The Fastest Unbound Stars in the Universe
DD RAFT VERSION A PRIL
3, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
THE FASTEST UNBOUND STARS IN THE UNIVERSE J AMES G UILLOCHON
AND A BRAHAM L OEB Draft version April 3, 2018
ABSTRACTThe discovery of hypervelocity stars (HVS) leaving our galaxy with speeds of nearly 10 km s − has providedstrong evidence toward the existence of a massive compact object at the galaxy’s center. HVS ejected via thedisruption of stellar binaries can occasionally yield a star with v ∞ (cid:46) km s − , here we show that thismechanism can be extended to massive black hole (MBH) mergers, where the secondary star is replaced by aMBH with mass M (cid:38) M (cid:12) . We find that stars that are originally bound to the secondary MBH are frequentlyejected with v ∞ > km s − , and occasionally with velocities ∼ km s − (one third the speed of light),for this reason we refer to stars ejected from these systems as “semi-relativistic” hypervelocity stars (SHS).Bound to no galaxy, the velocities of these stars are so great that they can cross a significant fraction of theobservable universe in the time since their ejection (several Gpc). We demonstrate that if a significant fractionof MBH mergers undergo a phase in which their orbital eccentricity is (cid:38) . v ∞ > km s − ) SHS may be aslarge as 10 Mpc − . Hundreds of the SHS will be giant stars that could be detected by future all-sky infraredsurveys such as WFIRST or Euclid and proper motion surveys such as
LSST , with spectroscopic follow-upbeing possible with
JWST . Subject headings: black hole physics — gravitation INTRODUCTION
Typical stellar velocities throughout the Milky Way area few hundred km s − . However, there are particular sub-populations of stars that are found to move at greater veloc-ities; these include the hypervelocity stars (HVS; The fastesthaving v (cid:46)
700 km s − , Brown et al. 2005; Brown 2011;Brown et al. 2014), runaway stars (e.g. Heber et al. 2008),and a small fraction of compact objects, with examples beingthe binary white dwarf LP400-22 (Kilic et al. 2013, v > − ) and the kicked pulsar PSR 2224+65 ( ≈ km s − ,Cordes et al. 1993). While these objects are moving quicklyas compared to most other stars in the galaxy, they still travelat speeds significantly below that observed for stars in ourown galactic center, where the velocity of the star with thecloset-known approach to the central black hole exceeds 10 km s − (Ghez et al. 2005), 3% the speed of light.In this paper we describe a mechanism by which binarymassive black hole (BMBH) mergers can liberate these tighltybound stars from their host black holes, resulting in semi-relativistic hypervelocity stars (SHS) that are capable of cross-ing large swaths of the observable universe and hence canserve as a new cosmological messenger (see also Loeb &Guillochon 2014). Predicated on numerical calculations ofSesana (2010) and Iwasawa et al. (2011) that suggest thatBMBHs may be excited to very large eccentricities prior tomerger, we argue that many stars originally bound to the less-massive of the two black holes (the secondary) can be ejectedin a manner closely resembling the Hills mechanism (Hills1988) for the production of HVS. As in the HVS mechanism,these stars can receive a significant speed boost above and be-yond their average orbital velocity, occasionally yielding starswith asymptotic velocities v ∞ nearing c . After several Gyr, [email protected] Harvard-Smithsonian Center for Astrophysics, The Institute for The-ory and Computation, 60 Garden Street, Cambridge, MA 02138, USA Einstein Fellow these stars evolve off the main sequence and become brightgiants that are potentially detectable by future all-sky infraredsurveys, and imaging surveys by the next generation of tele-scopes. We demonstrate that no mechanism aside from ec-centric merging BMBHs can accelerate a detectable numberof main-sequence (MS) stars with speeds in excess of ∼ km s − , and thus the detection of even a single star moving ata velocity greater than this value would suggest that a signifi-cant fraction of BMBH mergers proceed eccentrically.This mechanism is schematically depicted in Figure 1. Themerger of two galaxies (panel 1) results in the eventual mergerof the nuclear clusters hosting their MBHs (panel 2). Asthe secondary black hole scatters stars in orbit about the pri-mary, its eccentricity grows quickly to a value of order unity(panel 3). Once the eccentricity has been excited to a largevalue, the relative binding energy of the secondary to the pri-mary becomes small as compared to the specific binding en-ergy of stars in orbit about either the primary or the secondary,whose eccentricities are on average much lower. As the peri-apse distance r p , shrinks, the eccentric Hill radius around thesecondary also shrinks, r H , = r p , q − / , where q ≡ M / M ,resulting in the removal of all stars with apoapses compara-ble to this distance (panel 4). This process complements themechanism in which stars that are originally in orbit about theprimary occasionally enter the secondary’s Hill radius, result-ing in their acceleration to similar speeds (Yu & Tremaine2003; Holley-Bockelmann et al. 2005; Levin 2006; Sesanaet al. 2006). While we did not perform explicit calculationsof this complimentary channel, the mechanism we describehere is qualitatively similar in both the number and distribu-tion of SHS that are produced, which likely would increasethe number of SHS by a factor of a few.To quantify the number of SHS produced by mergingBMBHs, we perform Monte Carlo three-body scattering ex-periments in which random combinations of primary, sec-ondary, and tertiary parameters are drawn, and the outcomesare recorded (see Figure 2 for example output orbital evo- a r X i v : . [ a s t r o - ph . GA ] J un Guillochon, Loeb
Primary GalaxySecondary Galaxy Semi-RelativisticHypervelocity Star(s)( v > 0.03 c )Secondary NuclearClusterPrimary NuclearCluster PrimarySMBHSecondarySMBH Tightly-BoundStars F IG . 1.— Diagram of primary production channel for SHS. 1: Two galaxieswith central black holes merge. 2: Dynamical friction brings the two nuclearclusters and their host MBHs together. 3: The eccentricity of the secondaryMBH’s orbit about the primary is excited by asymmetrical scattering of starsthat originally orbited the primary MBH. A tighlty bound cluster of starsremains bound to the secondary. 4: With each passage of the secondary bythe primary, a fraction of stars are ejected as SHS. lutions). We then measure the outgoing velocity of ter-tiary objects that escape the system and calculate their subse-quent evolution from stellar isochrones in order to determinetheir detectability via current and future surveys. Addition-ally, we find that binary star systems (which we term semi-relativistic hypervelocity binaries, or SHB) are also capableof being accelerated by the same mechanism, which may en-able additional means for detection and identification of semi-relativistic stars. We find that approximately 10 objects mov-ing with velocity greater than 10 km s − exist out to the dis-tance of the Virgo cluster and ∼ within a Mpc of the MW.Roughly one SHS should be detectable per 100 square degreesfor a survey with limiting K-band magnitude of 24, a numberthat is too small to guarantee a discovery with current surveys,but will yield tens of detections with a next-generation all-skyinfrared survey such as WFIRST or Euclid .We find that the number of objects ejected is a strong func-tion of velocity, with the vast majority of SHS that wouldlikely be detected moving with velocity (cid:46) km s − . Fasterobjects would exist near the Milky Way, but these objects aremost likely to be low-mass MS dwarfs that would only bedetectable with deep imaging by either Hubble Space Tele-scope or James Webb Space Telescope ( JWST ) . Most of thedetectable stars will be red supergiants that evolved from starsslightly less than a solar mass, but the Doppler shift associatedwith such large speeds will make these objects appear bluerthan a similar star at rest, with color shifts of a few tenths ofa magnitude in the bluest JWST bands. Despite their aver-age distance of a Mpc, their large velocities will give them a http://wfirst.gsfc.nasa.gov http://hubblesite.org proper motion that is potentially detectable with LSST over10 years.In Section 2 we calculate the speed limits for MS stars ac-celerated via single, double, and triple object mechanisms,and show that only eccentric BMBH mergers can yield MSstars with v > km s − . In Section 4 we describe the setupof our numerical scattering experiments, and in Section 5 wedescribe the outcomes of these experiments. In Section 6 wedemonstrate that binary star systems can also be acceleratedby this same mechanism, and present an example ejection. InSection 7 we calculate the detectability of luminous SHS withvarious survey constraints. However, we find that identifica-tion may be somewhat more challenging, we discuss the prob-lem of identification and complications of our various modelassumptions in Section 8. alternative means of detection andidentification, Lastly, we summarize the scientific value ofSHS and SHB if they are detected in Section 9. SPEED LIMITS OF MS STARS
A single star can typically be accelerated to velocities up toits own escape velocity if it can asymmetrically eject a masscomparable to its own mass. Instances of tremendous massloss usually occur shortly before the birth of a compact object,but mass-loss episodes for stars that do not immediately formcompact objects do not result in large bulk velocities for thesurviving star. As an example, η Carinae, which lost tens ofsolar masses of ejecta a century ago in an explosive episode, ismoving radially toward the Sun at less than 10 km s − (Smith2004).When two objects interact with one another, the maximumvelocity in the system is set by the object with the lowest den-sity, as this object determines the minimum approach distancebetween the two objects before a collision or tidal disruptionoccurs. If the two objects are point-like and initially bound toone another, the two objects will remain bound indefinitely,assuming Newtonian dynamics. However, there are three pos-sibilities that enable the ejection of an object at a velocitycomparable to the maximum orbital velocity of the system:The tidal break-up of one of the two objects, effectively con-verting the system into a multi-body encounter (Faber et al.2005; Manukian et al. 2013), the destruction of one of the twostars via a supernova (runaway stars, Blaauw 1961), or thetidal break-up of a binary through interaction with a third ob-ject (Hills 1988; Agnor & Hamilton 2006; Hoffman & Loeb2007).In the first two cases, the outgoing velocity is limited to afraction of the velocity at pericenter, which is comparable tothe escape velocities from the surfaces of the two objects. Inthe last case, the outgoing velocity can be enhanced by thepresence of the third, more massive object (Hills 1988). Be-cause of this additional ingredient, this mechanism can pro-duce stars that are significantly faster than what is possiblewith a system comprised of only one or two objects.In the remainder of this section we focus upon the combina-tions of object types with MS stars and examine the maximumvelocities that can be produced for a given combination. Labelsubscripts for each object are assigned in order of descendingmass of the constituents, i.e. “1” will always refer to the mostmassive object in the system (the primary), “2” to the second-most massive (the secondary), and “3” to the least massive(the tertiary). Double subscripts (e.g. “12,” “23”) refer toquantities that jointly apply to both subscripted objects; as an emi-Relativistic Hypervelocity Stars 3 A B C DE F G HI J K L F IG . 2.— Twelve randomly selected trajectories of a stellar ejection resulting from the eccentric passage of the secondary MBH by the more-massive primaryMBH, with the semimajor axis of the tertiary restricted to 1 < log a , min / r IBCO , < .
