### Abstract

We present the first detailed physically-based
thermodynamic model of the Galaxy as obtained from high-resolution kinematical data
from the inner stellar halo. 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 fully virialized galactic system existing
in quasi-equilibrium with its surroundings. The conventional view is that the
HVS sample originates from chance gravitational encounters that have attained
enough speed to escape the Galaxy. We counter that the HVS population is
created by thermodynamic effects and link the observed mid-disk velocity peak with
the M-B most probable velocity V_{Peak} ≈ V_{mp}=432 km/s. Most
of the HVS population originates at the Galactic virialization radius of 23 kpc
with the current observed sample reproducing the M-B distribution with fidelity.

### Introduction

Hyper Velocity Stars (HVS) have become a valuable tool to constrain the total dynamic mass of the Milky Way. We investigate a thermodynamic origin for the HVS population and compare it against popular ejection (slingshot) mechanisms. We perform a combined analysis (HVS sample and stellar halo kinematics) and comfortably fit a Maxwell-Boltzmann distribution to these data sets, a result consistent with virialized Galaxy models. In this paper, we advance a physically-based approach conveniently termed the Rotation Curve-Spin Parameter (RC-SP) model to distinguish it from ɅCDM.

### The Virial 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 from extended rotation
velocity profiles, this paper strengthens the RC-SP model via precision
measures of inner stellar halo component dispersion velocities and HVS “escape velocity”
analysis. The RC-SP approach is based on two classical equations, the Virial
Theorem and Newton’s second law for circular motion. The Virial Theorem is
expressed below and includes the constraint which limits the theorem to
isolated, self-gravitating systems in “equilibrium” and is equally applicable
to dark matter halos or baryonic disks. Only two parameters are required to
determine the global “system” properties of galaxies, R_{Virial} and V_{Esc}
(where R_{Virial} ≈ R_{Disk}):

Although Newtonian dynamics ensures V_{Circ} = V_{Esc}/√2
at R_{Virial}, it cannot provide a defined V_{Circ }profile as
a function of radius. ɅCDM theory removes this difficulty as dark matter halos
conveniently have constant V_{Circ} (flat virial halo rotation)
permitting use of Newton’s second law for circular motion:

This substitution effectively
decouples galactic V_{Esc} (and by association, V_{Circ}) from any
significant baryonic influence. Rather than constraining V_{Circ} to a
theoretical value throughout R_{Virial}, we treat V_{Circ} as a*
direct observable,* now possible with the availability of accurate component
velocities measured between 6 and 30 kpc, a region spanning both the inner and
outer Galactic disks (King III 2015). In following sections we explore the
implications of non-flat circular velocity against the latest, most
sophisticated ɅCDM simulations. We leverage the unbound Hyper Velocity Star
(HVS) population to estimate Galactic V_{Esc} and provide a physical
(classical) origin and explanation for King’s recently discovered kinematic feature.

### Galactic Rotation – RC-SP versus ɅCDM Models

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) and the Keplerian decline beyond. Into the original figure, we have inserted a model of King’s velocity peak (gray dash) where observations are entirely missing. As shown below, all three ɅCDM rotation curves smoothly span this range, perhaps unaware of the recent discovery of this kinematic feature.

*Figure
**1**: 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 average rotation velocity
(blue dash) and the Sun’s velocity and position (red dot) is provided for
reference. Image source – Fig.1 (Bajkova 2016)*

In the above figure, all three ɅCDM models conflict with the Galaxy data especially at outer radii where it is evident that a conventional Keplerian decline provides a better fit than the flat rotation prediction (Sofue 2015) (Huang 2016). The gray dashed curve in the region of Bajkova’s missing data is based on a recently measured kinematical feature in the inner stellar halo. We challenge the notion this feature is a perturbative/ transient halo artifact, contending it is long-lived and thermodynamic in origin.

