A Theorem on Central Velocity Dispersions

It is shown that, if the tracer population is supported by a spherical dark halo with a core or a cusp diverging more slowly than that of a singular isothermal sphere (SIS), the logarithmic cusp slope Delta *g of the tracers must be given exactly by Delta *g = 2 Delta *b, where Delta *b is their vel...

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Bibliographic Details
Published in:The Astrophysical journal 2009-08, Vol.701 (2), p.1500-1505
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:It is shown that, if the tracer population is supported by a spherical dark halo with a core or a cusp diverging more slowly than that of a singular isothermal sphere (SIS), the logarithmic cusp slope Delta *g of the tracers must be given exactly by Delta *g = 2 Delta *b, where Delta *b is their velocity anisotropy parameter at the center unless the same tracers are dynamically cold at the center. If the halo cusp diverges faster than that of the SIS, the velocity dispersion of the tracers must diverge at the center too. In particular, if the logarithmic halo cusp slope is larger than two, the diverging velocity dispersion also traces the behavior of the potential. The implication of our theorem on projected quantities is also discussed. We argue that our theorem should be understood as a warning against interpreting results based on simplifying assumptions such as isotropy and spherical symmetry.
ISSN:0004-637X
1538-4357
DOI:10.1088/0004-637X/701/2/1500