# Debated Models for Galactic Rotation Curves: A Review and Mathematical Assessment

^{*}

## Abstract

**:**

## 1. Introduction and Background

_{in}:

#### 1.1. Evidence Independent of Rotation Cuve Modeling

#### 1.2. Types of Models

#### 1.3. Purpose and Goals

## 2. Available Data and Approximations Used in its Analysis

#### 2.1. Shapes from Images

#### 2.1.1. Density Contours of the Oblate Spheroid

^{2}= (1 − e

^{2})(r

^{2}− a

^{2}) in their equation 2.114 provides imaginary numbers for z inside the oblate body. Formulae for spheroids in their tables 2.1 and 2.2 are also incorrect, perhaps due to this mistake.

#### 2.1.2. Flat Oblate Spheroids are not Thin Disks

^{2}dr, and that for a spheroid is simply the latter times a constant geometric factor involving its ellipticity. Basically, spherical and cylindrical geometrical elements differ by a factor of r. Cylindrical geometry is ideal for the analysis of fields about long, thin wires [49], but fields around a flat disk are far more complicated (Section 3.1.5).

#### 2.2. Rotation Curves from Doppler Shifts

#### 2.2.1. General Approach in Data Analysis

_{0}is the (systemic) velocity of the center away from the Sun and V

_{C}is the circular velocity at a specified mean equatorial radius (r) of the ring. The azimuthal angle (θ) is defined by:

_{0}and y

_{0}depict the galactic center. An iterative procedure is required due to the forms of these equations [50]. A high degree of smoothing exists, as seen in the ring model of Figure 3d. Also, the desired parameter V

_{C}is multiplied by sin(i), which lumping induces ambiguity [55], unless the inclination is independently determined.

#### 2.2.2. Two Examples

#### 2.2.3. How the Geometrical Control on Density Contours Affects Velocity-Flux Systematics

_{C}for the radius being sampled (e.g., [53,57,58,59]; Figure 5a). This is untrue for both the oblate shape and the thin disk approximation in the regions even where elements are not superimposed, as follows:

_{eq}) would be the average of the highest and lowest velocities along the LOS (Figure 5b). However, as radius decreases, conditions become increasingly optically thick. Light is attenuated as it emerges from within the galaxy, in such a manner that emissions originating below the equatorial plane are highly attenuated, due to this light crossing large amounts of superjacent matter. The blue curve in Figure 5b sketches the expected attenuation pattern. The velocity associated with the maximum flux depends on the particulars of the gas distribution. The same holds for the velocity associated with the average flux. Instead, V

_{eq}will be represented by the average of the extrema detected, as long as some emitted light is received from the innermost ring crossing the LOS. Due to the asymmetry of the peaks (Figure 5a), using statistics overestimates |V

_{eq}|, if the thin disk geometry were a reasonable approximation.

_{eq}. Without attenuation, triangular profiles describe each of each of the top and bottom sections of the galaxy. Attenuation will make the termination less vertical, but as long as some light is received from the equatorial plane, the maximum velocity is the equatorial velocity, and an asymmetric flux-velocity profile is expected (red curve in Figure 5c). Attenuation will move the peak to lower velocities than the cutoff associated with the equatorial plane. How much depends on the gas distribution.

^{−1}, not ~185 km s

^{−1}from the position of the flux peak. This ~+10% correction is near the center of the RC curves. Data were not provided to the outside. However, a correction of < +2% is indicated for the outskirts by the flux profile of ESO 79-G14 [59] and the lower resolution data (Figure 5a) point to this systematic error decreasing with radius. Hence, RC are not as flat as currently reported.

_{C}| by variable amounts. We cannot reconstruct RC with the information reported in Doppler studies of galaxies. Re-analyzing raw data is needed.

_{eq}= V

_{C}are presumably being measured. Bergeman’s equations, here (3) and (4), should thus describe spiral galaxies except near the center and for the minor axis, where elements are superimposed. Another factor merits consideration.

#### 2.2.4. Why Multiple Spin Axes Should Exist

#### 2.2.5. Evidence for Multiple Spin Axes in Doppler Patterns

#### 2.2.6. Evidence for Multiple Spin Axes in Rotation Curves

#### 2.2.7. Summary and Prognosis

## 3. Forward Models of Galaxies that Presume Nested Orbits

#### 3.1. Synopsis and Evaluation of the Mathematical Underpinnings of Forward Orbital Models

^{2}for any particle in a circular path, or for a thin ring or thin cylindrical shell about the special axis. Forces around a central point or outside of a spherical distribution of matter are described by:

#### 3.1.1. Spheres vs. Point Masses

_{in}were positioned at the very center. Furthermore, Newton proved that shells of matter outside that radius exert no net force on the interior particle. This particle is equivalent to the test mass in an orbital problem. Hence, modeling a galaxy in the limiting case of a spherical mass distribution requires that M

_{in}grows with r up to the body radius a, whereupon growth stops. The result is Keplerian orbits when r > a, whereas for r < a, a velocity profile for a homogeneous object is linear, even if randomly oriented orbits are considered.

_{in}= 4πρ r

^{3}/3 applies. This substitution yields:

#### 3.1.2. Oblate Shapes

_{centrip}of its constituents from (6) but greatly affects the pull of gravity in its interior. Spin is a symmetry breaking mathematical operation that creates a special axis. A body with a tiny difference between its c and a axes is an oblate spheroid, which lacks the special spherical symmetry.

^{2}from (8), but moreover the lines of force around a flattened oblate only point toward the center along the r and z axes (Figure 9). Behavior of rounder bodies is similar albeit less pronounced [45]. Consequently, stable particle orbits around a spheroidal mass distribution are either polar ellipses or equatorial circles: These limitations underlie the restriction of the orbits of dwarf galaxies around the proximal Milky Way and Andromeda to certain planes [45], and of the orbital patterns inside spiral galaxies to a very few types: normal, counter-rotating, or polar rings.

_{in}grows with r up to radius a of the oblate body, whereupon growth stops. Consequently, flatter oblates have overall flatter rotation curves, whereas moderately round to nearly circular oblate spheroids (e→0), have essentially Keplerian behavior for r > a (Figure 8). A peak exists in RC for all ellipticities.

^{2}until great distance is attained, per (8), leading to a much different formulation for the dynamical mass from (9), see Section 3.1.4. Basically, transformation of a sphere into an oblate shape causes proportionally more material to lie near the equatorial zones. The strong forces in the equatorial plane cause rotational motions associated with axial symmetry to lie in the equatorial plane. Comparing Equations (1) and (9) shows that k is close to 2/3, not ~1 as previously assumed. The excessive mass in NOMs stems from assuming central forces that decrease inversely with r

^{2}plus using an overly large moment of inertia, that of a test point or ring.

#### 3.1.3. Approximate Formulae for Rings and Disks

#### 3.1.4. Geometry, Stability, and the Dynamical Mass

_{dyn}= rv

^{2}/G is that r >> a. For this reason, Burbidge’s [4] evaluation of dwarf satellite galaxy orbits using (1) yielded a reasonable mass for the Milky Way.

