Relation between central and mean density in a White Dwarf (polytropic model)

I'm currently studying the radius of a white dwarf and, in deriving some useful equations, I've seen my professor use the following relation: $$\bar\rho=\frac{\rho_c}{6}$$ when using a polytropic model $$P(r)=\rho(r)^\gamma$$ with $$\gamma=\frac{5}{3}$$, but I can't figure out why that holds...

I'm thinking maybe one can integrate $$\frac{1}{V}\int_0^R \rho 4\pi r^2 dr = \bar\rho$$ substituting the pressure where the density stands and using the initial conditions $$P(0)=P_c=\rho_c^{\frac{5}{3}}$$ and $$P(R)=\rho(R)=0$$, but I don't know what expression for $$P$$ I should be substituting there. Any help will be much appreciated!

• It has nothing to do with the poly tropic model. It has to do with how p varies radially. May 27 '21 at 17:08
• I haven't the slightest idea. May 27 '21 at 18:12
• researchgate.net/figure/… May 27 '21 at 18:18
• @ChetMiller it has everything to do with the adoption of a particular polytropic model. Given that you don't know where the result comes from, your initial comment is a bit puzzling. May 27 '21 at 18:40
• @ProfRob2 Very interesting. Thanks very much. May 27 '21 at 20:42

This is a standard result obtained by numerically solving the Lane-Emden equation for a polytrope with $$n=3/2$$ (equivalent to an adiabatic index of $$\gamma = 5/3$$).