25. Each panel is centered about the secondary MBH’s position, withthe yellow-red curve showing the trajectory of the primary MBH (yellow early times, red late times), and the green-purple curve showing the trajectory of theejected star (green early times, purple late times). The initial conditions of each encounter is drawn randomly for each panel, as described in the text. A varietyof outcomes is apparent, with the ejected stars leaving in many possible directions relative to the original encounter geometry. example a would refer to the semimajor axis of the primary-secondary system. Commas in the subscripts separate group-ings, for example r p , , would refer to the pericenter distanceof the secondary-tertiary system about the primary. The Hills Mechanism
It was found through numerical experiments by Sari et al.(2010) that the maximal velocity for outgoing objects in aHills encounter ( M (cid:29) M ) is v kick , max = 1 . v q / , (1)for the lowest-mass object in the system ( M < M ), where v is the escape velocity from the surface of the secondaryobject at a separation equal to the sum of the stellar radii, v = (cid:114) GM R + R , (2)and q , ≡ M / M ( M ≡ M + M ). The highest velocitiesare guaranteed for objects originating from binaries that are ator near contact, provided that the binary’s tidal disruption ra-dius r t , ≡ a q / , is greater than the periapse distance r p , , .If the incoming binary is not a contact binary to begin with,a value similar to that of Equation (1) can be achieved at adistance ∼ . q , shouldbe as large as possible, but keeping in mind that v tends todecrease with increasing q , for a fixed M . The followingsections describe how these two scalings compare to one an-other for specific object types. Stellar-mass Secondaries
The escape velocity from a MS star scales weakly with stel-lar mass, as R ∗ ∝∼ M . ∗ (Tout et al. 1996), and thus v esc , ∗ ∝∼ M . ∗ . Because the kick velocity scales as q / , ∝ M − / (as-suming M (cid:29) M ), there is no advantage to increasing themass of the secondary, and in fact the maximum kick velocity decreases slightly, v kick ∝∼ M − . .The mass of the black hole that yields the largest velocitiesis set by finding the largest black hole that will not swallowone (or both) of the stars whole, but still can disrupt the binary.For parabolic encounters, the critical distance at which an in-coming object will be swallowed is the “innermost bound cir-cular orbit” r IBCO (Bardeen et al. 1972), and for non-spinningblack holes this distance is twice the Schwarzschild radius, r IBCO = 4 r g = 4 GM / c . In principle, the binary separation a can be arbitrarily large and still experience a large kick, pro-vided that the two stars approach one another closely at peri-apse, but the probability of this occurring for a given passageis progressively smaller for increasing initial separations.To maximize Equation (1), we want to determine the mostmassive primary that will not swallow a given binary wholeto maximize the q , ratio, and thus we set the periapse dis-tance r p , = r IBCO , . If we presume that the binary is a contactbinary, a = ( R + R ) /
2, and assuming R ∗ = R (cid:12) ( M ∗ / M (cid:12) ) . ,we find the primary mass M that leads to the largest outgoingvelocity given a stellar mass, M , max = 4 × (cid:18) M M (cid:12) (cid:19) . M (cid:12) . (3)Combing this expression with Equation (1), we find the max-imal velocity possible for a star of a given mass, independentof M , v kick , max = 11,000 (cid:18) M M (cid:12) (cid:19) . km s − , (4)which scales extremely weakly with M . Any star movingfaster than this velocity is very unlikely to be produced by the Guillochon, Loebstandard Hills mechanism in which both objects are MS stars,as it would require that a binary with a wide separation (andthus v circ < σ ) passes by the central black hole, which is onlypossible if the binary is set onto a plunging orbit from beyondthe black hole’s sphere of influence. This limit is consistentwith numerical experiments (Ginsburg et al. 2012).By changing the secondary to a degenerate object, suchas a WD or a NS, M is limited to the Chandrasekhar mass M Ch (cid:39) . M (cid:12) . This means that either the tertiary is selectedto be somewhat less massive than the secondary, or that thecompact object becomes the tertiary, limiting the ejectionvelocity of the (now heavier) MS companion to 0 . v esc q / , (Sari et al. 2010). In either case, the maximum kick veloc-ity is smaller than what is possible with two MS stars, whosemasses can exceed M Ch , although only slightly given the weak M dependence of Equation (4).The only option for secondaries with masses larger than thatof the most massive stars ( ∼ M (cid:12) ) are black holes. If thesecondary is a stellar-mass black hole (SBH), M is not lim-ited by M Ch , and the denominator of Equation (2) ceases to de-pend on R as R (cid:29) R = r IBCO , for M (cid:46) M (cid:12) . However,as the secondary becomes more massive, and its separationis only determined by the size of the tertiary, the tertiary canbe tidally disrupted by the secondary, and this tidal disruptionradius r t , = R q / sets the minimum separation distance. In-corporating these changes, the kick velocity scales with M toa higher power, although still rather weakly, v kick , max = 20,000 (cid:18) M M (cid:12) (cid:19) / (cid:18) M M (cid:12) (cid:19) − . km s − , (5)where above we have scaled M to a 10 M (cid:12) , the mass ofa typical SBH. The expression above applies for M > M and M > . M (cid:12) . The most-massive stellar mass black holeknown may have a mass as large as 33 M (cid:12) (Silverman & Fil-ippenko 2008), yielding a v kick , max = 24,000 km s − = 0 . c for a solar mass star. Massive Secondaries
Black hole masses are known to range from a few to tens ofbillions of solar masses, with a paucity of black holes knownto have masses between 100 and 10 M (cid:12) . If this deficit is real,then it suggests that stars launched via the Hills mechanismwhere the secondary is a black hole have a bimodal distribu-tion of velocities. As we showed in the previous section, thelimit for MS binaries is ∼ . c , and this limit arises from thefact that MS stars of increasing mass have increasing size, andthus collide rather than resulting in an ejection. Black holesdo not have this limitation, so in principle the maximum ve-locity possible for a non-spinning black hole could be as largeas the velocity at the Schwarzschild radius of the secondary r IBCO , , the speed of light c .Equation (5) estimates the maximum kick velocity possi-ble so long as r IBCO , is small compared to R , but the pic-ture is of course complicated by general relativity, which cancause particles on orbits of finite energy to inspiral exteriorto r IBCO , , depending on the black hole’s spin. As a hyper-velocity ejection is typically the result of the tertiary beingplaced on a near-radial orbit about the secondary by the pri-mary, it approaches the secondary on a parabolic orbit, whichwould inspiral into the secondary at its IBCO at r IBGO , for anon-spinning black hole. As a result, the maximum possiblevelocity is set at this distance rather than the Schwarzschild radius r Sch , = 2 r g , . This suggests that the speed limit for SHSis c / γ ), and possiblymore for spinning black holes, which we do not consider here.This maximum speed is a factor ∼
10 times larger thanthe speed limit for MS binaries, and 5 times larger than thelimit for MS-SBH binaries. If the spectrum of SHS velocitieswere similar to that of HVS, in which the typical velocitiesare a few thousand km s − , it would suggest that the aver-age SHS’s speed could be a few 10 km s − . However, twoimportant factors influence the SHS velocity distribution: thesecondary’s binding energy to the primary, which determinesthe depth of the potential well that any ejected tertiary wouldneed to climb out of, and the distribution of orbits about thesecondary, which is quite different from the distribution of or-bits of stellar binaries. Importance of Secondary’s Orbital Energy
For the traditional HVS mechanism, the incoming binariesare deposited into the loss cone via kicks they receive typi-cally near apoapse, with the majority of these binaries orig-inating from the primary’s sphere of influence (Lightman &Shapiro 1977). At this distance, the binding energy to theblack hole is initially small ( ∼ σ ), and thus the final ve-locities of ejected stars are only reduced by a small amount(Kobayashi et al. 2012).For the SHS mechanism, the secondary’s orbital energy de-pends entirely on how the two black hole clusters merge. Thetraditional picture has been that dynamical friction drags thesecondary black hole and its surrounding stars into the out-skirts of the primary’s nuclear cluster. At this point, stars arescattered by the secondary with no preferred direction, result-ing in an inspiral in which the secondary’s orbit remains ap-proximately circular at all times, although eccentricity may beexcited for shallow stellar density profiles (Antonini & Mer-ritt 2012). This means that stars that are tidally removed fromthe secondary by the primary will only leave with a veloc-ity comparable to the local velocity dispersion, and will notescape the primary’s gravity.However, numerical N -body results (Baumgardt et al. 2006;Iwasawa et al. 2011; Khan et al. 2012) suggest that the sec-ondary’s orbit does not remain circular during its inspiral ow-ing to a preferential ejection of stars orbiting the primary inthe same direction as the secondary. They suggest that the or-bit can become extremely eccentric, with (1 − e ) ∼ M / M ∗ =10 − (ignoring general relativistic effects) for M = 10 M (cid:12) .Iwasawa et al. (2011) suggests that the only mechanism thatprevents a total plunge ( e = 1) is the random noise introducedby the discrete nature of the scattered stars, which suggeststhat more massive secondaries could result in even more ex-treme maximum eccentricities. If such behavior occurs in na-ture, the specific orbital energy of the secondary black holeis comparable to that of incoming stellar binaries in the HVSscenario, and thus the velocities of ejected stars will not bereduced much by their initial orbital energy. RATES
Scattering of Stars from the Primary Cluster
When a cluster contains a single MBH, there are two pri-mary mechanisms for ejecting stars at great velocities. Thefirst mechanism is the Hills mechanism (Hills 1988), de-scribed in Section 2.1. The second is the scattering of a star bya random passer-by star or compact object (Binney & Merri-field 1998; O’Leary & Loeb 2008). Although the initial con-emi-Relativistic Hypervelocity Stars 5figuration pre-encounter is different in the two cases, the setupof the system at the time of scattering is ultimately the same:Two stars that lie a distance from one another that is less thantheir mutual Hill radii. This similarity can be seen by com-paring expressions presented for the two mechanisms in Yu& Tremaine (2003); setting the distance of closest approachequal to the binary separation results in a velocity different of √
2, the difference in orbital velocity between a circular andparabolic orbit. The scaling of v with the masses of the twostars is identical for the two mechanisms.The literature concerning HVS produced via the mergers ofMBH has focused on the stars originally bound to the pri-mary (Quinlan 1996), more specifically, the scenario of abinary MBH in our own galactic center was considered forproducing HVS (Yu & Tremaine 2003; Sesana et al. 2007a).These papers focused on the analog of the scattering by a ran-dom passer-by, except in this case the passer-by is itself aMBH. As above for stars, the Hills mechanism and scatter-ing by a random passer-by are identical modulo a factor √ V / V = ( r H , / r , ) = M / q , = M , whereas in the Hills casethe mass liberated is simply the mass of stars originally boundto the secondary M . Removal of Secondary’s Cluster
