Astr Physics

( )

20 2006

1 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 . . . . . . . . . . . . . . . . . . . . 41.2.2

. . . . . . . . . . . . . . . . . . . . . . 51.3 . . . . . . . . . . . . . . 10

1.3.1 . . . . . . . . . . . . . . . . . . . . . 121.3.2 : . . . . . . . . . . . . . . 131.3.3 . . . . . . . 14

1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 192.1 . . . . . . . . . . . . . . . . . . . . . . 20

2.1.1 . . . . . . 202.1.2 . . . . . . . . . . . . . . . . . . . 212.1.3 - . . . . . . . . . . . . . . . . . . . 212.1.4 . . . . . . . . . . . . . . 22

2.2 . . . . . . . . . . . . . . . . . 242.3 . . . . . . . . . . . . . . . . . . 25

2.3.1 . . . . . . . . . . . . . . . . 252.3.2 . . . . . . . . . . . . . . . . . 26

2.4 . . . . . . . . . . . . . . . . . . . . . . . . . 272.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 312.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 333.1 . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Jeans . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 . . . . . . . . . . . . . . . . . . . . . 38

i

ii

3.4 . . . . . . . . 413.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 433.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 454.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 . . . . . . . . . . . . . . . . . . . . 454.3 . . . . . . . . . . . . . . . . . . 474.4 . . . . . . . . . . . 494.5

. . . . . . . . . . . . . . . . . . . . . . . . . 504.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 575.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2 . . . . . . . . . . . . 58

5.2.1 . . . . . . . . . . . . . . . . . 585.2.2 . . . . . . 59

5.3 . . . 605.3.1 . 605.3.2 . . . . . . . . . . . . . 62

5.4 . . . . . . . . . . . . . . . . . 635.4.1 . . . . . . . . . . . . . 635.4.2 ,

. . . . . . 645.5 . . . . . . . . . . . . . . . . . . . . . . . . 65

5.5.1 . . . . . . . . . . . . . . . . . 655.5.2 . . . . . . . . . . . . . . . . . . . . 66

5.6 . . . . . . . . . . . . . 685.6.1 . . . . . . 68

5.7 . . . . . . . . . . . . . . . . . . . . . . . . . . 715.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6 736.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.2 . . . . . . . . . . . . . . . . . 746.3 ; . . . . . . . . . . 77

6.3.1 Zeeman . . . . . . . . . . . . . . . . . . . . 776.3.2 . . . . . . . . . . . . . . . . . 81

6.4 (MHD) . . . . . . . . . . . . . . . . . 83

iii

6.5 V irial . . . . . . . . . . . . . . . . . . . . . . . . . 866.6 Alfven . . . . . . . . . . . . . . . . . . . . . . . . . 896.7 90

6.7.1 . . . . . . . . . . . . . 916.7.2 V irial . . . . . . . . . 92

6.8 . . . . . . . . . . . . . . . . . . . . . . . . . . 93

1

, .

;

., , , , , , .

. , , .

, .

.

; , .

.

1

2

1.1: ( , , ).

() () cads [ms

1] 344 104 3 105 104...5 [kgm3] 1.21 103 10+5 1021 [m2s1] 1.5 105 10+2 102 1017

1.2: SI.

1 Mm = 106m1 AU = 1.5 1011 m1 pc = 3.1 1016 m1M = 2.0 1030 kg1 yr = 3.2 107 s1 Myr = 106 yr

. 1AU( -) . kiloparsec (kpc). 10Kpc, , , (light year). (1yr = 3.16 107s) (M = 2 1030kgr).

, .

AGN SN BH WD (, )CV Cataclysmic Variable ( WD )

1.1 3

1.3: .

6 Mm 70 Mm 700 Mm (CV) 10 1000 Mm 15 kpc

1.4: .

G 6.673 1011 m3 kg1 s2 Stefan-Boltzmann kB 5.67 108 kg s3 K4 R 8, 314 m2 s2 K1 c 3 108 m s1 0 4 107 V s A1 m1

1.1

. , , . .

(,), () ( Lorentz ).

: ( ) (); : ` (L) (`

4

1.1: E

.

E ~w ~u(~x, t). ( . (1.1) )

`

` = (n)1

n . 1015cm2 n 1019cm3, ` 104cm. n 10cm3 ` 1014cm ( -), 1019cm. .

1.2

1.2.1

. ( (~x, t),

1.2 5

P (~x, t), T (~x, t), ~u(~x, t) .

. Euler (~x = ~x0) , Langrange ( ) . (~x(t)). Euller

dQ

dt=

(Q

t

)

~x0

(1.1)

LangrangeDQ(~x(t), t)

Dt=

(Q

t

)

~x0

+ ~u Q(~x(t), t). (1.2)

(convectivederivative).

2.1: V = xyz

D(V )Dt

= ( ~u)V. (1.3)

(V =.),

~u = 0.

2.2: (Ox)

~u = 0.

.

1.2.2

.

6

1.2: m 2mvx.

, , (yz) x. m 2px, (2pxn0vx), n vx ( . (1.2)). x 6 , (2pxn0vx) 6,

P =2nmv2x

6=

nmv2x3

.

32kBT =

mv2

2

PV = NkBT, (1.4)

N = nV V kB Boltzmann . . = M/V = Nm/V

1.2 7

P =

mkBT (1.5)

(, )

P =

mkBT =

R

T (1.6)

R . ( )

, n. ()

=A

Z + 1

.

