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

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  
\[p(0) \gt \frac{GM^2}{8 \pi R^4}\]
 

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