_{dyn}in the limit of r = a:

_{dyn,sphere}= av

^{2}/G). Considering e→1 underscores limitations of the thin disk geometry. From Maclaurin and Todhunter’s [40] equation describing the connection of ellipticity and angular velocity in a self-gravitating oblates, no rotation in expected. Simply put, the limiting case (e→1, which is a plane) cannot rotate because the mass would be infinite per Equation (14), and the energy would be infinite since a is infinite.

#### 3.1.5. Geometry, Coordinate Systems, the Theorem of Gauss, and Logarithmic Potentials

_{in}/a

^{2}for the force associated with the exact result for an oblate spheroid (8) or with the asymptotic brackets on force for a disk (11). This prefactor must accompany all approximations for the potential of a disk, since it exists in the exact formulae for the special z-axis of the disk (12), as well as in F(z) for the oblate (8). The prefactor of GmM

_{in}/a

^{2}originates in the symmetry breaking operation of spin which transforms spherical into cylindrical coordinates.

^{2}cosθdθdφdr). Considering Gauss’ theorem leads to the same conclusion [48].

#### 3.1.6. Toomre’s Mathematically Invalid Analysis of the Disk

- Flat disks (Figure 2f,g; Figure 11c) require a simple relationship between surface mass density (σ), full thickness (H), and thermodynamic density (ρ):$$\sigma (r)={\displaystyle {\int}_{-H/2}^{H/2}\rho (r,z)dz;\sigma (r)=H\rho (r),\mathrm{if}\mathrm{density}\mathrm{along}z}\mathrm{and}r\mathrm{involves}\mathrm{separable}\mathrm{functions}$$
- Poisson’s equation cannot be applied to a surface, as its use requires volume elements, per the discussion of Garland [66] (see his appendix), who based his work on Kellogg’s [62] 1925 edition. Similarly, MacMillan ([63] p. 124) states that the surface under consideration must be closed, which condition cannot be met by a plane. The above division by H = 0 is a simple explanation for application of (15) to a plane being a faux pas. Proof that an enclosed volume is required for (15) is straightforward per the theorem of Gauss [48].
- Toomre’s 2nd equation proposes a solution to (15) which includes an exponential function of the from exp(−k|z|) where k is a dummy index that is used subsequently in integration. In Toomre’s 4th equation and thereafter he sets z = 0. Obviously, his analysis is limited to the plane, which is invalid, as noted above.
- In his exponential function exp(−k|z|), k must be inversely proportional to some scale length, in accord with dimensional analysis and to provide a dimensionless argument kz. Because the relevant scale length along z is H and H = 0, k must equal some constant divided by H, and so k is infinite. Hence, k does not vary and cannot be used as the variable of integration, which invalidates Toomre’s analysis [63].
- Use of an integral formula for the potential is invalid independent of all other mathematical errors. Because all integrals can be recast as summations, the potential Toomre provided is a summation of simpler component potentials. However, Poisson’s equation is non-homogeneous. From Pinsky [67] (Chapter 1), solutions to non-homogeneous differential equations cannot be summed (i.e., superimposed), as in homogeneous equations such as that of Laplace, where ρ = 0 One can arrive at the finding that Toomre’s 2nd equation cannot solve Poisson’s equation from another perspective: Obviously, the exponential dependence on z in Toomre’s 2nd equation involves separation of the potential into a some function of r multiplied by another function of z. Whereas separation of variables is commonly used to solve homogenous differential equations, it is not possible to solve an inhomogeneous differential equation in this way, e.g., [67].
- Solving (15) using separation of variables is impossible, as revealed by inspection. Separation of variables for the potential requires that density also be a multiplication of two distinct functions, one of z and another of r. For this representation, the RHS of (17) holds at any given radius, and so the density does not depend on z. For this case, the potential cannot depend on z either. Toomre addressed this problem by setting z = 0, which prohibits solving Poisson’s equation.
- From another perspective, in “dropping” the z-dependence of the potential in Toomre’s 2nd equation from his 4th equation and beyond, Toomre assumed that density is independent of z (via Equations (15) and (17)), i.e., he actually assumed that density is constant along the z-axis. Zero is a constant. Coaxial cylinders (Figure 2) is actually the geometry described in [46].
- Due to the properties of the exponential function, Toomre’s component of the potential along z cannot reduce to the exact result for the special axis of a disk, which was known circa 1930 [63]:$${\psi}_{\mathrm{disk},\mathrm{ext},\mathrm{axial}}(z)=-2\frac{G{M}_{\mathrm{disk}}m}{{a}^{2}}\left[\sqrt{{z}^{2}+{a}^{2}}-z\right]$$Equations (11) and (18) respectively reduce to the correct, inverse square dependence of force with distance at great distance, and of potential with inverse distance, if their limits as a/z approach zero are properly evaluated. The exponential function does not reduce to this required functional dependence.

#### 3.1.7. Fundamental Mathematic Problems in Many Post-1998 NOMs Models

- Densities do not sum, as discussed in numerous books on thermodynamics. Importantly, addition of densities in Equation (17) is equivalent to summing solutions of individual differential equations. Use of linear superposition is indeed described in RC literature [58,69]. Again, Poisson’s equation is a non-homogeneous partial differential equation: it is well-known that solutions to such equations cannot be summed (e.g., [67]).
- It is immaterial what component is being summed: velocities, masses, densities (e.g., [30]), or v
^{2}, which is generally the case [29,58]. All are equally problematic. All such summations amount to linear superposition, which is allowable only for homogeneous differential equations, i.e., when ρ = 0 everywhere.

^{2}is permitted, since each of the different mass distributions can be represented by some effective point mass at the center. This approach stems from the force balance of Equation (1), and is a reasonable approximation for nearly round elliptical galaxies. Spirals cannot be treated in this way because the forces for this axially symmetric shape are not central and do not vary inversely with r

^{2}, as discussed in Section 3.1.2 and Section 3.1.3 and shown in Figure 9.

#### 3.1.8. Relativistic Orbital Models

_{0}), a reference mass (M

_{0}), and a reference radius (r

_{0}). The number of free parameters is reduced to two by the Newtonian orbital result of a

_{0}= GM

_{0}/r

_{0}

^{2}. Reference values of mass and radius are linked via the formulation for density. For this reason, Brownstein and Moffat [7] consider their fits to involve one parameter. However, the function assumed for the density is also a constraint, and masses for stars and gas were modeled separately.

^{2}for these two different mass distributions are summed and fit to RC.

^{2}= r

^{2}+ z

^{2}. As discussed above, velocities can sum only if the masses are spherically distributed and have central forces (i.e., going as 1/s

^{2}where the vectors point to the center and the velocities are tangential to the spherical shells. However, because substitution of r for s is not justified in Newtonian physics (Section 3.1.5), this substitution is equally questionable in general relativity.

#### 3.1.9. Modified Newtonian Orbital Dynamics

_{0}= GM

_{0}/r

_{0}

^{2}, as above. Because acceleration decreases as r increases, forces in MOND are non-central.

^{2}when more than one mass type is considered. For such a summation to be valid, requires central forces (spherical distributions) for all components.

