1 The Global Escape Velocity Profile and Virial Mass Estimate of The Milky Way Galaxy from Gaia Observations

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The Global Escape Velocity Profile and Virial Mass Estimate of

The Milky Way Galaxy from Gaia Observations

Jeffrey M. La Fortune

1081 N. Lake St. Neenah, WI 54956 forch2@gmail.com

17 October 2022

Abstract

Gaia HyperVelocity Star (HVS) kinematic observations favor a local escape velocity of ~700 km/s, nearly

forty-percent greater than conventional estimates. Combining HVS and dwarf galaxy satellite data reveal

the global escape velocity profile for the Galaxy smoothly traces an unbroken Keplerian decline from the

central bar to the most remote satellite galaxy. We reveal a robust upper bound in baryonic mass

discrepancy (maximal relative accelerations) linked to the virial theorem and obtain a fundamental and

universal mass discrepancy-acceleration relation for virialized compact cosmic objects.

Introduction

Understanding and quantifying the Galaxy’s escape velocity profile is paramount in determining its

mass. In this work, we employ Gaia’s precision kinematical data and the virial theorem to construct a

new mass model consistent with the underlying thermodynamics. We employ the Mass Discrepancy-

Acceleration Relation (MDAR) made popular by McGaugh, Lelli, and Schombert in their seminal paper

establishing the Radial Acceleration Relation (herein termed the MLS RAR) derived from the Spitzer

Photometry & Accurate Rotation Curves (SPARC) disk galaxy database (Lelli, 2016) (McGaugh, 2016).

MLS revealed a tight 1:1 correlation between baryonic and observed accelerations and near zero

dependence with other galactic parameters. Within the MOdified Gravitational Dynamics (MOND)

framework, the RAR is a natural outcome as is the Baryonic Tully-Fisher Relation (BTFR) (Milgrom, 1983).

In ΛCDM, adherence to these scaling relations is not as distinct and the RAR is often used as a check to

ensure consistency between dark matter halos and baryon phenomenology. An argument has been

made claiming NFW halo properties also naturally lead to the MLS RAR (Navarro, 2017).

While the MLS RAR is based on acceleration derived from ‘cold’ ordered motion disk kinematics, our

proposed scaling relation is derived from loosely bound or ‘hot’ Hyper-Velocity Stars (HVS) and the

Galaxy’s dwarf galaxy satellite population. This approach is an independent measure of galactic mass

without resorting to disk components to obtain it. We make extensive use of the MDAR to translate

kinematics to relative accelerations and mass discrepancies to draw out the virial scaling relation. We

then demonstrate the universal nature of this virial scaling relation by extending it to a sample of galaxy

clusters with previously determined RARs.

Many Galactic mass estimates have been conducted with results dependent on tracer selection, spatial

location, methodologies, and models (Gallo, 2022). Our approach employs tracers at the mutual

interface separating the thermodynamic system from its surroundings (near or at local escape

velocities). Our work associates the notion of ‘missing mass’ with the kinetic energy content of virialized,

stationary compact objects and find the virial theorem may offer a solution framework for this mass

discrepancy without ‘new’ physics.

The Mass Discrepancy/Acceleration-Virial Theorem Link

The MDAR provides a unique perspective and tool to explore galaxy models and baryonic scaling

relations utilizing observed (aObs=VObs2(r)/r) and baryonic (aBar=GMBar(<r)/r2) acceleration data. We make

use of the mass discrepancy parameter D=(aObs/aBar) and equivalent forms (VObs2/VBar2) and (MObs/MBar).

The total gravitational potential and virial mass are two properties crucial in the physical description for

any compact, gravitationally bound system. For a galaxy in thermodynamic equilibrium, the virial

theorem relates its time-averaged total potential (U) to kinetic (T) energies simply as 2T=-U. This

theorem also serves as the fundamental basis for many mass estimator models found in the literature

(An, 2011). For galaxies, disk properties provide input for the virial theorem and dynamic mass via

MDyn=VC2RDG-1 (circular velocity VC and radius RD). These same parameters are also used to calculate the

Baryonic Tully-Fisher Relation (BTFR), a principal tenet of MOND.

Likewise, galaxy disk properties and

especially rotation curves have provided a critical constraint in NFW dark matter halo model fits. For the

Galaxy, a new halo fitting constraint is introduced based on Gaia kinematics for sample sets not

associated with the disk.

Prior to Gaia, only a few extreme velocity stars were identified and those had uncertain or missing

proper motions and/or 3D velocities. For many years, black hole ‘ejection’ remained the leading

candidate as calculations demonstrated adequate velocity boost to launch stars beyond Galactic VEsc

(Hills, 1988) (Marchetti, 2022). Following this lead, additional ‘event’ driven ejection mechanisms have

been proposed, most requiring very specific (and rare) conditions that have the capacity to launch stars

to Galactic escape velocity. As data quality and numbers increased, proposed ejection mechanisms have

come to include substructure where close gravitational interactions are more likely. In addition to the

central black hole, HVS nurseries are thought to include star clusters, infalling satellites, stellar binaries,

accreting dwarf galaxies, and supernovae explosions. With Gaia’s precision, HVS orbits can now be back-

integrated in time with results indicating a sizeable fraction potentially originating from the disk; the

main stellar repository (Marchetti, 2021) (Irrgang, 2018).

