Density Profiles of 51 Galaxies from Parameter-Free Inverse

Article Density Proﬁles of 51 Galaxies from Parameter-Free Inverse Models of Their Measured Rotation Curves

Robert E. Criss * and Anne M. Hofmeister

Department of Earth and Planetary Science, Washington University, St Louis, MO 63130, USA; [email protected] * Correspondence: [email protected]; Tel.: +1-314-935-7440

Received: 2 February 2020; Accepted: 24 February 2020; Published: 26 February 2020

Abstract: Spiral galaxies and their rotation curves have key characteristics of differentially spinning objects. Oblate spheroid shapes are a consequence of spin and reasonably describe galaxies, indicating that their matter is distributed in gravitationally interacting homeoidal shells. Here, previously published equations describing differentially spinning oblate spheroids with radially varying density 3 are applied to 51 galaxies, mostly spirals. A constant volumetric density (ρ, kg m− ) is assumed for each thin homeoid in these formulae, after Newton, which is consistent with RCs being reported simply as a function of equatorial radius r. We construct parameter-free inverse models that uniquely specify mass inside any given r, and thus directly constrain ρ vs. r solely from velocity v (r) and galactic aspect ratios (assumed as 1:10 for spirals when data are unavailable). Except for their innermost zones, ρ is proven to be closely proportional to rn, where the statistical average of n for all 36 spirals studied is 1.80 0.40. Our values for interior densities compare closely with − ± independently measured baryon density in appropriate astronomical environments: for example, calculated ρ at galactic edges agrees with independently estimated ρ of intergalactic media (IGM). Our finding that central densities increase with galaxy size is consistent with behavior exhibited by diverse self-gravitating entities. Our calculated mass distributions are consistent with visible luminosity and require no non-baryonic component.

Keywords: inverse models; rotation curves; galactic density; galactic mass; galactic luminosity; Newtonian gravitation; dark matter

1. Introduction Hundreds of astrophysical studies concern the dynamics of galaxies, which are organized assemblies of stars, gas, and dust. Circular or elliptical motions are observed not only in these immense celestial bodies, but also for objects having many other scale lengths. Such rotational motions are classified either as orbits, which are governed by the potential exterior to an object that is often dominated by the effects of external bodies, or as axial spin, which is controlled by the interior potential of the object itself. These two gravitational potentials are mathematically distinct [1–3], matching only on the object’s surface [4]. Hence, the induced motions of spins and orbits differ, even though both are more or less circular. Orbital motions are familiar because many problems (e.g., Keplerian orbits of planets) are accurately described by the reduced 2 body problem, where a large, central mass controls motions of its small satellites, which are assumed to move independently of each other. However, even planetary orbits in our Solar System are not perfectly co-planar, and planet–planet perturbations are well known, which led to the discovery of Neptune. The velocity of such orbital systems projects to infinity at the central point. In great contrast, velocities inside spiral galaxies trend to the null value at the center.

Galaxies 2020, 8, 19; doi:10.3390/galaxies8010019 www.mdpi.com/journal/galaxies Galaxies 2020, 8, 19 2 of 33

ThatGalaxies the 20 motions20, 8, x FOR of PEER central REVIEW regions of spiral galaxies resemble spinning records is required for2 theof 33 organized motions of spin [5,6]. SpinSpin is is defined defined as as rotation rotation of of a a body body about about a a special special axis axis where where tangential tangential velocity velocity is is zero, zero, and and thatthat the the motions motions are are predicated predicated on on interactions interactions of of the the matter matter assembling assembling the the body body itself. itself. These These two two characteristicscharacteristics distinguish distinguish spin spin from from orbits. orbits. Spin Spin is is quantified quantified by by simple simple equations equations involving involving the the momentmoment of of inertia inertia (I ()I and) and angular angular velocity velocity [7 [7]]. Geometry. Geometry [ 8[8]] and and internal internal densification densification a ffaffeectctI. I For. For a a rigidrigid sphere sphere or or spheroid spheroid with with constant constant volumetric volumetric density density (ρ (),ρ), mass mass (M (M),), and and equatorial equatorial body body radius radius 2 (R(),R),I =I =2 MR2MR2/5./5. IfIfthis this rigid rigid spheroid spheroid is is self-gravitating self-gravitating and and conservative, conservative, the the Virial Virial theorem theorem (VT) (VT) providesprovides the the tangential tangential equatorial equatorial velocity velocit ofy of its its spin spin as: as:

223 GM vv3 GM at the equatorial perimeter ; = 0 along the polar axis , (1) v2 spin= at the equatorial perimeter; v2 = spin0 along the polar axis, (1) spin 2 2R R spin wherewhereG Gis is the the gravitational gravitational constant constant [9 [9]]. In. In contrast, contrast, orbiting orbiting bodies bodie ats at some some distance distancer fromr from a a massive massive centralcentral body body such such as as the the Sun Sun are are described described by: by:

2 GM in v GMin for a central point mass or spherical distribution (2) v2 orbit= for a central point mass or spherical distribution (2) orbit r r EquationEquation (2) (2) was was applied applied in in early early analyses analyses of of galaxies galaxies (e.g., (e.g., [10 [10]]).). While While these these equations equations are are deceptivelydeceptively similar, similar,M andM andR in R Equation in Equation (1) represent (1) represent the mass the and mass radius and of radius the object of the itself, object whereas itself, whereas Min and r in (2) represent the interior mass of the system and the distance of some external Min and r in (2) represent the interior mass of the system and the distance of some external object toobject the system’s to the centersystem’s of mass.center In of multi-body mass. In multi systems-body these systems diﬀerences these aredifferences extremely are important. extremely Clearly,important. considering Clearly, orbits considering (2) rather orbits than (2) spin rather (1) as than describing spin (1) the as coherentdescribing rotational the coherent motions rotational of the galaxymotions yields of the a much galaxy larger yields mass a m foruch equivalent larger mass values for equivalent of r and R .values The poor of r ﬁt and of theR. The orbital poor model fit of ofthe Equationorbital model (2) to rotationof Equation curves (2) (Figureto rotation1a) followscurves ( andFigure underlies 1a) follows the proposal and underlies of dark the matter proposal halos of [ 11dark]. matter halos [11].

(b) z s c

r a

FigureFigure 1. 1.Schematics. Schematics.(a) (a)Graph Graph comparing comparing spin spin of of a a rigid rigid top top (with (with null null central central velocities) velocities) to to Keplerian Keplerian orbitsorbits (which (which have have high high central central velocities) velocities) and and to to galactic galactic RC, RC, which which show show a gradationa gradation between between these these limitinglimiting cases. cases. Mass Mass components components cannot cannot be be summed summed to to produce produce RCs RCs [9 [9],], as as has has been been pursued pursued in in the the NewtonianNewtonian orbital orbital model model approach approach (e.g., (e.g. [,12 [12–14–14]]. (b). (b)Nesting Nesting of of similar, similar, homogeneous homogeneous homeoids homeoids to to formform a stratiﬁeda stratified oblate oblate body, body, with with axes axes labeled. labeled.

SpinSpin is is a a key key attribute attribute of of planets planets and and stars, stars, yet yet this this behavior behavior is is not not predicated predicated on on rigidity rigidity or or high high density,density, because because this this gravitational gravitational phenomenon phenomenon is is described described by by the the geometric geometric arguments arguments of of Newton Newton andand Laplace Laplace [15 [15]]. Regarding. Regarding rigidity, rigidity, even even layers layers in in the the dense dense and and highly highly viscous viscous Earth Earth are are decoupled, decoupled, asas demonstrated demonstrated by by westward westward drift drift of o Earth’sf Earth’s outermost outermost lithospheric lithospheric plates plates relative relative to to its its interior interior [ 16[16]]. . DecouplingDecoupling of of motions motions between between layers layers is is referred referred to to as as di differentialfferential rotation, rotation, and and may may also also exist exist for for Earth'sEarth's core core [17 [17,18,18]]. The. The precise precise term term is is di differentialfferential spin. spin. SpinSpin exists exists regardless regardless of theof statethe state of matter, of matter, or its density,or its density, since spin since also spin describes also describes gas giant gas planets, giant rarifiedplanets, stars rarified [19], stars and many[19], and atmospheric many atmospheric phenomena. phenomena. Hurricanes Hurricanes spin, yet thesespin, yet objects these are objects mostly are composedmostly composed of N2 and of ON22 gasand with O2 gas embedded with embedded water droplets, water droplets, and so and rigidity so rigidity is not requiredis not required for spin for (e.g.,spin [ 6 (e.g.,]). Galactic [6]). Galactic rotation rotation curves (RCs), curves which (RCs limit), which the dependence limit the dependence of velocity ofv to velocity only equatorial v to only equatorial radius r, implicitly assume zero velocity along the axis of rotation (Figure 1a). This description is consistent with spiral galaxies spinning (Figure 1b).

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Galaxies 2020, 8, x FOR PEER REVIEW 3 of 33 radius r, implicitly assume zero velocity along the axis of rotation (Figure1a). This description is consistentFor withspinning, spiral self galaxies-gravitating spinning objects, (Figure Maclaurin1b). quantitatively showed that the balance between centrifugalFor spinning, and gravitational self-gravitating forces objects, produces Maclaurin an oblate quantitatively spheroid, showed as deduced that theby balanceNewton between[20]. This centrifugaltheoretical and shape gravitational is observatio forcesnally produces established an oblate for spheroid, spinning as planets deduced and by several Newton stars [20]. and This is theoreticalconfirmed shape by edge is observationally-on images and established isophotes of for spiral spinning galaxies planets at wavelengths and several ranging stars and from is conﬁrmed the visible byto edge-on radio, see images e.g., Refs. and isophotes[21–25] and of Figure spiral galaxies2. Oblate at shapes wavelengths even describe ranging the from rarified, the visible distant to material radio, seeof e.g., spirals Refs. [21–25] and Figure2. Oblate shapes even describe the rariﬁed, distant material of spirals.

22 m

3109 (442 nm)

4594 (3.6 m) median of 30 spirals (radio)

Figure 2. Isophotes of edge-on galaxies at various wavelengths: (left) New General Catalog (NGG) 3109Figure in the 2. I B-band.sophotes Modifiedof edge-on after galaxies figure at 6 various in Carignan, wavelength C. 1985s: ,(left) Light New and General mass distribution Catalog (NGG) of the3109 magellanic-type in the B-band. spiral Modified NGC after 3109. figureAstrophys. 6 in Carignan, J. 299, 59–73 C. 1985 [24], with Light permissions; and mass distribution (middle) NCG of the 4594magellanic (Sombrero)-type isophotes spiral NGC at 3.6 3109.µm Astrophys. (circles are J. foreground 299, 59–73 stars).[24] with Modified permissions; after figure (middle) 1 in NCG Gadotti, 4594 D.A.(Sombrero) and Sánchez-Janssen, isophotes at 3.6 R. µm2012 (circles, Surprises are foreground in image stars). decomposition Modified after of edge-on figure 1 galaxies: in Gadotti, Does D.A. Sombreroand Sánchez have-Janssen, a (classical) R. 2012 bulge?, SurprisesMon. in Not. image R. Astron.decomposition Soc. 423 of, 877–888edge-on [galaxies:26], with Does permissions Sombrero fromhave Oxford a (classical) University bulge? Press Mon. on behalfNot. R. of Astron. the R. Astron.Soc. 423, Soc.; 877– (right)888 [26], Grey with shades permis= sionsthe L-band from Oxford from averagingUniversity 30 Press spirals, on usingbehalf the of the 22 µR.m Astron. contour Soc.; to represent (right) Grey the shades disk. Modified = the L-band after from figure averaging 6 in [21] 30 (Wiegert,spirals, using T. and the 9 others, 22 µm contour2015. CHANG-ES. to represent IV. the Radio disk. continuumModified after emission figure of6 in 35 [21] edge-on (Wiegert, galaxies T. and observed9 others, with 2015 the. CHANG Karl G. Jansky-ES. IV very. Radio large continuum array inD emission configuration—Data of 35 edge-on release galaxies 1. Astron.observed J., 150:81,with the doi:10.1088Karl G. Jansky/0004-6256 very /large150/3 array/81), with in D permissions.configuration—Data release 1. Astron. J., 150:81, doi:10.1088/0004- 6256/150/3/81), with permissions. That gaseous astronomical objects hold together during spin is obviated by behavior of planetary and stellarThat atmospheres.gaseous astronomical The gas objects molecules hold are together part of durin the sping spin of is the obviated star and by cannot behavior be of treated planetary as eachand having stellar independentatmospheres. motions. The gas Wemolecules refer to thisare part behavior of the as spin dynamical of the star cohesion, and cannot and note be thattreated this as conditioneach having does independent not require a motions. solid body, We but refer can to apply this behavior to interacting as dynamical “particles” cohesion, comprising and anote larger, that non-solidthis condition object, does such not as cars require in congested a solid body, traffic but or dropletscan apply in convectingto interacting clouds “particles” [5,6,27]. comprising a larger,Historically, non-solid oblate object, spheroids such as cars were in recognizedcongested traffic as being or droplets the required in convecting shape for clouds galaxies [5, [6,27]28,29. ]. For anHistorically, oblate spheroid oblate (Figure spheroids1b), the were minor recognized ( c) and major as being (a) axes the require are tiedd via shape the ellipticityfor galaxies (e): [28,29]. For an oblate spheroid (Figure 1b), the minor (c) and major (a) axes are tied via the ellipticity (e): 1 2 2 2 1 e = 1 c /a222 (3) e−1 c a  (3) This definition for the oblate spheroid surface can be used to relate the axial position (z) to This definition for the oblate spheroid surface can be used to relate the axial position (z) to cylindrical radius (r) for each surface of the nested homeoids, because these have the same shape cylindrical radius (r) for each surface of the nested homeoids, because these have the same shape factor e: factor e: z2 = 1 e2 a2 r2 (4) 2− 2− 2 2 z1  e a  r  (4) From (4), volumetric density ρ(z) depends on ρ(r), along with a and e, for oblate spheroids, which considerablyFrom (4), simplifies volumetric analyses density of RCsρ(z) depends from spin. on ρ(r), along with a and e, for oblate spheroids, which considerablGiven they above,simplifies we recentlyanalyses investigated of RCs from the spin. outer velocities of a spinning oblate spheroid galaxy by applyingGiven the the rigid above, body we approximation recently investigated while considering the outer various velocities internal of a spinning density variations oblate spheroid [5,6]. Becausegalaxy galaxies by applying are bound the systems, rigid body their approximationenergetics are specified while consideringby the Viral theorem various (VT) internal of Clausius. density variations [5,6]. Because galaxies are bound systems, their energetics are specified by the Viral theorem (VT) of Clausius. Emden’s [30] classical book applied the VT to dispersed nebulae and rainclouds. For density governed by a polytrope of index j, the mass interior to any given radius is:

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Emden’s [30] classical book applied the VT to dispersed nebulae and rainclouds. For density governed by a polytrope of index j, the mass interior to any given radius is:

!3 2 5 j r e M = v2 − (5) in tangent 3 5 G arcsin(e)

Galactic masses calculated from Equation (5) match the luminous mass for the 14 spirals with well-constrained data [5]. Applying Equation (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., the galaxy is spinning stably. Stability requires that each homeoidal shell (Figure1b) has constant density. Based on this finding, Criss and Hofmeister [ 27] addressed differential spin, which allows probing interior motions. Their inverse solution is: ! 1 2v ∂v v2 e ρ(r) = + (6) 6πG r ∂r r2 √1 e2arcsin(e) − This result was applied to Andromeda but is applicable to isolated galaxies of all morphological types. We made no assumptions other than Newtonian physics and conservation laws. Equation (6) is analytic and exact, has no free parameters, and allows direct and unambiguous extraction of density and mass profiles from RC. In the present paper, Section2 describes how spreadsheet data on rotation curves ( vtangent as a function of equatorial radius) can be used to compute mass and density as a function of radius, in an approach equivalent to Equation (6). Section3 applies our model to RC data to directly provide mass and density, each as a function of r, for 51 galaxies that range widely in size, RC pattern, and type (spiral, lenticular, irregular, spheroidal, elliptical, and polar ring), although the majority (36) explored are spirals. Although elliptical galaxies and other roundish types do not technically have RC, we include these types because Romanowsky et al. [31] and others have treated their velocity dispersions as RC. We test and verify our approach by comparing our deduced densities with independent measurements involving many astrophysical environments (Table1). Consistent trends emerge from our analysis and provide the physics underlying empirical luminosity-velocity relations (Section4). Neither dark matter nor non-Newtonian physics are required to explain galactic rotation curves (Section5).

