Chapter 2 Stellar Structure Equations 1 Mass conservation equation

( d 2 P
dr 2 )c
= −
( dρ Gm
+ ρ G dm
dr r 2 r 2 dr − 2ρGm r
)c
3
= − 4πGρ2 c
, (16)
3
Therefore near the centre
P (r) = P c − 2 3 πGρ2 cr 2 . (17)
For the luminosity F (r) we have F c = 0 (boundary condition), and
( d 2 F
dr 2 )c
( ) dF
= 4π ( r 2 ρq ) = 0, (18)
dr
c
c
(
= 4π 2rρq + r 2 dρ
dr q + r2 ρ dq )
= 0, (19)
dr
c
( d 3 F
dr 3 )c
= 4π [ 2ρq + r (...) + r 2 (..) ] c = 8πρ cq c . (20)
Therefore near the centre
F (r) = 4 3 πρ cq c r 3 . (21)
For the temperature T (r) we obtain
( ) dT
= − 3 ( kρ F
= −
dr
c
16πac T 3 r
)c
1 ( ) kρ 2 q
2 4ac T r = 0, (22)
3
( d 2 T
dr 2 )c
= − 3 [ ( )
kρ d F
+ F (
d kρ
= −
16πac T 3 dr r 2 r 2 dr T
)]c
1
3 4ac
k c ρ 2 cq c
. (23)
Tc
3
5