The Galactic Mass Discrepancy-Acceleration Relation (MDAR) and Implications for Galaxy Formation and Cosmology


We present the first spatially resolved Mass Discrepancy-Acceleration Relation (MDAR) for the Galaxy as determined from high-resolution stellar kinematics. The physically obtained MDAR exhibits a rich complexity with substructure reflecting the true dynamics of the Galactic system. The results contradict the presence of a massive extended dark matter halo in lieu of a baryonic solution. We compare the effective baryonic fraction as a function of radius (>200 kpc) demonstrating nearly all Galactic so called “missing mass” originates in and around the immediate vicinity of the baryonic disk. We argue that nearly all disk galaxies are a product of relativistic dynamics and baryonically derived gravitational forces existing during primordial assembly of the Galaxy. Our cosmological proposition is buttressed by recent stellar chemical analysis suggesting two-phase galactic assembly that uniquely matches our kinematic observations and theoretical considerations that are consistent with this formation proposal.


Over the years, two main theories have emerged to explain galactic rotation curves; the Lambda-Cold Dark Matter (ɅCDM) and MOdified Newtonian Dynamics (MOND) (Milgrom 1983). By characterizing the rotation curves in specific ways, each seeks to explain curves that appear to not be governed by classical Newtonian mechanics. Popular terms for this discrepancy are unconventional gravitation or missing mass. This paper takes a different route in solving this mystery. The solution consists of a simple visual inspection of the detailed Galactic circular rotation curve, including all kinematic substructure that ɅCDM and MOND routinely discount in their oversimplified galactic models. In this manner, we provide the first physically obtained radial MDAR profile for the Galaxy. Next, we compare the physically obtained MDAR against the MOND model, highlighting the differences and report significant differences in between what is observed in the Galactic setting and the ɅCDM prescription of galaxy formation and by extension, cosmology. We offer an alternative that is consistent with current knowledge of the Galaxy and can be verified via the scientific method.

The Rotation Curve-Spin Parameter (RC-SP) model advocated in this paper provides a solution wholly consistent with all observed Galactic features (La Fortune 2015b) (La Fortune 2015c). RC-SP integrates new data to strengthen a classical, physically-based description of the “Galactic state.” The intent of this paper is to present a physical model that accounts for all velocity data, not partial sets employed for computational and/or theoretical convenience. All figures in this paper have been constructed from the observed velocity data given in the Appendix.

The RC-SP model employs recently discovered kinematic substructure (features) directly obtained from stellar inner halo (King III 2015). One may question the use of the halo kinematics as a proxy for the Galactic disk, but it has been shown that any self-gravitating system in equilibrium having angular momentum results in “reflection symmetry” about its galactic plane (An 2016). A recent high-resolution rotation curve confirms this symmetry with the disk rotation curve duplicating stellar halo kinematics, including the significant velocity peak located at 23 kpc (Bajkova 2016).

Galactic Mass Discrepancy-Acceleration Relation (MDAR)

In this section we construct a detailed MDAR for the Galaxy using observed stellar velocities. To clean up the terminology, the RC-SP definition of “total” velocity is equivalent to that in MOND and ɅCDM (MTotal ≈ MDyn ≈ MDM). This is an excellent first order approximation between the two paradigms and allows direct comparison to the RC-SP-based findings. Under MOND, any total velocity that exceeds the baryonic expectation is a result of deviation from conventional Newtonian mechanics, reverting to a pseudo 1/r force law as local gravitational acceleration falls below a threshold value. ɅCDM accounts for these gravitational anomalies via the requisite amount and distribution of dark matter to reproduce observed Galactic rotation velocities. RC-SP contends total rotational velocities (VDyn) greater than the baryonic contribution are attributed to angular momentum and total energy associated with galaxy morphology and dynamics governed exclusively by Newtonian mechanics.

The MDAR relies on the measure of mass discrepancy which is the ratio between dynamic and baryonic mass, D = MDyn/Mbar.  We translate MOND velocity discrepancy values to equivalent baryonic fractions “fb” using the conversion formula, fb=1/D. A more practical derivation D uses the observed velocities in similar fashion in combination with the estimated baryonic velocity contribution, D = (VDyn/Vbar)2 (Pato 2015). This relation places tight constraints on the radial profile of Galactic missing mass that rigorously tests and calibrates theoretical expectation with observation.

Figure 1 below is a plot of mass discrepancy as a function of galactic radius with solid color points representing measured mass discrepancies for a wide variety of rotationally supported galaxies. In addition to McGaugh’s raw data, we include the generic MOND model (blue solid) and the RC-SP profile obtained from King’s observed velocity dispersions (black dash) for comparison.