Table 1. Volumetric densities of diﬀerent astrophysical environments and objects.

Density 3 3 3 Object or Region kg m− Msun pc− H Atoms cm− 1 9 10 12 Dispersed Solar System 5.2 10− 8 10 10 × 15 × 6 Molecular Cloud Core 2 10− 30,000 >10 ×18 16 4 Globular Cluster Core 7 10− –10− ~100–1500 0.4–6 10 × 17 × 3 Best Laboratory Vacuum 10− 150 6 10 18 × Globular Cluster ~7 10− ~0.5 ~4200 × 18 Giant Molecular Cloud ~10− ~15 ~600 2 21 Solar Neighborhood ~3 10− ~0.04 ~2 3 × 21 Interstellar Medium ~10− ~0.02 ~0.6 Intergalactic Medium 4 ~5 10 26 ~7 10 7 ~3 10 5 × − × − × − Baryons in Universe 5 ~10 26 ~10 7 ~6 10 6 − − × − 1 Out to the sphere with radius of 30 AU. 2 From data in Ledrew [32], including dispersed white dwarfs, which 3 4 6 agrees with Luyten’s [33] estimate. Ferrière [34] provides 10− to 10 atoms per cc for various components in the interstellar medium (ISM). 4 From Fang et al. [35], the intergalactic media (IGM) is 5 to 200 times denser than the cosmological models of the baryons: we use the average of 100 times. 5 Cosmological model depending on the Hubble constant [36]. Galaxies 2020, 8, 19 5 of 33

2. Methodology Our inverse model is based on the VT, which stems from a mathematical identity. An important consequence for a bound, conservative state is that average linear momentum cancels in all directions over its restricted space [9]. The axial spin of galaxies arising through gravity can thus be analyzed using the VT [5,6,27]. Some details are given below. This paper calculates volumetric density.

2.1. Inverse Models Inverse problems are fundamentally diﬀerent than forward (direct) problems (Table2). These are best understood via contrast to the latter which are much more familiar. Direct problems are solved by inserting known, or assumed, inputs, such as source characteristics, into a standard equation, formula or program, that returns a result. In contrast, many important problems in astronomy involve the opposite approach, which is to deduce the nature of a remote source from the nature of its output (e.g., [37]). The direct calculation of density proﬁles from observed RC data, as rendered possible by Equation (6), is a successful example of a solved inverse problem.

Table 2. Comparison of forward and inverse approaches to RC analysis. 1

Type Input Physical Model Output Forward density and shape 2 Newtonian orbits RC fits Forward density in a disk-like shape MOND 3 RC fits Inverse RC and c/a data Virial theorem 4 density vs. radius 1 After Groetsch [38] (Chapter 1). 2 Assumed and iterated until a fit is obtained. 3 MOdified Newtonian Dynamics includes a parameter in the force law that is varied to obtain a fit to RC data. 4 Applied to Newton’s and Maclaurin’s descriptions of spinning, self-gravitating bodies.

Mathematical solutions to inverse problems are not always possible [38], which has made this approach less familiar. The importance of inverse models to astronomy is underscored by Ambartsumian’s [39] determination of the proper velocity distribution of stars from observed radial velocity distribution.

2.2. Homeoid Properties and Their Consistency with Observations Diverse astrophysical environments are less dense than vacuums produced in the laboratory (Table1) and collisions are rare, and so friction is absent, and galaxy motions, shape, and distribution of mass are all gravity driven and linked. Hence, the nested spheroidal shells (Figure1b) each rotate independently at some angular velocity. Direct evidence for independent rotation of galactic shells is provided by the reduction of tangential velocities above the planes of spiral galaxies [40,41], slower rotation in the bulge than in the disk at any given r in an S0-E6 type [42], and examples of counter rotation and perpendicular rotation (polar rings) in a few spiral galaxies (e.g., [43]). As Newton discovered, ellipsoidal shells (homeoids) are equipotential [44] (p. 87) and therefore each has a characteristic density. Otherwise, forces along its surface would be unbalanced. Newton’s homeoid theorem shows that a test particle inside a homeoid experiences no net force. Constant density for individual homeoids, which is associated with gravitational stability [45,46] (p. 100–101), is appropriate for several reasons. First, v(r) are angular averages that do not describe evolutionary changes. Second, spiral galaxies vary greatly in appearance (star distribution) yet their RC curves are quite similar [47], permitting classification as a very few types (e.g., [48]). Thus, only the radial dependence of the density distribution is important. Third, heterogeneities in the images which correlate with locations of luminous stars do not reveal heterogeneity in density over a larger scale. For example, clouds in Earth's atmosphere are no more dense than dry surrounding air, even though water droplets have high density. An oblate spheroid is simply a flattened sphere that can be considered to represent a collection of nested homeoids. The self-potential of a spheroid differs from that of a sphere only through Galaxies 2020, 8, 19 6 of 33 the ellipticity. Based on Maclaurin’s geometrical constraints, Todhunter [20] determined that the self-gravitational potential of a spheroid (Figure1b) is a simple factor ( k) times the self-gravitational potential of a homogenous sphere of equivalent volume, where:

2 1/6 arcsin(e) 3 GM k = 1 e2 and U = in (7) − e g,sel f ,sphere −5 S From Newton’s homeoid theorems (e.g., [44]), rotating external homeoids are equipotential surfaces and exert no net force on matter inside, as is the case for spherical shells. Diﬀerentiating Equation (7) gives the potential for a homeoid of mass m that surrounds an interior mass Min:

GMinm GMinm arcsin(e) Ug,homeoid = k = (8) − seqv r e where the left hand side incorporates the radius of the sphere of equivalent volume for the oblate 2 1/6 spheroid of seqv = a(1-e ) , since the equatorial radius r which varies from 0 to a is most appropriate. Diﬀerentiating the well-known formula for I of an oblate spheroid which is:

2 8πρih 1/2 I = Ma2 = a5 1 e2 (9) oblate 5 15 − gives a simple result for the moment of inertia of a homeoid:

2 I = mr2 (10) homeoid 3 Existing complicated relationships in Sersic [49] reduce to Equation (10). For reference, homeoid volume (dV) and mass (m, which actually is dm) are, respectively:

1/2 dr dr dV = 4πr2dr 1 e2 = 3V and m = ρdV = 3ρV (11) − r r

Applying the Virial theorem to a series of nested coaxial homeoids results in (GMinm/r)arcsin(e)/e 2 2 = 2/3mr ω . Rearranging terms gives the equatorial velocity as:

3GM arcsin(e) v2 = in (12) homeoid 2r e From (12), the angular velocity of any spheroidal shell depends only on its size, ellipticity, and the mass interior to that homeoid, provided that the latter is distributed in a spheroidally symmetric manner (Figure1b, Figure2). This relationship, a consequence of Newton’s homeoid theorem, di ffers only from that for a spherical assemblage of particles by a geometric factor involving the ellipticity. Because e ranges from 0 to 1, the geometrical factor arcsin(e)/e ranges only from 1 to π/2 (~1.57). Thus, for flattened homeoids, v2 is ~ (2 0.4) GM /r, which is not greatly different than the relationship ± in for a spherical shell. Importantly, nested homeoids also spin about the same axis; the composite object is merely a flattened sphere. We reiterate that the shape of the surface of the homeoid shares the ellipticity with the oblate body, per Newton and Laplace.

2.3. Tie to Previous Conventions and Equations The RC literature uses the variable “r” to depict galactic distance in the equatorial plane. The preceding equations used r for cylindrical radius and s for spherical radius where appropriate (see Figure1b). The remainder of this report uses “ r” because we are concerned with the equatorial radii of galaxies. Calculating orbital velocities of a system of coaxial, rotating spheroidal shells of variable density is straightforward. The relevant mass is the mass of matter interior to a particular shell. Galaxies 2020, 8, 19 7 of 33

Note that n in the denominator of equation 15 of Criss and Hofmeister [27] should be replaced by 3. This typo did not affect other equations in [27]. This forward model is not being used here. According to Marr [50], inverse models for galactic spin which input measured velocities and output mass or density had not been constructed previously. Prior to detailed measurements of RC, Brandt [51] attempted to provide analytical expressions for mass as a function of radius by inverting the formulation for oblate spheroids of Burbidge et al. [29], which describe attraction to the exterior of the oblate (r > R). Setting e = 0 (spherical symmetry) in their formula and using constant ρ provides a Keplerian orbit, thereby confirming that Burbidge et al. [29] modelled satellite orbits exterior to an oblate body, rather than describing the organized motions inside. In exploring the case of e ~ 1, Brandt [51] arrived at Keplerian orbits in his equation number 25, which instead requires e = 0. This inconsistency resulted from his inversion including division by the factor (1-e2)1/2, which is 0 in the limit of e = 1. Division by 0 occurred early in Brandt's [51] derivation, specifically in his equation number 7, which voids his inverse model for orbits about an oblate. Such difficulties stem partially from the equations for forces and potentials exterior to an oblate body not being cast in particularly useful forms until 2018 [4]. More importantly, correct and simple equations for the gravitational potential and moment of inertia for a homeoid are presented for the first time as equations (8) and (10), above. These new equations are the basis of our inverse model.

2.4. Methodology for Inverse Modelling For spiral galaxies lacking well constrained e, we used c/a = 0.1 for which arcsin(e)/e = 1.4871 (Table3). These assumed values are consistent with edge-on spirals (excepting Sombrero with its large bulge, Figure2) and with compiled data on aspect ratios [ 52], recognizing that tilt makes the objects appear rounder. Moreover, assuming c/a = 0.1 has little eﬀect, since arcsin(e)/e neither varies greatly (Table3), nor has a huge e ﬀect, as shown by a few examples in Section3.

Table 3. Geometrical factors describing galactic shapes.

Shape or Type 1 Object c/a 1 Ellipticity Arcsin(e)/e Sphere NGC 2434, 4494 1 0 1.0000 E1 NGC 3379 0.98 0.2 1.0068 cE2, spheroidal M32; WLM 2 0.87 0.49 1.0451 S0 UGC 3993 0.75 0.66 1.0921 E6 NGC 2768, 4431 0.52 0.85 1.1953 Sombrero NGC 4594 0.4 0.917 1.2655 Irregular NGC 3034 0.38 0.925 1.2768 S0/a NGC 7286 0.36 0.93 1.2843 Flat oblate All other spirals 3 0.1 0.995 1.4781 Inﬁnite oblate Limiting value ~0 1 1.5708 1 Descriptions and data from [23]. 2 The other 7 dwarf irregulars and spheroidals were assumed to have the same c/a. 3 We used c/a = 0.1, based on the infrared contours of Figure2 that average 30 nearly edge-on spirals [ 21,22] and the generally accepted view that spirals are thin and ﬂat.

To convert reported values of angular diameter to kpc, we used the average distance in the NASA/IPAC Extragalactic Database (NED) (see Section 2.5). Next, we calculated Min in a spreadsheet from Equation (12) and then computed dM/dr. Density is obtained from a geometrical constraint on the homeoids: ∂M 1 ρ(r) = (13) ∂r 4πr2c/a Results for Andromeda from (6) and (13) are indistinguishable, although the former equation provides the most direct link between density and RC data. Galaxies 2020, 8, 19 8 of 33

2.5. Criteria for Selecting Galaxies and Data Available The database on RCs of spiral galaxies is very large, although skewed to SA and SAB classes [53]. In comparison, relatively few irregular, lenticular, elliptical and (rare) polar ring galaxies have been studied, although efforts have been directed towards LSB and dwarf spheroidal galaxies (e.g., [54]). Measurements on non-spiral types concern velocity dispersions, which have been treated as RCs to estimate masses. High resolution, recent studies were sought, particularly for the various morphologic spiral sub-types (a-d), as this sequence illustrates increasing gas content and decreasing bulge size, e.g., [14,55]. As described below, we analyzed 36 spirals and 15 other galaxy types, which encompass known morphologies, cover a wide range of galactic properties, and include both typical and unusual patterns for RC. Ellipticities are listed in Table3. Table4, which lists galaxy properties from [ 23], is divided into five subsections, which address different issues arising in analyses of RC. Each subsection lists galaxies by NCG number, if available. Types are discussed in [23]. For a consistent measure of size, we used the NED [23] tabulations of 2 distance and of r at 25 B-magnitude arcsec− and denote this isophote as the visual edge, for brevity. All distances are based on models which make some assumption about absolute luminosity. For an explanation see [56] (redshift section). The 1st subsection of Table4 consists of large and moderate sized spiral galaxies that encompass the main morphologies, have different attributes, and were measured in several studies, mainly due to their proximity (e.g., Andromeda, Triangulum, and Sombrero). In addition, the Milky Way provides a unique internal view of rotational data [57–60]. NGC 253 and 2599 have declines in v with r at large r [61]. Large Uppsala General Catalog (UGC) 2855 has data extending far beyond its visible disk [62]. Irregular NGC 3034 has lopsided rotation curves [63]. This galaxy is similar in size to NGC 4826, which counter-rotates, and to NGC 7793 [54]. The 2nd subsection consists of galaxies considered by Bottema and Pestaña [64] to have highly desirable characteristics for accurate analysis. Useful selection criteria include a well-established distance, symmetric RCs that extend beyond the optical disk, inclinations between 50 and 80◦, and a range of masses. We omitted 2 of the 12 galaxies from [64] because their rotation curves are similar to others in this grouping and/or have widely spaced data points. Of the 10 galaxies examined, 8 are spirals, NGC 4789a is irregular, and NGC 1560 has low surface brightness. The 3rd subsection lists all Messier objects studied by Sofue and collaborators using tabular data on Sofue’s website [65]. Not only are various types of spirals included, but the method of data analysis is consistent, and both centers of the galaxies and outer reaches were examined and merged in a consistent fashion. Sofue et al’s [53,66] measurements compare reasonably well with more recent studies [14] and have been used to evaluate models [67]. The 4th and 5th subsections cover galactic types other than spirals. We used high-resolution data or extended RC, if available, but for types such as ellipticals and spheroidals, velocity dispersion curves are the only data available (e.g., [68]).

