- Free-fall time
The free-fall time is the characteristic
timethat 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 formationto helioseismologyto supernovae -- in which gravity plays a dominant role.
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 from a
point sourceof mass which falls radially inward to it. Crucially, Kepler's Third Law depends only on the semi-major axisof 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 . 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 , or
To see that the semi-major axis is , 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 as one focus. In the case of a very small mass falling toward a very large mass , the center of mass is within the larger mass. The focus of an ellipse is increasingly off-center with increasing ellipticity. 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 . In other words, the ellipse becomes a line of length . The semi-major axis is half the width of the ellipse along the long axis, which in the degenerate case becomes .
If the free-falling body completed a full orbit, it would begin at distance from the point source mass , fall inward until it reached that point source, then turn around and journey back to its original position. In real systems, the point source mass isn't truly a point source and the infalling body eventually collides with some surface. Thus, it only completes half the orbit. But since the infalling part of the orbit is symmetric to the hypothetical outgoing portion of the orbit, we can simply divide the period of the full orbit by two to attain the free-fall time (the time along the infalling portion of the orbit).
Now, consider a case where the mass is not a point mass, but is distributed in a spherically-symmetric distribution about the center, with an average mass density of ,
where the volume of a sphere is :
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 from the center of the sphere depends only upon the total mass contained within . 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 can be expressed solely in terms of the total mass interior to it. In terms of the average density interior to ,
where the latter is in SI units.
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 write
Here we have estimated the numerical value for the free-fall time as roughly 35 minutes for a body of mean density 1 g/cm3.
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 ≈ 0.42.
Wikimedia Foundation. 2010.
Look at other dictionaries:
Free-fall — is motion with no acceleration other than that provided by gravity. Since this definition does not specify velocity, it also applies to objects initially moving upward. Although the definition specifically excludes all other forces such as… … Wikipedia
Free Fall Associates — was a computer game developer of the 1980s and early 1990s. It was founded in 1981 by Jon Freeman, his wife, game programmer Anne Westfall, and game designer Paul Reiche III. To start the new company, Freeman and Westfall left the computer game… … Wikipedia
Free Fall — Album par Jimmy Giuffre Sortie 1962 Durée 56:57 Genre jazz Label Columbia Records Critique All … Wikipédia en Français
Free Fall (Jimmy Giuffre album) — Infobox Album Name = Free Fall Type = studio Artist = Jimmy Giuffre as Jimmy Giuffre 3 Released = 1962 Recorded = Genre = Jazz Length = 56:57 Label = Columbia Records Producer = Reviews = * Allmusic Rating|4.5|5… … Wikipedia
free fall parachute — large parachute specially designed to allow the parachuter to stay in the air for a long period of time and to control his course … English contemporary dictionary
Time management — is commonly defined as the various means by which people effectively use their time and other closely related resources in order to make the most out of it. [The Concise Dictionary of Business Management, by David A. Statt, Taylor Francis Group… … Wikipedia
Free improvisation — Stylistic origins Free jazz Avant garde jazz 20th century classical music Serialism Aleatoric music Cultural origins Mid 1960s United States, UK, and Europe Typical instruments Varies … Wikipedia
Free-diving — Free diver with monofin, ascending. Freediving is any of various aquatic activities that share the practice of breath hold underwater diving. Examples include breathhold spear fishing, freedive photography, apnea competitions and, to a degree,… … Wikipedia
Free electron model — In three dimensions, the density of states of a gas of fermions is proportional to the square root of the kinetic energy of the particles. In solid state physics, the free electron model is a simple model for the behaviour of valence electrons in … Wikipedia
Fall (Jon Foreman EP) — Infobox Album Name = Fall Type = EP Artist = Jon Foreman Released = flagicon|USA November 27, 2007 Recorded = 2007 Genre = Acoustic Length = 21:57 Label = Lowercase People Records Credential Recordings Producer = Charlie Peacock Jon Foreman… … Wikipedia