### 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 advance a physically-based approach conveniently termed the Rotation Curve-Spin Parameter (RC-SP) model to distinguish it from ɅCDM. 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

The RC-SP solution employs a baryonic-based interpretation based on the spin parameter equation, and includes angular momentum and total energy associated with the Galactic “state” (Peebles 1971) (La Fortune 2016). In addition to obtaining galactic dynamics primarily via detailed, extended rotation velocity profiles, this paper strengthens the RC-SP model through high resolution inner stellar halo component dispersion velocities and HVS motion analysis. We assume no dark matter or modified gravitation law, relying on the classical notions of the virial theorem for self-bound gravitational baryonic-only systems:

The RC-SP parameters are; total Galactic dynamic mass (M_{Dyn}),
average rotation velocity (V), and disk radius (R). Properly parameterized, we
expect the Galaxy to obey Newtonian dynamics and find that it does through a
simple recasting of the previous equation:

The virial assumption assures various combinations of R
and V exist for any dynamic mass value. However, all combinations are not
physically possible. There is only a narrow window of permissible R-V
combinations that are specific and appropriate with respect to the galactic
setting. 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 M_{Dyn}
= 0.49x10^{12}M_{ʘ}, very close to 0.5x10^{12}M_{ʘ}
obtained from the RC-SP Keplerian model/curve fit. Figure 1 shows the effective/functional
R-V relation for a range of galaxies from small gas-dominated dwarf irregulars
to massive star-dominated spirals (green dash). The baryon content for M31
(solid circle) is depicted below (no change in R or V) determined from the mass
discrepancy (M_{Dyn}/M_{Bar}) or baryonic fraction.

*Figure
**1**: RC-SP Galactic Dynamic Mass-Velocity-Radius Relation. The
functional M _{Dyn}-R-V relationship (green dash) is the physically
permissible relation based on galactic morphological and kinematic properties. Estimated
M31 baryon mass; f_{b}=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 2016) for
details.*

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, illustrated
above for M31 as corresponding to f_{b}=0.17, the ɅCDM cosmic average.
We anticipate most dynamic mass must originate where most baryons reside, inside
40 kpc. 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. This composite rotation curve is reproduced 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). Beyond the disk, the rotation curve declines in accordance with Keplerian dynamics.

*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). 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 Bajkova’s data in the range of 20 to 24 kpc were excluded with the veracity of the data in doubt. We have interpolated the rotation curve within this short span from the available data as a velocity peak. We see than all three ɅCDM models do not accurately reproduce the Galaxy’s physical rotation curve. The fact that the most obvious kinematic substructure in the Galactic rotation curve has been excluded in ɅCDM models suggests this feature does not inform the dark matter model to any significant degree.

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. This profile is not
an artifact of the Bhattacharjee/Bajkova data, as it is also present in other
extended rotation curves of the Galaxy (Sofue 2015) (Huang 2016). This
disconnect between ɅCDM model and the physical rotation curves is obvious with
no reasonable physical explanation offered. Throughout this paper, M_{Dyn}
≈ M_{Total }- equivalent to ɅCDM total Galactic mass (M_{Bar} +
M_{DM}), where M_{DM} is the purported mass of the dark matter
halo.

### The Observed Kinematics of the Galactic Stellar Halo

With respect to the pronounced velocity spike at 23 kpc, we employ recent results of a high-resolution survey of component velocity dispersions observed within the inner 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) from King III.

