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 helio-seismology to supernovae—in which gravity plays a dominant role.

Derivation

Infall to a point source of gravity

It is relatively simple to derive the free-fall time by applying Kepler's Third Law of planetary motion to a degenerate elliptic orbit from a point source 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 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

we must examine properties of orbits as they become increasingly elliptical. Kepler's First Law states that an orbit is an ellipse with the center of mass the center of mass is within

(This formula also follows from the formula for the falling time as a function of position.)

This amounts to 64.6 days.

Infall of a spherically-symmetric distribution of mass

Let us assume that the only force acting is gravity. Then, as first demonstrated by Newton, and can easily be demonstrated using the divergence theorem The consequence of this result is that if one imagined breaking the sphere up into a series of concentric shells, each shell would collapse only subsequent to the shells interior to it, and no shells cross during collapse. As a result, the free-fall time of a test particle the free-fall time iswhere the latter is in SI units.

Applications

Here we have estimated the numerical value for the free-fall time as roughly 35 minutes for a celestial body of mean density 1 g/cm3.

Comparison

For an object falling from infinity in a capture orbit, the time it takes from a given position to fall to the central point mass is the same as the free-fall time, except for a constant

References

• Galactic dynamics adsabs.harvard.edu Binney, James; Tremaine, Scott. Princeton University Press, 1987.