#### 3.2. Comparison of Orbital Forward Models

#### 3.2.1. Allowable Number of Free Parameters

_{in}), also as a function of r. In forward models, the assumed mass distribution is the input, although this can be cast as density for a specific shape. In general, terms the forward computational approach is (after Groetsch [27]):

#### 3.2.2. Ambiguities in Force Laws

#### 3.2.3. Density Formulations for Disk Models Based on Central Forces

^{3}for the oblate (or ρr

^{2}for a disk), the existence of finite mass at the center requires that ρ increases at least as strongly as 1/r

^{3}at the center of an oblate (1/r

^{2}for a disk).

^{2}) for flattened shapes (Section 3.1). The apparent realism of the computed RC shape underscores the uncertainties inherent to forward modeling.

#### 3.2.4. Approximate Analytical Models of the Disk with a Single Density

^{1.06}per unit mass would be balanced by the centrifugal force ~v

^{2}/r, also per mass. The velocity inside such a disk would increase greatly outwards, stronger than r, as shown in our numerical calculation (Figure 10b). Hence, a hypothetical thin galaxy of constant density would spin with increasingly faster velocities on the outside than would a solid record with interior cohesive forces. Any concentration of mass towards the center would slow down the spin of the outer rings. Thus, rotation of the thin disk with some density function which decreases with r could, in principle, provide the basic shape of RC whereby velocities first increase with r, flatten, and then sometimes decrease roughly as 1/r. Figure 10b compares the numerical results for velocities in a thin disk with density that decreases exponentially outwards (as is commonly assumed) to v of a constant density disk and of a point source, all with the same total mass. Declining density provides an RC similar to those observed for galaxies, but with an undesirable edge effect.

#### 3.2.5. Forward Models of the Spinning, Oblate Shape

^{n}, where b was chosen for each value of n to provide a value for M

_{in}of 10

^{11}solar masses at an equatorial radius of 18 kpc, which approximates the Milky Way. For n = 0, velocity v depends linearly on r, because density is the same for all shells; this result is the same as the interior of a rigidly rotating sphere or spheroid of Figure 8. Interestingly, for a power law with n = −2, the equatorial velocity does not vary with horizontal distance, but instead remains constant at [6πGb ArcSin(e)/e]

^{½}. Figure 12b was similarly constructed.

_{2}, H and He atmosphere (Figure 1).

_{2}(e.g., [78]). Indices > 1 give model RC declining at large distance (Figure 13b) that are similar to many measurements. Indices near 2 or 3 are promising because these have flat, concentrated density near the center (Figure 13a), and so resemble a galaxy with a bulge.

#### 3.2.6. Summary

^{2}and the lines of force are only parallel to the special directions along the special directions (Figure 9 and Figure 10, [45]). Assuming central forces in NOMs models contributed to overestimating galactic mass. Forward models which are non-central, even with formulae that are unverified with experiments, provide reasonable masses and do not require NBDM haloes.

## 4. Inverse Models of Galactic Rotation

#### 4.1. Numerical Disk-Ring Models

#### 4.1.1. Mathematical Construct

_{test}r

^{2}, which is valid for a thin ring or hollow cylinder. The total mass M of the galaxy is related to the volumetric density (ρ) or the surface density (σ) by the following:

_{c}

^{2}/(MG), a characteristic velocity (v

_{c}) defined by the RC, and two elliptical integrals (K and E):

- Values for H are arbitrary, as stated by the authors, who assumed H = 0.01a.
- In this formulation, using σ(r) rather than ρ(r) eliminates the free parameter, H from (26). However, this simplification stems from assuming that density in (24) is independent of z, i.e., ρ(z,r) = ρ(0,r). Hence, the model actually describes rotating coaxial cylinders. As sketched in Figure 1 and Figure 2f, these can be tall, since H is unconstrained.
- The limit of q = a was applied in evaluating the integral of (26). This step assumes that material near the edge of the disk affects motions of material near the center. This behavior is unlike Newton’s analysis of self-gravitating spherical and spinning oblate bodies, which shows that only mass internal to the test mass controls its orbit. Evaluation over the entire disk or cylinder is needed because these shapes are not gravitationally stable (Figure 2e).

#### 4.1.2. Results of Ring-Disk Models

^{11}solar masses out of 10 kpc for NGC 4736. Luminosity in the visible is much lower, ~1 × 10

^{10}solar [42] for this nearly face-on spiral. The relatively large computed mass is connected with the assumption of disk geometry, whereby the thickness H is uniform from r = 0 to r = a, and in addition, ρ is assumed to the constant in the z-direction. For comparison, mass inside r = a, calculated for an ultrathin disk from the approximate formula of (13), is ~4 × 10

^{10}solar masses, see Figure 14b. This rough value confirms that the additional density (mass) above and below the equatorial plane influences the inversion.

#### 4.2. Numerical Mass Summations in the Equatorial Plane

#### 4.2.1. Mathematical Construct

_{q}for each radius (q) of interest:

_{r}. Since these point masses are spaced in area elements, σ vs. r is provided, and the result is unique. To obtain the proper spacing of point masses, i.e., mass per unit area, the authors accounted for circumference being proportional to the radius. Pavlovich et al. [10] provide a detailed diagram of their geometrical construction.

- An equatorial plane of finite radius a is modeled: so no mass exists above or below z = 0.
- Each orbit is affected by all the mass points in this plane: As discussed in Section 4.1.1, no theorem of Newton exists to guide evaluation of an integral (or summation) in a disk geometry.

#### 4.2.2. Results of Mass Summations in the Equatorial Plane

^{10}solar masses at 14.2 kpc for NGC 1808. Luminosity in the visible is lower, 5.9 × 10

^{9}solar [42], in part due to the tilted presentation, where the major axis appears to be 2.1× the minor axis. Because the area of an ellipse is πab, but the area of circle is πa

^{2}, the luminosity of a relevant, face-on presentation would be 1.2 × 10

^{10}solar. NGC 1808 is actively forming stars and so likely has substantial unconsolidated gas. For comparison, we calculated mass outside r = a, for an ultrathin disk from the approximate formula of (14) as ~4 × 10

^{10}solar masses, which is in good agreement with the matrix inversion of [11], see Figure 15b.

#### 4.3. Analytical Model of the Oblate

#### 4.3.1. Mathematical Construct

_{in}. A similar approach gives its moment of inertia:

#### 4.3.2. Results for Differential Spin

^{−1.8}. For all galaxies examined, ρ at the visual edge does not vary much, such that the average value of 1.1 × 10

^{−21}kg m

^{−3}matches ISM density. The visual edge, being an isophote, is defined by a certain concentration of luminous matter in the galaxy. Association of the visual edge with a certain value of density indicates that total mass correlates with star mass. Consistency exists: luminosity of the 51 galaxies studied linearly depends on the radius associated with the visible edge (Figure 14 in [15]). Further out, the density decreases to roughly IGM values. Near the centers of most galaxies, ρ is like that independently determined for molecular cloud cores, such that larger galaxies, are more concentrated at their centers.

_{2}) whose density becomes very low at great distance. Galaxies thus have “atmospheres.”

