Физика дисков: Монография

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Author: Морозов А.Г. | Хоперсков А.В.

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b dpf+i

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b dpffi

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z

w

z Z

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u

b dpf

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b I\<X &Y^joVYTZCc>T7]nT

XQ^T.X'RQ`\<`&`YTZ5

Φ0 `UW.X'YnX&]_XC[Z_a[<`<] c1q `'RQ`\<`oac`1ekU.`^.XQ^ y X −2 [<`o TVU.`[<`1eb ρg0 (z) = ρg0 (0)ch {z/h} YnX h = cs / 2 π G ρg0 (0) p#[Z X(m Y RZ']X'YbXC[ [<` σgo = 2 hρg0 (0) T c7R

X

`[Z'b7[Z&[\Lb

[Si

U.`[<XQ^ TR"]

s

kg h = 1.

X(m

b dpcc+i

m

C[<`&`,Z&``[^.`$wXQ^T7_\TY^.` [<`aUWT.m<]nT XQ^T.XwmXQZCc1`^.X ^.`[<`^c1`&` m YTZCc7R,Y]_ TC b XQ^T7_ c1`<]]XQc.[T\^j+fUW.`$.XQZCZ&`\G\>TQ`<]nTW.`\R<^^j+f RQ`\j+f m YTZCc7RIf`c7RI j\RLXC[Z_s^.XQUWTeXQ^Tekjel\,`cW.XQZ&[^.`Z&[T\<`<]n^.`\<`&` TZ']R k = kg X Z&``[\<XC[Z&[\Lb ys XQ&`,^R\jU.`<]n^.XQ^TTPb1Z']`\T7_ b dp¢wwi ρg0 b dpc3dSi kg h ' 1. ρ∗0

[LR
X

XQZ']nTU7]`[^.`Z&[ +I\<X"&Y^.`oc1`1ekU.`^.XQ^.[j

U7]`[^.`Z&[ u'RQ`\<`o c1`WW.XQc.[^.`yZ

m

T.[LRL[

X

[<`Y]_

TZCZ']X'Yn`\R<^T7_b1Z&[<`o

XQ&`mXQZCc1`^.X

Yn`U.`<]n^T.[<X&] ^.`&`b1Z']`\T7_

X

I^R

m

m

m

T.[<X&] ^.`UW.XQ\<`Zf1`QYT.[

T\<`Z&[T'RQ`\<`&`YTZCcR

^.`}[<`^cTeUWTt\jU.`<]n^.XQ^TTt^.XQc1`[<`W.`&`

c>\j\<`QYbPc1`[<`W.`&`VekjU.XQW.X'f1`QYTerp

7k#"!"!Q)

C#L$I"!)+( -

X

m

X

^.`Xb1Z']`\T.X Yn`<] ^.` ` XQ\TY^.` \<`I^Tc^7b7[ %Tb1Z']` s \T7_aUW.XQ^.XCmW.X XQ^T7_a\c7]RIYn`1eI \<X"& Y^.`&`SYTZCc7Rx\w\<`"eb XQ^^joVCWR<\T.[LRT `^^joU.`[<XQ^ TR"]pjd]_>`$. XQ^cT.C [<`&`,\c7]RIY RTZCU.`<] C bXIet\jWR XQ^T7_PY]_ s \<`"eb XQ^^.`oiU.`\<XQWf^.`Z&[^.`otU7]`[^.`Z&[Tg\}WRekc7RIfUW.`Z&[<XQo B Tf`QY^.`W.`QY ^j+fe`QYnX&]XQo>'RQ `\<`&`T I \<X"& Y^.`&`wYTZCc1`\ +[<`Yn`U.`<]n^T.[<X&]

σg1 = σg0 k 2 Φ1 (ω 2 − 4Ω2 − k 2 c2s )−1 ,

b dpc+¡i

σ∗1 = σ∗0 k 2 Φ1 (ω 2 − 4Ω2 − k 2 c2r )−1 I∗ ,

I\<X"&Y^.`&` Z

m

YTZCc7R

k∆∗ < 1

UWT m

[<``^

m

"ReXC[T.[ X

i

Z

T.[LRLXIe[<`^cTerp ]X'YbXC[

u

X m ys m y s V%`WerVdRc1`^.X ^7b [<`<] T^7b X b ZQerpU po7 w p¢7 w pc+i R'R Q`\jotYTZCcu\iZ&``[\<XC[Z&[\TTZ b d pc3dSi

I∗ = (1 + k ∆∗ )−1

YnX

b dpc

[<`

X

^.XQU.`ZCW.X'YZ&[\<XQ^^jeu\j

Y RLXC[s\<XQW^7b

y

I∗

TZCU.`<] Q`\R"]nZ_

k∆∗ 1

b WR
f1`[Q_\]R<\<X0w~V%`WerVdR
TZ']XQ^T.XIee`

R

^.`Pb7mX'YT.[ Z_}\G[<`1e

X

C[<`&`VUW.X'YnX&]R,^.XC`mQf1`CYTe`aU.`[<`1eb [<`Vb1Z']`\T.X b dpc3dSie` XC[,mjr[ a\c7] X X m XQ^.`acRZ']X'YbXC[T b dpc+¡i b dpc i TeXCXC[\TY σ∗0 σg0 z . b dpc i 2 2 cs cr \<X'Y_>c1` VWVST T.XQ^.[aR<^TQ `[W.`UTTI \<X"& Y^.`&`BYTZCcR ZSU.`1e`

s

eXC[

X

`

m

y

b dpc#fiU.`<]b

`a\<[<`W.`1elZ']b XQ\TY^.`

\TY

RLX

α = cr /cz ,

b dpc|+i

h ρg0 (0) 1. α 2 ∆∗ ρ∗0 (0)

b dpedx+i

b

Te

k∆∗ 1 I∗ ' 1/k∆∗ irb1Z']`\T.X σg1 σ∗1 ρ2g0 (0) h2 2 1. α2 ∆2∗ ρ∗0 (0)

/I\<X"&Y^j+fYTZCc7RIfxR"]R
`ZCc1`<] c7bS\ m

R<\<[<`1ehRL[T

X

m

m

m

^.`

UWT

b dped(fi

α > 1X

^.XQWR<\<XQ^Z&[\R

b dped(fi

7 #Lk-# &#L$I"!'"( -#

ys

yUWT\<X'YnXQ^^j+f\j1wX}`$.XQ^.`c\jr[<XQc7RLXC[gZ']X'Yb m

XC`mQf1`QYTekjeTxYn`Z&[LRL[<`

XCX}b7[\<XQW

^jesb1Z']`\T.XIesUWTeXQ^Te`Z&[TUWT.m<]nT

c1`&`,YTZCc7RY]_}TZCZ']X'Yn`\R<^T7_sCWR<\T.[LRT.`^^.`>^.X&b1Z&[<`o

m

YnXQ^T.Xp

XQ^T7_[<`^ s

T\j+f\<`"eb

XQ^To

\G'RQ`\j+fsU.`QYZCTZ&[<XIehRIf'R"]R
c7R z

X

bIYn`\L]XC[\<`W_

ys

b dpedx+i

TekTP^.XQWR<\<XQ^Z&[\Re

XQZ&[\<XPUWTeXQWRWR
+`<]n^ R σg0 /σ∗0 ' 0,1 cz /cs z y | # p [Z Y R ρg0 (0)/ρ∗0 (0) '

~RCbeXCXC[Z_

`cW.XQZ&[^.`Z&[T X.u X(z X

b dpedx+i+\jU.`<]n^7_

f X

y

cc

b1Z']` [Z_vp

k kg iS\<`" m eb XQ^To\'RQ`\<`1e YTZCcXU7]`ZCc1`o R"]R

§q§

®¬'³®¹Y¬

701054(0

%:M'JC8";.:<=<918

"RIY R

m

UWTTCb

m

b dped(fip

' ∆∗ /h ' 3 α = cr /cz ' 2 cc3d X 0,3 h/α2 ∆∗ < Z']X'Yn`\RL[<X&] ^.` ∼ 0,1 T

\T7_UWTeXQ^Te`Z&[Te`CYnX&]nTP[`^c`&`YTZCc7R\V`WeX X

X

x³

b

XQ^TTyZC\<`oZ&[\c1`W.`[c1`\<`<]n^.`\j+f

U°x®

M²¸

r®

¹ ´#®

¹Y¬

C8 8CN.D

Z&``[\<XC[Z&[\TTZ>UW.`\<X'YnXQ^^je\U pdpfpcTZCZ']X'Yn`\R<^T.XIe bTCb

m

XQ^T7_YTZCU.XQWZCT.`^^j+fgZC\<`oZ&[\

^Toi\U7]`ZCc`Z&[Ti[<`^c`&`

RQ`\<`&`>YTZCc7R

'RQ`QYT^RekTcT>\0C[`oe`QYnX&]nT>Zb

m

XC[<`1e

b dp

^.XC`Z&XQZCTeheXC[WT

i

fc3d X

X

|3dp

b dpf+i

X

m

U.`Z&[LR<\Te

^j+ft\<`"eb

s

X

Zf1`QY^jrXVb.WR<\^.XQ^T7_

b dpf¡iTeX

∂σ ∂(rσu) ∂(σv) + + = 0, ∂t r∂r r∂ϕ ∂u v ∂u v 2 ∂Φ 1 ∂p p d ln Ωz ∂u +u + − =− − −D , ∂t ∂r r ∂ϕ r ∂r σ ∂r σ dr ∂v ∂v v ∂v uv 1 ∂Φ 1 ∂p +u + + =− − , ∂t ∂r r ∂ϕ r r ∂ϕ σr ∂ϕ

y

[,\TY

b dpedSwi

b dpedxc+i

b dpeddSi

∂p ∂p ∂ru ∂v ∂p d ln(CΩz ) +u +v + γs p + = (γs − 1)pu , b dpedS¡i ∂t r∂r r∂ϕ ∂r r∂ϕ dr X X YnX u v Z WRIYTR"] ^R"_yTyRITeb7[LR"] ^R"_yc1`1ekU.`^.XQ^.[j ZCc1`W.`Z&[T'R"R σ p Z U.`\<XQWf^.`Z&[^jrXU7]`[^.`Z&[ yTY R<\L]XQ^T.X\ 'RQ`\<`1eYTZCc1X X RYTZCZCTUR# !

-

!$%" 2

8 "$ , %

& $%-(%" &&9C&+$%"% +" &&

" &

& % *" . &&9C 8 "" & 8 .&9C&/"

7. #Lk-#B!L$C m

[T\^jrX s

s

]XQ^ju`U7b

XQ^~dpedSip

jd]_T Cb

m

P!w . -Q,( -#

XQ^j

b TZCZ']X'Yn`\R<^T

y

YTZCZCTURL[T\^j+f

XQ^T7_YT^RekTcTekR"]nj+ft\<`"eb

s s

VWV%XQc.[<`\U.`ZC\L_

XQ^Toy]nT^.XIR<WTCbXIeZCTZ&[<XIeb

C[`&`UW.X'YZ&["R<\Tet\"f1`QY_ T.XS\0C[LbPZCTZ&[<XIebGU.XQW.XIeXQ^ s m ^jrXx\\TYnXxZ'beheWR<\^.`\<XQZC^j+f>T>\<`" eb XQ^^j+fP\<X&]nT T^ ÷

b dpedSwi

b dpedS¡ipjd]_

σ = σ0 (r) + σ1 (r, ϕ, t),

u = u1 (r, ϕ, t),

p = p0 (r) + p1 (r, ϕ, t),

m

W.X"Cb"] I[LRL[<XwU.`<]b

v = rΩ(r) + v1 (r, ϕ, t),

Φ = Φ0 (r) + Φ1 (r, ϕ, t).

b dped

u i

Te

∂u1 u1 ∂σ1 ∂v1 ∂σ1 + u1 σ00 = 0 , +Ω + σ0 + + ∂t ∂ϕ ∂r r r∂ϕ !

b dped

i

∂u1 ∂u1 ∂Φ1 1 ∂p1 σ1 p1 z b dped +Ω − 2Ωv1 = − − − g − D(ln Ωz )0 , i ∂t ∂ϕ ∂r σ0 ∂r σ0 σ0 ∂v1 ∂e2 1 ∂Φ1 1 ∂p1 ∂v1 b dped1|+i +Ω + u1 = − − , ∂t ∂ϕ 2Ω r ∂ϕ σ0 r ∂ϕ ! ∂p1 ∂v1 ∂u1 u1 ∂p1 +Ω + γ s p0 + + + u1 [p00 − (γs − 1)p0 (ln CΩz )0 ] = 0, ∂t ∂ϕ ∂r r r∂ϕ

w[WTfP`I^R

YnX

m

RLXC[UW.`TI\<`CY^7b

y

\"Yn`<] WRIYTR"] ^.`o>c``WYT^RL[j b dpffi

`[\<XC[Z&[\TTZb1Z']`\T.XIeWR<\^.`\<XQZCT7_

X

X

b dp¢¡+i

TP\KZ&`

p0 dΦ0 p00 − r Ω2 . g = − − D (ln Ωz )0 = σ0 σ0 dr jx`U.`<]n^Te

b dped

ZCTZ&[<XIeb

bR

b w7p¢¡c+iG]nT^.XIR<WT Q`\R<^^jeb.WR<\^.XQ^T.XIe

∆Φ1 = 4πGσ1 δ(z). s

R<\TZCTe`Z&[ \<`"eb

XQ^^j+f>\<X&]nT

T>`QY^.` .`QY^.`Z&[

y

m

b dp¢¡wi

T^

f1 = {σ1 , p1 , u1 , ...}

R
b dp¢¡c+i y

t \%ZC\L_ITKZ&`SZ&["RT.`^R ^.`Z&[ T\RLXIe`oR<\^.`\<XQZC^.`oe`QYnX&]nT>\VRITeb7[LR"] ^.`1e

`[dRITeb7[LR"] ^.`oc`` YT^RL[j

÷ i

b dp¢¡fi

ϕ

T\ .XIeXQ^T

^R

f1 = f (r) exp{−iωt + imϕ} . `Y RGZCTZ&[<XIehR ^j+f>\<X&]nT

m

b dped

i

÷b

dp¢¡+i m

T^>`U7b1ZCc7RLXIeT>Z

X

b dp¢¡wiU.X .XQoYnXC[>\

T.[LRLXIe

C = const i

b T^YnXQcZ

Of

−iˆ ω Σ/c2T + (u0 + u/r + imv/r) + u/Lσ = 0,

b dp¢¡:dSi

b}\<`"eb

−iˆ ω u − 2Ωv = −R0 − Φ0 − (1/Lσ + D/Lz )R + (1/Lp + D/Lz )Σ,

s

XQ^

b dp¢¡¡i b dp¢¡

u i

7 #Lk-# &#L$I"!'"( -#

−iˆ ω v + ∂e2 u/2Ω = −imR/r − imΦ/r, −iˆ ω (R − γs Σ) +

c2T S00 u

b dp¢¡

= 0,

b dp¢¡

i

z i

Φ0 m 2 Φ ∂ 2 Φ − 2 + 2 = 4 π G σ δ(z), Φ + b dp¢¡<|+i r r ∂z X X 2 ˆ = ω − mΩ X 1/L = σ00 /σ0 X 1/Lp = p00 /p0 X YnX Σ = cT σ1 /σ0 R = p1 /σ0 ω qσ 1/Lz = Ω0z /Ωz X S0 = ln(p0 Ωz1−γs /σ0γs ) X cT = p0 /σ0 Z TQ`[<X ekT m XQZCc7R"_ √ X cs = γs cT Z RIYTRLm1RL[T m XQZCcR"_ZCc` .`Z&[ I\Lb.cR.p ZCc1` .`Z&[ 0I\Lb.c7R ym s ZCc7] TeT UT\<X'YnXQ^^.`o\j1w XZCTZ&[<XIekj\<`" eb XQ^^jrXZCc` .`Z&[T p jd]_C[<`&`,T b dp¢¡ i+^RIf1`QYTe 00

m i∂e2 u + (R + Φ) v=− 2Ω ω ˆ rω ˆ X

T

U.`QYZ&["R<\L]_._

u

X

u

b dp

+i

m

C[<`V\j R XQ^T.Xx\ b dp¢¡ i U.`<]b Te 2mΩ i 0 (R + Φ) + ω ˆ [(ln p0 ΩD { u= 2 z )Σ − 2 ω ˆ − ∂e r

jwTVWV%X .XQ^ T 7b_

"RL[<XIe

.X"Cb"] I["RL[V\eXQZ&[XdZ

b dp

0 0 0 − (ln σ0 ΩD z ) R − R − Φ ]} .

u

+i

b dp X

σ0 r (R + Φ) + ln 2 ω ˆ − ∂e2 00

u

fiU.` u b dp f+ i \

!0

u

b dp

fi

RIYTR"] ^.`olc1`` YT^RL[<X}TlU.`QYZ&["R<\L]_._ X b dp¢¡¡i U T\<`QYTe b dp¢¡¡ic>\TYb 0

0

(R + Φ)0 + ln σ0 ΩD R0 − ln p0 ΩD Σ0 − z z

σ0 r 1 2mΩ 0 2mΩ + ln 2 − (R + Φ)  2 + r ω ˆ r rω ˆ ω ˆ − ∂e2  m2

!0  

!