### Observed Kinematics of the Galactic Stellar Halo

The kinematics of the stellar halo serves as a sensitive probe of Galactic dynamics especially within ɅCDM cosmology where the stellar and dark matter halos share the same space. We examine in detail the properties of the stellar halo recently obtained from King’s high precision component velocity dispersion survey spanning 6 to 30 kpc.

Our focus is a previously identified kinematic feature termed the “tangential dip.” New observations have resulted in this dip becoming a significant trough, creating more tension between galactic kinematics and ɅCDM and MOND model expectations. The main take-away is that this this stellar feature should not be discounted or ignored within any truly accurate model of the Galaxy.

Figure 2 shows King’s component velocity dispersion
results (R, φ, and θ). We have annotated King’s data with isotropic dispersion (black
dash) corresponding to a central baryonic mass M_{Bar} = 0.085x10^{12}M_{ʘ}
and average Galactic rotation velocity (blue dash). The lower right-hand panel
provides the anisotropy coefficient β, demonstrating the extremity of this “dip”
that translates to high velocity circular orbits.

*Figure
**2**: a., b., and c. panels provide observed velocity
dispersions for radial, polar and azimuthal components. Included with the
original data is the MW average disk rotation velocity V _{Circ} = 230
km/s (blue dash) and the isotropic baryonic ‘point mass’ curve (black dash). In
the lower right panel, Extreme depression in anisotropy coefficient, β = [1-(σ_{
θ}^{ 2}+σ_{ φ}^{ 2})/2σ_{ R}^{ 2}]
denotes a spike in highly tangential orbits near mid-disk. To date, this
kinematic substructure has not been incorporated into *

*ɅCDM models of the Galaxy. Image source - Fig. 10 (King III 2015)*

The dynamic mass distribution within the Galaxy is roughly traced as the net positive difference in velocity between the baryonic isotropic curve (black dash) and particular dispersion components. We find very little velocity support from the radial, with azimuthal and vertical components being dominant. This particular kinematic substructure cannot be reproduced within the context of the theoretical properties of dark matter halos.

In the next section, we next
construct a dynamical model that relies on this complex but subtle kinematic
substructure. We combine King’s results with those obtained from Hyper Velocity
Star (HVS) surveys to construct a physically consistent (kinematical,
dynamical, and thermodynamical) model of the Galaxy. Note we emphasis a single RC-SP
Galaxy model based on *observation* rather than simulation ad hoc “best
fits.”

### Hyper Velocity Star Orbital Parameters

In this section, we advance a thermodynamic origin for
the observed population distribution of Hypervelocity Stars (HVS). This unbound
stellar population of stars is receiving attention as a method to quantify the dynamic
(or dark matter halo) mass of the Galaxy. Currently, the HVS population is
thought to acquire extreme velocities through intense gravitational ejection
mechanisms deep inside the Galactic core (Tauris 2015) (Fragione 2016a) (Rossi 2016) (Fragione 2016b). Dynamic masses obtained by ejection mechanisms rely on “chance”
encounters (constrained to the Galactic center) and complex three-body
gravitational interactions to obtain a HVS model population. Figure 3 shows the
HVS sample space against four dark matter halo mass models (see inset). In this
figure, Fragione regarded stars V_{Obs} >275 km/s as “unbound” and a
HVS candidate. To this original figure, we include King’s summed component
velocities in quadrature (blue dash) with its measured peak velocity of 432
km/s at 23 kpc. Just beyond the peak, we find Galactic velocities plunge into a
conventional Keplerian decline beyond the baryonic disk equivalent to M_{Dyn}
= 0.5 x10^{12}M_{ʘ} and not the excessively high dark matter
halo masses depicted.