In the initial stages of a merger, the secondary retains agroup of stars within its Hill radius, whose distribution likelyresembles the original distribution of stars. Prior to merger,the radial distribution of these stars likely resembles n ( r ) ∝ r − / (Bahcall & Wolf 1976) exterior to the distance wherethe orbital velocity v orb = (cid:112) GM / r is less than the escape ve-locities from typical stars, v ∗ = (cid:112) GM ∗ / R ∗ . If the black holeis surrounded by a population of stellar mass black holes, thisdistribution can continue a factor of a few more in r beforeterminating, as stars would be capable of relaxing to higherbinding energies without colliding (O’Leary & Loeb 2008).In such systems, N ( a , e ) ∝ a / e (Merritt 2013). However, ifwe consider a scenario where the cluster orbits within a largercluster on an eccentric orbit, the boundary condition of thesecondary’s cluster is complicated. Not only does the regionwithin which the secondary dominates the dynamics changein size, the conditions at its boundary change with distancefrom the central black hole.If we assume that the cusp of stars around the primary fol-lows a power-law distribution with r , ρ ∝ r − α , where α < M of stars is contained (Matsub-ayashi et al. 2007), a s = a h , q α − , (6)where a h , ≡ GM /σ is the radius of the sphere of influenceof the primary. At this distance, the stellar density is ρ ( a s ) = 3 − α π q α − α M a , . (7)At the primary’s sphere of influence, the velocity dispersionis (Kormendy & Ho 2013) σ = 75 (cid:18) M × M (cid:12) (cid:19) . km , (8) and so the Hill radius of the secondary at this location is r H = 3 − / q / GM σ . (9)While the secondary’s eccentricity grows, the Hill radiusshrinks as the secondary’s periapse comes closer to the pri-mary. This results in the stripping of the outermost stars inorbit about the secondary. As the eccentric growth rate is slowcompared to the orbital period of stars around the secondary,stars are almost always removed when their apoapse is largerthan secondary’s eccentric Hill sphere at periapse, r H , e = (1 − e ) r H . (10)This means that stars with the smallest separations from thesecondary are removed in the final phases of the secondary’sinspiral. These stars are the ones that potentially have thelargest ejection velocities, so long as the secondary’s orbit re-mains eccentric during the inspiral. Eventually, every singlestar that was once bound to the secondary will experience oneof the following outcomes: become bound to the primary, beswallowed or tidally disrupted by one of the two black holes,or become unbound to both the primary and the secondary.This last possibility can potentially produce SHS. Justification for Single-scattering Approximation
The secondary black hole’s eccentricity increases slowly asa function of time, meaning that the secondary-tertiary sys-tem gradually becomes more and more prone to perturbationfrom the primary. In principle, the tertiary can be lost wellbefore a > r H , , meaning that the maximum kick velocitymay be limited (Sari et al. 2010). However, if the evolutionof the secondary’s eccentricity is rapid, the periapse distancemay change significantly from orbit to orbit, and the periapsedistance r p , for a particular three-body encounter should bedrawn randomly between zero and r H , e (i.e. “pinhole” scatter-ing, Lightman & Shapiro 1977). In such a situation, the firststrong interaction for a given tertiary may be at a periapse dis-tance that is equal to or even significantly deeper than the tidaldisruption radius.Iwasawa et al. (2011) presented an expression for the ec-centricity growth timescale T e as a function of the primary-secondary binary parameters, which we reproduce here usingour notational conventions, T e ∼ . × (cid:18) M M (cid:12) (cid:19) / (cid:18) M M (cid:12) (cid:19) − × (cid:18) a
10 pc (cid:19) / (cid:18) ρ M (cid:12) pc − (cid:19) − yr . (11)The main uncertainty in this expression the values of a and ρ . When two central black holes merge, their evolution is ini-tially governed by dynamical friction, which causes the lighterof the two black holes to sink rapidly toward the heavier (Dottiet al. 2012). During this phase, the orbit may acquire mod-erate eccentricities for shallow density profiles, but often re-mains circular (Antonini & Merritt 2012). This phase ceasesonce the secondary black hole reaches the stalling radius a s (Equation (6)), at which point a ceases to evolve for isotropicstellar distributions. At this point, the eccentricity of the sec-ondary begins to grow, and thus the appropriate values of a and ρ are determined at a distance a s from the primary. Guillochon, LoebSetting a = a s (Equation 6) and ρ = ρ ( a s ) (Equation 7) inEquation (11) and taking the ratio of this timescale to the sec-ondary’s orbital period P at a s , we find T e P = 0 . − α ) (cid:18) M M (cid:12) (cid:19) / q α − α − (12)The orbit of the secondary can be considered “plunging” if T e is comparable to P . Setting T e / P = 1 and solving for q wefind the maximum mass ratio that will have rapid eccentricityevolution q plunge , q plunge = 65 √ − α, (13)which is equal to 73 for α = 7 /
4. However, it has been shownthat near-equal mass MBHs ( q (cid:46)
3) do not show strong ec-centricity growth (Sesana 2010; Wang et al. 2014). As T e / P increases with increasing q , this suggests that there mustbe a critical q value for which eccentricity excitation is themost effective between these two extremes, however any massratio in between these two values should have P (cid:46) T e , andthus a single-scattering approximation in which r p , is ran-domly drawn is appropriate for these encounters. Similarities Between Primary and Secondary SHS
While the two scenarios described above are somewhat dif-ferent in their initial setup, the conditions required to producean SHS are remarkably similar. For stars that are originallybound to the primary (primary SHS), most of the encounterswith the secondary will occur at a distance from the secondarycomparable to the secondary’s Hill radius. Likewise, stars thatare original bound to the secondary (secondary SHS) are re-moved from the secondary when the Hill radius shrinks to asize comparable to their distance from the secondary. As weshow in Section 5, the resulting distribution of secondary SHSis very similar to that found for primary SHS. Additionally,the total number of primary and secondary SHS are approx-imately the same, with the total mass ejected being (cid:39) M inboth cases.If the eccentricity of the secondary is small, primary SHSwill only acquire an energy comparable to the secondary’s cir-cular velocity about the primary. The same is true for sec-ondary SHS; stars that are removed from a secondary on anear-circular orbit will also have velocities comparable to thesecondary’s circular velocity. The high-velocity tail is onlyaccessible for primary SHS when the secondary has a signifi-cantly eccentric orbit (Sesana et al. 2006), and as is describedin Section 2.2, the same is true for secondary SHS. Becausethese two mechanisms are the only channels that produce suchhigh-velocity stars, and the highest velocities are only accessi-ble in eccentric mergers, the discovery of SHS would suggestthat BMBH indeed merge eccentrically. BMBH SCATTERING EXPERIMENTS
To determine the rate of stellar ejections, we perform MonteCarlo scattering experiments using the same orbit integrationroutines in
Mathematica (version 9.0) of Manukian et al.(2013) used to investigate the production of “turbovelocity”stars. While this approach is only modestly scalable to mul-tiple processors on a single machine, it has the advantageof completely controllable numerical errors, at the expenseof increased computational cost. For this paper we performour scattering experiments with quadruple floating-point pre-cision ( (cid:39)
32 significant decimal digits), and restrict the max-imum relative error in the orbital energy E and the vectorial angular momenta J x , J y , and J z to (cid:39) − . This extreme pre-cision enables us to evaluate the back-reaction on the blackholes imparted by the ejection of the SHS, which is not possi-ble when the primary-secondary system is assumed to have afixed orbit. For the special case of the ejection of binary stars,described in Section 6, we employ octuple floating-point pre-cision ( (cid:39)
64 digits).We are ultimately interested in the typical black hole andstellar properties that produce SHS within a given velocityrange. As the mechanisms we describe here hinge upon MBHmergers at the centers of galaxies, we need to determine themerger rates of the galaxies themselves, and relate the prop-erties of those merging galaxies to the black holes that theyhost. The parameter space to be explored is quite large;known MBHs range over five orders of magnitude in mass,and the stars that orbit them possess a range of masses andorbital parameters. Additionally, the rate of BMBH mergersis very poorly quantified, especially for those black holes thatare separated by less than ∼
10 pc (a typical value for a stall ),which are difficult to resolve even in the radio (Burke-Spolaor2011). While the rates of merging MBHs is highly uncertain,the rates of dark matter halos are very well known throughvia results of dark matter-only simulations such as the Mil-lennium simulations (Boylan-Kolchin et al. 2009; Genel et al.2010; Fakhouri et al. 2010). We use these results as a startingpoint for determining the rate of MBH mergers.We use a standard Sheth-Tormen distribution for determin-ing dark matter halo masses M h , calculated through the on-line tool HMFcalc (Murray et al. 2013) using the defaultparameters, and taking z = 0. We draw 4 × primaries fromthis distribution, and then calculate the stellar mass M ∗ from M h via Equation (22) of Moster et al. (2010), M ∗ = M ∗ , (cid:0) M h / M (cid:1) γ (cid:104) + (cid:0) M h / M (cid:1) β (cid:105) ( γ − γ ) / β (14)where we have changed the variable notation slightly to matchour conventions, and the constants are as defined in that paper:log M ∗ , = 10 . M = 10 . γ = 7 . γ = 0 . β = 0 . M bulge / M ∗ for each galaxy and multiply the total stellarmass by this value to obtain the bulge mass M bulge . Finally,we use the bulge to black hole mass relation determined byMcConnell & Ma (2013) to obtain M , the mass of the pri-mary black hole in the system,log M = 8 . + .
05 log (cid:20) M bulge M (cid:12) (cid:21) , (15)adding 0.34 dex of scatter to the returned value, as reported inthat work.With the primary’s black hole mass now determined, wenow refer to the results of Fakhouri et al. (2010) in which theypresent the expected number of mergers for a halo of a givenmass M h , at a given redshift z with secondaries with a massratio ξ = M h , / M h , . We note with this definition that weare implicitly assuming the merger rate is defined by the massof the descendant halo rather than the mass of the progenitor http://hmf.icrar.org emi-Relativistic Hypervelocity Stars 7 Log q C D F H s t a r s r e m ov e d fr o m M L q = q p l un g e Rapid e evolution, e ® q = F IG . 3.— Cumulative distribution function of hypervelocity stars producedfrom mergers of a given q . Below a critical mass ratio q plunge (Equa-tion (13)), the cross-section of the secondary is small, reducing its dynam-ical drag and rate of eccentricity excitation (Iwasawa et al. 2011). Above q plunge , the secondary’s eccentricity evolves on a timescale comparable to anorbital period. For q (cid:46) halo, this introduces a slight bias toward higher masses for thehighest-mass halos (Genel et al. 2009).For each primary host halo, we integrate Equation (1) ofFakhouri et al. (2010),d N m d ξ d z = A (cid:18) M ( z )10 M (cid:12) (cid:19) α ξ β exp (cid:20)(cid:18) ξ ˜ ξ (cid:19) γ (cid:21) (1 + z ) η , (16)where A , α , β , γ , ˜ ξ and η are constants defined in that paper,from z = 0 to z = 10 and ξ = 10 − / M to ξ = 1 (see Sec-tion 3.2.1) to obtain the average number of mergers for a haloof that mass. The mass of a halo as a function of redshift M ( z )is determined by selecting a halo mass at z = 0 and integratingthe equation for the median growth rate of a halo, d M dz = 25 . M (cid:12) yr − (cid:18) M ( z )10 M (cid:12) (cid:19) . × (1 + . z ) (cid:113) Ω m (1 + z ) + Ω Λ , (17)the second line of Equation (2) of Fakhouri et al., where Ω m are Ω Λ are the standard cosmological parameters and aretaken to be Ω m = 0 .
27 and Ω Λ = 0 .