Table 4. List of studied galaxies with characteristics from the NASA/IPAC Extragalactic Database (NED [23]) and references for RC data.

Distance L L 1 Edge 2 NGC Other Name Type Notes Refs. vis 21 (Mpc) (LSun,vis) (LSun,21) (kpc) Well-Studied Spirals - Milky Way SBc inside view [57,58] 4 1010 18 × 224 Andromeda SA(s)b nearby, bulge [59] 0.78 3.19 1010 2300 23.3 × 253 SAB(s)c starburst [69] 3.16 1.88 1010 18 × 598 Triangulum SA(s)cd no bulge [14,70] 0.84 3.65 109 1110 9.45 × 925 SAB(s)d large H cloud [54] 8.6 6.50 109 2830 16.2 × 2599 UGC 4458 SAa high velocity [61] 64.2 5.01 1010 13,800 25.8 × 3034 M82 I0, edge-on3 starburst [63] 4.01 8.81 109 7.6 × 4594 Sombrero SA(s)a huge bulge [71,72] 10.0 6.10 1010 14.4 × Galaxies 2020, 8, 19 9 of 33

Table 4. Cont.

Distance L L 1 Edge 2 NGC Other Name Type Notes Refs. vis 21 (Mpc) (LSun,vis) (LSun,21) (kpc) 4826 M64-Evil Eye (R)SA(rs)ab counter rotating [54] 5.16 1.60 1010 7.8 × 7793 SA(s)d jets, bulge [54] 3.93 3.70 109 739 8.0 × - UGC 2855 SABc distant [62,67] 79 9.90 1010 15.8 × Spirals with Desirable Characteristics 1560 SA(s)d LSB, edge on [64] 3.28 7.80 108 21.5 5.56 × 2403 SAB(s)cd bright in HII [66] 3.48 4.81 109 14.15 × 2841 SA(r)b LINER, bulge [66] 17.6 7.43 1010 20.73 × 2903 SAB(rs)bc starburst [66] 8.70 2.15 1010 16.8 × 3109 SB(s)m small, edge on [64] 1.33 2.51 108 6.35 × 3198 SB(rs)c psuedobulge [65] 14.0 1.28 1010 20.3 × 4789a DDO 154 IB(s)m LSB, dwarf [64] 3.84 4.40 107 1.72 × 5585 SAB(s)d psuedobulge [64] 8.0 1.71 109 277 7.07 × 6503 SA(s)cd LINER [64] 5.52 2.42 109 6.42 × 7331 SA(s)b bulge [66] 14.3 5.32 1010 28,300 23.84 × Messier Spirals 1068 M77 (R)SA(rs)b Seyfert [66] 12.65 2.94 1010 16.5 × 3031 M81 SA(s)ab grand design [66] 3.71 2.04 1010 14.7 × 4192 M98 SAB(s)ab active nucleus [53] 15.61 2.05 1010 8620 23.8 × 4254 M99 SA(s)c star forming [53] 15.43 2.44 1010 13.5 × 4258 M106 SAB(rs)bc LINER [66] 7.46 1.87 1010 23.8 × 4303 M61 SAB(rs)bc supernovae [66] 12.0 1.83 1010 14.5 × 4321 M100 SAB(s)bc grand design [66] 16.56 4.46 1010 21.0 × 4501 M88 SA(rs)b Seyfert [53] 18.79 6.50 1010 19.3 × 4548 M91 SB(rs)b LINER [53] 16.36 2.78 1010 13.1 × 4569 M90 SAB(rs)ab LINER [66] 12.29 1.93 1010 17.0 × 4736 M94 (R)SA(r)ab LINER, ring [66] 5.02 9.85 109 10.2 × 5055 M63 Sunﬂower SA(rs)bc ﬂocculent [66] 8.0 1.73 1010 17.5 × 5194 M51 Whirlpool SABc has companion [66] 7.85 2.14 1010 12.8 × 5236 M83 SAB(s)c two nuclei [66] 4.8 3.20 1010 19.0 × 5457 M101 Pinwheel SAB(rs)cd star forming [66] 6.81 2.22 1010 27.3 × Dwarf Irregular and Spheroidal Galaxies - WLM Ib(s)m isolated LSB [73] 0.985 5.83 107 1.8 × - M81dWb Im star forming [74] 7.66 7.13 107 1.45 × - Holmberg II Im star forming [74] 3.30 4.70 108 92.4 4.43 × - Carina dSph, dE3 classical dSph [75–78] 0.103 2.95 105 0.45 × - Draco dSph, Epec classical dSph [75–78] 0.082 2.95 105 0.60 × - Fornax dSph, dE4 classical dSph [75–78] 0.139 1.30 107 1.42 × - Leo I dSph, dE3 classical dSph [75–78] 0.246 4.99 106 0.43 × Lenticular and Elliptical Galaxies - UGC3993 S0? bulge [61] 62 1.21 1010 15 × 7286 UGC12043 S0/a no bulge [61] 20.5 2.45 109 5.6 × 2768 S0-E6 edge Seyfert, jets [42] 19.3 2.80 1010 2570 22.8 × 3379 M105 E1 elliptical [31] 11.0 2.11 1010 138 8.5 × 2434 E0-1 elliptical [68] 26.9 2.38 1010 15.4 × 221 M32 cE2 compact dwarf [55] 0.775 2.61 108 0.183 1.0 × 4431 VCC1010 dE, SA0(r) dwarf elliptical [79] 16 9.87 108 4.06 × Polar Ring or Disk Galaxies 4560A The prototype S0/a pec? 2 disks [80] 46.4 9.67 109 16.5 ⊥ × 1 2 “21” refers to the 21 cm line of H atoms, where LSun was obtained at this wavelength. “Edge” denotes the 2 isophote which is at 25 B-mag arcsec− for almost all galaxies listed in the NED, which approximates the visible regions of the galaxy. 3 Appears to be a spiral in near-IR imaging.

3. Results: Mass and Density from Inverse Models of Rotation Curves Most of the visualizations of mass and density for the individual galaxies are placed in AppendixA. Results are summarized in Table4: Note: this broadside and after the references.

3.1. Detailed Analysis of the Milky Way Sofue [58] compiled available data on the Milky Way to provide a RC from 0.001 kpc to the outer 1 reaches, using the accepted value of v = 200 km s− near the Sun. Recent observations at greater distances [81,82] motivated revision of the outer RCs to higher velocities [59]. We combine v at low r from Sofue’s [58] compilation with v at high r from his [59] compilation and use the differences in values to gauge uncertainties. Assuming c/a = 1, 0.5, 0.1 or 0.01 causes the calculated values for Min to vary only by a factor of ~1.5 (Figure3a). Since e can vary only from 0 to 1, this range encompasses all reasonable values for e. Calculated Min values for flat shapes (c/a = 0.1 or 0.01) are similar at any r. Due to this finding and Galaxies 2020, 8, 19 10 of 33

the behavior observed for density (discussed below), we use the nominal aspect ratio (c/a = 0.1) for most spiral galaxies unless data exist (Table3), permitting us to concentrate on the e ﬀects of diﬀerent patterns for RC.

FigureFigure 3 3.. InverseInverse models models showing showing the the effect effect of of varying varying cc//aa overover a a large large range range for for an an internally internally probed probed galaxy,galaxy, thethe MilkyMilky Way.Way. Compiled Compiled RCs RC ares are used used to calculateto calculate (a) mass(a) mass inside inside and (andb) density (b) density at any at given any givenradius. radius. In part In ( apart), upper (a), upper box describes box describes line patterns line patterns associated associated with RCs with [58 RC,59,s66 [5]8 using,59,66 the] using left axisthe leftand axis two and additional two additional constraints constraints on v near on the v Sunnear [81 the,82 Sun] and [81 from,82] globularand from cluster globular data cluster [60]. Lowerdata [60 box]. Lowerlists aspect box ratioslists aspect assumed ratios in analyzingassumed in each analyzing dataset foreach mass, dataset shown for onmass, the rightshown axis. on Errorthe right bars axis. on v Errorfrom [bars58] areon shown.v from [5 Downturns8] are shown. in MDownturnsin and ρ indicate in Min and that ρ the indicate “edge” that of the the Galaxy “edge” is of gradual the Galaxy and isbetween gradual 18 and and between 30 kpc. In18( band), density 30 kpc. above In (b),r =density0.1 kpc above was fit r to= 0.1 a power kpc wa law,s fit as to shown. a power law, as shown. Notably, if v decreases weakly as r increases, calculated values for Min increase with r, as is 3 requiredNotably, by geometry. if v decreases This behavior weakly as largely r increases, arises because calculated volume values is a fortimes Min anincrease ellipticity with factor r, as foris requireda spheroid. by geometry. For a strong This decrease behavior in vlargely, the calculated arises because values volume of Min isdecrease a3 times withan ellipticityr. This unrealistic factor for abehavior spheroid. partially For a strong results decrease from large in v uncertainties, the calculated in valuesv measured of Min near decrease the termination with r. This of unrealistic the Milky behaviorWay, as shown partially in Figure results3a. from If the large edge uncertainties were sharp, inM inv measuredwould increase near withthe terminationr up to some of radius, the Milky and Way,remain as constantshown in thereafter.Figure 3a. If Our the resultsedge were indicate sharp, that Min the would galactic increase edge with is not r up sharp, to some but radius, grades a intond remainthe IGM, constant which thereafter. is consistent Our with results the knownindicate existence that the galactic of substantial edge is outlying not sharp, material, but grades such into as gas,the IGM,globular which clusters, is consistent or satellite with galaxies the known [83,84 ]. existence of substantial outlying material, such as gas, globularMost clusters, of the mass or satellite in the Milkygalaxies Way [83 (MW),84]. lies at r beyond the visible disk at 18 kpc. This behavior 3 occursMost because of the spheroidmass in the volume Milky is Waya times (MW) an lies ellipticity at r beyond factor the and visible underlies disk at velocity 18 kpc. This profiles behavior being occursflat at largebecauser. Luminosity spheroid volume from visible is a3 times wavelengths an ellipticity is therefore factor and derived underlies from velocity a much profiles smaller being mass, flatmainly at large that r inside. Luminosity the visible from edge visible (Tables wavelengths4 and5). is therefore derived from a much smaller mass, mainly that inside the visible edge (Tables 4,5).

Galaxies 2020, 8, x; doi: FOR PEER REVIEW www.mdpi.com/journal/galaxies Galaxies 2020, 8, 19 11 of 33

Table 5. Properties of galaxies extracted from rotation curves, with a focus on volumetric density.

Visible Edge Properties

Min Lvis/Min Density rturndown Lowest ρ ρ at 0.1 kpc Highest ρ 1 rmax Mmax Name 3 3 3 3 Power (MSun) (LSun/MSun) (kg m− ) (kpc) (kg m− ) (kg m− ) (kg m− ) (kpc) (MSun) Large Well-Studied Spirals Milky Way 1.21 1011 0.330 1.30 10 21 18–30 1.0 10 24 1.00 10 17 1.00 10 14 2.06 230 1.03 1012 × × − × − × − × − − × 224 1.44 1011 0.222 4.20 10 22 ? 2.0 10 24 4.00 10 17 4.00 10 17 2.11 285 7.03 1011 × × − × − × − × − − × 253 4.30 1010 0.437 1.00 10 21 ? 2.8 10 22 9.00 10 18 7.00 10 17 1.5, 2.0 12.68 6.90 1010 × × − × − × − × − − − × 598 1.52 1010 0.240 1.30 10 21 ? 8.0 10 22 1.80 10 17 1.80 10 17 1.57 15 2.60 1010 × × − × − × − × − − × 925 2.00 1010 0.325 8.60 10 22 13 8.6 10 22 3.00 10 20 0.9 13 2.03 1010 × × − × − × − − × 2599 1.90 1011 0.264 3.51 10 22 20? 3.1 10 23 3.54 10 17 3.54 10 17 2.36 56 3.52 1011 × × − × − × − × − − × 3034 1.00 1010 0.881 2.20 10 21 ? 3.0 10 21 3.70 10 18 1.00 10 16 1.5, 1.9 3.57 7.20 109 × × − × − × − × − − − × 4594 2.00 1011 0.305 3.30 10 21 ? 3.0 10 21 8.70 10 18 1.60 10 15 1.6 9.4 1.58 1011 × × − × − × − × − − × 4826 1.90 1010 0.842 4.50 10 21 20 1.0 10 22 1.00 10 17 2.00 10 18 1.92 21.7 5.69 1010 × × − × − × − × − − × 7793 8.80 109 0.421 1.20 10 21 6–7 5.0 10 23 6.40 10 19 6.40 10 19 1.68 6.82 8.96 109 × × − × − × − × − − × UGC 2885 1.17 1011 0.846 1.53 10 21 50 1.0 10 22 2.80 10 18 2.80 10 18 1.5 74.7 3.00 1011 × × − × − × − × − − × Spirals with Desirable Characteristics 1560 2.80 109 0.279 1.40 10 21 6 4.0 10 22 9.90 10 20 2.60 10 19 1.0 8.6 5.40 109 × × − × − × − × − − × 2403 2.90 1010 0.166 5.20 10 22 17 1.1 10 22 3.20 10 18 9.20 10 18 1.74 19.6 3.90 1010 × × − × − × − × − − × 2841 1.70 1011 0.437 8.02 10 22 12 5.0 10 23 4.00 10 17 2.85 10 17 2.03 74 5.90 1011 × × − × − × − × − − × 2903 6.70 1010 0.321 6.30 10 22 28 5.0 10 25 1.10 10 17 1.10 10 17 2.12 30 1.17 1011 × × − × − × − × − − × 3109 2.99 108 0.839 8.00 10 22 5.6 7.0 10 22 5.00 10 20 1.80 10 20 0.86 6.62 3.08 109 × × − × − × − × − − × 3198 6.00 1010 0.213 3.26 10 22 43 1.4 10 23 5.50 10 18 3.90 10 18 1.85 46.4 1.19 1011 × × − × − × − × − − × 4789a 1.88 108 0.234 3.90 10 21 4.8 3.0 10 23 2.00 10 19 5.00 10 20 1.41 2 7.40 1.97 109 × × − × − × − × − − × 5585 5.90 109 0.290 1.01 10 21 - 6.0 10 22 4.12 10 19 5.50 10 19 1.39 10 9.00 109 × × − × − × − × − − × 6503 9.58 109 0.253 1.84 10 21 17 1.2 10 22 1.10 10 17 2.20 10 19 2.09 19 2.9 1010 × × − × − × − × − − × 7331 1.47 1011 0.362 8.70 10 22 30 2.5 10 23 9.00 10 18 1.70 10 17 1.89 31.6 1.98 1011 × × − × − × − × − − × Galaxies 2020, 8, 19 12 of 33

Table 5. Cont.

