Chapter 2 Stellar Structure Equations 1 Mass conservation equation

Example. Derive the behavior of m(r),P (r), the luminosity F (r), defined
as dF/dr = 4πr 2 ρq, where q is the energy production rate, and T (r) near
the centre of a star by Taylor expansion for given composition and physical
properties at r = 0: ρ c , P c and T c . The temperature is given by dT/dr =
− (3/4ac) (kρ/T 3 ) (F/4πr 2 ), where k is the opacity and a is the radiation
constant.
If we adopt r as independent space variable, the Taylor expansion near
r = 0 for any function f(r) is
f = f c +
( ) df
r + 1 dr
c
2
( d 2 f
r
dr
)c 2 + 1 ( ) d 3 f
r 3 + ..., (10)
2 6 dr 3
and we retain only the first non-vanishing term besides f c . For the mass
m(r) we have m c = 0 (boundary condition), and from the mass continuity
equation we have
( d 3 m
( d 2 m
dr 3 )c
( ) dm
= 4π ( r 2 ρ ) = 0, (11)
dr
c
c
dr 2 )c
(
= 4π 2rρ + r 2 dρ )
= 0, (12)
dr
c
(
= 4π 2ρ + 4r dρ
dr + d2 ρ
r2 = 8πρ
dr
)c
2 c . (13)
Therefore near the centre
m(r) = 4 3 πρ cr 3 , (14)
as if the density were uniform and equal to the central value. For the pressure
P (r) we have
( ) dP
(
= − ρ gm ( ) 4πGρ
= −
dr
c
r
)c
2 r
= 0, (15)
2 3
c
4