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Gaia and the Galactic Center Origin of Hypervelocity Stars

We use new Gaia measurements to explore the origin of the highest velocity stars in the Hypervelocity Star Survey. The measurements reveal a clear pattern in the B-type stars. Halo stars dominate the sample at speeds about 100 km/s below Galactic escape velocity. Disk runaway stars have speeds up to 100 km/s above Galactic escape velocity, but most disk runaways are bound. Stars with speeds about 100 km/s above Galactic escape velocity originate from the Galactic center. Two bound stars may also originate from the Galactic center. Future Gaia measurements will enable a large, clean sample of Galactic center ejections for measuring the massive black hole ejection rate of hypervelocity stars, and for constraining the mass distribution of the Milky Way dark matter halo.

1. INTRODUCTION Hills (1988) first proposed that a 3-body exchange between a pair of stars and a massive black hole (MBH) can eject "hypervelocity stars" (HVSs) at 1000 km s −1 velocities from the Galactic center. We discovered the first HVS (Brown et al. 2005). This 3 M ⊙ main sequence B star moves with a Galactic rest frame velocity >670 km s −1 , about twice Galactic escape velocity at its current distance of 100 kpc. Only a gravitational interaction with a massive compact object can plausibly explain its motion.
The discovery of HVS1 inspired the HVS Survey, a targeted radial velocity survey of B-type stars that should not exist at faint magnitudes in the halo (Brown et al. 2006(Brown et al. , 2007b. Twenty one stars are significantly unbound in radial velocity alone ). The extreme velocities, the short-lived nature of the stars, their distribution in Galactic latitude, and their overall numbers match theoretical expectations for the Galactic center origin proposed by Hills (1988). However the measurements provide only an indirect link to the MBH.
Alternative origins for unbound stars include ejection from the Galactic disk through binary disrup-tion (Blaauw 1961;Poveda et al. 1967) and ejection from the Large Magellanic Cloud (Boubert & Evans 2016;Boubert et al. 2017). Galactic disk ejections are called "runaways" (Blaauw 1961;Greenstein & Sargent 1974). The disruption of a binary by a supernova, where the surviving star is released at the orbital velocity of the progenitor binary, can yield unbound runaways in extreme circumstances (e.g. Tauris 2015). The first observational example of an unbound runaway is the B star HD 271791 (Heber et al. 2008;Przybilla et al. 2008a). The first example of an unbound Large Magellanic Cloud ejection is the B star HE 0437−5439 (Edelmann et al. 2005;Przybilla et al. 2008b;Erkal et al. 2018). Compact objects like white dwarfs can have higher binary disruption ejection velocities than main sequence stars. The first observational examples are the unbound subdwarf O star US 708 (Hirsch et al. 2005;Justham et al. 2009;Wang & Han 2009;Geier et al. 2015), the white dwarf LP 40 − 365 (Vennes et al. 2017;Raddi et al. 2018), and three new white dwarfs found with Gaia (Shen et al. 2018).
The European Space Agency mission Gaia has begun a new era of precision astrometry. The trajectories of unbound stars hold the key to their origin. Measuring radial velocity with km s −1 precision is easy with modern spectroscopy; measuring tangential velocity, the product of distance and proper motion, is difficult. Known hypervelocity stars are at distances of 50 to 100 kpc; their expected proper motions are <1 mas yr −1 . Newly released Gaia Data Release 2 provides improved proper motions for many HVSs (Gaia Collaboration et al. 2018a). Here, we use Gaia measurements to determine the origin of stars from the HVS Survey (Brown et al. 2007b on the basis of their trajectory and velocity. In Section 2 we define the sample and compare Gaia proper motions with previous Hubble Space Telescope (HST) measurements. In Section 3 we evaluate the origin of these stars on the basis of computed trajectories and ejection velocities. The results are in Section 4, and we conclude in Section 5. We identify Galactic center HVSs, Galactic disk runaways, and Galactic halo stars with different but overlapping velocities; the highest velocity stars are probably Galactic center ejections.