#### 4.4. Comparison of Inverse Models

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- de Swart, J.G.; Bertone, G.; van Dongen, J. How dark matter came to matter. Nat. Astron.
**2017**, 1, 0059. [Google Scholar] [CrossRef][Green Version] - Rubin, V.C.; Ford, W.K. Rotation of the Andromeda nebula from a spectroscopic survey of emission regions. Astrophys. J.
**1970**, 159, 379. [Google Scholar] [CrossRef] - Faber, S.M.; Gallagher, J.S. Masses and mass-to-light ratios of galaxies. Ann. Rev. Astron. Astrophys.
**1979**, 17, 135–187. [Google Scholar] [CrossRef] - Burbidge, G. On the masses and relative velocities of galaxies. Astrophys. J.
**1975**, 196, L7–L10. [Google Scholar] [CrossRef] - Milgrom, M. A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. Astrophys. J.
**1983**, 270, 365–370. [Google Scholar] [CrossRef] - McGaugh, S.S. A tale of two paradigms, the mutual incommensurability of LCDM and MOND. Can. J. Phys.
**2015**, 93, 250–259. [Google Scholar] [CrossRef][Green Version] - Brownstein, J.R.; Moffat, J.W. Galaxy rotation curves without nonbaryonic dark matter. Astrophys. J.
**2006**, 636, 721–741. [Google Scholar] [CrossRef][Green Version] - Feng, J.Q.; Gallo, C.F. Modeling the Newtonian dynamics for rotation curve analysis of thin-diskgalaxies. Res. Astron. Astrophys.
**2011**, 11, 1429–1448. [Google Scholar] [CrossRef] - Feng, J.Q. Rotating Disk Galaxies without Dark Matter Based on Scientific Reasoning. Galaxies
**2020**, 8, 9. [Google Scholar] [CrossRef][Green Version] - Pavlovich, K.; Pavlovich, A.; Sipols, A. Newtonian explanation of galaxy rotation curves based on distribution of baryonic matter. arXiv
**2014**, arXiv:1406.2401P. [Google Scholar] - Sipols, A.; Pavlovich, A. Dark matter dogma: A study of 214 galaxies. Galaxies
**2020**, 8, 36. [Google Scholar] [CrossRef] - Marr, J.H. Galaxy rotation curves with lognormal density distribution. Mon. Not. R. Astron. Soc.
**2015**, 448, 3229. [Google Scholar] [CrossRef][Green Version] - Marr, J.H. Entropy and Mass Distribution in Disc Galaxies. Galaxies
**2020**, 8, 12. [Google Scholar] [CrossRef][Green Version] - Hofmeister, A.M.; Criss, R.E. The physics of galactic spin. Can. J. Phys.
**2017**, 95, 156–166. [Google Scholar] [CrossRef][Green Version] - Criss, R.E.; Hofmeister, A.M. Density Profiles of 51 Galaxies from Parameter-Free Inverse Models of Their Measured Rotation Curves. Galaxies
**2020**, 8, 19. [Google Scholar] [CrossRef][Green Version] - Forbes, D.A.; Lopez, E.D. On the Origin (and Evolution) of Baryonic Galaxy Halos. Galaxies
**2017**, 5, 23. Available online: https://www.mdpi.com/journal/galaxies/special_issues/baryonic_galaxy_halos (accessed on 2 April 2020). [CrossRef][Green Version] - Tumlinson, J.; Peeples, M.S.; Werk, J.K. The Circumgalactic Medium. Ann. Rev Astron. Astrophys.
**2017**, 55, 389–432. [Google Scholar] [CrossRef][Green Version] - Ackermann, M.; Albert, A.; Anderson, B.; Baldini, L.; Ballet, J.; Barbiellini, G.; Bastieri, D.; Bechtol, K.; Bellazzini, R.; Bissaldi, E.; et al. Dark matter constraints from observations of 25 Milky Way satellite galaxies with the Fermi Large Area Telescope. Phys. Rev. D
**2014**, 89, 042001. [Google Scholar] [CrossRef][Green Version] - DeVega, H.J.; Salucci, P.; Sanchez, N.G. Observational rotation curves and density profiles versus the Thomas-Fermi galaxy structure theory. Mon. Not. R. Astron. Soc.
**2014**, 442, 2717–2727. [Google Scholar] [CrossRef][Green Version] - Giagu, S. WIMP dark matter searches with the ATLAS detector at the LHC. Front. Phys.
**2019**, 7, 75. Available online: https://doi.org/10.3389/fphy.2019.00075 (accessed on 19 February 2020). [CrossRef] - Peccei, R.D.; Quinn, H.R. CP Conservation in the presence of pseudoparticles. Phys. Rev. Lett.
**1977**, 38, 1440. [Google Scholar] [CrossRef][Green Version] - Nagano, K.; Fujita, T.; Michimura, Y.; Obata, I. Axion dark matter search with interferometric gravitational wave detectors. Phys Rev. Lett.
**2019**, 123, 111301. [Google Scholar] [CrossRef] [PubMed][Green Version] - NRAO. National Radio Astronomy Observatory (see the gallery of images). Available online: https://public.nrao.edu/gallery/warped-disk-of-galaxy-ugc-3697-2/ (accessed on 30 May 2020).
- CHANG-ES. Continuum Halos in Nearby Galaxies- and EVLA Survey. Available online: http://www.queensu.ca/changes (accessed on 26 January 2020).
- Irwin, J.; Irwin, B.; Rainer; Benjamin, R.A.; Dettmar, R.-J.; English, J.; Heald, G.; Henriksen, R.N.; Johnson, M.; Krause, M.; et al. Continuum Halos in Nearby Galaxies: An EVLA Survey (CHANG-ES). I. Introduction to the Survey. Astronom. J.
**2012**, 144, 43. [Google Scholar] [CrossRef] - Wiegert, T.; Irwin, J.; Miskolczi, A.; Schmidt, P.; Carolina Mora, S.; Damas-Segovia, A.; Stein, Y.; English, J.; Rand, R.J.; Santistevan, I. CHANG-ES IV: Radio continuum emission of 35 edge-on galaxies observed with the Karl, G. Jansky very large array in D configuration—Data release 1. Astronom. J.
**2015**, 150, 81. [Google Scholar] [CrossRef] - Groetsch, C.W. Inverse Problems: Activities for Undergraduates; Cambridge University Press: Cambridge, UK, 1999. [Google Scholar]
- Ambartsumian, V. On the derivation of the frequency function of space velocities of the stars from the observed radial velocities. Mon. Not. Roy. Astron. Soc.
**1936**, 96, 172. [Google Scholar] [CrossRef][Green Version] - Diaconis, P. Theories of Data Analysis: From Magical Thinking Through Classical Statistics; John Wiley & Sons: New York, NY, USA, 2011. [Google Scholar]
- Jardel, J.R.; Gebhardt, K.; Shen, J.; Fisher, D.B.; Kormendy, J.; Kinzler, J.; Lauer, T.R.; Richstone, D.; Gültekin, K. Orbit-based dynamical models of the Sombrero galaxy (NGC 4594). Astrophys. J.
**2011**, 739, 21. [Google Scholar] [CrossRef][Green Version] - Kam, Z.S.; Carignan, C.; Chemin, L.; Amram, P.; Epinat, B. Kinematics and mass modelling of M33, Hα observations. Mon. Not. R. Astron. Soc.
**2015**, 449, 4048–4070. [Google Scholar] [CrossRef][Green Version] - McGaugh, S. Predictions and Outcomes for the Dynamics of Rotating Galaxies. Galaxies
**2020**, 8, 35. [Google Scholar] [CrossRef] - Lin, H.-N.; Li, M.-H.; Li, X.; Chang, Z. Galaxy rotation curves in the Grumiller’s modified gravity. Mon. Not. R. Astron. Soc.
**2013**, 430, 450–458. [Google Scholar] [CrossRef] - Scelza, G.; Stabile, A. Numerical analysis of galactic rotation curves. Astrophys. Space Sci.
**2015**, 357, 44. [Google Scholar] [CrossRef][Green Version] - Bottema, R.; Pestaña, J.L.G. The distribution of dark and luminous matter inferred from extended rotation curves. Mon. Not. R. Astron. Soc.
**2015**, 448, 2566–2593. [Google Scholar] [CrossRef][Green Version] - Sofue, Y.; Rubin, V.C. Rotation curves of spiral galaxies. Ann. Rev. Astron. Astrophys.
**2001**, 39, 137–174. [Google Scholar] [CrossRef][Green Version] - Sil’chenko, O.K.; Moiseev, A.V. Nature of nuclear rings in unbarred galaxies: NGC 7742 AND NGC 7217. Astronom. J.
**2006**, 131, 1336–1346. [Google Scholar] - Moulton, F.R. An Introduction to Celestial Mechanics; MacMillan: New York, NY, USA, 1914. [Google Scholar]
- Schmidt, M. A model of the distribution of mass in the galactic system. Bul. Astron. Inst. Neth.
**1956**, 13, 15. [Google Scholar] - Todhunter, I. A History of the Mathematical Theories of Attraction and Figure of the Earth; MacMillan and Co.: London, UK, 1873. [Google Scholar]
- Criss, R.E. Analytics of planetary rotation: Improved physics with implications for the shape and super-rotation of Earth’s Core. Earth Sci. Rev.
