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The Jeans length is James Jeans's calculation of the radius of a cloud at which pressure counterbalances gravity, given various characteristics of the cloud. It, as well as related parameters (see below), are used for basic models of both star formation and galaxy formation.
15kBT
λJ = √ —————
4πGμρ
Where:
If a cloud with the given temperature, density and molecular mass has greater than this radius, it collapses. The same occurs for a portion of a cloud, given all these values for that portion. This is a form of a relationship termed the Jeans criterion, a limit, which if exceeded, results in the collapse of a cloud, due to gravity overcoming pressure, a condition called a Jeans instability.
The above equation essentially states the case in which that energy-sum equals zero, i.e., the sum of the kinetic energy and gravitational potential energy (the latter of which is negative) is zero, given the characteristics of gases. Plugging some cloud's radius into the equation (in place of the Jeans length) and solving for the implied density yields the Jeans density, and multiplying that by the implied volume (to calculate a cloud mass) yields the Jeans mass: a cloud of those characteristics, except with a higher density (and mass) will collapse.
The name Jeans swindle has been coined for an apparent flaw in Jeans's derivation: that it ignores the density and mass of its surroundings. Such a swindle has been claimed regarding the use of the Jeans criterion to explain larger entities such as galaxies and larger. While simplification is necessary and ubiquitous in astrophysical equations, this absence appears to be required to assure the instability. However Jeans's model apparently works and effort has been expended on explaining why Jeans's model appears to work despite the apparent flaw.
Aside from the Jeans swindle, the above Jeans criterion also assumes a cloud of constant density. This would not be the case, but may be close and the criterion is a workable approximation.