TIDAL BREAKUP OF BINARY STARS AT THE GALACTIC CENTER. II.astrophysics.rit.edu/fantonini/tbbs2/ms.pdf · Tidal breakup of binary stars at the Galactic Center-II 3 that, at any radius,

Draft version August 31, 2010Preprint typeset using LATEX style emulateapj v. 08/13/06

TIDAL BREAKUP OF BINARY STARS AT THE GALACTIC CENTER. II.HYDRODYNAMIC SIMULATIONS

Fabio AntoniniDepartment of Physics and Center for Computational Relativity and Gravitation, Rochester Institute of Technology, 85 Lomb Memorial

Drive, Rochester, NY 14623, USA

James C. Lombardi Jr.Department of Physics, Allegheny College, 520 North Main Street, Meadville, PA 16335, USA

David MerrittDepartment of Physics and Center for Computational Relativity and Gravitation, Rochester Institute of Technology, 85 Lomb Memorial

Drive, Rochester, NY 14623, USA

Draft version August 31, 2010

ABSTRACTIn Paper I, we followed the evolution of binary stars as they orbited near the supermassive black

hole (SMBH) at the Galactic center, noting the cases in which the two stars would come close enoughtogether to collide. In this paper we replace the point-mass stars by fluid realizations, and use asmoothed-particle hydrodynamics (SPH) code to follow the close interactions. We model the binarycomponents as main-sequence stars with initial masses of 1, 3 and 6 Solar masses, and with chemicalcomposition profiles taken from stellar evolution codes. Outcomes of the close interactions includemergers, collisions that leave both stars intact, and ejection of one star at high velocity accompaniedby capture of the other star into a tight orbit around the SMBH. For the first time, we follow theevolution of the collision products for many (& 100) orbits around the SMBH. Stars that are initiallytoo small to be tidally disrupted by the SMBH can be puffed up by close encounters or collisions, withthe result that tidal stripping occurs in subsequent periapse passages. In these cases, mass loss occursepisodically, sometimes for hundreds of orbits before the star is completely disrupted. Repeated tidalflares, of either increasing or decreasing intensity, are a predicted consequence. In collisions involvinga low-mass and a high-mass star, the merger product acquires a high core hydrogen abundance fromthe smaller star, effectively resetting the nuclear evolution “clock” to a younger age. Elements likeLi, Be and B that can exist only in the outermost envelope of a star are severely depleted due toenvelope ejection during collisions and due to tidal forces from the SMBH. Tidal spin-up due to eithera collision or tidal torque by the SMBH at periapsis can explain the observed high rotational velocityof HVS 8. However, in the absence of collisions, tidal spin-up of stars is only important in a narrowrange of periapse distances, rt/2 . rper . rt with rt the tidal disruption radius. We discuss theimplications of these results for the formation of the S-stars and the hypervelocity stars.Subject headings: black hole physics-Galaxy:center-Galaxy:kinematics and dynamic

1. INTRODUCTIONTidal breakup of binary stars by the supermassive

black hole (SMBH) at the Galactic center (GC) has beeninvoked to explain a number of otherwise puzzling discov-eries, including the hypervelocity stars (HVSs) that areobserved in the halo of the Milky Way (Brown et al. 2005,2006, 2007, 2009), and the S-stars, apparently young,main-sequence stars in tight eccentric orbits around theSMBH (Eisenahauer et al. 2005; Gillessen et al. 2009).As first pointed out by J. Hills, close passage of a binarystar near a SMBH can result in an exchange interac-tion, such that one component of the binary is ejectedwith greater than escape velocity while the other staris scattered onto a tightly-bound orbit (Hills 1988; Yu& Tremaine 2003). The predictions of this model arebroadly consistent with the observed properties of boththe HVSs (Bromley et al. 2006) and the S-stars (Peretset al. 2009; Perets & Gualandris 2010).