**2019**, 192, 471–479. [Google Scholar] [CrossRef] - NASA/IPAC. Extragalactic Database. Available online: https://ned.ipac.caltech.edu/ (accessed on 1 February 2020).
- Binney, J.; Tremaine, S. Galactic Dynamics, 2nd ed.; Princeton University Press: Princeton, NJ, USA, 2008. [Google Scholar]
- Criss, R.E.; Hofmeister, A.M. Galactic density and evolution based on the virial theorem, energy minimization, and conservation of angular momentum. Galaxies
**2018**, 6, 115. [Google Scholar] [CrossRef][Green Version] - Hofmeister, A.M.; Criss, R.E.; Criss, E.M. Verified solutions for the gravitational attraction to an oblate spheroid: Implications for planet mass and satellite orbits. Planet. Space Sci.
**2018**, 152, 68–81. [Google Scholar] [CrossRef] - Toomre, A. On the distribution of matter within highly flattened galaxies. Astrophys. J.
**1963**, 138, 385–392. [Google Scholar] [CrossRef] - Perek, L. Heterogeneous spheroids with Gaussian and exponential density laws. Bull. Astron. Inst. Czechoslov.
**1958**, 9, 208–212. [Google Scholar] - 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. [Google Scholar] [CrossRef][Green Version] - Halliday, D.; Resnick, R. Physics; John Wiley and Sons: New York, NY, USA, 1966. [Google Scholar]
- Begeman, K.G. HI rotation curves of spiral galaxies. I. NGC 3198. Astron. Astrophys.
**1989**, 223, 47–60. [Google Scholar] - Sofue, Y. Dark halos of M 31 and the Milky Way. Publ. Astron. Soc. Jpn.
**2015**, 67, 759. [Google Scholar] [CrossRef][Green Version] - Sofue, Y. Rotation curve of Milky Way. Galaxies
**2020**, 8, 37. [Google Scholar] [CrossRef] - De Blok, W.J.G.; Walter, F.; Brinks, E.; Trachternach, C.; Oh, S.-H.; Kennicutt, R.C., Jr. High-resolution rotation curves and galaxy mass models from THINGS. Astrophys. J.
**2008**, 136, 2648–2719. [Google Scholar] [CrossRef] - De Blok, W.J.G. Is there a universal alternative to dark matter? Nat. Astron.
**2018**, 2, 615–616. [Google Scholar] [CrossRef] - Transtrum, M.K.; Machta, B.B.; Brown, K.S.; Daniels, B.C.; Myers, C.R.; Sethna, J.P. Perspective: Sloppiness and emergent theories in physics, biology, and beyond. J. Chem. Phys.
**2015**, 143, 010901. [Google Scholar] [CrossRef] - Koribalski, B.S.; Wang, J.; Kamphuis, P.; Westmeier, T.; Staveley-Smith, L.; Oh, S.H.; Lopez-Sanchez, A.R.; Wong, O.I.; Ott, J.; de Blok, W.J.G.; et al. The Local Volume HI Survey (LVHIS). Mon. Not. R. Astron. Soc.
**2018**, 478, 1611–1648. [Google Scholar] [CrossRef][Green Version] - Carignan, C. Light and mass distribution of the magellanic-type spiral NGC 3109. Astrophys. J.
**1985**, 299, 59–73. [Google Scholar] [CrossRef] - Chemin, L.; Carignan, C.; Foster, T. HI Kinematics and dynamics of Messier 31. Astrophys. J.
**2009**, 705, 1395–1415. [Google Scholar] [CrossRef] - Gentile, G.; Salucci, P.; Klein, U.; Vergani, D.; Kalberla, P. Mapping the inner regions of the polar disk galaxy NGC 4650A with MUSE. Mon. Not. R. Astron. Soc.
**2004**, 351, 903. [Google Scholar] [CrossRef][Green Version] - Iodice, E.; Coccato, L.; Combes, F.; de Zeeuw, T.; Arnaboldi, M.; Weilbacher, P.M.; Bacon, R.; Kuntschner, H.; Spavone, M. Mapping the inner regions of the polar disk galaxy NGC 4650A with MUSE. Astron. Astrophys.
**2015**, 583, A48. [Google Scholar] [CrossRef][Green Version] - Wiegert, T.; English, J. Kinematic classification of non-interacting spiral galaxies. New Astron.
**2014**, 26, 40–61. [Google Scholar] [CrossRef][Green Version] - Kellogg, O.D. Foundations of Potential Theory; Dover Publications: New York, NY, USA, 1953. [Google Scholar]
- MacMillan, W.D. The Theory of the Potential; McGraw-Hill: New York, NY, USA, 1930. [Google Scholar]
- Feng, J.Q.; Gallo, C.F. Mass distribution in rotating thin-disk galaxies according to Newtonian dynamics. Galaxies
**2014**, 2, 199–222. [Google Scholar] [CrossRef] - Evans, N.W.; Bowden, A. Extremely flat halos and the shape of the galaxy. Mon. Not. R. Astron. Soc.
**2014**, 43, 2–11. [Google Scholar] [CrossRef][Green Version] - Garland, G.D. The Earth’s Shape and Gravity; Pergamon Press: Oxford, UK, 1977. [Google Scholar]
- Pinsky, M.A. Introduction to Partial Differential Equations; McGraw-Hill: New York, NY, USA, 1984. [Google Scholar]
- Dehnen, W.; Binney, J. Mass models of the Milky Way. Mon. Not. R. Astron. Soc.
**1998**, 294, 429. [Google Scholar] [CrossRef][Green Version] - Ibata, R.; Lewis, G.F.; Martin, N.F.; Bellazzini, M.; Correnti, M. Does the Sagittarius stream constrain the Milky Way halo to be triaxial? Astrophys. J. Lett.
**2013**, 765, L155. [Google Scholar] [CrossRef][Green Version] - Sanders, R.H.; McGaugh, S.S. Modified Newtonian Dynamics as an Alternative to Dark Matter. Ann. Rev. Astron. Astrophys.
**2002**, 40, 263–317. [Google Scholar] [CrossRef][Green Version] - Disney, M.J. Modern Cosmology, Science or Folktale? Am. Sci.
**2007**, 95, 383–385. [Google Scholar] [CrossRef] - Van Albada, G.D.; Sancisi, R. Dark matter in spiral galaxies. Philos. Trans. R. Soc. Lond.
**1986**, A320, 447–464. [Google Scholar] - De Lorenzi, F.O.; Gerhard, L.; Coccato, M.; Arnaboldi, M.; Capaccioli, N.G.; Douglas, K.C.; Freeman, K.; Kuijken, M.R.; Merrifield, N.R.; Napolitano, E.; et al. Debattista, Dearth of dark matter or massive dark halo? Mass-shape-anisotropy degeneracies revealed by NMAGIC dynamical models of the elliptical galaxy NGC 3379. Mon. Not. R. Astron. Soc.
**2009**, 395, 76. [Google Scholar] [CrossRef][Green Version] - Boroson, T. The distribution of luminosity in spiral galaxies. Astrophys. J. Supp.
**1981**, 46, 177. [Google Scholar] [CrossRef] - Van Van der Kruit, P.C. The radial distribution of surface brightness in galactic disks. Astron. Astrophys.
**1987**, 173, 59–80. [Google Scholar] - Gallo, C.F.; Feng, J.Q. A thin-disk gravitational model for galactic rotation. In Proceedings of the 2nd Crisis Cosmology Conference, Washington, DC, USA, 7–11 September 2009; Volume 413, pp. 289–303. [Google Scholar]
- Emden, R. Gaskuglen—Anwendungen de Mechanischen Wärmetheorie; B.G. Teubner: Leipzig, Germany, 1907. [Google Scholar]
- Maron, S.H.; Prutton, C.F. Fundamental Principles of Physical Chemistry; Macmillan: New York, NY, USA, 1970. [Google Scholar]
- Romanowsky, A.J.; Douglas, N.G.; Arnaboldi, M.; Kuijken, K.; Merrifield, M.R.; Napolitano, N.R.; Capaccioli, M.; Freeman, K. A dearth of dark matter in ordinary elliptical galaxies. Science
**2003**, 301, 1696–1698. [Google Scholar] [CrossRef][Green Version] - Feng, J.Q.; Gallo, C.F. Deficient reasoning for dark matter in galaxies. Phys. Int.
**2015**, 6, 11–22. [Google Scholar] [CrossRef][Green Version] - Sofue, Y.; Tutui, Y.; Honma, M.; Tomita, A.; Takamiya, T.; Koda, J.; Takeda, Y. Central rotation curves of spiral galaxies. Astrophys. J.
**1999**, 523, 136–146. [Google Scholar] [CrossRef][Green Version] - Brandt, J.C. On the distribution of mass in galaxies. I. The large-scale structure of ordinary spirals with applications to M31. Astrophys. J.
**1960**, 131, 293–303. [Google Scholar] [CrossRef] - Hofmeister, A.M.; Criss, R.E. Spatial and symmetry constraints as the basis of the Virial Theorem and astrophysical implications. Can. J. Phys.
**2016**, 94, 380–388. [Google Scholar] [CrossRef] - Langley, S.P. The history of a doctrine. Am. J. Sci.
**1889**, 37, 1–23. [Google Scholar] [CrossRef] - Nicolis, G.; Prigogine, I. Self-Organization in Nonequilibrium Systems: From Dissipative Structures to Order through Fluctuations; J. Wiley and Sons: New York, NY, USA, 1977. [Google Scholar]
- Samuelsen, P.A.; Planck, M.; Kuhn, T.S.; Poincare, H. Science Makes Progress Funeral by Funeral. Available online: https://quoteinvestigator.com/2017/09/25/progress/ (accessed on 24 April 2020).

