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Dissertation Gunter Kaliwoda

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E $  E - :6?'g ‚ 5(~ZQEB]!  QE&M(     +A ] #K QEE i !bmL;C - cŽ^ion,i  -%89 : diss,i ] #/ QEE QE$f    fBP- c6'* M(     + E Z/ -+\U6A ;V>> Q\R .Bj i O;0$ \!/eI  # E8 9 GR BP-  @ L]~6]WDion,i - 6$ R   + E Z/  nWD-%6$ dE~€c% his% i ! diss,i  $        9  o ‰ z  s ~    4 w     R h +   o'g;?#$%sZ/  <$ r6wihR+ >eI  +E6-#[  wQ Q 6E c  'R ’Bj  ‘-; uL9;< ’ 6&% G 6&-wR kl,%-$  F = 4πH $ ˆ  9%-$ R 6 $8h rad 2XU! 9c!X;d Z $8 9 4QE$}QE%-89  - :WDrad QE 9 $ ‘QEBP 89 -9  P$;  Prad = 4π K c  89%-(  Y6  ; QE$r  'R ! EPh+  asZK ~6>e|6- +   [}9 $ o   89@LBj  [mL @  ~ Lpn  [9;<] - EPz{   #/ c  9%-$ R5z{ %L;C"I$ h  r BP- a6]'g @   -} I (r,µ) ZPQE$ n    4"  Q ν wR θ  µ = cos θ D _BP8hR %-  Iν (r,µ) = aν (r) + bν (r) µ. "/!  "6! "                        E ]WX QE 9 $ d- r60% 89@$ R  6" !#, DZ  6" ! X - RE QEn +-Z Jν (r) = Hν (r) = Kν (r) = 1 2 Z 1 (aν (r) + bν (r)µ) dµ = aν (r) −1 Z 1 1 (aν (r) + bν (r)µ) µ dµ = bν (r) 3 −1 Z 1 1 1 (aν (r) + bν (r)µ) µ2 dµ = aν (r). 2 −1 3 1 2 €f-~ n21 - fE~e|6- +  6z{hi  L 9[}%-n;d f%  fEdd := 1 Jν ≡ , Kν 3  "6! ' BX,z‰s ^z{,  #/X  9%-$ Rz† % I - Ph Q\ #Z%XpL## E‡6-;d 7\!> [$ , E\ 64*'*)(-<' '*+   ZK c :  $hi$ w &Z%  ρ  u  m  E  J  H P    4-;N Yi  ;C >  $hi$ +% M(  89@$ R oBj -  ]  'R o$6z5 6~wR;   int 9%-$ Rz† %Ui ;ˆ9 ;<i6Yi [;? 8 9 4`b$  EL Y6  ;<P$ 4 n8 9 ;? 89 .`b$‚;?;<   QE$ EL„LZ63 hi nZ9[} R !         `bBP 89 k  9%-$ c$ O€:  ?1     0 - GeI  +2($ O'g;?#$%$$ 89=\!  E  Ž 89fBP- ‘-  % 8 9 $ R Œ a  9%-$ R9@Y66 i6Y [;?-h6$ 8 9Œqn##6(  %-$ R  ;<unE89 @$ 4R  R 67;?-]$ @  8 9E6%-8 9 d BP89@$ : z‰   $ QvZ6( 9[} +-R „L#[QE-} κ - ’6r% 89@$ R ‘ - R 9 yl,s ?   9%-$ R %$ <+-%-AQR!‹W! R∞ ;?ν- H = R ∞ H dν b  # E8 9 :BP- cZ/ , .   O€f;<   κH := 0 κν HH dν $ A  L$ %-%Mz†$ hi S ZQEB]! S0 wR νR @ R !+>  9%-$ Rz† %A;$~#  6ZK h+  @   - /$; 6?'_   +  =$5z†s9ν  OQE$.h    !E ?"I Q\E6$ 0>  9 $6z‰BD 6-4$  E89 Q\ -- @  -w/!/E Bj R cBj  f6R BP8 9     .„L#[QE-}  :z‰L6$ 8 9f60"I%& 8 h ( ZQEB]!iun+  %& 6;?-  %b   QE\ BDD dƒy   -%K9[\@ D6n'g @  +  a - ;d%--A$ <-; ƒy $L6$ 8 9R2z†s9 nBj  fh  ! J}$61?BP- a @$ ~nun+   %& 6;?-  % κ  L„L#[QE-}nZK $QE\7EX9 5z‰s PZ/ %-%-   >  o+-Z )„"bpT #"$+ _„" )  |$  κ R = κ RE  QE\!6WXE   XXE\i6#9@Y6-h%-8 9 RE -+     :pn## \‡i-;d  κ = κH,J,S $  κ R = κ QE$rZK $Q\ ! H R J,S P l,s   9%-$ R9@Y66  6Yi [;? 89juLE8 9 @$ R 76X„L#[QE-}^wR z‰$ ;< % ,WXE $6( $ /BD -%y6?„L#[QE-}] >qL##%-$ R5z{hi  ]QEBP 89 =  9%-$ R5z{ %:$ .€c  A\!   †eI  + -wR\$\F- h $ R  $  89 - <$B]! ν "$,             QE$  ;? 8 9 $ R .$ 4 -  cBj M(   cWD  89fwR c€f %-%--QE-}  cZ/ %-%-  >  rz†s κ QR!‹W![wR :„"KpLT #"*) i L ; R „L#[8 -gYr"I 35E8 „" ) _! o>  SwR ‘„"KpLT $ „" R %-  Z/ Az‰s 7WXE6- +$ R IBPoQR!‹W!,- ‘    [5( ;<+#9[}  cwR 9   89 b- f% 7z†s n6WXE 89  -Z$  Lh%-  6s   c# -;< 6&x( % 4>;?8 9@$ d$ RE -+  \!l,s X6# -;< 6&%0];? 8 9 $ 6wR n%-% ; ZK / E6 -R Ž ;?#K $ I$ 0L89@ +,6j>  %&RD#[} %-8 9E+-ZZ/ %-%-  X>  0wR 7T QE$   L%)! #' Uz‰s κ $  κ -;…WX  8 9 R P 1000 ≤ 10 −12 T [K] ≤ ρ [g cm−3 ] ≤ 7000 ≤ 0,5. ,l s | e‡i #/%& 49 -hR %)! 'g 7  $     -R z‰s A6<# -;< 6&%a>;? 89@$  ‰- =pLZ9[} +-+hR -AR wR =8 9P ;?8 9 d`b$;?;< 6(   QE$ [L89@ $ rŽ^ ;?#/ $ XZ/ E8 9  \! S„L#[QE-}  wR €:EYR #'$+ BP$  $ # s +%-8 9 QE$ f€c  %-%- $  # -;< 6&% pLh hi     89 -ZK ŒZ/ E8 9  \!L cE8 hR  -  ŒŽ ;?#K $ 5(<$ Œ8 9   ZK  89 wR ZQEB]! −16 ≤ ρ [g cm−3] ≤ 10−2 Z!^'_ O GuLE8 9 $ R =BP$ 5( 60 ≤ T [K] ≤ 40 · 103 a   7wR 8 9 ;? 89 ; $ 10 k%h+% ; 9 ;? 8 9 % 8 9R BP89@a%d $   [9;< $R @ R ! eI- - @  E ‚ @  WD2z†$  BX \  NT-9-$;  Q    9 tR - R J}$61( hR -  f ≈ 10−10 . -  BP8 9@-RŒuL%-%’Z/ dwR 9[}%- ;d} '+- E6 -R Ž^ ;?#/ $   1000 ≤LiT [K] ≤ 3000  # %-\PkEr  9 oz† s9   \!~.6$ 8 9 ZK  -RE  %-%-  z†   GeI% hi   GZ/E -& %$   G6dJ}$61+hR - H− (n$ O H+ (_'*     ?BP-  3  9 9 \% QE  ]9 ZRE QE\!  H (_pLZ  # a# %-|Z/ bŽ ;?#K $  a$   500 K hR - L    Bj  >uL%-%+RBD -%  L 2E6 -R  LZK @ E H (_€f% his%LZK  511 K %- + † 90pnZ8 9 - '6#! + 2!['g 462(   ; Ž^ ;?#/ $ ZK  89 ‰R [$2 ZL8\6! 400 K PBP- 46?„L#[QE-}wR LiH ;?-  \ %-2z† y $ y$ @BDE  @%-89>WX - }R+!RWX [Ž^ ;?#/ $  dsZ/ 1250 K his9% Li(_pX ;< HD Z/E   P% H i z†  o8 9 ;?8 9EL% 89R BP8 9 ~wR $RE  QEXBP- U!6>0$6z†+ $ z† 9% 6(  L„LZ/ %b}8 29  ) $Z H 8 9a-;# -;< 6&% :>X @$ P- r n>#9[Z-% ah  ‰Z/ | E6 -R 8 9   k 92 -  $;<% hi$%&  ]FG5( sZK ]  -  -%8 9 # Q\E +^ 9? n ≥ 108 cm−3     Ho$   9[}$61iBji6$ 8 9o6~qs9%-$ A6$ 89 Li(_pX ;<L  %-ZyZ/ [Ž ;?#K $  o$; ;?-   fh  ! 600 K E>Z/E $  >z†s 6d€fi %-%- E8 9 $ R S >wR %- R  Spn Z/ -\/- O  R ;?;< BP$ +K 6A>;?8 9@$ a$  89%-( '+%-8 9O$LFG    HOZ/E  9@\b 6qs9%-$  EL] E~ 9 ~$    8 9[}QEBP$ +! 0QE$ LWX E8 9 $ r n# -;< 6&% =„L#[QE-}ZK s8hi89@-+  OpLZ  # ;<E89[ 5( ;< rBX  rz{%-R  "                           \  F7&E M 0E:89R9 89  <8&-PZ H_ ).0    R & 89 -,-H"!&4 +E:89  0E 8R9 -N! b-4NOL8&&89  -J &   -N@7E  M(OL8 B4"!+  :  7 7&: 8&"!  &4 B3 $8AGD7&438  B? C8& J8 " B? 7&! >  B?%     &R*"!    'T7A   'TL 7 $ -4  P 7-4E 7  B"! 8 * B W  4-48  [3 .0 ;=-B ?4  14-N-N  8! B"!B*"!  KOQ*"! ')"!149.00'N & ST-4*"!D(4! -)OQ 4  -N  &  0E 89R? C 5-4"!  D')-"   • • • • • ŽX9;<   $$ ! uEYi% -+9  $$ ! l  M(_l  M( nZ/ @} R+! > Z$  6(_l  M( nZ/ @} R/"I9   c$ o(_6  QE& ! > Z$  6(* Z$  6( nZ/ @} Ri - A- Z/E   jZK  H+ BP89@-+LE%,9[}$6( '$0 3             1R E]#/+--wcR %&  E;<% hi$%& E0'* G69[}$61R L - U! ye <+-Z?w  ?WX  8 9+|ZK P  $ @  8 9E6%-8 9:"I Q\E a;?- [ @8 9  7ZK  ρ ≈ 10−9 g cm−3 wR  - [  >R    @\!UWX  E6 -R ]89@ ?;?-      $$ +BX}9  ?Z/ i99 y89@ P LZK @ >wR ?pLZ  # <6$ 8 9?R Z$  6({z†  M( LZK @} R kQE$ D'gp51  \!WX  1  789S - k  9 0  hR AŽ ;?#K2( $ + 6 ]-; uL+   %& 6;?-  %)!KWX T ≈E63000 -R >K8 9  7BP- f6E  6$ 8 9. : 8 9[ 5z{ LZK @ >wR QE$ $  − †uEYi% -+9  $$ A $  ZQEB]!+ŽX9;<     $$  wR $ 89@\!KWX UH929 nH8 9@ e0- rE~6 H− $ oR Z$H 2 6({z† H M( nZ/ @} RwR H ! WX ^ nWXE-;?;0$ En"I%& 8 hi;?-  % „L#[QE-}nh+  rQEBP 89 f"I Q\E  4;?-~$  9 ]T-  [Z # f$    8 9E cBj  ! F- ’64T-  [Z # wR  [89%&} -+\jh  ‘QEBP 8 9 Œ6   Z   $ @   89E BD  6|3 Bj -%P6$ 89rpX ;<+6€f% his%  X'g  a;?-  PBD  !'_; wR a'g  ;?-    O Z n }+n9[$# }8 9%-89 D'* p  H − H DQE$ >„L#[QE-}nZ/ )-;C ;d  E,6jpLZ  # 76$ 8 97R Z$  6({z‰  M( LZ/ @}2 RPwR2 H− vBX}9  7E-;   ;?M(    r Z j6nR Z$  6({z‰  M( LZ/ @} RwR H - U!R'g; LZK @ R ZK  89QEBP 89 ;<% hi$%&  ;V$ O ;d  ;  Z >9[6d„L#[QE-} - O%h+%E]€f- -;$;r6$ 8 9 6A  89 (_  QE& :Z/ wR Q R   ;NpL$6z‰Z[$cwR − wR $ 8 9 >BP- U!