We present the first detailed physically-based thermodynamic model of the Galaxy as obtained from high-resolution kinematical surveys in the outer disk. We interpret the observed distribution of Hyper Velocity Stars (HVS) as a physical manifestation of the Maxwell-Boltzmann (M-B) probability distribution expected for a ‘system’ in thermodynamic quasi-equilibrium with its ‘surroundings.’ We counter the popular hypotheses of chance gravitational encounters and relate the HVS distribution directly to thermodynamic effects. The proposed Galactic M-B distribution is based on a single physical constraint, the most probable velocity, Vmp=432 km/s as obtained via radially binned component velocity dispersion profiles obtained in the inner stellar halo. The radius of maximal entropy is 23 kpc where most of the HVS population is “shed” (not ejected). This link between stellar halo dispersion characteristics and the HVS is physical evidence that the Galaxy is fully self-gravitating and exists in a state of thermodynamic and mechanical “quasi-equilibrium” as required for open virialized systems.
There has been an explosion within the astrophysics community to fit (and potentially explain) observed galactic rotation curves. Often, this analysis is conducted with a comparison between the data and theoretical models. This introduces bias in the physical interpretation of the rotation profile and other, more subtle kinematic clues. In this paper, we examine the kinematic evidence and compare the proposal, termed Rotation Curve-Spin Parameter (RC-SP) for the purposes of this discussion as to distinguish it from ɅCDM computer simulation models. We investigate the origin of dynamic mass through an analysis of component velocity dispersion data and link it to thermodynamic properties attributable to virialized self-gravitating systems. No ‘new physics’ are required, just an unbiased analysis of the kinematic features as presented by the most recent, high-resolution surveys available.
The Rotation Curve-Spin Parameter (RC-SP) Model of the Galaxy
Much has been published regarding both mainstream theories, but these rely on ‘new physics’ to accomplish their goals to fit (and explain) galactic rotation curves and in doing so, solve the problem of Galactic “missing mass.” An ‘old physics’ RC-SP solution has been previously advanced featuring a baryonic-based interpretation of the extended rotation curve as governed by the spin parameter equation, including all angular momentum and energy associated with the Galactic state (La Fortune 2016b). This approach takes a step back and infers galactic dynamics directly from detailed radial rotation velocity and includes analysis of high resolution component dispersion profiles and Hypervelocity star observations. The RC-SP model assumes no dark matter or modified gravitation law and relies on the Virial Theorem for gravitationally bound (baryonic) systems:
The parameters are: total Galactic dynamic mass (MDyn), rotation velocity (V), and disk radius (R) per the RC-SP prescription. Properly parameterized, the Galaxy (as well as most all other disk galaxies) obeys the Newtonian equation for circular velocity:
The virial assumption assures an R-V solution exists for any dynamic mass value (any complimentary R-V combination may satisfy the equation). We see, however, that this wide spectrum is physically constrained to a narrow window of permissible R-V combinations. For example, the Galaxy has an average disk velocity of 230 km/s and a disk radius of 40 kpc. Per the above equation, the estimated dynamic mass 0.49x1012Mʘ, very close to 0.5x1012Mʘ obtained from the RC-SP rotation curve fit template. This constraint is the result of angular momentum and total energy carried by the Galactic baryons (Peebles 1969).
The “functional R-V relation” from “dark” gas-dominated dwarf irregulars to “bright” massive star-dominated spirals is shown in Figure 1 below (green dash). Note that this relation can be configured for baryonic mass via MBar/MDyn associated with galactic “missing mass,” baryonic fraction, or mass discrepancy.