Visible Edge Properties

Min Lvis/Min Density rturndown Lowest ρ ρ at 0.1 kpc Highest ρ 1 rmax Mmax Name 3 3 3 3 Power (MSun) (LSun/MSun) (kg m− ) (kpc) (kg m− ) (kg m− ) (kg m− ) (kpc) (MSun) Messier Spirals 1068 5.00 1010 0.588 3.00 10 21 - 3.0 10 20 1.22 10 17 1.22 10 17 1.53 5.09 4.29 1010 × × − × − × − × − − × 3031 5.60 1010 0.364 4.50 10 22 - 9.3 10 24 6.70 10 18 9.60 10 18 2.0 23 7.00 1010 × × − × − × − × − − × 4192 9.00 1010 0.228 1.20 10 21 ? 6.0 10 22 1.66 10 17 2.70 10 17 1.74 17.7 8.65 1010 × × − × − × − × − − × 4254 >2x1010 <1 ~5x10 21 - ~2x10 21 3.00 10 18 3.00 10 17 1.4 1.9 4.50 109 − − × − × − − × 4258 1.10 1011 0.170 2.50 10 22 24 6.3 10 23 2.38 10 17 6.00 10 16 2.09 33.4 1.33 1011 × × − × − × − × − − × 4303 3.68 1010 0.497 6.81 10 22 - 5.0 10 22 2.44 10 17 5.00 10 17 1.96 18.6 5.00 1010 × × − × − × − × − − × 4321 1.68 1011 0.265 8.80 10 22 22 1.7 10 23 3.57 10 17 6.70 10 17 2.00 23.3 1.80 1011 × × − × − × − × − − × 4501 2.00 1011 0.325 4.80 10 21 - 6.8 10 21 1.36 10 17 3.60 10 17 1.46 8.32 9.54 1010 × × − × − × − × − − × 4548 3.80 1010 0.732 3.20 10 22 - 1.5 10 21 2.40 10 17 3.50 10 17 2.57 8.39 3.33 1010 × × − × − × − × − − × 4569 9.00 1010 0.214 2.20 10 21 - 3.7 10 21 3.78 10 17 6.39 10 17 1.55 11.8 8.27 1010 × × − × − × − × − − × 4736 1.77 1010 0.556 1.40 10 21 6.3 1.7 10 22 2.90 10 17 2.90 10 17 2.12 6.55 2.14 1010 × × − × − × − × − − × 5055 7.07 1010 0.245 5.21 10 22 16.5 2.4 10 23 1.25 10 17 2.63 10 17 2.19 40 1.36 1011 × × − × − × − × − − × 5194 4.00 1010 0.535 1.00 10 22 6.5 1.0 10 22 3.00 10 17 7.00 10 17 2.1 6.51 4.32 1010 × × − × − × − × − − × 5236 4.79 1010 0.668 1.66 10 22 19 1.0 10 23 2.85 10 17 4.48 10 16 2.42 19.90 4.83 1010 × × − × − × − × − − × 5457 5.26 1010 0.422 2.50 10 22 9 1.7 10 23 1.00 10 17 1.50 10 17 2.10 12.90 5.29 1010 × × − × − × − × − − × Dwarf Irregular and Spheroidal Galaxies WLM 8.63 107 0.676 9.20 10 23 2 2.3 10 23 - 6.60 10 22 1.9 2 2.0 1.08 108 × × − × − × − − × M81dWb 7.00 107 1.02 5.00 10 22 0.8 4.5 10 22 1.87 10 19 1.90 10 19 2.2 2 0.92 7.31 107 × × − × − × − × − − × Holmberg II 4.28 108 1.10 3.64 10 22 7 1.6 10 23 1.00 10 19 2.20 10 20 1.52 10.2 1.02 109 × × − × − × − × − − × Carina 3.38 106 0.087 6.50 10 22 - 1.4 10 22 8.70 10 21 3.00 10 20 1.59 2 0.88 6.88 106 × × − × − × − × − − × Draco 3.40 106 0.087 7.40 10 22 0.5 1.3 10 22 3.00 10 20 3.10 10 20 1.58 1.78 4.10 107 × × − × − × − × − − × Fornax 1.24 107 1.047 3.73 10 23 1.4 2.0 10 23 1.10 10 20 2.00 10 20 1.56 2 1.62 1.37 107 × × − × − × − × − − × Leo I 7.74 106 0.647 1.36 10 21 0.8 1.8 10 22 4.40 10 21 2.00 10 20 0.85 0.95 1.26 107 × × − × − × − × − − × Galaxies 2020, 8, 19 13 of 33

Table 5. Cont.

Visible Edge Properties

Min Lvis/Min Density rturndown Lowest ρ ρ at 0.1 kpc Highest ρ 1 rmax Mmax Name 3 3 3 3 Power (MSun) (LSun/MSun) (kg m− ) (kpc) (kg m− ) (kg m− ) (kg m− ) (kpc) (MSun) Lenticular and Elliptical Galaxies UGC 3993 1.78 1011 0.068 3.10 10 22 18? 4.0 10 23 1.00 10 17 8.00 10 19 2.07 54.6 5.40 1011 × × − × − × − × − − × 7286 5.90 109 0.415 4.60 10 22 ? 6.0 10 23 6.00 10 19 7.00 10 20 1.71 15.7 1.77 1010 × × − × − × − × − − × 2768 1.98 1010 1.41 1.00 10 21 10 4.0 10 22 2.75 10 20 2.75 10 20 0.61 21.8 1.96 1010 × × − × − × − × − − × 3379 1.10 1010 1.92 9.00 10 23 - 9.0 10 23 3.64 10 18 1.10 10 17 2.38 5.38 1.32 1010 × × − × − × − × − − × 2434 5.00 1010 0.476 1.00 10 22 - 5.0 10 22 2.95 10 18 2.95 10 18 2.02 8.22 4.54 1010 × × − × − × − × − − × 221 2.33 107 11.20 1.00 10 22 - 8.0 10 22 6.10 10 20 4.20 10 15 2.8 0.75 2.30 107 × × − × − × − × − − × 4431 3.00 109 0.33 1.30 10 21 2.6 1.3 10 21 3.00 10 20 3.00 10 20 0.8 2 3.68 2.88 109 × × − × − × − × − − × Polar Ring or Disk Galaxies 4650A 1.90 1010 0.51 1.0 10 23 18 1.0 10 22 1.28 10 19 5.00 10 19 0.87 13.0 1.60 1010 × × − × − × − × − − × 1 Power for inner or average regions; see ﬁgures. 2 Density was also ﬁt reasonably well by an exponential function. Galaxies 2020, 8, 19 14 of 33

Milky Way density (Figure 3b) was determined using extended RCs [ 58,59]. Calculated densities increaseMilky as Wayc/a decreases, density (Figure per Equation 3b) was (13).determined Density using follows extended a power RCs law[58,59]. over Calculated most of the densities Milky Way.increase Density as c/a falls decreases, off rapidly per Equation from 18 to (13). 30 kpc,Density consistent follows with a power the existence law over of most an edge, of the although Milky Way. the terminationDensity falls of off the ra galaxypidly from is not 18 particularly to 30 kpc, abrupt. consistent Near with the galacticthe existence center, of density an edge, depends although weakly the onterminationr (discussed of the further galaxy below). is not particularly abrupt. Near the galactic center, density depends weakly on r (discussedOur calculated further densities below). (Table 5) are consistent with independent measures (Table1). Near the MWOur galactic calculated center, densities density is (Table like that 5) are of molecularconsistent with cloud independent cores, whereas measures mid-galaxy (Table densities 1). Near arethe likeMW those galactic of typicalcenter, moleculardensity is clouds.like that In of the molecular Solar Neighborhood, cloud cores, whereas our results mid match-galaxy the densities sum of theare distributedlike those of star typical density molecular plus the clouds. ISM density In the Solar (cf. Tables Neighbo1 andrhood,5). Further our results out, the match density the sum grades of tothe lowerdistributed values. star Out density to 2000 plus kpc, theρ ISMremains density significantly (cf. Tables higher 1, 5). Further than that out, of thethe cosmologicallydensity grades to inferred lower densityvalues. (Table Out to1 ), 2000 which kpc, is consistentρ remains with significantly the Milky higher Way being than part that of of a the local cosmologically group of galaxies inferred that constitutesdensity (Tabl a concentratione 1), which is consistent of matter inwith the the universe Milky Way being part of a local group of galaxies that constitutesThe average a concentration density in of the matter innermost in the region universe of the MW varies more gradually than in the outer zonesThe (cf. average Figure4 ).density A power-law in the innermost fit presumes region a singularity of the MW at rvaries= 0, but more the gradually actual trend than at thein the smallest outer valueszones (cf. of Figurer is flatter. 4). A Thepower real-law trend fit presumes is neither a fit singularity by an exponential at r = 0, but nor the anyactual other trend simple at the function. smallest Therefore,values of rtwo is flatter. extrapolations The real weretrend madeis neither to estimate fit by an conditions exponential near nor the any center. other Extrapolating simple function. the massTherefore, trend suggeststwo extrapolations 1 star of Solar were mass made per to sphere estimate of 1 a.u.conditions radius. near Alternatively, the center. assuming Extrapolating a constant the densitymass trend of 100 suggests times the1 star highest of Solar density mass definedper sphere by of the 1 dataa.u. radius. provides Alternatively, 1 Solar mass assuming per 200 a.u. a constant radius. Averagingdensity of 100 these times two the estimates highest provides density adefined density by for the the data galactic provides center 1 similarSolar mass to that per of 200 the a.u. Dispersed radius. SolarAveraging System th (Tableese two1). estimates provides a density for the galactic center similar to that of the Dispersed Solar System (Table 1).

Figure 4. Expanded view of the Milky Way center.center. Mass and density are shown on left and right axes, respectively.respectively. TheThe ﬁtfit toto ρρ isis for forr r< < 0.20.2 kpc.kpc. CentralCentral massmass isis estimatedestimated byby extrapolatingextrapolating the massmass curvecurve (dotted(dotted line)line) andand byby assumingassuming aa highhigh densitydensity (dashed(dashed line).line). SeeSee FigureFigure3 3 for for data data sources. sources. 3.2. Well-Studied and/or Illustrative Spiral Galaxies 3.2. Well-Studied and/or Illustrative Spiral Galaxies Compiled velocity for Andromeda [59] varies with r in an “oscillatory” pattern, similar to data for Compiled velocity for Andromeda [59] varies with r in an “oscillatory” pattern, similar to data the Milky Way. Likewise, mass and density (Figure3 from [ 27]; ) resemble values for the Milky Way for the Milky Way. Likewise, mass and density (Figure 3 from [27]; Appendix Figure A1) resemble (Table5), but better conform to a power law. An edge was not obvious in either mass or density, even values for the Milky Way (Table 5), but better conform to a power law. An edge was not obvious in though the RC extends far beyond the visible edge at 23 kpc. either mass or density, even though the RC extends far beyond the visible edge at 23 kpc. Three RC datasets for Triangulum are available [14,65,66,70] and all agree (Figure5), providing Three RC datasets for Triangulum are available [14,65,66,70] and all agree (Figure 5), providing similar values for Min. We used the smoother data sets to calculate densities. RCs barely cross the similar values for Min. We used the smoother data sets to calculate densities. RCs barely cross the visible edge at 9.5 kpc, and thus do not deﬁne the density of its outer reaches. This galaxy is only half visible edge at 9.5 kpc, and thus do not define the density of its outer reaches. This galaxy is only half the size of MW and Andromeda yet has similar density from r = 1 to 10 kpc. the size of MW and Andromeda yet has similar density from r = 1 to 10 kpc.

Galaxies 2020, 8, x; doi: FOR PEER REVIEW www.mdpi.com/journal/galaxies Galaxies 2020, 8, 19 15 of 33

FigureFigure 5. 5.Comparison Comparison ofof threethree datasets on on Triangulum Triangulum to tocalculate calculate mass mass (a) and (a) density and density (b) vs. ( bradius.) vs. radius. RCsRC froms from Kam Kam et et al. al. [14 [14]] (grey (grey crosses) crosses) were were used used toto calculatecalculate massmass (grey diamonds). diamonds). RC RCss of of Sofue Sofue [61] (black[61] dashed (black dashed line) were line) used were to used calculate to calculate mass mass (heavy (heavy black black dotted dotted line) line) and and density density (black (black dotted line).dotted RCs line). of Corbelli RCs of Corbelli and Salucci and Salucci [70] (black[70] (black points points with with error error bars) bars) were were used to to calculate calculate mass mass for for c/a = 1 (solid line) and c/a = 0.1 (medium dashed line) and density for c/a = 0.1 (circles and solid c/a = 1 (solid line) and c/a = 0.1 (medium dashed line) and density for c/a = 0.1 (circles and solid line). line). Agreement is good and the masses extracted are similar. The fit to ρ is over all r of [70]. An Agreement is good and the masses extracted are similar. The fit to ρ is over all r of [70]. An abrupt drop abrupt drop off in density is not observed. RC data are limited to the visible disk. off in density is not observed. RC data are limited to the visible disk. The next few examples further probe how details in RCs affect calculations of mass and density. The next few examples further probe how details in RCs affect calculations of mass and density. In the remaining figures, both mass and velocity are plotted against the left axis, which is linear, by In the remaining figures,6 9 both mass and velocity are plotted against the left axis, which is linear, by scaling Min by 10 to 10 MSun, as appropriate. Density is depicted by the logarithmic right axis, which 6 9 scalingfacilitates Min by comparison 10 to 10 withMSun independent, as appropriate. measures Density (Table is 1) depicted and allows by us the to extrapolate logarithmic to right the central axis, which facilitatesregions comparisonwhere velocity with data independent are not available. measures (Table1) and allows us to extrapolate to the central regionsSombrero where velocity (Figure data 6) has are an not unusual available. shape. Jardel et al. [71] and Kormendy and Westphal [72] bothSombrero indicate (Figureinternal6 structure) has an near unusual 0.5 kpc, shape. but neither Jardel RC et al. extends [ 71] andbeyond Kormendy the visual and edge Westphal at 14.4 [72] bothkpc. indicate We use internal the more structure extended near data 0.5 set kpc, for butdensity neither calculations. RC extends A decrease beyond in the density visual exists edge at at 0.4 14.4 kpc. Wekpc, use but the given more the extended uncertainties, data set ρ follows for density a power calculations. law, despite A decreasethe zig-zag in pattern density for exists v(r). at 0.4 kpc, but given the uncertainties, ρ follows a power law, despite the zig-zag pattern for v(r).