00

+R ln σ0 ΩD z + 

2 ω ˆ − ∂e2 2mΩ D 0 +Σ + ln σ0 Ωz − rω ˆ c2T !0 2mΩ σ0 r D 0 D 00 D 0 ln p0 Ωz − − ln p0 Ωz − ln 2 ln p0 Ωz = 0. b dp u wi 2 ω ˆ − ∂e rω ˆ u z X m X `QYZ&[LR<\L]_._s[LR
σ0 r + ln 2 ω ˆ − ∂e2

S00

h

0

0

!0

ω ˆ R +Φ +

0 ln σ0 ΩD z

0 (ln σ0 ΩD z )R

−

0 (ln p0 ΩD z )Σ

i

2mΩ − (R + Φ) + r

ω ˆ 2 − ∂e2 (R − γs Σ) = 0 . +ω ˆ c2T TZ&[<XIehRb.WR<\^.XQ^To

b dp

u

wi

X

b dp

u

c+i

X

\eXQZ&[<XyZ}b.WR<\^.XQ^T.XIe

b dp

u

c+i

bR
" eb XQ^To\ge`QYnX&]nT'RQ`\<`&`yYTZCc7RZ}UW.`TI\<`<] ^jekTWR
X

7. #Lk-#B!L$C

7010{0wDM.AC?M'D.:<;.;.:
&;.J"?.:

?.8"=<;.AC;.D.Ax=KD m

~R

C[`oe`QYnX&]nT>ZC\L_#eX

y

XQZCc7b

Lc

Yb

d ln cs Lc = dr

.D.N.AQM'91:

e`QYnX&] p

^.XQo

!−1

2Lσ , (γs − 1)(1 + Lσ /Lz )

=

u

dSi

X

X

m

u

¡i

T.[LRL[

γ ≤ 5/3 [<`Y R Lc > Lσ pq[Z y Y R

`m<]R
}Z

X

b dp

b ZQerpU pdpfpfipkZ']nTiZ

Z

u

b dp

`UW.X'YnX&]_XC[Z_>Z&``[^.`$wXQ^T.XIe

Lσ T

U7]`ZCcTo U.`cR"RL[<X&] U.`<]nT.[W.`Uj Y]_tZCTZ&[<XIekj ehR
YnX

DM'9A

p0 = const . σ0γs Ωzγs −1

S0 = ln

P!w . -Q,( -#

γs ≤ 3/2 T Z']X'Yn`\RL[<X&] ^.` m y _ZC^.` [`KT Q^.[W.`UT XQZCc7R"_Pe`QYnX&] w ^.XU.`[T\<` .X T.[wY R<^^jei^RLm<] YnXQ^To y m U.`S'RQ`\<`1ebSYTZCc7bÈR"]RY R"] m s ^.XQo BToaR<^R"]nT\xWRekc7RIfVT Q ^.[W.`UT XQZCc1`oae`CYnX&]nTVZ D = 1 b m`<]XCX`m To \

b dpf m

Z&``[\<XC[Z&[\TTZ Xxm

Z']b

m

i

R^T

m

T Q ^.[W.`UT

X:ip

XQZCc`oe`QYnX&]nT>T

b dp

u

c+i+\jr[<XQc7RLXC[

R = γs Σ .

#KUWT\<`QYT.[b.WR<\^.XQ^T.X

b dp

+[LRZC\L_

σ0 r (Φ + γs Σ) + ln 2 ω ˆ − ∂e2 00

!0

  m2

(^R

m

1 2mΩ × 2 + r ω ˆ r

(Φ + γs Σ)0 +

!0

X

m

m

\

R
X

Z&[TC^7b7[sZ,U.`1e` Z']`\L]XQ^.`[XIe

s

y

q

2mΩ σ0 r + ln 2 rω ˆ ω ˆ − ∂e2

c1`u\<X&]nTcTU.`uZCWR<\^.XQ^T UWT.m<]nT

= 0.



Xm

[<`UW.`[Q_

ys

u+ i

XQ&`y'R#

m

^j+fi\<`"eb

s

XQ^Toymj]Yn`

ZyXQ&`[<`<]

f (r) ∝ exp(i kr r)

XQ^^.`Z&[ YTZCcRlTf7R<WR
s

T^.`o

Xm

[<`^R<W_7Yb

Zy\jU.`<]n^.XQ^T.XIe

Tlb1Z']`\T7_UWTeXQ^Te`Z&[T

q

ZZQerp b dp¢¡:dSi

kr r 1,

kr L 1,

b dp

uz i

L = min{|Lc |, |Lσ |, |Lz |, |∂ ln Ω/∂r|−1 } p X X XGm XC[W7bIY^.` `QY^R

\GWRIYTR"] ^.`1e^R
b dp¢wwiae`'b7[imjr[ \jU.`<]n^.XQ^j

XQ^T7_

b dp

\`UW.X'YnX&]XQ^TTlb1Z']`\T7_XQ&`CWR<\T.["RT.`^

^jrXehR
!0  

R<^R"]nT"RP\WRIYTR"] ^.`1e^R
`mZ&[<`I_[<X&] Z&[\<`1e y

i

ω ˆ 2 − ∂e2 Σ − (Φ + γs Σ) × c2T

T\<`Z&[Ti`[^.`ZCT.[<X&] ^.`}`Z&XQZCTeheXC[WT

uu

wi+c>\TYb

T.[<X&] ^joUW.`CW.XQZCZP\U.`^TehR<^TT¤VSTITcTgCWR<\T.[TW7b

Q`\<`&`yYTZCcRiT ^.`oyb1Z&[<`o

u

b dp

7 #Lk-# &#L$I"!'"( -# u+

b dp

X i

Y]_c1`[<`Wj+f\jU.`<]n^7_XC[Z_Pb1Z']`\T.X

q m |ˆ ω| k r = r2 kr2 + m2 , ∂e s

\`m X

Q`1e `[Q]nT

m

TCb

X

Xrm

u

|+i

|ˆ ω | ∂e p R\a^TI c1` R
RLX>^.XPm`<]XCX

Z&[TZCU.XQc.[WR.pg+[<`s`CWR<^T m

b dp

m

XQ^T.X

XQ^T7_}CWR<\T.["RT.`^^.`o

X

`QY^R
^.X&b1Z&[<`o

m

XIeb1Z']`\T.X

X

X

^.X,_\L]_XC[Z_

s

Z'b

XQZ&[\<XQ^^jeuY]_

T\<`Z&[T}'RQ`\<`&`YTZCc7R.pGjxXQoZ&[\T.[<X&] m

T\<`Z&[TKRQ`\<`&`rYTZCc7R`[^.`ZCT.[<X&] ^.``Z&XQZCTeheXC[WT ^j+f s 2 ω ˆ = 0 \sekT^TebeXYTZCU.XQWZCT.`^ \<`"eb XQ^To `UW.X'YnX&]_XC[Z_ T Vb1Z']`\T7_ 2 2 ˆ =ω ˆ (k) ( ^.`oscWT\<`o ω p jd]_sW.X wXQ^T7_s\<`UW.`ZQR`ma`UW.X'YnX&]XQ^TTsCWR<^ T j ^.`

CWR<^T Rb1Z&[<`o

XQZ&[c1`X

b1Z&[<`o

m

Xm

YbXC[}TZ f1`QYT.[ }TP[<`&` s

m

T\<`Z&[TYTZCc7RG`[^.`ZCT.[<X&] ^.`s^.XC`Z&XQZCTeheXC[WT [<`

\]

y

s

XQ^To}Z']X

m`o^.XC`QY^.`W.`QY^.`oiZCTZ&[<XIeX,Yn`<]

XQZ&[\<`\RL[ tc1`W.`[c1`\<`<]n^.`\jrXCWRIYT.XQ^.[^jXy\<`"eb

m

\<`"eb

^j+f

m

X

s

X

XQ^T7_

^jZ'b

ehR
ωg ∼ ω0 /k L ω0 YnX ω0 Z R

Y

T^Z&`\ZCc1`oTCWRIYT.XQ^.[^.`o\<XC[\<XQo

ω ˆ J (k) i m ^TI c1`

X

b ekT^TebehRyYTZCU.XQWZCT.`^^.`ocWT\<`o X

ω ˆJ ∼ ω ˆ g ∼ ∂e/k L ∂e pq%W.`1eX[<`&` YTZCU.XQWZCT.`^^jrX%ZC\<`oZ&[\R s R
ZCTekjoiT^.[<XQW.XQZ

b ZQerp

X

^R
X

U7b.^c.[j

CWRIYT.XQ^.[^.` Q ^.[W.`UTo^`o^.X&bZ&[`o

m

X

U.`ZC\L_

T\<`Z&[TGip

bIYnXIeU.` C[<`1ebWR
\<`"eb

s

^jrXa\<`<]n^.`\jrXa^.`1eXQWR>c1`[<`Wj+fbIYn`\L]XC[\<`W_

C[<`&`b1Z']`\T7_

U.`<]n^.XQ^TT

WRIYTR"] ^.`oc1``WYT^RL[<XK\ y

m

m

]XQ^j b dp

u+

s

XQ^T7_ y

XQ^^jrXG\<`<]n^Re~`ZCZ&mTT Xm

TgRITeb7[LR"]

R
[b1Z']`\T

y

b dp

u

ZVU.XQW\jekTUW.`TI\<`QY^jekTy`[

i`c7R Ij\R X

y

[Z_UW.XQ^.XCmW.X

|+i'p%WT\j

Φ T

σ

U.`

Te`GehR"]njekTU.`

u+

ZU.XQW\je ]XQ^.`1erp%` C[<`1eb `[<mWR

ZCU.`<] Cb_

"RL[<XIec1`W.`[c1`\<`<]n^.`\<`XW.X wXQ^T.X+b.WR<\^.XQ^T7_|bR

7. #Lk-#B!L$C

c1`&`BYTZCc7R

f

u c

X

P!w . -Q,( -#

dSw

Φ=−

2πGσ , k

b dp y

UWTf1`QYTeyc,TZCc1`1e`1ebBYTZCU.XQWZCT.`^^.`1ebKb.WR<\^.XQ^T

Y]_aT Q ^.[W.`UT

'RQ`\<`&`wYTZCc7R

m

fi

XQZCc1`&`

r dΩ2 ω ˆ −ω ˆ ∂e + − 2 π G σ0 k 1 − 2s 2 ∂e dr ! 2 Ωσ d 0 = 0, −2 s Ω (k c2s − 2 π G σ0 ) ln b dp wi 2 dr ∂e X ys m YnX s = m/k r Z UR<WReXC[W f7R<WR

2

k 2 c2s

2

σ0 = σ0 (0) (1 − r 2 /R2 )1/2

b dp

c+i

T>U.`<]nT.[W.`U^jetb.WR<\^.XQ^T.XIetZ&`Z&[<`I_^T7_

p0 = p0 (0) (1 − r 2 /R2 )3/2 ∝ σ03 ,

b dp

dSi

σ0 (0) = 2 R Ω20 /π 2 G p0 (0) = σ0 (0) Ω20 R2 (1 − β 2 )/3 p C[<`oge`QYnX m ]nTmX"K TZCU.`<] Q`\R<^T7_ q UWT.m<]nT XQ^T7_mj+]`PU.`<]b XQ^.`,YTZCU.XQWZCT.`^^.`X X ys s b.WR<\^.XQ^T.X `UTZCj\R XCXBZC\<`oZ&[\RaUW.`TI \<`<] ^j+f\<`"eb XQ^To\GU7]`ZCc1` YnX

Z&[TGYTZCcR

ω ˆ (ˆ ω 2 −4 Ω2 )+{ˆ ω [l (l+1)−m2 ]−2 m Ω} {Ω20 (1−β 2 )−4 Ω20 γlm } = 0,

b dp

¡i

YnX

(l + m)! (l − m)! u b dp ! ! #2 . i l + m l − m 22l+1 ! ! 2 2 X m X b dp dSi[LR
"

q

4 Ω20 γlm = 2 π G σ0 / (l2 − m2 ) (R2 − r2 ) .

jd]_}`UW.X'YnX&]XQ^T7_yR<^R"]`'RP\<`<]n^.`\<`&` m u X [<` f c dSw

Φ = −4 γlm R Ω20 σ

√

m

TZ']R

k =

q

q

kr2 + m2 /r2 "ReX

m

RLXIe

R2 − r2 /σ0 (0) ' −2 π G σ (R2 − r2 )/(l2 − m2 ).

