian systems. Applications to effective stability. Examples. In A.E. Roy, editor, Long-term dynamical behavior of natural and artificial N-body systems, 481-503, Kluwer Academic Publishers, Dordrecht, Holland, 1988. [U38] Simo, C : Estabilitat de sistemes hamiltonians, Memories de la Reial Academia de Ciencias i Arts de Barcelona 48:303-348, 1989. [U39] Simo, C : Analytical and numerical computation of invariant manifolds. In D. Benest et C. Froeschle, editors, Modern methods in celestial mechanics, 285-330, Editions Frontieres, 1990. [U40] Simo, C : Averaging under fast quasi-periodic forcing. In I. Seimenis, editor, Proceedings of the NATO-ARW Integrable and chaotic behavior in Hamiltonian Systems, Torun, Poland, 1993, 13-34, Plenum Pub. Co., New York, 1994.

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[U41] Simo, C : The use of time as a complex variable, Memories de la Reial Academia de Ciencias i Arts de Barcelona 55(5): 276-290, 1996. [U42] Simo, C : Effective Computations in Hamiltonian Dynamics. In Cent ans apres les Methodes Nouvelles de H. Poincare, Societe Mathematique de France, 1-23, 1996. [U43] Simo, C : Effective Computations in Celestial Mechanics and Astrodynamics. In Modern Methods of Analytical Mechanics and their Applications, Ed. V. V. Rumyantsev and A. V. Karapetyan, CISM Courses and Lectures Vol. 387:55-102, Springer, 1998. [U44] Simo, C : Invariant Curves of Perturbations of Nontwist Integrable Area Preserving Maps. Regular and Chaotic Dynamics 3:180-195, 1998. [U45] Simo, C. editor: Hamiltonian systems with three or more degrees of freedom, volume 533 of NATO ASI Ser. C: Math. Phys. Sci. KluwerAcad. Publ., Dordrecht, Holland, 1999. [U46] Simo, C , Gomez, G., Jorba, A. and Masdemont, J.: The Bicircular Model near the Triangular Libration Points if the RTBP. In From Newton to Chaos: Modern techniques for Understanding and Coping with Chaos in N-Body Dynamical Systems, Editor A.E.Roy, Plenum Press 1995, 343-370. [U47] C. Simo y T. Stuchi: Central Stable/Unstable Manifolds and the destruction of KAM tori in the planar Hill problem. Physica D 140:1-32, 2000.

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