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2.5 Goryachie zvezdy

Teper' rassmotrim drugoi predel'nyi sluchai, v kotorom glavnuyu rol' igraet davlenie izlucheniya.

Napomnim uravnenie sostoyaniya dlya ideal'nogo gaza

$\displaystyle P={{\cal{R}}T\over \mu}\,\rho,$

zdes' $\mu$ -- molekulyarnyi ves veshestva, t.e. velichina, kotoraya pokazyvaet, skol'ko edinic atomnogo vesa prihoditsya na odnu chasticu (otlichaite $\mu$ ot $\mu_e$ !). Naprimer, $\mu_H=1/2$ , no $\mu_{eH}=1$ , i $\mu_{He_2^4}=4/3,\;\mu_{eHe}=2$ i t.p. Pri vysokih temperaturah, kogda gaz polnost'yu ionizovan i yavlyaetsya odnoatomnym, teplovaya energiya gramma veshestva ravna

$\displaystyle E={3\over 2}\,{{\cal{R}}T\over \mu}.$

S drugoi storony, pri vysokih temperaturah v termodinamike zvezdy vse bol'shuyu rol' nachinaet igrat' davlenie izlucheniya. Dlya izlucheniya, nahodyashegosya v termodinamicheskom ravnovesii (plankovskogo izlucheniya), plotnost' energii odnoznachno opredelyaetsya temperaturoi (sm. nizhe, v razdele 3.3)

$\displaystyle \varepsilon_r=aT^4=7,56\cdot 10^{-15}\,T^4\;$ erg $\displaystyle /$ sm $\displaystyle ^3.$

Davlenie pri etom

$\displaystyle P_r=\varepsilon_r/3=2,52\cdot 10^{-15}\,T^4\;$ erg $\displaystyle /$ sm $\displaystyle ^3.$

V laboratornyh usloviyah izmenyayut ne plotnost' $\varepsilon_r$ , a potok luchistoi energii

(Pochemu?), kotoryi svyazan s $\varepsilon_r$ prostoi zavisimost'yu

$\displaystyle q=\varepsilon_r c/4.$

(Poluchite koefficient 1/4 v etoi formule.) Takim obrazom, $q=5,67\cdot 10^{-5}\, T^4=\sigma T^4$ (zakon Stefana-Bol'cmana). Po formule Einshteina plotnost' massy izlucheniya

$\displaystyle \rho_r=\varepsilon_r/c^2\;$ g $\displaystyle /$ sm $\displaystyle ^3.$

Imeetsya shirokaya oblast' astrofizicheskih uslovii, kogda davlenie i energiya izlucheniya i veshestva sravnimy, no plotnost' massy izlucheniya mnogo men'she plotnosti massy veshestva ( $\rho_r\ll\rho_m$ ). Vypishem teper' vyrazheniya dlya polnogo davleniya veshestva i izlucheniya

$\displaystyle P={{\cal{R}}T\over \mu}\,\rho+{aT^4\over 3}={8,3\cdot 10^7\,T\rho\over \mu}+2,5 \cdot 10^{-15}\,T^4.$

Rassmotrim model' zvezdy, v kotoroi svyaz' mezhdu plotnost'yu i temperaturoi daetsya formuloi

$\displaystyle T=\tau\rho^{1/3},$

gde $\tau$ -- postoyannyi mnozhitel'. Togda davlenie veshestva $P_m\sim \rho T\sim \rho^{4/3}$ i davlenie izlucheniya $P_r\sim T^4\sim \rho^{4/3}$ , t.e. v takoi modeli otnoshenie davleniya izlucheniya i veshestva postoyanno po zvezde i

$\displaystyle P={8,3\cdot 10^7\,\tau\over \mu}\,\rho^{4/3}\,(1+3\cdot 10^{-23}\,\mu\tau^3).$

Otmetim, chto v etom sluchae entropiya

menyaetsya (Kak? Rastet ili padaet naruzhu?). Vvedem velichinu

$\displaystyle y=\tau\;^3\surd\overline{3\cdot 10^{-23}\,\mu},$

togda

$\displaystyle P=2,7\cdot 10^{15}\,\mu^{-4/3}y[1+y^3]\rho^{4/3}.$

Parametr

imeet prostoi smysl:

Takim obrazom, $P=K_1\rho^{4/3}$ , i my imeem uzhe znakomoe nam uravnenie politropy . My znaem, chto v etom sluchae ravnovesie vozmozhno tol'ko pri odnom znachenii massy. Podstavlyaya v formulu (2.1), poluchim

$\displaystyle M=19M_\odot\mu^{-2}{[y(1+y^3)]}^{3/2}.$

Ispol'zuya etu formulu, mozhno ocenit' rol' davleniya izlucheniya dlya zvezdy dannoi massy (sm. tabl. 2, v kotoroi prinyato $\mu=0,5$ ).

Tablica 2.2: Rost davleniya v zavisimosti ot massy zvezdy ( $\mu =0.5$ )

	0.05	0.1	0.7	1	2	10
$M/M_\odot$	0.85	2.4	70	215	5800	$7.6\cdot10^7$
$1-\beta=P_r/(P_r+P_m)$	$10^{-4}$	$10^{-3}$	0.25	0.5	0.89	0.999

Iz tablicy 2 vidno, chto zvezda s massoi $215\;M_\odot$ yavlyaetsya granichnoi (). Kak pokazal A.Eddington, dlya zvezd s massoi poryadka $1\;M_\odot$ rol' davleniya izlucheniya prenebrezhima, a dlya zvezd s $M\sim 100\;M_\odot$ davlenie izlucheniya yavlyaetsya dominiruyushim.

Primenim teoremu viriala k postroennoi vyshe modeli. S uchetom izlucheniya teplovaya energiya zvezdy

$\displaystyle Q=\int \left({3\over 2}\,{{\cal{R}}T\rho\over \mu}+3P_r\right)dV.$

Po teoreme viriala gravitacionnaya energiya zvezdy

$\displaystyle U=-3\int PdV=-\int \left(3{{\cal{R}}T\rho\over \mu}+3P_r\right)dV.$

Polnaya energiya zvezdy ${\cal{E}}= Q+U$

$\displaystyle {\cal{E}}=-\int {3\over 2}\,{{\cal{R}}T\over \mu}\,\rho\,dV,$

t.e. zvezda gravitacionno svyazana, no eta svyaz' ravna tol'ko toi dole energii, kotoraya opredelyaetsya veshestvom

$\displaystyle {\cal{E}}={1\over 2}\;\beta \;U,$

poetomu pri $\beta\to 0\quad{\cal{E}}$ -- malo. V etoi modeli my iskusstvenno vveli politropu

, no entropiya ne postoyanna po zvezde (esli

const po zvezde, to pri $n=3,\;{\cal{E}}=0$ ). Podcherknem raznicu mezhdu pokazatelem adiabaty $\gamma$ i pokazatelem politropy

. Voz'mem odnoatomnyi nerelyativistskii gaz ( $\gamma=5/3$ ), dlya kotorogo $P\sim e^S\rho^{5/3}$ . Pust' raspredelenie entropii po zvezde opredelyaetsya zavisimost'yu $e^S\sim \rho^\alpha$ , togda davlenie i plotnost' svyazany sootnosheniem

$\displaystyle P\sim \rho^{{5\over 3}+\alpha}.$

Struktura zvezdy budet opredelyatsya pokazatelem politropy (zdes' $5/3+\alpha=1+1/n$ ), a ustoichivost' zavisit ot pokazatelya adiabaty, t.e. ot uprugosti veshestva (v nashei modeli $\gamma=5/3$ ).

Za schet raspredeleniya entropii my mozhem poluchit' ustoichivuyu zvezdu, naprimer, s . V rassmatrivaemoi vyshe modeli , no polnaya energiya etoi zvezdy ne ravnyalas' nulyu, tak kak model' neizentropichna. Ustoichivost' zvezdy opredelyaetsya ne raspredeleniem veshestva, a tem, kak ono vedet sebya pri szhatii (t.e. ego uprugost'yu!).

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