/WD  Ei( -R ’Ž ;?#KH 2 $  $ L89@  z)}%-%-?64„L#[QE-}<$ @ H d%-%4  Q\ Bj -%nE7hR -  pLZ  # ;<E89[ ;< f+-Z\[6>QE$ L„L#[QE-}nZ/ - R rBPs  ! F- a @$ c6]TU-  [Z # :;?-~Z/ s8 h689@-+\/}  L89fZ-REnWD-%r $%-89!bWX  E6 -R rŽ ;?#K $  r$ o8 9   4 +-ZP8 9a6$ 89r6T-  [Z # 4 - a"I%& \$ - r „L#[QE-}\ZK U9 9  489@  f;?-  L ZK 6z)%-%~6  D'*p066# #/  [%^QE$ 8 9  ]\!WX 99 LŽ^ ;?#/ $ n }+n6>pLZ  # 46$ 894R Z$  6({z†  M( nZ/ @} R wR H ; ;<   rQE$ „L#[QE-}~ZK )! ]- <35E ; WD  89f;?-    .pLZ  # ;<E8 9[  ;< : - rpnZZ-%6$  "6! [6 6f„L#[QE-}  ‘z‰s 6a"#' '')(*> ;? 8 9 $ G;?-<$ ’9 4TU-9-$; Q\ -+ †uL+   %& i( $ r"y%& 8 hi;?-  %)[% QE  En;?-~$ a9 ]TU-   2b - R  R ! ƒy . :  :QE$ >WX E8 9 $ 4 >„L#[QE-}>BP- 4z{%-R  ]L89@ 2(~$ cŽ^ ;?#/ $ 5( Z/  8 9:ZREE8 hi 60 ≤ T [K] ≤ 40 · 103 10−16 ≤ ρ [g cm−3 ] ≤ 10−2 . l,s rŽ ;?#K $  QEBP 8 9 4,6 K $  1010 K BP-  $4 FO   wR eI%6 M( R Ž$  -   #K%- \!K'_; 10 ye - Q\ %-  4BP- oz{%-R  ;d 'R 4wR R @ R  '                        log(κ [cm2 g-1]) 6 4 2 0 -2 -4 -6 -8 -10 -12 -14 -2 -4 -6 -8 log ρ [g cm ] -10 -12 5 4 -3 -14 3 -16 1 2 log(T [K])    \ ]\ 9GD7&438  B  ?  - 0 E:89R9 -% )E-4 B7&-4 *89DOL8 B4"!+  Q 8"! <8&->Z H_  • • >IuL+   %& 6;?-  % |„L#[QE-}IBP- z†s ,w &Z%DFO  DwR ρ $  T $ 7x( ZK %-%-   oFO   o- @  #/%- \6BjZ/  κ $   X XpL  [9;<LZ/ E8 9  XBP- U6  8 9O]>  Z = 0 nz‰s  :R R ZK R ρ( T ()FO -; 89 ;? 8 9 k% 8 9R BP89@ ZK21  \! >7"I%& 8h ;?-  %n <„L#[QE-}?h+  S _BjE  [%cQE$ κ z†s A - 8 9 ;? 89 % 8 9R BP89@ QE$;?;<  QE$ =i ]- GpLZ9[} +-+hR -wR =R Ah $ %-% S8 9 ;?M(  8 9 `b$;?;<   QE$ Z/E-;?;0BD  P6c$ a€fi %-%- E8 9 $ R  @5( ;?;< .BP- U!KE Aƒy R 9 BD  7Lz‰s "I%& 8 hi;?-  %y-;V R Q7QE$; un+   %& 6;?-  %/ ; +%-89 BD -%UEj899 D$;  -  %-- \  4`b$;?;< 9[ ?9[ 6(  %-\@ |Ey %&$ZD6~WX - }Rn | - Q\ %-  "I Q\E  ~R    @DQE$ 104,6 BP-  κ = κ RE  QE\! P R WX / 'g @  #/%&  $-  a9[  %-PEj 8 9$;ˆhi$Z 8 9] #%-- E\!ipL$6z^6E nFO ~ 89 RE  %-%-\y 7 89@< $ <6- @  #/%-     'R †64„L#[QE-} BD   2|    $89f  4pLZ% -$ R 4  -d - U! O„L#[QE-}4Eo# -;< 6&% >Eh+  FO  .   9;< ~6. -  WX  8 9 wR  gBD "$0=  'R  6 $ R QEBP 8 9 −14 2 g−1 $  106 cm2 g−1 -; l[%-%4wR κ ZQEB]! 10−12 cm2 g−1 $  108 cm2 g−1 Z/10  κ cm [sZK  #[   BP]- f fpLZZ-%6$ R "6#! R' P $   "6#! '?QE$r  9 r\! '$"             log(κ [cm2 g-1]) 8 6 4 2 0 -2 -4 -6 -8 -10 -12 -2 -4 -6 -8 log ρ [g cm ] -10 -12 5 4 -3 -14 3 -16 1 2 log(T [K])    K\   J*89 :" B  34  - 0E 89R? C -D 6E-4 B7&-4 *896OL8 $*4"!1  :8"! <8-QZ `_  l,s 7Bj -  R 9 rey- Q\ %-9 -  ‘  P$6zL6oL   kwR k€:EYR #'$+ PwR BPE  ,-  a};0%-8 9:Z/ s8 h68 9 -+ ."I Q\E  : vBPf6cR [$cƒy R 9 Bj :ZK  WX2( E89 @$ o „L#[QE-}L -%-%- LZKE8 9  Z/ 4BP- U!eI- . $ oz†s n6>$    8 9E6%-89 eI _BP8h %-$ ?wR "#~' '')(^$ "#~'_(*     oj6Lpn |$ ;? 89@$  BP$ t;?-  %  -  … [%-Yi 8 9 l,-5( z{ ;< % )% 89@$   "6! )  z†s  uL+   %& 6;?-  %VwR >-.% R [}9  $  RE  QE\!6€fi %-%+66;?-~6E  4„L#[QE-} Bj   rR E89  ~Bj 5( κR = κP = κPopI (ρ, T )  h    ’% c;?-?eI-  89 } h $ R Œ%AŽE;<i %-%z‰s ?",#')(_"I  +    rRE  9 BD  !Uƒy ]%-% ;VZK >  ; +%-8 9 6E> -  .ƒy +% 8 9= ey @_BP8h %-$ R SwR ."I  (     [6$6z,$    8 9E6%-89AFO  >;?-~ c  9%-$ R5z{ %rBDE8 9  %-BP- hR ! κ−2 = ;?- 1 1 T010 · 2 + 2 10 10 κice T0 + T κice,ev + κ2dust,abs 1 1 + 2 + 2 κdust,ev + κ2mol + κ2H− κatom + κ2e− T0 = 3000 κice = 2 · 10−4 · T 2 κice,ev = 1016 · T −7 κdust,abs = 0.1 · T 0.5 κdust,ev = 2 · 1081 · ρ · T −24 2     > -2-  / 0 V ) $  ! $ κmol = 10−8 · ρ2/3 · T 3 κH− = 10−36 · ρ1/3 · T 10 κatom = 1.5 · 1020 · ρ · T −5/2 κe− = 0.348 '$'  "6! )                        log(κ [cm2 g-1]) 2 0 -2 -4 -6 -8 -10 -2 -4 -6 -8 log ρ [g cm ] -10 -12 5 4 -3 3 -14 2 -16 1 log(T [K])   K\    G)71*89 : B  ? - 0E 8R9 -F7&E MNOL8 B4"!+  $ 8"!WOQ"!+ 9 i8 9  ]Ž^ ;?#/ $ n9   89@\!               d`b$ +% 89@$ :  QE] $8 h/Ž^ ;?#/ $ $ .89@ P+-%-   "6! + kB T ρ = Ntot kB T. P = mH µ ¯     ~      P  ? ;       %  0  f €    %  i h  $ & %    R P B   8 9 \  b      ~  ;?- µ ¯ X >E; Qv9% ~Ž^ -%89 † -  89%-( '+%-8 9:eI% hi    N = n tot i i nq  Q\   .  #K QEE 2H/ hi-wR>pn Qv9%eI% h    Zeff,i pX ;?R BP8 9@  #/ QEE i Ai z{%-R  ;d 'R fRE8 9  Z/ rBj  fh   Ci = ni Ntot µ ¯= i X Ci (1 + Zeff,i ) i '*) Ai !−1 .  "6!              T^$6$;< . >%)! #+$" 60pL  [9;<7 - EL\% G>EnR E89@5z{ -+\ z{  c6 8 9   −6 −3 8 9 oBjE  %-8 9 sZ/  8 9  -  \XZ/ 9 9  FO   Œ    Œ 8 9 5( \%e10H/ hi g+Ucm BP7QR!‹W! $8 hi  $6z5U6   D`b$ 5( +% 89@$ Az‰s y - d  - E|FG   Hb($  - <  - E|J %--$;?@yQE$ jƒy 5z‰s+$ i$y   n $8hz†s 125 ≤ T [K] ≤ 1,15 · 106 10−6 ≤ ρ [g cm−3 ] ≤ 5623 -   #K%- jBj  dh+  !RpL$6zK6E ~FO  XBPs  dQEBX |6~Z/ ?R [  @  oeH/ hi +- 5( Z/E   6L $8h  ‚ fwR H Z/ ^ nWXE-;?;0$ EL $8 h6nZ/ s8 h68 9 -+\! 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J %--$;DQEBD ~XQEBj -9[}$61R  ]eI% ;< @\bZK P8 9 ;?8 9a-  \ A PE\6ZRE  9 wR; - 6-  hi  WX - 0QE$ H (_WD-%6$ RZ/  |89 ;? 8 9 oeI @gBP8 hi%-$ ?Ey# -;< 6&x( % c] EP# hi 8 9rhR - un%-2%># %-\!                  rFG   HU8 9 ;?fZK +-  d;?-  6&-wR uL hR;Z- [ ŒwR "y    $  eI% hi      / -   H+ + e− → H + γ, 0      %0   -OV 3$    z ≈ 2500  $ +$0                   BDZK /6n"I9   9[$# }89%-8 9a6$ 8 9oun hR;0Z- [ - o R  + `b$ } LZR R2( Z/ aBj  b ? ~6$ 89f]  >FG   H ;<> 8 9 nBPE n   ~BP$  ! >9[}$61R r$ ‘BP8 9@-R  c€c% his%n-; z† s9 mL -wR $;  H !y  $BP$ ]wR 4€f8EvBj %-% # ' wR RE 8 9%&R  H + e − → H− + γ H− + H → H 2 + e − . E  nuL\h   #6z{o~$%-@   " PZK  2! nZ/ ~6E  aFO <~ diss - H (_pn   -%wR f ≈ 10−3    8 9@Z[ \! z ≈ 100 2 H     \  K  X B   >  6    A  y e @      9  $  f   R @ 9 \  /  [ 9    $  #     }  8  9 %   8 G 9 6A  #8 97ƒy  :wR H †6 HD 2 WD-%6$ d  QEL _BDnz† s9 ~ - 2BjZ/ ^E~ - >BD -  uL\h  a+-Z  2 H+ + D → D + + H D+ + H2 → HD + H+ D + H2 → HD + H. +                    ]Q TVdWn Q  TQ ^EX6T 'g  n> #9[ -  'x(_q  #K   'R> X$    9 4sZ/ ~6]uL\h   + H+ 2 + H2 → H3 + H HD+ + H2 → H2 D+ + H HD + H+ → H2 D+ + H. 2  npLZZ[$aRE 8 9 9 6$ 8946  QE&-wRun hR;Z- [  − H+ → H2 + H 3 +e − H+ → H+H+H 3 +e H2 D+ + e − → H 2 + D H2 D+ + e− → HD + H H2 D+ + e− → H + H + D.           PTU-9-$;<89 ;?  E = 5,39 eV ,  QE|$6z†+ $ < Ihi% -  dŽ^ -%89 68 9@ >%% QE   - !UA%--9-$;?9[%--R ion S #/ QEE]-   +  −  $  LiH+ !K7TU-9-$;<89 ;? Z/ +-  @L;?