**Figure 1.**Distributions of baryonic matter in edge-on spiral galaxies: (

**a**) Visible image of ultrathin UCC3697 (whitish). The blue overlay images the surrounding neutral H atom gas, detected using the L-band, which includes emissions at 21 cm. Thin red arrow indicates the z-axis. Yellow X indicates a secondary spin axis, pointing into the page with the sense shown by the curved yellow arrows. This secondary spin explains the form of the warp (see Section 2.2 ). Publicly available from NRAO/AUI/NSF [23]; (

**b**) Radio contours of NGC5907 in the C- and L-bands, with an optical image, are publicly available at https://www.queensu.ca/changes/. The L-band range. The CHANG-ES project [24] is described in Irwin et al. 2012, AJ, 144, 43 [25] and details about this first data release are provided by Wiegert et al. 2015, AJ, 150, 81 [26].

**Figure 2.**Geometry of spiral galaxies, as approximated in oblate spheroid and disk models: (

**a**) Visual image of nearly face-on NGC 7742, type SA(r), by the Hubble Heritage Team (AURA/STScI/NASA), and publicly available from http://hubblesite.org/newscenter/archive/releases/1998/28/image/a/). This particular ring galaxy is counter-rotating [37]; (

**b**) Schematic of graded density in the equatorial plane; (

**c**) Geometry of a spinning, oblate spheroid; (

**d**) Side view of homeoids of constant density and shape which nest to form the oblate; (

**e**) Geometry and gravitational forces (white arrows) for a disk of finite thickness and radius. Although the horizontal gravitational forces would be balanced by the centrifugal pseudo-force (solid arrow), the force along z is unopposed at any finite distance above the equatorial plane; (

**f**) The basis of ring models: perspective view of concentric cylinders of various density (gray) rotating about the z-axis. Density and velocity gradients are co-linear; (

**g**) Perspective view of stacked disks rotating about the z-axis, which describes density varying vertically; (

**h**) Side view of the assembly of a thick disk with graded density. Nested hollow pillboxes are the required shape for isodensity contours in thick disk geometry.