Sample
We study 42 radial velocity outliers from the Hypervelocity Star Survey (Brown et al. 2007b. We include all stars with heliocentric radial velocity transformed to the Galactic frame v rf >+275 km s −1 , v rf = v helio +11.1 cos l cos b+247.24 sin l cos b+7.25 sin b, (1) where l and b are Galactic longitude and latitude, respectively, and we assume the Sun is moving with respect to the local standard of rest as measured by Schönrich et al. (2010) and the Galactic disk circular velocity is 235 km s −1 (Reid et al. 2014). We choose v rf >+275 km s −1 because of the significant absence of negative velocity outliers in the HVS Survey. The four most negative velocity stars have −300 < v rf < −275 km s −1 , implying that the 42 stars with v rf > +275 km s −1 are a relatively clean sample of ejected stars with minimal halo star contamination.
The HVS Survey targeted stars selected by color  with no kinematic selection. The spectroscopy is 99% complete. Stellar atmosphere fits establish that the majority of unbound stars are main sequence B stars (Brown et al. 2014, 2015, andreferences therein). The bound stars are also probably main sequence B stars on the basis of their velocity distribution. The absence of −300 km s −1 stars in the HVS Survey implies that the +300 km s −1 stars must have lifetimes less than their ∼1 Gyr orbital turnaround time (Brown et al. 2007a;Kollmeier & Gould 2007;Yu & Madau 2007). Thus they are B stars, and we calculate stellar distances using Padova mainsequence tracks (Girardi et al. 2004;Marigo et al. 2008;Bressan et al. 2012) with an estimated precision of 15%.
We transform heliocentric distances to the Galactic frame assuming that the Sun is located 8 kpc from the Galactic center (Camarillo et al. 2018).
The HVS Survey exclusively samples the stellar halo. The 17 < g < 20.25 apparent magnitude limit corresponds to approximately 25 < R < 120 kpc in Galactocentric radial distance. For completeness, we include five stars from the bright 15 < g < 19.5 portion of the initial HVS Survey (Brown et al. 2007b). The bright stars are nearby 10 < R < 30 kpc and bound, but four have HST proper motion measurements for comparison with Gaia. Table 1 lists the sample of 42 stars with their coordinates and apparent g-band SDSS magnitude. We sort the table by decreasing v rf . We refer to bound objects with the letter B followed by their target number in the HVS Survey, and unbound objects by their published HVS number. We also list the east-west and north-south components of proper motion, (µ α , µ δ ), obtained from three sources.

Proper Motions
Gaia Data Release 2 contains proper motions for 39 of the 42 stars listed in Table 1. The three missing stars (HVS14, HVS23, and B149) have too few Gaia measurements for a robust solution. We drop them from further consideration. The remaining 39 velocity outliers have median Gaia proper motion error ±0.73 mas yr −1 . The brightest stars have errors of only ±0.11 mas yr −1 . Brown et al. (2015) publish independent proper motion measurements made with HST for 14 stars. The HST proper motions were measured relative to background galaxies. They thus provide a test of Gaia's absolute reference frame. Figure 1 (upper panel) plots the difference between Gaia and HST proper motion divided by the errors summed in quadrature. Clipping two >3σ outliers (see below), the average difference ∆µ α (Gaia − HST ) = −0.09±0.29 mas yr −1 and ∆µ δ (Gaia−HST ) = −0.15± 0.25 mas yr −1 consistent with zero.

HST Comparison
The comparison also reveals that the most problematic HST measurements are the brightest stars. For bright stars, Brown et al. (2015) paired short and long exposures to tie the stars to faint background galaxies. This approach likely introduced additional error. We add ±0.5 mas yr −1 in quadrature to the published HST error for those objects (B434, B485, B711, B711, HVS7, HVS8). The median HST proper motion error for the 14 stars is then ±0.91 mas yr −1 ; the best-measured star has an error of ±0.34 mas yr −1 .  The HST measurements highlight the value of obtaining pointed observations with long exposure times. Although Gaia errors are 3 times better than HST errors for bright stars, HST errors are 4 times better than Gaia errors for faint stars like HVS1. Errors are comparable in size at g ≃ 18.5 mag. A weighted mean would maximize the information from HST and Gaia, however we do not want to add measurements that include possible systematic error.
We adopt Gaia proper motions for the seven bright g < 18 stars with σ HST > 2σ Gaia . This subset includes all stars observed with paired short+long exposures in the HST program. We adopt a weighted mean for the three 18 < g < 19 stars where σ HST and σ Gaia are within a factor of two (HVS4, HVS6, and HVS9). We adopt the HST proper motions for the four g > 19 stars with σ Gaia > 2σ HST (HVS1, HVS10, HVS12, HVS13).