-~ nun hR;0Z- [ fwR LiLi Li Li LiH Li+ + e− → Li + γ, BD -  0 ;d EnT-9-$;C R Z/ ]un\hi  4-   Li+ + H− → Li + H Li+ + H → LiH+ + γ LiH+ + e− → Li + H LiH+ + H → Li + H+ 2, +$"   BX}9   Li+                 6$ 8946]uL\h   Li + γ → Li+ + e− Li + H+ → Li+ + H Li + H+ → Li+ + H2 2 Li + H+ → Li+ + H + γ LiH+ + H → Li+ + H2     9 :h  !bpn$ R'  LsZK 6Z-R :un\hi  fBP-  QE& :ZR Z[$  LiH+ $8 9:6$ 8 9."I9 i6  ( LiH+ + γ → Li+ + H LiH+ + γ → Li + H+ BP- a6$ 89a 6&-wR8 9;< ~R Z-% ~$ o6$ 8 9a#9 i 8 9;< LZQEB]!iR R 6(   --RL $ %- :ZR Z[$  Li− Li + e− → Li− + γ Li− + γ → Li + e− Li− + H+ → Li + H. €f-~ neI    9@$ wR f ;d  4TU-9-$;ChR  @ 0TU-9-$;?9@Y66 4R Z-% nBj   Li + H → LiH + γ Li + H− → LiH + e− , D6$ 8 9o"I9 i6  QE& 6T^6$ R$$8 9o$ dpL$$ 89 \h   oQ\  jBj 5(  rh+   LiH + γ → Li + H LiH + H+ → LiH+ + H LiH + H → LiH + H2 LiH + H+ → Li+ + H2 .  dJ$#  @   9 $ RBj wR TU-9-$;?9@Y66  %&}$6z‰J}$61+hR -~wR  X 9 PR -  ~WX - ?QE$ L„L#[QE-} †\!€:EYR  #'$+  ^3E 89 89@nQE$4wR  [8 9%&} LiH -R ! pL 6z) RBj  7 LqL Q\ @   G 8 9 ;? 89 = #K QEEvBP ]- : f€c  %-%M( E89 @$ R Z/ @$QE7Bj  - G- SŽ,ZK %-%  '6! $6z†R %-  \!P- RE;0L10  % hi  8 94  $ %\! 2 +$'                   D  89  13'T-J - V07 R 1-"8C47&  Q - "! B4"!  E:R /  <7+-4"! +   -)27&-    ,D-4:?IX #J-4  -  ,  +"a  F89  #/ QEE  c i6=H ci T^6$ 1R− 9[%-$  H H+ H− H2 H+ 2 H+ 3 e−  qn Q\   5· 10−4  P 1 · 10−15 1 · 10−10 1 · 10−15 1 · 10−15 1 · 10−10  8:'  99 +*-.! $ = >-BAC8:9;=$>-<'  9;%DA,AC-<'*9;- 3  *') * 3 >*-.!iAC8&9;=$>*-  ' 3  5 8&= 1 #&'*) +-<9  %(9-<9  %(*? +*8&- Lp %-%-R ;< - jpL$ R 0QE$ ^9 ;? 89 Aey @_BP8h %-$ ] - EI> E^QE$;d89 0^ 8 9@;( %-89 Ej$6z|hR;?#%2‡`b$‚;?;< # %K6-wR  DJ -Q ($ qs9%-# Q\E  L hR;?;0\! eI- =BjE  %-8 9 7l[hi  \^ ]QE$ pLZhis9%-$ fE0>E>z†s9 \U]6d6&Z[ 89eI‡ ( #[  <EmL -wR $;<\!E DBD wR I yeI @    9@$ ]wR ?€c% h s% ? I;?-    "I Q\E\!@npLZhis9%-$ 0ED] E|6$ 89e‡6#[  E|mL -wR $;<j # %-D$6z X`K -5( h+%&f ?    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LBj  K$6zI6E>FO  ] 9[}%-~;d c6]qs9%Mz†$ h  r# €f% his%  - 4€c% his% X  W (X) = 1 X (EvJ − Ev0 J 0 )nvJ A(vJ → v 0 J 0 ) nX 0 0  W (X) [erg s−1 ]. z†s  '6!#" vJ,v J >Z/  -  E > E 6?eI  + O >un 5()ƒ-Z  5(*`b$}  vJ $  v0 J 0  6Ž^ -%8 9vJ 689@ v0J-;…`b$  vJ $  A(vJ → v0 J 0) 6 LZ/ @ RBD9  89 - %-8 96( nvJ hR -~wR; `b$  vJ [89 v0 J 0 ! 7mn;< QE$ a- O6A"y x‡i]$6z†Bj 6-r$ :hR;?#%2‡UUA 5z†  >  Z/ O >eI- 6( Z/ QE 9@$ OEd  9%-$ R5z{ %E<6 ; +%-8 9 ?R [$aq>  @ < ?"I Q\E QEBP8 9  .Z/   -%--+  GpX ;< O$ O€c% his% UU! 9!Kz‰s X3EE]pX ;tZQEB]!€c% h s%I;0s   G%M( %L  9%-$ R(^$    '+sZ/ @ RBD9  89 - %-8 9hR -   ‰F- hi$ R @$ 8 9 -  yZK h+  @   - ![eIZK ?- r6i8 9 BP- +$ R5(X$ run sZ/ @} R]QE$aZ/ s8 h689@-R ! pL%-   [-w4QE$ch  fQE$ WXE-;?;0$ o qs9%Mz†$ h  .$R @$QEBD  K\BP - dpLZ 89 - "6#! '6! hi$ QPZKE 89  ZK 6>$   9%-$ R5z{ %% 8 9@$ oz†s L 06!U  9%-$ R;<;< @ )% 8 9 $ 6( R  "6#! + ZQEB]!  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P$ R +!  \ \ , - +$%  √1 (↑↓ − ↓↑) !, #- @BD %-% 6z‰$ hi ’0R R =ƒy 5( I = 0  Q o H2 $ 8 9 $ k Ž^ -%89 Œ Y 2;?;<  8 9X64uL QE$} r9[Z/ ‘#/+--wR ", -}\K6>un   $[ @  Qv9%, -;?;LR >FO    J = 0,2,4, . . . ! Ž^ -#%  I = 1 P;QSX xmPˆo H2  ↑↑    √1 (↑↓ 2 + ↓↑) ↓↓ >60pLZ} 0QEBP 8 9 $  $ @  8 9E6%-8 9 K +  ']- U9[} +y6PpLZ 9J%-$= >w0 R→ 2H= 510 $8 9?KwR; ƒyJ =9[}1%- →I3wR= 845 para Z+E ortho QE$ Z/  }+-; 9 ;? 89 <% 8 9R BP8 9  g2 = I+1 = 3 ) #-  -h 2H!2'g 0l,%\BD H 2‘"I- \$ E0l    jBP- .R Q\ -+\^ ]gƒy 9[}%-I 0wR1 Hortho QE$ Hpara -;N# -;< 6&% 2 >~;?-n `K -PhR - EBD RnhR  nR BDE  4  - a;0$ \2! 8 9  pL$BX9%-  R %-  -  $ .  9%-$ RsZ/ @} Rk;?- ∆J = ±2  %&$Z\U! 9! 6r  9%-$ R$;0BD 6%-$  ortho > 0wR Z/  !I>0 9()#[ x( para  ƒy 9[}%- Lh  4Z/ ~6$ 8 9f H\2h -wRA↔  H 'R2 >;?- ∆JH = i±1  H+ wR }  nBD   s a Hortho + H ↔ Hpara +H 2 2 + H+ Hortho + H+ ↔ Hpara 2 2 ZK  T & 1000 K ZK  T ∼ 100 K. 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" #   " "# %$ '&"  )*!( + , -" # /. *"   # )* (% + ,!( 0 ,*   - & *" ( " 1% "  2 0 3% -"   "(  # % 0 0 %. *+"  -"  *" 4 5) 5* () + *0 6 * # )  ##7 +% '&"  = k m n      kilmn,l nj,l nk,l nl,l · ∆Vl  = :;9 8 D9 + Eint (T, ρ, ni ) − Eint dEint (T, ρ, ni )/dT  = Tneu =Talt −  :;9 8 F9 5* () E-" 1 " )# +% '&"  0 =(ml − ml−1 ) − ∆Vl ρl         XXX l = n    j kijk,l nj,l nk,l · ∆Vl + m + Ediss,i  XX l 5* () +% &"' *"  C + X k         j  ;:9 8 9B '! δ(∆Vl ρl Eth,l ) ad ad 2 = − (fl Eth,l − fl−1 Eth,l−1 ) − Pl (rl2 ul − rl−1 ul−1 ) − 4πκl ρl (Jl − Sl )∆Vl δt   ul − ul−1 ul−1/2 2 2 − µQ,l ρl ∆Vl − 3 rl − rl−1 rl−1/2   XX X XXX − Eion,i  kijk,l nj,l nk,l · ∆Vl + kilmn,l nj,l nk,l nl,l · ∆Vl      = *A " + "+ % '&"  )5!* ( + 4πGml ∆ml 4π δ(∆ml ul ) ad 2 + Hl ∆ml = − (fl+1/2 uad κ l+1/2 − fl−1/2 ul−1/2 ) − rl (Pl+1 − Pl ) − δt c l+1/2 rl2      2 ul+1 − ul ul+1/2 ul − ul−1 ul−1/2 3 − − µQ,l−1 ρl−1 − rl3 µQ,l ρl − rl−1 3rl rl+1 − rl rl+1/2 rl − rl−1 rl−1/2 :;9 8 @9 ?" > )*"+ + +% '&"  5!)*( + :;9 8 <9 δ(∆Vl ρl ) 2 ad = − (rl2 ul ρad l − rl−1 ul−1 ρl−1 ) δt                   )! )        (        % "   ,        *     + *  )    -       1     #    1      $ 0  /            2    ! !  !    /  ,   * )  ,   !  !   $ "         /    7           * '       0   7      $   %                     ,           !                 +   * +     (     '       !     *    "    6     (       +  .        /                5    /  ,  -      . $      $    "  # 0    4                .        ! !   !  * )    !    #  0           3  .        # ni,l mAt,i mu    '     $ &                #         !              kilmn,l n l k j   1 2 1 δ(∆Vl+1/2 Hl ) ad 2 ad ad 2 = − (rl+1/2 uad l+1/2 Hl+1 − rl−1/2 ul−1/2 Hl ) − rl (Kl+1 − Kl ) c δt c   3Kl+1/2 − Jl+1/2 1 2 2 ad ∆Vl+1/2 − Hl (rl+1/2 uad − l+1/2 − rl−1/2 ul−1/2 ) − κl Hl ∆ml rl c   ! ad fl Jlad fl−1 Jl−1 1 2 2 − Hl−1 ) − Kl (rl2 ul − rl−1 ul−1 ) − (rl2 Hl − rl−1 ad ad c ρl ρl−1 ul−1/2 1 + (3Kl − Jl ) ∆Vl − κl ρl (Jl − Sl )∆Vl c rl−1/2                   $ "  /         ,  i mu          mAt,i k j i l m  i X 0 =ρl − l XXX δ(∆Vl ni,l ) X X ad 2 rel ad = kijk,l nj,l nk,l · ∆Vl + kilmn,l nj,l nk,l nl,l · ∆Vl − (rl2 urel l nl − rl−1 ul−1 nl−1 ). δt m n       i l     1 δ(∆Vl Jl ) 1 =− c δt c            MIKa5?!$Q.#% :%><.".!$5?!'#%"/S!'!'&)R Ka*-!'c>(+#%b *-:%,.<.010/%"!'!'TJ13!'24>0U5% "6$!V7.9(F8;W;:=><>#?#?/?/@5?7. 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PJ@9L B)L#M `m }&V‘ "*;!+#%#=ÅÇ! *-IJ!@.!$!'#0.E˜IJ7.!$TJ#%:%!'<..1"5=WM*-#%>#=.!'[#F2424005%5%ºœbM#%‘}Wu!'GH5!$#?EF/?IJ7.!'TJ1"!'!'.IM5=/S*-R>>>B*;#=24#˜0*-5%Z&ÊG»!$8-#?7./?TJIJ!'!'5%<9#?*-/%10!'/%š!'c¢3R;7.2£.!'X`IJ/ !'\.!' 5"Œ>!@W;‰f5%.!$!'#h5?*-7.&@./¤I O rN Mtot X :VG_H =GIH L< S*\IS*>RQ g>LE^ >RQ G_OPYFQ YKADC':=RQ;O bDCF ^DG L< O T_O0TIA*CK>RQ;O H;< W S!CKGIAKA(AKLSFQ >L >RQ 1>LE^ >X Q G_OPYKQ=Y*ADC7:=J< HO CF ^DG L< O T_O \,SKA*>LSK< AH;>LSFQ >LRQN>RAZOPH;^FQP>LW S*>LA*CK> >RQP>LRQ.