Galactic Rotation Described with Bulge+Disk Gravitational Models
aa r X i v : . [ a s t r o - ph ] A p r Galactic Rotation Described with Bulge+Disk Gravitational Models
C. F. Gallo and James Q. FengSuperconix Inc, 2440 Lisbon Ave, Lake Elmo MN 55042October 29, 2018
Abstract
Observations reveal that mature spiral galaxies consist of stars, gases and plasma approximatelydistributed in a thin disk of circular shape, usually with a central bulge. The rotation velocitiesquickly increase from the galactic center and then achieve a constant velocity from the core to theperiphery. The basic dynamic behavior of a mature spiral galaxy, such as the Milky Way, is welldescribed by simple models balancing Newtonian gravitational forces against the centrifugal forcesassociated with a rotating thin axisymmetric disk. In this research, we investigate the effects ofadding central bulges to thin disk gravitational models.Even with the addition of substantial central bulges, all the critical essential features of ourthin disk gravitational models are preserved. (1) Balancing Newtonian gravitational and centrifu-gal forces at every point within the disk yields computed radial mass distributions that describethe measured rotation velocity profiles of mature spiral galaxies successfully. (2) There is no needfor gravity deviations or “massive peripheral spherical halos of mysterious Dark Matter”. (3) Thecalculated total galactic masses are in good agreement with star count data. (4) The addition ofcentral bulges increases the calculated total galactic masses, possibly more consistent with the pres-ence of galactic gases, dust, grains, lumps, planets and plasma in addition to stars. (5) Comparedwith the light distribution, our mass distributions within the disk are larger out toward the galacticperiphery which is cooler with lower opactiy/emissivity (and thus darker). This is apparent fromedge-on views of galaxies which display a dark disk-line against a much brighter galactic halo.
Telescopic images of mature spiral galaxies reveal most of the stars, gas and plasma reside in anapproximately circular disk that is very thin compared with its radius, often with the presenceof a central bulge. The data on galactic rotational velocity profiles (Refs.[1]-[6]) of mature spiralgalaxies are characterized by a rapid increase from the galactic center, reaching a nearly constantvelocity from the outer core to the outer periphery. These basic measured features may be idealizedas V ( r ) = 1 − e − r/R c , (1)where V ( r ) denotes the dimensionless rotational velocity measured in units of maximum asymptoticrotational velocity V and r the radial coordinate from the galactic center. The parameter R c is a description from the data of the various “core” radii of different galaxies. Typical galactic1rotational profiles described by (1) are displayed in Fig 1. As indicated by the measurement data,the rotation velocity typically rises linearly from the galactic center (as if the local mass was inrigid body rotation), and then reach an approximately constant (flat) velocity out to the galacticperiphery.The observed galactic rotation curves (Eq(1) and Fig.1) can not (Ref.[7])be explained by simplyapplying the so-called orbital velocity law , derived for a spherically symmetric gravitational fieldapplicable to the Keplerian rotation of our solar-planet system (where most mass is located at thecenter), but not to galaxies with substantial mass distributed in a disk-like shape. In fact, thegalactic mass distribution calculated by the orbital velocity law applied to these constant (flat)galactic rotation curves yields an increasing mass density with radius , contrary to the measuredgalactic luminosity curves which decrease exponentially with radius. For a thin rotating galactic disk, we impose a balance between the Newtonian gravitational forcesand centrifugal forces at each and every point. Because the gravitational field of a thin disk is notspherically symmetric, the orbital velocity law is not applicable. As illustrated by Feng & Gallo[8] [9], an axisymmetric thin disk gravitational model successfully describes the basic rotationaldynamics of mature spiral galaxies with a mass density decreasing from the center to periphery.And the calculated total galactic masses are in good agreement with star count data.For simplicity, the observed central bulge in mature spiral galaxies was not considered in previousidealized thin disk gravitational models by Feng & Gallo [8] [9]. In this paper, consistent withobservations, we add the gravitational effects of central spherical bulges to thin disk models. We donot address the mechanism(s) maintaining the spherical shape against gravitational and centrifugalforces in this publication [10]. As observed, the central spherical bulge is implicitly assumed torotate at the same radial speeds as the disk (in cylindrical coordinates), but