Electronic address: [email protected]

The origin of the binary progenitors of the HVSs isnot clear. One possibility is that the binaries originatedat distances of a few parsecs from the GC and weresubsequently scattered inward by “massive-perturbers”(Perets et al. 2007). In this scenario, most of the bina-ries will lie on either unbound or weakly-bound orbitswith respect to the SMBH, and they will encounter itonly once before being scattered onto different orbits. Al-ternatively, the binaries may form closer to the SMBH,perhaps in the young (or an older) stellar disk that is ob-served between ∼ 0.04 pc and ∼ 0.5 pc from the SMBH(Nayakshin & Cuadra 2005; Paumard et al. 2006; Nayak-shin & Cuadra 2007; Levin 2007). In this case, the bina-ries would be bound to the SMBH and would encounterit many times before being disrupted.

If the finite sizes of stars are taken into account, anumber of outcomes are possible in addition to simplebinary disruption. The two stars can collide, resultingin a merger if the relative velocity is less than stellar es-cape velocities (Ginsburg & Loeb 2007). Since the radiusof tidal disruption of single stars by the SMBH is com-

2 Antonini, Lombardi & Merritt

Fig. 1.— Evaporation time of binaries vs. galactocentric radiusfor different values of the binary semimajor-axis a0. Solid curvesshow the evaporation time for the density model of equation (2)with γ = 0.5 while the dashed curves correspond to the corelessmodel with slope γ = 1.8. The filled grey region gives the ages ofthe S-stars (Eisenahauer et al. 2005).

parable to the binary disruption radius, stars can alsobe tidally disrupted by the SMBH, either before or aftertheir close interaction with each other.

In Paper I (Antonini et al. 2010) we presented the re-sults of a large number of N -body integrations of point-mass binary stars on eccentric orbits around the GCSMBH. In many cases, the trajectories of the two starswere found to imply a physical collision, assuming thatthe unperturbed stars had radii similar to those of nor-mal main-sequence stars of the same mass. The proba-bility of physical encounters was found to increase sig-nificantly if the binaries were allowed to complete manyorbits about the SMBH. In some cases, one or both starsalso passed close enough to the SMBH that gravitationaltides would be expected to significantly affect their in-ternal structure.

In this paper, we use smoothed-particle hydrodynam-ics (SPH) simulations to study the binaries from PaperI that approached closely enough to physically interact.The N -body simulations of Paper I were first used toidentify initial conditions that resulted in close interac-tions between the two stars. The point-mass stars werethen realized as macroscopic, fluid-dynamical models andintegrated forward in the gravitational field of the SMBHusing an SPH algorithm. As in Paper I, we followed thetrajectories for multiple orbits around the SMBH, allow-ing us, for the first time, to investigate the consequencesof repeated tidal interactions with the SMBH.

In §2 we briefly discuss time scales for binary disrup-tion at the GC. Our initial conditions and numericalmethods are described in §3 and the results in §4. Someobservable consequences are presented in §5. §6 sums up.

2. THE SURVIVAL TIME OF BINARIES AT THEGALACTIC CENTER

In a dense environment, binaries may evaporate due todynamical interactions with field stars if

|E|/(Mbσ2) . 1, (1)with E the internal orbital energy of the binary, Mbthe binary mass, and σ the one-dimensional velocity dis-persion of the stellar background. In principle, becausemost of the binaries at the GC are expected to be “soft”,|E| . Mbσ2, and because the binary evaporation timetev is a function of the distance from the SMBH, thevariation of tev with galactocentric radius can be usedto constrain the origin of the HVSs (Perets 2009). Ifthe evaporation time at some radius is shorter than thelifetime of a typical main-sequence star, the stellar popu-lation in this region would be dominated by isolated (i.e.single) stars.

Here we show that the survival time of binaries atgalactocentric distances r < 0.1pc is likely to be com-parable to the typical main-sequence lifetimes of moststars in this region. We also show that, within a radiusof r ∼ 0.3pc, tev becomes essentially independent of ra-dius.

