= Solution
Write the polytropic law as $p=K\rho^{1+1/n}$, with constant $K$ and $n>0$. Its <pressure> potential, or <polytropic enthalpy> relative to zero density, is
$$
H(\rho)=\int_0^\rho\frac1{\widetilde\rho}\frac{dp}{d\widetilde\rho}\,d\widetilde\rho=(n+1)K\rho^{1/n}.
$$
Hydrostatic balance is $dH/dr=-d\psi/dr$. If the surface has zero density and potential $\psi_0$, integration gives $H=\psi_0-\psi$. At the centre, $\Delta\psi\equiv\psi_0-\psi_c=(n+1)K\rho_c^{1/n}>0$. Therefore
$$
\theta=\frac H{\Delta\psi}=\left(\frac\rho{\rho_c}\right)^{1/n},\qquad r=\alpha\xi,\qquad\alpha^2=\frac{\Delta\psi}{4\pi G\rho_c}.
$$
Insert $\psi=\psi_0-\Delta\psi\theta$ and $\rho=\rho_c\theta^n$ into the spherical <Poisson equation for Newtonian gravity>:
$$
\frac1{r^2}\frac d{dr}\left(r^2\frac{d\psi}{dr}\right)=4\pi G\rho\quad\Longrightarrow\quad\boxed{\frac1{\xi^2}\frac d{d\xi}\left(\xi^2\frac{d\theta}{d\xi}\right)=-\theta^n}.
$$
The central conditions are $\theta(0)=1$ and $\theta'(0)=0$; the latter follows from a regular spherically symmetric centre. This reproduces the supplied coordinate and potential scaling rather than introducing an unrelated normalization.
For the sound-speed task, the <adiabatic sound speed> of a perfect polytropic gas is
$$
c_s^2=\frac{dp}{d\rho}=\gamma K\rho^{\gamma-1}=(\gamma-1)H=(\gamma-1)(\psi_0-\psi),\qquad n=\frac1{\gamma-1}.
$$
A <diatomic gas> with active rotational but frozen vibrational degrees of freedom has $\gamma=7/5$ and $n=5/2$. With the requested surface-potential gauge $\psi_0=0$,
$$
\boxed{c_s=\sqrt{-\frac25\psi}}.
$$
The interior potential is then negative. Choosing zero potential at the surface is legitimate; for a finite isolated sphere it is a different additive gauge from setting potential to zero at infinity. Both choices cannot generally be imposed simultaneously.
For the mass-radius task, let $\xi_1$ be the first zero of the regular <Lane-Emden function> and $Y_n=-\xi_1^2\theta'(\xi_1)$. A finite-radius <polytrope> has
$$
R=\alpha\xi_1,\qquad M=4\pi\alpha^3\rho_cY_n,\qquad\frac{GM}{R}=\Delta\psi\frac{Y_n}{\xi_1}=(n+1)\frac{Y_n}{\xi_1}\frac{R^*T_c}{\mu}.
$$
Thus
$$
\boxed{M=\frac{(n+1)Y_n}{\xi_1}\frac{R^*T_c}{G\mu}\,R}.
$$
\b[At fixed central <temperature>, fixed composition and fixed <polytropic index>, $M\propto R$.] These are <homologous polytropes at fixed central temperature>. They do not in general have constant $K$; imposing constant <entropy> instead gives a different polytropic mass-radius sequence. If $n$ or $\mu$ varies between stars, the displayed dimensionless structure factor and composition factor must be retained: equal $T_c$ alone is not a universal exact proportionality for arbitrary stellar models.
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