### Jeans length

(critical radius below which gravity will collapse a cloud)

The Jeans length is James Jeans's mathematical formula for the radius of a cloud at which pressure balances gravity.

```       15kBT
λJ = √ —————
4πGμρ
```

Where:

If a cloud with the given temperature, density and molecular mass has greater than this radius, it collapses (and the same goes for any sub-region of the cloud). 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. Basically, it means the sum of the kinetic energy and gravitational potential energy (the latter of which is negative) is less than zero.

The above equation essentially states when that sum equals zero (i.e., the limit beyond which a Jeans instability occurs) given the characteristics of gases. Solving the equation for mass indicates the Jeans mass (given values for the other parameters) and solving for the density similarly yields the Jeans density.

The name Jeans swindle has been coined for an apparent flaw in Jeans's derivation, that ignored some effects by dropping a term in the underlying equation that would account for the surrounding density. It applies to 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 produce the instability. However Jeans's model apparently works and effort has been expended on explaining why Jeans's model appears to work despite the 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.

(astrophysics,clouds,limit)