Free-fall time (Infall of a spherically-symmetric distribution of mass)

Description

The free-fall time is the characteristic time that would take a body to collapse under its own gravitational attraction, if no other forces existed to oppose the collapse. To derive the free-fall time we apply the Kepler’s Third Law of planetary motion to a degenerate elliptic orbit. Consider a point mass “m” to a distance “R” from a point source of mass “M” which falls radially inward to it. Crucially, Kepler’s Third Law depends only on the semi-major axis of the orbit, and does not depend on the eccentricity. Consider a case where the mass M is not a point mass, but is distributed in a spherically-symmetric distribution about the center and the only force acting is gravity. The free-fall time of a massless particle at R can be expressed of the total mass M interior to it, or in terms of the average density interior to R.

Related formulas

Variables

tffFreefall time (s)
πpi
GNewtonian constant of gravitation
ρThe average mass density (kg/m3)