Vol. 12, No. 7, 2019

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A systematic development of Jeans' criterion with rotation for gravitational instabilities

Kohl Gill, David J. Wollkind and Bonni J. Dichone

Vol. 12 (2019), No. 7, 1099–1108

An inviscid fluid model of a self-gravitating infinite expanse of a uniformly rotating adiabatic gas cloud consisting of the continuity, Euler’s, and Poisson’s equations for that situation is considered. There exists a static homogeneous density solution to this model relating that equilibrium density to the uniform rotation. A systematic linear stability analysis of this exact solution then yields a gravitational instability criterion equivalent to that developed by Sir James Jeans in the absence of rotation instead of the slightly more complicated stability behavior deduced by Subrahmanyan Chandrasekhar for this model with rotation, both of which suffered from the same deficiency in that neither of them actually examined whether their perturbation analysis was of an exact solution. For the former case, it was not and, for the latter, the equilibrium density and uniform rotation were erroneously assumed to be independent instead of related to each other. Then this gravitational instability criterion is employed in the form of Jeans’ length to show that there is very good agreement between this theoretical prediction and the actual mean distance of separation of stars formed in the outer arms of the spiral galaxy Andromeda M31. Further, the uniform rotation determined from the exact solution relation to equilibrium density and the corresponding rotational velocity for a reference radial distance are consistent with the spectroscopic measurements of Andromeda and the observational data of the spiral Milky Way galaxy.

Andromeda and Milky Way star formation, Jeans' self-gravitational instabilities, rotating adiabatic inviscid gas dynamics, astrophysics
Mathematical Subject Classification 2010
Primary: 35B36, 35Q85, 76E07, 76E99
Received: 3 January 2018
Revised: 22 May 2019
Accepted: 22 May 2019
Published: 12 October 2019

Communicated by Martin J. Bohner
Kohl Gill
Department of Mathematics
Washington State University
Pullman, WA
United States
David J. Wollkind
Department of Mathematics
Washington State University
Pullman, WA
United States
Bonni J. Dichone
Department of Mathematics
Gonzaga University
Spokane, WA
United States