GPS1 Comparison
We compare Gaia proper motions with the Gaia-PanStarrs1-SDSS catalog (GPS1, Tian et al. 2017). GPS1 proper motions are based on astrometric positions from ground-based PanStarrs1, SDSS, and 2MASS  Despite the longer time baseline of ground-based observations, σ GPS1 ≃ 5σ Gaia . Thus we do not use GPS1 values here. The median GPS1 proper motion error for the 33 stars is ±2.69 mas yr −1 ; the best-measured star has an error of ±1.68 mas yr −1 .

ANALYSIS
We evaluate the probability of origin on the basis of computed trajectories and ejection velocities. We select a gravitational potential model, trace the trajectory of each star backwards in time, and calculate the required ejection velocity from that position in the Milky Way.
We estimate likelihood from the density distribution of simulated trajectories as they cross the Galactic plane.

Gravitational Potential Model
To address the origin of HVS ejections from the Galactic center, we require a gravitational potential model that fits observed mass measurements from the Galactic center to the outer halo. We adopt the three component bugle-disk-halo model of Kenyon et al. (2014). The model has a flat 235 km s −1 rotation curve and a 10 12 M ⊙ halo mass consistent with recent Gaia measurements from the orbits of globular clusters and dwarf galaxies (Gaia Collaboration et al. 2018b;Fritz et al. 2018;Posti & Helmi 2018;Watkins et al. 2018).
The results are insensitive to the choice of potential model because the stars are on nearly radial trajectories. However, the choice of potential model determines whether the stars are bound or unbound.
We determine effective Galactic escape velocity, v esc , by dropping a test particle from rest at the virial radius. At the Solar circle R = 8 kpc, v esc = 580 km s −1 consistent with Solar neighborhood escape velocity measurements (Smith et al. 2007;Piffl et al. 2014). At the radius of influence of the supermassive black hole, v esc ≥ 900 km s −1 (Kenyon et al. 2008). Only a gravitational interaction with the supermassive black hole can eject a main sequence star at >900 km s −1 (Hills 1988). At the median R = 55 kpc depth of the HVS Survey sample, v esc = 350 km s −1 .

Trajectory Calculation
Calculations are straightforward for the nearly radial trajectories of the stars. We start from the measured position and velocity of each star, and integrate its trajectory backward in time through the Galactic potential. We record where each trajectory crosses the Galactic plane, and its distance from the Galactic center. We also record the ejection velocity required to launch the star from the disk-crossing location, given the angle of the trajectory relative to Galactic rotation. Figure 2 plots the proper motion measurement for each star. Because we know position and radial velocity, a proper motion in Figure 2 corresponds to a unique trajectory through the Galaxy given a gravitational potential model. The blue ellipses in Figure 2 are the loci of proper motions with trajectories that cross the Galactic plane at R = 20 kpc, the edge of the Milky Way disk. The red circle is the region of minimum Galactic disk ejection velocity, the region where Galactic disk ejections can most easily escape the gravitational pull of the Milky Way. A Galactic Center trajectory is marked by the large black dot. Measurement errors broaden the range of possible trajectories. Thus we draw final conclusions from the density distribution of possible trajectories. For each star, we perform 10 6 Monte Carlo trajectory calculations assuming that measurement errors are normally distributed. Distance and proper motion are the dominant sources of error. We then measure the density of simulated trajectories as they cross the Galactic disk plane.
We evaluate likelihood of origin based on the 0.3173 (1σ), 0.0455 (2σ), and 0.0027 (3σ) thresholds of the trajectory distribution at the Galactic plane. This approach is valid for testing the origin of unbound stars that cross the Galactic plane only once, or bound stars with lifetimes less than their orbital turn-around time; this approach is invalid for long-lived stars that cross the plane more than once. The measurements provide trajectory constraints for about half of the sample, namely for the stars with R < 60 kpc.