` \ > \I^KOP.\,E  1>L< WXSK>LADCF Y*AD a(C( $ OP< > > g>LE^ >RQ G_OPYFQ0\ZCK>RQY*EU,> >LSKQ;O f  $ a \,EE O $ AKa < W S5O []\IQ a f ¾Ò  "   #d"    (     &    N 9*;:=<›^*;#%#%!'>!'5?<9*-10/?7..IV>*-5¤GXJ!'0º^*-/%!'5?N*-1cQ.KL!'5F.!'šE˜7.TM>!'5=*->š!'0 O†W`.!'5˜*-7>#?Õ>"!'TJ!' + rN = Rtot = const uN = 0 = const ‘f.!$!'#5¢3E˜5%7.W;TJ/%!'WM.#?/%5=!'*-5?>>\#Sb KL!'IM5%!'.6$/3>*;#]/%!'IM5=*-/?"W;>#?IJ!'K."!'/ !$# <9*->.!'10/3#?":%<[.":=<]/37.&Æ "!˜ÀFKa!'5¤Õq8;:%<>! TJº!'..5=!'*-5>B8-7.!'TJ]!'/%5%#%#?,./%5?!'"_:%<]‡Y/†:%.<9!'*-&¶1"!+K.24"#‚05%6$7.XJ5‚!'‹J0>W;5?!+1"!'Åc/?!'6$7>/%!':%\<`/?‡YX`:=5=<9*±*-G»/˜1"!!'Ÿ5?6S!'7.IM/S¡Çb>!'‘}5?6S"!!'7.EFIMK>/%!'#?/?c5=R;*-7.<.&107.. I\"!†Q.IJKa!$!'W;54&\.!'!'/?º5?"#%E˜:%<>7 ! O …3!'5% Q..`7..Iˆ‹J!'5?5?0.IJ!'5?/%!'¿‡`/?5=*-<.107..IJ#%#?/?5%W;&ºb ‘}"[email protected]!'5%!+Š}*-> KL!$ 0.IM7..IVG Q.5 "!@‡`/?5=*-<.107..IJ#%&\W;&\!']/%!u1N*-7./%!'/ mN = Mtot = const. N −1 1 2 B(Trad ) − (HN + 41 B(Trad )) gEdd 2 2 0 = − (rN HN − rN −1 HN −1 ) − ρN κP (JN − SN ), 0 =JN − 2£W;KL!'Ç.!'5F3. 0.IM/%W; GH*-XY/%W;5 HŸ ³ b ´`¡ &[5%!$0#?/§7.10./?"!'!'È5%!'c^›R9W;KL&\!'`!'`7./%/?!'6$/2†!'5%7..>!'› 7.>R£  "!¿*-7>#.!'5ÅǕM#?7..I.!'5º‡`/?5=*-<.107..IJ#%/?5=*->#?,LW;5?/?IM1"!'":=<]7..I ŸH³ b¨³J¡  "!F(f05%:%<.<>W*)LOž¢1N*->:%X`OžŒ.7..XY/?"W;š.!$#£‡`/?5=*-<.107..IJ#¤GH!'1".#SR`0ˆ>*;#† "!AW;10XJ!˜!'0.IJ!'Ka!'/?/%!'/£"#?/SRM"#?/Sb gEdd = ˜ N + 1 B(Trad ) H 4 1 ˜ JN − B(Trad ) 2 J˜ ˜ H B(Trad ) = σ 4 T π rad  M5B #EGQ @2A ? 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"IJ"!+!V!'º‰f"!#%!$0#=>^*-B&@*;0#%/?$ #%&V!-s*;.*-#=!$Ka#%#Ž![!'10<].1"!˜¼ !' 5@® 5%b0‰WMҎ#?*;6$/=7>#?*-2†#=/?"*-W;#%&[10:=XJ<>&\!M!'Rœ_!' .("IJ![!P!'G²^5?*;>#%*;# #?#%b/S#%bJA«!uEF.8-7  `7. ."Iš!™a#?0*-><.œ1"!'Ra]2†2†!'!'":=5?<>/%!@!'C.!$ #†"!B<`¼Y‹J W;5% WM#%/=*-/?"#%#?:%/=<>*-5?!'X›º‹J(W;!'>5?!'>0#†9*-G»Q.>5Ž.!'*-5Z101"!f*-Kc^_RaW`2£.*;!'#+101"!F*-‹JW;.!'58-<.‘f.!P10Ñ>":=.<>0!'/?"5FW;‰}C5% •;"TJ!$!'#%!'5O ‰}5%•;TJ!+10"!'IM/Sb Tc = 105 K 104 Tinit = 200 K ninit = cm−3 Mτ   8 14" " ,4&)(K4 74!#" ; &>,48 !#1=! &  f2£‰ !'!'5?IJ/%!'!'>#%/=.*-!'>5+‡`/?5?"7.!$#%XY!$/?#B7.5?E˜‹;K>*-5?#%N:=*-<.K..1"0!'/?c/%#\bc‡`#?"0!Z>U247.^›5%.WY!'._!'1010 5%7.!$:%5%<.:=<.`IJ7.!P.G IJQ.!'<.©5?/S&[Ra7.0/B&7..]!'/%!'_5%#=:%0<. 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"7.W;.I„.® !'b05}ÒS¯B^_6$W`7º.!'#%10!'1"<>!@!'<9š*-"KL#?!'/SR 0› "!u*-10^›1"!'šWY.Œ!'8-101"10!@1"!'º&[09/}*-<><>•;!'<>6$7º!'5%IM!'51"!'"E˜:=!'> ":%<`/%!Z!'0>!}IJ!'5?0.IJ!'5%!Z‰!$#=*-&@/?&V*;#%#=!f7.>„>*-&[0/!'0>!'§XY1"!'0>!'5%!'›‰f!$#=*-&Z/?5=*; 07>#Sb E˜7>:=<”!'5?"!$5%!'#%!¥":%<`^›/˜WY<9.*-/S!'10bq105%™q!$7º:%<. `"7.!$#%.!'IJ&!' ™L!'240/?7.,.5%7...!'X` /˜Ka.!$!'!'5>3.]!'/S/ž24Ru"#%:=W;X`K9107.*-.1"{I§#? 0">!«§™L#%!'"!]/?^›5=*-WY10/%.!'!'&[101"!M,aR>!'*-5=*-K./?IJ7.!P5O #%!'<>!'º‹JW;š.!'5˜Ås*-IJ!+.!$#EF7.TJ!'.5=*; 07>#SR..":%<`/˜‹JW;>!'09*->.!'5}6$7§7.]/%!'5%#%:=<>!'".!'cb Ó b  QBLB)@ L >A EC@ Q  0@9L0M5Q EC@   L0A ECB)LE PL B)L A EC@9D @ B EGJ J5L#M 5B 2Q E E IB  ÉMÉ k m k m `m goMp goSp m ninit = ninit = 106 cm−3 104 k Tinit = 200 K ninit = 105 cm−3 cm−3 105 K 6oMp m goMp lk m Um k -4 -6  #   N   ! " I -8 log ρ [g cm ] 6 κ=const κPopI κPopIII 5 #  (   log T [K] -10 -3 κ=const κPopI κPopIII -12 -14 4 -16 3 -18  #  I -( &   !  #      -20 -22 2 10 11 12 13 14 15 16 log r [cm] 17 18 19 10 -10 log Mr / MO· u [km/s]      #    #   d    -20 -25 -30 -40 -45 10 11 12 13 14 15 16 17 18 14 15 16 log r [cm] 17 18 19 17 18 19 κ=const κPopI κPopIII 19 10 11 12 13 14 log r [cm] 15 16 log r [cm]             Tinit = 200 K                                             ninit = 104 cm−3                                 !            &   κ=const κPopI κPopIII -35 13 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -5 -15 12 M¯S¯Ò 0 11 -4 -6 -8 log ρ [g cm-3] 6 ninit=1045 ninit=106 ninit=10 ninit=1045 ninit=106 ninit=10 5 log T [K] -10 -12 -14 4 -16 3 -18 -20 2 18 19 10 0 log Mr / MO· -20 -25 -30 ninit=1045 ninit=10 ninit=106 12 13 200 K 14 15 16 log r [cm]   11 .       10  19       $             17 18 19 -8 18   17 19 -4    14 15 16 log r [cm]   13          10 cm −3  12       6 * 10 cm 11                10   -45 18 -6  -40 17 -2      -35 14 15 16 log r [cm] 0  u [km/s] -15 13 ninit=1045 ninit=106 ninit=10 2 -10 −3 12 4 -5 5 11     % $  " 17 104 cm−3 * 14 15 16 log r [cm]                      13   12  * 11  Ò S¯Ò 10         #  !  !   N  4      &  -22 -4 -6 -3 ninit=1045 ninit=106 ninit=10 5 -10 #  (   log T [K]  #   N   ! " I -8 log ρ [g cm ] 6 ninit=1045 ninit=106 ninit=10 -12 -14 4 -16 3 -18  #  I -( &   !  #      -20 -22 2 10 11 12 13 14 15 16 17 18 19 10 11 12 13 14 log r [cm] 2 -10 0 log Mr / MO· u [km/s] -5 -15 -20 12 13 14 15 16 18 19 -8 17 18 19 10 11 12 13 14 15                                                             −3 104 cm−3 #    # 200 K                   !                                               #      10 cm 17 -6 log r [cm]          10 cm 6 −3 16 -4               5 19 ninit=1045 ninit=106 ninit=10 log r [cm]       &   11 18 -2 ninit=1045 ninit=10 ninit=106 10 17 M½S¯Ò      #    #   d    4 -30 16 log r [cm] 0 -25 15 10 cm 5 * −3 10 cm 6 −3 -20 -25 14 15 16 log r [cm] 17 17 -14 -18 -22 18 18 19 -15 ninit=1045 ninit=10 ninit=106 19 10 0 4 -5 2 10 11 11 12 13 12 14 15 16 log r [cm] ninit=1045 ninit=106 ninit=10 0 -2 -4 -6 -8 13 200 K 14 15 16 log r [cm] 17 18 19 17 18 19     % $  " 5 104 cm−3         #  !  !   N  4      &  6 * -12 log T [K] -10  -10 log Mr / MO· 14 15 16 log r [cm] $                               .            13  log ρ [g cm-3] -8  $ 13  12    ! 11 12           10 11   10             u [km/s] -6 ninit=1045 ninit=106 ninit=10 *                                 MÓS¯Ò -4 ninit=1045 ninit=106 ninit=10 4 -16 3 -20 2 -4 -6 -10 -3 Tinit=100 K Tinit=200 K Tinit=400 K 5 #  (   log T [K]  #   N   ! " I -8 log ρ [g cm ] 6 Tinit=100 K Tinit=200 K Tinit=400 K -12 -14 4 3 -16 -18 2  #  I -( &   !  #      -20 -22 1 10 11 12 13 14 15 16 17 18 19 10 11 12 13 14 log r [cm] u [km/s] log Mr / MO· 12 13 14 15 16 17 18 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 19 10 11 12 13 14 log r [cm] 19 16 17 18 19 15 log r [cm]                                                     100 K 200 K #  #     104 cm−3                                             !                                                    