X

WR<\^T\R"_

7 #Lk-# &#L$I"!'"( -#

C[<`G\jWR

UWTf1`QYTegc>\j\<`QYb

X(m

W.X wXQ^T.XIeb.WR<\^.XQ^T7_bR
XQ^T.XZ [<`

qq

b dp

fi

X

d ln(σ0 )/k 2 dr = −r/k 2 (R2 − r2 ) = −r/(l2 − m2 ),

T>U.` C[<`1ebab.WR<\^.XQ^T.X

b dp

¡ire`

XC[mjr[ "R
d ln(σ0 ) 2mΩ (k c2s −2 π G σ0 ) = 0, b dp i kr dr ys b dp wiw\}UW.X'YnX&]X Ω = ` ] p [<` YnXQZ&[\<XQ^^.`yZ&`\URIY R XIeZab.WR<\^.XQ^T.XIe +[<`[zVSRc`WW.XQc.[^.`Z&[T u X UWT.m<]nT XQ^T7_ b dp |+i TZCU.`<] Q `\R<^^.`&`Y]_ q R<^R"]nT" RKYTZCU.XQWZCT.`^^j+f s Xm X ZC\<`oZ&[\\<`"eb XQ^ToS\&`WR& Yn`m`<]XCXhZ']` ^.`o XIe b dp dSi e`QYnX&]nTS'RQ `\<`&` YTZCc7RZSUW.`TI\<`<] ^jekTWR
70103/{0+:LH;

:M&M
jwTZCU.XQWZCT.`^^.`Xb.WR<\^.XQ^T.X b dp wih`UTZCj\RLXC[w[WTP\<XC[\TGc1`<]XCm1R<^Toa'R m y Q`\<`&`YTZCcR.pk Z']nTUW.XQ^.XCmW.X s^.XC`QY^.`W.`QY^.`Z&[ YTZCc7RsTYTVWV%XQW.XQ^ T y s X X·m

#

R"] ^.`Z&[ XQ&`P\WR XQ^T7_ [<`>^.XC[W7bIY^.`Gb7mX'YT.[$Z_ [`aY\<XT,^Tf}CWR<\T m y [LRT.`^^jrX b Y T^Z&`\ZCcT.X:ihTVTf R]nT^.XQo^.`&`GU.` ω m ˆ = 0 ip6`Q_\L]XQ^T.X>[W.XC[ <XQ&`[TUR\<`"eb s XQ^TotZC\L_"R<^.` Z^.XC`CY Z']b RLX ω y y s ^.`W.`QY^.`Z&[ YTZCcRaTYTVWV%XQW.XQ^ TR"] ^.`Z&[ XQ&`G\WR XQ^T7_ b UW.`I_\L]XQ^T.X X m ZY\T&`\<`oPb.UW7b.&`Z&[T^.XC`QY^.`W.`CY^.`osZCW.X'Yj0i TTf RZC\<`m`QY^.`&`\ b dp wi m ]XQ^.`\ fc3d b1Z']`\T7_Pm1R"]R<^ZQRB]nT^.XQo^.`&`,U.` ω 2 m Ω (k c2s − 2 π G σ0 ) d ln(2 Ω σ0 /∂e2 )/dr \<X&]nT T^ TeX [VeXQZ&[<`V\,RL[e`Z)V%XQWRIfT>`c1XIR<^ RIf u#X U7]R<^.XC[ b \^7b7[W.XQ^^T.XGCWR<\T.[LR T.`^^jrXG\< `<]n^j T\<`<]n^j~`ZCZ&mT>Z ZQerp w X X |+w#À i U7]R" eX b YW.XQoV%`\jrXB\<`<]n^j Z ZQerp ffw#À i I \<X"& Y^.`1eYTZCc1X b ZQerpn]p wiT Z']RLm`GYTVWV%XQW.XQ^ TR"] ^je b n 1 Y]_

7 x n L

L!QC#L$I"!'"K

#"!Q

ω ˆg = ω ˆ R gal = kϕ = m/r p

YnX

"S( -.#

2 kϕ dΩ/dr , kr2 + kϕ2 + 4 Ω2 /c2s

b dp

|+i

RL[e`Z)V%XQWRIfsU7]R<^.XC["R
^j+f>\<`"eb

XQ^To~v`ZCZ&mT

q`^cW.XC[^jrX &XC`1eXC[WT

y

[TcYn`<]

u

TeXCXC[\TY 2 kϕ (dΩz /R dθ) . ω ˆR = − 2 kθ + kϕ2 + 4 Ω2z /c2s w

XQ^T7_UR<WReXC[W.`\

X

X

^j

`

m

m

XQ\TY^.`

X

XQZCcTegZ&`<]nT.[<`^Rel~`ZCZ&mT X

\j

m

z

b dp ys

f7R<WR
b.U.`1e_^7b7[j+f\j1wXP\TfW.XQ\j+fZ&[W7b.c.[Lb.W

[LR<W^jeR<^.[T Tc7]`^T

m

I^R

X

+i

TfuYT^RekTc7bT

R<^R"]`CT

m

^j+fU7]R<^.X

\'RQ`\j+fVYTZCc7RIf>'R"]R

TZ']_[ Z_\g^.X&]nT^.XQo^.`o[<XC`WTT p

ZC\L_ITZ

C[TeT^.[<XQW.XQZC^RU.`Ujr[c7R}UW_ne`&`}U.XQW.XQ^.`ZQRsW.X"CbI] I[LRL[<`\[<XC`WTTtZ&`<]nT.[<` m X ^.`\t~`ZCZ&mTu^R eX&]nc1`ou\<`QYnX ^RZ']b R
m

m

T.X YTZCU.XQWZCT.`^^j+fBZC\<`oZ&[\U7]R<^.XC[LR<W^j+fKTB'R"]RZCU.XQc.[WR k ∼ 1/ρ ip m c7 R "R<^^.`X,`mZ&[<`Q_[<X&

] Z&[\<`sU.`QY XQWcT\RLXC[^.XC`mQf1`QYTe`Z&[ \j_\L]XQ^T7_ m

ys

VWVXQc.[T\^.`&`>T^.XQZQRe`U.`QY R<\L]_ XQ&`Z_eX'f7R<^T"ehRV\<`QmTw\<`" e` ^j+fP[LR\U pdpc.p¢¡7p Yn`Z&[LRL[<`

^.`

§¦6§ ° ´+¯® ´#®!ª¬'ª Uª ¬G®´#®

XQZ ehRbeXQZ&[^.`-&YnXQZ d ^R
b.W.`\^.XB`mZ'b

X

U°Sª

r®

¹h¬

\dc1`[<`W.`1ey^Rdc7R

Y RLXC[Z_}eX'f7R<^T"elCWR<\T.["RT.`^^.`o}^.X&b1Z&[<`o

m

m

XQZ&[\<XQ^^.`1e

T\<`Z&[T'RQ`\<`&`

YTZCc7R.p

703/1054(0

MH:<=
?.8"=
jwT^RekTc7bGehR"]nj+fG\<` "eb

XC[BYTZCU.XQWZCT.`^^.`XSb.WR<\^.XQ^T.X

s

8 C:<=<:
M'J":
XQ^Toa\BT Q ^.[W.`UT b dp

wi

X

c1`[<`W.`Xwe`

m

ω ˆ 3 + p(k) ω ˆ + q(k) = 0.

XQZCc1`oPe`QYnX&]nTa`UW.X'YnX&]_ ^.`

"R
b dp

z

fi

7 #Lk-# &#L$I"!'"( -# m

T\<`Z&[T'RQ`\<`&`YTZCcR y `[^.`ZCT.[<X&] ^.`c1`W.`[c1`\<`<]n^.`\j+f\<`" eb XQ^ToyZC\<`QYT.[Z_c b1Z']`\T `[Z'b7[ Z&[\T7_c1`1ekU7]XQcZC^j+f>c1`W^.XQoGb C [<`&`b.WR<\^.XQ^T7_ ` C[<`1ebw\<`UW.`Z`mb1Z']`\TTKCWR<\T.[LR T.`^^.`oBb1Z&[<`o s

D(k) = (p/3)3 + (q/2)2 ≤ 0.

z

b dp

wi

m jd]_>Z&[LRLmT7]nT "R TT>'R Q`\<`&`wYTZCc7RB`[^.`ZCT.[<X&] ^.`,`Z&XQZCTeheXC[WT ^j+f s m X0m s 0 0 ∂ iG\<` "eb XQ^ToYn`Z&[LRL[<` ^.` [<`mj dd(f p m m y s

0

b m =

c = c = πGσ / e C[<`1e T\jekT`c7RI j\R [Z_s\<`"eb XQ^T7_Z k = k0 =

RLXKehR<WCT^R"] ^.`,b1Z&[<`o G σ0 /c20 X T>UW T C[<`1e

Z']b

π

ω 2 = ∂e2 + k02 c20 − 2 π G σ0 k0 = 0. m

j

m

TZ']_._>\<X&]nT

D(k0 )

T^7b

y

cs = c0

3



U.`

b dp

z

wi

X

U.`<]b

m

RLXIe

2

(Ω2 )0  ∂ ln(2 Ω σ0 /∂e2 )  D(k0 ) = −2 s2 r + s Ω ∂e2 ≥0 3 k0 ∂r 

Y]_V]

UWT

mj+fG^.X%\<`IWR
ys

y

z

b dp

c+i

TfGZW.`Z&[<`1eiWRIYTR"] ^.`oGc1``WYT^RL[j!Vb.^c To

Ω(r) p`ZCc1`<] c7bT^RLm<] YnXQ^Toa^.XZ']X'YbXC[wZ'b s XQZ&[\<`\R<^T7_V\xU7]`ZCcTfV'R# ym X X ]R 0 b "RTZCc7] XQ^T.XIe mjr[ Se` XC[ .XQ^.[WR"] m X z ^j+f RT b dp c+i\jr[<XQc7RLXC[P\R ^jo}W.X"CbI] I[LRL[ nY]_ s m Z&[LRLmT7]nT" R TT^.XC`Z&XQZCTeheXC[WT ^j+f b s 6= 0iS\<`" eb XQ^Toy\'RQ `\<`1euYTZCc1X Xm c0 X \<X&]nT m T^R cs p ^.XC`mQf1`QYTehRBm`<] K R"_ XIe m m m cs = c1 (s) X ^.XC`mQf1`QYTeb y Y]_ j TZ']nTe ZC^R R"]R \<X&]nT T^7b m s Z&[LRLmT7]nT" R TT ^.XC`Z&XQZCTeheXC[WT ^j+f \<`" eb XQ^To \ e`QYnX&]nT YTZCc7R Z ∂ ln(2Ωσ0 /∂e2 )/∂r = 0 p C [<`1eZ']b m RLX ZC\<`m`QY^jo m ]XQ^x\ b dp wi `mWR s RLXC[Z_ ˆ X UWT m XIe \d^7b"] d TBYTZCU.XQWZCT.`^^.`Xb.WR<\^.XQ^T.XZ&[LR<^.`\T.[Z_,c\RIYWRL[^jeyU.` ω m X ˆ m ]XQ^`[Z'b7[Z&[\LbXC["pÈ WR<^T ]nT^.XQo^joVU.` ω R%b1Z&[<`o T\<`Z&[TY]_YTZCc7R `UT X ZCj\RLXIe`&`>["RYTZCU.XQWZCT.`^^.`olcWT\<`oGip m Xm m s jU.`<]n^.XQ^T.X!C [TfabZ']`\Toa`I ^R RLXC[ [`wY]_Vb1Z&[<`o T\<`Z&[TP\<`" eb XQ^ToaZ s < 1 ^.XC`mQf1`QYTe` VSTcZCTW.`\R<^^je

q

cs ≥ c1 (s) = c0 1 − 2 s2 r(Ω2 )0 /∂e2 , R>Y]nT^R\<`<]n^jehR<WCT^R"] ^.`sb1Z&[<`o

m

T\j+f

`UW.X'YnX&]_XC[Z_>Z&``[^.`$wXQ^T.XIe

b UWT

cs = c1 (s) ix\<`"eb

k = k1 (s) = k0 [1 − 2 s2 r(Ω2 )0 /∂e2 ]−1 . [Z

y

Y R,\TY^.`

X'm

[<`Y]_b7mj\R

XC[Z_G\xY]nT^^.`\<`<]n^.`\Lb s

XQ^To>\<X&]nT

m

T^R

y

ys

Tf

Ω(r)

CWR<^T Rb1Z&[<`o

z

b dp s

XQ^To

b dp m

dSi

z

¡i

T\<`Z&[TsZY\T'R#

`m<]R
cs = c1 (s) \<`IWR

7 x n L

m

XQW.XQoYnXIei[<XQU.XQW c>U.`<]b

`UW.X'YnX&]XQ^^.`oGZC\L_IT m

b1Z&[<`o

L!QC#L$I"!'"K

T\<`Z&[TPYn`<]

dD(k)/dk = 0

#"!Q y

XQ^T

"S( -.#

m

b1Z']`\T7_Pb1Z&[<`o

T

b ZQerpU pxw7pedpc+ip

UR<WReXC[W7b

m

j

TZ']XQ^T7_,UW.`\<`QYTe

2Ω s ∂e kT

µ=

X#m

T\<`Z&[T\,e`QYnX&]_7fGmX"

m Ω(r) p( XQ\TY^.` [<`\ C[<`1eZ']b RLXSCWR<^T R X ^R`UW.X'YnX&]_[ Z_TwZ&`\<`c7b.U^.`Z&[TGb1Z']`\To D(k) = 0

σ0 (r)

m

\d]R<\^.`1e}U.`W_7Yc1XU.`

d ln(2 Ω σ0 /∂e2 ) dr

b dp

zu i

kT = ∂e2 /2 π G σ0 = k0 /2 i%T TZCU.`<] Cb_\>c7R m XQZ&[\<X,^R m R"] ^.`&`UWT z m D = 0 m<]nT XQ^T7_W.X w XQ^T.X b dp dSip W.X"CbI] I[LRL[<XyT b1Z']`\T7_ U.`<]b RLXIe m s s fc3d cWT.[<XQWTo>b1Z&[<`o T\<`Z&[T\<`"eb XQ^ToPZ "RIY R<^^je b YnX

cs ≥ csmin (s) ' c0 [1 − 2 s2 r(Ω2 )0 /∂e2 ]1/2 (1 + 0,6 µ2/3 ), R,Txb1Z']`\T7_

dD/dk = 0

cs = c2 (s) ir\<`"eb

s

Y]nT^7b\<`<]n^jehR<WCT^R"] ^.`,b1Z&[<`o Z

m

b dp

\'RQ`\<`1e

i

b UWT

T\j+f

XQ^To

k = k2 (s) ' k0 [1 − 2 s2 r(Ω2 )0 /∂e2 ]−1 (1 − 0,6 µ2/3 ). ~X"Cb"] Q[LRL[

z+

b dp

zz

iv` I^R

m

RLXC[

X3m

[<`dehR<WCT^R"] ^.`b1Z&[<`o

YTZCc1XZ>W.`Z&[<`1eZ&[<XQU.XQ^TTfg^.XC`Z&XQZCTeheXC[WT

m m

b dp s

T\jrX\<`"eb

zz i

XQ^T7_

^.`Z&[TZ&[LR<^.`\L_[Z_ s

\Z&Xm`<]XCX>Y]nT^^.`\<`<]n^.`\jekT pqjd]_lZ&[LRLmT7]nT"RTTl[LR
s

Xm

XIeY]_}Z&[LRLmT7]nT"RTTs`Z&XQZCTeheXC[WT

XQ^To p(` C[<`1ebaXQZ&[<XQZ&[\<XQ^^.`U.`Z&[LR<\T.[ ,\<`UW.`Z`\<`"e` m z + XQZCc1`&`,UWTeXQ^.XQ^T7_cWT.[<XQWT7_bZ&[<`o T\<`Z&[T b dp ip

m

kZ']nTs`CWR<^T X

b `m<]R
[<`GY]_b1Z&[<`o

m

m

T.[ Z_s[<`<] c1`G`Z&XQZCTeheXC[WT

m

^jekT

s = 0 i\<`"eb

b

m

s

^.`Z&[T XQ^T

'RQ`\<`&`,YTZCc7RG^.XC`mQf1`QYTe`P\jU.`<]n^.XQ^T.X,^.XQWR#

T\<`Z&[T

cs ≥ c0 = π G σ0 /∂e p .XQ^T\R"_C[LbV\<X&]nT m T^7b,\wZ&`< ]n^.X m ^.`o,`cW.XQZ&[ u X m m 2 b σ0 ' 10 ^.`Z&[T 'R"]R
U pYfp¢w7p¢wip

W.`Z&[<`1eZ&[<XQU.XQ^T^.XC`Z&XQZCTeheXC[WT m

m

^.`Z&[T\<`"eb

s

XQ^To^.XC`mCf1`QYTehR"_

Y]_KU.`QY R<\L]XQ^T7_TfB^.X&b1Z&[<`o T\<`Z&[TwYTZCU.XQWZCT7_KZCc1`W.`Z&[<XQoB'RQ`\j+fw`m<]RUW.X'YnX&]X |s| → 1

