A Cusp Slope-Central Anisotropy Theorem

For a wide class of self-gravitating systems, we show that if the density is cusped like r super(-g) near the center, then the limiting value of the anisotropy parameter b = 1 - < v super(2) sub(t)>/(2< v super(2) sub(r)>) at the center cannot be greater than 7/2. Here < v super(2) su...

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Bibliographic Details
Published in:The Astrophysical journal 2006-05, Vol.642 (2), p.752-758
Main Authors: An, Jin H, Evans, N. Wyn
Format: Article
Language:English
Subjects:
Astronomy
Earth, ocean, space
Exact sciences and technology
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Summary:For a wide class of self-gravitating systems, we show that if the density is cusped like r super(-g) near the center, then the limiting value of the anisotropy parameter b = 1 - < v super(2) sub(t)>/(2< v super(2) sub(r)>) at the center cannot be greater than 7/2. Here < v super(2) sub(r)> and < v super(2) sub(t)> are the radial and tangential velocity second moments. This follows from the nonnegativity of the phase-space density. We compare this theorem to other proposed relations between the cusp slope and the central anisotropy to clarify their applicabilities and underlying assumptions. The extension of this theorem to tracer populations in an externally imposed potential is also derived. In particular, for stars moving in the vicinity of a central black hole, this reduces to g . b + %, indicating that an isotropic system in Keplerian potential should be cusped at least as steep as r super(-1/2). Similar limits have been noticed before for specific forms of the distribution function, but here we establish this as a general result.
ISSN:0004-637X
1538-4357
DOI:10.1086/501040