Beyond ∼ 1 pc from Sgr A∗, the mass density de-termined from the stellar kinematics follows ρ ∼ r−β ,1.5 . β . 2 (e.g. Oh et al. 2009). At smaller radii, num-ber counts of the dominant (old) stellar population nearthe GC suggest a space density that is weakly rising, orfalling, toward the SMBH, inside a core of radius ∼ 0.5pc(Buchholz et al. 2009; Do et al. 2009; Bartko et al. 2010).Approximating the mass density as a broken power-law,

ρ(r) = ρ0

(r

r0

)−γ [1 +

(r

r0

)2](γ−β)/2(2)

with r0 = 0.3pc and β = 1.8, and setting ρ0 = 1.3 ×106M⊙pc−3 gives a good fit to the space density outsidethe core (e.g. Merritt 2010). At smaller radii, the uncer-tainties in ρ are represented by the poorly determinedvalue of γ.

The evaporation time is given by (Binney & Tremaine2006) :

tev =Mbσ

M 16√

πρa0 lnΛ, (3)

where lnΛ is the Coulomb logarithm, M the mass of thefield stars, a0 the binary semimajor-axis, and σ is calcu-lated from the Jeans equation,

ρ(r)σ(r)2 = G∫ ∞

r

dr′r′−2 [M• + M⋆(< r′)] ρ(r′), (4)

with M⋆(< r) the total mass in stars within r, and M•the mass of the central black hole. Hereafter, we adoptM• = 4 × 106 M⊙ (Ghez et al. 2008; Gillessen et al.2009). In Figure 1 we plot the evaporation time of bina-ries in the density model of equation (2) as a function ofgalactocentric radius, assuming Mb = 2M , ln Λ = 15 andtwo different values of the internal slope: γ = 0.5 repre-sentative of the observed distribution and γ = 1.8 whichcorresponds approximately to a relaxed system around aSMBH (Bahcall & Wolf 1976).

From the figure it is clear that the survival time ofbinaries would be greatly increased if the distribution ofstars at the GC has a low density core (for comparison,see also Figure 1 in Perets [2009]). The figure also shows

Tidal breakup of binary stars at the Galactic Center-II 3

that, at any radius, for a0 . 1AU, this time is largerthan the typical lifetime of the the S-stars. Therefore, itcannot be excluded that the S-stars were initially part ofbinary systems originating at galactocentric distances offew tens of milliparsecs.

We finally note that, in the context of this paper, itmight be more appropriate to compare tev with the timescale required to drive the eccentricities of the stars awayfrom their initial values to produce quasi-radial orbits.At the GC this time can be of the order of few Myr(Löckmann et al. 2008; Madigan et al. 2008; Merritt etal. 2009; Perets et al. 2009; Fujii et al. 2010), which ismuch shorter than the typical evaporation time scale ofbinaries with a0 ≤ 0.2AU.

3. INITIAL CONDITIONS AND NUMERICALMETHOD

3.1. Orbital initial conditionsIn Paper I we used the high-accuracy numerical inte-

grator ARCHAIN (Mikkola & Merritt 2008; Mikkola &Merritt 2006) to study the dynamics of main-sequencebinary stars on highly elliptical orbits whose periapsideslay close to the SMBH. We determined the final orbitalproperties of both ejected and bound stars. Initial condi-tions consisted of equal-mass binaries on circular relativeorbits with random orientations, initial separations a0 inthe range 0.05 - 0.2AU, and individual stellar masses Mof 3 M⊙ − 6M⊙. The binaries were given a tangentialinitial velocity in the range 4 km s−1 - 85 km s−1 andinitial distances of d = 0.01 − 0.1 pc from the SMBH.Using the mass-radius relation R/R⊙ = (M/ M⊙)

0.75

(Hansen et al. 2004), we could assign a physical dimen-sion to the particles and investigate the probability ofstellar collisions and mergers. In detail, we defined theminimum impact parameter for a collision as 2R, and thetwo stars were assumed to coalesce when their relativevelocity upon collision was lower than the escape veloc-ity from their surface. The binary separations adoptedin this work were close to the extremes of the intervalwithin which HVSs can be produced: small semimajor-axes, a0 . 0.05AU, result in contact binaries, while fora0 > 0.2AU few stars would be ejected with velocitiessufficient to escape the Galaxy (Gualandris et al. 2005).

