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 ?