400 K 18 Tinit=100 K Tinit=200 K Tinit=400 K      &   11 17 log r [cm] Tinit=100 K Tinit=200 K Tinit=400 K 10 16 -´S¯Ò      #    #   d    0 -5 -10 -15 -20 -25 -30 -35 -40 -45 -50 15    100 K 200 K      10 11 12 12 400 K 13 13 14 15 16 log r [cm] -25 -30 14 15 16 log r [cm] 17 17 -12 -14 -18 -22 18 18 19 -15 -20 Tinit=100 K Tinit=200 K Tinit=400 K 19 10 0 -5 10 11 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 11 12 12 13 13 14 15 16 log r [cm] Tinit=100 K Tinit=200 K Tinit=400 K 14 15 16 log r [cm] 17 18 19 17 18 19 104 cm−3         #  !  !   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H;Ra QPHOP\#U,>  >RQ;O >RQ CK>RA $ f C AB ÒMÒSÉ C log r [cm]  −8 0 −10 −5 −15 −4 −20 −10 −40                                                                                               "                     r       Mr                    #   u 16 14 12 10    $   "       !                                #         '               '                                                                         (  # #     & %                        +              *    " "          )                                               N   " I        !          #    #   d         &   2 8 6 20 18 16 14 12 10 8 6 #    ρ ∼ r−2 −2 log Mr / MO•   0 14 12 10  I  4 20 −35 log r [cm] log r [cm] M¯S½Ò −6 τR≤2/3 τR≥2/3 u<0.1⋅vff −30 T τR≤2/3 τR≥2/3 u<0.1⋅vff 5 16 18 log r [cm] 20 −25 log T [K] 5 log ρ [g cm−3] −15 −20 3   (  !  I -( &   !  2 −25 8 6 20 18 16 14 12 10 8 6 ρ r −5 18 u [km/s] 0 4 −10 τR≤2/3 τR≥2/3 u<0.1⋅vff 6 τR≤2/3 τR≥2/3 u<0.1⋅vff −2 r~ρ 5 -6 τR=2/3 5000 10000 -2 2 -3 1 -4 0 -5 -1 R Mτ =2/3 R • Mτ =2/3 -7 15000 20000 Alter [a] 25000 -2 -3 30000         #  !  !   N  4      &      % $  " log M [MO• ] 25000  ( 0 15000 20000 Alter [a]    **  ) ,     - 30000 10000  -2  15 5000  %  0 $   '  $ 0   #    17 1 log Teff [K] "   -3   -2  -1 • 16 log M [MO• /a] log L [LO• ] 0  25000     20000  15000 Alter [a]    10000   log L [LO• ] 1    2       5000 $ 0     !      log r [cm] 2 %            3                                !             (     . *)      "   *                                  /       (       )    0      .    "   "          Ò S½Ò 4 7 6 5 4 3 2 1 0 -1 -3 LEdd 30000 3 4 0 0 14 Mτ =2/3 R • Mτ =2/3 R τR=2/3      '         "       +             !                                                 #                        & %       !                                      !                                                         #     & %                                                                        #   & %     N   " I        !          #    #   d         &   M½S½Ò −1 −1 25695 25695 25695 25696 25696 25696 25696 25696 Alter [a] 13 25695 25695 25695 25696 25696 25696 25696 25696 Alter [a] log M [MO• ]  log Teff [K]   •  I  log M [MO• /a] LEdd 2 25695 25695 25695 25696 25696 25696 25696 25696 Alter [a] 2 log L [LO• ] 3 log L [LO• ] 3   (  !  I -( &   !  log r [cm] 4 4 "$%   &   4N! !#   1 0 −2 −4 −2 −6 2 −1 log cH log cH 0 −3 −8 −10 −4 −12 τR<=2/3 τR>=2/3 u<0.1⋅vff −5 −6 6 8 10 12 14 16 τR<=2/3 τR>=2/3 u<0.1⋅vff −14 −16 18 20 6 8 10 log r [cm] −5 log cH− log cH+ −20 τR<=2/3 τR>=2/3 u<0.1⋅vff −35 −40 6 8 10 12 14 16 −18 18 20 6 8 10 −5 −5 −10 −10 −15 −15 + 3 0 −20 −25 −40 6 8 10 12 14 16 log r [cm] 18 20 d2Z b H H2 −25 τR<=2/3 τR>=2/3 u<0.1⋅vff −40 18 20 6  2- 8 10 12 14 16 log r [cm]   " $D*+$/ ^ $&  5#)\]S$F8hQ5#,S  /! )*+- -,-/G$E3-2 O 16 −20 −35  2  !&=5#)M c$ /=4Y T-$&!#"+-/ 4& F/8`/ Q4E   /  - g; !  "4&4&!#4F$D2C,-/!&32- 14 −30 τR<=2/3 τR>=2/3 u<0.1⋅vff −35 12 log r [cm] log cH + 2 log cH 20 τR<=2/3 τR>=2/3 u<0.1⋅vff −16 0 −30 ^ / =@  18 −12 log r [cm]  20 −14 −30  18 −10 −15 −25  16 −8 −10  14 −6 0   12 log r [cm] !DI^  =@ ÒS½MÓ !i0("-  "- ,2S2!D "+$&J  O H+ H−  J$P$D /!& ^ / =@   "-4D4P/!D4P$&2 C4&  b S$D!D"+$& 2-/ +O +  S$&!D .2- H2 H3 −20 −8 log r [cm]   −15 −4 −20 0 −10 −35                                                            "                     r       Mr                    #   u 16 14 12 10                                     $   "       !                                   #         '               '                                                                         (  # #     & %         +         *    "    )                                                                           #       ρ ∼ r−2 −2 −10     N   " I        !          #    #   d         &   −5  2   0 14 12 10  I  4 8 6 20 18 16 14 12 10 8 6 T τR≤2/3 τR≥2/3 u<0.1⋅vff 5 20 log r [cm] 16 18 −30 log r [cm] log r [cm] -´S½Ò −6 τR≤2/3 τR≥2/3 u<0.1⋅vff −25 log T [K] log ρ [g cm−3]   (  !  I -( &   !  2 −25 20 log Mr / MO• 3 −15 8 6 20 18 16 14 12 10 8 6 4 −5 18 u [km/s] 0 ρ r −10 τR≤2/3 τR≥2/3 u<0.1⋅vff 5 τR≤2/3 τR≥2/3 u<0.1⋅vff −2 r~ρ 5                              0.5 Alter [a] 0 −1 −2 −2 −3 −3 −4 −4 −5 τR=2/3 R Mτ =2/3 R • Mτ =2/3 1.0 1.5 Alter [a] 2.0 −5 −6 2.5     % $  " −1         #  !  !   N  4      &  2.0 log M [MO• ] 1.0 1.5 Alter [a]     (      * )    ,      $ τR=2/3  2.5 0.5  %  $&  '  2.0 −6 0.0 $ 13   #    2 3 log Teff [K] −2 0.0 "   −2   −1  log L [LO• ] 0  • 12 log M [MO• /a] log L [LO• ] 1  1.5       !    1.0   0.5       11 0.0  log r [cm] 4 $     !              2   %            !              (        *  )                 .         "     *       !                             .  /       "  (     )   0  "   M³S½Ò 3 4 3 2 1 0 −1 LEdd 2.5   &    d ##    !   I " N  I  !(&-I !( 1 0 −2 −4 −2 −6 2 −1 log cH log cH 0 −3 −8 −10 −4 −12 τR<=2/3 τR>=2/3 u<0.1⋅vff −5 −6 6 8 10 12 14 16 log r [cm] −16 18 20 6 8 10 −5 log cH− −20 τR<=2/3 τR>=2/3 u<0.1⋅vff −35 −40 6 8 10 12 14 16 −18 18 20 6 8 10 −5 −10 −10 −15 −15 + 3 −5 log cH + 2 log cH 0 −20 −25 −40 6 8   - 10  14 16 log r [cm] 2  !&=5#)M G$    /=4S( T $&!#"+H4&  12  14 18 20 −25 τR<=2/3 τR>=2/3 u<0.1⋅vff −35 −40 18  20 d2ZPZ&Z b  6 8 10 12 14 O H H2  2- 16 log r [cm]  " $D*+$/ ^ /G$D Q5#)i\]S$F8hQ5#,M   ! F/8`/ Q4] L()*+,-$a3-2 !`0"- Q"+ ,-2M2-!#  "%$D9  "4&4&!#4F$D2 C,-/!&32- 16 −20 −30 τR<=2/3 τR>=2/3 u<0.1⋅vff −35 12 log r [cm] 0 −30 ^ / =@  20 τR<=2/3 τR>=2/3 u<0.1⋅vff −16 log r [cm]  /- g; ! 18 −12 −14 −30  20 −10 −15 −25  18 −8 −10   12 14 16 log r [cm] −6 0 log cH+ τR<=2/3 τR>=2/3 u<0.1⋅vff −14 !DI^  =@ ÒS½M® O H+ H−  J$P$D /!& ^ / =@   "-4D4P/!D4P$&2 C4Pb S$D!D"+$& 2-/ O + +  S$&!D .2- H2 H3 "$%   &   4N! !#        " "%& 7335(2&I* 8 #  %‹J#EF!'W;1"<>5?#Z10!'"C‰f!'IM!'2†/SK.Rc!'7.5%>5?.*;/%!'#%#fÈ#%/?7c7.Ÿ»b¨*>E˜ba.K>Q.!\#%Ka:=!'<.!'0>.50!$!'/?#+0/u>¢®J!V5%¡PW;ÀFRœ/%2£KLWM#?!'!'/%5¤01D!'Õq5?8;<.>:="#@!'