Modelling Pressure at in a Star

We can simply model a star as made up of shells of gas held in equilibrium by gravity pulling+ the shell in, at gas pressure pushing the shell out - or rather a greater gas pressure acting on the inside of the shell over the outside.
\[\frac{Gmdm}{r^2}= -4 \pi r^2 dp\]

  is the distance from the centre of the star to the shell of mass  
  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  
  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  
  at the centre of the by taking  
  in the integral to give  
\[p(0) \gt \frac{GM^2}{8 \pi R^4}\]

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