Chapter 2 Stellar Structure Equations 1 Mass conservation equation

(Remark : You can check above equation yourself and B n contains all n-
dependent coefficients. In table 2 of Supplementary notes, you can see that
B n varies very slowly with n. )
For a given M and P c we obtain
( ) 3/4
ρ c,1.5 B3
= . (42)
ρ c,3 B 1.5
The central density of the polytropic star is given in terms of the mean
density as
M
ρ c = D n ¯ρ = D n
4πR 3 /3 , (43)
where
[ ( ) ] −1
3 dθ
D n = −
, (44)
ξ 1 dξ
ξ 1
are some constants. For a given M we obtain the ratio of radii R(n) as
R (1.5)
R (3)
=
=
(
D1.5
(
D1.5
) 1/3 ) 1/3 ( ) 1/4
ρ c,3
B1.5
=
D 3 ρ c,1.5 D 3 B 3
( ) 1/3 ( ) 1/4 5.991 0.206
< 1, (45)
54.81 0.157
and therefore
R(3) > R (1.5) . (46)
This result makes sense because for a large polytropic index, the star is stiffer
and hence its pressure can resist a bigger gravity so the star should be larger
for the same mass.
Example. Capella is a binary star discovered in 1899, with a known
orbital period, which enables the determination of the mass and radius of the
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