9 9 FigureFigure 6. Inverse6. Inverse models models ofof interesting spirals. spirals. Velocity Velocity and and Min/(10MinM/(10Sun) MareSun both) are plotted both on plotted the left on the leftaxis. axis. (left (left) )Sombrero Sombrero galaxy, galaxy, showing showing that that disparities disparities between between RC measurements RC measurements have little have effect. little effect. Heavy solid line = RC of [71] was used to calculate Min (light solid) and ρ (open diamonds). Heavy Heavy solid line = RC of [71] was used to calculate Min (light solid) and ρ (open diamonds). Heavy brokenbroken lines lines= RC = RC of [of72 [72]] used used to calculateto calculate mass mass (light (light broken broken line) line) and and density density (dots (dots and and fit). fit). The The visual visual edge is beyond the measurements; (right) Counter-rotating NGC-4826. RCs from deBlok et al. edge is beyond the measurements; (right) Counter-rotating NGC-4826. RCs from deBlok et al. [54]. [54]. Results for mass and density are close to similar sized normally rotating NGC 7793 (Figure A3). GalaxiesResults 2020 for, 8, x; mass doi: FOR and PEER density REVIEW are close to similar sized normally rotatingwww.mdpi.com/journal/ NGC 7793 (Figure A3galaxies). We We assumed that v = 0 at 4 kpc, midway between the nearest counter-rotating data points. The breadth assumed that v = 0 at 4 kpc, midway between the nearest counter-rotating data points. The breadth of of the v = 0 region affects the mass calculation, but little alters the density profile. the v = 0 region affects the mass calculation, but little alters the density profile. NGC 2599 (Figure A2) has very high velocities near the center, but does not show the expected flat trend observed for many large spirals. Instead velocities decline further out. Nonetheless, the density profile is remarkably similar to that of Andromeda (Figure A1) for which the RC is oscillatory but is overall fairly flat. Several other large galaxies have visual edge densities close to that of the ISM, whereas the calculated ρ near galactic centers depends on the specific velocity profile. For example, studies of NGC 253 (Figure A2) have RCs that significantly differ at low r [66,69]. The calculated mass and density are affected at low r, but beyond a few kpc the differences are small. Hence, the total mass of the galaxy is indicated by the magnitude of v at high r. The fits differ in detail, due to the contrasts in v at low r, but the differences are not huge. The large spiral NGC 925 has low velocity (Figure A2) which indicates low density at its center. Moderate to strong differences in velocity for irregular NCG 3034 (Figure A3) give similar calculated masses. The differences in v are more apparent in the densities; however, the effect is not large. Therefore, when asymmetries exist, averaging for more symmetric rotation curves should provide well-constrained mass and density. Large UGC 2855 has an extended RC [62,66]. A drop off in density exists at high r (Figure A2). The power-law fit is good over an extended region, but density is flatter at small r and steeper at large r. Densities are similar to those of the Milky Way and Andromeda. Moderately large NGC 7793 (Figure A3) has lower central density (Table 5) than the aforementioned large galaxies. The pattern resembles that of the similar sized Evil Eye (Figure 6). Counter-rotation does not strongly affect the calculated density. Because the Virial theorem involves energies, the direction of rotation is not important to our calculations.

3.3. “Representative” Galaxies Large galaxies (visual edge >15 kpc) among the set examined by Bottema and Pestaña [64] (Figure A4) have similar mass and density profiles. Significantly smaller galaxies have much lower central density (Figure A5) like Triangulum (Figure 5). Otherwise, the profiles are similar to those of very large to moderately sized spirals. Even smaller dwarf DDO-154 has very low ρ at the center (Figure 7a).

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NGC 2599 (Figure A2) has very high velocities near the center, but does not show the expected flat trend observed for many large spirals. Instead velocities decline further out. Nonetheless, the density profile is remarkably similar to that of Andromeda (Figure A1) for which the RC is oscillatory but is overall fairly flat. Several other large galaxies have visual edge densities close to that of the ISM, whereas the calculated ρ near galactic centers depends on the specific velocity profile. For example, studies of NGC 253 (Figure A2) have RCs that significantly differ at low r [66,69]. The calculated mass and density are affected at low r, but beyond a few kpc the differences are small. Hence, the total mass of the galaxy is indicated by the magnitude of v at high r. The fits differ in detail, due to the contrasts in v at low r, but the differences are not huge. The large spiral NGC 925 has low velocity (Figure A2) which indicates low density at its center. Moderate to strong differences in velocity for irregular NCG 3034 (Figure A3) give similar calculated masses. The differences in v are more apparent in the densities; however, the effect is not large. Therefore, when asymmetries exist, averaging for more symmetric rotation curves should provide well-constrained mass and density. Large UGC 2855 has an extended RC [62,66]. A drop off in density exists at high r (Figure A2). The power-law fit is good over an extended region, but density is flatter at small r and steeper at large r. Densities are similar to those of the Milky Way and Andromeda. Moderately large NGC 7793 (Figure A3) has lower central density (Table5) than the aforementioned large galaxies. The pattern resembles that of the similar sized Evil Eye (Figure6). Counter-rotation does not strongly affect the calculated density. Because the Virial theorem involves energies, the direction of rotation is not important to our calculations.

3.3. “Representative” Galaxies Large galaxies (visual edge >15 kpc) among the set examined by Bottema and Pestaña [64] (Figure A4) have similar mass and density profiles. Significantly smaller galaxies have much lower central density (Figure A5) like Triangulum (Figure5). Otherwise, the profiles are similar to those of very large to moderately sized spirals. Even smaller dwarf DDO-154 has very low ρ at the center

Galaxies(Figure 20720a)., 8, x FOR PEER REVIEW 2 of 33

Figure 7. Small galaxies: (a) Low surface brightness, dwarf irregular galaxy NGC 4789a. RCs from Figure 7. Small galaxies: (a) Low surface brightness, dwarf irregular galaxy NGC 4789a. RCs from Bottema and Pestaña [64]. Min ratioed to 106 6MSun Two fits to density are shown: solid = power law Bottema and Pestaña [64]. Min ratioed to 10 MSun Two ﬁts to density are shown: solid = power law and dots = exponential; (b) Compact dwarf elliptical M32. RCs from Howley et al. [55]. Min ratioed and dots = exponential; (b) Compact dwarf elliptical M32. RCs from Howley et al. [55]. Min ratioed to to 106 6 MSun. Density above 0.6 kpc is described by a power law, but not an exponential. 10 MSun. Density above 0.6 kpc is described by a power law, but not an exponential.

3.4.3.4. Messier Messier Galaxies, Galaxies, including including the the Virgo Virgo Cluster Cluster TheThe Messier Messier exa examplesmples are are strongly strongly skewed skewed to to large large galaxies. galaxies. Despite Despite the the variety variety of of RC RCss [53 [53,6,611],], calculatedcalculated mass mass and and density density are are quite quite similar similar ( (FigureFigure A A66; ;Table Table 5).).

3.5. Dwarf Galaxies Rotation curves of the dwarfs Holmberg II, WLM, and M81dwb [73,74,85] (Figure A7) provide trends similar to those of DDO 154 (Figure 7a). For two of the four dwarfs, density decreases roughly exponentially with radius. The main difference between the dwarf irregulars and the largest spirals is the centrally concentrated density of the latter. Dwarf spheroidals are smaller than irregulars and their even lower velocities [75–78] lead to lower mass and density (Figure A8). Central densities are very low (Table 5).

3.6. Lenticular Galaxies RC of lenticular galaxies (e.g., [61]) resemble those of spiral galaxies, yielding similar mass and density profiles (Figure A9). As observed for the spirals, central density is small for small galaxies. The disk of NGC 2769 has lower density than its bulge, which is consistent with depiction of the density structure in terms of shells (Figure 1b). This galaxy is nearly elliptical.

3.7. Elliptical Galaxies Elliptical galaxies have ambiguous orientations, precluding clear definition of their rotation curves, and so astronomers present their raw data as velocity dispersions. It must be recognized that uncertainties are large compared to spirals, and differential rotation between shells may involve different axial orientations. For the large ellipticals, the results are similar to spirals of like size (Figure A10). Smaller ellipticals [79] show more variety. High ρ calculated at the center of NGC 221 (Figure 7b) may or may not be typical and no other data exist for this compact type. Yet, measurements of v for NGC 221 are available for very small radii and do set a constraint on its interior (Table 5).

Galaxies 2020, 8, 19 17 of 33

3.5. Dwarf Galaxies Rotation curves of the dwarfs Holmberg II, WLM, and M81dwb [73,74,85] (Figure A7) provide trends similar to those of DDO 154 (Figure7a). For two of the four dwarfs, density decreases roughly exponentially with radius. The main diﬀerence between the dwarf irregulars and the largest spirals is the centrally concentrated density of the latter. Dwarf spheroidals are smaller than irregulars and their even lower velocities [75–78] lead to lower mass and density (Figure A8). Central densities are very low (Table5).

3.6. Lenticular Galaxies RC of lenticular galaxies (e.g., [61]) resemble those of spiral galaxies, yielding similar mass and density proﬁles (Figure A9). As observed for the spirals, central density is small for small galaxies. The disk of NGC 2769 has lower density than its bulge, which is consistent with depiction of the density structure in terms of shells (Figure1b). This galaxy is nearly elliptical.

3.7. Elliptical Galaxies Elliptical galaxies have ambiguous orientations, precluding clear definition of their rotation curves, and so astronomers present their raw data as velocity dispersions. It must be recognized that uncertainties are large compared to spirals, and differential rotation between shells may involve different axial orientations. For the large ellipticals, the results are similar to spirals of like size (Figure A10). Smaller ellipticals [79] show more variety. High ρ calculated at the center of NGC 221 (Figure7b) may or may not be typical and no other data exist for this compact type. Yet, measurements of v for NGC 221 are available for very small radii and do set a constraint on its interior (Table5).

3.8. A Polar Ring Galaxy Accurate RC data are available for only one polar ring galaxy, NGC4650a. Recent data of Iodice et al. [80] were analyzed (Figure8). At small r, RCs of receding and approaching limbs of both the central discoid and the orthogonal polar discoid are all identical within uncertainty. A smooth variation of velocity across the entire object is seen, showing a common control. Velocities differ at large r between the limbs, as has been observed in some other spirals. Because data from the receding polar limb are Galaxies 2020, 8, x FOR PEER REVIEW 3 of 33 much less certain, we do not use this in our extraction. However, the fit for all data is not much different 3 thanpolar that limb excluding are much the less receding certain, polar we do limb not use (Figure this 8ina). our The extraction. data are However, well described the fit byfor allv = dataAr is-B r and resemblenot much trends different for many than spirals that excluding (see Appendix the recedingA). polar limb (Figure 8a). The data are well described by v = Ar3-Br and resemble trends for many spirals (see Appendix A).

Figure 8. The prototype polar ring galaxy NGC 4650a. RC raw data from Iodice et al. [80] are similar to Figure 8. The prototype polar ring galaxy NGC 4650a. RC raw data from Iodice et al. [80] are similar previous reports. (a) Comparison of RC data from different sectors and fits; see text. (b) The fit that to previous reports. (a) Comparison of RC data from different sectors and fits; see text. (b) The fit that 8 excluded the uncertain dataset to provide Min/(108 MSun) and ρ. excluded the uncertain dataset to provide Min/(10 MSun) and ρ.

The consistent, gradual change in v implies gradual changes in Min and ρ with r. Trends in extracted mass and density (Figure 8b) are similar to other spirals but less dense, more like the small galaxies (Figure 7). Density for NGC 4650a over all r does not conform to an exponential function or a power law, but a definite outwards decline is seen, which is more pronounced towards the edge. Galactic dynamics of this polar ring galaxy differ little from that of spirals of similar size, other than the tilt of the inner discoid. Tilt angles are high, ~90 ° (NGCs 4650a and 4282) or ~73 ° (SPRC-7) [86].

4. Discussion Profiles of density and mass vs. radius can be uniquely extracted from RC. These profiles are similar for all the galaxies examined, despite the wide range in shapes, sizes, and types (Table 4) and the variety of RC patterns exhibited (Figures 3–8, Figures A1–A10). Density tends to fall off as a power law, whereas values of Min rise strongly with radius. If velocity data are available near the galactic center (e.g., the Milky Way), then the extractions show that density at low r depends more weakly on r than it does at high r. If velocity data are available at great distance, density at very high r sometimes steeply declines with radius, defining a physical “edge.” In some cases, the visual and physical edges coincide (cf. Tables 4,5). Scatter existing in the calculated densities partly results from uncertainties in velocities, which can be substantial as suggested in Figure 3 and by comparing results of different studies of the same object (Figures 4–6). Despite the uncertainties, clear and consistent trends of extracted density with r exist. Our calculations (Figures 3–8, Figures A1–A10) show that the rotation curves are flat where the density dependence on radius approximately follows r−1.8 (Table 5). The following subsections compare our extracted Min and ρ with several measures that are entirely independent of RC determinations.

4.1. Trends in Density with Galaxy Size and Morphology 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 (cf. Tables 1,5). 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 (Figure 9, Figure 10a) indicates that total mass correlates with star mass. For

Galaxies 2020, 8, 19 18 of 33

The consistent, gradual change in v implies gradual changes in Min and ρ with r. Trends in extracted mass and density (Figure8b) are similar to other spirals but less dense, more like the small galaxies (Figure7). Density for NGC 4650a over all r does not conform to an exponential function or a power law, but a deﬁnite outwards decline is seen, which is more pronounced towards the edge. Galactic dynamics of this polar ring galaxy diﬀer little from that of spirals of similar size, other than the tilt of the inner discoid. Tilt angles are high, ~90 ◦ (NGCs 4650a and 4282) or ~73 ◦ (SPRC-7) [86].