**Figure 3.**Pictorial essay of an RC extraction by deBlok et al. [53] for a tilted, intermediate sized spiral galaxy from high-resolution study of the 21 cm band using the Very Long Array: (

**a**) Detailed, visual image of NGC 2403, type SAB(s)cd, by Göran Nilsson and The Liverpool Telescope group, which is publicly available at https://commons.wikimedia.org/w/index.php?curid=63317782. Red lines link this image with the Doppler measurements; (

**b**) Image at 3.6 μm from IRAC Spitzer, showing the multiple spiral arms, which is on the same scale as the Doppler measurements; (

**c**) Velocity field derived from fitting Hermite polynomials to the natural-weighted data cube, where the thick line indicates the systemic velocity and the difference between contours is listed; (

**d**) The thin disk model of the Doppler measurements with the same grayscales and contours as the data; (

**e**) Extracted RC (black curve) along the major axis with the inclinations stated on a velocity-position diagram (gray). Excellent agreement is obtained between the approaching and receding limbs. Panels (

**b**) to (

**e**) are modified after de Blok et al. (2008) High-resolution rotation curves and galaxy mass models from THINGS. Astrophys. J. 136, 2648–2719 [53] with permissions by the AAS; (

**f**) Color version of the Doppler measurements with superimposed rotation curves. The vertical dashed line is approximately at 13 kpc. Reproduced with permissions from Springer Nature: Nature Astronomy 2, 615-616, Is there a universal alternative to dark matter? De Blok, W.J.G., https://doi.org/10.1038/s41550-018-0547-4, copyright August 1, 2018 [54].

**Figure 4.**Pictorial essay of an RC extraction for a giant spiral galaxy with an unusually large baryonic atmosphere and a twisted velocity field: (

**a**) Detailed, visual image of M83, which is type SAB(s)c with a double nucleus, by William Blair, NASA, ESA, and the Hubble Heritage Team (STScI/AURA), publicly available at http://www.spacetelescope.org/images/heic1403a/. Red lines link this image with the area mapped in Doppler measurements. Yellow arrow marks one possible 2nd axis of spin, from symmetry of the arms; (

**b**) Integrated HI distribution overlaid onto the B-band image of M83; (

**c**) The mean velocity field, plotted on the same scale; (

**d**) Position-velocity diagram along the major axis with the inclinations stated; (

**e**) The minor axis. Panels (

**b**) to (

**e**) are a high-resolution study of the 21 cm band which used the Australia Telescope Compact Array and are modified after supplement figure A50 in “The Local Volume HI Survey (LVHIS)” by B.S. Koribalski et al. (2018) Mon. Not. R. Astron. Soc. 478, 1611–1648 [56].

**Figure 5.**Velocity distributions in a spiral galaxy: (

**a**) Measured profile, modified after C. Carignan, (1985) Light and mass distribution of the magellanic-type spiral NGC 3109, Astrophys. J. 299, 59–73 [57], with permissions by the AAS. In this panel only, velocity increases to the right. Blue line shows the expected equatorial velocity for a disk. Red and green lines = the expected equatorial velocity for an oblate; (

**b**) Schematic constructing the expected profile (upper section) for a thin disk consisting of rotating rings, where the lower section shows a vertical slice through the galaxy indicating the systematic changes in density and velocity with equatorial radius and a line of sight (LOS). Light lines show the flux-velocity correlations inside the disk, where the colored curve estimates attenuation of the emitted flux along the LOS. For a ring-disk geometry, the equatorial velocity would be midway between the minimum and maximum velocities; (

**c**) Schematic constructing the expected profile (upper section) for an oblate shape consisting of rotating homeoids (lower section). The top and bottom velocity-flux correlations are identical, but are offset for clarity. These will sum, but emissions from the bottom are attenuated more. The red curve approximates the attenuated sum. The equatorial velocity would be the maximum observed, if resolution is very high, but more likely a small tail will exist, representing beam smearing.

**Figure 6.**Indication of multiple spin axes: (

**a**) Schematics of a single rotation of a tilted spiral (top) and a simple secondary rotation for a face-on spiral; (

**b**) Wire diagrams of the oblate spheroid and Jacobi’s triaxial ellipsoid, with spinning about the short axes added to images created by Ag2gaeh and publicly available at https://en.wikipedia.org/wiki/Ellipsoid#/media/File:Ellipsoide.svg under a Creative Commons Attribution Share Alike 4.0 International license; (

**c**) The mean velocity field of Circinus, showing superimposed spins around two axis (black arrows), which are nearly parallel as presented to the LOS, modified after figure 13b in “The Local Volume HI Survey (LVHIS)” by B.S. Koribalski et al. (2018) Mon. Not. R. Astron. Soc. 478, 1611–1648 [56].

**Figure 7.**Evaluation of RC for possible multiple spin axes: (

**a**) Schematic of how systematic errors will affect rotation curves. Blue = typical measured RC with a flat trend. Red arrows = corrections for asymmetric peaks in flux-velocity profiles. Green arrows = corrections for a 2nd spin axis. Both effects make the slopes of rotation curves decrease more strongly at high radius, as shown in the orange and green RC; (

**b**) Simplified representation of commonly observed rotation curve types, after the classification scheme used by Wiegert and English [61] and others. Gray rectangle = the galactic region used to determine the dV/dr; (

**c**) Histogram of the list of non-interacting galaxies in Table 3 of [61]. Types with a ring are indicated by (r) and an ellipse. The box shows the morphological types. The double arrow indicates that dV/dr being negative correlates with axial symmetry, but anti-correlates with symmetry lowering elements such as bars.

**Figure 8.**Forward models of rotation curves for a test particle orbits, comparing a large, central point mass (dot-dashed curve) to orbits both in and around oblate bodies with homogeneous density but varying ellipticity (various black curves). Gray curve = a sphere, for which e = 0: this same pattern was obtained for Coulombic forces in and around a sphere with uniformly distributed charge (e.g., figure 28–9 in [49]). The dashed curve represents an axial ratio of c/a = 0.2, whereas the dotted curve depicts c/a = 0.1. For variable density, the maximum would be rounded rather than a corner, and thus would resemble the idealized RC curves in Figure 7b. Please note that this approach assumes orbiting points or rings, not co-rotating spheroidal shells.

**Figure 9.**Lines of force (red) superimposed on ellipsoidal contours of constant potential (purple or blue): (

**a**) Sphere (black circle) with c/a = 1, where the lines of force are perfectly radial and the orthogonal equipotential surfaces are spherical; (

**b**) Oblate body (black ellipse) with c/a = 0.1, as in spiral galaxies, where the lines of force are not radial and are only perpendicular to the body for the special axial directions. White arrow = spin axis. Equipotential contours are ellipses which become rounder with distance. Contours near the oblate are too finely spaced to be shown.

**Figure 10.**Energetics of ultrathin disks: (

**a**) Numerical calculation of work (dots) done by a test particle (the ant) as it crawls across a disk surface with constant density. Nearly 10

^{6}point masses were used in our integration. Light gray curve and inset shows a two-parameter fit. Dotted curve shows a rough power law. Dark gray curve shows a numerical calculation for the work done at r > a by an external test particle (ant in spacesuit); (

**b**) Numerical calculations of velocity inside and outside an ultrathin disk for homogeneous density (solid) and for an exponentially decaying density but with the same mass (dots), showing a discontinuity at r = a. Keplerian orbits (dot-dashes) match these curves at several body radii, as expected. Weakening the exponential decline (moving mass outwards) would provide an RC less steep at the center, but steeper on the outside, moving the peak at r = a closer to the RC for constant density. Strengthening the exponential (concentrating mass inwards) would provide an RC that is steeper at the center and flatter in the middle, such that the peak at r = a is weaker, while remaining higher than the Keplerian orbit al constraint for r > a.