this feature is notexplicitly addressed because it does not affect the computational results. Only the gravitationaleffects of this assumed central bulge and the disk are computed in solving for the rotating diskmass distributions.In detail, it is assumed the bulge has a spherically symmetric mass density decreasing with radialdistance (in spherical coordinates) via a Gaussian function e − β r where β is a positive adjustableparameter. This Gaussian function is convenient since changing one parameter β allows us tovary the size of the spherical bulge relative to the disk to examine the effects of the bulge size ongalactic disk rotation. Our final generic results are not sensitive to the details of this Gaussianassumption. Our model of the entire galaxy consists of a variable superposition of two components:an axisymmetric thin disk and a spherically symmetric central bulge. Both the size and mass of thebulge relative to the disk are independently varied to examine the effects of the bulge on galacticdisk rotation. Similar to the treatment of Feng & Gallo [8] [9], the equation of force balance in an axisymmetricthin disk including a spherically symmetric central bulge is written as Z (cid:20)Z π (ˆ r cos φ − r ) dφ (ˆ r + r − rr cos φ ) / (cid:21) ρ (ˆ r ) h ˆ rd ˆ r − M b r R r e − β ˆ r ˆ r d ˆ r R e − β ˆ r ˆ r d ˆ r + A V ( r ) r = 0 , (2)where all the variables are made dimensionless by measuring lengths (e.g., the radial coordinate r , the radial coordinate as the variable of integration ˆ r , and the thickness of disk h ) in units ofthe outermost galactic radius R g , mass of bulge ( M b ) in units of the total galatic mass ( M g ), diskmass density ( ρ ) in units of M g /R g , and rotational velocity [ V ( r )] in units of the maximum galacticrotational velocity V . The disk thickness h is assumed to be constant and small in comparisonwith the galactic radius R g . Actually the physically meaningful quantity here is the combinedvariable ( ρ h ) that represents the effective surface mass density on the thin disk. As long as thedisk thickness h is much smaller than R g , its mathematical effect is inconsequential to the value of( ρ h ). The gravitational forces of the finite series of concentric rings is described by the first term(double integral) while the centrifugal forces are described by the third term. The second termrepresents the effects of the spherically symmetric central buldge.In (2), we call the dimensionless paremeter A “galactic rotation parameter”, given by A ≡ V R g M g G , (3)where G denotes the gravitational constant, R g is the outermost galactic radius, and V is themaximum asymptotic rotational velocity.As described by Feng & Gallo [8] [9], both ρ ( r ) and A can be determined from a given V ( r ) anda given (but varied) value of M b by solving an equation system including (2) and a conservationconstraint for constant total mass of the galaxy M g , e.g.,2 π Z ρ (ˆ r ) h ˆ rd ˆ r = 1 − M b . (4)Assuming a bell-shape mass density distribution for the spherically symmetric bulge describedby a Gaussian function, the bulge mass density is given by ρ b ( r ) = M b π R e − β ˆ r ˆ r d ˆ r e − βr , (5)where β is a positive adjustable parameter. The bulge mass density is assumed to end at the galaxyrim ( r = 1), so that 4 π Z ρ b (ˆ r )ˆ r d ˆ r = M b . Figure 2 illustrates several bulge mass density distributions ρ b ( r ) at various values of β and M b .Note the bulge mass density ρ b ( r ) is spherically symmetric, whereas the disk mass density ρ ( r ) isonly axisymmetric and r denotes the radial disk coordinate in the cylindrical coordinate systemused in the present computations. To facilitate numerical computation, we discretize the governing equations (2) and (4) by dividingthe one-dimensional problem domain [0 ,
1] into a finite number of line segments called (linear)elements. As described by Feng & Gallo [8], each element covers a subdomain confined by two endnodes, e.g., element i corresponds to the subdomain [ r i , r i +1 ] where r i and r i +1 are the nodal valuesof r at nodes i and i + 1, respectively. With each of the N − ,
1] in the ξ -domain (i.e., the computational domain), N independent residual equationscan be obtained from the collocation procedure, i.e., N − X n =1 Z (cid:20) E ( m i )ˆ r ( ξ ) − r i − K ( m i )ˆ r ( ξ ) + r i (cid:21) ρ ( ξ ) h ˆ r ( ξ ) d ˆ rdξ dξ + 12 " AV ( r i ) − M b r R r i e − β ˆ r ˆ r d ˆ r R e − β ˆ r ˆ r d ˆ r = 0 , (6)where K ( m ) and E ( m ) denotes the complete elliptic integrals of the first kind and second kind,with m i ( ξ ) ≡ r ( ξ ) r i [ˆ r ( ξ ) + r i ] . (7)The N residual equations (6) can be used to compute either the N nodal values of V ( r i ) fromgiven distribution of ρ ( r i ) or the distribution of ρ ( r i ) from a given set of V ( r i ), with given values of A and h . Without loss of generality, the value of h is assumed to be 0 .