In order to treat also the case of unequal-mass binaries,we extended the work of Paper I to include an additionalset of ∼ 2000 integrations of binaries with componentmasses M2 = 1 − 3M⊙ and M1 = 6 M⊙. Initial condi-tions and results of these new runs are summarized inAppendix A. In the following, for the case of unequalmass binaries, we distinguish between the primary andsecondary stars’ quantities using the subscripts 1 and 2respectively.

The orbital initial conditions that we adopt in the SPHsimulations correspond to binaries that enter well withintheir tidal disruption radius rbt, approximated by theexpression (Miller et al. 2005):

rbt ∼(

M•Mb

)1/3a0, (5)

with Mb = M1+M2 the total mass of the binary. There-fore, the binaries in all of our simulations are stronglyperturbed at the first periapse passage. Moreover, wefocus on cases where the tidal perturbations from the

SMBH on the single stars are expected to be signifi-cant. This corresponds to binaries with periapsides thatlie close to the tidal disruption radius of a single star, or

rt ∼(

M•M

)1/3R. (6)

We selected three types of initial condition that canbe classified according to the final outcome of the SPHsimulations.1. Stellar collision (without merger). In some cases,gravitational perturbations from the SMBH can lead toa physical collision between the two stars. If the relativevelocity at collision is sufficiently high, the stars can sur-vive the interaction and avoid a merger. Soon after thecollision, one star can be ejected at high velocity.2. Stellar merger. If the two stars collide with a relativevelocity at impact smaller than approximately the escapevelocity from their surfaces, coalescence occurs, resultingin the formation of a new, more massive star. The mergerremnant will remain bound to the SMBH unless extrememass loss occurs during coalescence.3. (Clean) ejection of a HVS. The tidal breakup of thebinary by the SMBH results in the ejection of a star atvery high velocity. The former companion of the ejectedstar loses energy in the process and is deposited onto atight orbit around the SMBH. Here, “clean” means thatthe member stars of the binary do not collide with eachother during the process of ejection.

For the sake of simplicity, in all the SPH simulations wechose the initial apoapsis of the external binary orbit tobe d = 2000 AU ≈ 0.01 pc (except for case H8 which hasd = 1700 AU; see Table 1). However, starting the SPHsimulations from this distance would greatly increase therequired computational time. The stars were thereforeinitially placed at a point of the orbit corresponding to amuch smaller distance from the SMBH: r0 = 5rbt. Thischoice for r0 was motivated by the fact that at thesedistances, the tidal forces from the SMBH are still tooweak to significantly influence the internal structure ofthe stars. In addition, since r0 is considerably larger thanrbt, at the initial time the internal binary eccentricity isstill close to zero (e . 10−2). As in Paper I, the plane ofthe binary’s orbit about the SMBH is the x − z plane.

3.2. SPH numerical techniques and initial conditionsSPH is a Lagrangian method in which the fluid is rep-

resented by a finite number of fluid elements or “parti-cles.” Associated with each particle i are, for example,its position ri, velocity vi, and mass mi. Each parti-cle also carries a purely numerical smoothing length hithat determines the local spatial resolution and is usedin the calculation of fluid properties such as accelerationand density. For a recent review of SPH, see Rosswog(2009). The SPH code used in this work is presented inGaburov, Lombardi, & Portegies Zwart (2010), with theaugmentation that the analytic solution to the Keplertwo-body problem can be used to advance a star boundto the SMBH through those portions of the orbit whenhydrodynamic effects are negligible (see §4.4).

In our simulations, the SMBH is a compact object par-ticle that interacts gravitationally, but not hydrodynam-ically, with the rest of the system. The gravity of theSMBH is softened according to a density profile defined


TABLE 1

Summary of the SPH simulations.

