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Fizicheskie osnovy stroeniya i evolyucii zvezd

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6.2 Sootnoshenie massa-svetimost'

Vid uravnenii pozvolyaet naiti ryad sootnoshenii dlya integral'nyh harakteristik zvezd (massa, svetimost', radius), ne reshaya chislenno sistemu. Eto mozhno sdelat' v tom sluchae, kogda neprozrachnost' $\varkappa$ i energovydelenie $\varepsilon$ stepennym obrazom zavisyat ot plotnosti i temperatury. Na samom dele eti zavisimosti razlichny dlya razlichnyh sloev, i, bolee togo, v nekotoryh sluchayah voobshe nel'zya primenyat' stepennuyu approksimaciyu. Tem ne menee dlya shirokogo klassa ``gladkih'' modelei poluchayushiesya sootnosheniya podtverzhdayutsya pryamym chislennym raschetom.

V vyrazheniyah dlya neprozrachnosti budem pol'zovat'sya pri nizkih temperaturah zakonom Kramersa:

$\displaystyle \varkappa=\varkappa_0\rho T^{-7/2}\;,$

a pri vysokih -- tomsonovskom neprozrachnost'yu:

$\displaystyle \varkappa=\varkappa_{\mbox{\sc t}}=0,4\;\mbox{sm}^2/\mbox{g}.$

Pri ocenkah budem proizvodnye zamenyat' na otnosheniya tipa:

$\displaystyle {dM\over{dr}}\sim {M\over R}\;,\quad{dP\over{dr}}\sim {P\over R}$ i t. d. $\displaystyle$

Togda pervye tri uravneniya dayut sleduyushie tri sootnosheniya:

$\displaystyle {dM_r\over{dr}}=4\pi r^2\rho\to M=R^3\rho_c,$

$\displaystyle {dP\over{dr}}=-{GM_r\over{r^2}}\rho\to P={GM\over R}\rho=GM^{2/3}\rho^{4/3}\;,$

$\displaystyle L_r=-4\pi r^2D{d(aT^4)\over{dr}}\to L\sim R{T^4\over{\varkappa\rho}}\sim RT^{15/2} \rho^{-2}.$

Pri takih ocenkah chislennye koefficienty opuskayutsya. Ispol'zuya pervoe sootnoshenie, poluchim dlya svetimosti

$\displaystyle L\sim M^{1/3}T^{15/2}\rho^{-7/3},$

a uravnenie sostoyaniya ideal'nogo gaza $P=\Re\rho T$ daet

$\displaystyle T\sim GM^{2/3}\rho^{1/3}.$

Takim obrazom,

$\displaystyle L\sim G^{15/2}M^{16/3}\rho^{1/6}.$

Analogichno zamenyaem uravnenie dlya energii:

$\displaystyle {dL_r\over{dr}}=4\pi r^2\rho\varepsilon \to L=R^3\rho\varepsilon=M\varepsilon.$

Dlya energovydeleniya imeem vyrazhenie (approksimiruyushie tochnye formuly 5.5):

$\displaystyle \varepsilon=\rho T^{nu}\;,$

gde $\nu=4$ pri $T\sim 13\cdot 10^6$ K (v sluchae pp-cikla) i $\nu=15\div 20$ pri $T\sim 20\cdot 10^6$ K (CNO-cikl). Dal'she ogranichimsya sluchaem pp-cikla ( $\nu=4$ ). Togda

$\displaystyle L\sim M\;\rho\;T^4\sim G^4M^{11/3}\rho^{7/3}.$

Isklyuchaya plotnost' iz dvuh vyrazhenii dlya

okonchatel'no poluchim

$\displaystyle L\sim G^{7,8}M^{5,5}\;,$

$\displaystyle R\sim M^{0,07}\;,$

$\displaystyle T_{\mbox{ef}}\sim M^{1,3}.$

Otmetim, chto chislennye raschety dayut $L\sim M^4$ (dlya zvezd s massoi poryadka 1 $M_{\odot}$ ). Sleduet obratit' vnimanie na sil'nuyu zavisimost' svetimosti ot postoyannoi tyagoteniya $L\sim G^{7,8}$ . Iz geologii izvestno, chto svetimost' Solnca ne menyalas', po krainei mere na protyazhenii poslednih treh milliardov let. Eto govorit o tom, chto postoyannaya tyagoteniya ne mogla sil'no menyat'sya s vozrastom Vselennoi.

V sluchae tomsonovskoi neprozrachnosti (vysokie temperatury)

$\displaystyle \varkappa=\varkappa_{\mbox{\sc t}},$

$\displaystyle L\sim RT^4\rho^{-1}\varkappa^{-1}\sim M^{1/3}T^4\rho^{-4/3}\sim M^{1/3}(T/ \rho^{1/3})^4\sim G^4M^{3}, %% *** check in the book 11/3$

t. e. energootvod ne zavisit ot $\rho$ . Dlya energovydeleniya pri bol'shih temperaturah imeem

$\displaystyle L=M\rho T^{15}$ (CNO-cikl), $\displaystyle$

chto daet

$\displaystyle \rho\sim M^{-4/3}G^{-11/6},$

$\displaystyle R\sim M^{7/9}.$

Privedem tablicu 3, poluchennuyu putem chislennyh raschetov dlya verhnei chasti glavnoi posledovatel'nosti ( $M>M_\odot,\;Z=0,02$ ).

Tablica 3.
Modeli zvezd glavnoi posledovatel'nosti

$M/M_\odot$	2,5	5	10
$L/L_\odot$	20	300	3000
$R/R_\odot$	1,6	2,4	3,6
$\rho_c\,[$ g sm $^{-3}]$	48	20	8

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