= Solution
A fully ionized <helium> nucleus supplies two <Electrons> for approximately $4m_p$ of <mass>, so $n_e=\rho/(2m_p)$. Filling all <Electron> <momentum> states up to the <Fermi momentum> gives
$$
n_e=\int_0^{p_0}\frac{8\pi p^2}{h^3}dp=\frac{8\pi p_0^3}{3h^3},\qquad
p_0=\left(\frac{3h^3\rho}{16\pi m_p}\right)^{1/3}.
$$
For a nonrelativistic <Electron>, the speed is $p/m_e$. The three mean squared Cartesian components of <momentum> are equal by rotational symmetry and sum to $p^2$, so each is $p^2/3$. This is the <isotropic tensor integral>. Thus the <momentum flux>, or <electron degeneracy pressure>, is
$$
P_e=\frac13\int_0^{p_0}\frac{p^2}{m_e}\frac{8\pi p^2}{h^3}dp
=\frac{8\pi p_0^5}{15m_eh^3}.
$$
Substituting the expression for $p_0$ and simplifying the powers of two gives
$$
\boxed{P_e=K\rho^{5/3},\qquad
K=\left(\frac3{2\pi}\right)^{2/3}\frac{h^2}{40m_em_p^{5/3}}.}
$$
Here $h$ is the <Planck constant>. This is the pure-helium specialization of <nonrelativistic electron-degeneracy pressure in a hydrogen-helium mixture>.
For a cold <helium white dwarf>, neglect ion thermal <pressure> and use this <electron degeneracy pressure> as the support against <Newtonian gravity>. Combining <hydrostatic equilibrium> with <mass conservation> gives
$$
\frac1{r^2}\frac d{dr}\left(\frac{r^2}{\rho}\frac{dP}{dr}\right)=-4\pi G\rho.
$$
Set $\rho=\rho_c\theta^{3/2}$ and $r=\alpha\xi$, where $\alpha^2=5K\rho_c^{-1/3}/(8\pi G)$. Direct substitution gives the <Lane-Emden equation>
$$
\frac1{\xi^2}\frac d{d\xi}(\xi^2\theta')=-\theta^{3/2},\qquad\theta(0)=1,\quad\theta'(0)=0.
$$
Its <dimensionless> solution is independent of $\rho_c$. Let $\xi_1$ be its first zero and $\omega_1=-\xi_1^2\theta'(\xi_1)=\int_0^{\xi_1}\xi^2\theta^{3/2}d\xi$. Integrating <mass conservation> gives
$$
R=\alpha\xi_1,\qquad M=4\pi\alpha^3\rho_c\omega_1.
$$
Eliminating $\rho_c$ yields the <polytropic mass-radius relation>
$$
\boxed{R=AM^{-1/3},\qquad
A=\frac{5K}{8\pi G}\,\xi_1(4\pi\omega_1)^{1/3}.}
$$
Thus a more massive <nonrelativistic white dwarf> is smaller at fixed <stellar composition>. The assumptions require $p_0\ll m_ec$ and $k_BT\ll p_0^2/(2m_e)$ in the region providing most support; the law is not extended into the relativistic or thermally supported regimes.
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