Run M1(M⊙) M2(M⊙) a0(au) rper(au) λ1 (λ2) ζ1(ζ2) SPH N -bodyC1 3 3 0.05 1.50 1.36 0.668 Collision+HVS Collision+HVSC2 3 3 0.2 2.67 2.43 1.19 Collision+HVS Collision+HVSC3 6 6 0.05 1.50 1.07 0.526 Collision+HVS MergerC4 6 6 0.2 2.04 1.46 0.716 Collision+HVS Collision+HVSC5 6 3 0.1 5.61 4.01 (5.09) 1.96 (2.49) Collision+HVS (primary) Collision+HVS (primary)

M1 3 3 0.05 5.05 4.59 2.25 Merger MergerM2 3 3 0.05 1.50 1.36 0.668 Merger MergerM3 3 3 0.05 4.18 3.79 1.86 Merger Collision+HVSM4 3 3 0.2 4.18 3.79 1.86 Merger MergerM5 3 3 0.2 38.2 34.7 17.0 Merger MergerM6 6 6 0.05 2.67 1.91 0.935 Merger Collision+HVSM7 6 6 0.05 2.67 1.91 0.935 Merger MergerM8 6 6 0.05 4.18 2.99 1.46 Merger MergerM9 6 6 0.05 5.05 3.62 1.77 Merger MergerM10 6 6 0.2 2.04 1.46 0.716 Merger MergerM11 6 6 0.2 20.4 14.6 7.13 Merger MergerM12 6 3 0.1 8.08 5.78 (7.34) 2.83 (3.59) Merger MergerM13 6 1 0.1 3.48 2.49 (5.28) 1.22 (2.59) Merger Merger

H1 3 3 0.05 1.50 1.36 0.668 HVS HVSH2 3 3 0.05 7.07 6.42 3.14 HVS HVSH3 3 3 0.2 2.04 1.86 0.909 HVS HVSH4 6 6 0.05 1.50 1.07 0.526 HVS HVSH5 6 6 0.2 2.04 1.46 0.716 HVS HVSH6 6 6 0.2 0.60 0.429 0.210 HVS HVSH7 6 1 0.1 1.44 1.03 (2.19) 0.503 (1.07) HVS (secondary) HVS (secondary)H8 6 1 0.1 0.795 0.569 (1.21) 0.278 (0.592) HVS (secondary) HVS (secondary)

by the standard SPH cubic spline kernel with a constantsmoothing length h• = 20R⊙. This approach has the ad-vantage that the treatment of gravity is unsoftened forseparations r > 2h•. We note that h• is small comparedto the periapsis separation rper in all cases, so our code isable to follow bound stars around the black hole for manyorbits without introducing spurious secular effects fromgravitational softening. The SMBH is allowed to movein response to gravitational pulls. However, because the4 × 106 M⊙ SMBH is much more massive than any ofthe binaries being considered, the SMBH always remainsvery near the center of mass of the system, which we taketo be the origin.

Before simulating the interaction of a binary with theSMBH, we must first prepare an SPH model for eachbinary component in isolation. To compute stellar struc-ture and composition profiles, we use the TWIN stellarevolution code (Eggleton 1971; Glebbeek & Pols 2008;Glebbeek 2008) from the MUSE software environment(Portegies Zwart et al. 2009). We evolve main-sequencestars with initial helium abundance Y = 0.28 and metal-licity Z = 0.02. The 3 M⊙ star is evolved to an age of50 Myr, yielding the same 2.15R⊙ (0.01 au) radius as inthe corresponding models of Paper I. The 1 and 6M⊙stars are each evolved to 18.2 Myr, yielding stellar radiiof 0.891 and 3.44R⊙ respectively. Figure 2 shows the re-sulting composition profiles for our 1, 3, and 6 M⊙ stars,colored red, green, and blue respectively.

Initially, we place the SPH particles on a hexagonalclose packed lattice, with particles extending out to a dis-tance only a few smoothing lengths less than the full stel-

Fig. 2.— Fractional chemical abundances (by mass) versus en-closed mass fraction m/M for our M = 1M⊙ (red curves), 3M⊙(green curves), and 6M⊙ (blue curves) stars, as calculated by theTWIN stellar evolution code.

lar radius. After the initial particle parameters have beenassigned according to the desired profiles from TWIN, weallow the SPH fluid to evolve into hydrostatic equilib-


Fig. 3.— Radial profiles of the SPH model for a 6M⊙ star.The frames in the left column show profiles of pressure P , densityρ, temperature T (in Kelvin), and mean molecular weight µ inunits of the proton mass mp, with the dashed curve representingresults the TWIN evolution code and dots representing particledata from our SPH model. The right column provides additionalSPH particle data: individual SPH particle mass mi, smoothinglength hi, number of neighbours NN , and radial component of thehydrodynamic acceleration ahydro (upper data) and gravitationalacceleration g (lower data).

rium. During this calculation, we include artificial vis-cosity in the acceleration equation only, with no addi-tional drag force on the particles. This approach allowsus to model the desired parent stars very accurately. Asan example, Figure 3 plots both desired profiles and SPHparticle data for the 6M⊙ star. The structure and com-position profiles of the SPH model closely follow the de-sired TWIN profiles. Our relaxed models remain staticand stable when left to evolve dynamically in isolation.