7 #Lk-# &#L$I"!'"( -#

e`

XC[dmjr[ wTZCU.`<] Q`\R<^.`%Y]_,`$.XQ^cT,\<XQWf^.XQoCWR<^T j^.XC`mQf1`QYTe`oBY]_ m

m

cs T\<`Z&[T'RQ`\<`&`wYTZCc7R\<X&]nT T^j v v u u u2 + n u 2 t sup{csmin (s)} = cˆs ' c0 1 + 0,6 t

b1Z&[<`o

n d ln σ0 + 2 − n kT r kT dr

2−n

! 2/3 , b dp z |+i

Ω ∝ r−n p .XQ^Te m n = 1 T (d ln σ0 /dr)−1 ' \<X&]nT T^7b ^ R.pr`<]R<'R"_ cˆs X \`cW.XQm Z&[^.`Z&[Tt+`<]n√ −4 cUc cˆs ' c0 3(1 + 0,07) ' 7,5 ce Zp(+``[\<XC[Z&[\Lb y ¡<| U.`<]b RLXIe s (1) R"_ C[<`oa`$.XQ^c1X`QY^.`1eXQW^R"_,YTZCU.XQWZCT7_GZCc1`W.`Z&[<XQoV`m<]R
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y

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y

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m

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U.`QYZCTZ&[<XIehReW_7Y RsYW7b

[<`

XP\W.XIe_YTZCU.XQWZCT7_

ZCc1`W.`Z&[<XQo'RQ`\j+f>`m<]R
YTR"] ^.`oc1``WYT^RL[j m

b "RsTZCc7]

ym

X

u

pY` C[<`1eb m XQZ&[<XQZ&[\<XQ^^.`VU.`Ujr[LRL[$Z_UW.`\<XQWT.[ aCTU.`[<X"CbP`GehR<WCT^R"] ^.`o>b1Z&[<`o T\<` X Z&[T'RQ`\j+ftU.`QYZCTZ&[<XIe'R"]R \\TYnX ∼ sup{csmin (s)} = c obs < σ0 ∼ σ ˆ0 (cs , n, Lσ ) p R`$. XQ^cT \<XQWf^.XQo CWR# X X s ^T jtU.`\<XQWf^.`Z&[^.`oU7]`[^.`Z&[T b T Z']X'Yn`\RL[<X&] ^.` `m XQoKehR
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σ ˆ0 &:GGB13YB&YB1H*B 9'G&:$*B&;

A

965L!>QZ@53:\%G~

!E'A

2

σ0

$4G '3:9'A'$

(6<*5?>@5A*B$'A3

Z13bBGI'A'$DNOG`Z965qWL3YbHI'3YB5

UP R GB!u

H2 CfZ/*HY5!; A A'E'ADC Z3 r =S DU?0V /2 r = 8 ( [5!Xl5c<5?>@5A*B$'A*5E'/'3sZ&YB51?>3`965

σ H2 ' 1,5 σHI BGI' A5\%$ W A'E'AR%$ W A'E'ARs( Γ−10 r = 10 Γ−6 r = 6 E'G&s>3sZ9'3:\h&s>6HI653 G'3:9'A5 E'/'3sZ&YB51!>6;'3YBb13:/M*9'$'~ σ ˆ E'/'3sZ3s>CQE'G&:AG!>A*H=1 !YBG~ 0GL!>@5&YB$rLG!>Xl5!; I65&YB

\x5&:&:2wi5A>pI'3:965g10\%G!>3:A*H`>6;'/'9'2QMGL!>@5A5?MOCiZ'>6;lAG!u BG/'2QM

cobs s

9'3:&:AG!>AGc\x3Y9'X83

U.`QYZCTZ&[<XIetU7]`ZCcTf>'R"]R
X ^T7_UW.`Z&[<XQo wXQo>W.XIR"]nT"RTTC[<`oTYnXQT Q z m 2/3 µ ∝ b dp + \ | i b `m
E'/'$'9*;*BG:Gc`965iI'3:9'$*;

Xm

ReXC[Te " [`,U.` ∝ d ln(2Ω σ0 /∂e2 )/dr UWR<\c7R b dp wi_i`cRIj\RLXC[Z_ \YTZCU.XQWZCT.`^^.`1e b.WR<\^.XQ^TT eXQ^ w XT7]nT U.`W_7Yc7R X y −1 (kT Lσ ) ∼ c0 /∂eLσ 1 TtU.` C[<`1ebi\}U.XQW\<`1e XQ^TTtX Z']X'Yb q UWT.m<]nT m m X(m −n X XC[,UW.XQ^.XCmW.X p `Y RZ%b XC[<`1eg[<`&` [<` ∂e ' 2(2 − n) Ω UWT Ω ∝ r m z U.`<]b RLXIetT b dp dSi √ obs √ obs 2 c 2 cs Ω 2 − n 2 − n Ω s √ √ σ0 < σ ˆ ' . ' b dp |+i 0 ∼ πG πG 2+n 2 − n + 2s2 n m XQW\jrX`$. XQ^cT \<X&]nT T^j σˆ0 \W_7YnX'R"]RZ m T.[LR"]nTZ [<` cs U.`Z&[<`Q_n^^jekT X ym m X b n = 1i RTZCc7] " XQ^T.XIegZ']b R"_ YnX C [<`_\^.`V^.Xw\jU.`<]n^7_XC[Z_ n = 3/2 b z y fip+~X"C b"] I [LRL [j ZCWR<\^.XQ^T7_^RLm<] Y RLXIe`o σ0 Tg`$. XQ^cT σˆ0 UWT\<X'YnX Xqm ^j ^RyWTZpJdpfp TY^.` [<`yY]_lm`<] B T^Z&[\Ry'R"]R R\^.X B ^Tf RU.`<]b

7 #Lk-# &#L$I"!'"( -#

\<`<]n^RIf>U7]`[^.`Z&[T p

[eXC[TesX

$

s

X+`QY^.`Yn`\<`<] ^.`%^.XC`

¡f p+["R>` .XQ^c7R>^R"]R<'RLXC[>`CWR<^T X X

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]R
m

XC[Z_\<`"e`

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T m

m

y

XQ&`iehR
X

`$.XQ^cTlc1`[<`Wj+f

m

r ≤ r0

V0 = V(r0 ) p

XIe

"R<\TZ_[`[UWT ZQRekjeU.`I_\L]_

XQ^T7_lT^V%`WehRTT`yWR
Bc7R"]njWR
Mgas (r0 ) YnX

b dp |+i

UW.`m<]XIeb\jrm`WR

b dp |+i Z']X'YbXC[K`CWR<^T

XQ^T7_

XQ^T.Xr`$.XQ^cT

XQ^T7_y^R>U7]`[^.`Z&[ s'RQ`\<`&`aYTZCc7R>'R#

^7_[<`&`Yn`WR

TY R<^^.`XUWT7]`

m

< ∼

√ cobs V 0 r0 2 2 s √ , G 2+n

T.[j\R"_[<`[VdR
Xhm

[<`\RQ`\j+f

X X

b dp |#fi

U.`QYZCTZ&[<XIehRIf

'R"]R
&]XIeXQ^.[<`\S[C_ X&]XCX\<`QYn`W.`CY R b \S`ZC^.`\^.`1e MH ' 0,7 gas X T X U.XQW.X'f1`QY_tciXQZ&[<XQZ&[\<XQ^^jeY]_ Z &X&]nT7_'iwTi[<XIe ZQRekje X m UWT7]` XQ^To>X'YT^T RegT"eXQW.XQ^T7_ T b dp |#fiU.`<]b RLXIe [TcVZ&`CYnXQW

T.[Z_VUWTeXQW^.`[c

MH /M

< ∼

2,3 · 106 r0 V0 ' 6,7 · 105 V0 r00 ∆,

0 V0 T "eXQW_XC[Z_i\ce Z r0 Z \cUc r0 Z \b.]`\j+fekT^7b7[LRIf kWRZ&[<`W.`^j w

YnX

MH /M ' 2,36 · 105 SH ∆2 ,

SH = SH (r0 ) Z U.`[<`c\G]nT^TT HI X \GX'YT^T RIfS^ · ce Z X TC]b m RLX X XQ^T7_,Y]_ H / ekjoV\wUW.X'YnX&]RIf r0 p.WR<\^T\R"_aUWT\<X'YnXQ^^jrX\j1wX\jWR m U.`<]b RLXIe`$. XQ^c7bPWR
∆

< ∼

∆max ' 2,83 V0 r00 /SH .

~X"Cb"] Q[LRL[jZCWR<\^.XQ^T7_xUW.X'YnX&] ^.`od`$.XQ^cT

b dp |+wi

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ys

X

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XQ^TTf1`W.`$w`s`UTZCj\RLXC[ X r > ^.` Z&`Z&[<`I_^T.XVRQ`\<`oU.`CYZCTZ&[<XIekjÈR"]R`m<]R
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X

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Ω(r) X

^.X>`CWR<^T

m

T\R"_Z b1Z']`\T.XIe

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m

UWT.m<]nT

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U.`<]b

eX

\

b dp

u

wi

X

b dp

u

b dp

^.`Z&[T

T\<`Z&[T>[LRdp w7p¢w

m

m

^je>\<`"eb

b1Z']`\T.X s

u

dSi

X

X

XQ^Toi\

σ0 (r) T0 (r) X TgTZCZ']X'YbXIe

|3dp

UWTeXQ^Te`Z&[T

XQ^T7_neG\'RQ`\<`1e,YTZCc1X

q

b dp

u

|+i

c+i+c\TYb

2 m Ω d ln(2 Ω σ0 /∂e2 ) 2 m2 dΩ2 − k + Φ+R − rω ˆ dr r ∂e2 dr ω ˆ 2 − ∂e2 2 m Ω d ln p0 Ωz d ln σ0 Ωz − Σ+ Σ = 0, −R c2T rω ˆ dr dr ω ˆ 2 − ∂e2 p0 2mΩ d ln γs γs −1 Φ + R − [R − γs Σ] = 0, rω ˆ dr c2T σ 0 Ωz Σ = c2T σ/σ0 X R = p/σ0 p m T.[j\R"_ "RL[<XIe\jr[<XQc7R ys b y T b X m RLXIeTZCc1`1e`X%YTZCU.XQWZCT.`^^.`X%b.WR<\^.XQ^T.X Yb Φ T Σ U.`<]b

XQ^^.`X

XQ^T7_wcx^.XC`Z&XQZCTeheXC[WT

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YnX

m

T Q ^.[W.`UT

s

XCXGZC\<`oZ&[\Rs\<`"eb

2

4

ω ˆ −ω ˆ

2

∂e

2

+

k 2 c2s

− 2 π G σ0 k

2/γ

1−

b dp |c+i

b dp |3dSi dp

#

fivZC\L_

2 2 r dΩ − 2s 2 ∂e dr

d 2Ωp0 s 2(1−γs )/γs 2πGσ0 d 2Ω ln 2 ln 2 σ0 − Ωz − γs dr ∂e σo kc2T dr ∂e 2sΩkc2T 2 d ln p0 Ωz 2πGσ0 d ln σ0 Ωz d p0 b dp |+¡i = 0. − − ln ∂e dr kc2T dr dr σ0γs Ωzγs −1 −2 s Ω ω ˆ kc2T

X

7 #Lk-# &#L$I"!'"( -#

#%$'&(e+'( )*( Q3YB1$jAG!>3:L.59'$'~

:/651$*B$'/*H7l3,`Gs5?u

`G1G:GhZ$'&:A5j1w\%G`Z3s>$&

]

d ln(Ω)/d ln(r) = −1 ] η = d ln(cs )/d ln(σ0 ) = 0,3 cs = 1,87 c0 =] ] ] 1,87 πGσ0 /∂e γs WaY= 1,5 s/k = 0,05 ] T Lσ Z3:&: 2 s/kT r = 0,02 ] kT =Rs( g∂eH9'/2π G σ0 A*B$'/'965!; −1 L = |d ln σ /dr| s = m/kr σ 0 A'/'$'1.5!; $'9'A'/'3:\%3:9*B :/65?Z$'3:9*B9'G!u!:9*B./'GE'$'~'96G~ 9'3YH&YBG~'I6$'1G&YB$

XC[W7bIY^.`B\TYnXC[ b dp |+¡i}U.XQW.X'f1`QYT.[

Xxm

Q^.[W.`UT

m

[<`B\wUW.X'YnX&]XT Q ^.[W.`UT

I\Lb.c7R b

XQZCc1`&`YTZCc7R

m

T√ ^j

cs ∝ X

\s`m

s

T0 ip

η= s

XQZCc`&`dYTZCc7R b dp

XIe

Z']b

`[Q]nT m

m

T0

b ZQerp b dp

U T7]`

u

dSi_i

XQ^To

p 0 ∝ σ 0 T0 i T.X`[ab.U.`1e_^7b7[<`&`>UW.X'YnX&]RaT" b

RLXGZ&`Z&[<`I_^T.XVYTZCcR`UTZCj\RLXC[X

d ln cT d ln T0 = , d ln σ0 2 d ln σ0

ToP\BYTZCU.XQWZCT.`^^.`XSb.WR<\^.XQ^T.X

wipjd]_

Y R<\L]XQ^T7_gct[<XIekU.X RL[Lb .X

`QYT^>UR<WReXC[W

\"f1`QY_

m

\YTZCU.XQWZCT.`^^.`Xgb.WR<\^.XQ^T.X

bIYn`m^.`U.X .XQo.[Tg`[\<X&]nT T7]nT}ZCc1` .`Z&[T

W&:\(6ED(6+'( )*( +R

b dp |+¡i

m

XQW.X"x\<X&]nT

d ln σ0 p0 d ln γs = 2 (η − η0 ), dr σ0 dr

m

b dp |

s

X u i

T^7b b dp |

i

η0 = (γs −1)/2 b UWT η = η0 YTZCcT Q^.[W.`UT m XQ^\%Z']b m RLX Ωz = ` ] ip Xm m SZC^.` [<``$. XQ^c7R^.XC`mCf1`QYTe`oSY]_KCWR<\T.[LRT.`^^.`owbZ&[<`o T\<`Z&[TK'R# m z Q`\<`&`xYTZCc7RB\<X&]nT T^j cˆs b ZQerp b dp |+i_ihYn`<] ^R " R<\TZ&XC[ B `[UR<WReXC[WR η p m m ` C[<`1eb,XQZ&[XQZ&[\<XQ^^.`KU.`Z&[LR<\T.[ \<`UW.`Z 7UWTPc7Rb1Z&[<`o T\<`Z&[TPYTZCc7R\<X&]nT T^R cˆs (η) ekT^TehR"] ^R X [<` C [<`}mUR<WReXC[W.`\'RQ `\<`oGU.`CYZCTZ&[<XIekjZ&`]R

7 x n L ys

L!QC#L$I"!'"K

#"!Q

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^RWTZp#dpc+ipGSR η1 m m m X [<` UWTim`< ] B Tf I ^R XQ^T7_7f cs TZ X"" RLXC[^.X&b1Z&[<`o T\<` Z&[ `mCWR<\T.[LR Q T.`^^.`o b i\<XC[\L_nekT p c7R" R<^^jo>U.XQ W.X'f1`QY ys m `UW.X'YnX&]_ ToVCWR<^T 7 bb1Z&[<`o T\<`Z&[TaZC\L_I T`[BUR<Wju\<XC[\<XQo b i UWT X y η < η1 cUR<W.X b iGUWT η > η1 UWT\<`QYT.[cU.`I_\L]XQ^T ekT^TebehR m \lVb.^c T.`^R"] ^.`o" R<\TZCTe`Z&[TP^.XC`mCf1`QYTe`oVY]_Gb1Z&[<`o T\<`Z&[T>'RQ `\<`&` m cˆs (η) UWT η = η1 ' 1/2 p YTZCc7R\<X&]nT T^j m m %W.`\<X'YnXIe[<XQU.XQW UWT.m<]nT XQ^^.`Xi\j TZ']XQ^T.Xy\<X&]nT T^ η1 T cˆs (η1 ) m b U.`Z']X'Y^.XQo9Z c7R<cu`$. XQ^cT\<XQW\\TYnX z ∆(k,ˆ ω) = ω ˆ 4 + p(k) ω ˆ 2 + q(k) ω ˆ + d(k) = 0. b dp | i `m<]RIY R

