Differentiating the Tolman-Oppenheimer-Volkoff (TOV) pressure equation

1. Apr 14, 2013

S.H.2013

1. The problem statement, all variables and given/known data

I need to differentiate the pressure P(r) equation directly below, for the case where the density is constant (i.e. ρ(r) = ρc), to show that it is of the same form as the dP/dr equation further below:

2. Relevant equations

P(r) = rhoc×c2×( (√(1-2βr2/R2) - √(1-2β))/(√(1-2β) - √(1-2βr2/R2)) )

β = GM/Rc2

dP/dr = -G[( (ρ(r) + P(r)/c2)×(m(r) + 4∏r^3P(r)/c2) )/ r(r - 2Gm(r)/c2)]

dm/dr = 4r2ρ(r)

m(r) = M(r/R)3

ρ(r) = ρc

3. The attempt at a solution

I tried to differentiate the P(r) using the quotient rule (combined with chain rule) and substituted in for β and m(r) but got stuck at a point and don't know what to do next.

The point that I got stuck at is when:

dP/dr = 2GMrc2/R3×{ [3(1-2GM/Rc2)1/2/(1-2GMr2/c2R3)1/2 - 1] - [1 - (1-2GM/Rc2)1/2/(1-2GMr2/R3c2)1/2 ] } / { [10-18GM/Rc2-2GMr2/R3c2-6(1-2GM/Rc2)1/2(1-2GMr2/R3c2)1/2] }