PX282 - A0 - hydrostatic equilibrium

a key aspects of stars and planets
balance between gravity and pressure gradient
basically, a body is massive enough that gravity can overcome its rigid body forces, and is round
consider a shell within the planet/star with a thickness, $d r$ , and a mass, $d m$ , at a radius, $r$
consider a small element within the shell having a mass, $M_{r}$ (a cylinder, with axis perpendicular to the surface, faces aligned along the surfaces of the shell)
gravitational force,

\begin{matrix} (1) & F_{G} = \frac{G M_{r} d m}{r^{2}} \end{matrix}

where, $d m = ρ (r) d r d A$ , $ρ (r)$ is the density, $d A$ is the area of the cylinder's circular surface

\begin{matrix} (2) & F_{P} = ρ (r) d A - ρ (r + d r) d A = - d P d A \end{matrix}

\frac{d P}{d r} = - \frac{G M_{r} ρ (r)}{r^{2}}

can be used to solve for the interior structure of a star/planet
- eg: $ρ (r)$ , if $P (ρ)$ is known, ie: the equation of state