73. The number of merg-ers N m is then used as an input to a Poisson distribution, fromwhich we draw the number of mergers experienced by thathalo N merger . Then, we draw N merger realizations of ξ over thesame range defined above using Equation (16), where we as-sume that ξ has no dependence on z or M . This enables acalculation of the halo mass of the secondary M = ξ M ,from which we calculate the secondary’s black hole mass M using the same procedure used above to determine M .Because mergers of near-equal mass will likely not resultin large eccentricities, and because secondaries with too smallof a mass will not have rapid eccentricity evolution (see Sec-tion 3.2.1), we eliminate all mergers for which q < q > q plunge . This restricts our calculation to systems in whichthe secondary’s eccentricity grows to large values on a shorttimescale and our single-scattering approximation is applica-ble. As can be seen in Figure 3, this reduces the total numberof SHS produced to about one third the amount that would beproduced if all SBH mergers were eccentric and plunging.With sample of merger events now defined, three-body en- counters are drawn from the sample of mergers with the prob-ability of a draw being proportional to M , as the total numberof stars within the secondary’s sphere of influence scales di-rectly with its mass, M ∗ , = 2 M . This implies that the mostmassive mergers can potentially produce the most SHS, forthe simple reason that more massive black holes are orbitedby more stars. The properties of the tertiary (the star in thesystem) are assumed to not depend on either black hole; itsmass M is drawn from a Kroupa distribution (Kroupa 2001),and its orbital parameters relative to the secondary are drawnpresuming a thermal ( P ( e ) ∼ e ), isotropic distribution of orbitsabout the secondary.A system of identically massive gravitating bodies in or-bit about a dominant central object will relax via two-body interactions to a radial distribution n ( r ) ∝ r − γ , where γ = 7 /
4, corresponding to a distribution in semimajor axisof P ( a ) ∝ a − γ ∝ a / (Bahcall & Wolf 1976). Two-bodyrelaxation becomes ineffective interior to a distance a relax inwhich the timescale for star-star collisions is shorter than therelaxation timescale. If a population of stellar mass blackholes are present in the cores of nuclear clusters, a relax fora solar-type star is determined by where the escape velocityfrom the stellar mass black hole is equal to the local Keplervelocity about the secondary, a relax = ( M / M bh ) R (cid:12) , where weassume M bh = 15 M (cid:12) (O’Leary & Loeb 2008). The distribu-tion of low-mass MS stars is poorly known interior to a relax ,as we are only able to resolve individual MS stars within a relax for our own galactic center. For these stars, for whichonly high-mass MS stars ( M ∗ (cid:38) M (cid:12) ) with short lifetimes aredetectable (Ghez et al. 2005), the distribution with distancefrom the black hole r is very shallow, P ( r ) ∝ r − . , implying P ( a ) ∝ a . . However, the surface brightness distribution ofthe unresolved stars in the galactic center, which is thought tobe dominated by K-dwarf stars, is slightly steeper, P ( r ) ∝ r − (Yusef-Zadeh et al. 2012), implying P ( a ) ∝ a . Because it turnsout that most of the observed SHS are giants that evolved fromrelatively low-mass MS stars, we use the power-law impliedby the unresolved population, P ( a ) ∝ a , for a < a relax .Given these functional forms, the tertiary’s semimajor axisdistribution is defined as a broken power-law, with a cutoffcorresponding to the maximum of the tidal radius of the ter-tiary and the Schwarzschild radius of the secondary a dest ≡ max( r IBCO , , q / R ), and a cutoff excluding objects in whichthe gravitational wave merger timescale τ GW , of the tertiaryabout the secondary (see Peters 1964) is less than the orbitalperiod of the secondary about the primary P , P ( a ) ∝ (cid:40) a / : a > a relax a : a < a relax , (18) P ( a ) = 0 if a < a dest ∨ τ GW , < P . Each simulation is started at a time t = t p , − P , where t p , is the time of the secondary’s periapse about the primaryand P is the orbital period of the tertiary about the sec-ondary; as the encounters are all close to parabolic this times-pan is sufficiently long to ensure that the secondary-tertiarysystem is unperturbed by the primary at t = 0. The choice ofan integer number of orbital periods also ensures that effectsof the phase of the orbit, defined by the initial mean anomaly M , can be directly related to the experiment outcomes.Rather than draw tertiaries directly from Equation (18),which would result in a distribution in which low-energy starsare better sampled than high-energy stars, we perform a num- Guillochon, Loebber of independent samples within small bins of a , keepingin mind that the outcomes of each experiment needs to be nor-malized later by integrating Equation (18) over the range ofsampled a . We select a bin size of 0.25 dex in log ˜ a , wherewe define ˜ a ≡ a / r IBCO , , the ratio of the tertiary’s semima-jor axis to the IBCO of the secondary, with our bins spanning1 ≤ ˜ a ≤ . 4096 systems are then drawn for each bin; with28 bins this means that our results are derived from 114,688independent three-body experiments.Lastly, we must select a proper periapse distance for eachencounter. For circular orbits, the region of stability is well-defined by the Jacobi constant, but this expression ceases tobe constant when generalized to an elliptical orbit. Numer-ical experiments have shown that triple systems will eventu-ally lose one of their components at periapse distances thatare significantly greater than the tidal disruption radius of theinner binary r H , / a = 3 − / q / (Mardling & Aarseth 1999;Mardling & Aarseth 2001), r p , a < . (cid:20) (1 + q ) 1 + e (1 − e ) / (cid:21) / (cid:18) − . i (cid:19) (19) ≡ r p , , crit a , where i is the inclination of the orbital plane of thesecondary-tertiary system relative to the orbital plane of theprimary-secondary system, in degrees. Because stability isguaranteed for systems with greater r p , , this expression setsthe maximum possible r p , for which ejection is possible, andthus we only draw r p , values that are less than this limit. Asthe secondary is in a plunging orbit in which its orbital an-gular momentum changes by order unity in an orbital period,we draw periapse distances from the same distribution as isused for “pinhole” scattering (Lightman & Shapiro 1977), e.g. P ( r p , ) ∝ constant. r p , , crit / r H , (Equation (19)) ranges be-tween 4.8 to 6.4 for 1 ≤ q ≤ for a circular orbit, butincreases rapidly as e →
1, growing to 10 for e values atthe time of the final phases of the plunge. However, Equation(19) only specifies that a component of the triple system willbe lost eventually , which could potentially be after many thou-sands of orbits. As an example, the Sun-Earth-Moon system,for which all three components are approximately circular andcoplanar ( e (cid:39) i (cid:39) r p , / r p , , crit (cid:39) . r p , (cid:39) r p , , crit to almost never be lost in a single orbit, andsystems for which r p , (cid:39) r H , to usually become unboundwith one passage.As with the tertiary’s orbit about the secondary, the sec-ondary’s orbit can also evolve by the emission of gravitationalwaves. Once this occurs, the secondary’s orbit will circular-ize, and the production of SHS during the eccentric phase ofthe black hole merger will cease, with the emitted stars hav-ing velocities no greater than the orbital velocity of the sec-ondary black hole (see Section 3.3). Assuming that e ∼ r p , min at which T e /τ GW , = 1(for α = 7 /
4) is r p , min r g , (cid:39) (cid:20) (1 + q ) q / (cid:21) / (cid:18) M M (cid:12) (cid:19) − / , (20)corresponding to SHS that orbit the primary with v / c = 0 . M , . Such stars are likely ejected at thesevelocities even from circular orbits (Sesana et al. 2006), comparable to the fastest stars produced during the eccentricmerger phase which we detail in the next section. Addition-ally, if the secondary’s orbit is truly plunging, its periapsedistance may change by order unity over a single orbitalperiod, this would yield at least a few orbits of moderate ec-centricity for r p < r p , min . In conjunction, these two effects willlikely result in the production of SHS at a slightly reducedrate relative to what we present here (as we have ignored theeffects of GW emission from the primary-secondary system),but for only the stars of the very highest velocities. RESULTS
In this section we present the results of our scattering ex-periments, and focus solely on the kinematic properties ofSHS that result, neglecting observational prospects, which wecover in Section 7.
Fates of Tertiaries
Our setup involve the passage of a two-body system (thetertiary in orbit about the secondary) by a more-massive pri-mary on a near-parabolic orbit. Even for the most massivestellar tertiaries, the ratio of the tertiary mass to the sec-ondary mass is minuscule, and therefore its presence doesnot significantly alter the trajectory of the primary or sec-ondary MBHs. And because the orbit of the secondary is as-sumed to be plunging, the semimajor axis of the secondary isalways significantly larger than its periapse distance, result-ing in trajectories that are nearly identical in shape. Thus, ifthe semimajor axis distribution of stars about the secondarywere a power-law, and no mechanism for stellar destructionexisted, our results would not depend on ˜ a . It is the in-clusion of the break in the power-law likely arising fromstellar collisions, the tertiary’s tidal disruption radius, andthe secondary’s Schwarzschild radius that eliminate this self-similarity, which effects the shares of outcomes at different ˜ a . The production of SHS typically occurs when the orbit ofa tertiary is dramatically altered by the tidal potential of theprimary. The shrinkage of the secondary’s Hill radius is notinstantaneous, with the timescale being comparable to thesecondary-tertiary orbital period P , and this means that thetertiary usually executes at least one orbit about the secondaryduring the encounter (this is evident in many of the examplesshown in Figure 2). During this time, the periapse distance ofthe tertiary about the secondary can change by factors of orderunity, which can result in higher orbital velocities at the timeof ejection, or can result in the tertiary’s destruction when itcrosses its tidal radius or the distance of the secondary’s laststable bound orbit.The top panel of Figure 4 shows the outcomes of our scat-tering experiments as a function of ˜ a min . For ˜ a ∼
1, these starsare already within a few r IBCO , of the secondary, and thusonly a moderate perturbation by the primary is required to po-tentially knock them into the secondary. As the initial orbitsabout the secondary are thermal, changes in orbital angularmoment of order unity are required to unbind them from thesecondary, and this means that those encounters that wouldbe likely to produce an SHS are also likely to result in thetertiary’s destruction. As a result, small ˜ a is characterizedby encounters in which the tertiary is usually destroyed, andSHS production is relatively rare ( <
1% of stars within thissemimajor axis bin). This also reduces the number of objectsthat become bound to the primary after the encounter, as thisemi-Relativistic Hypervelocity Stars 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . % % % % % % % % % % % Single Orbit
Log a Ž min Swallowed by M Swallowed by M Disrupted by M Bound to M Bound to M SHS . . . . . . . . . . . . . . . . . . . . . . . . . . . . % % % % % % % % % % % Cumulative
Log a Ž min Swallowed by M Swallowed by M Disrupted by M Bound to M SHS F IG . 4.— Outcomes from individual scattering experiments as a functionof semimajor axis bin, denoted by the minimum value of each bin log a min .The top bar chart shows the direct outcome from each scattering experiment,and includes a “bound binaries” category in which the tertiary remains boundto the secondary after the encounter. The bottom bar chart assumes that anyscattering event that did not result in either the removal or destruction of thetertiary will continue to be subject to repeated close periapse passages, even-tually removing the star from its orbit about the secondary, with the probabil-ity of each outcome being the same as the single orbit case. outcome also requires changes in the tertiary’s orbit of orderunity.Because the difference in mass between the primary andsecondary is usually a factor of 10, the primary’s IBCO r IBCO , and the periapse distance of the secondary r p , are compara-ble in size, occasionally resulting in the primary destroyingthe tertiary when it wanders into its IBCO. Because the typ-ical mass of the primaries are 10 M (cid:12) , tidal disruption by theprimary is extremely rare, as r IBCO , is usually several timeslarger than the tidal radius of MS stars. The secondary, beingabout an order of magnitude less massive, is at a mass wherethe IBCO and tidal radius are comparable, and thus tidal dis-ruptions are more likely to be the means of destruction.As ˜ a increases to ∼