2.3: X , Y Z (Am = 2Zm, Zm >> 1)

=1

2X + 3Y4 +Z2

.

. m . Maxwell, Boltzman vx vx + dvx ( [;, ] ) ( )

Maxwell-BoltzmannfM (vx) =

(m

2kBT

)1/2exp

[mv2x2KT

]. (1.7)

8

. fM

fM (vx)dvx = 1.

.

P =

2nv2xfM (vx)dvx = nkBT =N

VkBT. (1.8)

fM (v) =(

m

2kBT

)3/2e

(mv2

2kBT

). (1.9)

< v2 >=< v2x + v2y + v2z >= 3 < v2x >

< v2 >=

v2fM (v)d3v = 4

0v2fM (v2dv)

=3kBT

m

( )

U =12mn < v2 >=

32nkBT.

Vrms =

< v2 > =(

3kBTm

)1/2.

Vm =

2kBT

m. (1.10)

1.2 9

( Vm )

< v >= 4

0vfm(v)(v2dv) =

(8kBTm

)1/2.

: , . dQ , , dU dV dW = PdV . U = 32NkBT dU = (3/2)NkBdT.

dU = dQ + dW = dQ PdV. (1.11)

CV =(

U

T

)

V

=(

Q

T

)

V

=32NkB.

CP =(

Q

T

)

P

.

(1.11)

(Q

T

)

P

= CV + P(

V

T

)

P

= CV + NkB (1.12)

CP = CV + NkB.

dQ S, dQ = TdS.

10

(dS = dQ = 0).

.

CV dT = PdV = NkBT dVV

. (1.13)

. (1.13) ln(TCV V NK)=

TVNkBCV = TV 1 = . (1.14)

= CV +NkBCV =CPCV

= 5/3. . (2.13)

P = k . (1.15)

= 1, .

c2s =P

. (1.16)

c2s = P/, , , c2s = RT/ = P/.

.

1.3

, , , ,

D~u

Dt= P.

1.3 11

~F ( )

D~u

Dt= P + ~F (1.17)

Euler 1.5,

1.5: V m, q, r, V , ~g, ~B, ~, , K .

~Fg = m~g Lorentz ~Fm = q~u ~B/cCoriolis FC = 2m~ ~u Fcf = m2r ~FD = 6K~u ~Fb = V ~g

. Lorentz, Coriolis .

(r)

~F = Gm(r)(r)r2

er = Ug (1.18)

Ug ,

m(r) = r

04r2(r)dr

r .

12

1.3.1

: m = V V , ( ),

D(V )Dt

= D(V )

Dt+ (V )

D

Dt= 0

. (1.3)

(V )D

Dt+ (V ) ~u = 0

t+ (~u) = 0. (1.19)

. (1.19) .

: dm = V . . (1.17) (V ~u)

(V )~uD~u

Dt= (~F ~u ~u P )V

D

Dt(12u2V ) = (~F ~u ~u P )V.

D(PV )Dt

= (V )DP

Dt+ P

D(V )Dt

= (V )(

P

t+ ~u P

)+ P ( ~u)V

D

Dt

[12u2 + P

]=

(~F ~u + P

t+ P ~u

). (1.20)

1.3 13

~F =Ug

D

Dt

[12u2 +

P

+ Ug

]=

(P

t Ug

t

). (1.21)

,

(12u2 +

P

+ Ug) = . (1.22)

Bernouli.

1.3.2 :

. . .

Fb = g

g 10m/s2 = i e .

= TT

.

60oC 300K / = 0.2. 1kgrm3 , 43 R

3g Mg 500kgr

43

R3 = M

R =(

3M4

)1/3= 9m.

14

100Km. , ( ) . 10 Krakatau, . . t =

2Hg = 140s.

.

mdv

dt= mg FD.

FD = 6Kv, = 2 105kgm1s1 , K , m v . -

0 = 43

r3Kpg 6Kv

p = 4 103Kgm3 . K = 1m

v =22Kpg

9 4 104m/s.

t = H/v = 10 .

1.3.3

: . .

1.3 15

. .

Oz (x, y, z) (x, y, z) .

x = x cos + y siny = x sin + y cosz = z

. = (~ = (0, 0, )) x, y, z xrot, yrot, zrot

xrot = x + (~ ~r)xyrot = y + (~ ~r)yzrot = z.

~rrot = ~r + ~ ~r.

(rrot)2 = (xrot)2 + (yrot)

2 + (zrot)2)

= (x)2 + (y)2 + (z)2 = (r)2.

Lagrange

L =m

2(r)2 Ug = m2 (rrot)

2 Ug = m2 (r2 + ~ ~r)2 Ug,

L =m

2[(x y)2 + (y x) + z2] Ug (1.23)

16

Ug . Langrange

m(x y)m(y + x)my + Ugx

= 0 (1.24)

m(y + x) + m(x y) + mx + Ugy

= 0 (1.25)

mz +Ugz

= 0 (1.26)

.(1.24)-(1.26) )

m~r = m~r ~ + 2m~r ~ + m~ (~r ~) Ug~r

. (1.27)

. (1.27) , , Coriolis (FC = 2m~r ~) (Fcf = m~(~r ~). . ( [;]).

(. (1.19) ) (. (1.17) (x, y, z) (r, , z)( . (1.3) ).

. , ( ).

(r, ) z u(r, ) = ur(r, )er + (u(r, ) + r)e

v = ur r

+ (u + r)1r

.

1.3 17

1.3:

= er r

+er

.