01 as comparable with thatobserved for the Milky Way galaxy. If the constraint equation (4) is also used with a discretizedform 2 π N − X n =1 Z ρ ( ξ ) h ˆ r ( ξ ) d ˆ rdξ dξ − M b = 0 , (8)the value of A can also be determined as part of the numerical solution.These generally applicable equations are conveniently used for computing variables, even whenanalytical formulas are available for some special cases. Hence, a unified treatment for all cases isestablished for convenient comparison and analysis. Moreover, as discussed by Feng & Gallo [8],imposing a boundary condition at the galactic center r = 0 for continuity of derivative of ρ , i.e., indiscretized form ρ ( r ) = ρ ( r ) , (9)is desirable for obtaining high-quality numerical solutions.With the adjustable parameters such as R c , β , and M b specified and mathematical singularitiesproperly treated, linear equations (6) and (8) for N + 1 unknowns can be solved with a standardmatrix solver, e.g., by Gauss elimination [11]. Here the effects of bulge parameters on galactic disk mass distributions compatible with measuredrotational velocities are explored. Attention is focused on the Milky Way galaxy which has arotation velocity profile (1) closely represented in Figure 1 with R c = 0 .
015 (cf. Feng & Gallo [8][9]). β = 100 butVarious Bulge Masses M b Figure 3 shows the disk mass density distributions computed with constant β = 100 but variousbulge masses ( M b = 0, 0 .
1, 0 .
2, and 0 . R c = 0 .
015 according to (1). Note that M b = 0 represents a thin disk withouta bulge. With increasing bulge mass M b , a localized decrease of disk mass density appears wherethe bulge mass has significant density around the galatic center. A local minimum develops around r = 0 . M b ≥ .
2. When the value of M b is further increased, the nodal values of disk massdensity ρ around local minimum may become negative which is physically unacceptable. Thus, ourcomputational results demonstrate an upper limit for the bulge mass M b corresponding to a givenbulge size as characterized by the value of β . This is consistent with reality. In all bulge + diskcases, the measured galactic rotation profiles are accurately reproduced.Another noteworthy feature in Figure 3 is that the bulge influence on the disk mass distributiondiminishes beyond r = 0 .
