White Dwarfs Mass-radius relation for different chemical compositions The standard non-relativistic mass-radius relation of low mass white dwarfs is this :
\begin{equation}\tag{1}
R(M) = \frac{\mathcal{C}}{M^{\frac{1}{3}}},
\end{equation}
where $\mathcal{C}$ is a constant that may (AFAIK) depend on the chemical composition of the star (He, C, N, O, ...).
I would like to know that constant (empirical or theoretical value) for several chemical compositions (especially the He, C, N, O white dwarfs), so I could plot the curves on a $R-M$ diagram.
 A: The relationship you are looking for is
$$ \left(\frac{R}{R_{\odot}}\right) = 0.013\left(\frac{\mu_e}{2}\right)^{-5/3}\left(\frac{M}{M_{\odot}}\right)^{-1/3},$$
where $\mu_e$ is the number of electrons per mass unit in the gas, which of course is composition dependent.
For "standard" white dwarf stars made of carbon or oxygen, then $\mu_e=2$ (e.g. an ionised carbon atom has 12 mass units and 6 electrons). For He $\mu_e=2$, for H $\mu_e=1$ and if such a thing as an iron white dwarf existed, then $\mu_e =56/26$ and it would be smaller than a "standard" white dwarf of the same mass, because there are fewer electrons to provide the degeneracy pressure.
Thus all the compositions you mention in your question would have the same mass-radius relationship if the white dwarfs were supported by ideal electron degeneracy  pressure. Of course, this crude relationship does not capture all of the composition-dependent phenomena in the real (non-ideal) mass-radius relationship. Coulomb corrections to the equation of state become larger for larger atomic numbers, making the gas more compressible and the stars smaller. At low masses then the atmospheric composition plays an important role at finite temperatures. Finally, the inverse beta decay threshold density is different for different nuclei. This causes an instability at different masses for varying compositions (though this is always in the highly relativistic regime).