10, the number of objects destroyedby the secondary decreases significantly, as now orbits can beperturbed without much threat of destruction. The outcomeof most encounters in this regime is that the tertiary remainsbound to the secondary, although often with an alteration ofits orbital parameters. SHS production in this range begins tobecome significant, with a few percent of stars becoming SHSper orbit. Because the orbit of the secondary is so eccentricwhen the most tighlty bound stars are perturbed, the specificorbital energy of the secondary about the primary is small,and as a result the outgoing velocity is mostly determined bythe kick received via the Hills’ mechanism.For ˜ a greater than a few 10 the fraction of stars that becomebound to the primary increases to become comparable to thefraction of SHS produced. This can be understood by com-paring the specific binding energies of the primary-secondarysystem to the secondary-tertiary system. In the plunging sce-nario, the semimajor axis a is nearly constant for all ˜ a forfixed primary and secondary masses, so long as r p , (cid:28) a . Using Equation (8) we rewrite Equation (6) in terms of M and q , and find that the ratio of the specific binding energyof the primary-secondary system (cid:15) to the secondary-tertiarysystem (cid:15) is (cid:15) (cid:15) = 2 . (cid:18) ˜ a (cid:19) (cid:18) M M (cid:12) (cid:19) / (cid:16) q (cid:17) − / , (21)showing that indeed the two energies become comparable for ˜ a ∼ .In the bottom panel of Figure 4 we show the fraction of finaloutcomes under the presumption that the per-orbit probabili-ties are unaffected by the stars’ orbital histories. Because allother outcomes are mutually exclusive (a star swallowed byone of the MBHs cannot also be ejected), the cumulative en-counter outcome is simply equal to the relative contributionsof each outcome in the single case with the “bound to M ”case being removed. Antonini et al. (2010) performed sim-ulations of repeated encounters of binaries with a MBH andfound that the rate of ejection did not increase significantlywhen considering multiple orbits where the periapse is heldconstant. However, in the case of a plunging secondary, theperiapse distance shrinks on a timescale comparable to the or-bital period, and thus the periapse does not remain constantfrom orbit to orbit. This makes stars progressively easier toremove on subsequent encounters as the secondary plungesdeeper within the primary’s potential. Properties of SHS
As described in the previous section, the fate of a star inorbit about a plunging secondary is dependent on its dis-tance from the secondary ˜ a and its vulnerability to destruc-tion, which depends mostly on the IBCO of the secondaryand the tidal radius of the tertiary. If the star is ultimatelyejected from the system, the distribution of outgoing veloci-ties v ∞ is also likely to depend on ˜ a for two simple reasons:They are orbiting with larger velocities about the secondaryto begin with, and some stars that would be ejected at a par-ticular velocity may instead be destroyed by passing too closeto the secondary. Velocity Distribution
All stars that become unbound to the secondary inevitablyacquire a positive excess velocity relative to it, but this ex-cess velocity is not necessarily sufficient for the star to escapethe primary as the secondary is bound to it. This leads tothe production of a population of stars that remain bound to M ( (cid:15) < M ( (cid:15) > (cid:15) can be ejected with velocities thatare arbitrarily close to zero, even if the time-averaged orbitalvelocity of the tertiary about the secondary (cid:104) v (cid:105) (prior to re-moval) is significant.In Figure 5 we show the output distributions of v ∞ as afunction of ˜ a , with each histogram summing across the ex-periments drawn from the expected triple configurations, asdescribed in Section 4. As each histogram is produced viaan identical number of experiments, the small number of sys-tems contributing to the distributions at both large and small ˜ a demonstrates that producing SHS for these extreme semi-major axis values is difficult. The output distributions are also0 Guillochon, Loeb - - - - - @ v ¥ D H km s - L P D F + o ff s e t Log @ v ¥ (cid:144) c D Log a Ž min F IG . 5.— Probability distribution functions of the asymptotic velocity v ∞ of stars ejected from merging BMBHs. Each histogram shows the outcome ofa set of scattering experiments for a restricted range of initial semimajor axesof the tertiary about the secondary, a min < a < a max , where a min and a max are spaced logarithmically in intervals of 0.25, with purple corresponding to a min = r IBCO , and red corresponding to a max = 10 r IBCO , . Each histogramshows the results of an independent scattering experiment composed of 4096systems, where the parameter combinations are drawn as described in Sec-tion 4, and are plotted normalized to the bin with the most systems withineach histogram. visibly Gaussian, and centered about a particular v ∞ for each ˜ a (Figure 6, top panel), although this value is somewhat inexcess of (cid:104) v (cid:105) . This excess is mostly due to the fact thatthe secondary is in motion at the time of the tertiary’s escape,which gives an additional kick to the tertiary on top of its time-averaged velocity, scaling as q / , (Hills 1988). Even thoughthe orientation and phase of the binary play a large role indetermining v ∞ for a particular system, the fit to datapointsclearly shows this relationship (Figure 6, bottom panel).The combined probability distribution determined by prop-erly normalizing the distributions presented in Figure 5 isshown in Figure 7. We find that the slope of the total numberof objects n is similar to that found for stars that are originallybound to the primary and are scattered by the incoming sec-ondary (see Figure 1 of Sesana et al. 2007a) for objects with v ∞ ∼ several thousand km s − , but that the slope steepens - - - - @ v ¥ (cid:144)X v \D C D F - - - - - - - - @ q D L og @ v ¥ D H k m s - L L og @ v ¥ (cid:144) c D v ¥ µ q (cid:144) F IG . 6.— The top panel shows cumulative distribution functions of the ratioof v ∞ to the average velocity of the star’s orbit about the secondary prior toits ejection, (cid:104) v (cid:105) , demonstrating the close relationship between the tertiary’soriginal binding energy to the secondary and the mean ejection velocity. Theblack vertical dashed line shows what the mean value would be for q = 1,whereas the black vertical dotted line shows the enhancement to the meanvalue owing to the average mass ratio q (cid:39) v ∞ is shown as a function of q , with each lineshowing the least-squares linear fit within a given semimajor axis bin, color-coded to the particular scattering experiment labeled in Figure 5. The thickblack line segment shows the expected v ∞ ∝ q / relationship; consistentwith our linear fits, but with tremendous scatter. For extreme values of ˜ a the slope is less well-defined (as an example log ˜ a min = 6 .
75 includes onlytwo systems), as our scattering experiments only produced a few SHS withinthese velocity bins. for higher velocities. This is likely because we assume thatthe radial profile of stars flattens out interior to the distanceat which stars are frequently destroyed by stellar collisions.When compared to the local density of HVS, which we calcu-late assuming an average velocity of 3,000 km s − (Bromleyet al. 2006) and a production rate of 10 − yr − (Yu & Tremaine2003), a nearly equal number of SHS occupy a volume within1 Mpc of the MW as HVS. This distribution does not takeinto account the fact that SHS must climb out of the potentialwells of their host galaxies, and as we will describe in Sec-tion 7, this effect can significantly reduce the local populationof low-velocity SHS. Dependence on Initial Orientation and Phase
In the left panel of Figure 8 we show the CDF for the z -component of the secondary-tertiary system’s angular mo-mentum vector (cid:126) J , where z is defined to be perpendicular tothe primary-secondary’s orbital plane. Even though we tendemi-Relativistic Hypervelocity Stars 11 - - - - - - - - @ v ¥ D H km (cid:144) s L L og @ n d e x - D H M p c - L Log @ v ¥ (cid:144) c D n HVS,MW H r MW < L n HVS,MW H r MW < L n HVS,MW H r MW <
10 Mpc L n µ v ¥- F IG . 7.— Volume density n as a function of v ∞ , ignoring the additionalvelocity required to escape the host galaxies’ potentials. The dashed blackline shows the slope found for the MBH mechanism (Sesana et al. 2007a;Sherwin et al. 2008). The dotted blue lines show the average density of HVSproduced by the MW within a certain distance of the MW r MW , where thedistances are as labeled in the figure, where we have assumed that HVS havevelocities equal to 3,000 km s − . to draw smaller r p , (Equation (19)) for retrograde systems(i.e. J z < J z >
0. The trend is most pro-nounced for the fastest SHS, with 75% of SHS being pro-duced from prograde systems when ˜ a <
10. As ˜ a increases to ∼ , the number of systems ejected depends less on whetherthe system is prograde or retrograde; we speculate that this isbecause the primary-secondary orbit becomes more circularwith increasing ˜ a , and thus each secondary-tertiary pair orbitsabout one another a few times (as opposed to roughly once ina parabolic encounter) while in the vicinity of the primary’stidal field, averaging out the tertiary’s orbital motion.In addition to a dependence on the angles of the two sys-tems to one another, the kick received can also depend on thetertiary’s position within its orbit near the time of periapse.In our simulations, each run was performed an exact integermultiple of the secondary-tertiary system’s orbit period priorto periapse, with the initial mean anomaly M being drawnuniformly, where M = 0 ◦ (or 360 ◦ ) corresponds to the ter-tiary being at periapse. As a result, M , in conjunction withthe orientation of the orbit, directly relates to the position ofthe tertiary when the secondary reaches periapse, although notperfectly as the orbit is subject to some perturbation by theprimary over the course of the orbit. The right panel of Fig-ure 8 shows that ejections are disfavored when the tertiary hasrecently crossed periapse and is on its way toward apoapse(0 ◦ < M < ◦ ), and are favored when the tertiary is re-turning to periapse (180 ◦ < M < ◦ ). This is as expected;when the tertiary approaches periapse it is in the process ofaccelerating, so the additional acceleration provided by theprimary in this phase can unbind the tertiary to a greater de-gree than the case where the tertiary is deccelerating. Dependence on the Masses of the Three Bodies
Figure 9 shows CDFs for the masses of the primary, sec-ondary, and tertiary bodies. Immediately apparent in the pri-mary and secondary distributions (top two panels) is that mostof the large ˜ a systems originate from low-mass black holes,this simply reflects the fact that the sphere of influence issmaller than ˜ ar IBCO , for large black holes. For ˜ a in which - - - - J z (cid:144)È J È C D F M M Retrograde M M Prograde M H deg. L C D F M M On way out M M On way in F IG . 8.— Cumulative distribution functions of the z -component of the ini-tial angular momentum vector (cid:126) J of the secondary-tertiary orbit relative to theprimary-secondary plane, and initial mean anomaly M of stars that woundup being ejected from their BMBH system. Each CDF is color-coded to theparticular scattering experiment labeled in Figure 5. The solid black curvesshows the input distributions assumed for each variable, with J z being ran-domly drawn from a 2-sphere and M being drawn from a uniform distri-bution. The dashed vertical line shows a separation of retrograde versus pro-grade orbits in the left panel, and orbits in which the tertiary is moving towardapoapse versus periapse in the right panel. all black hole masses are included, a slight preference to-ward higher black hole mass is apparent for both the primaryand the secondary as compared to the input distributions, withthe exception of large ˜ a which can only originate from smallblack holes for which the sphere of influence is much largerthan the IBCO.For the tertiary, whose mass is tremendously smaller thanthe primary and secondary, is for all intents and purposes a testparticle; no statistically significant dependence on its mass isapparent (Figure 9, lower panel). One might expect that theinclusion of the tidal disruption radius would preferentiallydestroy stars with a lower density, but the vast majority ofsystems that produce SHS have masses in excess of 10 M (cid:12) ,a mass for which the IBCO lies exterior to the tidal radius formost MS stars, as is apparent from the small fraction of tidaldisruptions accounted for in Figure 4. Dependence on Eccentricity of the Tertiary
The stars that orbit our own galactic center exhibit a widerange of eccentricities, but the inner cluster of stars seems tobe well-described by a thermal distribution (Gillessen et al.2009). While some stars were likely deposited via the disrup-tion of stellar binaries resulting in much higher initial eccen-tricities (Madigan et al. 2011), we assumed that the stars inorbit about the secondary are drawn from a thermal distribu-tion because of the strong relaxation effect of the repeatedly2 Guillochon, Loeb @ M (cid:144) M Ÿ D C D F @ M (cid:144) M Ÿ D C D F - - @ M (cid:144) M Ÿ D C D F F IG . 9.— Cumulative distribution functions of the primary, secondary, andtertiary masses, and the ratio of the secondary to the primary mass 1 / q .Each CDF is color-coded to the particular scattering experiment labeled inFigure 5. The thick black curve in each panel shows the initially assumeddistributions for each mass, as described in Section 4. passage of the primary black hole (Perets et al. 2007), whichis likely to destroy any pre-existing coherent structures in an-gular momentum space.Figure 10 shows the cumulative distribution function ofSHS as a function of the secondary-tertiary eccentricity e .While our experiments for large ˜ a show no statistically sig-nificant deviation from the input distribution, small ˜ a doesshow that more-circular systems are preferably accelerated.The reason for this is that tertiaries in highly eccentric orbitsare more likely to be destroyed after being perturbed by theprimary, as their periapses are initially closer to the swallow-ing radius r IBCO , . For systems that are not swallowed, we findno statistically significant trend between e and v ∞ . There-fore, eccentricity distributions that are biased toward highereccentricities than thermal are likely to result in a reductionof the number of SHS produced. e C D F F IG . 10.— Cumulative distribution function of the initial eccentricity ofthe star about the secondary e . Each CDF is color-coded to the particularscattering experiment labeled in Figure 5, and the thick black curve showsthe initially assumed thermal distribution, P ( e ) ∝ e . Beaming of Outgoing SHS Trajectories
