Free-fall time

Remarks

Check: a shell starting at radius $r_0$ feels the attraction of the mass $M = \frac{4\pi}{3} r_0^3 \rho$ inside it, which stays constant because shells do not cross. By Newton's law of gravitation the shell moves on a degenerate Kepler orbit with semi-major axis $r_0/2$, and it reaches the centre after half a period, $\frac{\pi}{2} \sqrt{r_0^3 / (2GM)} = \sqrt{3\pi / (32 G \rho)}$, independent of $r_0$. Example: for $\rho = 1.4 \times 10^3$ kg per cubic metre, close to the mean density of the Sun, and $G \approx 6.67 \times 10^{-11}$ in SI units, the formula gives about $1.8 \times 10^3$ seconds. Non-example: a star in hydrostatic equilibrium has pressure, so it does not collapse in the time $t_{\text{ff}}$ or at all. Still, $t_{\text{ff}}$ is used as its dynamical time scale.