~u = urr

+

r

(u + r) +

ur

r

er . (er/ = e).

r+

1r

r(rur) +

1r

((u + r)) = 0 (1.28)

~u ~u = ur ~ur

+u + r

r

~u

( ).

[~u ~u]r = ur urr

+(u + r)

r

ur

(u + r)2

r

18

[~u ~u] = ur ((u + r)r

+ur(u + r)

r

+(u + r)

r

(u + r)

(r, , z)

urt

+ ururr

+(u + r)

r

ur

(u + r)2

r

= Fr Pr

(1.29)

ut

+ ur((u + r)

r+

ur(u + r)r

+(u + r)

r

u

= F 1r

P

(1.30)

r .

1.4

.

1. (. 1.19 )

t+(~u) = 0.

2. (. 1.17)

D~u

Dt= P + ~F .

3. ( )

P = C

P = RT .

2

( . 2.1) . .. ( LS ), , . .

2.1:

19

20

.

2.1

, ,

P = ~Fg, (2.1)

.

2.1.1

2.1: 0 = 1.4gr/cm3 .

,

dP

dr= GM0

R2

dP |0Pc= GM0

R2dr |R0

Pc = GM0R

= 2.7 1015dynes/cm2

Pc = 1.41017dynes cm2. , . ( . 1.5)

Pc =(

0kBm

)Tc (2.2)

2.1 21

(=0.62)

Tc = 1.4 107K.

2.1.2

(. (2.1 . ~Fg = ~g, ~g . . (2.1)

dP

dz= g

P = c2s

c2sd

dz= g

1

d

dz= g/c2s

(z) = 0ez/H (2.3)

H = c2s/g.

2.1.3 -

,

Pg =Fg

4r2=

R0

G(4r20dr)(43r

30)4r2r2

= 23G2R2 = 3

8GM2

R4

22

( )

P = nkBT =MkBT

43R

3mH.

38

GM2

R4=

MkBT43R

3mH

M 2kBTGmH

R (2.4)

M = 2 1033gr M. .

2.1.4

. , . dr, r (r),

m(r)Gr2

4r2(r)dr

4R2dP

dP .

dP

dr= Gm(r)

r2, (2.5)

m(r) = r

04(r)2(r)dr. (2.6)

2.1 23

= ce

(1 r

Rc

)

ce . . (2.6)

m(r) =43

r3ce r4

Rcce (2.7)

M = m(Rc) =R3c3

ce

m(r) = M[4r3

R3c 3r

4

R4c

]. (2.8)

m(r) . (2.5)

P = Pce G2ce[23r2 7

9r3

Rc+

r4

4R2c

](2.9)

Pce P (Rc) = 0

Pce =536

G2ceR2c =

5M2G4R4c

.

r

P (r) =5G2ceR

2c

36

[1 24r

2

5R2c+

28r3

5R3c 9r

4

5R4c

]. (2.10)

(P (r) = nkbT (r)) r

T (r) =mHP (r)kB(r)

=536

GmHkB

ceR2

[1 +

r

R 19r

2

5R2+

9r3

5R3

].

. (2.5) (2.6) ( Bowers and Deeming, Vol. 1, Chapter 7 ).

24

2.2

= csk ( ). (P = P0+P1, =0 + 1, u = u1 P0 = ~F

1t

+ 0 u1 = 0 (2.11)

u1t

+10P1 = 0. (2.12)

P = c2s

P1 =(

dP

d

)

0

1 = c2s1 (2.13)

. (2.11), (2.12) (2.13)

21t2

c2s21 = 0 (2.14)

. cs.

1 = 10ei(~k~rt)

.

= csk (2.15)

(). ( )

cads =(

5P3

)1/2=

(5kBT3mH

)1/2(2.16)

cs = 10(

T104K

)1/2 Kms .

2.3 25

2.3

2.3.1

( . 6.18)

W = 16220G

3

R0

r4dr = 35

GM2

R.

2.2: 6.18 Virial, L = 3.0 1033ergs/sec.

dW

dt 3

5GM2

R2dR

dt

Virial W = 2U ,

dW

dt 3

5GM2

R2dR

dt 2L

( (U ) (L)

dR

dt 2.38 105cm/sec

, . , .

2.3: R0 >> R R0 R, ;

26

E = W +U , Virial W = 2U , E = U .

U = 12W =

12

(35

GM2

R

) 1048ergs

tk =1048ergs

1033ergs/sec 107yrs

, 109 .

2.3.2

,

41H1 2 He4 + 2e+ + 2e + Energy.

41H1 4(1.0078amu) 2He4 4.0026amu (1 amu(atomic mass unit)=1/12 12C). M = 0.0286amu,

0.02864.0312

= 0.71%

. (0.1M)

E = 0.71 (0.1M)c2 = 1.27 1051ergs

tn = t0L = 1010yrs

.

2.4 27

2.4

r V = r S

[P (r + r) P (r)] S + GM(r)(r)Srr2

= 0

dP (r)dr

= Gm(r)r2

(r).

M(r + r)M(r) = 4r2(r)r

dM

dr= 4r2(r)

.

E = U + Ug

Virial 2U + Ug = 0

E = U = Ug2

28

. dL(r)dr 4r2(r)(r),

dL(r)dt

= 4r2(r)(r)

r , A = 4r2 r Prad = 13T

4,

dPraddr

dr4r2

[m2/kgr] L(r)/cdr.