R"_aZC\<`oZ&[\RekTa\<`<]n^G~`ZCZ&mT X

b ZQerpU pxdp¢w7pc+iZ

CWRIYT.XQ^.[^R"_

b \<XC[\

Xm η = η1 T UWT\<X'YnXQ^^j+f\j1wXiWRU.XQW.XQ\R"] ^jekT>[<` c7RekTVb.^c TT ∆(k,ˆ ω iT X Z']X'Yn`\RL[<X&] ^.` X \^TfGYn`<] ^j\jU.`<]n^7_[ Z_Pb1Z']`\T7_ ∂∆ ∂∆ = 0, = 0 , ∆(A) = ∆(B) = 0 . b dp ||+i ∂k A,B ∂ω ˆ A,B X ~X w XQ^T.XIe C [<`oZCTZ&[<XIekj m

7 #Lk-# &#L$I"!'"( -#

2 kT Lσ 1 iST \UW.X'YnX&]X,`CY^.`W.`QY^.`&` b λ → ∞ i X ^.`PYTVWV%XQW.XQ^ TR"] ^.` s ys Ω(r) ∝ r−n YTZCcRW.X w \WR R XQ&`Z_ XQ^T.XxZCTZ&[<XIekj b dp ||+iTeXCXC[\TY q

cˆs = c0 (2 + n)/(2 − n) ,

b dpf+i

k1,2 = kJ = π G σ0 /c2s ,

UWT.m<]nT

q

b dpf#fi

ω ˆ 1,2 = ± −p(k0 )/2 = ω ˆ 01,2 .

b dpf+wi

λ 1 iVW.X wXQ^T7_uT δ1,2 X 1,2 Z ehR"]njrXS\<X&]nT b

XQ^TTZ']RLm`^.XC`QY^.`W.`CY^.`&`yYTZCc7R X

k1,2 = kJ + δ1,2 ω ˆ 1,2 = ω01,2 + 1,2 YnX m `Y RT b dp ||+i+U.`<]b RLXIe 1,2 = q(kJ )/4 p(kJ ),

\TYnX

δ1,2 = 1/ˆ ω01,2

"RL[<XIeC[T,W.X"Cb"] I[LRL[jl\

`QYZ&[LR<\L]_._

XQ^.[LRekT

b dp |

z

i+\K[<`

m

dq dk

X

cRIf>c7R\<XC[\<XQoVYn`<]

\TYTe

q(kJ ) = 0,

Λ = n Lσ /r p

m

~R
m

Z']X'Yn`\RL[<X&] ^.`

1/2

XQ^^jrXW.X"CbI] I[LRL[jtZ+[<`

^.`o`cW.XQZ&[^.`Z&[TÈR"]R
X

Ybc1` VWVST T

cˆs (η1 )

T.X

b dpf+¡i

UWR
m

XQZCcTP^.X`[Q]nT

m

,

b dpf

u i

W.XQ^T7_KTfBUWT7]` XQ^To I λ = 2 kT Lσ ' 35 Λ ' 0,4 X T m

cT

RLXC[Z_G`[w`UW.X'YnX&]_XIe`oVZ&``[

i

p X

cˆs (η0 ) b YnXQoZ&[\T.[<X&] ^.` X [ˆ cs (η0 )/ˆ cs (η1 ) − 1) < u 0,05) p.`aUW.X'YZCc7RIj\RLXIehR"_sZ&``[^.`$wXQ^T.XIe b dpf i η1 = η0 + γs (1 − Λ)/4 γs = 1,5 T Λ ' 0,4 \<X&]nT m T^R η1 ' 2η0 ' 1/2 X.m [<`^.XZ&`]Rc1`<]XCm1R<^To>'RQ `\<`&`wYTZCc7R.p ^.`$wXQ^T.XIe

b dp

|+iS\<X&]nT

m

ys

b dpf



X

γs 1 (3 − Λ) − , 4 2

 π G σ0  Λ(2 + n)/(2 − n) q cˆs (η1 ) '  ∂e Λ − 2(1 − Λ)(5 − 3Λ)/(2 − n) 

YnX

T^jSp

RLXIetTZCc1`1ekjroW.X"CbI] I[LRL[

η1 '

R
[<`KeX

J

m

XIe\

b dpf3dSi

p2 (kJ ) = 4 d(kJ ) + 2 (dq/dk)2 /(d2 p/dk 2 )|k=k ,

TZCU.`<] Cb_>c1`[<`WjrXwU.`<]b

Xxm

.

^j\jU.`<]n^7_[ Z_>Z']X'Yb

Z&``[^.`$wXQ^T7_

m

b dpfc+i

−1 2 d p   dk 2 |k=kJ

!

b dp ||+i

s

T^j

7 x n L

L!QC#L$I"!'"K

703/100F?.8 D.AC;.J";.: b

YT.XQ^.[^R"_ s

\<`"eb

"S( -.#

2 &;.J"?.: .D.O.;.8"EG;.AM'J":
qSR

#"!Q

i\<XC[\T y

XQ^.`\UW.X'Yj+Yb c1`<]XCm1R<^To

[Z_y^TIc1`

m

s

XIeWR&YnX&]X

X

Q^.[W.`UTo^R"_

RQ`\<`&`VYTZCc7RGUWT]

y

b

#

i+TPCWR

mj+fsY]nT^RIf

\<`<]n^

|ˆ ωII, III | ∼ ∂e/k Lσ ∂e Z ZQerp s =X WTZpYdpc+ip [<` Xa\W.XIe_yCWR<\T.["RT.`^^jrXG\<`"eb XQ^T7_ b i!"R>UW.X'YnX m y m ]RekTibIc1`o Q`^j \<`<]n^.`\j+f TZ&X&]gZ k ' kJ f7R<WRm

R
b

2 2Ωσ0 d ln σ0 2 kJ d 2 2 2 k cT + cs k − ln 2 + 8 s Ω 2 × × − η0 ) dr 1 + k h dr ∂e ∂e d ln σ0 2 2 kJ z 2 2 b dpf × (η − η0 ) 2 k cT (η − η0 ) + cs k − = 0. dr 1+kh

4 k c2T (η

m

XC[W7bIY^.`\TYnXC[

Xm

[<`b.WR<\^.XQ^T.X

b dpf

z

i

i UW.X'YZCc7R Ij\RLXC[WR I\T.[T.X^.X&b1Z&[<`o

T\<`Z&[TUWT>\jU.`<]n^.XQ^TTPb1Z']`\T7_

8 γs α(η − η0 )(Λ − zˆ) − 16 zˆ(η − η0 )2 + α2 γs2 (1 + Λ)2 < 0,

b dpf|+i

Λ = d ln(2 Ω/∂e2 )/d ln σ0 α = 1 − 2 kJ /k(1 + k h) zˆ = α(k cs /∂e)2 (1 − 2 s2 r dΩ2 /∂e2 dr) p(È WR<^T j`m<]R k1 X Z&[T È ) ^.X&b1Z&[<`o TZ&X&] √ T\<`Z&[T]X T.[\UW.X'YnX&] RIf\<`<]n^.`\j+f k1 = ( 1 + 8 kJ h − 1)/2 h Z U.`<]` T.[<X&] ^joc1`W.XQ^ YnX b.WR<\^.XQ^T7_ YnX

7 #Lk-# &#L$I"!'"( -#

+'( +'( gL!>@5&YB

#%$'&(

9'3YH&YBG~'I'$61G&YB.$ W

k; η

RE'G

Wai5!XbB/'$M.G1.5965.R

Rs(

1

E>G&YAG&YB$

s5!`G1G:GrZ$'&:A5

E65/65\%3YB/65\

1tGA'/'3:&YB9'G&YB.$ W

:/65?Z$'3:9*B9'G!u!:9*B./'GE'$'~'96G~ <*5?>@5A*B$'A'$

QG!>965t1tE'/'3sZE'G!>G?kl3:9'$'$ γ = 3/2 x&YBG~'I'$'1G~ ;'1!>6;'3YB&;=>$*X8j\%sG`Z3s>j&

η0 = 1/4 WE'G!>Gik3:9'$'3JBGI'A'$ i51$'&:$*BwGBj&YB3:E'3:9'$ η9'3:G=&:3:η&:0$'\%\%3YB/6$'I'9'G&YB.$1G`\FHη78s3:9'$'~R

k − 2 kJ /(1 + kh) = 0 m

b UWT

kJ h → 0

\<X&]nT

m

T^R

k1 → 2 kJ ip[eX X T\<`Z&[T p ` U.XQW\j+f

m

`ZC^.`\^jrX>`[Q]nT T.[X&] ^jrXsZC\<`oZ&[\RUÈ ) ^.X&b1Z&[<`o z X m m Z&`]Re`'b7[Kmjr[ K[<`<] c1`^.XC`Z&XQZCTeheXC[WT ^jrX

[Te

s = m/kr 6= 0 i \<`"eb s XQ^T7_vp ` \<[<`Wj+f X È ) ^.X&b1Z&[<`o m T\<`Z&[ e` XC[KWR" s ys m \T\RL[ Z_TK\[\<XQW

v u

]X'Yn`\RL[<X&] ^.`

X

2 s cs u 1 t ω = ±i (η − η0 )(η + ). L σ γs 2

b dpfffi m

\wWR
T\<`Z&[ KTeXCXC[KeXQZ&[<`

UWT

η > η0

(a)

b dpffwi

η < −1/2.

(b)

b dpffc+i

T>UWT

[<`mjU.`^7_[ }VSTIT

m

XQZCcTo

^jrXrCWRIYT.XQ^.[jWR<\^.`\<XQZC^j+f

ZQekjZ']

C[Tf}`CWR<^T

m

XQ^To

Q^.[W.`UTTTdY R<\L]XQ^T7_

m

X

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XQW.X"rWRIYTR"] ^joKCWR#

YT.XQ^.[,U7]`[^.`Z&[T

d ln σ0 c2T dS0 (η − η0 ) = , dr T0 η 0 dr

(a)

b dpffdSi

d ln σ0 dp0 = σ0 c2T (1 + 2 η) . dr dr

( )

b dpff¡i

7 x n L

L!QC#L$I"!'"K

b dpffdSiw\TY^.`

\<XC[Z&[\LbXC[tb1Z']`\T

Xrm

#"!Q

m

[<`}b1Z']`\T.XP^.X&b1Z&[<`o

y

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,

b dpffwix\[<`

T\<`Z&[T

m

^.`Z&[TgZ&``[

\<`I^Tc^.`\<XQ^T7_c1`^\<XQc TT\g^.XC`QY^.`W.`QY^.`oZCW.X'YnXZ

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cWR
X

y

[Z_

X

X

^RPTf

U.XQWTV%XQWTTGi%\PZ&``[\<XC[Z&[\TTZZ&`QYnXQW

c7R
X

R<^T.XIeU p.dpc.pf

b1Z']`\T.XIeCWR<\T.[LRT.`^^.`owb1Z&[<`o

Rx`[^.`$wXQ^T.XSTfaCWRIYT.XQ^.[<`\tZb1Z']`\T.XIeb1Z&[<`o

s

m

m

m

T\<`Z&[TwYTZ5

T\<`Z&[TPCWRIYT.XQ^.[^j+fGT

Q^.[W.`UTo^j+f\<`"eb XQ^To p C[<`1etZ']b RLX%Yn`<] ^.`Vmjr[ η = Lσ /Lc 1 p X m y ` \TYTe`1eb TeXQ^^.`>U.`C[<`o}UWT T^.X,UWT}`mZ'b YnXQ^TTY R<^^j+f ^RLm<] m YnXQ^ToU.`V\<X&]nT T^.XdYTZCU.XQWZCTTZCc1`W.`Z&[<XQo'RQ `\j+fP`m<]R
703/10 {0+:

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:M&M < D

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]RU.`Ujr["RL[$Z_

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m

T\<`Z&[T

b dp

|+i

\

'RQ`\j+f

'R#

^R
|+¡#poXC[W7bIY^.`t\TYnXC[ m

Xom

[<`tUWT\jU.`<]n^.X m

Q^.[W.`UT XQZCc1`1e s b η 6= η0 inYTZCc1XYTZCU.XQWZCT.`^^jrX ZC\<`oZ&[\RrWRb.WR<\^.XQ^T.XIe ZCUWR<\<X'Y m m X X ]nT\jeu[<`<] c1`\GT Q^.[W.`UT XQZCc1`1eZ']b RLXp+[<`[}VdRRI Teb7[LbR<^.`1ehR"]nTT}cWT\j+f ^TTbZ']`\T7_gWR
cT9È ) ^.X&b1Z&[<`o

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T\<`Z&[T

b dpf|+iB\}^.XQT

7 #Lk-# &#L$I"!'"( -#

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1/65!783:9'$*;

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5

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123AG9Z3:9'&`5 '$'$

9'$'$ 14E'/'GB$'1GE'G!>G?kT9'G\rGB '3:9*B/65TZ$'&:A*54965?u E'/651!>3:9'$'$D] '$'A>G9'$'I'3:&:A'$'~b$ 59*B$ '$'A>G!u 9'$'I'3:&:A'$'~1$M*/'$D] /65&:&YBG?;'9'$'3bGB '3:9*B/65Z$'&:A5 c ZG '3:9*B/65c1$M*/'3:1G~&YB/*HA*BH*/'2R

V

s

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XQ^T7_

r

/

b WTZpdp¢¡ip6+[TKR<^.`1ehR"]nTTBe`

^.`T^.[<XQWUW.XC[TW`\RL[ %c7R
ehR
X

Z&`mZ&[\<XQ^^jrX

b Tc7]`^GivT7]nTSmjr[ UW.`[T\<`U.`<]` ^j s y X ekT b R<^.[T Tc7]`^Gi+\WR XQ^T YTZCc7Rx'R"]R
\WR

s

XQ^T7_dc1`[<`Wj+fwe`'b7[Z&`\URIY RL[

r

K R TekT\<X&]nT T^7bGY T^Z&`\ZCc1`&`aehR^R,ekR
m

XYm

b dpff

u i

h > 0 Tb1Z']`\T7_b1Z&[<`o s y T\<`Z&[TCWR<\T.[LRT.`^^.`o\<XC[\Tc1`<]XCm1R<^ToYTVWV%XQW.XQ^ TR"] ^.`\WR R s X u Xom zm > −1 p`Z']X'Y^.XCX XQ&`Z_l'RQ`\<`&` YTZCc7R T b dpff i,\jr[<XQc7RLXC[ [<` m Xrm m `I^R RLXC[ [<` \}CWR<\T.[LRT.`^^.`}b1Z&[<`o T\j+ftRQ `\j+fYTZCc7RIfU.`<]`ZQR>È ) m s ^.X&b1Z&[<`o T\<`Z&[TVU.`wUR<WReXC[W7b η b ZQerp b dpff+i_im

b

XC[<`1e[<`&`}VdR
[`

km < k J

\aYTZCc7RIf

Z

7 x n L

L!QC#L$I"!'"K

#"!Q

"S( -.#

#%$'&(*+'( *( D 51$'&Y$'\xG&YB $

GBlE65/65\%3YB/'G1

Λcrit R 0 2 2 x = (c0 /cs ) (1 − 2 s r Ω2 /∂e2

#%$'&(y+'( (g`G!>$'9'$'$

9'$ 65?M

s Ω/kJ Lσ1 9'3YH.&YBG~'I'$'1G&YB.$ $

R

$'9'A'/'3:\%3:9*BG1

m

k < k1

W1

3sZ$u

xu

A'/*HE'9'G\5!&YXbB5L9'G~ \%G`Z3s>6;MZ$'&:AG1 N

0,3 Λcrit] = 0,633 Λ = −0,75 Λ = −1 `m<]R
δ = kJ h

&

kJ h =

Λ = −0,6

< ]*

UWT>\jU.`<]n^.XQ^TTPb1Z']`\T7_

Λ2 > Λ2crit = −ˆ zm .

