= Solution
Since $R=a\xi_1$, the scaling in part (b) gives
$$
\rho_c^{(1-n)/n}
=\frac{4\pi G}{(n+1)K}\frac{R^2}{\xi_1^2}.
$$
Substitute this into $M=4\pi(R/\xi_1)^3\rho_cY_n$. After collecting powers,
$$
\boxed{M=A_nK^{n/(n-1)}R^{(3-n)/(1-n)},}
$$
where
$$
\boxed{
A_n=4\pi Y_n
\left(\frac{n+1}{4\pi G}\right)^{n/(n-1)}
\xi_1^{(3-n)/(n-1)}.}
$$
This is the <polytropic mass-radius relation>.
An $n=3$ polytrope models a <white dwarf> whose electrons are an ultrarelativistic degenerate gas. For $n=3$, the radius exponent vanishes and the mass is independent of both $R$ and $\rho_c$. This fixed limiting value is the <Chandrasekhar mass>: increasing the central density can no longer support a larger mass, so a cold nonrotating white dwarf above the limit cannot remain in hydrostatic equilibrium.
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