2, where the bulge reaches its effective edge (cf. Figure 2). M b and β Figure 4 shows the computed disk mass density distributions for various combinations of bulgemasses M b and β . For reference, the unlabled curve is for M b = 0 which represents a thin diskwithout a bulge. In all cases, the appropriate calculated galactic rotation parameters A are indi-cated. All these curves are for the Milky Way galaxy with R c = 0 . The measured rotational velocity profiles V ( r ) includes knowledge of maximum rotational velocity V and galactic radius R g . With the computed value of the galactic rotation parameter A , the totalgalactic mass M g can be calculated as M g = V R g A G . (10)To check viability we investigate the idealized rotational velocity profile V ( r ) of our own MilkyWay galaxy shown in Figure 1 with core radius R c = 0 . R c = 0 . V = 2 . × ( m/s ), and R g = 10 (light-years) = 9 . × ( m ). In Table 1, the total galactic mass M g of the Milky Waygalaxy is calculated for a wide range of bulge masses M b and bulge sizes β , and the correspondingcomputed values of galactic rotation parameter A . The total mass of the Milky Way galaxy is thendetermined from (10) to be in the range M g = 2 . − . × (solar-mass) . These values are in very good agreement with Milky Way star counts of 100 billion. In Table 1, notethat a large increase in bulge mass M b only yields a small increase (10 percent) in the total galacticmass M g over the disk-only case (no bulge) ( M b = 0). However, these larger galactic masses maypossibly be more compatible with reality since the galaxies also contain gases, dust, grains, lumps,planets and plasma, all in addition to stars.However, we emphasize the essential physics of galactic rotation is gravitationally controlled bythe ordinary baryonic matter within thin galactic disks. The central bulge has minor effects becausethe comparatively small amount of matter in the outer regions of the disk are gravitationally moreeffective in controlling the rotational dynamics of the galactic periphery. To theoretically describe the measured rotational velocity curves of spiral galaxies, there are threevery different approaches and conclusions.(1) Ordinary Baryonic Matter. We assume Newtonian gravity/dynamics and computationallysolve for mass distributions that successfully duplicate the measured rotational velocities. Thesemass distributions decrease roughly exponentially from the galactic center in the central core, butthen decrease more slowly (inversely with radius) towards the periphery. This decrease is slowerthan the measured light distribution. Thus there is ordinary baryonic matter within the galacticdisk distributed towards the cooler periphery with lower emissivity/opacity and therefore darker.Our view is consistent with edge-views of galaxies which exhibit a dark disk line against a much M b β A M g (solar-mass)0 – 1.57 2.84 × × × × × × × × × Table 1: For the Milky Way galaxy, the total galactic mass M g (in units of solar-mass) is calculated(10) from data and various bulge parameters. From the measured galactic rotation curve, R c =0 . R g = 9 . × (m), and V = 2 . × (m/s). A range of values of bulge parameters ( M b , β ) are examined along with the corresponding computed values of the galactic rotation parameter A . The computed total galactic mass M g increases only 10 percent over the disk-only case (nobulge) ( M b = 0) in spite of large increases of the examined bulge mass M b .brighter galactic halo. There are no mysteries in this rational scenario based on verified physics(Refs.[12]-[34]).(2) Dark Matter. By contrast, others inaccurately assume the galactic mass distributions followthe measured light distributions (approximately exponential), and then the measured rotationalvelocity curves are not duplicated. But this assumption of a simple direct relationship betweenlight intensity and mass is very inaccurate because it is not based on sound physical principles.This so-called Mass/Light ratio is inaccurate since both the temperature and opacity/emissivity areimportant but ignored variables. These deficiencies are clear from edge-on views of spiral galaxieswhere a dark galactic line is obvious against a bright galactic background. revealing the substantialradial temperature gradient across the galaxy. There is no simple direct relationship between massand light, and such an assumption is grossly over-simplified (Refs.[35]-[60]).With this inaccurate assumption, the discrepancy between measured and calculated velocityprofiles are particularly severe beyond the galactic core. To alleviate this discrepancy, speculationsare invoked re “massive peripheral spherical halos of mysterious Dark Matter” But no significantmatter has been detected in this untenable unstable gravitational halo distribution. This spec-ulated Dark Matter is “mysterious” since it does not interact with electromagnetic fields (light)nor ordinary matter except through gravity. This Dark Matter must have other abnormal (non-baryonic) properties to maintain its peripheral spherical shape against the galactic rotation andgravitational attraction of ordinary matter. Many unverified “mysteries” are invoked as solutionsto real physical phenomena (Refs.