Problem

Solve equation (2.16) and obtain equations (2.23) and (2.24).

Figure: Free-fall. x-axis and y-axis represent $\cos^2 \eta$ and $\eta+\sin 2\eta /2$.

In the present case, at $t=0$, since $d r/d t=0$ the energy is negative. Equation (2.19) shows us $r$ becomes equal to zero (the gas collapses) if $\eta=\pi/2$ as well as $\eta=0$ at $t=0$. Equation (2.21) indicates it occurs at the epoch of

$$t=t_{\rm ff} = \left(\frac{R^3}{2GM_r(R)}\right)^{1/2}\frac{\pi}{2},$$

$$= \left(\frac{3\pi}{32G\bar{\rho}}\right)^{1/2},$$ (2.26)

where $\bar{\rho}$ represents the average density inside of $r$, that is $M_r/(4\pi r^3/3)$. This is called ``free-fall time'' of the gas. This gives the time-scale for the gas with density $\bar{\rho}$ to collapse. In the actual interstellar space, the gas pressure is not negligible. However, $t_{\rm ff}$ gives a typical time-scale for a gas cloud to collapse and to form stars in it.