<>5@!MX;Rc*-!'5%..#%!'/?&)&V>*-*;¢1"#@#@<>!'W;!'5%/%0#?WM/?#?&V/?,.<>*-<9!'108-05?IJ5%&[!'!V."#%5=:=*;5?<Ð 5%!'0*-7>":=K.#„<>IJ!'Ÿ²!$‰}C#=:%1"<.‹J!'"W;1":%WM <`#%7.#%!'.>I_!$#B® b0‰fÒ ¡P!'RaK.‹J01"IJ!'.!P5¤! 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I " N  14 τR≤2/3 τR≥2/3 12 u<0.1⋅vff 18 14 8 6 4 2 2 0 10 12 14 log r [cm] 16 18 τR≤2/3 τR≥2/3 30 u<0.1⋅vff 6 8 10 12 14 log r [cm] 16 40 18 35 τR≤2/3 τR≥2/3 30 u<0.1⋅vff 20 3 35 20 M [MO / (cs, lokal /G)] 8 3 25 20 • 15 10 • • 8 6 40 • 10 4 6 M [MO / (cs, lokal /G)] u<0.1⋅vff 12 • • • M [MO /a] 10 0 5 8 10          $&!#"+Q4F$`  0("-  4 • • 20 15 10 0 6 • 25 5 0 / τR≤2/3 τR≥2/3 16 • M [MO /a] 16 I  !(&-I !( r 12 14 log r [cm] 16 18 20 2  !&=5#)M -/HTV!h / '"-4D4P/"+,M,M!&$& 2-  TV!(   1/4D"+ $&cKM   6  d2-Z&ZPZ b[7"-  @d 8 10 12 14 log r [cm] 16 18 20   , 4D@U   ;2Z b[7"+  !D/5#)S$D4#@AM( T b  2- @]  8i 3  S$&/@aV ! ˙ = 4π r2 ρ u M M /a  Q"%$D24P!D/5#)S  cs,lokal /G ‘'5FW;101-,‹ '5?1N8-7 G»/‹JW -7.T   ;:=<   cb K/‘   B)G-‘ I  "":% 1< ›0- Œ.T .0%Iu5%²W-2Kb Ñ  43 ‹ '4  b`N8-×;7¯ G !  † Œ  Î5G %0 I &'(*, ! 0IM X  ! "SˆR#&# (G$%% 0! I  &6')(+"*,')(7 %! 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I (  ‹  " W  q¡Pb @ Tinit = 200 K ninit = 104 cm−3 ninit = 105 cm−3 ninit = 106 cm−3 BB C 7C C -B A CKLH;>XQ e K< CFY*AKUL LY#H;>RS*>LAcb H;LXQPHOP>RQNY*A*C >L< OP>RQV>RQPAU,> K< CF>RO $ SDG >LAgb*CK=LA *G >XO G_H.H;W S T_WXSK>RQ.GIY*H;UI>L^KQ T_UIO bDG >RQ.CK>RA*A*\5W SdH;B @ &C ?C ÒS½MÉ B AB 2 −25 log r [cm] −8 −10 κPopI κPopIII −2 −4 −6   &        N       &       16 14 12 10 8 6 20 18 16 14 12 10 8 6 log r [cm]    log r [cm] M¯SÓÒ 0 20 2 2 0 −2 −4 −6 −8 −10 −12 −14 −16 −18 −20 14 12 10 8 6 20 18 16 14 12 10 8 6                                                                                                                                          !                            #  t = 105 K κPopI κPopIII 4 18 log r [cm] 16 20 log M [MO• ] 3 −15 log T [K] −3 log ρ [g cm ] −10 18 u [km/s] 0 4 −5 κPopI κPopIII 5 κPopI κPopIII 5 −20    '                     −15 −20 −25 ninit=1034 ninit=10 ninit=105 16 104 cm−3 105 cm−3 18 20 106 cm−3 5 4 0 −5 2 8 10 0 ninit=1034 ninit=105 ninit=10 6 8 10     % $  " 6 Tinit = 200 K 6 5 12 14 log r [cm] 12 14 log r [cm] 16 −2 −4 −6 −8 −10 16 18 20 18 20         #  !  !   N  4      &                                              12 14 log r [cm] 20 $ −40 18  −25  −10 log T [K] ninit=1034 ninit=105 ninit=10          −10 log M [MO• ] −5       −35 16 −30  12 14 log r [cm]  log ρ [g cm−3] 0 *  10                      . 8 $ 10 * 8    ! 6  u [km/s] 6              Ò SÓÒ 5 ninit=1034 ninit=105 ninit=10 4 −15 −20 3 2 5 0 log ρ [g cm ] 6 ninit=1034 ninit=105 ninit=10 5 log T [K] −5 −3 ninit=1034 ninit=105 ninit=10 −10 4 −15 3 −20 −25 2 8 10 12 14 16 18 20 6 8 10 12 log r [cm] 4 0 2 −5 −15 −20 −25 6 8 10 12 14 18 20 −4 −6 −8 −10 16 18 20 6 8 10 12    log r [cm] 14 16 log r [cm]  "                                                                                                        !                                                                  # #         106 cm−3  200 K     104 cm−3 105 cm−3 20 −2 ninit=1034 ninit=10 ninit=105 −40 18 ninit=1034 ninit=105 ninit=10 0 log M [MO• ] u [km/s] −10 −35 16 log r [cm] 5 −30 14 M½SÓÒ   &        N       &       6           "$%   &   4N! !#   ‘À@"È%<&B) 0%! h¡) ! 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"f¸6&'()0% .X 8 .I ÒSÓMÓ 5·10−10 f H2 H2 f H2 5·10−4 Tinit=100 K Tinit=200 K Tinit=400 K 4 log T [K] −3 log ρ [g cm ] 5 Tinit=100 K Tinit=200 K Tinit=400 K 3 2 8 10 12 14 16 18 20 8 10 12 14 log r [cm] 4 0  −5 −10 12 −6 14 −8 −10 16 18 20 8 10 12 14 log r [cm]  104 cm−3                                                                                                   !                                                           # #       400 K 16 log r [cm]             100 200 20 −4 Tinit=100 K Tinit=200 K Tinit=400 K 10 18 −2  - & !  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"  "'(+F/K  ?Q '=;X "'(+0.I *C  !  š X %  cb ÒSÓM¾ "$%   &   4N! !#   1 0 0 −2 −4 2 −6 log cH log cH −1 −2 −8 −10 −3 −12 −4 −14 −5 −16 6 8 10 12 14 16 log r [cm] 18 20 0 6 8 10 6 8 10 12 14 16 log r [cm] 18 20 16 18 20 16 18 20 0 −2 −10 −4 −6 log cH− log cH+ −20 −30 −40 −8 −10 −12 −14 −50 −16 −60 −18 6 8 10 12 14 16 18 20 log r [cm] 12 14 log r [cm] 0 0 −5 −10 −30 −20 + 3 −15 log cH log cH + 2 −10 −20 −40 −40 −60 −45 −70 −50 6    8 10      −30 −35 −50 4& $   −25   12 14 16 18 log r [cm] 20  2M  !D/5#)M HJ$ fH = 5·10 !&V −4 6 d2Z b  R  ?@ 8i 8 O 2/!&L^  =@ O H+ H− $P$& !DX^ 12   " $D*+$   S$&!#4D5#)=  5#)/!(T "+S4 / 4&"-J/ b  !&2-$& −10 R   =@P@ -( T- $D!D"--/ fH =.5·10 2- /S$&!#"%$D2  =@ O + H+ 2 H3 ÒSÓMÉ 14 log r [cm] 2 ,-$h3-2I /!0("  "- ,2M2-!# "%$&E  2 ()*+ H H2 \] 2M  6 10  S$D!DH^  6 !  "-4D4P/!D4P$&2  4&  F/8`/ Q4 ,!D.32- / =@(TV!i "41 T "+4Pb9  "4   = J (!          (# # %  % I #(( 1 0 −2 −4 −2 −6 2 −1 log cH log cH 0 −3 −8 −10 −4 −12 −5 −14 −6 −16 6 8 10 12 14 16 log r [cm] 18 20 0 6 8 10 6 8 10 12 14 16 log r [cm] 18 20 16 18 20 16 18 20 0 −2 −10 −4 −6 log cH− log cH+ −20 −30 −40 −8 −10 −12 −14 −50 −16 −60 −18 6 8 10 12 14 16 18 20 12 log r [cm] 14 log r [cm] 0 0 −5 −10 −10 −15 + 3 log cH log cH + 2 −20 −30 −40 −20 −25 −30 −35 −40 −50 −45 −60 −50 6   10     4& $   8   12 14 16 18 log r [cm] 20  2M  !D/5#)M HJ$ fH = 5 · 10 -!DV −4 R 6 d2Z b  ?@d 8i 8 10 12 14 log r [cm]   " $D*+$  fH = 5 ·  10  S$&!#4D5#)=  5#)/!(T "+S4 / 4&"-J/ b  !&2-$& −10 R   =@&@AM( T $&!#"+4P   2 ,-$h3-2I /!0("  "- ,2M2-!# "%$&E  2 ()*+  .  2- /S$&!#"%$D2  6 !   O 2/!&L^ H H2 \] 2M  6  =@ O H+ H− $P$& !DX^  =@ O + H+ 2 H3 Ò'´]¯  S$D!DH^  "-4D4P/!D4P$&2   F/8`/ Q4 ,!D.32- / =@(TV!i "41 T "+4Pb9  "4            # “".  <+‹*   ›< ¢+!   <4A A)A-,¦‡Ÿ   ')(q ¡ J ("& ' (!   /u. ¸6 ! '(+ 8 " ( +*C ")# 8 ˆ‰%$ ! '& ! "()$ ! 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YA'b /$ 2 2E" !UG2>9Q!"($&/# 2 9Qc2E/ %B!(.X &c4 2>(%'c%&72 89 AB2EY=f%B 2#"FB!S &d/$71$"A'%B2Ej!"e"2G9 4582EA'%'1$A*&g16%'<>%'n"/ 7\16& %B"$82><%B! 2 : 2  89  )2;$& \ J*\ $89Q"U&h/9Q-ic" 2E 1$162E V%$Go2E=3j1$ 2EFDn/ 2ED(FX$&! 2 "U2E(%B%^ !(#"%B1$!2>#"2E`I2E?;pkl& #S7VW9Q("$9Q/$A'A'" A/$m" YG1$2> ! 2 J!"C2EAB1$(?;%B2>& 72 c#Y9Q2E<><" / , -= 2 !1$2E/ .U4%'2q1$2E Q%'A'A'%B 2 (&!2`rN2>!!"!2E% $/$" o1$ 2 0s  %G2E"2E(-i2E%B#2c%B(&0%'oEt  %2G2Cuwv@,Exl<>/Zy1$2E/ . z {#| }~Y€‚`ƒ‚„P…!†‡!