Superimposed is an ideal isotropic model, as individual
components) corresponding to a total baryonic ‘point’ mass M_{Bar} =
8.5x10^{10}M_{ʘ}. (M_{Bar}/M_{Dyn}) ≈ f_{b}
= 0.17 (black dash). The average Galactic rotation velocity, 230 km/s is
included for completeness (blue dash). Panels 3a., 3b., and 3c., are observed
component stellar halo velocity dispersions and panel 3d. is the associated anisotropy
coefficient, β. From a purely qualitative/visual standpoint, any net positive
deviation between the isotropic curves 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. The
average disk rotation velocity is 230 km/s (blue dash). A spherical ‘point
mass’ isotropic curve (black dash) corresponds to the baryonic content of the
Galaxy. Lower right panel, orbital character of the halo is defined by the
anisotropy coefficient, β = [1-(σ ^{2}_{Azimuthal-θ}+σ^{2}_{Polar-φ})/2σ^{2}_{Radial-R}]
(solid black). β=1 for perfectly radial orbits, β=0 for the isotropic case and
β→ -∞ describing pure circular motion. Note in panel d., β≈-25 signifying
highly circular orbits. 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 at peak
velocity. We confirm that the kinematic break in the
Bhattacharjee/Bajkova rotation curve is attributable to exceedingly high
velocity circular orbits which create high uncertainties in the data around 23
kpc. 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

We advance a thermodynamic origin for the observed population distribution of Hypervelocity Stars (HVS) that has received recent attention for a method to quantify the mass of the dark matter halo. HVS stars are those that exceed the local escape velocity of the Galaxy. Currently, the HVS population is thought to acquire their extreme velocity through intense gravitational interaction (ejection) with central black holes or exploding supernovae (Tauris 2015) (Fragione 2016a) (Rossi 2016) (Fragione 2016b).

We don’t subscribe to chance encounters as sourcing the HVS
population, but rather thermodynamics considerations which are related to the
velocity peak at 23 kpc. We believe the observed HVS sample of is a
manifestation of a thermodynamically open system with the HVS population characterizing
mass exchange with surroundings to maintain quasi-equilibrium. Figure 4 below
employs King’s velocity dispersion data (blue dash) for a comparison between RC-SP
and dark matter halo simulations (see insert, Fig. 4), including the observed HVS
sample as a function of velocity and radius. We link King III’s velocity peak (432
km/s) as the local Galactic escape velocity (V_{esc}) near 23 kpc.

*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 (432 km/s) at 23 kpc for a dynamic mass M _{Dyn}=0.5x10^{12}M_{ʘ}.
Average rotation velocity and Keplerian decline included for perspective, as
labeled (black dash). Image source – Fig 2 (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} (black 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 M _{ʘ}) HVS distribution (gray histograms) from Brown 2014. The
alternative RC-SP/Maxwell-Boltzmann probability distribution (blue dash) is based
on most probable velocity V_{mp}=432 km/s (black dash) with arbitrary
vertical scale. Image source - Fig. 3 (Rossi
2016)*

In Figure 5, we cannot discriminate between models
(ejection or thermodynamic) using HVS velocities less than V_{mp}. Therefore,
we need to look beyond V_{mp} and include lower stellar mass ~0.9M_{ʘ}
G/K dwarfs shown in Figure 6 below. As expected, the G/K dwarfs have greater
net velocity and accurately trace the upper velocity tail of the M-B
distribution (blue dash).

*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. The M-B average velocity is near the oft
cited Vesc≈550 km/s escape velocity value for the MW. Figure corrected/updated
Feb17. 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 quite
well. In defining this distribution, we also need to account for the stellar
universal Initial Mass Function (IMF) which is a distribution of stellar mass (Baldry 2003) (Offner 2014). Galactic IMF distributions exhibit a peak in stellar mass
between 0.2M_{ʘ} to 4M_{ʘ}. We would expect to have a
population of HVS in the mass range 1-2 M_{ʘ} leading to a fairly
conventional M-B distribution. Perhaps the dearth of HVS observed in this mass
range is 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, but is expected to be populated with lower
velocity HVS with masses >3M_{ʘ}.

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 the “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 with peak velocity exhibiting very circular orbits, β≈-25.

### Conclusions

Recent data suggests the Galaxy is an open thermodynamic “system”
in quasi-equilibrium with its external “surroundings.” Under this model, we
employ observed kinematics of the latest sample of ‘unbound’ stars (Hyper
Velocity Stars) to construct a fit to 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. 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 equivalent 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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