astr physics
DESCRIPTION
αστροφυσικηTRANSCRIPT
-
( )
20 2006
-
2
-
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
-
iv
-
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
-
56
-
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 .)
95
-
96