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

Abstract

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.

Density Profiles of 51 Galaxies from Parameter-Free Inverse Models of their Measured Differential Rotation

“…In great contrast, velocities inside spiral galaxies trend to the null value at the center. That the motions of central regions of spiral galaxies resemble spinning records is required for the organized motions of spin [5][6].…”

“…Spin exists regardless of the state of matter, or its density, since spin also describes gas giant planets, rarified stars [19], and many atmospheric phenomena. Hurricanes spin, yet these objects are mostly composed of N2 and O2 gas with embedded water droplets, and so rigidity is not required for spin (e.g., [6]). Galactic rotation curves (RC) which limit the dependence of velocity v to only equatorial radius r, implicitly assume zero velocity along the axis of rotation (Fig.…”

“…The gas molecules are part of the spin of the star, and cannot be treated as each having independent motions. We refer to this behavior as dynamical cohesion, and note that this condition does not require a solid body, but can apply to interacting "particles" comprising a larger, non-solid object, such as cars in congested traffic or droplets in convecting clouds [5,6,27].…”

“…From (4), volumetric density (z) depends on (r), along with a and e, for oblate spheroids, which considerably simplifies analyses of RC from spin. Given the above, we recently investigated the outer velocities of a spinning oblate spheroid galaxy by applying the rigid body approximation while considering various internal density variations [5][6]. Because galaxies are bound systems, their energetics are specified by the Viral theorem (VT) of Clausius.…”

“…Applying Eq. (5) with j =2, appropriate for mixtures of H and H2 gas, to all 356 galaxies with both RC and luminosity data provides a 1:1 trend [6]. Because RC data depict an average condition at an instant of time, the motions must be modeled as being steady-state, i.e.…”