In the center-of-mass frame of the triple system, the motionof the secondary and tertiary both follow a nearly-parabolicelliptical orbit about the primary. Because the tertiary is un-bound from the secondary near periapse, its position at thispoint defines the apex of a cone of outgoing trajectories. Theunbound objects leave the system on hyperbolic orbits, wheretheir eccentricity is directly related to the excess orbital ve-locity and the secondary’s periapse distance, e = (cid:115) + v ∞ r p , GM . (22)As r p , = a q / , and v ∞ (cid:39) v q / (cid:39) GM q / / a , Equation(22) becomes e (cid:39) (cid:113) + q − / . (23)This in turn is related to the angle of the outgoing velocity vec-tor by φ = π/ − arccos(1 / e ), where φ is measured from a vec-tor parallel to the secondary’s motion at periapse (Figure 11,upper left panel) and is equal to π/ q (cid:39) φ (cid:39) ◦ .In the lower left panel of Figure 11 we show histograms of φ (longitude), as defined above, and θ , the azimuthal angle(latitude). We find that indeed the average ejection φ has avalue of around 40 ◦ , although with some considerable scatter.Given our isotropic angle distribution assumption, it is notsurprising that our θ distribution is symmetrical about zero,and with a scatter comparable to the average φ value, withlatitudes near the orbital plane being preferred (Sesana et al.2006). The fraction of solid angle that the SHS are beamedinto is rather small, with approximately half the SHS emanat-ing from ∼
5% the area of the full sphere (Figure 11, rightpanel).If the secondary’s orbital evolution is plunging, this impliesthat many stars are lost from the secondary nearly simultane-ously, perhaps even on the same orbit. If this is the case, thebeaming described above would cause stars from these mergerevents to be confined to a small cone, which would enhancethe number of SHS originating from particular merger eventsin which the cone is pointed at us, at the expense of manymerger events beaming their stars in directions away from theEarth. And because the majority of SHS are produced by theemi-Relativistic Hypervelocity Stars 13 - - -
50 0 50 100 1500.000.020.040.060.080.10 Θ , Φ H deg L P D F F H Θ , Φ L F IG . 11.— Distribution of velocity vector angles for SHS. Cumulative fraction F ( θ , φ ) of ejected stars with v ∞ >
0. The diagram in the upper left showsthe positions of the three objects (primary black hole, labeled M , secondary black hole, labeled M , and tertiary star, labeled SHS) shortly after the ejectionof an SHS, where φ is the polar angle and φ = 0 is defined to be the angle of M ’s velocity vector at periapse, and the azimuthal angle θ is measured from theorbital plane. The single-parameter histograms in the bottom-left panel and the 2D histogram mapped to the surface of a sphere show that SHS are preferentially"beamed" in a cone centered at θ = π / φ = 0 (see Zier & Biermann 2001; Sesana et al. 2007b).F IG . 12.— Trajectories of SHS within 2 Mpc of the MW under the pre-sumption that all originate from a single merger event 3 Gyr ago in M87. Theplot shows the present locations of SHS in a Hammer projection of galacticcoordinates, where the thin black lines trace their path back to their galaxy oforigin. Deflection by either intervening substructure or the MW itself is nottaken into account. most-massive black hole mergers, this may mean that a selectfew merging MBHs would be responsible for the lion’s shareof the local SHS distribution.In Figure 12 we show an illustrative example where a MBHmerger occurred in M87 at a time 3 Gyr prior to the presentand whose cone was beamed directly at the MW. For this fig-ure, we assume that stars within a distance 2 Mpc of the MWare detectable, and additionally that this one merger event isresponsible for all SHS in this velocity range, as might be ex-pected if the beaming of SHS is ubiquitous. The figure showsthat if SHS are indeed beamed, and the three-dimensional ve-locity vectors of multiple SHS can be determined, those starswith similar total velocities are likely to originate from thesame galaxy, whose location can be determined by tracingtheir paths back to a common origin. This prospect is dis-cussed in more detail in Section 8.1. SEMI-RELATIVISTIC HYPERVELOCITY BINARIES
A unique feature of the SHS ejection process is that the ter-tiary need not be a single star, but itself can be replaced bya binary. For tight binaries, the tidal disruption radius of thebinary is only slightly larger than the tidal disruption radius of the star, and because binaries have a chance to survive evenwhen passing deep within a tidal potential (Sari et al. 2010),they can sometimes survive even in extremely strong tidal en-counters.The survival of a multiple system undergoing the Hillsmechanism has been studied in the context of triple stellarsystems (Perets 2009) and for planetary systems orbiting oneof the two stars (Ginsburg et al. 2012). Hypervelocity binarieshave also been shown to be produced by a chance scatteringvia the inspiral of an intermediate mass black hole, similarto the mechanism we describe here, but where the binary isoriginally bound to the primary MBH (Levin 2006; Lu et al.2007; Sesana et al. 2009). In these papers, either the primary-secondary system is treated as being on fixed orbits (Lu et al.2007), or the tertiary-quartary binary is not modeled explicitly(Sesana et al. 2009). Here we present the first set of simula-tions in which no approximations are made, and the orbits ofall four bodies are solved for explicitly, however it should benoted that the assumption that the primary-secondary systemremains unperturbed is a perfectly valid one if one is onlyconcerned with the fate of the outgoing stellar binary. Byevolving the orbits of all four bodies explicitly, we are ableto calculate the energy lost by the primary and secondary inthe ejection, as we do in the three-body case, which enablesus to guarantee that the total energy and angular momentumof the system is conserved.We show the production of an example semi-relativistic hy-pervelocity binary (SHB) in Figure 13. The orbit of the stellarbinary, which we refer to as the tertiary-quartary system, isselected such that its periapse is beyond the distance specifiedby Equation (19), where all variable indices are incrementedby one (as we are considering two objects in orbit about thesecondary). While satisfying this criterion guarantees that thesystem will not be disrupted, it doesn’t prohibit secular evolu-tion of the orbit of the secondary-tertiary-quartary triple sys-tem, and the upper right panel of Figure 13 shows that a cleareccentric Kozai excitation (Kozai 1962; Lithwick & Naoz2011; Naoz et al. 2013; Li et al. 2014) is taking place priorto the secondary’s periapse about the primary. Generically,4 Guillochon, Loeb P r e - p e r i a p s e ( A ) P o s t- p er i a p s e ( B ) AB F IG . 13.— Example of a BMBH encounter in which an SHB is produced. The left panel shows the orbital trajectory of the secondary (purple path) about theprimary (black dot), and the tertiary-quartary’s path about the secondary (orange path). Two regions are highlighted in the left panel and shown as insets in thetertiary-quartary frame, where the tertiary is colored orange and the quartary is colored blue: A pre-periapse phase (A) and a post-periapse phase (B). Beforeperiapse, the tertiary-quartary system is on an eccentric orbit about the secondary and demonstrates a complicated eccentric Kozai excitation in which eccentricityand inclination are exchanged. The action of the secondary passing the primary results in the unbinding of the tertiary-quartary system from the secondary, atwhich point the eccentric Kozai ceases to be driven, and the stellar binary settles into a stable orbit. the orbits of stars (and binaries) about an MBH have non-zeroeccentricities, and so long as the two stars are themselves notinteracting tidally, they too will have a range of eccentricities.As a result, eccentric Kozai excitations of varying magnitudeare likely common pre-ejection states for SHB, especially forthe fastest examples where the minimum separation is limitedby the physical size of the stars in the binary.Because more SHS are produced by the mergers of near-equal mass black holes, the separation between the tertiary-quartary and the secondary is often times comparable to theprimary-secondary distance at periapse (see Figure 2), mean-ing that the tertiary-quartary system can in some cases be dis-rupted by the primary if it lies close to it at periapse. In theexample presented in Figure 13, the tertiary-quartary systemis further from the primary than the secondary, being on the“outside track,” and thus experiences tidal forces upon it fromthe primary that are weaker than those imposed by the closersecondary, but this is not guaranteed for all encounters. Justas in the single star case, the tertiary-quartary system can alsobe driven closer to the secondary, which can also disrupt it;again our example shows a case in which this does not occur.Figure 4 showed that very few single stars were tidally dis-rupted by either black hole (most of those that were destroyedwere swallowed whole), this is likely to be different for SHBthat have a tidal radius that is at least a factor of a few largerthan that of single stars.After the secondary reaches periapse, and if the tertiary-quartary system survives the tidal forces of both black holes,the system can be ejected at velocities that are comparable toSHS velocities. Of course since the stellar binary leaves thevicinity of both black holes, it settles into a simple ellipticalorbit that is subject to no external perturbation. The final ec-centricity and semi-major axis distributions of these objectsis not calculated here, but is likely to be more extreme rela-tive to binaries formed via other processes. As a result, such objects may merge more-frequently than the average stellarbinary, and perhaps the population of SHB may be somewhatenhanced in merger products relative to the field. OBSERVATIONS
Once accelerated, SHS travel across the universe with lit-tle influence from self-gravitating structures, with the escapevelocities from the most massive structures rarely exceedinga few 10 km s − (as an example, the Phoenix cluster, one ofthe most massive known clusters, has an escape velocity of ∼ − , McDonald et al. 2012). As star formationactivity peaks at z ∼
2, and low-mass stars are more commonthan high-mass, the stars that survive their journey such thatthey are potentially observable are those with lifetimes in ex-cess of ∼
10 Gyr, similar to the MS lifetime of the Sun. Starswith masses (cid:46) M (cid:12) can remain on the MS for far longer thana Hubble time, but can be significantly dimmer than the Sunat solar metallicity. Those stars that evolve into giants are sig-nificantly brighter and thus easier to detect, however the giantphases are typically short ( (cid:46) The unbound SHS population
To determine the population of SHS that would be detectedby a survey, we use the results from our SHS scattering exper-iments as inputs into a second Monte Carlo calculation. Weassume that the initial velocity distribution of SHS is univer-sal and has no dependence on system properties, as suggestedby our scattering experiments in which most of the depen-dence is on ˜ a and q , both of which are scale-free. Randompositions are drawn within a sphere of radius r Virgo = 16 Mpc,emi-Relativistic Hypervelocity Stars 15 t H Gyr L P r ob a b ilit y P H t ej L P H t age L F IG . 14.— Probability distribution of times of ejection t ej (blue) and ages t age (gold) of SHS. The derivation of these distributions is described in thetext. where r Virgo is approximately the distance to the Virgo clus-ter, the maximum distance to which future surveys will beable to detect single stars. We assume that the distributionof SHS is isotropic, an assumption that is increasingly validfor increasing v ∞ ; as an example, the distance traveled byan SHS that was ejected 10 yr ago exceeds 50 Mpc for v ∞ (cid:39) × km s − . SHS emitted with velocities lower thanthis value are likely to be somewhat concentrated toward themassive galaxies within which the majority are produced, wedo not consider this effect here.Initial velocities are drawn from the probability distributiondisplayed in Figure 7, but as previously stated, these velocitiesare only relative to the binary black hole system, and do notaccount for deceleration in galactic potentials. To determinethe actual velocity at infinity, individual stars are assigned toindividual merger events with a probability proportional to thesecondary stellar mass in each merger. Then, the combinednuclear stellar masses and combined halo masses are used todetermine the additional energy each SHS requires to escapeits host galaxy. To determine the additional energy required toescape the nuclear cluster, we assume that the black hole dom-inates interior to the sphere of influence, and that the addi-tional energy required to escape the nuclear cluster can be de-termined from the cluster’s velocity dispersion, v nuc , esc = 2 σ ,where σ is defined as in Equation (8). We then use a rela-tion for the maximum circular velocity of a halo (Klypin et al.2011), multiplied by √