Figure 1: The Galactic MDAR as a function of radius for a wide range of galaxies (colored points) and the generic MOND model (solid blue). The RC-SP relation (black dash) is obtained from King. The generic MOND model continually increases well beyond the Galactic disk to self-described “indefinite radii” while the RC-SP model asymptotically approaches D  = 5.9 just beyond the baryonic outer disk. The horizontal portion of the RC-SP depicts a conventional Keplerian decline. Image source - Fig. 5 (McGaugh 2014)

In Figure 1 above, we find the Galactic RC-SP “D-R relation traces several of the curves obtained from McGaugh, including some very complex substructure. MOND attributes kinematic substructure to baryonic mass and distribution governed by unconventional gravitational dynamics. It appears highly unlikely these features arise exclusively due to MOND dynamics alone. ɅCDM on the other hand has no mechanism to reproduce rich substructure hampered by dark matter halo theoretical constraints.

The RC-SP profile exhibits a local peak at 23 kpc equivalent to D = 12.1, consistent with McGaugh’s upper limit. The most significant disparity between RC-SP and the generic MOND model occurs in the region beyond the baryonic disk. The generic MOND curve continually rises to “indeterminate radii” eliminating the dependence on radial distance from the baryonic source resulting in an asymptotically flat rotation curve. The horizontal RC-SP to the right reflects a Keplerian velocity profile with the Galaxy behaving as a point mass beyond 70 kpc. Within the inner disk (<10 kpc), the RC-SP curve agrees with recent findings indicating an absence of dark matter (D = 1) where Newtonian dynamics dominate (Lelli 2014). It is not surprising that the radial acceleration obtained by galactic rotation curves tightly correlate with the baryonic distribution potentially being the “end product of galaxy formation” that also would need to include angular momentum and total energy consideration (McGaugh 2016).

As with the Baryonic Tully-fisher Relationship (BTFR), MOND employs the MDAR to challenge ɅCDM models (McGaugh 2004) (Di Cintio 2015). Unfortunately, MOND misrepresents the MDAR and BTFR as evidence of non-Newtonian dynamics. Rather, the BTFR has been revealed to be an analytical function expressing the relationships between radius, rotational velocity, and dynamic mass consistent for a wide range of galaxies, from small dark irregular dwarfs to massive bright spirals (La Fortune 2016a).

Galactic Total (Dynamic) Mass

This section compares the RC-SP model against the latest and most “precise” Galactic ɅCDM total mass estimates provided to date (Eadie 2016). Figure 2 is adapted from Eadie depicting a best estimate of dark matter halo mass and distribution. The figure has been extended from 150 to 350 kpc to highlight the differences between ɅCDM and RC-SP radial (dynamic) mass profiles.

Figure 2: Eadie’s total Galactic Mass estimate (dark gray shaded curve) compared to RC-SP MDyn (blue dash) obtained from spatially resolved component velocity dispersions. RC-SP dynamic mass peaks at 23 kpc corresponding to a local stellar velocity Vpeak = 432 km/s (D = 12.1) equivalent to and ɅCDM baryonic fraction, fb= 0.08. This peak velocity corresponds to a local dynamic mass, MDyn 1 x1012Mʘ. We include the RC-SP total baryonic contribution to show the relationship between MDyn (blue dash) and Mbar (red dash). Net mass discrepancy for the total system is: 0.5 x1012Mʘ/0.085 x1012Mʘ equivalent to D = 5.9, fb = 0.17. Image source - Fig. 4 (Eadie 2016)

The RC-SP dynamic and baryonic mass as a function of Galactic radius are represented by the blue dashed and red dashed curves, respectively. Eadie’s dark matter halo estimate is depicted by the arcing the gray band covering the entire radial distance in the figure. Based on Eadie’s analysis, the precise mass estimate of the Galaxy is 0.481 x1012Mʘ <125 kpc (gray data point with white boarder) consistent with the RC-SP total Galactic dynamic mass 0.5 x1012Mʘ. The peak in dynamic mass at 23 kpc has been shown to be a thermodynamic artifact (La Fortune 2016b). In the next section, the Galaxy’s ɅCDM’s theoretical baryonic fraction profiles are compared to “effective” baryonic fractions from highly-resolved component velocity dispersions.

Galactic “Effective” Baryon Fraction

We now turn to the radial profile of the Galaxy with regard to its baryon fraction (fb), ɅCDM’s measure of mass discrepancy (Bland-Hawthorn 2016). Figure 3, the dark matter baryonic fraction profile is represented by the broad tri-color curves obtained in cosmological simulations that also dictate Galactic formation and evolution. The RC-SP mass discrepancies depicted in Figure 1 are converted to an “effective baryonic fraction via the standard conversion formula, fb ≈1/D (Vbar/VDyn)2.

