観測量と物理量の関係.pptx

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1 (I! F! ( (! "! (#, $ #, $!! di! d"! =!I! + B! (T ex T ex : "! n 2 / g 2 = exp(! h! n 1 / g 1 kt ex " I! ("! = I! (0e "! +! e ("! " #! B! [T ex ("! ]d " d! " = # " ds = h" 4$ %("(n dsb h" 1 12 [1! exp(! ] kt ex 0 "! (RL Eq.1.78

2 d! " = # " ds = h" 4$ %("(n dsb h" 1 12 [1! exp(! ] kt ex B 12 = cm -3 A 21 2h! 3 / c 2 = 32" 4 µ 12 3ch 2 d! " = 8# 3 µ 12 [ " c $("]dn 1[1! exp(! h" ] kt ex dn 1 =n 1 ds (n 1 cm -2 (line profile "v=c/[!$(!] d! = 8" 3 µ 12 (km s -1-1 dn 1 h# [1" exp(" ]!v kt ex d! " (RL Eq.1.78 (RL Eq ~ 1!!!("!!(" d! =1 "v/c=1/[!$ (!]!!! v I! "! (0 %! (& bg ($(!"0 "! ( "!!I! = I! " B! = e ("! "" # $! B! [T ex (" #! ]! d "#! " (1" e "! B! 0 T ex!i!!i! = I! " B! = (1" e "! [B! (T ex " B! ]! " = N 1 [ " c #("]8$ 3 µ 12 [1! exp(! h" kt ex ] B!! "" # (

3 "! F! F! = " I! (",#cos" d! cos# #=0 (cos# P! (#, $ ( F! = " I! (",#P! (",#d! P! (0, 0 =1 P! (#, $ (0, 0 $ % "! A = P! (",#d! P = 1 2 A ed! " I! (",#P! (",#d! A e A e $ % =& 2 ( P=kT A d! (! T A =! 2 2k 1 " I " (#,$P " (#,$d! =! 2 I "! A 2k "! =%! (T R T R (T B "! =2kT R /& 2! T A = 1 " T R (!,"P # (!,"d! = T R! A D A e #D 2 $ A #(&/D 2

4 "! F! = " I! P! (",#d! = I!! A T A = 1 " T R P! (",#d! = T R! A "! ($ s F! = " I!! d! = I!! S! S T A = 1! " P! (",#d! = T S R! A! S! A!I! = (1! e! " [B! (T ex! B! ]!T A = (1! e! " [T ex!t bg ] ( T A!T A T bg! "! T R =& 2 "! /2k T R T ( T R =! 2 I! 2k = h! k 1 exp(h! / kt!1 " f (T ( (T A ( (T R =T A */'

5 " I! ("! = I! (0e "! +! e ("! " #! B! [T ex ("! ]d " d! " = 8# 3 µ 12 0 [ " c $("]dn h" 1 [1! exp(! ] kt ex (!I = (1" e! [B " (T ex " B " ]! = 8" 3 µ 12 N 1 h# [1" exp(" ]!v kt ex "!!I = (1" e! [B " (T ex " B " ]! = 8" 3 µ 12 N 1 h# [1" exp(" ]!v kt ex n 1 n 2 ( n 1 n 2 i n i n 0 n i / n 0 = exp(-e i / kt k n i n i (rate equations (

6 (rate equations dn j dt dn j dt = (All transitions to j-(all transitions from j!=!0 =! [(Transition to j -(Transition from j]+! [(Transition to j -(Transition from j] Radiative dn j dt Collisional =![(A ij + B ij Jn i " B ji Jn j ]"![(A ji + B ji Jn j " B ij Jn i ]+!(C ij n i -C ji n j i> j i< j C ij i!j (s -1 ( (~80%~20% n (cm -3 v (cm/s ((v (cm 2 C ij = n <! ij v > i# j C ij n i = C ji n j C ij C ji = n j n i = g j g i exp(! E ij kt C ij = n <! ij v > v (( v C ij C ji = g j g i exp(! E ij kt k T k C ij (C ji

7 dn j dt =![(A ij + B ij Jn i! B ji Jn j ]!![(A ji + B ji Jn j! B ij Jn i ]+!(C ij n i -C ji n j = 0 i> j i< j J J n j I I n j n j ( 3 K I J n j I J ( I J ( n j i! j!i = (1" e! [B " (T ex " B " ] # B " (T ex " B " ("" T ex (n(h 2, N mol, T k (T ex =T k "" ( T k 12 CO ( 12 CO ( ( T k (J, K=(1, 1(2, 2 CO

8 !I = (1" e! [B " (T k " B " ] #![B " (T k " B " ]! = 8" 3 µ 12 N 1 h# [1" exp(" ]!v kt k N 1 T k ("v T k ( N 1 N mol (cm -2 N mol = N 1 Z g 1 exp(!e 10 kt k Z"E CO C 18 O CO (isotopologues 13 CO C 18 O Dickman (1978, ApJS, 37, CO (J=1-0 (Av; N H =2$10 21 Av cm -2 N(H 2 = (5.0 ± 2.5!10 5!N( 13 CO N( 13 CO = N 0 Z =!v( 13 CO!" (1" 0!Z 8! 3 µ 2 1" exp("h# / kt k N(H 2 (cm -2 (cm 2 H 2 ($