ˆY‰3€‚`„P†Š€‚`ƒŒ‹†Ž\`‡‘R‡!’P‚† “T%B23¢;/2E$< <>/£R2E%'& 9%¥¤T9816%'/ /$1 I = I (r, µ, t) 1$2 =ž*/(”!2E%'&(/$$•G(%B!"F&(/$$• µ = cos θ H θ %B(&=1$2 §¦_%'$¨G2EAU<>-0%B!#"ν2Eo1$2 N9Q/t(-=^%(&!i•G2 (%B!";r&!2E&!2E ©T˜ C9QAB2E`/$1‚1$2 ª¤*%B!";&(/$$•31$2>¡«"˜8&!˜8¬M®­ j (r) %B(&ª16%B2§%B!˜8&!˜8XY2§4#g%B!(%'—@%'&#^Q&L­ κ (r) /$1 (%'1`16%B25¯J!2>¢;/2E$%B2E;&!2EC%9 ¥¤T9816%'/ κ (r) Ÿr Φ (µ, µ ) %B(&±16%B2g«"U98#2E6¯²/$$¨@&(%B˜8l­l16%B2g16%B2\¦e9Q"!!#"2E%'$A'%B!"$¨G2E%'&\2E%'2 `?;&!2E/$/$$•³9Q/q1$ 2  ¤*%B!"F&(/$$• µ %'³16%B2q¤P%B#"F&(/$$• µ 9Q$•7%'”$&LŸR?;%B23%B!&P!˜\˜ g%B2 (&L­U198!0¯²´ *9QA'AB2q¤*%B!";&(/$$•G2E µ ν ν ν Str ν ν 0 0 •7%'A'&LŸ 1 2 Z 1 Φ(µ, α) dα = 1 −1 ,EµF¶ Abs ν             !#"  R B A z r p θ R $&%'%'(*)*+-,/.'01$32*465/798;:=A@B)*)*,'.'0C+-@B: pzD E %F@B.'@G2 ©398!"1$2 *s§9%(%–9Q”$AB2EF&#9Q‘¯J˜ C9Q&(%B˜8 H[ž`Ÿ+)GM H[ž`IŸ HGM p (r, µ) → (r,p), p = r 1 − µ2 ( Iν+ (r,p,t) µ ≥ 0 Iν (r, µ, t) → Iν− (r,p,t) µ < 0 -0%!1 b AB2E%B!"F/$$• H[ž‚Ÿ',OMi%'™<>-i2E% •G2E&!2E$F&!2 b AB2E%B!";/$$•G2E H[ž‚Ÿ µ;M H[ž‚Ÿ+¶GM = (κ + κ )[I − S ] ¯[´c2E%'2DXY˜7(%'&(%'—G2C/$1e2E%'2D2E•F9Q&(%'—G2KJ*˜ gXY˜82EF&!2 I ”$<>-qŸ I 9Q/6¯[•G2>(XU9QA'&!2El­ª-i2E%'A=16/ !!" 16%B20/$U9Q”$"U^Q$•7%'•G2*s§9%(%–9Q”$AB2 p $%B#"F&§<>-0%B#!"2E µ > 0 /$1 µ < 0 /$;&!2 !!!"$%B2>1$2E]-i2 !1$2E ¨89Q$lŸ “32 .«9%#9%D2E&!2  H²—G2 (•7AB2E%B#"2‚ž.”$”$%'AB16/$$•ož`Ÿ',OM0-0%!1f?;&!˜8œ7XU9%#9%D2E&!2 ±•G2EU9Q$F&L­R2 *¨79Q$™¯[´  2E%'2E™•G2E•G2E”Y2E2E³p ¤T9816%'/ r ¦ 2 (&!2c<>-0%B!!"2E 0 /$1 r 9Q$2E"D2ElŸ : VU9QA'Aª%B!˜8&!˜8XR2 3?;&!2E/$/$$•fH MN/$1Z% AB˜8¨89QAB2E³&("2 D˜;16š@U9%g%B!!"2E b AB2E%B#"6› •G2E-0%B#"F&T¨89Q$™16%B2 L./2EA'A ¯[/$$¨@&(%B֘8 (µ,S µ¯[˜8)A'•G=2E11$2 C9QœG2Ef•G2>!!"(%B2E”Y2E™-= 2 !1$2El 1 ∂Iν+ ∂I + +µ ν c ∂t ∂r 1 ∂Iν− ∂Iν− − +µ c ∂t ∂r + = −(κAbs + κStr ν ν )[Iν − Sν ] Abs ν Str ν − ν + ν − 0 ν ν 1 Sν = Abs κν + κStr ν X κAbs,i Bνi ν + κStr,i Jν ν i  ! . H[ž‚ŸIMGM N.%B 2 §(%'1 /$1 1$2 §ž.”!˜8X$&(%B˜8(¨G˜@2®°D<>%B2EF&N1$2 i?;XR2E<>%B2> ”$<>-qŸG1$2 !2ECrN2E%'&#9Q• <>/=4Bg%B!(κ%'—@%'&#^Q&§/$1 κκ B/$1 κ J 1$2 N?;&!2E/$¨G˜@2®°D<>%B2EF&01$2 =?;XY2E<>i%B2> i ”$<>-qŸ;1$2 !2E rN2E% › &#9Q• <>/ T?;& !2E/(& #9Q"$A'/$$•Ÿ J %B(&P198P;/$A'A'&!2 Of˜ D2EF&T1$2 T?;&#9Q"$A'/$$•G(%'F&!2E!%'&#^Q&LŸRž./$œG2 !1$2  -= 2 !1$2E˜;#" 198\ 2 !(&!2]/$1 <>-i2E%'&!2?;&#9Q"$A'/$$•GD˜ D2E;& H ”$<>-qŸ K ”R2ES8&(%'•7&LŸ§?;%B2](%'1 ,E/µ M Abs,i ν Abs,i ν Str,i ν i ν Str,i ν ν ν ν ν  !      ! C      !1  &  !!  " 9#C   ¯J˜8A'•G2E1$2 C9QœG2Ef1$2®y$%B2 (&L H[ž‚Ÿwv7M H[ž‚Ÿ GM H[ž‚Ÿ GM Z 1 1 + Jν = (I + Iν− ) dµ 2 0 ν Z 1 1 + Hν = (I − Iν− ) µ dµ 2 0 ν Z 1 1 + Kν = (I + Iν− ) µ2 dµ. 2 0 ν z { *`ƒ †ƒ‚‡!‚„P€`‚„P† €‚‚ƒ z  *`„ I†~WW† V˜ `/$A'%B2 (&@C9Q™16%B2 b AB2E%B!"F/$$•G2E³¯[´N2E%'2c(X$"U^%(%B!!"2!N*´$A'AB2g%'&P$2E#9Q1 /$1Zž*/6› œG2E #9Q1 r /$F&! 2 \1$2 ‚ž.$U9Q"D27­198!‚(%B!" 16%B2D¦ ˜8A'¨G2]%' 2E%'2  H²%:Ÿ+96Ÿª16%'$•7/$$• ¯[´  0 < p < r , H[ž‚Ÿ',LKGM I (r ,p,t) = w (p,t) -=˜8”R2E% w (p,t) 16%B2`9Q/6¯51$2Efž*/$œG2E #9Q1³& !2 Y2E1$2\?;& #9Q"$A'/$$• ”Y2>!#"!2E%'”$&LŸWV¡´ .1$2EfVU9QA'A %B#˜Q› & !˜8X —G 2 !16´$$;&! 2 g?@#"F-N%9 (<>¨G S (XR 2 !(& #9Q"$A'/$$• 1$ 2  ª 2 gXR 2 #9Q&(/  T -0% !1 w (p,t) 16/ !!"1$2E ž./!1 (/!¨ H[ž‚Ÿ',7,OM w (p,t) = W B (T ) g%'&*1$ 2  s5 2 !16´$$F/$$•G‘¯:9Q¨;&! ˜  W ”Y2>!#"(%B2E”R2ElŸ V¡´ *1$2E™$2E #9Q1 `/!0•G2EA'&!2E ¯²´  r 6 p 6 r H[ž`Ÿ',L)GM I (p,p,t) − I (p,p,t) =0 ¯[´  0 6 p 6 r H[ž`Ÿ';, HGM I (r ,p,t) − I (r ,p,t) =h (p,t) ¯:9QA'ABo2>Z2E%'2³!#"(%B2E”R2E³-0% !1d­$•7%'”$&LŸ h (p,t)      min max − ν max ν max  ν  ext ν + ν + ν ν ext − ν − ν min min ν min max ν min ν z { € †~Y‡ UˆY‰‚† I†~Y†ˆY‰`3€‚`„ ¦ 2E$ 16%B2L./2EA'A ¯[/$$¨;&(%B˜8 S H²COf˜@1$2EA'AB¬M®­W#˜L-0%B2 16%B2 O9Q&!2 (%B2>2E%'•G2E!!"U9 ¯[&!2El­*16%B2 κ /$1 κ ¯[2>(&(AB2E•G2El­=”Y2E¨89Q$;&Z(%'1d­=%B(&Z2> DS8•7A'%B!" 198 b AB2E%B!";/$$•G!(š6(&!2 ™­ª198`16/!!" 16%B2 b AB2E%B#"F/$$•G2E H[ž`Ÿ µ;M®­=H[ž`Ÿ+¶GM®­ 16%B2C¤.9Q16”Y2>16%'$•7/$$•G2E H[ž‚Ÿ',LKGME­PH[ž‚Ÿ',L)GM‚/$1 H[ž`Ÿ';, HGM>­!˜L-0%B2™16%B2Z“32®y$%'&(%B˜82E H[ž`ŸIMGME­PH[ž‚Ÿwv7M`/$1 H[ž‚Ÿ',7,OM`”R2>!&(% c& -0% !1d­8<>/\ABS7!2ElŸ O³%'&1$2Eg"$%B 2 !16/ !#"C2 ("U9QA'&!2E2EDF&!2E(%'&#^Q&!2Eg¨79Q$g/lŸ+96Ÿ81$2 4§1$16%'$•7&!˜86¯J9Q¨@&!˜  ”Y 2 !2>!"$2E&.-i 2 !1$2El­ g%'&.1$ 2  16%BC2 O³ ˜ D2E;&!2E$•7AB2E%B!";/$$•G2Em•G2>!!"$AB˜7!#2E™-i 2 !1$2ElŸ V¡´ 516%B20rN2E1$ 2 516%B(;¨ !2E&!2Egs§%9 (%–9Q”$AB2ED-i 2 !1$2Eg¯J˜8A'•G2E1$2PrN2E¢;/2E$< ν  l = 1, . . . , L Ÿ ,Eµ;v  ! #" $ &% ' ν Abs,i ν Str ν j 1 k 1 n l min x J max min k max     !     ! C    b AB2E%B#"F/$$•G2EmH[ž`Ÿ µ;Mª/$1³H[ž`Ÿ+¶GMª-=2 !1$2ED¯[´16%B2*V!2>¢F/2E$< ν /$1g2E%'2Eg¯[2>!&!2E ?;&!˜8œ7XU9%#9%D2E&!2  ”Y2E& #98!"F&!2E&LŸW©T98#"1$ 2  #9Q‘¯[ ˜ C9Q&(%B˜8³%'1983 J.˜; ˜ !16%'U9Q&!2E(š6(&!2  p H[ž`Ÿ',Eµ;M 1 α = (µr ± ct) 2 2 ("U^QA'?& C9Q  g%'& κ (α , α ) = κ + κ  H[ž`Ÿ',L¶GM ∂I =−κ I −S ∂α  H[ž‚Ÿ';, MGM ∂I = κ I −S . ∂α H[ž`Ÿ',Ov7M O³%'&P1$2Ež.”$¨;´ (<>/$$•G2E H[ž‚Ÿ', GM 1 t = t − (r µ − r µ ) , c H[ž‚Ÿ', GM 1 t = t − (r µ − r µ ), c Z ”$<>-qŸ ∂x = σ (α , α ), H[ž‚Ÿ+)7KGM x (r, t) = σ (y, α ) dy ∂α Z ”$<>-qŸ ∂x = −σ (α , α ), H[ž‚Ÿ+)6,OM σ (α , y) dy x (r, t) = − ∂α H[ž‚Ÿ+)7)GM ∆τ = x (r , t ) /$1f1$2Emr=2E˜8(&§—G˜8 ”$%B /$1Z—G˜8 b AB2E%B!";/$$•H[ž‚Ÿ';, MGMN2EF&(A–9Q$•]1$ 2 J3/ (—G2 α >˜8α(&0—G˜8 (r ,t(r) ”$%B,t (r) ,t )(r ,t ) Z H[ž`Ÿ+-) HGM ˆ I = I exp(∆τ ) + S(α (x ), α ) exp(−x ) dx Z H[ž‚Ÿ+)Qµ;M ˆ I = I exp(−∆τ ) + S(α , α (x )) exp(−x ) dx .  k ± e + Abs − Str + e + + − e − − + n j j n j+1 j+1 j−1 j−1 − α+ (r,t) + α+ (r j−1 e + − e − + ± j − j − j+1 e + − ± j∓1 − + + j−1 − n ± j∓1 + j + − α− (rj+1 ,t− ) ± e + ,t+ ) α− (r,t) − j j ∆τ + + j−1 j+1 + + − ± + − + j n j n + 0 ∆τ − − + − − − − 0 “T%B2>*2E;&!(X (%B!";&T1$ 2 .h;&!2E•.#9Q&(%B˜8³1$2  b AB2E%B#"F/$$•G2EH[ž`Ÿ µ;MN/$1 H[ž`Ÿ+¶GMN2E;&(A–9Q$•C%'"!2  i"U9%#9Q¨F› &!2 (%B(&(%'¨G2ElŸ .”R2 N16%B2 b AB2E%B!";/$$•G2E H[ž`Ÿ+)-HGM5/$1mH[ž`Ÿ+)Qµ;M5¨GS8$2E™9QAB!˜‚16%B2TF&!2E!%'&#^Q&!2E I 9% ¤T9816%'/ r <>/ 3£R2E%'& t ”Y2 !2>!"$2E&T-i2 !1$2El­W—G˜ #9Q/(•G2>!2E&(<>&T16%B2 L./2EA'A ¯[/$$¨;&(%B˜8 S /$116%B2q6› &!2E(%'&#^Q&!2E Iˆ (%'1™<>/f2E%'2  £R2E%'&(X$/$$¨@&.—G˜  t ”Y2E¨89Q$;&LŸW“T%B2ck S7(/$$• 1$2>. c´#!2Ej9Q/D1$ 2  —G ˜ #9Q$•G2E•F9Q$•G2E2E £R2E%'&‘› !#"(%'&(& L × J × K ¦ 2 (&!2c¯[´  I %  ?;XR2E%B#" 2 ±•G2E"U9QA'&!2E-i 2 !1$2El­d1%9 (´$”Y 2 T"$%'U9Q/3%B!&T2E%'2 9Q¨@!!"#^Q$¨@&!2>3£R2E%'&(%';&! 2 ‘› —79QA'Ai 2 !2E%B#"F”U%9 LŸž*/`16%B2>!2E b (´$1$2Ee-0% !1 2>`˜Q¯[&q—G ˜ (•G2E/³”R2>!#"#^Q$¨G2ElŸW16%B2>! 2  VU9QA'A -= 2 !1$2E³;/ .16%B2‚9Q¨;&(/2EA'AB2E ›:¦ 2 (&!2q”Y2ES8&(%'•7&L­ /  16%B2 µ›hh;&!2E.