The binaries in our simulations are then created simplyby shifting two stellar models to the appropriate initialposition and velocity provided by the N-body code. Webegin with binary components irrotational in the inertialframe, which allows us to more easily study any rotationimparted during the subsequent interaction. Unless oth-erwise noted, our simulations employ N ≈ 4 × 104 SPHparticles: such a particle number provides an appropriatebalance between resolution and the need to sometimesfollow the hydrodynamics for many hundreds of orbits.

3.3. Timescale considerations and orbital advancementBecause of shock heating in collisions and mergers, in

addition to tidal heating during the periapse passage, thebound stars are out of thermal equilibrium and largerthan a normal main-sequence star of the same mass.The global thermal readjustment of the bound stars pro-ceeds on a thermal timescale tthermal ≈ U/L, where Uis the total internal energy in the star and L is its lu-minosity. The SPH simulations confirm that the inter-nal energy U of the bound star after one periapsis pas-sage is comparable to the total internal energy of the

star(s) from which the bound star came and typicallyU ≈ (2 − 7) × 1049erg, with the larger values generallycorresponding to more massive stars. For weak collisionsand clean ejections of HVSs, the luminosity of the boundstar will be comparable to the value it had in the ini-tial binary: the thermal timescale in such cases is thenroughly 105 to 107 years. Guided by calculations of bluestragglers (Sills et al. 1997), we estimate that the lumi-nosity of a bound star produced by a merger or strongcollision may be up to ∼ 100 times larger than that of amain-sequence star of the same mass. Thus, the luminos-ity of our most massive merger products could be brieflyas large as ∼ 106L⊙ ≈ 4×1039erg s−1, so that the globalthermal timescale tthermal & 600 years (although the lo-cal thermal timescale in the outer layers of the star couldbe less). We conclude that thermal adjustment over anorbital period is small and often completely negligible,and we therefore do not attempt to model the thermalrelaxation here.

Although the orbital period is small compared to thethermal timescale, it is large compared to the hydrody-namical timescale, which is about an hour. Following thefull hydrodynamics of a multiple orbit encounter wouldtherefore not be practical. What we do instead is wait forthe star(s) to move sufficiently far away from the blackhole and then advance any bound star around most ofits Kepler two body orbit. At the same time, we removefrom the simulation any HVS, any ejecta, and any gasthat has become bound to the SMBH. As long as the pe-riapse passages are treated hydrodynamically, our resultsare not sensitive to precisely which portion of the orbit istreated in the two body approximation. In practice, wewait at least 200 hours after periapse, and at least 200hours after the merger or ionization of a binary, beforemeasuring the orbital elements and implementing the twobody analytic solution. The orbital advancement is per-formed such that the distance from the black hole to thebound star is unchanged but that the objects are nowapproaching one another. We preserve the orientationsof the orbit and spin of the bound star during the orbitaladvancement.

4. RESULTSThe first important result of the SPH simulations per-

formed in this work is that their qualitative outcomeagrees very well with that of the N-body integrations de-vised in Paper I. Among a total of 26 simulations, only in3 cases did the N-body approach fail to match the resultsof the hydrodynamic calculations.