2, to define the escape velocity fromthe merged halo, v halo , esc = 930 (cid:18) M + M M (cid:12) (cid:19) . km s − , (24)where we have adopted a Hubble parameter h = 0 . a ej , where a ej = (1 + z ej ) − is the cosmolog-ical scale factor at the time the star was ejected (Peebles 1993;Weinberg 2008), where z ej is the redshift of the source galaxy.This effect is quite significant for SHS as a significant fractionof merger activity occurs at high redshift (see Equation (16)).This makes SHS rather unique: aside from individual parti-cles such as photons, neutrinos, and cosmic rays, SHS are the only objects that are not bound gravitationally, and thus arepotentially useful as sub-relativistic probes of cosmology, thedetails of how these stars may be used in this context are pre-sented in a companion paper (Loeb & Guillochon 2014). Sec-ondly, stars a given distance from the MW will follow Hub-ble’s law, giving them an additional component of recessionvelocity. This has the effect of making blueshifted SHS ap-pear to move more slowly and redshifted SHS more quicklythan an SHS at zero distance from the MW. The effect is rela-tively minor for the majority of SHS that are only a few Mpcfrom the MW, but in principle can be significant for brightstars that are visible out to the distance of Virgo, where theHubble velocity is ∼ km s − .Taking these effects into account, the velocity of an SHS (cid:126) v SHS is then given by (cid:126) v SHS = v H ˆ r + (cid:126) v p (25) (cid:126) v p = (cid:126) v ∞ | (cid:126) v ∞ | (cid:113) | (cid:126) v ∞ | − v , esc − v , esc + z ej , (26)where (cid:126) v p is the star’s peculiar velocity relative to the Hubbleflow and where the Hubble velocity v H = H r for r (cid:28) c / H .Stars for which | (cid:126) v p | evaluates to an imaginary number are con-sidered to remain bound to their host halos and are removedfrom the sample.Given the velocity distribution of nearby SHS, the age ofeach star must be derived to determine what stage of its life itis in when it reaches us. SHS are composed of stars that werein existence at the time of each black hole merger, and there-fore the redshift of formation z form of a given SHS should berandomly drawn using the star formation history of the galaxyup to the redshift of ejection z ej . The time of ejection is de-termined by randomly drawing redshift values in proportionto the growth rate as a function of redshift, given in Equation(17), where we consider a halo mass of 10 M (cid:12) for simplic-ity (the difference in the relative growth rate is minuscule fordifferent halo masses, scaling as M . ). For the star forma-tion history as a function of z we use the dust-corrected starformation rates presented in Bouwens et al. (2012), and draw z form between z ej and z max , where we set z max = 10. The result-ing age distribution is shown in Figure 14. Because most starsform at z ∼
2, the typical age of an SHS is (cid:38) yr.Once the local velocity distribution and age are determined,the distribution of stars that can be detected by a given sur-vey can be determined by assigning a mass to each star andthen using a stellar evolution track to evaluate its luminosityand color for its age, for this purpose we use the PARSEC1.2S isochrones (Bressan et al. 2012; Chen et al. 2014) avail-able through the web-based tool CMD 2.7 . Again, we adopta Kroupa mass function of stars. Given a mass, if a star’sage is found to be greater than the maximum age withinthe PARSEC track corresponding to that mass (which do notinclude post-giant phases), we assume that the star has be-come a compact object, where stars with M ∗ < M (cid:12) becomewhite dwarfs, 8 M (cid:12) < M ∗ < M (cid:12) become neutron stars, and M ∗ > M (cid:12) become black holes.The resulting distributions of stars that lie within a volumeconstrained to the distance of the Virgo cluster r Virgo = 16 Mpcis shown in Figure 15. The pink histogram reproduces thehistogram presented in Figure 7, where only the black hole’spotential is taken into consideration, whereas the orange-red http://stev.oapd.inaf.it/cgi-bin/cmd - - - - - - - @ v SHS
D H km (cid:144) s L L og @ N H r < r V i r go L d e x - D Log @ v (cid:144) c D Initial Dist.Unbound Dist.Blueshifted Dist. E - ELT (cid:144)
GMT (cid:144)
TMT
Phot.
JWST
Phot.
JWST
Spec.
Euclid (cid:144)
WFIRST
Phot.
LSST
Det. Motion F IG . 15.— Number of SHS satisfying various criteria at a distance r less than that to the Virgo cluster ( r Virgo = 16 Mpc). The pink histogram shows theinitial distribution of SHS resulting from MBH mergers, where we have multiplied the total by a factor 2 to account for both primary and secondary SHS (seeSection 3.3), and only considers the gravitational influence of the black holes themselves. The orange-red histogram shows the population of unbound SHS afteraccounting for the fraction of SHS that do not move fast enough to escape their host halos, resulting in a significant depletion of SHS for v ∞ (cid:46) km s − , andalso accounts for the slowdown due to the expansion of the universe. The blue histogram only shows those stars that show a blueshift relative to the MW. Theremaining histograms show the distributions expected to be detected by the various listed surveys, the assumed parameters of which are described in the text. Thevertical dotted black line shows v ∞ = 11,000 km s − , the speed limit for the HVS mechanism with a binary star system and a single MBH, whereas the verticaldashed black line shows v ∞ = 24,000 km s − , the speed limit for HVS produced from SBH-star systems, all objects to the right of this line can only be producedby merging MBHs (see Section 2.1.1). The solid black line shows where the number count of objects within a specified distribution equals one, whereas thecolored points along this line show the velocity for which the probability of detection for velocities greater than the marked value exceeds unity. histogram accounts for the slowdowns associated with the nu-clear cluster, host galaxy, cosmology, and redshift, as speci-fied by Equation (25). Interestingly, this Figure shows thatthe fastest star out to r Virgo is likely moving with v SHS (cid:39) c / γ = 1 .
06. While the distri-bution does extend to these extreme velocities, the vast major-ity travel at a more pedestrian value of 1,000 km s − , similarto the escape velocities of their host galaxies. Of these stars,less than half are blueshifted toward the milky way due to theredshift associated with the Hubble flow.We consider several different kinds of future astronomicalsurveys and the yields of SHS that they are expected to detect,these estimates are largely based on the proposed capabilitiesfor each survey and are only intended to roughly estimate thedetected SHS distributions. For LSST , we consider SHS thatsatisfy both a magnitude and proper motion cut; stars musthave I-band magnitudes < 25, with proper motion in excessof the limits presented in Ivezi´c et al. (2008), ∼ − .We include two all-sky spaced-based IR surveys of roughlysimilar capability, Euclid (Laureijs et al. 2011) and
WFIRST (Green et al. 2012), for both we require an SHS to have H-band magnitude < 24 to be detected with no proper motionconstraint, and assume that both surveys will cover ∼ JWST ’s instruments will have very smallfields of view (a few arcminutes), it will image some partsof the sky very deeply, potentially enabling the detection ofSHS at the distance of Virgo; for this survey we presumethat
JWST will be in operation for 10 years and that any SHSbright enough to collected photometrically (K < 29) or spec-troscopically (K < 26) would be considered detected. The next generation of large telescopes such as
E-ELT (Gilmozzi& Spyromilio 2007), GMT (Johns et al. 2012), and TMT (Macintosh et al. 2006) are also likely to have slightly largerfields of view (but still sub-degree) for their instruments; wesimilarly assume that these instruments will be in operationfor a decade and that objects with K < 29 are detected.We find that the fastest star in the volume out to Virgo likelytravels with v ∞ (cid:39) − , with the fastest blueshiftedstar traveling with velocity 70,000 km s − . With the abovesurvey parameters in mind, the fastest star detectable with E-ELT / GMT / TMT travels with v ∞ (cid:39) − , the fasteststar detectable with JWST travels with v ∞ (cid:39) − , thefastest star for which JWST spectroscopy is possible travelswith v ∞ (cid:39) − , the fastest star likely to be detectedwith Euclid/WFIRST travels at 4,000 km s − , and the fasteststar that will have detectable proper motion in LSST travelswith v ∞ (cid:39) − .Figure 16 shows the stages for stars that are detectable withvarious surveys. Because the majority of stars in the universehave masses less than a solar mass, a large fraction of starshave yet to evolve off the MS. Around 10% of stars haveevolved beyond the giant branches to become a white dwarf,neutron star, or black hole, with most of these objects beingwhite dwarfs (left-most column of Figure 16). The remaining ∼
1% are luminous giants, these are the stars that dominatethe samples of surveys given the advanced age of SHS that emi-Relativistic Hypervelocity Stars 17lie near the MW (four right columns of Figure 16). Red giantbranch stars contribute about half of the detectable sample,with the shorter-lived (but brighter) giant branches contribut-ing a significant fraction of the total. Palladino et al. (2012)identified a sample of candidate red giants with distances of300 kpc – 2 Mpc within SDSS and suggested that some ofthese stars may have a hypervelocity origin. While a largefraction of these stars are likely halo stars (Bochanski et al.2014), these candidates are at distances for which we expectthe SHS population to dominate HVS produced by the MW(see Figure 7).The fact that these evolved stages that are detectable onlycomprise a small fraction of the total population means thatthe number of detectable SHS is significantly reduced relativeto the total number of SHS in the vicinity of the MW. Fig-ure 15 also shows the expected velocity distributions of SHSwithin each survey’s sample of detected SHS, the maximumvelocity that is likely to be detected is ∼ − , abouta factor of five slower than the fastest star expected within r Virgo . DISCUSSION
Detection versus Identification
While a number of SHS are likely to be detected by fu-ture surveys, definitive identification of SHS from photom-etry alone is likely to be challenging. Indeed, there will bemany red, point-like objects in any selected imaging field, notjust from MW dwarfs, but also from unresolved high-redshiftbackground galaxies. This high chance of confusion is likelyto hamper efforts to identify SHS that are stationary on thesky, i.e. those that are far from the MW.The most straight-forward means of identification is to lookfor objects with high proper motions, and this will be proba-bly doable with
LSST . As described in the previous section,the vast majority of detectable SHS, regardless of the sur-vey, are likely to be detected having velocities on the orderof a few thousand km s − . This means that their velocitiesare in fact quite similar to that of HVS produced by our owngalaxy, or by nearby galaxies such as Andromeda (Sherwinet al. 2008). For these objects in which the velocities aresimilar to local HVS, the main identifying feature will be thefact that their velocity vectors are unlikely to originate fromeither our galactic center or Andromeda. Additionally, SHSare likely to outnumber the local HVS population at distancesgreater than a Mpc from the MW (see Figure 7).However, we showed in Section 5.2.5 that if the merger isplunging, many of the stars originally bound to the BMBHare likely to be beamed into a narrow cone. If this is thecase for most BMBH, it would imply that the local densityof SHS likely originates from only a couple BMBH merg-ers, and that several SHS would have trajectories that wouldpoint back to a common source galaxy (Holley-Bockelmannet al. 2005). In Figure 12 we assumed that the central Virgocluster galaxy M87 experienced a BMBH merger 3 Gyr prior,where a large fraction of that event’s SHS were beamed to-ward the MW. In such a scenario the trajectories of the SHSpoint back toward M87, potentially enabling its identificationas the source galaxy. Because M87 is relatively close to us,all stars launched in this hypothetical merger event that aremoving faster than 5,900 km s − have already passed beyondthe 2 Mpc distance limit set in that figure, whereas those starsthat are moving slower than 4,600 km s − have yet to reachus. This limited range in velocities, in conjunction with the CompactObjects All LSST E - ELT (cid:144)
GMT (cid:144)
TMT JWST Euclid (cid:144)
WFIRST0 % % % % % % % % % % % Black HoleNeutron StarWhite DwarfAGBHBRGBSGBMS F IG . 16.— Fraction of stellar types existent within a sphere with radius r Virgo detectable with various telescopes. The first and second columns showthe population of objects within this sphere, regardless of detectability, withthe first column only showing the compact objects, and the second columnshowing all stellar types, which is dominated by low-mass stars that are stillundergoing gravitational contraction. The subsequent four columns show thefraction of stellar types that are likely to be detected photometrically by vari-ous telescopes. The vast majority of detectable objects are evolved stars. - - - -
20 Log Λ H Þ L L og L Λ (cid:144) L Ÿ M3III - - - Λ H Þ L A B ob s - A B r e s t F IG . 17.— Comparison of rest-frame red giant spectra (in black) toblueshifted spectra (in blue) corresponding to the fastest star likely to be de-tected by E-ELT / GMT / TMT , v SHS = 0 . c (cid:39) − . The rest-framespectrum is drawn from the publicly available library of Pickles (1998) andshows a M3III giant. As the input spectra only extend to 25,000 Å, fluxesredward of this value are extrapolated assuming a Rayleigh-Jeans law. In-set within the figure is the difference between the observed AB magnitudeAB obs for individual JWST filters versus the magnitude AB rest that would beobserved if the star were at rest. spatial distribution of SHS and their velocity vectors, can po-tentially be used to associate families of SHS with particulargalaxies and merger events.For stars that move faster than a few thousand km s − , theirrarity means that their distance from the MW can be largeenough that proper motion will not be detectable. For thesestars, the best hope for identification is to happen to take aspectrum of an SHS. Because SHS have a high probabilityof being blueshifted toward the Milky Way (whereas mostextended objects will be redshifted), a spectrum showing ablueshift on the order of a few thousand km s − or greaterwould permit their immediate identification. Figure 17 showsthat the spectral shift associated with the fastest star that islikely to be detected by a thirty-meter class telescope wouldbe extremely obvious. If only photometry is available, thesestars will also have a measurable shift in color on the order ofa few tenths of a magnitude, however given that these stars aremost likely to be giant branch stars, this small shift in coloris likely difficult to disentangle from color changes associatedwith normal giant evolution.8 Guillochon, Loeb Other Means of Detection
So far we have only considered single, luminous stars inour analysis as these are likely to be the most common SHS.However, there are several other ways to either detect themdirectly or to infer their presence. We describe a few possibil-ities in this section.