L(r)cdr = dPraddr

dr4r2

L(r) = 4r24c3

T 3dT

dr(2.17)

1 2 ( . 2.2). 1(r) P1(r) 1 r+r, 2(r), P2(r). (r), P (r).

P2 = P1(2

rho1)

( ).

2.4 29

2.2:

P2 = P2 = P (r + r) ' P1 +(

dP

dr

)dr

2 = 1

(P2P1

) 1

' 1(

1 +1P

dP

drdr

) 1

2 ' 1 + 1P

dP

drdr

2 2 > 0

dP

dr d

dr> 0. (2.18)

(P =kT/mp) =

P

T

dT

dP () = 4 < > (1 r/R). () () () ()

32

2. ()

f(r) = P (r) + GM2(r)/8r4

( P (r) r M(r) r) . () .

3. , dr/dt = 5r4. , , , () () () .

4. (39.52AU ) ,

5. M 0=.,

6. () , V = V (r) Poisson

1r2

d

dr

(r2

dV

dr

)= 4G

1r

d2

dr2(rV ) = 4G

() (=.) () () = rho0er, =.

3

. . .

.

3.1

Rc . . .

r = GMr2

(3.1)

r ,

M =43

R3c0

33

34

0 . . (3.1)

r

Rc= 4

3G0

(r

Rc

)2. (3.2)

( = r/Rc, = t/t0, t0 =[(4/3)G0]1/2)

d2

d2= 2. (3.3)

122 = 1 1. (3.4)

= 0 = 1. . (3.4) , = cos2 (3.2)

22 cos2 cos2 = 0. (3.5) . (3.5)

12 +

14

sin 2 = 21/2. (3.6)

. (3.1) . (3.6). . (3.6) = /2( = 0)

tff =

332G0

. (3.7)

tff . 21020Kgr/m3, tff 106 .

. . (3.1)

3.2 Jeans 35

3.1: . (3.6).

Rct2ff

4G03

Rc

tff

R3cGM

. (3.8)

. (3.7).

3.2 Jeans

. , , , P0, 0,0, u0 = 0 ( ).

P = P0 + P1 = 0 + 1u = u1 = 0 + 1

36

0 ~u1 = 1t

(3.9)

~u1t

= (1 + P1/0) (3.10)21 = 4G1 (3.11)

P1 = c2s1 (3.12)

(3.9), (3.10)

t( ~u1) = 21

(c2s0

)21 (3.13)

t( ~u1) = 1

0

21t2

. (3.14)

21 . (3.11) (3.13) (3.14)

(2 1

c2s

2

t2+

4G0c2s

)1 = 0. (3.15)

. (3.15)

(2 1

c2s

2

t2

)1 = 0 (3.16)

P0 = ~F . 3.15 3.16

1 = 10ei(~k~rt)

( ~k ) . (3.15)

D(~k, ) = 2 k2c2s + 4G0 = 0 (3.17) . (3.16)

D(~k, ) = 2 k2c2s = 0. (3.18)

3.2 Jeans 37

3.2: . k kJ k > kJ = kcs.

D(~k, ) D(~k, ) = 0 . . (3.17) , k2 < 4G0/c2s 2 < 0 ( = i)

1 = 10etei~k~r.

Jeans

J =2cs4G0

=2kJ

(3.19)

. 1020Kgr/m3, cs = 1Km/s J = 20pc.

. (3.2) (1) k

38

, (2) k >> kJ = kcs.

3.3

VR(r) ( . (3.3).

=VR(R + r) VR(R r)

r.

R = 150pc

3.3: V (R) . .

, = 2 1015sec1. Fc = m2r ,

GMcR2c

2Rc

3.3 39

=(

McG

R3c

)1/2(3.20)

. (3.20)

>32

4G= c = 3.6 1062gr/cm3

. c = 1023g/cm3

. 5 ( 2 1024g/cm3). , , 5-10 .

3.1 Fc = m2r

d2r

dt2= GM

r2+

L2

r3(3.21)

L = r2= . . (3.4) . (3.21) L2/(GMcRc) = 0.1

. (3.4) req = L2/GMc . (3.4).

40

3.4: .

3.1: , , 10Km. , 25 .

: , L = I,

I =25MR2

25MR2 =

25MR2ff

Tf =(

RfR

)2T = 0.1msec. (3.22)

3.4 41

3.4

T .

3.5:

, , , Rc, . Virial ( . (6.16) ),

12

d2I

dt2= 2U + Ug 4R3Pext (3.23)

.

Ug = 3M2c G

5Rc

( . (6.18). (Pext). c = Mc/(4R3c)

U =32

MkBT

mH.

42

12

d2I

dt2= 0

. (3.23)

Pext(R) =3kBTMc4mHR3c

3M2c G

20R4c. (3.24)

. (3.6)

3.6:

Rc.

RJ =[

3

1516

kBT

mHcG

]1/2(3.25)

Jeans. R > RJ , RJ . , .

W = U + Ug =3MckBT2mH

3M2c G

5Rc(3.26)

3.5 43

( ) R < Rcr =(2/

3)RJ . . (3.24) Rcr

Pmax 3.14(

kBT

mH

)4 ( 1M2c G

2

). (3.27)

Pmax. . (3.23), Pext > Pmax.

Ug =35GM2

[1

Rf 1

R

].

.

3.5

. .

: 0 = 1024g/cm3, B = 3G = 8 1016 2 1015sec1.

1. ( ) .

2. , , .

44

3.

4.

5. = kcs ( ).

. . .

3.6

1. tff = [3/32G0]1/2 0 = 2 1024grcm3 4 1023grcm3. tff .