Table 1 reports the chosen initial conditions as well as aqualitative description of the final outcome of the SMBH-binary interaction in both SPH and N -body simulations.The strength of the SMBH-stars interaction is parame-terized by the dimensionless quantity: λ = rper/rt. Ingeneral, stars on orbits meeting the condition λ < 1are tidally disrupted (Luminet & Carter 1986; Evans &Kochanek 1989). However, even when tidal disruptiondoes not occur, we expect that mass will be stripped fromthe outer regions of any star passing within its Rochelimit (Paczynski 1971). In the table, ζ gives the Rochelobe radius (evaluated at periapsis) in units of the stellarradius:

ζ =0.49q2/3

0.6q2/3 + ln(1 + q1/3)× rper

R, (7)


with q = M/M• (Eggleton 1983). Although this formulawas derived under the assumption of circular orbit, it hasbeen shown to work reasonably well even for eccentric bi-naries, if used at periapsis (Regös et al. 2005). Note thatbecause q


TABLE 2

Ejection velocity vej of the HVS, orbital semimajor-axis a and eccentricity e of the captured star, and distance of closest approach between the twostars (r0) in the SPH (N-body) calculations. The quantity ∆Mb/Mb gives the fraction of mass lost from the binary, while ∆M•/Mb is the fractionof the mass lost from the binary that remains bound to the SMBH. All quantities are evaluated after the first periapsis passage, once the stars haveretreated far from the SMBH.

Run vej a e r0/(R1 + R2) ∆Mb/Mb ∆M•/Mb(km/s) (au)

C1 4608(5681) 124(101) 0.988(0.985) 0.23(0.32) −1.33 × 10−2 7.19 × 10−3

C2 3878(3964) 159(157) 0.983(0.983) 0.49(0.50) −2.80 × 10−3 2.06 × 10−3

C3 3335(4840) 190(117) 0.991(0.987) 0.38(0.45) −1.24 × 10−2 6.73 × 10−3

C4 1467(1554) 380(377) 0.995(0.995) 0.67(0.87) −1.29 × 10−3 7.08 × 10−4

C5 1764(2367) 212(163) 0.974(0.965) 0.42(0.61) −2.16 × 10−3 1.53 × 10−3

TABLE 3

Same as Table 2 but for stellar mergers after the first periapsis passage. Here the quantities in parentheses refer to the initial orbit of the binarycenter of mass.

Run a e ∆Mb/Mb ∆M•/Mb J

(au)

M1 1060(1000) 0.995(0.995) −2.57 × 10−2 2.57 × 10−2 0.153M2 984(1000) 0.999(0.999) −1.61 × 10−2 1.26 × 10−2 0.167M3 997(1000) 0.996(0.996) −5.07 × 10−2 2.71 × 10−2 0.103M4 1010(1000) 0.996(0.996) −4.26 × 10−2 4.24 × 10−2 0.259M5 1020(1020) 0.958(0.963) −2.16 × 10−2 1.86 × 10−2 0.222M6 996(1000) 0.997(0.997) −5.60 × 10−2 2.93 × 10−2 0.0666M7 1180(1000) 0.997(0.997) −2.50 × 10−2 2.37 × 10−2 0.242M8 1030(1000) 0.996(0.996) −4.00 × 10−2 2.06 × 10−2 0.249M9 1000(1000) 0.995(0.995) −2.93 × 10−2 2.65 × 10−2 0.238M10 1020(1000) 0.998(0.998) −6.09 × 10−2 5.99 × 10−2 0.108M11 1010(1010) 0.980(0.982) −4.38 × 10−2 2.51 × 10−2 0.131M12 1020(1010) 0.992(0.991) −1.80 × 10−2 1.23 × 10−2 0.181M13 1060(1000) 0.997(0.994) −2.20 × 10−2 1.63 × 10−2 0.0766

TABLE 4

Same as table 2 but for simulations in which the stars do not collide and one member becomes a HVS. Jcaptured gives the spin of the stars thatremain bound to the SMBH, while Jejected refers to the ejected stars.

