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So the equation for the pressure within a non-rotating, spherical gas cloud of radius R and uniform density $\rho$ ( if it is in hydrostatic equilibrium) is :

$$P(r)=P_c-\tfrac{2\pi}{3}G\rho^2r^2$$

starting from the hydrostatic equation in spherical co-ordinates

$$\tfrac{dP}{dr}=-\tfrac{GM(r)}{r^2}\rho(r)$$ we find that as $M(r)=\tfrac{4\pi \rho r^3}{3}$, P(r) is:

$P(r)=\int_0^r\tfrac{-4G\pi \rho^2 r}{3}dr=\tfrac{-2G\pi \rho^2 r^2}{3}$

But obviously this doesnt have the $P_c$ term we wanted. So my question is the following :

i) should I have have used $P(r)=\int\tfrac{-4G\pi \rho^2 r}{3}dr$ instead and treated P_c as a constant of integration ?

ii) if this is the correct method then what is the explanation for not using limits in our integrand ?

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you forgot to integrate $dP$: $\int_0^r dP=P(r)-P(0)$

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