Ejection Velocity
The ejection velocity required to explain the present position and motion of each star provides another constraint on its origin. In the absence of a massive black hole, the speed limit for ejection from a stellar binary is set by the finite sizes of the stars (Leonard 1991). The orbital velocity of an equal-mass pair of stars separated by their radii is equal to the escape velocity from the surface of the stars. Because stars on the main sequence have a quasi-linear relation between mass and radius, most stars in the Milky Way share a common escape velocity from their surface of about 600 km s −1 . To achieve higher binary orbital speeds, main sequence stars would have to orbit inside each other, which is impossible. A 600 km s −1 speed limit is optimistic; the speed can only be lower if mass transfer, tidal heating, or binary evolution are taken into account (e.g. Fregeau et al. 2004;Renzo et al. 2018).
The ejection velocities required for the fastest stars in the HVS Survey exceed 600 km s −1 . Thus the ejection velocities demand a Galactic center origin. For stars near Galactic escape velocity, however, there is finite region of the Milky Way disk where the stars can be ejected at <600 km s −1 . The region is set by the Milky Way gravitational potential and the rotation of the Milky Way disk: the ejection velocity minimum is located in the outer disk, at the position where the disk rotation vector points in the direction of ejection (Bromley et al. 2009). Convolved with the power-law distribution of runaway ejection velocities (Portegies Zwart 2000;Perets & Subr 2012;Renzo et al. 2018), the most probable disk runaway origin location is  this region of minimum ejection velocity. We mark the minimum disk ejection velocity region for each star in Figure 2 with a red ellipse.

CONSTRAINTS ON ORIGIN
We identify three classes of objects with distinct but overlapping velocity distributions in our sample: 1) Galactic center HVSs, 2) Galactic disk runaways, and 3) Galactic halo stars. Figure 3 and Figure 4 summarize the results. Figure 3 plots v rf versus Galactic radial distance R. The dashed line is the Galactic escape velocity curve. The symbol color indicates the likely origin of each star. Figure 4 groups the origins together and plots them relative to Galactic escape velocity, v rf − v esc . A total of 20 objects have robust constraints.

Galactic Center Hypervelocity Stars
A Galactic center origin is statistically preferred for all of the fastest stars with v rf > +500 km/s (HVS1, HVS4, HVS5, and HVS6). The trajectories currently provide 2σ constraints. The velocity itself provides an additional physical constraint for the unbound stars: the minimum Galactic disk ejection velocity is comparable to the escape velocity from the surface of the stars, a severe challenge to disk ejection. We identify seven probable Galactic center HVS ejections.
Two bound stars, B537 and B598, have trajectories that point directly from the Galactic center, and reject the Galactic disk origin hypothesis at >3σ significance. To better understand these objects, Figure 5 plots the probability contours where these two objects cross the Galactic plane in Cartesian coordinates. The red circles in Figure 5 mark the regions of minimum Galactic disk ejection velocity, excluded at 3σ confidence. The red lines in Figure 5 mark the locus of trajectories with 500 km s −1 Galactic disk ejection velocities. Thus it is possible that B537 and B598 are high speed Galactic disk ejections. A factor of 2 improvement in proper motion error would exclude this possibility for B598. It is also possible that B537 and B598 are halo stars on very radial orbits. High resolution spectroscopy can determine whether these are metal-poor halo stars or main sequence B stars.
Extrapolating these results to the unconstrained half of the sample, we expect that about half of the unbound stars in the HVS Survey are Galactic center ejections. Brown et al. (2014) thus overestimate the Galactic center ejection rate of by a factor of two. The revised ejection rate of unbound 2.5-4 M ⊙ HVSs from the Galactic center is ∼7 × 10 −7 yr −1 , in better agreement with theoretical ejection rate calculations for the MBH ejection scenario (Zhang et al. 2013).