• #9Q&(%B˜89Q/‘¯[´$" !2EZ<>/]¨GS8$2El­6-N9801$2E³?;XR2E%B#" 2 (”R2>1%9 ‘¯I1 #98(&(%B!#" !2>16/$<>%B 2 (&LŸ ,Eµ   ± j n ± n ± ±  !      ! C      !1  &  !!  " 9#C   “T%B2P;/D2 (%B!!"2TF&!2E•.#9Q&(%B˜8Z2E%'2 iV$/$$¨@&(%B˜8 f (x) ­;1$2>!!2ED¦ 2 (&!2T9Q 16%B(¨;!2E&!2Eo?;&(´$&(/2EA'% g%'$%B 2 !2E ”Y2>(&!2E"F&‚%'1$ 2 \F&!2E.• #9Q&(%B˜8´$”Y 2 \9QA'AB2 ?;&!˜8œ7XU%9 #%9 D2E&! 2 g%'& N.%'A ¯[2±1$ 2 P!˜8•Ÿ pz    ªŸ ž.”$”$%'AB16/$$•qž`Ÿ',i!#"$$%'&(&!2El­Y!˜;198#P16%B2>!12 J3/$•G2EA'f2E%'2`?@!"U%9 .¨G˜8$#2 b 2 #981$2C9Q 1$2Eež.”(&#^Q1$2E Ÿ ?@2E$¨ !2>#"F&`<>/ 16%B2>! 2  b 2 #981$2E -0% !1m2E%'2]?@!"U%9  XU%9 #9QA'AB2EAB 2  b 2 #981$2E™•G2EAB2E•7&LŸ4%'³«/$$¨;& pA ­$1$ 2 *9Q/6¯2E%' 2 *16%B2>! 2  b 2 #981$2EA'%B2E•7&L­$¨89Q$™16/ !#" ž.$•F9Q”R2g1$2>c?;&!˜8œ7XU%9 #%9 D2E&! 2 ! p /$1m1$ 2  J.˜; ˜ !16%'U9Q&!2 z 2E%'1$2E/$&(%'•™!"U%9 #9Q¨@&! 2 (%B(%B 2 (&c-= 2 !1$2El B4 ±"U9Q&q1$2E ž.”( Q1 —G ˜  ?@!"$$%'&(&(X$/$$¨@& ­Y1$2>!!2Emž.”( Q1—G ˜  £R2EF& (/  p ”Y2E& #^Q•7&LŸ “3 2 ?#9816%–9QAB2qž.”( Q1Z—Gz˜8 A —G ˜  £R2E;& (/  B%B(&P! ˜ g%'& p H[ž`Ÿ+-) MGM r = p +z 4%'2 b 2 #981$2016/ !!"C1$2E «/$$¨;& A /$1g198i£R2EF& (/  !!"$2E%B1$2E&§16%B2 b 2 #981$27­F9Q/6¯Y1$ 2  A A'%B2E•7&L­ /$;&! 2 .1$ 2  ¦_%'$¨G2EA θ ­$1$ 2 *16/ !!"³16%B2qrN2E<>%B2E"F/$$• r p H[ž`Ÿ+)Gv7M cos θ = 1 − z ”Y2>(&(%  c&*-0% !1dŸ :  V˜8A'•G2E1$2E -i 2 !1$2EeF/ q˜@!" «/$$¨@&!2 A ­Y16%B2C?@#"$$%'&(&(X$/$$¨;&!2 2E%' 2 q1$ 2 T¨G˜8$!20?@!"$$%'&(&(X$/$$¨@&!2N-i 2 !1$2E —G˜8 1 ”$%B K − (k − 1) 16/ !#"$F/  D 2 (%B 2 (&LŸY“T%B2 z?› J*˜@ ˜ !16%'U9Q&!2c16%B2># 2 *«/$$¨@&!2±%B(& q H[ž`Ÿ+) GM z = p − p , i = 1, . . . , K − (k − 1). “3 2  #9816%–9QAB2 ž*”( Q1m1$2>`«/$$¨@&!2> g%'&c1$2®a i —G ˜  £R2EF& (/  %B(& p Ÿlž*/6¯[.• (/$1 1$ 2  ?;;š  D2E& (%B2q1$2>0B« !˜8”$AB 2 D c´!!2EZ;/ P«/$$¨@&!2c9Q/6¯¡1$ 2 0XY˜7(%'&(%'—G2E z›hž0!"#2±”R2E& #98#"F&!2E&P-= 2 ‘› 1$2ElŸ ©./$ -0% !1I16%B2f?;& #9Q"$A'/$$•G(%'F&!2E(%'&#^Q&]A–^Q$•GC2E%'2>C”R2>!&(%  c&!2E ?;& #9Q"$ABL­52E%' C9QAT9Q/!-=%^ (&!  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H +H→H+H+H )6Ÿ µFK7K84›K K6Ÿ+K7K7K84873K7K ›)6Ÿ',L)7K84873KQµ r =k H +H →H +H  µ Ÿ I H 7 8 K  4 # › L , K 6 K + Ÿ 7 K 7 K 8 K 9 4 3 7 7 K K # › 7 , + Ÿ 7 K 7 ) 8 K 8 4 3 7 7 K ¶ 6 H + Ÿ 7 ¶ 7 K 8 K  4  › 6 K , r =k H +e → H+H+e µŸ+K7K7K84›#,L) ,7Ÿ+K7K7K84973K7K › 6Ÿwv8¶7K84873-K H r =k H +e →H+e +e 6 ¶ Ÿ I 7 H 7 K 8 K  4  › 7 ) K 7 , + Ÿ 7 K 7 K 8 K 9 4 3 7 7 K K › 6 w Ÿ 8 v 7 ¶ 8 K 8 4 3 7 K H r =k H +H →H+H+e :-;=A@CB=>EDGFGHJILKJF%MON+BC<H-P+QSRT@?>UBVHXWTBVYZ>[B [\ & !O]5^ .0$&- 1 18_  `ba -/')X] %c^ .0 '- %c] % &edea - %f^ -/.0 $(?^9g ' 1ihkj  Xl ( ')(kjEm %'+n 1&-o^8pqh & ( 5r  - % 1$%'[j -o^ .0$% &ts - +*% &&5^ .0(kr 1% & $%C^9u (?^ %c^ (*5^Ua - 'Sv % &t*v (*&&Tdea % && $-/%xwy  $&+  %C^zu (k^ %C^ &$-/.0 16a -/% -/&  - %c^ %' !'S] % - 1{yi] %0'$-/%6g}| (  - 1~n 1Vdi^[h &  %'&by5] %'€%- j $& xwy r &iv 1-oh &$% &f] %'% .)&%01{a - ') 2     + + − − 2 2 2 2 − − − − − − − − + − + − −      − + 2  − 2 − −  −  − ,L¶-H $ $  $ H2 zK%@9QSR-B Y @kQSR-B 9B=>ED €BC< T:-;=A@CB=>EDGFGHJI ¤.2L9Q¨;&(%B˜8 ž 6 Ÿ 7 H K84›#,L¶ H +H →H +e µŸ',L)7¶84›) H+H+H→H +H ¶6Ÿ',L-¶ M84A› H7K H+H+H → H +H 6Ÿ',L)7¶84›K H +H →H +H+H 6Ÿ -M7K84›#,L¶ H+H →H +e +H ,7Ÿ',L)7¶84›K H+H →H +e µŸ+K7K7K84›K H +H →H+H 6 ) Ÿ+K 7K84873KQµ H +γ →H+e ,7Ÿ+)7K7K84›K H +e →H+H ,7Ÿ+K7¶7K84›KGv H +H →H +H 6 ) Ÿ+)QµFK84›K H +H →H +H µŸ+K7K7K84›K H +e →H +H ,7IŸ M7K7K84›KGv H +e →H+H+H 7 , Ÿ µFK7K84873KGv H+γ →H +e )6Ÿwv8K7K84›#,LK H +H →H +e ,7Ÿ+K7K7K84›K H +H →H +e ,7Ÿ µFK7K84›KGv H +H →H+H+H ,7Ÿ+K7K7K84›)7K H +H →H +γ ,7Ÿ+¶7K7K84›#,Ov H +H→H +γ ,7Ÿwv8)7¶84›KGv H +H →H +H+H   + 2 + −  T¤ 9Q&!2E$¨G˜;2®°D<>%B2E;&!2E r “ ›AH6Ÿ+)7K7K84›K6, K6Ÿ+K7K7K84873K7K ›#,7Ÿ+K7K7K84873K7K K6Ÿ+K7K7K84873K7K ›#,7Ÿ+K7K7K84973K7K K6Ÿ+K7K7K84873K7K ›¶6Ÿ+K7K7K84›K6, ›¶6Ÿ+)7K7K84973KQµ ›AM6Ÿ+K7K7K84873K-H ¶6Ÿ+K7K7K84›K6, ›#,7Ÿ+¶ 7K84873K7¶ ›#,7Ÿ+K7K7K84›K6, K6Ÿ+K7K7K84873K7K ›¶6Ÿ+K7K7K84›K6, K6Ÿ+K7K7K84973K7K )6Ÿ';, H7K84973K7K 6Ÿ 7)7K84873-K H ›:µŸ+K7K7K84›K6, K6Ÿ+K7K7K84973K7K ›¶6Ÿ+K7K7K84›K6, K6Ÿ+K7K7K84873K7K µŸ+)7K7K84›K7) µIŸ M-M7K84873KQµ ›¶6Ÿ+K7K7K84›K6, K6Ÿ+K7K7K84873K7K ›¶6Ÿ+K7K7K84›K6, K6Ÿ+K7K7K84873K7K 6Ÿ 7K7K84›K6, ,7Ÿ+¶Gv8K84873K7¶ ›:µŸ µ 7K84›K6, H6Ÿ',L)7K84873KQµ ›:µŸ+K7K7K84›K6, K6Ÿ+K7K7K84873K7K ›¶6Ÿ+K7K7K84›K6, K6Ÿ+K7K7K84873K7K K6Ÿ+K7K7K84873K7K K6Ÿ+K7K7K84873K7K ,7Ÿ 7K7K84873K7K ›)6Ÿ+K7K7K84873K6, ›¶6Ÿ+K7K7K84›K6, K6Ÿ+K7K7K84973K7K   − 2 2 2 + 2 2 + − + 2 + + 2 2 + 2 + 3 2 − − − 2 + 3 2 − − − − − + − − − − 2 − + 2 + 2 + 2 + 3 + 3 2 2 + + 3 + 2 − − − + 3 + 3 2 L./2EA'AB2 «60*t «i?$? «i?$? «i?$? «i?$? ?@kl“ ?@kl“ kª?@“T% ?@kl“ ?@kl“ ?@kl“ ?@kl“ ?@k “ kª?@“T% kª?@“ ?@kl“ ?@k “ ?@kl“ ?@kl“T% ?@kl“ T“ %B2±4%'$"2E%'&T1$2>P¤.9Q&!2E$¨G˜@2®°D<>%B2EF&!2E k %B!& cm s ”$<>-qŸ cm s ”Y2E%l“!2E%'&!2E%'AB!"2E!2L9Q¨@&(%B˜Q› 2ElŸ “T%B2Z¤.9Q&!2E$¨G˜@2®°D<>%B2EF&!2E_"U9Q”Y2E 2E%'2 ‘2oU98#" L./2EA'AB2ZAB2E%B#"F&C/$;&!2 !!!"$%B2>16A'%B#"2V˜ ™­ T %B(& (&!2E&!.16%B2 b 98(&! 2 gXY 2 #9Q&(/ *%' J±2EA'—;%'lŸr9!˜  ™­ =˜8X$X$%:­Wk 9%!!˜8 H[r =kªMqu x -0%!1 k ¯J˜8A'•G2E6› 1$ 2 C9QœG2E³”R 2 !2>#"$2E&L 3 −1 6 −1  kBCL =A · T exp  C T  kBCL5 =A · T B exp  C T  kBCL6 =A · T B exp  C T  ·  C T  · TD B B kBCL9 =A · T exp ·  1+  T 105 D !−1 1+  T 106 D !−1 · (T /103 )−0,2 “T%B2g¤.9Q&!2E$¨G˜@2®°D<>%B2EF&!2E Q9 /q«/$š2E&c9QA:Ÿu µ xh­d«¡9QA'A–96­?69QA'XR2E&!2 L­ ?; Q"$AB2  u µ/HQx*H[«i?$?$Mq?; QE%'A:­ kl2EX$Xl­W“±9QA'•F%9 (˜™u+¶ Qxh­lu+¶ =x H:?@k “±MN/$1™kl2EX$Xl­R?; QE%'A:­U“39QA'•F9%(˜³uIH7) x=H[kª?@“qM=A–9Q/$&!2E¯[˜8A'•G2E6› ,L¶Qµ    C           1$2 C9QœG2El kPUY kPSS kSLD,LSD kSLDi,LSDi  B   C =A · exp − T     C D B =A · T exp − 1 − exp − T T   B  T T exp − =A · 300 C     C T B exp − =A · 300 T T 300 ,L¶7¶ ,L¶-M     /   “T%B!";&!2®›i/$1 b 2>!!";-0%'16%'•7¨G2E%'&!(—G2 (&!2E%'A'/$$•Z2E%'2 *¨G˜8A'A–9Q”$%B2 !2E1$2Ef¦ ˜8A'¨G2 HJ9Q/ k %9 !!˜8 u+) x²M®Ÿ ŸqŸcŸ±ŸqŸ±ŸqŸ±ŸqŸqŸ±ŸcŸ±ŸqŸqŸ±ŸqŸ±ŸqŸ±ŸcŸqŸ±ŸqŸ±ŸqŸ±ŸqŸcŸ±ŸqŸ±ŸqŸ ,Eµ 4;&h-0%B!¨@A'/$$•G(X$"U98!2E³2E%'2 =X!˜8&!˜7!&!2EA'A–9%!2EZ¦ ˜8A'¨G2\HJ9Q/x2@`/$¨89Q% ©*%B!"$%ªu µF)Qx²M®Ÿg,Ov ?;& (/$¨@&(/ §2E%' 2 §X (% D ˜ !16%–9QAB2E]X !˜8&!˜7(&!2EA'A–%9 !2E]¦ ˜8A'¨G20-=^Q"!2E1]1$2>&J.˜8A'A–9QX!2>L­ 9Q/P¤*%'XU%9 D˜8F&(%ª2E&*9QA:Ÿ u+¶7K xhŸ ŸqŸqŸ±ŸcŸ±ŸqŸqŸ±ŸqŸ±ŸqŸ±ŸcŸqŸ±ŸqŸ±ŸqŸ±ŸqŸcŸ±ŸqŸ±ŸqŸ )7K ¤*˜7#!2EA–9Q1@›5/$1™«A–9Q#¨g%'&(&!2EA¡1$ 2 *«ª˜8X#!:›c2.XU9Q<>%'&#^Q&T9Q/3 Of9LšG 2 `Iu H7¶ xhŸ ŸqŸ±ŸqŸ H7K ¤*˜7#!2EA–9Q1 g%'&(&!2EA]1$ 2 2.XU9Q<>%'&#^Q&¯²´ 16%B2IX (% D ˜ !16%–9QAB2 b 98 g%B!!";/$$•U98!" 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