Free-fall time

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. As such, it plays a fundamental role in setting the timescale for a wide variety of astrophysical processes—from star formation to helioseismology to supernovae—in which gravity plays a dominant role.

It is relatively simple to derive the free-fall time by applying nothing more than Kepler's Third Law of planetary motion to a degenerate elliptic orbit. Consider a point mass ${\displaystyle m}$ a distance ${\displaystyle R}$ from a point source of mass ${\displaystyle M}$ 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. A purely radial trajectory is an example of a degenerate ellipse with an eccentricity of 1 and semi-major axis ${\displaystyle R/2}$. Therefore, the time it would take a body to fall inward, turn around, and return to its original position is the same as the period of a circular orbit of radius ${\displaystyle R/2}$, or

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