2. r = 0.1pc, = 4 1023grcm3 = 2 1015s1 (30Kms1)

3. E = U + W Jeans

R2c = (4/3)R2J

4. () T = 50K 0 = 8.4 1022grcm3. () T = 150K 0 = 2 1016grcm3. ,

4

4.1

. ( ) . (jets) . , . - (Gamma Ray Bursts) .

4.2

R . .

. :

1. f(v). v > v, v

45

46

.

J =12n0(t)

v

f(v)v(v2dv) (4.1)

Maxwellian v , . , :

kT =GM

R(4.2)

T , . T 107K, 106K .

2.

dP (r)dr

= GMR

(r) (4.3)

P = nkBT = mpkBT

dP (r)dr

= GMR

mp

kBTP

dP (r)dr

= P

= (GMmp/kBTR)

P (r) = P0e(1R/r) (4.4)

r , P = P0e P (r ) = 105dynes/cm2

4.3 47

Pc = 1012dynes/cm2. ( )

,

dM = 4r2(r)dr

M = 4r2v(r)(r)

1AU n0 = 7 cm3, 0 = n0mp v(r = 1AU) = 400km/sec M = 31014M/. .

4.3

1.

t+(~u) = 0 (4.5)

2.

(

t+ ~u ~u

)~u = P Ug (4.6)

3.

t(E) +

[~u

(E +

1P

GM

R

)]= 0 (4.7)

E = 12v2,

.

. ,

48

1r2

d

dr(r2v) = 0 (4.8)

vdv

dr= c2s

dln

dr GM

r2(4.9)

(4.8)

dln

dr= 2

r dln|v|

dr

(5.2)

(v2 c2s)dln|v|

dr=

2c2sr GM

r2

v > cs

r =GM

2c2s

( r = 0.1AU . (4.7) ( )

B(r) =12v2(r) +

1c2s

GM

r(4.10)

r = r, v(r) = cs GM/r = 2c2s, ( = 5/3), B(r = r) = c2s. B(r = R) = GM/R ( v(r ) >> cs) B(r ) = v20/2, B = B(r )B(r = R) = 1/2v20 +GM/R

. , . .

4.4 49

4.4

(compact object ) , .

M = 4r2(r)v(r) = const. (4.11)

(r) = (mp + me)n0 ( n0 ) v(r) .

( ) V

L = V n20 (4.12)

= T mec2( , T ) .

12v2(r) =

GM

r(4.13)

(4.11) (4.13) n0

n0 =M

4r2(me + mp)(2GM/r)1/2

(4.12) (mp >> me),

L =(

43r3

)[M2

(4r2)2m2p2GM

r

]T mec

2 =M2T mec

2

12m2p2GM(4.14)

(4.14) L Eddington

50

LE 4GMcT

M2T mec

2

24m2pGM

M2E =962G2M2mp

T mec(4.15)

L

LG=

[T mec

2

24mpGM

]M2

GMMr

=(

T mec2r

24mpG2M2

)M =

4cME

M

ME= 104

M

ME(4.16)

.

4.5

(r)t

+1r2

r(r2(r)v) = 0 (4.17)

[v

t+ v

v

r

]+

r

(kBT

mp

)+

GM

r= 0 (4.18)

(E)t

+1r2

2

r2

[r2

(5kBT2mp

+ E GMr

)]= (, T ) (4.19)

E = 12v2 + , (, T )

(, T ) = 2T 1/2 (4.17), (4.18) (4.19)

v(r, t),(r, t) T (r, t).

4.6 51

4.6

1. (

t = 0

) ,

() r () r 1AU cs = 100km/sec() r

2. (heat flux)

~F = T

= 0(T/T0)5/2 T0, 0 .() ~F = 0

T = T0(r/r0)2/7

() P = kBT

P = P0exp

[7r05H0

[(r

r0

)5/7 1

]]

H0 (scale height)() .

3. , Euler

t+ (~v) = 0

~v

t+ ~v ~v +p = 0

t+ ~v + p

~v = 0

:

52

t(vi) =

xj(vivj + ijp)

t(12v2 + ) =

xj

[vj

(12v2 + + p

)](4.20)

.

4. , ,

t+

x(vx) = 0

vxt

+ vxvxx

= c2s

x(4.21)

() = vx = v0() vx ei(kxt) = (k).() vx x. .

5. 4,

6. 4, .

7. ( ) , : , . .() . (, )

4.6 53

ud

dr(12u2 +

1p

GM

r) = (r)

udu

dr= 1

dp

dr GM

r2

dM

dt= 4(r)u(r)r2 (4.22)

u , , p , , ( ) ( ). dMdt = const. ( , )() , ( )

(u2 c2) 1u

= something

c2 =p

( u < c u > c r ):

c2s = u2s =

GM

2rs+

12

(ru

)

s ( ) rs ( )() ,

|dMdt| =

rsrm

4r2(r)dr/[GM

rm+

5 34( 1)

GM

rs+

+ 14

(ru

)

s

]

54

() r > rm FUV , FUV ( , ). rm:

rm

dr 1

|dMdt| = 4r

3mFUV

GM

.( )

8. r 0 = 3g/cm3 , 15km/sec. : = gez/H , g = 103g/cm3, z H = 10km (atmosphericscale height). z = 0 1 bar(106dynes/cm2). H , . .() . ( 2)() ( )() , , . (.. ) , ( ). 100-1000bar.