4. Discussion Profiles of density and mass vs. radius can be uniquely extracted from RC. These profiles are similar for all the galaxies examined, despite the wide range in shapes, sizes, and types (Table4) and the variety of RC patterns exhibited (Figures3–8, Figures A1–A10). Density tends to fall o ff as a power law, whereas values of Min rise strongly with radius. If velocity data are available near the galactic center (e.g., the Milky Way), then the extractions show that density at low r depends more weakly on r than it does at high r. If velocity data are available at great distance, density at very high r sometimes steeply declines with radius, defining a physical “edge.” In some cases, the visual and physical edges coincide (cf. Tables4 and5). Scatter existing in the calculated densities partly results from uncertainties in velocities, which can be substantial as suggested in Figure3 and by comparing results of di fferent studies of the same object (Figures4–6). Despite the uncertainties, clear and consistent trends of extracted density with r exist. Our calculations (Figures3–8, Figures A1–A10) show that the rotation curves are flat where the 1.8 density dependence on radius approximately follows r− (Table5). The following subsections compare our extracted Min and ρ with several measures that are entirely independent of RC determinations.

4.1. Trends in Density with Galaxy Size and Morphology 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 (cf. Tables1 and5). 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 (Figure9, Figure 10a) indicates that total mass correlates with star mass. For spirals, density at the visual edge decreases with galaxy size (Figure9a), ostensibly because larger galaxies have greater attractive power, thus producing more gradational outer reaches. For non-spiral morphologies, the trend at the visual edge is rather flat (Figure9b). Density at the visual edge for different spiral morphologies (types SA, SAB and SB) follow similar trends (not shown) to those in Figure9a. Spiral type is not important to RC patterns [ 47] and thus not to extracted parameters. Rings or bars seem not to have a strong effect on density. However, tilt of the interior does have an effect: the polar-ring galaxy is less dense than others of its size. Essentially, this shape has more volume than the ordinary spirals, which are closer to being 2D objects than the obviously 3D ring type with mass well out of the equatorial plane. The increase in density in the interior (at 0.1 kpc) with galaxy size is consistent with gravitation. For some small galaxies, the density may depend exponentially on radius (Figures8b, A3, A5, A7, A8 and A9). However, exponential trends neither fit all small galaxies, nor any of our large galaxies (Figure A6). GalaxiesGalaxies 20 202020, ,8 8, ,x x FOR FOR PEER PEER REVIEW REVIEW 44 ofof 3333

spirals,spirals, densitydensity atat thethe visualvisual edgeedge decreasesdecreases withwith galaxygalaxy sizesize ((FigureFigure 99a),a), ostensiblyostensibly becausebecause largerlarger Galaxiesgalaxiesgalaxies2020 have, have8, 19 greater greater attractive attractive power, power, thus thus producing producing more more gradational gradational outer outer reaches. reaches. For For19 nonof non 33-- spiralspiral morphologies,morphologies, thethe tretrendnd atat thethe visualvisual edgeedge isis ratherrather flatflat ((FigureFigure 99b).b).

FigureFigureFigure 9. 9 9.Various .Various Various measures measures measures for for for the the the density density density of of of galaxies galaxies galaxies compared compared compared to to to their their their size, size, size, set set set to to to the the the visual visual visual edge edge edge at at at 25 B-magnitude arcsec−2−22. (a) Spiral and similar galaxies. M82 is included since near-IR images [23] 2525 B-magnitude B-magnitude arcsecarcsec− .(. (a) SpiralSpiral andand similarsimilar galaxies.galaxies. M82M82 isis includedincluded sincesince near-IRnear-IR imagesimages [23[23]] suggestsuggestsuggest it itit is isis a a spiral.a spiral.spiral. Highest HighestHighest and andand lowest lowestlowest densities densitiesdensities are areare depicted, depidepicted,cted, but butbut these thesethese values valuesvalues depend dependdepend somewhat somewhatsomewhat on smallestonon smallest smallest and largestandand largestlargest radii radii exploredradii explored explored in each inin RCeach each study. RC RC study. Forstudy. reference, For For reference, reference, we include we we include ρincludemeasured ρρ measuredmeasured at r = 0.1 at at kpc rr == and0.10.1 kpc ρkpcmeasured and and ρ ρ measured measured at the visual at at the the edge. visual visual The edge. edge. polar The The ring polar polar galaxy ring ring has galaxy galaxy lower has hasρ thanlower lower ordinary ρ ρ than than ordinary ordinary spirals of spirals spirals similar of of size.similarsimilar (b) size. Morphologiessize. ((bb)) MorphologiesMorphologies other than otherother spiral. thanthan Fits spiral.spiral. to ellipticals FitsFits toto ellipticals (lines)ellipticals roughly (lines)(lines) also roughlyroughly describe alsoalso the describedescribe lenticular thethe andlenticularlenticular spheroidal and and spheroidal classesspheroidal and classes areclasses similar and and are toare ﬁts similar similar for spirals. to to fits fits for for spirals. spirals.

FigureFigureFigure 10. 10. 10. Density DensityDensity trends trends inin spiralspiral galaxies:galaxies: ( (aa)) RadiusRadius of of of thethe the turndownturndown turndown in in in densitydensity density on on on the the the position position position of of of 0.1 kpc. Density at 0.1 kpc strongly increases with the size of both spiral and elliptical galaxies TableTable 0.0. kpc.kpc. DensityDensity atat 0.10.1 kpckpc stronglystrongly increasesincreases withwith thethe sizesize ofof bothboth spiralspiral andand ellipticalelliptical galaxiesgalaxies (Figure9a,b). The slopes of the regression lines for these types are similar, despite the disparity in the (Figure(Figure 9a 9a,b).,b). The The slopes slopes of of the the regression regression lines lines for for these these types types are are similar, similar, despite despite the the disparity disparity in in the the number of samples (36 spirals vs. 4 ellipticals), and disparities in central density between ellipticals numbernumber ofof samplessamples (36(36 spiralsspirals vsvs. . 44 ellipticals),ellipticals), andand disparitiesdisparities inin centralcentral densitydensity betweenbetween ellipticalsellipticals and spirals. The three lenticular and eight dwarf galaxies measured also fall on or near these trends. andand spirals.spirals. The The threethree lenticularlenticular and and eighteight dwarf dwarf galaxies galaxies measured measured also also fall fall on on or or near near these these trends. trends. The polar-ring type is less dense, due to its shape, with considerable matter out of the plane. Notably, TheThe polar polar--ringring type type is is less less dense, dense, due due to to its its shape, shape, with with considerable considerable matter matter out out of of the the plane. plane. Notably, Notably, the smallest galaxies, the dwarf spheroidals, have densities near the crossing of the trends and show thethe smallest smallest galaxies, galaxies, the the dwarf dwarf spheroidals, spheroidals, have have densities densities near near the the crossing crossing of of the the trends trends and and show show little variation in ρ with r. Interestingly, increasing tightness of the spiral arms is associated with higher littlelittle variationvariation inin ρρ withwith rr. . Interestingly,Interestingly, increasingincreasing tightnesstightness ofof thethe spiralspiral armsarms isis associatedassociated withwith ρ at 0.1 kpc (Figure 10b). This ﬁnding is consistent with spiral arms being concentrations of stars. higherhigher ρρ atat 0.10.1 kpckpc (Fi(Figuregure 10b).10b). ThisThis findingfinding isis consistentconsistent withwith spiralspiral armsarms beingbeing concentrationsconcentrations ofof 4.2. Comparisonstars.stars. of Extracted Density with Independently Known Densities

OurDensityDensity calculations at at the the visual visualof galactic edge edge mass for for different anddifferent density spiral spiral are morphologiessupported morphologies by consistent (types (types SA, SA, trends SAB SAB and and conﬁrmed SB) SB) follow follow bysimilarsimilar independent trendstrends (not(not measures shown)shown) of toto density thosethose inin (Tables FigureFigure1 9a.9a.and SpiralSpiral5). Near typetype the isis not centersnot importantimportant of most toto galaxies, RCRC patternspatterns density [4[477]] and isand likethusthus that not not of to to molecularextracted extracted parameters. cloudparameters. cores, Rings whereasRings or or bars atbars the seem seem middle not not to ofto have largehave a a galaxies, strong strong effect effect densities on on density. density. are like However, However, those of typicaltilttilt of of giantthe the interior interior molecular does does clouds have have (GMCs). an an effect: effect: For the the some polar polar large--ringring galaxies, galaxy galaxy inneris is less less densities dense dense than than exceed others others those of of of its its GMC size. size. cores. Yet, even in this inner zone the calculated densities are lower than ρ associated with evenly distributing the mass inside our Solar System (Table1). Galaxies 2020, 8, 19 20 of 33

At the radius of the Milky Way associated with our Sun, our calculated values for ρ match the sum of the Solar Neighborhood density, calculated from the mass and distance of proximal stars, plus the ISM density (Tables1 and5). Further out in the Milky Way, the density grades to lower values, but even out to ~1500 kpc, the radius of the local group, ρ remains significantly higher than cosmological values. Our finding is consistent with the known existence of globular clusters, gas, and satellite galaxies surrounding the Milky Way, and with a large amount of material surrounding Andromeda, constituting ~30% of total mass [87]. Furthermore, at 400 kpc from centers of each of the Milky Way and Andromeda we obtain the same value for ρ of ~1.5 10 24 kg m 3. This equivalence is consistent × − − with separation of their centers by 780 kpc. We next estimate IGM density by two approaches. The local group would have a density of 1 10 26 kg m 3, if its total galaxy mass (Table5) were uniformly redistributed out to a radius of 1.6 Mpc. × − − Alternatively, the IGM for the local group can be determined by extrapolating the trends in Figures3 and A1. RC data on the Milky Way projected to 1.6 Mpc provide an upper limit of 1 10 25 kg m 3. × − − These two estimates are compatible with independent estimates of the IGM (Table1). When plotted against galaxy size, the trends for ρ at 0.1 kpc and at the visual edge diverge, 21 3 suggesting an average density of ~10− kg m− for all galaxies. The trends for the lowest and highest densities calculated are scattered, but their tendency to diverge (Figure8a) also supports this “average” density. This crude average resembles that of the ISM and Solar Neighborhood (Table1). 14 3 The highest central density, 10− kg m− , was calculated for the Milky Way at 0.001 kpc and for a compact dwarf ellipsoid at 0.0003 kpc (Table5). Yet, this value is far lower than the distributed ρ of the Solar System (Table1). The trend for the central Milky Way (Figure4c) suggests that cores of large galaxies may reach 1 star per cubic a.u. Rotation curves begin too far from galactic centers to precisely constrain how central ρ depends on r. Exponential forms have been inferred from luminosity data (e.g., [88]). However, measured luminosity near the center is often more complicated than an exponential form and luminosity profiles generally end at r below where RCs are collected. The densities we calculate for some small galaxies may have an exponential dependence. Velocities at small r are needed to better quantify mass distribution near the center. Our calculations are consistent with ordinary matter being concentrated at galactic centers, but not inordinately so.

4.3. Dependence of Galactic Mass on Size The calculated mass within the visual edge increases with galaxy size raised to the ~2.7 power (Figure 11a). A power of 3 is compatible with constant ρ, or with any trend where a common central density declines to a consistent value at the visual edge. Our result is consistent with a gradual increase in the average density of increasingly large galaxies. The dependence of ρ on r is weaker inside 0.1 kpc than suggested by the power law fits extrapolated from values at larger r. Density may be nearly constant in the innermost zones of typical galaxies, where RCs cannot be determined. The total mass of a galaxy is larger than Min at the visual edge, because the latter is an arbitrary cutoff, albeit a consistent one. The maximum mass extracted depends linearly on the radius corresponding to this mass (Figure 11b). However, the maximum mass could not be defined when the RC data were terminated close to the visual edge. Due to uncertainties in v at large r, it is difficult to ascertain where the physical edge of a galaxy is located, if indeed such exists. Spiral galaxies grade into the gas of the IGM, whereas ellipticals and some dwarf galaxies have sharper edges (Figures3–8; Figures A1–A10). These different behaviors are attributed to spirals having a much higher proportion of gas and dust (e.g., [89]). Galaxies 2020, 8, 19 21 of 33 Galaxies 2020, 8, x FOR PEER REVIEW 6 of 33

FigureFigure 11. 11. GalaxyGalaxy mass–size mass–size relationships. relationships. ( a ()a Dependence) Dependence of galaxyof galaxy mass mass on the on position the position of the of visible the visibleedge. (edge.b) Dependence (b) Dependence of the of largest the largest mass calculatedmass calculated for each for galaxy each galaxy on the on radius the radius where where the largest the largestmass was mass calculated. was calculated. In most In cases, most this cases radius, this is radius the outer is the cuto outerﬀ of availablecutoff of available data, but somedata, exceptionsbut some exceptionsexist. Fits areexist. to Fits certain are types,to certain as listed types, in as the listed inset. in the inset.

4.4. Relationships of Extracted Parameters with Luminosity The total mass of a galaxy is larger than Min at the visual edge, because the latter is an arbitrary cutoff,Density albeit dependsa consistent on the one visible. The luminosity maximum in a mass manner extracted that is similar depends to the linearly dependence on the of radiusdensity corresponon size (cf.ding Figures to this9 and mass 12). (Figure Density 1 depends1b). However, on radius the tomaximum some power mass (Figures could 3not–8), be whereby defined density when theis more RC data concentrated were terminated to the centers close to of the larger visual galaxies, edge. whichDue to is uncertainties consistent with in v known at large concentration r, it is difficult of toluminosity ascertain where towards the the physical center. edge of a galaxy is located, if indeed such exists. Spiral galaxies grade into the gas of the IGM, whereas ellipticals and some dwarf galaxies have sharper edges (Figures 3– 8; A1–A10). These different behaviors are attributed to spirals having a much higher proportion of gas and dust (e.g., [89]).

4.4. Relationships of Extracted Parameters with Luminosity Density depends on the visible luminosity in a manner that is similar to the dependence of density on size (cf. Figures 9 and 12). Density depends on radius to some power (Figures 3–8), whereby density is more concentrated to the centers of larger galaxies, which is consistent with known concentration of luminosity towards the center.