In related HVS work, Li also points to galactic substructure as having the necessary conditions to launch

stars to hyper-velocities (Li, Q-Z, 2022). Supporting Li’s assertion are complementary studies finding very

few, if any, emanating from the black hole. In his work, Marchetti discovered a sizable HVS fraction with

trajectories having no obvious connection to the Galaxy, deeming these of ‘extragalactic’ origin. We

propose these orphans are fully relaxed stars on highly randomized orbits that no longer reflect initial

conditions/point of origin. The bulk of the HVS population may be better described as a time-averaged

probability distribution related to open-system thermodynamic processes that support the Galaxy’s

current quiescent quasi-equilibrium state. The ultimate bulk speeds and orbital properties of the HVS

population are highly chaotic making it very difficult to predict initial ‘launch’ conditions or place of

origin with high confidence.

Utilizing the ‘scaling’ model, we have previously introduced a generalized form (gMVR) for the BTFR eliminating a0 from the

zero-point expression. Our relation is exactly calculatable from available data as VC4=(πG2DΣDyn)-1MBar for disk parameters: mass

discrepancy (D), and disk dynamic surface mass density (ΣDyn=Mʘ/πRD2) (La Fortune, 2021). Plotting the acceleration ansatz

(πGDΣDyn) [LT-2], In D-ΣDyn space, regression analysis reveals the presence of a common acceleration scale close to a0. Rather

than a universal value attributed to MOND, measured accelerations are dependent on individual galactic properties.

Milky Way Galaxy MDAR Template and ‘Scaling’ Model

The Milky Way Galaxy has become a treasure-trove of kinematic information all due to the Gaia mission

(Prusti, 2016) (Brown, 2018) (Brown, 2021). In this work we leverage this precision data and analyze the

motions of HyperVelocity Stars (HVS) occupying the inner stellar halo and the dwarf satellite galaxy

population surrounding the disk (Du, 2019) (Fritz, 2018).

In Figure 1, the MDAR provides a template linking accelerations with associated mass discrepancies

based on our treatment of the observed kinematics as a virial phenomenon. The basic virial ‘scaling’

model consists of two dynamics. The first (green dash) is disk dynamic mass (MDyn) and virial energies

2T=-U. The second (red dash) provides a virial mass (MVir) estimate based on local escape velocity,

constant mass discrepancy D=12.1, and T=-U.

To simplify the model for our example, we fix dynamic

mass to the average ΛCDM cosmic baryon fraction fb=0.17 (D≈1/fb), acknowledging the wide diversity

exhibited in the local disk galaxy population. The ‘scaling’ model fully encloses all dynamic mass

(MDyn=0.5x1012 Mʘ) and baryon mass (MBar=0.085x1012 Mʘ) within the Galaxy’s 80 kpc diameter disk.

Local escape velocities correspond to a virial mass MVir=1.0x1012 Mʘ (MBar x 12.1). This estimate is in

close agreement with many past surveys – see Figure 8 (Bird, 2022). In Figure 1 below, we probe the

inner halo with Du’s HVS sample (blue cross) and outer/extended regions with (high-quality) dwarf

satellite galaxy data (blue points) from Fritz. We also present the MLS RAR (gray solid) and our dark

matter halo fit based on the simple NFW model (gray dash) and total acceleration with baryon support

(black solid).

We make use of Du’s HVS selection criteria and sophisticated statistics linking MDAR mass discrepancies

with an object’s probability of being gravitationally bound to the Galaxy. In Du’s HVS analysis, local

escape velocity is defined where the probability of HVS being unbound to the Galaxy is fifty-percent

(PUB≈50%). This escape velocity is consistent for the Galaxy’s dynamic mass D=5.9, when it should be

fixed to virial mass D=12.1. In revising Du’s definition, our local escape velocity matches the global

dynamic traced by the dwarf satellite galaxy sample. This revision effectively doubles Galactic mass and

the expected increase in escape velocities.

In this scenario, nearly all satellite galaxies in our sample can

be considered loosely bound, but long-lived residents of the Galactic system. In support, we cite a

complementary survey consisting of HVS (VGC>450 kms-1) obtained by SDSS/APOGEE finding most are

gravitationally bound halo stars (Quispe-Huaynasi, 2022).

In Figure 1 we observe most tightly bound HVS align along constant mass discrepancy D=5.9, the

Galaxy’s dynamic mass. Projecting this to lower accelerations shows Du’s escape velocity reference

corresponds to the disk’s circular velocity at edge (open red circle). In order to maintain a reasonable

escape velocity (VEsc=√2VC) at this radius, the decline is shallower than Keplerian and closer in form to a

NFW dark matter halo.

This mass discrepancy value was established several years ago from a ‘pre-Gaia’ survey of the stellar halo. This value has been

used in previous versions of the ‘scaling’ model and remains an accurate reference point in this age of Gaia (King III, 2015) (La

Fortune, 2019). D=12.1 is an arbitrary value but universal for any virialized structure. The effective or observed accelerations

may be lower than this expectation due to intrinsic dispersion and the strong upper cut-off.

Deason recently obtained a local escape velocity VEsc=528 kms-1 which is very close to Williams VEsc =521 kms-1 (Deason, 2019)

(Williams, 2017). In detail, we find that Deason’s total Galactic mass estimate is almost double Williams suggesting this is still an

unsettled area of interest.

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1TheGlobalEscapeVelocityProfileandVirialMassEstimateofTheMilkyWayGalaxyfromGaiaObservationsJeffreyM.LaFortune1081N.LakeSt.Neenah,WI54956forch2@gmail.com17October2022AbstractGaiaHyperVelocityStar(HVS)kinematicobservationsfavoralocalescapevelocityof~700km/s,nearlyforty-percentgreaterthanconventionales...

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