Change in Radius of star during core contraction

[zsyed94]

Homework Statement

We can make a rough estimate of how much the envelope of a red giant should expand as a result of the contraction of its core based on conservation of energy. We will consider a star of mass M and initial radius R, with a core of mass Mc and radius Rc. We will focus on the phase when there are no nuclear reactions in the helium core, and when hydrogen burning in the shell above it either does not occur or occurs too slowly to make a significant contribution to the energy budget.

d. Now suppose that the core contracts from its initial radius Rc,0 to a smaller radius Rc,1. This causes the envelope to expand from its initial radius R0 to a new radius R1. Assuming that the total energy content of the star is conserved in the process, compute R1/R0 in terms of R0, Rc,0, Rc,1, Mc, and M. For numerical convenience, you may set all the α factors equal to 1.

Homework Equations

E = Ω/2 = αGm^2/r, with m = M_c and r=R_c for the star's core, m = M-M_c and r = R for the envelope, assume R>>R_c.

Gravitational Potential Energy = Gm₁m₂/R

The Attempt at a Solution

With conversation of energy, I tried

G(M_c)^2/R_c,0 + G(M-M_c)^2/R₀ + GM_c(M-M_c)/R₀ = G(M_c)/R_c,1 + G(M-M_c)/R₁ + GM_c(M-M_c)/R₁

Essentially, I have the total energy of the core + energy of the envelope + gravitational potential energy due to attraction between the core and envelope is the total energy of the star. I used the virial theorem, since we neglect radiation pressure/nuclear production, to get E = Ω/2. It should be algebra from here, but I am getting an overly complicated answer and am wondering if I missed something.

 

[mfb]

It should be easy to solve the equation for R1, and everything else is known. Then divide by R0 and you are done. Well, cancel G and see if something else can be simplified.

There are two squares missing on the right side.