**Figure 11.**Lines of force in cylindrical geometry, showing how these depend on the aspect ratio of the central body (dark gray). The light gray cylinders are imaginary shapes constructed to evaluate the divergence of the lines of force, per Gauss’ method: (

**a**) Line source, where force lines need only be computed in the radial direction; (

**b**) Stubby body, showing edge effects and bending force lines, which are only perfectly straight along the z and r axes; (

**c**) Thin disk. Away from the edges, flux is essentially vertical, and analytic (Equation (11)), with no discontinuities.

**Figure 12.**Simulated RC for oblate bodies with varying internal density, calculated for c/a = 0.1 = (1-e

^{2})

^{½}and M

_{in}= 10

^{11}M

_{sun}at 18 kpc. An edge is not assumed: (

**a**) Power law; (

**b**) Formulae resembling that of Binney and Tremaine [43], which these authors used to cancel terms and make the integral tractable. The case of n = −3/2 has the same shape as the result provided by Figure 2.13 in Ref. [43]), which to our knowledge is the only analytical solution heretofore available for an oblate spheroid.

**Figure 13.**Polytropic model for galaxies: (

**a**) Density for polytropes (black patterns, with indices labeled, which are normalized to the central density as a function of scaled radius normalized to unity at the center and zero at the surface (r = a). Index n = 0 for constant density is not shown. Gray shows linear and exponential densities for comparison. Reproduction of Figure 4b from Hofmeister and Criss [14], which is open access under a Creative Commons Attribution 4.0 International License: (

**b**) Rotation curves for differentially rotating homeoids inside an oblate with the same ellipticity. During the normalization factors involve the ellipticity cancel.

**Figure 14.**Inverse analysis of NGC 4736 of Feng and Gallo [64], based on measured RC of De Blok et al. [53], shown as a black dotted line: (

**a**) Results for σ from numerical evaluation of (24) by [64]. Dashed line = least squares fit below 1 kpc, which equals 1/r within the uncertainty of the selected cutoff; (

**b**) Mass obtained form (25), which integration smooths σ. Black dashed line = a third order polynomial obtained from a least squares fit. Heavy lines in the lower right corner show mass computed for a thin, constant density disk from (13), using two approximations for RC. Average v = 159 km s

^{−1}. Thin dash-dotted line shows the extrapolation used. Equation (13) has a singularity in v at r = a, but merges with the point mass approximation at very large r.

**Figure 15.**Inverse analysis of NGC 1808 by Sipols and Pavlovich [11], based on measured RC of Sofue et al. [81], shown as a heavy dotted line: (

**a**) Results for σ from matrix inversion of (29) by [11]. Dashed line = least squares fit below 0.8 kpc. Thin line = fit from 2 to 12 kpc; (

**b**) Mass obtained form (26), which integration smooths σ. Black dashed line = the least squares fit listed in the box. For comparison, mass was computed for a thin, constant density disk from (13) using extrapolated velocity (thin dash-dotted line). Equation (13) has a singularity in v at r = a, while merging with the point mass approximation at large r.

Type | Input and Shape | Physical Model | Output | Examples |
---|---|---|---|---|

Forward | Surface density; 4 shapes ^{2} | Orbits under central forces ^{3} | Simulated RC with NBDM | e.g., [30,31] |

Forward | Surface density; disk ^{2} | Orbits under central forces ^{3} | Simulated RC | [12,13] |

Forward | Surface density; disk ^{2} | MOND ^{4} | Simulated RC | [5,6,32] |

Forward | Surface density; disk ^{2} | General relativity ^{4} | Simulated RC | [7,33,34] |

Inverse | RC; disk | Elliptical integrals ^{5} | Surface density vs. radius | [8,9] |

Inverse | RC; equatorial plane | Point mass attractions ^{6} | Surface density vs. radius | [10,11] |

Inverse | RC; spheroid | Virial theorem for spin ^{7} | Volume density vs. radius | [14,15] |

^{1}“Disk” refers to a coin shape with a finite height. “Shapes” refers to combining point, spherical, and disk shapes, where different densities or masses describe the shapes considered. Surface density (σ) is related to luminosity. Volumetric density (ρ) is the thermodynamic quantity.

^{2}Assumed and iterated until a fit is obtained.

^{3}Central forces are characterized not only by a 1/r

^{2}dependence but also by all vectors being directed to the center.

^{4}MOdified Newtonian Dynamics and general relativity calculations both include an acceleration parameter in a non-central force law, which can be varied for optimal fitting (see e.g., [7,35]).

^{5}Numerical model that examines synthetic and measured RC.

^{6}Newton’s law is used to describe attractions between pairs of points in the equatorial plane, which are then summed or integrated.

^{7}This formulation is constrained by the gravitational potential and moments of inertia of homeoidal shells inside spinning, self-gravitating oblate spheroids.

Model | Component Geometry ^{1} | Force Constants | Number of Parameters ^{4} | Limitations | References or Figures |
---|---|---|---|---|---|

NOMs | Disk + NBDM halo + bulge + black hole ^{2, 3} | G | ≥8 | Densities often summed ^{2} | [29,30,35,36,51,54,55,56,57,58,59,60,69] ^{6} |

Log-Normal | Disk | G | ≥2 | No mass at center | [12,13] |

MOND | Disk ^{3} | G + a | ≥3 | Ad hoc force law ^{5} | [5,6,32,70] |

Relativity | Disk ^{3} | G + a | ≥3 | Flexible force law ^{5} | [7,33,34] |

Numerical | Ultrathin disk | G | ≥2 | Disks are unstable | Figure 10 |

Spheroid | Homeoid | G | ≥2 | Complex mathematics | Figure 8, Figure 12 and Figure 13 |

Point mass | point | G | 1 | Unrealistic for galaxies | Figure 10 |

^{1}Densities or masses are assumed for each component, each of which has a shape that requires one descriptive parameter, i.e., e = 0 for the sphere, finite e for the spheroid, H/a for the coin-shaped disk, or H = 0 for the equatorial plane.

^{2}Summations of elements are restricted (e.g., all components must act effectively as point mass) for this approach to be mathematically correct.

^{3}Some models assume that the disk has two components, gas, and stars, with different mass distributions, which requires more parameters.

^{4}The minimum number is needed to describe each shape for the case of a constant density.

^{5}Recent studies involve complicated mathematics: nonetheless, equivalence between r and s is assumed.

^{6}Many additional studies use the NOMs approach. Examples of component behavior are shown in the figures listed.

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Hofmeister, A.M.; Criss, R.E. Debated Models for Galactic Rotation Curves: A Review and Mathematical Assessment. *Galaxies* **2020**, *8*, 47.
https://doi.org/10.3390/galaxies8020047

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Hofmeister AM, Criss RE. Debated Models for Galactic Rotation Curves: A Review and Mathematical Assessment. *Galaxies*. 2020; 8(2):47.
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Hofmeister, Anne M., and Robert E. Criss. 2020. "Debated Models for Galactic Rotation Curves: A Review and Mathematical Assessment" *Galaxies* 8, no. 2: 47.
https://doi.org/10.3390/galaxies8020047