Question #84100

A spherical protostar of mass $M$ and instantaneous radius $R$ is undergoing a slow, quasi-static gravitational contraction. The density profile of the protostar is non-uniform and given by $\rho(r) = \frac{c}{r}$, where $c$ is a positive constant and $r$ is the radial distance from the center. The protostar contracts such that its radius decreases at a constant rate $v$ (i.e., $\frac{dR}{dt} = -v$). Assuming the protostar remains in hydrostatic equilibrium such that the Virial theorem ($2K + U = 0$) is always satisfied (where $K$ is the total internal kinetic energy and $U$ is the gravitational self-energy), and that all the mechanical energy lost during contraction is radiated away from its surface as a perfect blackbody, find the instantaneous surface temperature $T$ of the protostar. (Given: $G$ is the universal gravitational constant, $\sigma$ is the Stefan-Boltzmann constant).