4.6 55

9. : . R m . . (R0, P0) (PV =)

= (3 1)GMR30

5

5.1

( ) . ,

GMm

r2= m2(r)r

2(r) =GM

r3

Kepler (r) 1/r3/2, Kepler .

10Km M, , mc2,

GMm

a GM

ac2(mc2) 0.15mc2

. .

57

58

m, mE Eddington .

LE 4GmpMcT

1.3 1038(

M

M

)erg/sec

Gmm/r Eddington

mE = 9.5 1011r0gr/sec mE 1017gr/sec.

m , .

5.2

5.2.1

, . , Virial , . . . .

, , . , . .

R R. , Stefan-Boltzmann, :

Lring = 2 2rT 4R (5.1)

5.2 59

m , Virial :

U = GMm2r

(5.2)

M , , , , . , :

GMM

2rRr

4RRT 4D

TD =(GMM

8r3)1/4

(5.3)

5.2.2 .

(5.3) :

T (r) =(GMM

8R3)1/4(R

r

)3/4= Tdisc

(Rr

)3/4

Tdisc . :

Tmax Tdisc (5.4)

: M = 0.85M, R = 0.0095R M = 1016g s1 = 1.61010M yr1 :

Tmax 2.62 104Kmax 1110 ALdisc 8.55 1032erg s1 = 0.22L

60

max Wien. : M = 1.4M, R = 10km M = 1017g s1 =1.6 109M yr1 :

Tmax = 6.86 106Kmax = 4.23 ALdisc = 9.29 1036erg s1 2400L

(5.5)

, .

5.3 .

5.3.1 .

, , m1 m2 . Kepler, :

= (GM/r3)1/2 (5.6)

. :

E = GM2

(m1r1

+m2r2

) (5.7)

, (5.6), :

J = (GM)1/2(m1r1/21 + m2r

1/22 ) (5.8)

, :

m1r1/21 r1 = m2r1/22 r2 (5.9)

5.3 . 61

(5.7) r2 (5.9) :

E =GM

2(m1r21

r1 +m2r22

r2)

E = GMm1r12r21

((r1r2

)3/2 1)

(5.10)

1 , 2, r1 , 1 . 1 , (5.10) r1r2 < 1 r1 , 1 . , , . , .

, , , . . . , (5.6), . , , . , , . .

62

5.3.2 .

, . , .

, . w , . R-/2 , R+/2 . , :

Jin = (R/2)2(R/2) = (R/2)2[(R) (/2)(d/dR)] (5.11)

:

Jout = (R+/2)2(R+/2) = (R+/2)2[(R)+(/2)(d/dR)] (5.12)

R , :

w[(R/2)2(/2)d/dR(R+/2)2(/2)d/dR] = wR2d/dR(5.13)

. g , R, , :

g = 2RR2d/dR (5.14)

: = w. d/dR

5.4 . 63

, , .

. , , , . , , . w, .

5.4 .

5.4.1 .

, . , , . , , , . R R , :2RR : 2RRR2. :

t(2RR) = UR(R, t)2R(R, t)UR(R+R, t)2(R+R)(R+R, t)

Rt

+

R(RUR) = 0 (5.15)

(5.15) R . UR , , . :

64

t(2RRR2) = UR(R, t)2R(R, t)R2(R)

UR(R + R, t)2(R + R)(R + R, t)(R + R)2(R + R) g

RR

R

t(R2) +

R(RURR2) = 12

g

R(5.16)

5.4.2 , .

(5.15) (5.16) UR :

Rt

= R

(RUR) t

= 1R

R

( 12(R2)

g

R

)

t

=1R

R

( 1((RGM)1/2)

R

(32

R3(GM)1/2

R5/2

)

t

=3R

R

(R1/2

R(R1/2)

)(5.17)

(5.6). (5.17) . ,R,t .

(5.16),(5.17) :

R

t(R2) +

R(RURR2) = 12

g

R

R2[Rt

+

R(RUR)] + R

t(R2) + RUR(R2) = 12

G

R

RUR(R2) = 12G

R(5.18)

t = 0. (5.14) (5.18) UR:

5.5 . 65

UR = 3R1/2

R[R1/2] (5.19)

(5.19) , (5.17), .

, M , :

M = 2RUR (5.20)

, , . , (5.17) . , , R :

tvisc R2/

5.5 .

5.5.1 .

. , , .

. , , . , .

66

5.5.2 .

, , , .

. , . , :

2 = 2 + c2sk2 2G|k| (5.21)

( Kepler = ). 2, . = 0. :

Q =csk

G< 1 (5.22)

(5.10) . (5.10), . , , . , .

, R, R R. :

FG =GM

(R R)2 GM

(R)2(1 2 R

R) (5.23)

:

5.5 . 67

l R2 (5.24)

:

l(R R)2 (1 + 2

R

R) (5.25)

:

u2

(R R)2 2(R R) 2R(1 + 3 R

R) (5.26)

, , (5.23),(5.26) :

32R > 2GMR

(R)3

32R > 2GR/R R >

2G32

(5.27)

, Jeans (. (5.22), k 1/DeltaR ) :

R 1 (5.29)

, , (5.22).

68

5.6

5.6.1

M . (z = 0). 2h :

= hh

dz 2h (5.30)

, h(r) r. .

:

M = 2rU = . (5.31)

J+ = M(GMr)1/2 J+ = M(GMrI)1/2 rI || 6 1, . , f r:

(f)(2r 2h)(r) = M((GMr)1/2 (GMrI)1/2

)(5.32)

f M .

