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Implications of Geometry and the Theorem of Gauss on Newtonian Gravitational Systems and a Caveat Regarding Poisson’s Equation

Department of Earth and Planetary Science, Washington University, St. Louis, MO 63130, USA
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Galaxies 2017, 5(4), 89; https://doi.org/10.3390/galaxies5040089
Received: 30 August 2017 / Revised: 29 September 2017 / Accepted: 21 November 2017 / Published: 29 November 2017
Galactic mass consistent with luminous mass is obtained by fitting rotation curves (RC = tangential velocities vs. equatorial radius r) using Newtonian force models, or can be unambiguously calculated from RC data using a model based on spin. In contrast, mass exceeding luminous mass is obtained from multi-parameter fits using potentials associated with test particles orbiting in a disk around a central mass. To understand this disparity, we explore the premises of these mainstream disk potential models utilizing the theorem of Gauss, thermodynamic concepts of Gibbs, the findings of Newton and Maclaurin, and well-established techniques and results from analytical mathematics. Mainstream models assume that galactic density in the axial (z) and r directions varies independently: we show that this is untrue for self-gravitating objects. Mathematics and thermodynamic principles each show that modifying Poisson’s equation by summing densities is in error. Neither do mainstream models differentiate between interior and exterior potentials, which is required by potential theory and has been recognized in seminal astronomical literature. The theorem of Gauss shows that: (1) density in Poisson’s equation must be averaged over the interior volume; (2) logarithmic gravitational potentials implicitly assume that mass forms a long, line source along the z axis, unlike any astronomical object; and (3) gravitational stability for three-dimensional shapes is limited to oblate spheroids or extremely tall cylinders, whereas other shapes are prone to collapse. Our findings suggest a mechanism for the formation of the flattened Solar System and of spiral galaxies from gas clouds. The theorem of Gauss offers many advantages over Poisson’s equation in analyzing astronomical problems because mass, not density, is the key parameter. View Full-Text
Keywords: theorem of Gauss; Poisson’s equation; self-potential; instability; symmetry; geometry theorem of Gauss; Poisson’s equation; self-potential; instability; symmetry; geometry
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Hofmeister, A.M.; Criss, R.E. Implications of Geometry and the Theorem of Gauss on Newtonian Gravitational Systems and a Caveat Regarding Poisson’s Equation. Galaxies 2017, 5, 89.

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