Mass Distribution in Rotating Thin-Disk Galaxies According to Newtonian Dynamics

Abstract

An accurate computational method is presented for determining the mass distribution in a mature spiral galaxy from a given rotation curve by applying Newtonian dynamics for an axisymmetrically rotating thin disk of finite size with or without a central spherical bulge. The governing integral equation for mass distribution is transformed via a boundary-element method into a linear algebra matrix equation that can be solved numerically for rotation curves with a wide range of shapes. To illustrate the effectiveness of this computational method, mass distributions in several mature spiral galaxies are determined from their measured rotation curves. All the surface mass density profiles predicted by our model exhibit approximately a common exponential law of decay, quantitatively consistent with the observed surface brightness distributions. When a central spherical bulge is present, the mass distribution in the galaxy is altered in such a way that the periphery mass density is reduced, while more mass appears toward the galactic center. By extending the computational domain beyond the galactic edge, we can determine the rotation velocity outside the cut-off radius, which appears to continuously decrease and to gradually approach the Keplerian rotation velocity out over twice the cut-off radius. An examination of circular orbit stability suggests that galaxies with flat or rising rotation velocities are more stable than those with declining rotation velocities especially in the region near the galactic edge. Our results demonstrate the fact that Newtonian dynamics can be adequate for describing the observed rotation behavior of mature spiral galaxies.

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

“…The absence of energy along the special axis is not accounted for in orbital models, which leads to excess mass and the inference of non-baryonic matter. Such matter has not been detected despite expenditures of billions of dollars over several decades, motivating alternative models [15,[23][24][25][27][28][29]. Spin models give a reasonable mass for 14 large galaxies; see the table in [10].…”

“…If the problem to be addressed involves multiple masses, then the effect of their combined forces on some external test mass involves integrating over several surfaces and summing, as illustrated in Figure 2e. For RC, the problem is that both the mass inside (M in ) and mass outside some equatorial radius can affect the particle orbiting at that radius (Figure 2b), as shown by force models [27]. Unless the mass outside the radius of interest is distributed spherically, or spheroidally if rotating, it contributes a force in addition to that represented by (10) and must be considered with another surface and volume.…”

“…It is valid to use this approximation in force calculations of distributions [27][28][29], but applying Poisson's equation to the equatorial plane is invalid, because this surface does not enclose a volume, e.g., [54].…”

The physics of galactic spin

“…Subsequent application of Milgrom's modified Newtonian dynamics (MOND) formulation demonstrated that dark matter is not needed for diverse galaxy types [11,12]. Concurrently, NOMs became more sophisticated, but with two rare exceptions [2,13], provide large amounts of dark matter, ϳ75% for the Milky Way and many others [4,14,15] and up to 99.9% for dwarf galaxies [16]. Because these ambiguous forward models provide dark matter exceeding cosmological estimates of ϳ27% [17], this component is arbitrarily minimized during fitting [3,5,6,18].…”

“…In particular, that centers of spiral galaxies spin like tops is unexplained [3], although this trend can be reproduced with fitting parameters. Lastly, recent force-fitting models [2,13] that depart from mainstream NOMs by accounting for effects of mass outside the orbit of interest, do not require dark matter.…”