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2.2 Osnovnye parametry politropy

Pri dannom $\xi_1$ radius zvezdy $R=\alpha \xi_1$ ,

$\displaystyle \alpha={\left[(n+1)K\rho_c^{{1\over n}-1}/4\pi \,G\right]}^{1/2}.$

Vvedem bezrazmernuyu velichinu $\mu_1=\int\limits_0^{\xi_1} \Theta^n \xi^2 d\xi$ . Togda massa zvezdy

$\displaystyle M=4\pi \int\limits_0^R \rho \,r^2dr=4\pi \alpha^3 \,\rho_c \mu_1,$

otsyuda legko poluchit' tochnuyu svyaz' mezhdu central'noi plotnost'yu i massoi zvezdy

$\displaystyle \rho_c=\lambda=(4\pi)^{n\over {3-n}}{\left({M\over \mu_1}\right)}^{2n\over {3-n}} {\left({G\over {(n+1)K}}\right)}^{3n\over {3-n}}.$

Dlya davleniya zvezdy v centre imeem

$\displaystyle P_c=p_1GM^{2/3} \,\rho_c^{4/3},$ gde $\displaystyle \;p_1=(4\pi)^{1/3}/{(n+1)\mu_1^{2/3}}.$

Zametim, chto

(pri dannyh $M, \;\rho_c$ ) ne zavisit ot

. Etot rezul'tat estestven, esli vspomnit', chto

$\displaystyle P\sim {GM^2\over R^4}\sim GM^{2/3} \,\rho_c^{4/3}.$

No raspredelenie davleniya po zvezde zavisit ot

, poetomu vyrazhenie dlya

soderzhit strukturnyi mnozhitel'

. Vvedem eshe mnozhitel'

takim obrazom, chtoby

$\displaystyle R=R_1(n){[K^n \,G^{-n} \,M^{1-n}]}^{1/(3-n)}.$

V ryade sluchaev (naprimer, v zadachah vrasheniya) vazhen moment inercii zvezdy, vyrazhenie dlya kotorogo zapishem v vide

$\displaystyle I=I_1MR^2.$

V tablice 1 my privodim znacheniya vvedennyh nami strukturnyh velichin dlya naibolee vazhnyh znachenii

. Vyshe (razdely 1.7; 1.8) byli vvedeny vyrazheniya dlya polnoi, gravitacionnoi i teplovoi energii zvezdy. Integriruya eti vyrazheniya dlya politropnyh sharov, mozhno poluchit' sleduyushie sootnosheniya:

$\displaystyle {\cal{E}}=-{{3-n}\over {5-n}} \,{GM^2\over R}, \quad U=-{3\over {5-n}} \,{GM^2 \over R},$

$\displaystyle Q={n\over {5-n}} \,{GM^2\over R}.$

Tablica 1.

	$\xi_1$	$\mu_1$	$\rho_c/\rho_{cp}$
0	2.45	4.90	1.00	0.806	0.400	0.602
0.5	2.75	3.79	1.84	0.638	--	0.832
1	3.14	3.14	3.29	0.542	0.261	1.253
1.5	3.65	2.71	5.99	0.478	0.205	2.35
2	4.35	2.41	11.4	0.431	0.155	7.53
2.5	5.36	2.19	23.4	0.394	0.112	186
3	6.90	2.02	54.2	0.364	0.075	--
4	14.97	1.80	622	0.315	--	0.0517
5	$\infty$	1.73	$\infty$	0.269	0

Odnako est' bolee izyashnyi sposob vyvoda etih vyrazhenii s ispol'zovaniem soobrazhenii razmernosti i variacionnogo principa.

Zapishem vyrazhenie dlya polnoi energii ${\cal{E}}$ v vide ${\cal{E}}=-aGM^2/R$ , gde zaranee ne izvestno, i podstavim zavisimost' , togda ${\cal{E}}\sim GM^{2-{ {1-n}\over {3-n}}} \sim GM^{{5-n}\over {3-n}}$ , ili v differencial'noi forme $d{\cal{E}}={{5-n}\over {3-n}}GM^{{{5-n}\over {3-n}}-1}\,dM$ $={{5-n}\over {3-n}}{{\cal{E}}\over M} \,dM$ . Zdes' $d{\cal{E}}$ est' raznost' energii dvuh ravnovesnyh zvezd, massy kotoryh razlichayutsya na .

No my mozhem izmenit' ${\cal{E}}$ , dobavlyaya massu na poverhnost' zvezdy (t.e. pri ). Togda $d{\cal{E}}=-{GM\over R} \,dM$ , tak kak vnutrennyaya energiya kuska ravna nulyu, i izmenilas' tol'ko gravitacionnaya energiya. Poluchennaya konfiguraciya ne yavlyaetsya ravnovesnoi. Tem ne menee soglasno variacionnomu principu s tochnost'yu do izmenenie energii zvezdy bezrazlichno k tomu, kakim obrazom menyaetsya massa zvezdy. Poetomu $d{\cal{E}}_1=d{\cal{E}}_2$ , otkuda

$\displaystyle {{5-n}\over {3-n}} \,{{\cal{E}}\over M}=-\; {GM\over R}$ i $\displaystyle \quad {\cal{E}}= -{{3-n}\over {5-n}} \,{GM^2\over R}.$

S drugoi storony, po teoreme viriala ${Q\over U}=-\;{n\over 3}$ ,

$\displaystyle {\cal{E}}=Q+U={{3-n}\over 3}U={{n-3}\over n}Q.$

Takim obrazom,

$\displaystyle Q={n\over {5-n}} \,{GM^2\over R}, \quad U=-{3\over {5-n}} \,{GM^2\over R}.$

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