## Modelling Pressure at in a Star

\[\frac{Gmdm}{r^2}= -4 \pi r^2 dp\]

where

\[r\]

is the distance from the centre of the star to the shell of mass \[dm\]

and \[m\]

is the mass inside the sphere with this radius, and \[4 \pi r^2\]

is the surface area of one side of the shell, and \[dp\]

is the pressure difference between the inside and outside of the shell. The negative sign indicates the forces are opposed.We can write the above equation as

\[dp= - \frac{Gn}{4 \pi r^4} dm\]

then integration gives \[p(M)-p(0)= - \int^M_0 \frac{Gm}{r^4} dm\]

We can obtain a lower limit for the pressure

\[P(0)\]

at the centre of the by taking \[p(M)=0\]

at \[r=R\]

in the integral to give \[p(0) \gt \frac{GM^2}{8 \pi R^4}\]