Figure 3: Comparison between radial ɅCDM and RC-SP “effective” baryonic fraction profiles. The models are equivalent in the region of the outer disk, but substantially deviate on either side. The divergent nature of the profiles suggests two disparate galaxy formation scenarios. Image source - Fig.17 (Bland-Hawthorn 2016)

In the above figure, it is clear that RC-SP and ɅCDM radial profiles describe two different representations of the Galaxy’s baryonic fraction. Rather than the featureless, smoothly declining profile from Bland-Hawthorn, the RC-SP curve exhibits a classical “step” function occurring at the boundary of the inner and outer disks. In addition, there are two main discrepancies between the RC-SP and the dark matter profiles shown in Figure 2. The ɅCDM model requires complete saturation of the inner disk with a dark matter component and a lazy decline extending to extreme radial distances, hallmarks of a weakly interacting stochastic dark matter accretion process. Per ɅCDM, as dark matter accumulates, baryons are attracted to these massive gravitational wells and receive through this process the requisite amount and distribution of angular momentum and total energy that characterize galactic properties currently observed. Even in the total absence of baryons, dark matter assembly would have proceeded with no significant changes to the current cosmic framework, including large scale structure of the local universe.

Galactic Formation and Dynamic Evolution

As shown in the above figure, the physically obtained “effective” baryonic fraction profile suggests an alternative galactic formation process that is inconsistent with ɅCDM cosmology. The RC-SP profile suggests that the inner and outer disks were constructed in a two-phase process with the inner disk forming first, followed closely by formation of the outer disk. The significant “step” in dynamics between these two disk components also suggests the disks were subject to different environmental conditions during each of their formation phases. This particular formation process has gained traction based on extensive chemo-analysis of the thick (inner) and thin (outer) stellar disks (Haywood 2013) (Haywood 2015) (Snaith 2014).

Up to this point, the paper has focused on empirical data to support RC-SP Galactic dynamics. In addition to observational science, new theoretical perspectives are emerging relying on thermo-statistical process explanations that produce galaxies with a defined inner and outer disk (Tello-Ortiz 2016). Although this work is limited to non-relativistic systems, there appears to be a causal connection with RC-SP physical evidence and Tello-Ortiz’s general findings; an initial phase where self-gravitation leads to a diffuse spatially extended “nebula” that is then followed by catastrophic “gravitational collapse.” This second phase creates a compact dense central core while simultaneously displacing matter into the immediate outer region. This theoretical treatment is somewhat analogous to the RC-SP galactic formation process briefly summarized in an introductory paper (La Fortune 2015a).

As suggested by Tello-Ortiz a more sophisticated treatment incorporating relativistic conditions underway, as this is required to fully appreciate physical processes driving early-era cosmic structure formation. Based on the physical evidence, any future conceptual work should include provisions for a time-varying (declining) energy density and sound treatment of angular momentum/energy where conventional conservation laws may not apply.

Relativistic process “relics” are still apparent today in the morphology with kinematics of disk galaxies; kinematics governed by a pseudo 1/r force law, flat rotation curves and vertical symmetry about galactic planes. This is in direct contradiction to the spatially isotropic and homogenous structures demanded by ɅCDM. Although well beyond the scope of this paper, evidence for a non-ɅCDM cosmology in the early universe has recently been confirmed within the local universe (Tekhanovich 2016) (Baryshev 2016). These surveys, extending to 300 Mpc reveal the two dimensional “planarity” in the galaxy distribution in large scale structure not anticipated or supported by ɅCDM cosmology.


The proposed Rotation Curve-Spin Parameter (RC-SP) model is a first-order approach using classical principles to physically describe and interpret Galaxy morphology and kinematics. The proposed model includes the following elements that differentiate it from the two mainstream paradigms:

·         A physical (baryonic) basis and interpretation of Galactic morphology and kinematics based on classical mechanics and conventional thermodynamic principles

·         An origin and physical interpretation for the Galactic the Mass Discrepancy-Acceleration (as well as the Baryonic Tully Fisher) relationships

·         A physical interpretation of the detailed Galactic rotation profile, including observed significant kinematic substructure

·         A Galactic formation process consistent with latest kinematic high-resolution kinematical and stellar chemo-analysis of the inner and outer disks

·         A cosmology harmonizing Galactic and large scale spatial structure and organization of the local universe

·         Does not require new gravitational dynamics or undetected matter to solve the Galactic missing mass problem

·         Amenable to the scientific method to verify/falsify proposed interpretations and results

The above claims are “necessary preconditions” that ɅCDM and MOND models have not been able to meet. In order to remain viable contenders, both mainstream paradigms either avoid or severely downplay the need to incorporate all observed data and represent exception-based science that will remain at odds with any new or “surprising” findings. We need to reset our initiatives to incorporate classical mechanics and basic thermodynamics principles under highly relativistic conditions to make true progress in describing the “Galactic state” and other large scale structures inhabiting the local universe. Thanks to the Winnower for the opportunity to share our perspective to a wider audience.




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