FigureFigure 12. 12. DependenceDependence of of density density at at various various galactic galactic positions positions on on the the visible visible luminosity luminosity of of the the galaxy: galaxy: (a(a) )Data Data for for spirals spirals and and the the polar polar-ring-ring galaxy galaxy;; (b (b) Other) Other galaxy galaxy types, types, with with a awider wider range range of of luminosity. luminosity. TheThe dwarf dwarf spheroidals spheroidals show show a a flat ﬂat dependence dependence (dash (dash-dotted-dotted curve). curve). For For the the smallest smallest galaxies, galaxies, the the 21 3 averageaverage density density is is ~10 ~10−21− kgkg m− m3, −i.e.,, i.e., where where trends trends for for the the large large spirals spirals converge. converge.

Mass is proportional to the visible luminosity (Figure 13a). If we compare Min at the visual edge to measured Lvis, then Lvis/Medge is ~0.4 in Solar units for all types of galaxies, on average. This ratio is halved if the maximum mass is used (Figure 13a), which is consistent with most galaxy mass residing at high r and young stars being at distance, because these require gas to form. For consistency, the measured luminosity in these figures are those reported in the NED [23]. Luminosity only measures surface brightness. Therefore, galactic mass should lie above the M = L line in Figure 13a, as observed for almost all galaxies.

Figure 13. Dependence of calculated parameters on measured visible luminosity of our galaxy sample. Least squares fits are shown: (a) Mass of all galaxy types. The maximum mass is affected by the termination of velocity measurements with distance. The visual edge provides a more consistent measure of galaxy size in relationship to luminosity. Power law fits give exponents very close to unity: the linear fits shown are similar. Grey line indicates a 1:1 correspondence; (b) Power (−n) for fits to density vs. radius. The least squares fit for spirals (solid curve) shows that density is more concentrated in the centers of larger galaxies. The polar-ring galaxy is less concentrated than normal spirals of similar size.

The relationship of luminosity to mass is affected by several additional factors. The distribution of star types is important, because luminosity for main sequence stars is mstar4 (e.g., [90]). A second

Galaxies 2020, 8, x; doi: FOR PEER REVIEW www.mdpi.com/journal/galaxies

Figure 12. Dependence of density at various galactic positions on the visible luminosity of the galaxy: (a) Data for spirals and the polar-ring galaxy; (b) Other galaxy types, with a wider range of luminosity. Galaxies 2020, 8, 19 22 of 33 The dwarf spheroidals show a flat dependence (dash-dotted curve). For the smallest galaxies, the average density is ~10−21 kg m−3, i.e., where trends for the large spirals converge. Mass is proportional to the visible luminosity (Figure 13a). If we compare Min at the visual edge to measuredMass is Lproportionalvis, then Lvis/ toM edgethe visibleis ~0.4 inluminosity Solar units (Figure for all 1 types3a). If of we galaxies, compare on M average.in at the Thisvisual ratio edge is halvedto measured if the maximumLvis, then L massvis/Medge is usedis ~0.4 (Figure in Solar 13 a),units which for all is consistenttypes of galaxies, with most on galaxyaverage. mass This residing ratio is athalved high ifr andthe maximum young stars mass being is used at distance, (Figure 1 because3a), which these is consistent require gas with to form.most galaxy For consistency, mass residing the measuredat high r and luminosity young stars in these being ﬁgures at distance, are those because reported these in therequire NED gas [23 ].to Luminosityform. For consistency, only measures the surfacemeasured brightness. luminosit Therefore,y in these galacticfigures massare those should reported lie above in the the NEDM = L[23]line. Luminosity in Figure 13 onlya, as measures observed forsurface almost brightness. all galaxies. Therefore, galactic mass should lie above the M = L line in Figure 13a, as observed for almost all galaxies.

FigureFigure 13. 13.Dependence Dependence of of calculated calculated parameters parameters on measured on measured visible visible luminosity luminosity of our galaxy of our sample. galaxy Leastsample squares. Least squares fits are shown:fits are shown (a) Mass: (a) of Mass all galaxyof all galaxy types. types. The maximumThe maximum mass mass is aff isected affected by the by terminationthe termination of velocity of velocity measurements measurements with with distance. distance. The The visual visual edge edge provides provides a a moremore consistentconsistent measuremeasure ofof galaxygalaxy sizesize in relationship to luminosity. Power law fitsfits give exponents very close to unity: the linear fits shown are similar. Grey line indicates a 1:1 correspondence; (b) Power ( n) for fits to the linear fits shown are similar. Grey line indicates a 1:1 correspondence; (b) Power (−−n) for fits to densitydensity vs. vs. radius. radius. The The least least squares squares fit for fit spirals for (solidspirals curve) (solid shows curve) that shows density that is more density concentrated is more inconcentrated the centers in of the larger centers galaxies. of larger The galaxies. polar-ring The galaxy polar-ring is less galaxy concentrated is less concentrated than normal than spirals normal of similarspirals size.of similar size.

TheThe relationshiprelationship ofof luminosityluminosity toto massmass isis aaffectedffected byby severalseveral additionaladditional factors.factors. TheThe distributiondistribution 4 of star types is important, because luminosity for main sequence stars is mstar (e.g., [90]). A second of star types is important, because luminosity for main sequence stars is mstar4 (e.g., [90]). A second factor is the number of white dwarfs and other dense stars which contribute significantly to mass but negligiblyGalaxies 2020 to, 8, luminosity:x; doi: FOR PEER Ledrew’s REVIEW [32] independent assessment of the Solarwww.mdpi.com/journal/ Neighborhood providesgalaxies Lvis/M = 0.8, which reflects combined main sequence stars and white dwarfs. The third factor is presence of H atoms, gas, and dust, which will further reduce Lvis/M of a galaxy. The average Lvis/M = 0.4 determined here is compatible with the Solar Neighborhood value of [32] because the later compares star mass to gas mass when scattering is unimportant to the measurement, and conversely for the former. Note that overall, the elliptical, lenticular, and dwarf galaxies have higher Lvis/M than the spirals, which is consistent with the spirals having considerably more gas (e.g., [89]). The fourth factor is size (Figure 14). Size has an effect because bigger galaxies are denser, as shown above, and thus pack more stars into a smaller volume. With this in mind, M32 owes its relatively high luminosity to being compact. Galaxies 2020, 8, x FOR PEER REVIEW 2 of 33 factor is the number of white dwarfs and other dense stars which contribute significantly to mass but negligibly to luminosity: Ledrew’s [32] independent assessment of the Solar Neighborhood provides Lvis/M = 0.8, which reflects combined main sequence stars and white dwarfs. The third factor is presence of H atoms, gas, and dust, which will further reduce Lvis/M of a galaxy. The average Lvis/M = 0.4 determined here is compatible with the Solar Neighborhood value of [32] because the later compares star mass to gas mass when scattering is unimportant to the measurement, and conversely for the former. Note that overall, the elliptical, lenticular, and dwarf galaxies have higher Lvis/M than the spirals, which is consistent with the spirals having considerably more gas (e.g., [89]). The fourth factor is size (Figure 14). Size has an effect because bigger galaxies are denser, as shown above, and thusGalaxies pack2020 , more8, 19 stars into a smaller volume. With this in mind, M32 owes its relatively23 high of 33 luminosity to being compact.

Figure 14. Figure 14. DependenceDependence of of luminosity luminosity in in the the visible visible and and at at 21 21 cm cm wavelengths wavelengths on on galaxy galaxy size, size, as as determined from the visual edge at 25 B-magnitude arcsec 2. Data compiled in Table4. Fits exclude determined from the visual edge at 25 B-magnitude arcsec−2. Data compiled in Table 4. Fits exclude the compact ellipsoid, M32. the compact ellipsoid, M32. 4.5. Non-Baryonic or Non-Luminous Matter? 4.5. Non-Baryonic or Non-Luminous Matter? In addition to the above correlations, independent measurements of the visible luminosity of the 51In galaxiesaddition andto the of above their visualcorrelations, size from independent the NED measurements [23] are linearly of correlatedthe visible luminosity (Figure 14). of This the 51correlation galaxies neither and of depends their visual on the size angle from of the inclination NED [23] nor are on morphology,linearly correlated except ( thatFigure the 1 compact4). This correlationellipsoid (NGC neither 221) depends is far brighter on the forangle its of size inclination than the othernor on 50 morphology, galaxies. Likewise, except that luminosity the compact at the ellipsoidradio (21 (NGC cm) emission 221) is far line brighter of atomic for H its follows size than a parallel the other trend 50 with galaxies the visual. Likewise, radius, luminosity again excepting at the radioNGC (21 221 cm) (Table emission5; Figure line 14 of). Similaratomic H power follows laws a parallel suggest trend that the with star the mass visual and radius, the H massagain in excepting a galaxy NGCare correlated, 221 (Table which 5; Figure is consistent 14). Similar with power luminous laws stars suggest being that mostly the hydrogen.star mass and the H mass in a galaxyThe are solid correlated, body approximation which is consistent to spin with provides luminous mass stars at thebeing visible mostly edge hydrogen. that is consistent with luminosityThe solid [5,6 body]. Here, approximation we deduce densities to spin fromprovides detailed mass velocity at the datavisible without edge thisthat approximationis consistent with and luminosityfind that these [5,6] densities. Here, we are deduce consistent densities with independent from detailed measures velocity of data baryonic without matter. this Weapproximation have shown andthat find flat RCsthat arethese expected densities for are a rotating consistent galaxy with which independent becomes measures more dilute of withbaryonic extent. matter. The flatnessWe have of shownthe RC that occurs flat where RCs are the expected decrease infor density a rotati withng galaxyr offsets which the growth becomes in volumemore dilute with withr: a largeextent. halo The of flatnessdark matter of the is RC unnecessary. occurs where the decrease in density with r offsets the growth in volume with r: a large Thehalo inner of dark part matter of typical is unnecessary. rotation curves features an increase in v as r increases. A linear increase in velocityThe inner indicates part of that typical density rotation is nearly curves constant feature ass an r increases increase in thisv as innermostr increases. zone. A linear increase in velocityThe mid-point indicates ofthat the density Milky is Way nearly has constant ~1 gas mass as r increases to ~4 star in masses, this innermost determined zone. by comparing measuredThe mid densities-point of to the those Milky calculated Way has at 8~1 kpc. gas If mass this ratioto ~4 representsstar masses, the determined entire Milky by Way, comparing then its measuredL/M should densities be about to 2, those if the calculated surface brightness at 8 kpc. includesIf this ratio all emittedrepresents light, the which entire is Milky clearly Way, not thethen case. its LImportantly,/M should be gas about increases 2, if tothe the surface outside brightness of spirals, includes as does mass.all emitted Thus, light,L/M ofwhich the whole is clearly galaxy not must the case.be reduced Importantly, from that gas inferred increases from to the Solar outside Neighborhood of spirals, propertiesas does mass. and Thus, is compatible L/M of the with whole the averagegalaxy must be reduced from that inferred from Solar Neighborhood properties and is compatible with the L/M = 0.4 at the edge determined here. average L/M = 0.4 at the edge determined here. The connections found here are compatible with independent assessments of galactic structure by Disney et al. [91] who showed that one parameter governs galactic structure. This key parameter is baryonic mass, which is directly connected with galaxy size (Figure 11).

5. Conclusions Evaluation of galactic rotation curves is greatly simpliﬁed by the combined use of Newton’s homeoid theorems, Clausius’s Virial theorem, and Maclaurin’s gravitational self-potentials. These powerful theorems can be applied to galaxies which possess the expected spheroidal shape of a spinning self-gravitating body (Figure1b). Existing orbital models include errors in mathematical physics [6] and cannot explain the rotational curves for galaxies based on detectable matter. Galaxies 2020, 8, 19 24 of 33

Previous efforts are forward models of RC, which presume the cause, i.e., the density structure for each of multiple shapes is specified. Rotation curves are then calculated from the resulting multicomponent mass distributions (e.g., 12–14,41,54,55,64,68,70,74,86). In contrast, the present paper applies the inverse model of [27] for differentially spinning, variable density oblate spheroids to 51 different galaxies having detailed RC data. Our simple and exact equations for the homeoid allow density and mass distributions of individual galaxies to be directly and uniquely calculated from their measured RC and aspect ratio. Neither fitting parameters nor deconvolution are required. Groetsch [38] clarifies the differences between forward and inverse models and provides many examples of the latter. Calculated densities are almost independent of galaxy type. Calculated densities in the inner parts of galaxies approximate those in GMC cores, whereas those closer to the middle zones approximate those of average GMCs (cf. Tables1 and5). Densities calculated from RCs for the Milky Way near the Sun’s position are consistent with the independently determined distributed density of proximal stars. Slightly further out, extracted densities match ISM determinations, which is reasonable. In the outermost zones, density grades into, but is higher than, previous estimates of IGM values. We calculate 26 3 IGM density for the local group as >10− kg m− . Almost all galaxies are described by density depending on radius to a power n. The statistical average for the powers of the 36 individual spirals is n = 1.80 0.40. The 15 galaxies with other − ± morphologies have similar average of n = 1.63 0.63. Including the polar ring galaxy in the spiral − ± category negligibly affects these values. Importantly, our results explain the empirically determined dependence of luminosity on velocity for each of the spiral and elliptical types (the Tully–Fisher and Faber–Jackson relationships). Our extracted masses are consistent with measured luminosity, given the presence of significant amounts of gas and some scattering. Our luminosity to mass ratios are consistent with the independent assessment of the Solar Neighborhood by Ledrew [32]. Our results for ellipticals and spirals differ only in detail, despite velocity dispersions being relevant in the first case and actual rotation curves in the other. Without friction, homeoids can rotate independently, as exemplified by counter-rotating spirals (e.g., NGC 4826) and polar-ring galaxies (e.g., NGC 4650a). For the latter case, the outer, less-dense region is tilted with respect to the inner, dense region, yet the RC curve is continuous, consistent with self-gravitation. Unlike flat spirals which must have co-axially spinning homeoids, in ellipticals, the rotation axes of homeoids can point in various directions, producing velocity dispersions. We have shown that rotation curves of 51 galaxies covering diverse types and sizes can be explained by Newtonian physics without invoking non-baryonic matter.

Author Contributions: Both authors contributed more or less equally to all aspects and components of this research. All authors have read and agreed to the published version of the manuscript. Funding: This research was partially funded by the National Aeronautics and Space Administration, grant number NNX13AB32A—3971105. Acknowledgments: This research has made use of publicly available data: (1) the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration and (2) data from the CHANG-ES project (Continuum Halos in Nearby Galaxies) involving the Karl G. Jansky Very Large Array. Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Appendix A Analyses of individual galaxies are organized after Table4 and labeled by NCG number. Similar sized galaxies are grouped together. Except for Figure A1, velocity (dashed, sometimes with various symbols) and Min/(XMSun) (dotted curve) are both plotted on the left axis, where the value of X that was selected for scaling is indicated (e.g., 106 to 109). Density (various symbols) is plotted against the right axis. Power ﬁt to density is shown as a solid line. Exponential ﬁts are shown as widely GalaxiesGalaxies 20 202020, 8, ,8 x, xFOR FOR PEER PEER REVIEW REVIEW 4 4of of 33 33

ConflictsConflicts of of Interest: Interest: The The authors authors declare declare no no conflict conflict of of interest interest. .The The funders funders had had no no role role in in the the desi designgn of of the the study;study; in in the the collection, collection, analyses, analyses, or or interpretation interpretation of of data; data; in in the the writing writing of of the the manuscript, manuscript, or or in in the the decision decision to to publishpublish the the results results. .