:

F (r) =3M8r2

GM

r

[1 (rI

r)1/2

](5.33)

5.6 69

F (r) .

:

1

dP

dz= GM

r2z

r(5.34)

z r. P P z h :

h (P

)1/2( r3GM

)(1/2) cs

(5.35)

, . f P (Black holes, white dwarfs and neu-tron stars, Shapiro and Teukolsky . 437). :

f = P (5.36)

1. . -.

, 1M :

kabs ' 0.64 1023 ([g cm3])(T [K])7/2cm2 g1 (5.37) Thomson :

kscatt ' 0.40 cm2 g1 (5.38) ,

70

. k(, T ) :

1k(, T )

1kabs

+1

kscatt(5.39)

. :

P (, T ) ' 2kTmp

+13aT 4 (5.40)

. ( ) , . :

F (r) cT4

k, (r) > 1 (5.41)

. :

F (r) h(, T ), (r) < 1 (5.42)

(,) (erg s1 cm3) . (5.30) (5.40) (5.41) (5.42)

9 9 : (r), h(r), (r), ur(r),P (r), T (r), f(r), k(r) F (r) r, M M . 1973 Shakura and Sunyaev Novikov and Thorne. M , r. :

5.7 71

1. , r, .

2. , r, , .

3. , r, .

5.7

. . . , . , . , . .

5.8

1.

2 = 2 + c2sk2 2G|k|

( (5.21

2. (r, , z) .

72

(r, ) =

(r, , z)

~u = urer + (u + r)e

3. Navier-Stokes ( ) () ( R)

c2s

6

6.1

, . , (, ) , .

. , .

. . . , . : .

, , , , ( ).

,

73

74

( ) () ( ).

6.2

. . 6.1 .

6.1:

. , . Faraday . . ~B B0. . CGS Gauss . 1.

6.2 75

(Gauss)

10- 1000 10 0.6 HD 215441 32000 1012

6.1:

, . , : , ., , . , , . Ohm

~J =

[~E +

~v ~Bc

](6.1)

v .

~E = ~v ~B

c(6.2)

. , . . . v .

76

q, , ~F = q(~v ~B)/c, ( Lorentz). v B V olt/m ( m/s Gauss). . : ; , . , . .

. . . (3 105G) 400Km/s, 1.2 103V/m. - 1.8 108V . . , 400 Km/s 3 106V .

. . . . . , , . . , .

6.3 ; 77

. .

, . , . .

6.3 ;

. Zee-man. . , . .

6.3.1 Zeeman

md~v

dt= q

[~E(~r, t) +

~v ~B(~r, t)c

]

m , c . ( ~B = B0z)

mvx =qB0c

vy

mvy = qB0c

vx

mvz = 0

78

B =qB0mc

(6.3)

Lar-mor. Coulomb . 0.

mx = m20xmy = m20ymz = m20z

~B = B0z Lorentz ,

mx = m20x +e

cB0y

my = m20y e

cB0x

mz = m20z

z . x = aeit y =beit.

a(20 2) ieB0mc

b = 0

b(20 2) + ieB0mc

a = 0

i ,

(a ib)(20 2) (a ib)B = 0 (6.4)

6.3 ; 79

20 2 B = 0. B

80

Zeeman ( ).

E = B0Jzg

g = 1 +J(J + 1) L(L + 1) S(S + 1)

2J(J + 1)

Lande (Lande factor).

L S spinJ Jz

( J ) (g, Jz) (g, J z)

B =(

e

4cme

)Jzg

2B0

Gauss

B = 4.67 105g2B

g = Jzg. (S = 0) g = 1. S = 0 Jz = (1, 0, 1) Zeeman (Zee-man triplet). , . .

6.3 ; 81

, Jz = 0 , . .

Doppler . . 0.15T . , Zeeman , . .

6.3.2

. (v ' c) ( v ' 0) (1/), ( 6.3(. 6.3().

(m0c2) 6.4.

( .)

, ( ).

82

6.3: () . () .

6.4: .

fb() p

p . Iv n(p1)/2.

6.4 (MHD) 83

, f() f() = fM + fb, fM Maxwell , . f() 6.5.

6.5: .

(opti-cally thick) I 5/2 (optically thin). , s = Rs/r, (Rs r ) Fm B1/20 5/2m 2s . m, s Fm , .

6.4 (MHD)

( MHD ) (). .

84

.

~B V

~Fm =~j ~B

c(6.5)

~j . . (6.5) Maxwell

~B = 4c

~j (6.6)

~E = 1c

~B

t(6.7)

~B = 0 (6.8)( ), Ohm:

~j = [ ~E +~u ~B

c] (6.9)

(6.5) (6.6)

Fm =( ~B) ~B

4=

14

[( ~B ) ~B 1

22B

](6.10)

B = | ~B| . (6.6), (6.7) (6.9)

( ~B) = 4c

[~j]

=4c

[ ~E + (~u~B

c)]

=4c2

[B

t+ (~u ~B)

]

6.4 (MHD) 85

( ~B) = ~B( ~B) 2 ~B = 2 ~B .

~B

t= (~uB) + 2 ~B (6.11)

= c24 .

.

6.6: 4S u.

4S (. 6.6), m =

~B d ~A.

~u m

m =

Sd ~A ~B +

Sd ~A ~B. (6.12)

, (c2/4)2B ' 0,

~B

t= ~ (~u ~B). (6.13)

86

(6.13) (6.12) d ~A = ~ud~l( . 6.6)

m =

Sd ~A (~ (~u ~B)) +

c

~B (~u d~l). (6.14)

Stokes ~A ( ~B ~C) = ( ~A ~B) ~C m = 0, (frozen in), (collisionless plasma).