Accreting and Merging SHB
As described in Section 6, the mechanism for acceleratingSHS can also produce SHB. These binaries are likely to haveshort orbital periods (otherwise they would not have survivedthe acceleration process), and thus are likely to eventually un-dergo mass transfer, followed potentially by a merger. If oneof the two stars is a compact object this mass transfer can illu-minate the binary to the Eddington limit ( ∼ ergs s − forsolar mass accretor), potentially making the system visible tovery large distances. Likewise, if the two stars merge aroundthe time they are close to the MW, they may form a blue strag-gler with a luminosity comparable to a giant that would po-tentially be more detectable with optical surveys. However,both X-ray binaries and blue stragglers are rather rare in thefield, and given that generic SHB consisting of two MS starsare rare in the first place, it implies that SHB would have topreferentially form such systems to have an appreciable prob-ability of detection. Bow Shocks
An additional means of detection may be via the bowshocks driven into the ISM by these high-velocity stars. Itis already known that fast-moving stars in star-forming re-gions can drive very strong and spatially large bow-shocksinto their environments (Meyer et al. 2014), and thus this maybe a promising way to detect the presence of a fast-movingstar. However, the SHS are unlikely to be significantly con-centrated toward the galactic plane where the densest phasesof the ISM exist (although there may be a local density en-hancement of ∼
100 for lower-velocity SHS due to gravita-tional focusing, see Sherwin et al. 2008), and therefore theirbow shock signatures are most likely to be confined to regionsof fairly low density, resulting in luminosities that are unlikelyto exceed the luminosities of the stars themselves.
Isolated Local Group Pulsars
While neutron stars are likely to only be (cid:46)
1% of all SHS,this means that roughly 10 would lie within 1 Mpc of theMW. Some fraction of these objects may be pulsars that arepotentially detectable with SKA (Smits et al. 2009; Lazio2013). Intrahalo Light (IHL)
There are about 10 Mpc − stars in the local universe,and with the SHS density being approximately 10 Mpc − ,SHS make up about 10 − of all stars in the universe at z = 0.Because they travel great distances from their source galax-ies (Figure 18), SHS are expected to have a nearly isotropicdistribution, and thus will also occupy the vast voids be-tween galaxy clusters that are otherwise unlikely to host manystars. This population may be detectable via their contribu-tion to IHL, which we estimate to yield ∼ − nw m − sr − at 1 micron. This is lower than the average IHL value of ∼ − sr − (Cooray et al. 2012), but is comparable to the minimum value of ∼ − nw m − sr − seen in some re-gions of the sky (Zemcov et al. 2014). This suggests that SHSmay indeed set the floor value for the IHL in voids. Supernovae from SHS and SHB
Far-flung supernovae that occur in no known host galaxyare potentially another way of inferring the presence of SHSand SHB. For SNe II, which occur only for stars with massesgreater than ∼ M (cid:12) , the range of the ejection distances is lim-ited, but potentially can be distinguished. An 8 M (cid:12) star travel-ing at 3,000 km s − will travel at most 300 kpc before death, atthese distances it is possible that the supernovae occurred in aunresolved dwarf galaxy. Given the local density of SHS andpresuming that the per-star supernova rate is the same as theMW ( ∼ − Mpc − yr − ), ∼
100 yr − SNe II will occur in thevolume out to z ∼ ∼ yr − ).Normal HVS can also contribute to these off-center Type IIevents as they eject stars at similar speeds. For stars movingat 30,000 km s − , about one Type II will occur at a distance ∼ z = 1.For SNe Ia, the rates are reduced somewhat by the fact thatthe progenitors are binary stars, of which an unknown fractionsurvives the ejection process (see Section 6). By definition,high velocity Type Ia progenitors cannot be generated by theHVS process, which only accelerates single stars, and insteadmust either come from a triple system disruption or the SHBmechanism described in this paper. A key difference betweenthese systems and Type II progenitors is that they can poten-tially live much longer before the supernova occurs, and thuscan be much further from their source galaxies. If double-degenerate systems are responsible for a significant fraction ofIa events, or the ejection occurs before the donor star evolvesoff the MS in the single-degenerate phase (to avoid havinga fragile giant in the SHB), their travel distance is only lim-ited by the Hubble time, and thus these supernovae can occurin truly isolated environments, perhaps even at the centers ofvoids where very few stars reside.Given that LSST is expected to detect a few 10 supernovaeper year (Abell et al. 2009), and presuming the above rates,we predict that LSST should find a few isolated supernovae(with d (cid:38)
100 kpc from their origin galaxy) that originatefrom the SHS/SHB population per year. Zinn et al. (2011)found several candidate supernovae at distances of kpc fromtheir source galaxies that likely originated from stars withvery short progenitor lifetimes. They inferred that the progen-itor stars must have traveled with speeds in excess of severalhundred km s − to get so far away from their source galaxies.However, it should be noted that a contaminating foregroundof both Type Ia and Type II would exist in galaxy clustersdue to tidal stripping of stars from member galaxies (Maozet al. 2005). Therefore, the best place to look for supernovaeoriginating from SHS/SHB are around galaxies that are not inclusters, or in the voids between galaxies. Complications
Aside from checking whether stars cross the IBCO of ei-ther black hole, our scattering experiments do not includethe effects of general relativity, which can be quite large forthose stars that we predict are launched with velocities of c / - -
505 Log d H Mpc L L og @ N H r < r V i r go L d e x - D d = P a r t i c l e H o r i z o n Log v H km L F IG . 18.— Histograms of distance d traveled by SHS from their point of origin for SHS moving at various velocities. The different colored lines show differentvelocity ranges (as labeled), while the line-styles refer to different criteria for detection: The dotted lines show all objects, the dashed lines show objects detectablephotometrically with JWST , and the solid lines show objects detectable with
Euclid / WFIRST . The sharp drop-off in the last bin shows where a large fraction ofstars are swallowed whole rather than being ejected as SHS. locities. As shown in Figure 4, the fraction of stars that comewith a few IBCO radii of either black hole and avoid beingswallowed or tidally disrupted is quite small.We assumed that all merging MBHs within the mass ratiorange specified merge eccentrically, and that the evolution ofthe secondary’s orbit is plunging. While both numerical sim-ulation and Fokker-Planck approaches have found that largeeccentricities can be excited, net rotation of the primary’s nu-clear cluster can suppress the build-up of eccentricity for asmany as half of all systems (Sesana et al. 2011; Dotti et al.2012).For the fastest SHS ( v > km s − ), the velocity distribu-tion is likely quite sensitive to the radial distribution of starsinterior to the distance at which star-star collisions becomecommon (see Section 4). In this paper we assumed that thisdistribution is defined based on rather limited observations ofthe MW’s nuclear cluster, but this observed distribution is notnecessarily generally applicable to the ∼ M (cid:12) black holesresponsible for producing the majority of SHS. Because two-body relaxation is ineffective at replenishing stars interior tothe radius where collisions become common (The results ofwhich would produce luminous transients, see Balberg et al.2013), it is largely resonant interactions (Hamers et al. 2014)and binary disruptions (Perets et al. 2009) that populate thisregion. Additionally, the secondary’s nuclear cluster is con-tinuously subjected to very strong perturbations from the pri-mary that are likely to alter the orbits of all stars within it,even if they do not immediately unbind. CONCLUSIONS
We have demonstrated in this paper that SHS (and SHB)are stars that can be accelerated to speeds in excess of 10 km s − , a speed which is difficult (if not impossible) to pro-duce via any other astrophysical mechanism. If discovered,they would be unique tracers of the eccentric mergers of su-permassive black holes. There is some observational evidencethat such mergers do occur (Valtonen et al. 2008; Batcheldoret al. 2010), but this evidence will likely remain inconclusive until gravitational waves are detected from them. Because thelarge velocities found here are contingent upon merging su-permassive black holes possessing significant eccentricities,the discovery of even one SHS or SHB would suggest thateccentric MBH mergers are common.While identifying these stars may be challenging, we haveshown that many of them are likely to be detected by futureastrophysical surveys, and that they may be discovered viaseveral other direct and indirect means.These fastest stars are also one of the few natural phenom-ena that are likely to cross the vast chasms of empty space be-tween galaxies. Figure 18 shows that even the slowest SHS,which move at a few 10 km s − , will have traveled tens ofMpc from their source galaxy by the time they reach the MW,and that the fastest, although quite rare, can travel nearly 10Gpc since being ejected. This makes them potentially pow-erful probes of cosmological expansion, which we detail in acompanion paper (Loeb & Guillochon 2014). Given the ratesof production we have calculated here, it is very likely that astar that has traveled a distance of over 1 Gpc lies betweenthe MW and the Virgo cluster. Such a star would be by farthe fastest, and most-traveled, luminous object in our localneighborhood.We are thankful for fruitful discussions with F. Antonini,K. Batygin, A. Bogdan, W. Brown, C. Conroy, S. Genel,I. Ginsburg, P. Groot, M. Holman, S. Naoz, E. Ramirez-Ruiz,R. Sari, A. Sesana, D. Sijacki, J. Strader, and Y. Levin. Weare especially grateful to M. C. Miller for extended discus-sions regarding the mechanism presented here, and to ourreferee who provided a more direct derivation of the maxi-mum speed for hypervelocity stars. This work was supportedby Einstein grant PF3-140108 (J. G.) and NSF grant AST-1312034 (A. L.). Much of this paper was written at the AspenCenter for Physics (NSF Grant REFERENCESAbell, P. A., Allison, J., Anderson, S. F., et al. 2009, ArXiv e-prints, 0912,201 Agnor, C. B., & Hamilton, D. P. 2006, Nature, 441, 1920 Guillochon, Loeb