### Abstract

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, V_{mp}=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.

### Introduction

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 (M_{Dyn}),
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 0.49x10^{12}M_{ʘ}, very close
to 0.5x10^{12}M_{ʘ} 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 M_{Bar}/M_{Dyn} 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 M_{Dyn}=0.5x10^{12}M_{ʘ}, in
agreement with a very precise Bayesian estimate of total Galactic mass, M_{Total}
= 0.522x10^{12}M_{ʘ} < 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, M_{Dyn}
≈ M_{Total }and is equivalent to ɅCDM-based total Galactic mass (M_{Bar}
+ M_{DM}).

### 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 M_{bar}
= 8.5x10^{10}M_{ʘ}. 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-(σ^{2}_{Azimuthal-θ}+σ^{2}_{Polar-φ})/2σ^{2}_{Radial-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 (V_{esc}).

*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 (V_{mp}). This is shown in
Figure 4 as the intersection of the physical velocity peak and the calculated
escape velocity (V_{esc}). It is not coincidental that this peak velocity
complements the estimated escape velocity for a point mass equivalent to a
dynamic mass M_{Dyn }= 0.5x10^{12}M_{ʘ} (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
3.5M_{ʘ}.

*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 V_{mp},
we cannot discriminate between the models. Therefore, we need to look beyond V_{mp}
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 ~0.9M_{ʘ} G/K dwarfs, we expect them
to have a higher net velocity than late-B stars. It is in this velocity range
(>V_{mp}) 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 V_{mp},
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 0.2M_{ʘ} to 4M_{ʘ}
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 M_{Dyn} = 0.5x10^{12}M_{ʘ},
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 M_{ʘ }(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.

### Conclusions

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, V_{mp}=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.

### Appendix

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