I am trying to calculate the gravitational potential energy, W, defined as:
$W = -\frac{1}{2}\int\rho(r)\Phi(r)d^{3}r$
for an isothermal sphere. I am given that the density profile varies with r as:
$\rho \propto r^{-2}$
and that $\Phi(r)$ is defined as:
$\Phi(r)=\Phi_{0} + \eta ln(r)$
where $\eta$ is a constant. Given it is a spherical system I set up the following equation.
$W = -\frac{1}{2}\int^{r_{0}}_{0}4\pi k(\Phi_{0} + \eta ln(r))dr$
Where k is a constant of proportionality. I am not sure how to calculate this integral without ending up with infinities due to $ln(0)$ values. I am also told that the sphere can be considered to be truncated at $r_{0}$, does this mean I can somehow ignore the $ln(r)$ term in the integral?