Figure 1: RC-SP Galactic Dynamic Mass-Velocity-Radius Relation. The functional MDyn-R-V relationship (green dash) is the only physically permissible relation based on galactic morphological and kinematic properties. Estimated M31 baryon mass; fb=0.17 (filled circle) positioned vertically below dynamic mass (no change in R or V). Individual galaxy data: And IV - Irregular dwarf (open triangle) (Karachentsev 2015), UCG 6818 and UGC 9211 - gas-rich galaxies (McGaugh 2011). See (La Fortune 2016b) for details regarding physical properties (MDyn, V and R) for the MW and M31 (open circles) and the two gas-rich galaxies (squares)
The issue of “missing mass” is the difference between dynamic mass and the estimated baryonic contribution. Since RC-SP is based on a constant baryonic fraction/mass discrepancy for rotationally supported galaxies, the baryon content is just a simple fraction of the dynamic mass, as illustrated by the solid black data point vertically below the open circle for M31 (corresponding to fb=0.17, the ɅCDM cosmic average).In order to respect baryonic-based physics, the Galaxy’s dynamic mass must originate where baryonic influence is the greatest – inside 40 kpc, the edge of the Galactic outer disk. This places the burden on RC-SP to demonstrate this expectation in very physical terms, beginning with the Galactic rotation curve.
The Galactic Rotation Curve
In this section, we examine a well-cited rotation curve from Bhattacharjee augmented with data from Bajkova and Bobylev shown in Figure 2. Included are three ɅCDM models, labeled 1, 2, and 3 from the original figure (Bajkova 2016). The RC-SP Galactic rotation curve fit is shown by the black dash. This curve fit is based on observed velocities within the disk (R≤40 kpc). The rotation curve is modeled as an ideal Keplerian decline beyond the disk.
Figure 2: Galactic rotation curve from Bhattacharjee, Bobylev and Bajkova. Panel (a) illustrates the Obs/RC-SP integrated rotation profile (black dash). Prominent kinematic features include a spike in rotation to escape velocity (gray dashed) and a Keplerian decline beyond the Galactic disk. ɅCDM models are identified 1, 2 and 3 (see Bajkova ref. for details). Panel (b) is a semi-log version of panel (a) indicating observed velocity spike occurs “within” the Galactic outer disk. This view provides the specific R-V combination (open circle) corresponding to the RC-SP Galactic dynamic mass. This mass is obtained from the fit (Keplerian decline) beyond 40 kpc. The (Pato 2015) average rotation velocity (blue dash) and the Sun’s velocity and position (red dot) is provided for reference. Image source – Fig.1 (Bajkova 2016)
Perhaps the most prominent feature in Figure 2 is the abrupt spike in rotation velocity at ~23 kpc. Note that data in the 20 to 24 kpc range were excluded, as the veracity of this data was in doubt. We have interpolated the rotation curve within this short span from the available data. We have selected the peak in velocity being near 432 km/s followed by steep decline to more “reasonable” values. Since discovery, ɅCDM proponents have downplayed this exceptional kinematic feature perhaps due theoretical constraints on the properties of dark matter. The fact that the most obvious kinematic feature in the Galactic rotation curve has been excluded in ɅCDM models (shown in Figure 2) suggests a failure to comprehend its physical meaning within the context of the operating paradigm.
In Figure 2, we find that all three ɅCDM models significantly deviate from the observed Galactic velocity profile. This deviation is especially apparent at outer radii where extended, flat rotation profiles are de rigueur for ɅCDM. Rather than flat rotation, the Galaxy exhibits a “break-point” at 40 kpc from a flat profile to one traced by a Keplerian decline equivalent to the RC-SP dynamic mass MDyn=0.5x1012Mʘ, in agreement with a very precise Bayesian estimate of total Galactic mass, MTotal = 0.522x1012Mʘ < 125 kpc (Eadie 2016). This Keplerian decline is not an artifact of the Bhattacharjee profile, as it is also present in other extended rotation curves of the Galaxy (Sofue 2015) (Huang 2016). What has eluded explanation to date is a physical interpretation of this kinematic feature by Bhattacharjee, Bobylev and Bajkova. Throughout this paper, MDyn ≈ MTotal and is equivalent to ɅCDM-based total Galactic mass (MBar + MDM).
