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Toomre Q parameter

(Q, Q parameter)
(stability criterion for differentially rotating disks)

The Toomre Q parameter (Q parameter) is a value used as part of a criterion for the stability of differentially rotating disks, and constructed so that Q < 1 implies instability. The instability in this case is a tendency for the higher-density locations on the disk to gain density. The letter Q is certainly used in analogy to an oscillator's Q factor, i.e., quality factor. The Toomre Q parameter was originally devised for disks of stars, such as in a disk galaxy, but has since found use for modeling accretion disks and protoplanetary disks. Form for the latter:

    csκ
Q = ———
    πGΣ

The condition of Q > 1 is known as Toomre's stability criteria, analogous to the Jeans criterion but including the effects of the disk's rotation. It specifically includes the effects of self-gravity of the disk material. For a gaseous disk, it is often labeled Qgas, and for molecular clouds, the assumption is that >1 means no star formation, i.e., a stable disk, and <1 means star formation.

The parameter and criteria are based upon the assumption that the disk is of a homogeneous substance. Stability criteria can be more complex given more than one substance, e.g., a disk of both a gas an dust. In such cases, sometimes a higher Q threshold (e.g., 3) is used to accommodate this.


(fluid dynamics,mathematics,measure,limit,rotation)
Further reading:
https://en.wikipedia.org/wiki/Toomre%27s_stability_criterion
https://ui.adsabs.harvard.edu/abs/1964ApJ...139.1217T/
http://ui.adsabs.harvard.edu/abs/1997AstL...23..483P/

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