Finding the minimum mass of a star consisting entirely of helium

[Stammer91]

Homework Statement

Consider a star that consists entirely of helium. Make an estimate of the minimum mass for which such a star can ignite helium using the following assumptions:
(i) helium ignites at a temperature of T_c = 10e8 K.
(ii) assume that the critical mass can be determined by the condidion that the ideal gas pressure and the electron degeneracy pressure are equally important in the star at the moment of ignition.
(iii) use the homology relations for the pressure and the density. you can assumed the solar central pressure, P_c,solar = 10e17 g/(cm*s²) and solar central density, ρ_c,solar = 60 g/cm³

Homework Equations

Necessary polytrope: P_c=Kρ_c^{1+[itex]\frac{1}{n}[/itex]}
Ideal Gas Law: PV=Nk_BT
Homology relations: [itex]P=\frac{M^2}{R^4}[/itex] and [itex]\rho=\frac{M}{R^3}[/itex]

The Attempt at a Solution

I know I first need to solve the polytrope where n=1/2; this will produce the constant, K. I also know the homology relations will give you scalings of the system, and that the ideal gas law will incorporate temperature, T, when the pressure is replaced with the ideal gas equation. The part I am stuck at involves assumption (ii) in deriving the initial equation that I will manipulate with the above statements.

 

[Asim Riaz]

How do I use the condition that the ideal gas pressure and electron degeneracy pressure are equally important at the time of ignition? Can someone nudge me in the right direction?