The Observed Kinematics of the Galactic Stellar Halo
With regard to the origin of the pronounced velocity spike at 23 kpc, we simply match the results of another Galactic survey of high-resolution component velocity dispersions observed in the inner portion of the stellar halo (King III 2015). This halo, while not significantly massive, serves as a sensitive indicator of galactic kinematics in and around the disk. Figure 3 shows component (R, θ, and φ) dispersion results from 6 to 30 kpc (red and green data). Superimposed is an ideal isotropic model (black dash, as individual components) corresponding to a total baryonic ‘point’ mass Mbar = 8.5x1010Mʘ. The average Galactic rotation velocity, 230 km/s is included for perspective (blue dash). In the figure below, panels, a., b., and c., are observed component stellar halo velocity dispersion as a function of radius, and d., the observed anisotropy coefficient, β. From a purely qualitative/visual standpoint, any net positive deviation between the isotropic curve and dispersion velocities is evidence of Galactic “missing mass.”
Figure 3: a., b., and c. panels provide observed velocity dispersions for radial, polar and azimuthal components, respectively. Included are the average disk rotation velocity (blue dash) and a spherical isotropic curve (black dash) for comparison. Lower right panel, orbital character of the halo is defined by the anisotropy coefficient, β (solid black) with β=1 for perfectly radial orbits, β=0 for the isotropic case and β→ -∞ describing pure circular motion. Image source - Fig. 10 (King III 2015)
Comparing panels, a., b., and c., we note that only the radial component (a.) traces the isotropic curve and thus does not significantly participate in velocity support. This kinematic signature suggests nearly all velocity support is tangential in origin but is short-lived as the dispersion components quickly return to more isotropic orbits. Panel d. illustrates the degree of increased circularity near 23 kpc with extremely negative anisotropy coefficients, β = [1-(σ2Azimuthal-θ+σ2Polar-φ)/2σ2Radial-R]. We confirm that the kinematic disruption in the Bhattacharjee profile is attributable to exceedingly high velocity circular orbits. In the next section we explore a physical mechanism for this kinematic feature as observed above in Galactic rotation and component velocity dispersion profiles.
Hyper Velocity Stars and Thermodynamic Considerations
In order to confirm the thermodynamic postulate, we investigate the observed distribution of Hypervelocity Stars (HVS) and defined as those stars able to leave the gravitational confines of the Galaxy. Most popular theories rely on intense gravitational interaction between candidate HVS stars and centrally located black holes or exploding supernovae (Tauris 2015) (G. L. Fragione 2016a) (Rossi 2016) (G. C.-D. Fragione 2016b). Therefore, the HVS population reflects the outcome between ejection velocities and the gravitational influence of the massive dark matter halo. Proponents claim by simply coupling random events with ɅCDM, the total mass of the Galaxy and the nature of HVS population distribution can be inferred.
Countering these chance ejections and low probability interactive encounters, we believe the HVS population has its origin in the global thermodynamic properties of the Galaxy. We show this population provides insight how the Galactic “system” remains in thermodynamic quasi-equilibrium with its “surroundings” for over the last 10+ billion years. Using King’s velocity dispersion data, we compare the RC-SP expectation against dark matter halo simulations (Fragione, see insert) against the observed HVS population as function of Galactic radius. In Figure 4 below, we show King’s velocity dispersion (peaked blue-dash) velocity profile and designate that upper physical constraint (432 km/s) as the local Galactic escape velocity (Vesc).
Figure 4: Fragione’s ɅCDM escape velocity models (see inset for details) is contrasted against the RC-SP profile (blue dot) as a function of Galactic radius. Quadrature summed velocity dispersion components; R, θ, and φ (blue dash). Note the observed total dispersion peak velocity coincides with an escape velocity (435 km/s) at 23 kpc. This “R-V” intersection highlighted in blue is the Newtonian escape velocity curve for a dynamic mass MDyn=0.5x1012Mʘ. Average rotation velocity and Keplerian decline included for perspective, as labeled (black dash). Image source – Fig 2 (G. L. Fragione 2016a)
We link the physical properties of the data to the thermodynamic model by assigning this the peak velocity to the Maxwell-Boltzmann distribution as the most probable velocity (Vmp). This is shown in Figure 4 as the intersection of the physical velocity peak and the calculated escape velocity (Vesc). It is not coincidental that this peak velocity complements the estimated escape velocity for a point mass equivalent to a dynamic mass MDyn = 0.5x1012Mʘ (red short-dash). From these observations, a thermodynamic interpretation is physically discernable from the observed HVS population.
