Self-gravitating equilibrium profile using density profile and external pressure

Mathematically, we are solving the equation of hydrostatic equilibrium \nabla P(r) = -\rho(r) \nabla \phi(r) subject to the constraint that P(R)=P_0. Since \phi(r) = \frac{G M(r)}{r} we can rewrite the equation for HSE as \frac{dP}{dr}=\frac{GM(r)\rho(r)}{r^2} and we can express the enclosed mass as M(r)=4\pi\displaystyle\int_0^r{\rho(r') r'^2dr'}