Free-fall time
From Biocrawler, the free encyclopedia.
The free-fall time is the characteristic time it 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.
Derivation
It is relatively simple to derive the free-fall time using nothing more than Kepler's Third Law of planetary motion. Consider a body 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 eccentricity zero and semi-major axis R. Therefore, the time it would take for the body to fall inward, go out to the opposite side, and return to its original position is the same as a circular orbit of radius R about M, or
torbit = 2π / (GM)1 / 2R3 / 2
The free-fall time time is the time the body takes to fall inward on the first quarter of this orbit and by symmetry, is clearly one-fourth the total orbital time --
tff = 1 / 4torbit = π / (2(GM)1 / 2)R3 / 2
Now, consider a case where the mass M is not a point mass, but is distributed in a spherically-symmetric distribution about the center, with an average mass density of ρ
ρ = M / (4 / 3πR3)
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 force of gravity at any given distance R from the center of the sphere depends only upon the total mass contained within R. 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 at R can be expressed solely in terms of the total mass M interior to it. In terms of the average density interior to R,
tff = (3π / 16Gρ)1 / 2
Often astrophysicists are interested only in order-of-magnitude estimates, in which case the free-fall time may be written approximately as
Applications
The free-fall time is a very useful estimate of the relevant timescale for a number of astrophysical processes. To get a sense of its application, we may
Here we have estimated the numerical value for the free-fall time as roughly one hour for a body of mean density 1 gm/cm^3.