To generate the first-order M-B distribution, we apply the dimensionless parameter and working equation (Wu 2014):
The raw data used to generate the M-B distribution is given in the Appendix. We can grasp the shape of the M-B distribution compared to two ejection models from Rossi in Figure 5 below. The shaded histograms provide the observed distribution of “late-B stars” having a mean mass near 3.5Mʘ.
Figure 5: HVS probability density functions per black hole ejection schemes (solid black and red dash). Observed “unbound” late B-star (~3.5 Mʘ) HVS distribution (gray histograms) from Brown 2014. The alternative RC-SP/Maxwell-Boltzmann probability distribution (blue dash) is based on most probable velocity Vmp=432 km/s (black dash) with arbitrary vertical scale for comparison. Image source - Fig. 3 (Rossi 2016)
In Figure 5 above, for HVS velocities less than Vmp, we cannot discriminate between the models. Therefore, we need to look beyond Vmp that best fits all the data that is available. As with the thermodynamic interpretation, ejection mechanisms expect low mass stars to attain greater velocities than their higher mass counterparts. We find that when a second star type is included, and in this case ~0.9Mʘ G/K dwarfs, we expect them to have a higher net velocity than late-B stars. It is in this velocity range (>Vmp) where the models can be distinguished. Figure 6 below shows the HVS distribution when G/K dwarfs are added to the previous figure.
Figure 6: Velocity distribution of HVS late B-type (blue histograms) and G/K-dwarfs (green histograms) with respect to the Galactic rest frame. The combined data traces the M-B probability distribution (blue dash) for a most probable velocity, Vmp= 432 km/s. The M-B vertical scale is matched to the data for comparative purposes. Image source - Fig. 1 (Tauris 2015)
In the above figure, for velocities beyond Vmp, the addition of the G/K dwarf sample fits the overall M-B distribution. In defining the distribution, the stellar universal Initial Mass Function (IMF) must also be accounted for (Baldry 2003) (Offner 2014). Galactic IMF distributions exhibit a narrow peak in stellar mass between 0.2Mʘ to 4Mʘ leading to a fairly conventional M-B relationship. Based on this assumption, the dearth of stars observed this mass range is either severely under sampled or reveals a more subtle dynamic effect not yet fully understood. In the above figure, the absence of HVS below 350 km/s is due to arbitrary truncation of the data. Based on the RC-SP dynamic mass MDyn = 0.5x1012Mʘ, the minimum escape velocity at 120 kpc is estimated slightly below 200 km/s in agreement with the escape velocity in Figure 4. Unfortunately, HVS data below 350 km/s was arbitrarily truncated but is expected to be populated with ‘unbound’ stars having masses greater than 3 Mʘ (a declining IMF at higher stellar masses may also be responsible for some reduction of the sample in this velocity range as well). The Galactic M-B distribution is a holistic manifestation of Galactic thermodynamics - an open system existing in quasi-equilibrium with its surroundings.
A recent article also discussed the applicability of the Maxwell-Boltzmann principle to describe galaxies in a dynamic state of “near-equilibrium” (Struck 2016). They concluded that the M-B model disagreed with an ad hoc constraint of constant velocity dispersion, β≈0 for the entire galactic disk. Of course, the M-B model cannot meet this constraint, it’s physically impossible for it to do so. Going forward, we would be remiss not to incorporate kinematic features like the one presented, rather than wholesale “erasure” in oversimplified and/or underdetermined theoretical models.
We used currently available Hyper Velocity star data to construct a thermodynamic-based on the Maxwell-Boltzmann probability distribution. The thermodynamic model fits the data very well and allows for further interpretation of the relevant physics as governed by the interplay between mass, morphology, angular momentum and total Galactic energy. Recent data suggests the Galaxy is an open thermodynamic “system” in quasi-equilibrium with its external “surroundings.” Through an analysis of component velocity dispersions and hypervelocity star observations, we conclude the radius of Galactic maximal entropy occurs at 23 kpc. A strong peak in total velocity is the local escape velocity and Maxwell-Boltzmann most probable velocity, Vmp=432 km/s. Thanks to the Winnower for open access publishing, the research community and private communication that make this information available to the larger audience. This paper is dedicated to my dad.
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