Free-fall time (radial trajectory of an ellipse with an eccentricity of 1 and semi-major axis R/2)

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. A purely radial trajectory is an example of a degenerate ellipse with an eccentricity of 1 and semi-major axis R/2. In the limiting case of a degenerate ellipse with an eccentricity of 1, the orbit extends from the initial position of the infalling object® to the point source of mass M. In other words, the ellipse becomes a line of length R. The semi-major axis is half the width of the ellipse along the long axis, which in the degenerate case becomes 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 R/2.

Related formulas

Variables

torbitFree-fall time (in s) (dimensionless)
πpi
RThe distance from the point source (in m) (dimensionless)
GNewtonian constant of gravitation:6.67384e-11 m^3/(kg*s^2) (dimensionless)
MThe mass of the point source (in kg) (dimensionless)
mThe point mass (in kg) (dimensionless)