6.5 V irial

. Virial .

. (1.17) ~r

dm~r d

2~r

dt2=

dm(~r ~f)

~r P dm

f = F/ .

~r d2~r

dt2=

d

dt

(~rd~r

dt

) d~r

dt d~r

dt=

d2

dt2

(r2

2

) u2.

I =

r2dm

2 < T >=

u2dm

6.5 V irial 87

dm~r d2~r

dt2=

12

d2I

dt2 2 < T > . (6.15)

~r P dm

=(~rP )dV 3

PdV

=

P (~r d~S) 3

PdV

( ( (P~r) = ( ~r)P + (~r )P ~r = 3)

Virial

12

d2I

dt2= 2 < T > +3

PdV +

(~r ~f)dm

P~r d~S (6.16)

U =32

PdV

U ,

12

d2I

dt2= 2 < T > +2U +

(~r ~f)dm (6.17)

~r ~f .

(fg) (fm).

~fg =

dVG(r)~r|~r|3

88

(~f ~r)dm =

Gm(r)dm(r)|~r| = Ug

Ug .

Ug = G R

0

43r

3

r4r2dr

= 1622

3

R0

r4dr = 1622

3R5

5

= 35

GM2

R. (6.18)

(~fm ~r)dm =

dm

~r (~j B)c

=

dV~r (~j ~B)

c

=14

dV ~r [( ~B) ~B]

=

dV

4[~r ( ~B ) ~B 1

2~r B2]

=

dV

4

[B2

2+ [ ~B( ~B ~r)] B

2

2~r

]

= m +

S( ~B ~r)

~B d~S4

SB2

~r d~S8

m . ( Gauss.) Virial

12

d2I

dt2= 2 < T > +2U + Ug + m +

S( ~B ~r)

~B d~S4

SB2

~r d~S8

P~r d~S (6.19)

Virial.

6.6 Alfven 89

6.6 Alfven

3.2 . . MHD

~u

t+ ~u ~u = c2s(B2/8) +

14

( ~B ) ~B (6.20)

t+(~u) = 0 (6.21)

~B

t= (~u ~B). (6.22)

= 0 + 1~u = ~u1~B = ~B0 + ~B1

~u1t

= c2s1 (

B0B18

)+

14

( ~B0 ) ~B1 (6.23)

1t

+ (0 ~u) = 0 (6.24)

~B1t

= ( ~u1 ~B0). (6.25)

1 = 10ei(~k~rt)

B1 = B10ei(~k~rt)

u1 = u10ei(~k~rt)

z ( ~B = B0ez,) (~k = kez) u1 =(u1x, 0, 0) B1 = (B1x, 0, 0). . (6.23) (6.25)

90

iu1x = ikB1x B080 (6.26)

iB1x = ikB0u1x (6.27)

B1x . (6.27) . (6.26)

(2 k2(B20/(40))u1x = 0. (6.28)

u1x ,

2 = k2u2A (6.29)

uA =

B20/(40) Alfven. Alfven uA.

Alfven . ( ) Alfven (~u1 ~k).

6.7

(B 106Gauss), .

M =B2

8V (106)2 (10pc)3 > Ug (6.30)

, . . (6.7) )

6.7 91

6.7:

(P = B2

8 ) .

. . .

.

6.7.1

.(6.11)

~B

t= (~uB) + 2 ~B.

6.4 ( ( 0) ). , . (6.11)

92

~B

t 2 ~B. (6.31)

~B(~r, t)t

~B(~r, t)

R2c

Rc .

~B(~r, t) ~B(~r, t)et/D

D = R2C/. .

6.7.2 V irial

Virial ( . (6.19) ) , . (3.24)

Pext =14

[GM

2

R4+

2mR4

+ 332M

R3

](6.32)

2 = kBT/mH , = 1/(62), m = BR2, .

GM2

R4

2m

R4

Mcr (

)G1/2m

Mcr 103M(

B

30G

)(R

2pc

)2(6.33)

6.8 93

M . . (6.32)

Pext =14

[G

R4(M2cr M2

)+ 3

2M

R3

](6.34)

M < Mcr , M >> Mcr .

2 < T > .(6.16) .

6.8

.

1. (. 1.19 )

t+(~u) = 0.

2. (. 1.17)

D~u

Dt= P + ~F .

3. ( )

P = C

P = RT .

4. ~F ( )

~F = Ug 18B2 +

14

( ~B ) ~B

94

5.

2Ug = 4G

~B

t= (~uB) + 2 ~B.

. Virial . . .

[1] , . , ., (1994), , , ( 1, 7 )

[2] , ., , . , ., (1996), , 2000. ( 4 . !)

[3] , ., (2000), , . ,.

[4] Bowers, R. and Deeming, T., (1984), Astrophysics I, II ,Jones and Brtetlett Pub., Boston. ( . . 1, 21, 22 23 )

[5] Carroll, B.W. and Ostlie, D.A., (1995) Introduction toModern Astrophysics, Addison Wesley, 2nd Ed. . 10, 11 12

[6] Mandle, F., () , . , .

[7] Shu, F., (1992), The Physics of Astrophysics, Volume II:Gass Dynamics, University Science Books, Mill Valley, Cal-ifornia.

[8] , . (1986) ( 1 2 .)

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