C[<`&`VbZ']`\T7_s`m<]R
m

b dpff

i

k< k1 WR'RQ ` X > \<`1eYTZCcX{È7R"]RTeXCXIebZ&[<`o T\<`Z&[ b ZQerpvWTZpYdped ^Rc`[<`W.`1eY]nT^^.`\<`<] m ^.`\R"_>U.`QYn`m<]RCWR<\T.[LR T.`^^.`Kb1Z&[<`o T\j+f b ^.`^.XSZ']nT B c1`1e U.X W.XQCW.XC[j+f i 'RQ`\j+faU.`QYZCTZ&[<XIe'R"]R 0 i X W.`^j RYTZCc7R^.XGZ']nT B c1`1e UW.X'YnX&]XQ^T.X Ω(r) T" y σ0 (r) b LΩ ' r/n < Lσ /Λcrit ip eX'Y]XQ^^.`,U.`,ZCWR<\^.XQ^T ZSWRWR\<`<]n^ Z']b

RLXK^.XQ\jU.`<]n^.XQ^T7_

X

T\<`Z&[TUWT

7 #Lk-# &#L$I"!'"( -#

I^R

m

Λ < 0 b σ0 (r) \eXQZ&[<XZ Ω(r) U.`<]R<'RLXC[Z_ XGm ys b7mj\R XQocU.XQWTV%XQWTT>YTZCc7RVb.^c T.XQoGip TY^.` [<`G\,T^.[<XQW.XQZ'b XQo m < ^R Λcrit LΩ ip R\<`<]n^y~`ZCZ&mTsYn`RekU7]nT.[LbIYj (˜ X X m X YTe` ` XQ\TY^.` \W.XIe_GU.`W_7YcR 20 ÷ 30 `m`W.`[<`\xYTZCc7R.p R
m<]

y

§ @§

XQ^T7_7f

Y RLXIekj+f>Z&[W7b.c.[Lb.W[TURZ&`<]nT.[<`^.`\a~`ZCZ&mT p

® ®²¬G´#®Gµ|°}°x¹´µ

%WT,W.X BXQ^TT,\<`UW.`ZQRS`wCWR<\T.[LR T.`^^.`o,b1Z&[<`o

m

T\<`Z&[Ta'RQ`\<`&`%YTZ5 m

c7RTy`UW.X'YnX&]XQ^TTyZCU.XQc.[WRc1`<]XCm1R<^Toy\XQ&`U7]`ZCc1`Z&[Tb ^j+f

m

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UWT.m<]nT

XQ^TT^.XQZ'b

m

s

XQZ&[\<XQ^ pg+[<`

e`

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^.`}UW.`T7]]

m

y

Z&[WTW.`\RL[ Z']X'Yb XQoS`$.XQ^c1`o p X&]nT T^Rf7R<WR`cW.XQZ&[^.`

m ν Z e`<]XQc7bI]_W^R"_icT^.XIehRL[T XQZCc7R"_\L_Ic1`Z&[ xip [<` m m −15 −1 2 ν kJ p Xx\W.XIe_QUT Tc7]nT XQZCc7R"_ RTZCZ']X'Yn`\R<^T.X VWV%XQc.[<`\ ZC\L_" R<^^j+ftZGb XC [<`1eYTZCZCT m X X URL[T\^j+f ]XQ^.`\ e` XC[yUWT\<XQZ&[Tgcg\R ^jeW.X"C b"] Q [LRL[LRerp ` U.XQW\j+f Xm s U.`[<`1eb [<`^.XQc`[`WjXi[TUj \<`" eb XQ^To \U7]`ZCc1`Z&[T 'RQ `\<`&`YTZCcR X X e`'b7[u`m<]RIY RL[ `[WT RL[<X&] ^.`o Q ^.XQWCT.XQo dSw T Z']X'Yn`\RL[<X&] ^.` mjr[ m m YTZCZCTURL[T\^.`^.X&b1Z&[<`o T\jekT Y R Xt\CWR<\T.[LR T.`^^.`b1Z&[<`o T\<`1e YTZ5 Z&[Tt+`<]n^ R

b &YnXQZ

7 7 Cf #OQ!) %-Q)

c1Xp

` \<[<`Wj+f

X

b

m

m

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]XQ^.`\GU.`I\<`<]_XC[G\GUWT^ TU.X`UW.X X

YnX&]nT.[ b.W.`\<XQ^ WR<\^.`\<XQZC^j+fLV]b.c.[LbRTo

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^.`wb

m

y

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u

|

p R
X

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eX&]nc1`1ehR
b ZQerpU p'¡7pfpfip

.8 . D.Dd;.8 ?.8"=
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T.[jr\RL[$K\

b [Lb .m

|

X

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c1`W.`[c1`\<`<]n^.`\j+fw`Z&XQZCTeheXC[WT

m

s

^j+fw\<`"eb

m

TekZ_

@BAC;.D7E

TCb

m

XQ^To p)jd]_x[LR
'R<We`^TcTu]nT^.XIR<WTQ`\R<^^j+fb.WR<\^.XQ^To'RQ`CYT^RekTcTZb m y z YTZCZCTURL[T\^j+f ]XQ^.`\TeX [,\TY fc#f b Z pZ b d¢p ¡¡i b d¢p ¡ i_i Vb.W <X

s

XQ^T.XIe m

−iωσ + iσ0 k u = 0 ,

−iωv +

2

XC[<`1e z

b dpff

−iωu − 2 Ω v = −ik(Φ + p/σ0 ) − λ k 2 u , ∂e

XQ^To

i

b dpffQ|+i

u = −ν k 2 v ,

b dpfw+i

2Ω −iωT0 S = 2 ν r Ω0 i k v − χ k 2 T cσ ,

b dpfwfi

λ = 4 ν/3 + µ ν X µ Z U.XQW\R"_T\<[<`WR"_ycT^.XIehRL[T m XQZCcT.Xa\L_Ic`Z&[T S Z\<`" eb s XQ^T.X1Q^.[ .`UTT χ Zc` VWVST T.XQ^.[x[<XIekU.XQWRL[Lb.W.`UW.`\<`QY^.`Z&[T cσ ZbIYnX&] ^R"_P[<XQU7]`XIekc1`Z&[ ,UWT>U.`Z&[<`I_^^.`oU7]`[^.`Z&[T p z TZ&[<XIehR b dpff i b dpfwfi Yn`<] ^Rmjr[ Yn`U.`<]n^.XQ^RKb.WR<\^.XQ^T.XIe9bR[<XQWe`CYT^RekT XQZCcTekTZ&``[^.`$w XQ^T7_nekT σ p b dpfwwi , = β σ0 T + σ0 σ0 c2T YnX

S = cp cp

YnX

∂S β = ∂p ~X K R"_UWT\<X'YnXQ^^7b

y

!2

T

=

cp − c σ . T0 σ02 c2T

\j1wXwZCTZ&[<XIebsR"]n&XCmWR
YTZCU.XQWZCT.`^^.`Xb.WR<\^.XQ^T.X eb

b dpfwc+i

ZbIYnX&] ^R"_P[<XQU7]`XIekc1`Z&[ ,UWTU.`Z&[<`Q_^^.`1eiY R<\L]XQ^TT>T

2

s

T + β p, T0

X

`UTZCj\R

ys

m

XQZCcTfPb.WR<\^.XQ^To

b dpfw:dSi X

U.`<]b

m

Te

XCXSZC\<`oZ&[\RBWR
XQ^To

ω 4 + i(ν + λ + χ)k 2 ω 3 − {(ν λ + ν χ + λ χ)k 4 + ωJ2 } ω 2 −  

− i ν λ χ k 6 + χ k 2 ωT2



d ln(Ω)  + ν k 2 (ωJ2 − ∂e2 ) + ν k 2 ∂e2 (γs − 1) ω+ d ln(r) 

7 #Lk-# &#L$I"!'"( -#

+ ν χ k 4 (k 2 c2T − 2 π G σ0 k) = 0,

b dpfw¡i

ωJ2 = k 2 c2s −2 π G σ0 k+ ∂e2 ωT2 = ωJ2 −k 2 (c2s −c2T ) p mX"&YTZCZCTURL[T\^.`1e m UWT.m<]nT XQ^TTT b dpfw¡iU.`<]b RLXIeuYTZCU.XQWZCT.`^^.`Xb.WR<\^.XQ^T.X,`Z&XQZCTekeXC[ m s 2 2 WT ^j+fCWR<\T.[LR T.`^^j+f\<`" eb XQ^To ω = ωJ (k) p s j_ZC^Te[<XQU.XQW \L]nT7_^T.XGYTZCZCTURTTt^R C[Tt\<`"eb XQ^T7_vp Z&``[ 2 ν˜ k |ωJ | X YnX \<XC[Z&[\TTZUWT\<X'YnXQ^^jekT\j1w X `$. XQ^c7RekTU.`<]R<RLXIe ν˜ = max{ν, λ, χ} p `Y R,\]nT^.XQo^.`1eU.`,YTZCZCTURL[T\^jelc1` VWVST T.XQ^.[LRe m ω = ω + δω |δω | |ω | b J J J J i UWT.m<]nT XQ^TTU.`<]b RLXIe YnX

d ln Ω ik 2 2 2 2 2 2 2 δωJ = − 2 ν ∂e + λωJ + χk (cs − cT ) − ν ∂e (γs − 1) . 2ωJ d ln r

u

b dpfw

i

c2s > c2T X γs > 1 T\KYTZCcRIfU7]`ZCcTf'R"]Rb1Z&[<`o T\<`1elYTZCc1X b ωJ > 0 iY T^Z&`\ZCcT.X,\<`"eb X y X m 2 ^T7_"RL[Lb1fR [ RK\CWR<\T.[LR T.`^^.`V^.X&b1Z&[<`o T\<`1eYTZCc1X b ωJ < 0 iTZCUjr[j y y y \R [GYn`U.`<]n^T.[<X&] ^7b YnXQZ&[LRLmT7]nT" R T T" " R,YTZCZCTURL[T\^j+f VWV%XQc.[<`\p X s +[<`[%W.X"C b"] I [LRL[c7R XC[Z_wXQZ&[<XQZ&[\<XQ^^je [LR

`ZCc1`<] c7b

C[<`[GW.X"Cb"] I[LRL[G^.XC[W7bIY^.`PU.`<]b s \j+f b k kJ ir\<`" eb XQ^To p T

m

s

m

T.[ PT

XCX+YTZCU.XQWZCT.`^^.`Xb.WR<\^.XQ^T.X

b dpfw

u

i\aUW.X'YnX&]XKc`W.`[c1`\<`<]n^.`

b dpfw¡iZb.WR<\^.XQ^T.X

m

XC[\<XQW.[<`oVZ&[<X Xm

mX"&YTZCZCTURL[T\^.`1e UWT.m<]nT X ^TT,T^.XQ&`dZ']X'YbXC[ Q [<`wcW.`1eX s s 2 2 ω = ωJ (k) Z'b XQZ&[\LbXC[GX B Y T^Z&`\ZCcTf X Y\R b Q^.[W.`UTo^j+fGi%[TUR>\<`" s m X |ω| ∼ ν˜ k 2 |ωJ | X eb XQ^TotZ ω = 0 p b XC[<`1e YTZCZCTU R TT U.`<]R<'R"_ U.XQ^TU.`

Y]_

ωi

ωp

C[Tfg\<`"eb

s

XQ^TotT

b dpfw¡iKU.`<]b

m

RLXIeb.UW.`

s

XQ^^.`X

b c\RIYWRL[^.`XU.`

YTZCU.XQWZCT.`^^.`X%b.WR<\^.XQ^T.X

k2 2 d ln Ω 2 2 2 ω + iω 2 χ ωT + ν(ωJ − ∂e ) + ν ∂e (γs − 1) − ωJ d ln r 2

−

ν χ k4 2 (ωT − ∂e2 ) = 0. 2 ωJ

m y b kZ']nTUW.XQ^.XCmW.X [<XQU7]`UW.`\<`QY^.`Z&[ s ys X z y b [<X&] ^.`,\WR R TekZ_ [<`,T d p fw i Z']X'Yb

b dpfw

z i

χ/ν → 0idTyZ m T.[LRL[ YTZCcy [\<XQX WYn`z [W.X"C b"] Q[LRL[jWRLm`[ ff¡ ¡<|

ωD = iν k 2 (∂e2 − ωJ2 )/ωJ2

7 7 Cf #OQ!) %-Q) X

m

X

m

ωJ2 > 0 iRQ`\<`1e YTZCc1Xte` m XC[WRI\T\RL[ Z_YTZCZCTURL[T\^R"_^.X&b1Z&[<`o T\<`Z&[ \u`m<]R
I^R

T

T.[

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b

T\<`1e

0 < k < 2 k J γs . m

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+[<`[W.X

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e`QYnX&]nT>CWR<\T.[TW7b

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Xm

m

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m

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m

z

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s

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X

m

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T

X

X

Z']X'Yn`\RL[<X&] ^.`

k = kJ eb

fd1| p

s

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[<XQW_XC[PZQekjZ']

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m

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b dpfc+i

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m
XQ^T7_ s

|ωJ (kJ )|3 < ν˜kJ2 ∂e2 ,

s

m

XC[jrW.X'f[TU.`\

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01 &(%&

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b dpfw¡i m

"

b mX Sb

X

W.X w XQ^T7_

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XC[LRBYTVWV%XQW.XQ^ TR"] ^.`Z&[T\WR

ω1 ' i ω0 ; ω2,3

s

T

|ωJ3 (kJ )| ν˜ kJ2 ∂e2

(

ω0 = ν

Z

kJ2 ∂e2

"

γs − 1 χ 1+ γs ν !

m

C [<`&`BZCU.XQc.[WRwW.X wXQ^ToV^.X&b1Z&[<`o ω = ω1 T>Tf>T^cW.XIeXQ^.[

y

[\TY

XQ^T7_aYTZCc7R+i

√ ± 3−i kJ2 (2γs − 1) , ' ω0 ; ω4 ' −i χ χ 2 γs + (γs − 1) ν

YnX

TeX

T\jekT,_\L]_

y

#)1/3

.

b dpfc+wi

[Z_V[<`<] c1`K\<`"eb

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b dpfc#fi

ys

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s

s

XQ^T7_

XQ^T7_nerp

ν kJ2 /∂e ∼ l/h 1 b l Z Y]nT^RZC\<`m`QY^.`&`UW.`mXQ'R m R
(

ω1 ' i ν

kJ2 ∂e2

"

b dpfw¡iU.`<]b

m

T

RLXIe

d ln Ω γs − 1 χ − (γs − 1) 1+ γs ν d ln r !