*Figure
**3**: Hypervelocity star sample population R-V space for the
MW compared against RC-SP radial escape velocity profile (bold blue dotted) and
four dark matter halo models from Fragione (various black - see inset for
details). King III quadrature summed velocity dispersion components (blue dashed
spike).As the MW is a virialized system, we assign the MW’s 432 km/s peak as
the Maxwell-Boltzmann “most probable” velocity. RC-SP parameterized average
disk and Keplerian decline (both black dashed) as labeled. Image source – Fig 2
(Fragione 2016a)*

The above figure highlights an issue which has been
plaguing ɅCDM cosmology since inception, the tremendous insensitivity between
halo properties and observation. Due to this lack of connection and high uncertainty
between the dark matter and baryonic constituents, a particular halo model with
a halo mass between 1.2 – 1.7 x10^{12}M_{ʘ} could only be
“favored” over the others. In effect, halo mass uncertainty is equivalent to
the RC-SP dynamic mass of the entire Galaxy. From Figure 3, we certainly
observe a link between V_{Peak} at 23 kpc as the virial radius of
origin for the HVS population. As such, the HVS sample should be distributed
based on thermodynamic considerations. The next step is to assign a *physical
mechanism* responsible for this particular profile for the HVS sample
population – the Maxwell-Boltzmann probability distribution.

### A Thermodynamic Solution to Explain HVS Sample/Population Statistics

In this section we focus on the Maxwell-Boltzmann form and
define peak velocity equivalent to the most probable velocity V_{mp} =
432 km/s within the distribution with (Wu 2014):

Figure 5 below compares Rossi’s expectation for the HVS velocity distributions (red dash and solid black) based on the gravitational ejection model. As it appears, the dearth of data beyond the peak indicates a narrow distribution at high velocities directly attributable to the very deep gravitational well of the dark matter halo. The M-B distribution (blue dash) shows a significant high velocity tail should be present.

*Figure
**4**: 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)*

As shown above, Rossi contends that the linear decline in
HVS distribution in the low velocity tail is expected. We find the M-B
distribution (blue dash) in this region is actually more linear than either
ejection model, but no true discrimination between models is possible < 432
kms^{-1}.

At the high velocity end of the distribution, Rossi contends the steep decline in the model is real with HVS becoming increasing rare at higher velocities. We contend it is the high velocity tail of the HVS distribution that distinguishes the M-B solution over chance gravitational encounters. In Figure 6 below, we expand the HVS sample distribution beyond ≈400 km/s by including lower stellar mass G/K dwarfs (Tauris 2015). In the figure below, the absence of HVS < 350 km/s is due to arbitrary truncation of the data.

*Figure
**5**: 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, V _{mp}= 432 km/s. The M-B vertical scale is
matched to the data for comparative purposes. The M-B average velocity is V_{Avg}
= 487 km/s and root-mean-squared V_{rms}=529 km/s. This latter value is
near the oft cited global Galactic escape velocity of 550 km/s. Image source - Fig.
1 (Tauris 2015)*

We find HVS stellar mass tracks well with the M-B
solution, with lighter G/K dwarfs exhibit greater net velocity than heavier
late B-stars, accurately tracing the overall M-B distribution and the magnitude
of escape velocities in relation to King’s velocity dispersion results. Of
course, the stellar universal Initial Mass Function (IMF) needs to be
considered as it directly influences the mass of the star that could become a
HVS candidate, i.e., the IMF exhibits a peak in stellar mass between 0.2M_{ʘ}
to 4M_{ʘ} (Baldry 2003) (Offner 2014). We would expect the M-B
distribution would become fully “occupied” but this (to date) is not the case. Either
the missing data is due to severe under-sampling or a more subtle effect not
yet fully understood.

This thermodynamic interpretation is physically consistent with a virialized, quasi- equilibrium system in highly ordered motion (Struck 2016). Struck interpreted this ordered motion as “free energy” that will be thermalized in the future. We contend ordered galactic motion is “potential energy” contributing to total dynamic mass today.

### 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 inner stellar halo and the latest sampling of Hyper Velocity Stars to indicate a thermodynamic origin without imposing any deviation from classical mechanics. 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 – King’s Stellar Inner Halo Velocity Dispersion

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