Disk Runaways
We estimate the likelihood of Galactic disk origin by averaging the trajectory density over a 5 kpc radius region centered on the position of minimum Galactic disk ejection velocity. This metric can be pessimistic for the nearest bound objects, but it is well-matched to the average distance and proper motion errors of the sample. We identify nine disk runaways with trajectories inconsistent with the Galactic center hypothesis at ≥3σ significance, and statistically consistent with a Galactic disk ejection.
Disk runaways and Galactic center HVSs have comparable frequency at speeds around the Galactic escape Figure 5. Density of simulated trajectories (magenta region = 1σ, cyan region = 2σ, yellow region = 3σ) where bound HVS candidates B537 and B598 cross the Galactic plane, in Cartesian coordinates. The Galactic Center (black dot) is the origin most consistent with the measurements. Trajectories from the region of minimum Galactic disk ejection velocity (red circle) are excluded at 3σ confidence, however 500 km s −1 Galactic disk ejections (red line) are allowed. velocity ( Figure 4). The fastest disk runaways have v rf − v esc ≃ +100 km s −1 . However, most disk runaways are bound.
Interestingly, the three unbound disk runaways are spectroscopically unusual. HVS8 is one of the fastest rotators with v sin i ≃ 300 km s −1 (López-Morales & Bonanos 2008; Brown et al. 2014;Irrgang et al. 2018). HVS7 and HVS17 are chemically peculiar B stars (Przybilla et al. 2008c;Brown et al. 2013). HVS7 is 10-100 times underabundant in He and in CNO relative to the Sun, and 100-10,000 time over-abundant in iron group and rareearth elements (Przybilla et al. 2008c). It is difficult to say whether abundance patterns are linked to a supernova binary disruption origin, however, because the abundance patterns of HVS7, HVS17, and HD 271791 (Przybilla et al. 2008a) differ significantly.
The frequency of unbound runaways is linked to their ejection rate. Theoretical ejection models predict a power-law distribution of ejection velocities (Portegies Zwart 2000;Perets & Subr 2012;Renzo et al. 2018). In these models, >99% of runaways are launched from the disk with <200 km s −1 speeds. Simulated distributions of runaways in the Milky Way therefore predict a flattened spatial distribution of runaways with a scale height comparable to the thick disk (Bromley et al. 2009). We expect that magnitude-limited surveys of less luminous types of stars than B stars will find many more runaways near the disk.

Halo stars
Halo stars first appear at v rf < +300 km s −1 . There are four bound objects with trajectories that signifi-cantly reject both Galactic center and Galactic disk origins, and which cross the disk in the direction opposite Galactic rotation. These stars are likely halo stars, presumably hot blue horizontal branch stars with temperatures and surface gravities similar to main sequence B stars. The number of halo stars is consistent with the number of −300 < v rf < −275 km s −1 negative velocity outliers in the HVS Survey. At even lower v rf < 250 km s −1 velocities, halo stars completely dominate the HVS Survey .

CONCLUSIONS
Gaia proper motions enable distinction between true Galactic center HVSs and other high velocity stars. We examine the probable origin for B-type stars from the HVS Survey. Twenty objects have robust constraints. Halo stars dominate the sample at bound speeds, v rf − v esc < −100 km s −1 . We identify nine disk runaways with trajectories that significantly reject the Galactic center hypothesis. The fastest disk runaways have v rf − v esc ≃ +100 km s −1 , but the majority are bound. We identify seven probable Galactic center HVS ejections. The Galactic center ejections dominate the sample at speeds significantly above Galactic escape velocity.
A clean sample of HVSs is important for constraining the Galactic center ejection mechanism. With a sample of 50 HVSs, for example, discrimination between a single and a binary MBH ejection scenario might be possible (Sesana et al. 2007). Different MBH ejection mechanisms predict different spatial distributions of HVSs on the sky (Levin 2006;Abadi et al. 2009;Zhang et al. 2013;Boubert & Evans 2016). These distributions can be tested with larger samples of HVSs.
A well-defined sample of HVSs will also enable measurement of the ejection rate of stars encountering the MBH. Tidal disruption events may be closely related to the encounters that produce HVSs (Bromley et al. 2012).
Gaia end-of-mission proper motion errors should improve by a factor of 3. There are a dozen objects in our sample with 1-2σ trajectory constraints that will have >3σ constraints in only a few years. For the faintest stars, however, Gaia cannot compete with pointed observations. Next generation missions like the proposed Theia mission (Theia Collaboration et al. 2017) are needed to measure HVS proper motions with uncertainties significantly below 0.1 mas yr −1 .
With uncertainties below 0.1 mas yr −1 , HVSs become important tools for measuring the Milky Way mass distribution (Gnedin et al. 2005;Yu & Madau 2007). HVSs integrate the gravitational potential from the very center of the Milky Way to its outermost regions. Thus HVSs can complement studies possible now, that use halo star streams (Bonaca & Hogg 2018), globular clusters (Posti & Helmi 2018;Watkins et al. 2018), or dwarf galaxies (Fritz et al. 2018) to constrain the Milky Way dark matter halo. Any deviation of a HVS's trajectory from the Galactic center measures the Milky Way's nonspherical mass distribution, independent of any other technique.

Facility: Gaia
This work has made use of data from the European Space Agency mission Gaia, processed by the Gaia Data Processing and Analysis Consortium (DPAC). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This research has made use of NASA's Astrophysics Data System. This research was supported in part by the Smithsonian Institution.