AppendixAppendix A A AnalysesAnalyses of of individual individual galaxies galaxies are are organized organized after after Table Table 4 4 and and labeled labeled by by NCG NCG number. number. Similar Similar sizedsized gala galaxxiesies are are grouped grouped together. together. Except Except for for Figure Figure A1, A1, velocity velocity (dashed, (dashed, sometimes sometimes with with various various Galaxies 2020 8 , , 19 in Sun 25 of 33 symbols)symbols) and and M Min/(/(XXMMSun) )(dotted (dotted curve) curve) are are both both plotted plotted on on the the left left axis, axis, where where the the value value of of X X that that 6 9 waswas selected selected for for scaling scaling is is indicated indicated (e.g., (e.g., 10 10 6to to 10 10).9). Density Density ( various(various symbols) symbols) is is plotted plotted against against the the right axis. Power fit to density is shown as a solid line. Exponential fits are shown as widely spaced spacedright dotted axis. Power lines. Resultsfit to density of ﬁtting is shown are as listed. a solid For line. size Exponential comparisons, fits are we shown use the as widely visual spaced edge at 25 dotted lines. Results of fitting are listed. For size comparisons, we use the visual edge at 25 B- dotted lines. Results-2 of fitting are listed. For size comparisons, we use the visual edge at 25 B- B-magnitude arcsec -2 provided by the NED [23] and illustrate this with an arrow. magmagnitudenitude arcsec arcsec- 2provided provided by by the the NED NED [23] [23] and and illustrate illustrate this this with with an an arrow arrow. .

FigureFigureFigure A1. A Andromeda.A1.1. Andromeda Andromeda Inverse. .Inverse Inverse modelsmodels models showing showing the the effect effect eﬀect of of ofvarying varying varying c /ca/ acfor /fora foran an externally an externally externally probed probed probed galaxy,galaxy,galaxy, with with with the the mostthe most most extensive extensive extensive RCs RC RCs available. savailable. available. Compiled Compiled RC RC RCss sof of ofSofue Sofue Sofue [5 [59 [9]59 ]are are] areused used used to to calculate calculate to calculate:: :( a(a) ) (a) Min and; (b) density vs. radius. Labels on curves list the aspect ratios considered. Min and;Min and (b); density(b) density vs. vs. radius. radius. Labels Labels on on curvescurves list the the aspect aspect ratio ratioss considered considered..

Galaxies 2020, 8, x FOR PEER REVIEW 5 of 33

FigureFigure A2. LargeA2. Large spirals: spiral NGCs: NGC 253 253 RCs RCs from from SofueSofue et al. al. [6 [666] ]and and derived derived M Mandand ρ areρ arecolored colored grey. grey. RCs fromRCs from Hlavacek-Larrondo Hlavacek-Larrondo et et al. al. [69 [6]9 are] are black black curves.curves. Although Although velocity velocity data data differ diff substantially,er substantially, similarsimilar mass mass and and density density are are obtained obtained near near the the visiblevisible edge. Differences Differences at atsmall small r affectr aff ectthe thefits to fits ρ;to ρ; NCGNCG 925 has925 lowhas low velocities velocities which which produce produce low low density;density; NGC NGC 2599 2599 RC RCss from from Noordemeer Noordemeer et al. et [6 al.1]. [61]. This spiralThis spiral has both has highboth v highand v a and strong a strong decline decline in v with in v radius.with radius. The increase The increase in v at in low v atr lowwas r assumed; was assumed; UCG 2855 RCs below 13.7 kpc from Sofue et al. [66], and RCs above 13.7 kpc from Roelfsema UCG 2855 RCs below 13.7 kpc from Sofue et al. [66], and RCs above 13.7 kpc from Roelfsema and and Allen [62]. The merge produces unrealistically high density over a short interval but does not Allen [62]. The merge produces unrealistically high density over a short interval but does not seem seem to affect the fit. A drop off in density exists at high r, whereas the trend is flatter than the fit near to affect the fit. A drop off in density exists at high r, whereas the trend is flatter than the fit near the the galaxy center. galaxy center.

Figure A3. Comparison of moderate-sized galaxies: NGC 3034 is the irregular M82 galaxy. RCs from Greco et al. [63]. An abrupt drop off in density is not observed, because RC data are limited to the visible galaxy. NGC 7793 RCs from deBlok et al. [54] have fairly low velocity, and lower density.

GalaxiesGalaxies 20202020,, 88,, xx FORFOR PEERPEER REVIEWREVIEW 55 ofof 3333

FigureFigure A2.A2. LargeLarge spiraspirallss:: NGCNGC 253253 RCRCss fromfrom SofueSofue etet al.al. [6[666]] andand derivedderived MM andand ρρ areare coloredcolored grey.grey. RCRCss fromfrom HlavacekHlavacek--LarrondoLarrondo etet al.al. [6[699]] areare blackblack curvescurves.. AlthoughAlthough velocityvelocity datadata differdiffer substantially,substantially, similarsimilar massmass andand densitydensity areare obtainedobtained nearnear thethe visiblevisible edge.edge. DifferencesDifferences atat smallsmall rr affectaffect thethe fitsfits toto ρρ;; NCGNCG 925925 hashas lowlow velocitiesvelocities whichwhich produceproduce lowlow densitydensity;; NGCNGC 25992599 RCRCss fromfrom NoordemeerNoordemeer etet al.al. [[6611]].. ThisThis spiralspiral has has both both high high vv andand a a strong strong decline decline in in vv withwith radius. radius. The The increaseincrease in in vv atat low low rr waswas assumedassumed;; UCGUCG 28552855 RCRCss belowbelow 13.713.7 kpckpc fromfrom SofueSofue etet al.al. [6[666],], andand RCRCss aboveabove 13.713.7 kpckpc fromfrom RoelfsemaRoelfsema andand AllenAllen [62[62]].. TheThe mergemerge producesproduces unrealisticallyunrealistically highhigh densitydensity overover aa shortshort intervalinterval butbut doesdoes notnot Galaxiesseemseem2020 toto, 8 affect,affect 19 thethe fit.fit. AA dropdrop offoff inin densitydensity existsexists atat highhigh rr,, whereaswhereas thethe trendtrend isis flatterflatter thanthan thethe fitfit nearnear26 of 33 thethe galaxygalaxy centercenter..

FigureFigure A3. A3. ComparisonComparison ofof moderatemoderate moderate-sized--sizedsized galaxiesgalaxies galaxies::: NGCNGC 30343034 isis thethe irregularirregular M82M82 galaxygalaxy galaxy... RCRC RCsss from from GrecoGreco et et al. al. [63 [[6363]].].. An An abruptabrupt drop drop offoff oﬀ inin density density is is not not observed observed,observed,, because because RCRC data data are are limited limited to to the the visiblevisible galaxy. galaxy. NGC NGC 7793 7793 RC RCsRCss from from deBlokdeBlok etet al.al. [[55454]4]] havehahaveve fairlyfairly lowlow velocity,velocity, and andand lower lowerlower density. density.density.

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FigureFigureFigure A4. A4. A4. High High-velocityHigh-velocity-velocity large large large spirals spirals spirals from from from the the the compil compilation compilationation of of of Bottema Bottema and and Pestaña Pestaña [[64 64 [64].]. Inner Inner Inner densitiesdensitiesdensities for forfor similar similarsimilar visible visiblevisible edges edgesedges depend dependdepend on on on velocities velocities. velocities. .

FigureFigureFigure A A5.A5.5. ThreeThree Three small smallsmall spirals spirals:spirals: RC: RCRC data datadata from fromfrom Bottema BottemaBottema and andand Pestaña PestañaPestaña [64 [[6464]]..] .

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Figure A4. High-velocity large spirals from the compilation of Bottema and Pestaña [64]. Inner densities for similar visible edges depend on velocities.

Galaxies 2020, 8, 19 27 of 33 Figure A5. Three small spirals: RC data from Bottema and Pestaña [64].

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Figure A6. A6. Spirals in Messier’s CatCatalogalog studied by Sofue et al. al. [ [5353,,6666].]. These large and and luminous luminous galaxies have a variety of RC patterns.patterns. TabularTabular datadata werewere downloadeddownloaded fromfrom [[6565].].

Figure A7. Irregular galaxies: For Holmberg II, squares and dashed line = RCs from Oh et al. [74]. Grey diamonds = RCs from Bureau and Carnigan [85], who state their velocities are not accurate beyond 10 kpc, and so this dataset was not used in the calculations. Dotted line = calculated mass. Circles = calculated density. Two different fits are shown as solid line and widely spaced dotted curve; WLM is unusual due to its isolation RCs from Leaman et al. [73]; M81dWb RCs from Oh et al. [74].

GalaxiesGalaxies 20 202020, 8, ,8 x, xFOR FOR PEER PEER REVIEW REVIEW 7 7of of 33 33

GalaxiesFigureFigure2020 A6., A6.8, 19Spirals Spirals in in Messier’s Messier’s Cat Catalogalog studied studied by by Sofue Sofue et et al. al. [53 [53,6,66].6 ]. These These large large and and luminous luminous 28 of 33 galaxiesgalaxies have have a avariety variety of of RC RC patterns patterns. Tabul. Tabularar data data were were downloaded downloaded from from [65 [65]. ].

Figure A7. Irregular galaxies: For Holmberg II, squares and dashed line = RCs from Oh et al. [74]. FigureFigure A7. A7. IrregularIrregular galaxies:galaxies: ForFor HolmbergHolmberg II, II, squares squares and and dashed dashed line line= RCs = RC froms from Oh etOh al. et [74 al.]. [ Grey74]. Grey diamonds = RCs from Bureau and Carnigan [85], who state their velocities are not accurate Greydiamonds diamonds= RCs = fromRCs from Bureau Bureau and Carnigan and Carnigan [85], who [85], state who their state velocities their velocities are not are accurate not accurate beyond beyondbeyond10 kpc, 10 and10 kpc, kpc, so and thisand so dataset so this this dataset wasdataset not was usedwas not not in used theused calculations. in in the the calculations. calculations. Dotted lineDotted Dotted= calculated line line = =calculate calculate mass.d Circlesd mass. mass. = CirclesCirclescalculated = =calculated calculated density. density. density. Two di ﬀTwo Twoerent different different ﬁts are fits shown fits are are shown as shown solid as lineas solid solid and line line widely and and widely spacedwidely spaced dottedspaced dotted curve; dotted curve WLM curve; is; WLMWLMunusual is is unusual unusual due to due its due isolation to to its its isolation isolation RCs from RC RCs Leaman sfrom from Leaman Leaman et al. [73 et et]; al. M81dWbal. [ 73[73]; ];M81dWb M81dWb RCs from RC RCs Oh sfrom from et al. Oh Oh [74 et ].et al. al. [ 74[74]. ].

Galaxies 2020, 8, x FOR PEER REVIEW 8 of 33

Figure A8. Dwarf spheroidal galaxies. RCs presented by Salucci et al. [75] were digitized. Original data from Walker et al. [76.77] and Mateo et al. [78]. Fornax is larger, more like the dwarf irregulars in Figure A7. For Draco, we compare extractions from the published data with those from the same data, but smoothed, and find little effect on the results. For Leo I, we compare extractions from the published data with those for an interpolated data set. Interpolating at r below the lowest measurement is equivocal and the choice of v affects interior density.

Figure A8. Dwarf spheroidal galaxies. RCs presented by Salucci et al. [75] were digitized. Original Figure A8. Dwarf spheroidal galaxies. RCs presented by Salucci et al. [75] were digitized. Original data from Walker et al. [76.77] and Mateo et al. [78]. Fornax is larger, more like the dwarf irregulars data from Walker et al. [76.77] and Mateo et al. [78]. Fornax is larger, more like the dwarf irregulars in Figure A7. For Draco, we compare extractions from the published data with those from the same in Figure A7. For Draco, we compare extractions from the published data with those from the same data, but smoothed, and find little effect on the results. For Leo I, we compare extractions from the data, but smoothed, and find little effect on the results. For Leo I, we compare extractions from the published data with those for an interpolated data set. Interpolating at r below the lowest Galaxiespublished2020, 8, 19 data with those for an interpolated data set. Interpolating at r below the lowest29 of 33 measurement is equivocal and the choice of v affects interior density. measurement is equivocal and the choice of v affects interior density.

Figure A9. Lenticular galaxies. RCs of UGC 3993 and NGC 7786 from Noordemeer et al. [61]. RCs of FigureFigure A9.A9. LenticularLenticular galaxies.galaxies. RCsRCs ofof UGCUGC 39933993 andand NGCNGC 77867786 fromfrom NoordemeerNoordemeer etet al.al. [[6161].]. RCsRCs ofof NGC 2768 from Forbes et al. [42]. NGCNGC 27682768 fromfrom ForbesForbes etet al.al. [[4242].].

FigureFigure A10.A10. EllipticalElliptical galaxies.galaxies. RCsRCs of nearly spherical NGC 2434 from Rix etet al.al. [[6688].]. RCsRCs ofof dwarfdwarf Figure A10. Elliptical galaxies. RCs of nearly spherical NGC 2434 from Rix et al. [68]. RCs of dwarf NGCNGC 44314431 fromfrom TolubaToluba etet al.al. [[7799].]. RCsRCs ofof NCGNCG 33793379 fromfrom RomanowskyRomanowsky etet al.al. [[31].31]. NGC 4431 from Toluba et al. [79]. RCs of NCG 3379 from Romanowsky et al. [31]. ReferencesReferences References 1.1. Kellogg, O.D.O.D. FoundationsFoundations ofof PotentialPotential TheoryTheory;; DoverDover Publications:Publications: NewNew York,York, NY,NY, USA,USA, 1953.1953. 1. Kellogg, O.D. Foundations of Potential Theory; Dover Publications: New York, NY, USA, 1953. 2.2. Schmidt, M.M. AA modelmodel ofof thethe distributiondistribution ofof massmass inin thethe galacticgalactic system.system. Bul.Bul. Astron. Inst. Neth Neth.. 19561956,, 1313,, 15. 15. 2. Schmidt, M. 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