1/h

#)1/3

.

b dpfcc+i

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eb

σ

YnX s

s

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XQ^To

2/3

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s

Z\<` "eb m

m

fd1|

1 − χ(γs − 1)/ν γs σ 2 , [1 + χ(γs − 1)/ν γs ]1/3 σ0

b dpfc3dSi

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ys

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Xm

ω1

R

b dpfc+¡i

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m

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y

y

u

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m

s

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y

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−iωu − ∂ev = −ik(R + Φ) − λk 2 u + f1Z&[ ,

b dpfc

−iωv + ∂eu = −νk 2 v + f2Z&[ ,

b dpfc

−iωΣ + iku = 0 ,

b dpfc

−iωS = −χk 2 cσ τ + f0Z&[ ,

u i

z

i i

b dpfc|+i

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7 #Lk-# &#L$I"!'"( -#

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E˙ = σ0

Z

XQ^T.XIe

Z

{u f1 + v f2 + T f0 } dS = −

Xi

∂Fi dS , ∂t

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s

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X

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b dpfdx+i

|

R

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αij (k, ω) m

YnX

Z[<XQ^ Q`WP`m`m s

s

m

XQ^^.`o>\<`ZCUWTTe

b dp fd(fi

T\<`Z&[T p

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σ σ0

*

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e`\

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T^j

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m

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nYnXQZ V ^.X

^.`

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\<X&]nT

i T0 ∗ (αnl − αln ) , 3 (2π) ω

hXl Xn ik,ω = ZCU.XQc.[WR"] ^jo

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ZCU.XQc.[WR"] ^.`o

m

m

TZ']RVU.XQW\jo

xbe`\ y U7]`[^.`Z&[ XQZCcTf

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m

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y

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b dpfdxc+i

s

]XQ^jie` m

RL[<X&] ^j+f XQZCcTf

xb

RLXIe

σ σ0

!2 + T r

T0 (2 + γs ) = 4π 2 σ0 c2s

Z

2 + γs ~ eik~r d~k = T0 δ(~r) . σ0 c2s m

xbe`\ Z']X'YbXC[SWR\<`" e` ^.`&`P\ m l p RehR
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k = kJ

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m

RLXIe

σ σ0

*

J0 (x)

YnX

r

r

m

C[<`&`s`mZ&[<`I_

XC[<`1e

b dpfdS¡i

W.`QY R.pJSR

X

T^.[<XQCWTW.`\R<^T.X

G T0 1 π ' 4 − Y0 (r/LT ) + H0 (r/LT ) , cT r 2 LT (

)

LT = c2T /2π G σ0 X Y0 (x) X H0 (x)

YnX

b

T0 ∂e2 kJ J0 (kJ r) , σ0 c2s ωJ (kJ )

'

!2 +GR2

σ σ0

!2 +GR1

Vb.^c T7_ XQZCZ&X&]_iU.XQW\<`&` m b d pfdxc+i ]XQ^RwY RLXC[

Z

U.`Z']X'Y^.XQ&`\

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YnX

u

b dpfd

i

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mc2s X YnX m

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m

Zf7R<WR

 *  

σ σ0

αT = 

jd]_^.XC[<XQWe`QYT^RekT

m



αGR1 =  b dpfd

u

SR


X

σ σ0

c7R

m

 !2 +T 1/2   

b dpfc#fire`

 !2 +GR1 1/2 

1 m ' l σ0

 

LT ∼ ρ

X

1/2

m ∂e  ' σ0 ωJ (kJ )ρ2 ehR
.

b dpfd



σ0

 

b.W.`\<XQ^ WR<\^.`\<XQZC^j+f.xbe`\\Z&`<]n^.X ekj¤È7R"]R
m

XQZCcTf

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αGR2 =  m

(m/σ0 )1/2 ' . (ρ5 l)1/6

^.XC[<XQWe`QYT^RekT

XQZ&[\<XaUWTeXQWRUWTeXQ^.XQ^T7_iTC]`

ehR
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i

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b dpfdS¡i%f7R<WR
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m

T.[LRL[

X+m

m

'

ρ l σ0

.

b UWT

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z i

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XQ^^.`oi\j1wXV[<XC`WTTy`$.XQ^Te

^.`o>`cW.XQZ&[^.`Z&[T'RQ`\<`o>U.`CYZCTZ&[<X

[<`SYTZCU.XQWZCT7_ZCc1`W.`Z&[<XQo'RQ`\j+fK`m<]R
cs ' 9 ce Z X Tf f7R<WR`cW.X XQZ&[^.`Z&m [T+`<]n^ R ∂e ' 10 −1 ρ = cs /∂e ' 300 Z σ0 ' 10 h ' 70 ÷ 80 Uc U.`<]b RLXIe Uc X0m Uc TZ']`g`m<]R
YTZCcR

7 #Lk-# &#L$I"!'"( -#

m

y

r ' 0,7 rm ip ` Y RuZ&X m XQ^T.Xi\"R

b U7b1Z&[

Y RLXIe`&`

αT ∼ 2 · 10−1 ;

jZ&`cTob.W.`\<XQ^

xbe`\

X

UW.X'YZCc7RIj\RLXIekjo

[<`HV]b.c.[LbRT.`^^jXZ&[W7b.c.[Lb.Wj

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y

αGR ∼ 5 · 10−2 .

^ToGi

R<WR
b [<XQWe`QYT^RekT

Y]nT^jZC\<`m`QY^.`&`}UW.`mXQ'R`m<]R
T[LR
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m

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m

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m

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b ZQerpWTZp.fpfip

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T`[

b dpf¡wi

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Ω1 + Ω 2 Ω2 − Ω 1 r−R + arctg Ω(r) = 2 π L UWT

!

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L/R → 0 p

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m

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s

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d˜ p 2mΩ γs − 1 0 = − D(ln Ωz ) p˜ + dr rω ˆ γs 



γs − 1 + σ0 ω ˆ 2 − ∂e2 − D 2 (ln Ωz )0 (ln CΩz )0 c2s  ξ, γs dξ p˜ = dr σ0 YnX

ξ

Z

b dpf¡:dSi

1 m2 2 m Ω 1 γs − 1  + − − (ln CΩz )0 ξ, − 2 2 2 r ω ˆ cs rω ˆ r γs 



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s

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s

b dpf¡¡i

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m u = dξ/dt = ∂ξ/∂t + Ω ∂ξ/∂ϕ = −iˆ ωξ p Z']b RLX X C Ωz = const `GTI\<XQZ&[^.`1eb bX R<\^.XQ^T7_ b dpf¡:dSi b dpf¡¡iUT\<`QYy _[Pcf` .`$B .X"Cb"] I["RL[Lb fwc fw:d p R R R r < R i X U.`<]R<'R"_Z&``[\<XC[Z&[\<XQ^^.` Ω(r) = Ω1,2 p%WT C [<`1eZCTZ&[<XIehRb.WR<\ R"] ^.`o

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71 n (#Lk Q- n L

b dpf¡:dSi

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X

L!Q

p˜(r) = p1,2

b dpf¡¡i+ZC\<`QYT.[Z_cP`QY^.`1ebVY]_

2 ˆ 1,2 − 4 Ω21,2 m2  d2 p1,2 dp1,2  ω + + − 2  p1,2 = 0 , dr2 r dr  c2s r 

s

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ξ



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~X w XQ^T7_C[TfPb.WR<\^.XQ^ToaYn`<]

u

b dpf¡

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m

(˜ p + σ0 R Ω2 ξ)|r=R−0 = (˜ p + σ0 R Ω2 ξ)|r=R+0 .

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m

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m

m

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y

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s

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3

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= P0 (R) + P00 (R)ξ + p˜(R)/σ0 ≡ P0 (R) + RΩ2 ξ + p˜(R)/σ0 ,

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r→ 0

r → ∞ plb

T

m

b dpf¡

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s

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u

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p1 = A Im (k1 r) , ξ1 =

m

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m y

p2 = B Km (k2 r) ,

A 2 m Ω1 0 Im (k1 r) − k1 Im (k1 r) , 2 2 σ0 c s k1 rω ˆ1 (

)

(

2 k1,2 = {4 Ω21,2 − (ω − m Ω1,2 )2 }/c2s ,

[,\TY u

b dpf

+i u

b dpf

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XQ^jUWT

)

b dpf

Re(k1,2 ) > 0,

b dpf

fi u

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u

c+i

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m

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u u b b d pf +i Z&`]R
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fwc

k12 α2 YnX

−

k22 α1

÷

r=R

(Ω21 − Ω22 ) + α1 α2 = 0, c2s

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α2 =

b dpf

2 m Ω2 K 0 (k2 R) , − k2 R m ω − m Ω2 Km (k2 R)

"

u

b dpf

b dpf

dSi

u

¡i

uu

i

jd]_}^R<]_7Y^.`Z&[T}UW.X'YZ&[LR<\L]XQ^T7_}W.X Cb"] I[LRL[<`\Gm
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q = Ω2 /Ω1 T

M∗ = R Ω1 /cs p

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b dpf

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71 n (#Lk Q- n L

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X

U7]nT.[LbIY

U.`<]b

m

RLXIe

√ 2 2 2 (1 − q) Ω 1 m −1 2 σ ˜ (r, ϕ) ∝ Re{Km (k2 r)eimϕ } ∝ r−m cos mϕ + r . 8 c2s

b1Z']`\T7_aU.`Z&[<`I_^Z&[\RtVdRIjl\<`"eb y

ys

s

XQ^T7_V\TY^.`

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ω = ω (0) =

I^R

m

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b dpf

+i

7 #Lk-# &#L$I"!'"( -#

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T

b dpf

u

dSi+Z']X'YbXC[

ω (1) ' −

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b dpf

fi

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b

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)

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q = 0,7

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M = R Ω1 |1 − q|/cs =S Pi|1_.,?−V q|M∗ Z'>6;\%G`Z2 q = 1,3 m=1

E'/'$

q < 1 ip'` C[<`1ebP^.Xd_\L]_XC[Z_GbIYT\T.[<X&] ^jeg[<`[{VdRTQ `mWR XQ^R " R<\TZCTe`Z&[ } T^cW.XIeXQ^.[LR}^.X&b1Z&[<`o T\<`Z&[T `UTZCj\RLXIe`o u m b dp f dSi YTZCU.XQWZCT.`^^je b.WR<\^.XQ^T.XIe UWT I ^R XQ^T7_7f UR<WReXC[WR q = Ω2 /Ω1 = 0,7 T q = 1,3 Y]_ge`QYj m = 1 p TY^.` Xm [<`UWT M∗ 1 m X T^cW.XIeXQ^.[j\K`m`TfGZ']b R"_7f b q = 0,7 q = 1,3 i m<]nTI cTVYW7b.ScVYW7b.'b ^.` m s > UWT M∗ ∼ 1 TfBWRC ]nT T.X+`c7RI j\RLXC[Z_w\<XQZ ehRZ'b XQZ&[\<XQ^^je "UWT q = 1,3 X \<`Q mdqpÈ RPUWT q = 0,7 `ZC^.`\^.`o\c7]RIYi\>T^cW.XIeXQ^.[ m ^.X&b1Z&[<`o T\<`Z&[TGY RLXC[VeX'f7R<^T" eT oKp s ~R
b

m

U.`<]b

RLXIe

Ω1 (1 − q) σ ˜ ∝ cos m ϕ + (r − R) × 2 cs

   

y

2 r−R exp M∗ Ω1 (1 − q ) 2 cs , × r−R 2   exp − M∗ Ω1 (1 − q ) 2 cs , Xm

Y R\TY^.` [{V%`Web

y

[Z

m

(r < R) (r > R) .

b dpf

wi

s

[<`,^.X&b1Z&[<`o T\jrXxU.` b dpf +i+\<`"eb XQ^T7_PU7]`[^.`Z&[TTeX g VZCUTWR"]XQo p R<S[LR

UWR<\L]XQ^TT>`UW.X'YnX&]_XC[Z_>Z&``[^.`$wXQ^T.XIe

S = m λs = RBb.]`\R"_>ZCc1`W.`Z&[ XCXd\WR

s

4 π cs 2 π m cs ' , |Re(ω − m Ω)| Ω1 (1 − q)

b dpf

c+i

XQ^T7_

Ωp =

Re(ω) Ω1 = (1 + q) . m 2

b dpf

dSi

7 #Lk-# &#L$I"!'"( -#

[<`

s

C[Tf\<`"eb

XK\W.XIe_ReU7]nT.[
bIY R"]XQ^T.XIel`[PWRIWj\R{Z

b dpf

Z&`]R
XQ^ToPYn`\<`<] ^.`VmjZ&[W.`ab7mj\RLXC[aZ

#

wirf7R<WR
^T7_RekU7]nT.[LbIYj

2 cs cs ' Im(ω) M∗ Ω1 (1 − q 2 ) m T>UWT M∗ 1 \<X&]nT T^R Ls S p m behekTW7bXIe U.`<]b XQ^^jrXW.X"C bI] I[LRL[jSpXQ^.[W.`mX X f7R<WR
^T7_tU7]`[^.`Z&[TtUW.X'YZ&[LR<\L]_

y

[

Z&`m`oi`[Z&["R

Z&[<`I_[<X&] Z&[\<`>\j]_7YT.[P\<XQZ ehR"RehR<^ W.X w XQ^T7_,UW.`m<]XIekjUW.`TZf1`

m

ys

b dpf

m

^R"_V^.X&bZ&[<`o Y RLXIekjrXX

y

¡i

T\<`Z&[ s

\<`"eb

X

T.XZCUTWR"]nT pJ`Z']X'Y^.XCXa`m

T\jeuZB[<`

m

cT

IW.XQ^T7_}\<`"e`

^.`&`

YnXQ^T7_aZCUTWR"] ^.`&`SbQ`WRd'R"]R
\WRekc7RIfBWR
]XQo

Ω .XQ^ ^R"_^.X&b1Z&[<`o T\<`Z&[ %&XQ^.XQWTW7bXC[Z']nT Bc1`1esc1`W.`[cT.Xr`[W.X"IcTBZCUTWR# Ls S iT}^.X,\j+YnX&]_XC[PU.`\<X&]nT m T^.X,T^cW.XIeXQ^.[LRPc7R
cW.XC[^7b

m

y

e`QYbp

[<`

WRIWj\<`1e

[<`TZCZ']X'Yn`\R<^T.X m`<]XCX W.XIR"]nTZ&[T

m

^j+f

Ω \j+YnX&]nT.[wc7R \jrXr^T#BT.Xe`CYj b \jZ BT.Xe`QYjZ m ∼ R/L YnX L Z BTWT^RWR"ehRIcT X Ω ^.Xrm

Xm

X+\W.XIe_d_ZC^.`

70 100%AM'J":
]b

%UW.X'YnX&]nTey\L]nT7_^T.X m

XQ^^jrXw\j1wXdW.X"Cb"] I[LRL[jSp

]R
|r − R| < L

ZCU.`<] CbXIe[LR
M∗ 1

I^R

m

XQ^T7_

XB[<`[ VdR
Z&[W7b.c.[Lb.WnR>^.X&b1Z&[<`o

W_7Yc1XdU.`aehR"]`o>\<X&]nT

e`

m

s

Ω

^RxU.`

[<`a\`m

[<`G\V^R
T\j+f\<`"eb

s

m

RLX

XQ^To}\>WRIYTR"] ^.`1e^R
|kR| = Ω (r) \dU.XQW.X'f1`QY^.`o`m<]R
Λ1/2 1/M∗

Ω1 M ∗ Λ 2 {m (1 − q)2 + M∗2 (1 + q 2 )2 }, ω = ω (0) − i 4

^.`,U.`<]b

T.[$

T^.X

XQ^TTU.`<]R<'RLXIe

Xm

Λ = L/R 1 `Z'b XQZ&[\L]_XC[Z_ Ω(r < R − L) = Ω1 Yn` Ω(r > R + L) = Ω2 p

^TTl_\L]_XC[Z_uc1`W.`[c1`\<`<]n^.`\<`o

jd]_K]nT^.XQo^.`oVR

^.`ehR"]nje

Xm

m

WRIWj\RSb.]`\<`oVZCc1`W.`Z&[T

U.XQW\<`1etUWT.m<]nT

m

ZwYn`Z&[LRL[<`

U7]R<\^jo,U.XQW.X'f1`QYV`[

WR" ehRIcT

BD.?.D.;

b dpf

u

fw:d

i

b dpf

i

71 n (#Lk Q- n L

ω (0)