[35]-[60]).(3) Modified Gravity. Possible deviations from Newtonian gravtiy/dynamics have been pro-posed, but there is no independent experimental evidence of such deviations. Our use of Newto-nian gravity/dynamics with sound computational techniques has proven successful to explain theobserved flat rotation curves (Refs.[61]).Conclusion. We conclude our approach utilizing Newtonian gravity/dynamics and computation-ally solving for the ordinary baryonic mass distributions within the galactic disk simulates realityand agrees with data. Our approach yields higher total galactic masses in agreement with star counts. Concurrently,our mass density distributions also yield more mass distributed out towards the cooler galacticperiphery. We wish to conjecture about the nature of some of this ordinary baryonic matter whichappears dark. In addition to stars, this material is some combination of dust, grains, lumps, planets,plasma and gases.Consider hydrogen gas. Since the temperature is higher in the galactic core and cooler towardsthe galactic periphery, we expect more ionized hydrogen (plasma) in the hotter core, then moreatomic hydrogen away from the core, and finally more molecular hydrogen out towards the coolerperiphery. This molecular hydrogen is often ignored, although measurements have revealed itspresence (Ref.[62]). However, quantitative estimates of its density vary widely. We note thatmolecular hydrogen is a naturally dark material since it has very low emission and absorptioncoefficients due to its high molecular structural symmetry. Combined with its presence towards thecooler periphery, we qualitatively believe this molecular hydrogen (Refs.[63][64]) is one componentof the ordinary baryonic dark matter in the disk periphery we have found from analysis of galacticrotation profiles.
Our Disk and Bulge+Disk gravitational models do not address many important features such asspiral structure, plasma effects, galactic formation, galactic evolution, galactic jets, black holes,relativistic effects, galactic clusters, etc..It is well known (Ref.[7]) that the internal gravitational behavior of a thin disk is much differentthan a sphere. This distinctly different behavior of disks enables our models to describe the rota-tional dynamics of mature spiral galaxies, and their total galactic masses, even with the additionof substantial central spherical bulges. The reason is that matter in the outer disk periphery isgravitationally closer and stronger in controlling the rotation in those outer regions, even comparedwith the substantial central spherical bulges which are more distant and gravitationally weaker.Our gravitational models have finite radial extent. Beyond the galactic radius, we assume thedensity has dropped to the inter-galactic level, which is approximately spherically symmetric andthus no longer affects the galactic dynamics. We mention this because some others (Refs.[7][35]),have taken the relevant integrals to infinity, which we think is inappropriate.In our approaches, we balance the gravitational forces against the centrifugal forces at eachand every point within the disk. Thus, our solutions for the mass distributions and total galacticmass satisfy the rotational velocity measurements and ensure stability within the same contextas similar calculations for our Solar System and Earth satellites. Some previous authors obtainsolutions that are not gravitationally stable because they obtain incorrect mass distributions andincorrect galactic masses and do not satisfy the measured rotational profiles. Thus, their solutionsare unstable, whereas our solutions are stable within the Newtonian context.Plasma effects are certainly active in the formation and evolution of galaxies from the origi-nal hot plasma (Refs.[65]-[67]). However, for mature spiral galaxies, the free plasma density hasdropped to levels sufficiently low that plasma does not affect the predominantly gravitational galac-tic dynamics. This is evidenced in our own Solar System in which gravitational dynamics dominateeven with the observed effects of solar wind, coronal mass ejections, auroras, comet tails, etc. Theplasma in our Sun is stabilized by gravitational forces, even though plasma effects are very activewithin the Sun itself. Since our Solar System is approximately 1/3 distance from the center of ourMilky Way galaxy, we have our Solar System evidence for the dominance of gravitational forceswithin our own Milky Way galaxy, at least at this radial distance and beyond to the periphery. Weexpect plasma phenomena to be more active in the hotter central galactic core.Summarizing, both our Disk and Bulge+Disk models are sufficient to describe the rotationaldynamics of mature spiral galaxies and their total galactic masses. Disk models with an additionalcentral Bulge yield higher total galactic masses.