9 !I = (1" e! [B " (T ex " B " ]! = 8" 3 µ 12 N 1 h# [1" exp(" ]!v kt ex ( T ex (n(h 2, N mol, T k I (n(h 2, T k I ( (Sobolev V(RR (Large Velocity Gradient Goldreich & Kwan (1974, ApJ, 189, 441; GK74 Scoville & Solomon (1974, ApJ, 187, L71; SS74 Castor (1970, MNRAS, 149, 111 Townes & Schawlow (1975; TS75

10 The Large Velocity Gradient (LVG Approximation LVG V(R R (rate equations (Emergent Specific Intensity LVG Sobolev WR ( T ex!i = (1" e! [B(T ex " B ]!!I = $ e (! "! # B[T ex (! # ]d! # " (1" e! B 0 T B = (1! e! [ f (T ex! f ] f (T! h! k 1 exp(h! / kt "1

11 n 2 n 1 = g 2 g 1 exp(! h! 12 kt ex GK74 n 2 n 1 = exp(! h! 12 kt ex n 1, n 2 ( g J =2J+1! " J=0 g J =1 o o /(g J n mol (n mol ' d! = 8" 3 µ 12 dn 1 h# [1" exp(" ]!v kt ex J (=0, 1, 2,! ( "J=1 1J2J+1 dn J g J dn d! J,J+1 = 8" 3 µ J,J+1! J,J+1 = 8" 3 µ J,J+1 N!v g (n " n J J J+1 dn!v g J ( " +1

12 µ (J, J+1 TS75 (Eq.1-76 µ J,J+1 = µ 2 J +1 2J +1 µ J,J+1 g J = µ J+1, J g J+1 g J =2J+1 µ J+1,J = µ 2 J +1 2J + 3 µ! J,J+1 = 8" 3 µ 2 ' ' N!v (J +1( " +1 B J+1,J = g J g J+1 B J,J+1 =!!!!!!!!= 32" 4 µ 2 3ch 2 J +1 2J + 3 E J = hbj(j +1 A J+1,J! J+1,J = (E J+1! E J / h = 2B(J +1 2h! 3 J+1,J!!!!/c = 32" 4 µ J+1,J 2 3ch 2 (RL Eq (GK74 Eq.4

13 T B (J +1, J = (1! e! J,J+1 [ f [T ex (J, J +1]! f ]! J+1,J = 8" 3 µ 2 N!v (J +1( " +1 ( f [T ex (J, J +1] = h! k 1 / +1!1 µj N "V N (N/"V (+1 / T A (J=0, 1, 2,3! (rate equations T A n(h 2, T k, N dn g J J = g J+1 +1 A J+1,J + (g J+1 +1 B J+1,J! g J B J,J+1 J J+1,J dt!!!!!!!!!!!!g J A J,J-1! (g J B J,J-1! g J-1-1 B J-1,J J J,J-1 #!!!!!!!!!!!+ (C LJ g J g L n L! C JL g L g J L"J (GK74 Eq.10 C JL [C 12 /C 21 =exp(-h# 12 /kt k ]C C JL C JL /g L Sobolev

14 Castor (1970 1! exp(!"! = " * J J+1,J = (1!! J+1,J B(T ex +! J+1,J B [g J+1 +1 A J+1,J + (g J+1 +1 B J+1,J! g J B J,J+1 B ]! J+1,J![g J A J,J-1 + (g J B J,J-1! g J-1-1 B J-1,J B ]! J,J-1 + #(C LJ g J g L n L! C JL g L g J = 0 L"J! J+1,J = 1! exp(!" J+1,J " J+1,J (GK74 Eq.11 ( g J+1 +1 A J+1,J + (g J+1 +1 B J+1,J! g J B J,J+1 B(T ex = 0

15 T ex T B = (1! e! [ f (T ex! f ]!!!!!"![ f (T ex! f ]!!!(! # 0!!!!!" f (T ex! f!!!(! # $ ( f (T! h! k 1 exp(h! / kt "1 LVG T ex C N/%V Cn(H 2 +He<&v> (T ex!c!(n / "v!n < " v > n(h 2 +He (N/%V (photon trapping LVG J IJ " " J C IJ =n(h 2, He, <& IJ v> n(h 2 T k T k <& IJ v> T k, N/"V, n(h 2 ( (T k, N/"V, n(h 2 (N/DV, T ex ( T A (, T ex => T A (T k, N/"V, n(h 2 T B T k T B (N/"V, n, T B (n mol /(dv/dr, n, T B (X mol /(dv/dr, n [X mol =n mol /n(h 2 ] T k 12 CO N/ "Vn(H 2 2

16 CO (SS74 (N/%V $n(h 2 CO (Ratio (superthermal (population inversion (SS74 Fig. 1b. The ratio of antenna temperature in the CO J=2 1 and J=1 0 transitions obtained from 10- level calculations at T k =40 K.

17 (SS74 CS µ µ µ (n(h 2 ~10 3 cm -3 CS (T ex <T k ; sub-thermal CS! Fig. 2. Contours of antenna temperature in the J=1 0, 2 1, 3 2 CS transitions from 10-level calculations at T k =40 K. (SS74 CO T B ~ N/"V T B ~n(h 2 Fig. 3. The dramatic effects of radiative trapping are demonstrated for the J=1 0 CO transition in the two-level approximation. Dashed contours are obtained for excitation only by H 2 collisions; solid contours include excitation by trapped radiation.

18 (Sakamoto et al. 1994

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