Past exam of the formation of Structure in the Universe course of the University of Cambridge / 2015 / Paper 1 / 2 / b / Solution

For inside-out collapse of a singular isothermal sphere, the expansion front is at $r_f=a_{T}t$. The original mass swept by this front isThus the only time-independent accretion scale is $a_T^3/G$. The front-sweeping estimate has coefficient two, but gas takes time to fall from the front to the object; it is not an exact central accretion coefficient. Writefor fixed mean particle mass $\mu m_H$. Determining the dimensionless coefficient $m_0$ requires the collapse solution; it is not fixed by front propagation alone and is not needed for the requested temperature scaling.^[ref]

Well inside the front, where the point mass dominates and infall is approximately pressureless and locally steady, $|v_r|\simeq \sqrt{2G{M}_{\star }/r}$. Conserving the radial mass flux gives the free-fall density profile around a growing point mass:The inner slope is $-3/2$, versus $-2$ in the undisturbed outer cloud. For constant ${\dot{M}}_{\star }$ and negligible initial object mass, $M_{\star }\propto t$ and the inner normalization varies as $t^{-1/2}$. These are inner asymptotes, not an assertion of an abrupt exact power-law join at the front.