Even with the addition of substantial central galactic bulges, all the critical essential featuresof our thin disk gravitational models are preserved. (1) Balancing Newtonian gravitational andcentrifugal forces at every point within the disk yields computed radial mass distributions thatdescribe the measured rotation velocity profiles of mature spiral galaxies successfully. (2) There isno need for gravity deviations or “massive peripheral spherical halos of mysterious Dark Matter”.(3) The calculated total galactic masses are in good agreement with star count data. (4) Theaddition of central bulges increases the calculated total galactic masses, possibly more consistentwith the presence of galactic gases, dust, grains, lumps, planets and plasma in addition to stars. (5)Compared with the light distribution, our mass distributions within the disk are larger out towardthe galactic periphery which is cooler with lower opactiy/emissivity (and thus darker). This isapparent from edge-on views of galaxies which display a dark disk-line against a much brightergalactic halo.Most previous research assumes a galactic density decreasing exponentially with radius outto the galactic periphery, analogous to the measured light distribution. But this assumption (byothers) is inaccurate since both the temperature and opacity/emissity are important but ignoredvariables. There is no simple relationship between mass and light. These prior models do NOTdescribe the measured velocity profiles, and speculations are invoked re halos of mysterious DarkMatter or gravitational deviations to compensate. The Dark Matter must have “mysterious” (non-baryonic) properties because there is no evidence of its existence and it is not responding to grav-itational, centrifugal and electromagnetic forces in any known manner. By contrast, our resultsindicate no massive peripheral spherical halos of mysterious Dark Matter and no deviations fromsimple gravity. Our total galactic mass determinations are also in reasonable agreement with data.The controversy is summarized as follows.We believe there is ordinary baryonic matter within the galactic disc distributed more towardsthe galactic periphery which is cooler with lower opacity/emissivity (and therefore darker).Others believe there are massive peripheral spherical halos of mysterious Dark Matter surround-ing the galaxies.
10 Acknowledgements
We gratefully acknowledge Louis Marmet, Ken Nicholson and Michel Mizony whose intuition andcomputational techniques convinced us that galactic rotation could be described by suitable massdistributions of ordinary baryonic matter within galactic disks. Anthony Peratt originally sparkedour interest with his plasma dynamical calculations re the formation and evolution of galaxies. AriBrynjolfsson has energetically supported our efforts.
References -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-0.100.10.20.30.40.50.60.70.80.911.1 rV(r) 0.0150.030.05Rc = 0.1
Figure 1: Typical Galactic Rotational Velocity Profiles V ( r ) idealized (1) from measurements for R c = 0 . .
03, 0 .
05, and 0 . -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6-20020406080100120140160180200220 r b u l g e m a ss d e n s i t y beta = 10, Mb = 0.5100, 0.25200, 0.15500, 0.1 Figure 2: Bulge mass density distributions ρ b ( r ) from (5) are displayed for reasonable bulge pa-rameters we have examined ( β = 10, 100, 200, and 500 with M b = 0 .
5, 0 .
25, 0 .
15, and 0 . -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.1110100100010000 r d i s k m a ss d e n s i t y Mb = 0, A = 1.57030.1, 1.55910.2, 1.54780.25, 1.5422
Figure 3: Disk mass density ρ ( r ) computed with β = 100 and various bulge masses. M b = 0, 0 . .
2, and 0 .
25. Note the case of M b = 0 represents a thin disk without a bulge. All these curvesare for the Milky Way galaxy with R c = 0 . A are indicated. In all these bulge + disk cases, the measured galactic rotationprofiles are accurately reproduced.7 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.1110100100010000 r d i s k m a ss d e n s i t y Figure 4: Disk mass density ρ ( r ) computed for various combinations of bulge parameters. For M b = 0 . β = 500, and A = 1 .
57. For M b = 0 . β = 200, and A = 1 .
56. For M b = 0 . β = 100,and A = 1 .
54. For M b = 0 . β = 10, and A = 1 .
39. For comparison, the curve without labelis for M b = 0 which represents a disk without a bulge. In all cases, the appropriate calculatedgalactic rotation parameters A are indicated. The constant value of R c = 0 ..