Solution
= Solution
At the centre, spherical symmetry and regularity require a finite central density $\rho(0)=\rho_c$ and
$$
\left.\frac{dP}{dr}\right|_{r=0}=0.
$$
At the stellar surface $r=R$, the pressure vanishes:
$$
\boxed{P(R)=0.}
$$
For a <stellar polytrope> these become the dimensionless conditions $\theta(0)=1$, $\theta'(0)=0$, and the surface is the first zero $\theta(\xi_1)=0$; specifying $\rho_c$ fixes the overall scale.