YnX ys

[Z

y

`UW.X'YnX&]_XC[Z_ Y Rw\TY^.`

XSm

b dpf

L!Q

+ip

[<`K\wUWT.m<]nT

XQ^TT

m X ih\<X&]nT T^R `UW.X'YnX X b dpf c+i_i ^.X,T "eXQ^7_XC[Z_vphY^R

b dpf

u

#

Re(ω)

"RZdehR"]nje m y m X'm s TZ']`1eZCUTWR"]XQo>`c7RI j\R [Z_m`<]XCXx^.X&b1Z&[<`o T\jekT XIe\<`" eb XQ^T7_Z X s m 1 p ` \<[<`Wj+f `m XCX b T `ZC^.`\^.`X,T" " R M∗ 1 ibeXQ^ w XQ^T.XT^ m cW.XIeXQ^.[LRx\wZ&``[\<XC[Z&[\TTaZ b dpf ¡ib.\<X&]nT T\RLXC[Sf7R<WRWRIYTR"] ^.`1e^RcWT\<`o>\WR XQ^T7_>ZCTZ&[<XIebVb.WR<\^.XQ^To b dpf¡:dSi b dpf¡¡i m m e` ^.`W.X K RL[ TZ']XQ^^.`^Ry c7R
CWR<^T

m

R"_Z&[<XQU.XQ^

&#L$I"!'"( -.#

^j+fPb1Z']`\T7_7f

∂ ln(˜ p) (0) = i k2 ; ∂r |r→∞ YnX

ξ|r=0 = 0,

b dpf

1q (0) (ω − m Ω2 )2 − 4 Ω22 ; Im(k2 ) > 0. b dpf |+i cs Xm X s X SZC^.` [<` U.`<]R<'R"_WRIYTR"] ^.`XZQeX XQ^T.X ξ = 0 \.XQ^.[W.XwYTZCc7R ekj TZ5 ym c7] RLXIetTwWR^R 2 iZ&[LRLmT7]nTI TW7b [Z_ U.`<]n^.`Z&[ UWT > > Λ ∼ 1/m p m = 2 Z&[LRLmT7]nTI TW7bXC[Z_ UWT Λ ∼ 0,45 b \UW.X'YnX&]X `QY R q → 0 ip+[<`[BW.X"C b"] I ["RL[B^.XC[W7bQY^.`KU.`^7_[ 1\<`w\WR s R ys XQoZ_,^.XQZ TehRLXIe`o m s X(m TYc1`Z&[TPY]_WR
k2 =

7 #Lk-# &#L$I"!'"( -# +'( _*( gL!>@5&YB$

#%$'&(

/'G1.59'$*;

ZG\%$'9'$u

/65!Y>$'I'9'2QM

W

\%GiZ

RE'G

$'9u

m = 2,1 3,E4,>G. .&:.AG&YB$ A'/'3:\%3:9*BH

E65?u ]

/65\%3YB/'G1

M∗ N = RΩ1 /c s E'/'$ q = Ω2 /Ω1 ] Λ = L/R = 0,02 E'/'$ (0$'&s>@5h1:/65?u Λ =BG0,1 9'$'I'9'2QM I'A5?M A'/'$'12QM $'9'A'/'3:\%3:9*B.2

13sZ$'9'$ 65?M

Ω1

#%$'&(6+'( Pid*(-0/'$'123[\x5/:$'965?>9'G~JH.&YBG~'I'$'1G&YB$\%G!u Z2

ClA5!kZ@5!;

BGI'A5

AGBG/'2QM

WE65/65

`965?u

m = 2RR0GE'/'3sZ3s>6;'3YBKE'/'Go[$>tA'/'$'1G~h1/65!783su q, Λ 9'$*;mW+'( P?U).RsCfZ'>6;=AGBG/'GYG E'/'$ &:GGB!u Im(ω) = +0( 0/'Go[$>$ 13YB&YB1H7l3:\ `965?I'3:9'$'$ E65/65\%3YB/65 M∗ C!&:GGB13YB&YB.1H7l$'3BGI'A5\ C &:\( 1/ V (r) ∗ 1, 2, 3 /'$'&(6+'( PP I'3:9'$'~W

#%$'&(+'( PP(.-0/'$'1231/65!783:9'$*;tWGA'/'3:&YB9'G&YB.

19*HB/'3:9u

9'3,:G

:G/'L.5.RsC?ZGE*H&:A*578$'3 1G`LHkZ3:9'$'3QZ1H.M*/*HA*51!u &:E'$'/65?>9'G: G H`G/65 &\x5?>2\j$'9'A'/'3:\%3:9*BG\ W&:\( /'$'&('+'( Pid.RE'/'$ ( Q G1GA*HE'9'G&YBcE65/65\%3YB/'G1 M∗ = 10 Z'>6; Y ! B$MA'/'$'12QM 9'G:G

{Λ; q; nmin = min(d ln Ω/d ln r)} $'\%3:3YBJ&s>3sZ6H7l$'3l`965iI'3:9'$*;DN%P ,

m

RLX%\eXQ^ xb

mT7]nTITW7b

ys

y

{0,29; 0,2; −1,7}

Λ

Z&[<`W.`^7bGU.`KUR<WReXC[W7b

Teg\L]nT7_^T.XIetZ

])

]

{0,36; 0,1; −1,8} {0,24; 0,3; −1,5}

`m
X

U.` \TYTe`1eb

X

Z&[LR#

TehRLXIe`Z&[TZCW.X'YjSp

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`m<]R
\<X'YnXQ^jg[WTacWT\jrX\WR

s

XQ^T7_

f7R<WR
b dpf¡c+i X

b \d`cW.XQZ&[^.`Z&[Ta\^7b7[W.XQ^^.XQ&`w&`W.m1R+i m ys

`mXQZCU.X

T\R

TekTP\<`Qm
YnXQ^T.X%Y\Lb1fW7b

XGm

X

M∗ = 10 ZwehR"]njegT^cW.XIeXQ^.[<`1erp TY^.` [<` s y m Y]_s\<`Qm
71 n (#Lk Q- n L

L!Q

&#L$I"!'"( -.#

#%$'&(+'( P?,(D51$'&:$'\%G&YB\x5&su /65?Z$65?>9'G:G

XbB5L.5 9'$*;

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HL21.5?u

1G:\%H783Y9u

9'G~{E>GB9'G&YB$

GB

E65?u

Ls A'/'$'1G~ /65\%3YB/65 Λ = L/R 1/65!783:9'$*;W+'( P?U).RZ'>6;j\%G`Z2

m=2

"

#%$'&(

+'( Pi)*( QG`\FH783:9'965!;

E>GB9'G&YB 1

S P`_.,?V

Z$'&:A3

σ ˜ (r)

E'G13:/M*9'G&YB965!;

9'3YH.&YBG~'I'$'1G~w\%GiZ2

&mA'/'$'1G~ ]

]

m=2

W+'( P?U).RsN

1/65!783:9'$*; ]

q = 0,1 ] Λ = 0,335 ] M∗ = 20] Ls = 2 R q = 0,1 Λ = 0,385 M∗ = 5 Ls = 6R

y

XQ^^.`owY\Lb.&`W.m`Z&[ X(u#X'X#z X' p z#R<XGc T.XrXGc WT\jrX\WR ^juYn`\<`<] ^.`BTW.`c1` ¡ d d1| ¡p Z']RLm`\jWR

s

]

XQ^T7_BWR
Λ = L/R Re(ω) Λ M Re(ω) 0 1 " # Λ Ω = Re(ω)/m q M Λ ^jolehR R bIYn`\L]XC[\<`WT.[<X&] ^.``UTZCj\RLXC[Z_ Z&``[^.`$w XQ^T.XIe b ZCW pZ b dpf ¡i_i cs z . Ls ' b dpf +i Im(ω) dptW.`Z&[<`1eUR<WReXC[WR UW.`TZf1`QYT.[w^.XQc`[`W.`XbeXQ^ wXQ^T.X m u m −2 < b ZQerp b1Z']`\T.X b dpf ph% WTYn`Z&[LRL[<` ^.`sehR"]`1e i_i`[Q]nT T.X ∼ ∗ m `[ I^R XQ^To b d pf ++ i ^.XxUW.XQ\j KRLXC[ft \Z&``[\<XC[Z&[\TTZdW.X CbI] I[L R m b [<`1e d pf i p· `GUWTc1`^.X ^j+f C[<`[ VWV%XQc.[GZ&[LR<^.`\T.[Z_ "ReXC[^je T ys UWT\<`QYT.[,c>Z&``[\<XC[Z&[\Lb TegW.`Z&[Lb KR<'RKZCUTWR"]nT b ZQerp b d pf +i_iTGbeXQ^ y s b ZQerp b wXQ^T b.]`\<`o>ZCc`W.`Z&[T>XCXd\WR XQ^T7_ d pf dSi_ i p p X X ∗ ¡7p `V\Z&XQoTZCZ']X'Yn`\R<^^.`o>`m<]R

R
X

Z'b

W.`Z&[<`1eUR<WReXC[WR

s

XQZ&[\<XQ^^.`XbeXQ^ wXQ^T.XT^cW.XIeXQ^.[LRw^.X&b1Z&[<`o

Λ

UWT\<`QYT.[Bc,\<XQZ ehR!"ReXC[^.`1ebWR
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y

m

T\<`Z&[TGZ `m<]R
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B 2

$ ( E %

$%.

.0 2 ÷

.&

"*

"

Λ

> ∼

0,2

. (%""%

Re(ω)

&(% % &* "

7 #Lk-# &#L$I"!'"( -#

r R `mj m ^.`af7R<WR
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b dpf

z

fi

wi

g = −p00 /σ0 p `CYnX&] ^R"_VcWT\R"_V\WR s XQ^T7_`UTZCj\R"]RY]_ m = 0 U.`<]b m Te   ! 2 2 dξ ω − ∂e 1 d 1 z b dpf c+i r + − 2 ξ = 0 , 2 r dr dr cs r YnX

z

b dpf

s

ξ Yn`<] 0 ξ(r → ∞) = 0 p YnXSZQeX

XQ^T.X

^.`bQYn`\L]XC[\<`W_[ CWR<^T ek^.`

m

R"_

^jeb1Z']`\T7_ne m C[<`b.WR<\^.XQ^T.XG^R rξ ∗ b I^R `c

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y

Y Rs\TY^.`

Xm

[<`

^.X&bZ&[<`o

TeXC[ PeXQZ&[<`V[<`<] c1`P\V[<`1euZ']b

m

RLX

m

X

T\<`Z&[

XQZ']nTsZ'b

U.`<]b

m

RLXIe

b dpf

z

dSi

ω 2 < 0 iwe` XC[ ∆r = r2 − XQZ&[\LbXC[aT^.[<XQW\R"]

e`QYj s

X

ξ(r = 0) = m `I ^R RLXC[

m = 0

b

71 n (#Lk Q- n L

r1 6= 0 X

b

\^7b7[WTc`[<`W.`&`

`Z']X'Y^.XCXde`

m`<] BT^Z&[\R

jd]_

X

)

&#L$I"!'"( -.#

r1 < r < r2 ia\<X&]nT

XC[,TeXC[ ,eXQZ&[<`

nmin

L!Q

XQZ']nT (

m

∂e

T^R

2

`[WT RL[<X&] ^R.p

d ln Ω < −2 . = min r d ln r

'R"]R
)

Z

Y\Lb.&`W.mjekT

b dpf

cWT\jekT

\WR

s

z

¡i

XQ^T7_

nmin > −2 Te`QY R m = 0 \^Tf\<`Qme`QY R m = 2 ^.X&b1Z&[<`o [<`,U.`c7R"R"]nT T\RTPUWT nmin > −2 m X X WR

"R
ym

y

XQ^T.XtWR
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Zib

m

XC[<`1e

\<`"e{b

&XQ^.XQWTW7bXIe`o

UWT\<X'YnXQ^^.`o\<`\\<X'YnXQ^TT

s

XQ^ToCWR<\T

ZCUTWR"] ^.`oyZ&[W7b.c

cY R<^^.`1ebWR&YnX&]b`$.XQ^cT

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7 #Lk-# &#L$I"!'"( -#

70 10 {0 918CN.:<9

7H:<J";.:M'J"D

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WRIWj\R

'RQ`\<`&`wYTZCc7RK\`cW.XQZ&[^.`Z&[T ^.X&b1Z&[<`o

m

T\<`Z&[TTy\<`Qm

\L]nT7_^TTW.X"Ic1`&`T"eXQ^.XQ^T7_iU7]`[^.`Z&[T

Ω

Y RLXIekj+fyZ&[W7b.c.[Lb.W p%W.X

WR\j1wXG`QY^.`W.`QY^jrXe`QYnX&]nTZ m

XQ^.[W.`mX . YnXa\Z&XQ&`"ReXC[Te

^RUR<WReXC[Wj

^.`o

Xhm

[<`

~b.]`\<`oZCc`W.`Z&[Tmj]nT

dS0 /dr ≡ 0 ipx%WT^R"]nT TT,ZCc7R m c7R%U7]`[^.`Z&[TV^.XC`mQf1` m X YTe`PTZf1`CYT.[ PT^.XQT Q^.[W.`UT XQZCcTf e`QYnX&]XQo p(jxXQoZ&[\T.[<X&] ^.` Y]_}WR" T Q ^.[W.`UT

^.XQ\<`"e`

Q^.[W.`UT Z&`ZCc7R

m

X

b

XQZCcTekT

Wj\^.`oe`CYnX&]nT>Z

m

b dpf¡wi+T

σ0 (r) = σ01 + (σ02 − σ01 ) Θ(r − R) m

b dpf

^.`V`CY^.`\W.XIeXQ^^.`XB\jrU.`<]n^.XQ^T.XwbZ']`\ToWR<\^.`\<XQZCT7_

S0 =

` ]

b

^.`Z&[T

p0

c1`1eGY R<\L]XQ^T7_

b dp

Y]_

[W.XCm
y

m

u

dSi_ip ReXC[Te m m

[%^R"]nT

T7_BZCc7R m

Xm

[<`

X

z

|+i

"

b dpffiTT

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∂S ∂S z b ¡7pf i +u σT = Fν − Frad + Fconv + F∂e + Y, ∂t ∂r s X YnX S Z bIYnX&] ^R"_ Q^.[W.`UT7_\<X XQZ&[\R.p UWR<\RGZ&[<`I_[>Z']R<'RLXIekjrX `UTZCj ys ytX s \R T.X-Q^.XQWCT c1`[<`WR"_U.`Z&[U7]` RIYT b T7]nT>b1f`QYT.[VZ ^.XCX:i "RKX'YT^T 7b\W.XIeXQ^T px^.XQWCT7_\j+YnX&]_XC[Z_\Z']X'YZ&[\T.Xx\L_I c1`oGYTZCZCT UR TT ! 2 Wrϕ dΩ 2 9 b ¡7pfQ|+i =νσ r Fν = = ν σ Ω2 . σν dr 4 m m ]b XQ^T.XsZU.`\<XQWf^.`Z&[Tl`UTZCj\RLXC[Z_ ]XQ^.`1e Frad p C WRekc7RIfue`CYnX&]nT m m `U.[T XQZCcT>[<`<]nZ&[<`&` ~jZb XC[<`1eY\Lb1f>XQ&`Z&[<`W.`^e` ^.` " R
Frad

4acT4 . = 3 σk

b ¡7p¢w+i

1&Qh ' L# ,( -"!# s#"--

^.XQWCT7_ge`

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b ZQerp

X

U.XQW.XQ^.`ZCT.[ Z_tc1`^\<XQc T.XQo\}\<XQW.[Tc7R"] ^.`1e^R
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s

ZQe

2

XQ^T.X

/,

fi A Z

Z

gf f ' 1 f e

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m X

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ZQe

b `mWRL[^.`Xx[<`We` I^.`XxT C]b

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2

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c.p

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2

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,

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