Run vej a e ∆Mb/Mb ∆M•/Mb Jcaptured(Jejected)

(km/s) (au)

H1 3883(3910) 160(160) 0.991(0.991) −2.230 × 10−4 1.221 × 10−4 0.211 × 10−2(0.188 × 10−2)H2 2026(2160) 309(308) 0.983(0.974) 0 0 0.268 × 10−3(0.238 × 10−3)H3 1082(1100) 423(421) 0.991(0.983) 0 0 0.153 × 10−3(0.729 × 10−4)H4 4221(4252) 142(142) 0.995(0.990) −1.019 × 10−2 5.401 × 10−3 0.194 × 10−1(0.147 × 10−1)H5 2121(2155) 306(307) 0.974(0.993) −8.454 × 10−5 4.759 × 10−5 0.149 × 10−2(0.168 × 10−2)H6 1637(947.6) 174(444) 0.988(0.999) −0.9914 0.9914 0.445 × 10−1(0.379 × 10−1)H7 2570(2574) 677(674) 0.983(0.978) −1.335 × 10−2 6.911 × 10−3 0.223 × 10−1(0.348 × 10−3)H8 2494(2497) 619(619) 0.991(0.990) −0.4835 0.2458 0.845 × 10−1(0.274 × 10−1)

4.2. MergersStellar collisions due to either binary evolution or dy-

namical interactions are thought to be the main forma-tion channel of blue stragglers in star clusters (Collieret al. 1984; Leonard 1989; Mateo et al. 1990). Simi-lar processes have been proposed in the past to explainthe puzzling presence of the young massive stars ob-served at galactocentric distances of few mpcs, wherestar formation is thought to be strongly inhibited by theSMBH tides (Genzel et al. 2003; Eisenahauer et al. 2005).In paper I, we showed that gravitational encounters in-volving stellar binaries and the SMBH lead, for a widerange of orbital parameters, to a stellar collision and thatamong the collisional products, stellar coalescence occursin more than 80% of the cases. In this section we study

this latter outcome and investigate the properties of theresulting stars to clarify whether they would be expectedto posses features commonly associated with the S-starpopulation.

Table 3 gives the orbital parameters (i.e., eccentricityand semimajor-axis) of the merger products in our sim-ulations as well as the mass ejected from the binary andthe fraction of mass captured by the SMBH after thefirst periapsis passage. The table shows that the mergerremnants lie on a orbit very close to the initial orbit ofthe center of mass of the binary around the black hole,implying only a small effect of the mass-loss on the dy-namical evolution of the stars. The mass ejected fromthe binary after the first periapsis passage is typicallylarger than that found in collisions that do not end up


Fig. 4.— Temporal evolution of the dimensionless spin parameterJ defined in equation (9) at times near the first periapsis passage.Dashed lines give J for the stars captured by the SMBH, while thecontinue lines correspond to the ejected stars. Time is scaled inorder to have t = 0 at the moment of the closest approach of thebinary with the SMBH.

with a merger (see Table 2) and is of order ∼ 10−2 timesthe initial mass of the binary. In many cases most of themass ejected from the binary during the merger remainedbound to the SMBH. This is a quite different situation re-spect to that found in Section 4.1, where approximatelyhalf of the mass ejected from the binary remained un-bound to the black hole; in these previous runs one ofthe two stars is always found on an escaping trajectoryand, consequently, the debris associated with such a starwill also tend to escape the SMBH. The last column inthe table gives the dimensionless spin parameter definedin equation (9) of the final merger products that showvery large spins, some of them close to the “break-up”value (i.e., J = 1).

In principle it is possible that, as a consequence of themass loss occurring near periapsis, the resulting mergerproduct gains orbital energy and escapes the SMBH (seefor instance Faber et al. [2005]). Although we do notexclude this outcome for a different set of initial condi-tions, in our simulations this mechanism does not pro-duce HVSs, and in all cases the merger remnant is still ona bound orbit around the SMBH. Another more impor-tant consideration is that, subsequent to the merger, thetidal heating results in some degree of expansion and aweakly bound configuration for the merger product. Be-cause the tidal radius of the newly formed star is muchlarger than that of its progenitors, the star will succes-sively lose more mass with each periapsis passage andeventually be torn apart by the SMBH tides. And infact, as it is shown below, this is the final outcome ofsome of our simulations.

An example of merger is displayed in Figure 7, whichinvolves a binary with a0 = 0.2 au and equal-mass com-ponents of masses M = 3M⊙ (run M4). In this run the

stars collide for the first time after ∼ 4 days from the timecorresponding to the periapsis passage; the internal pe-riapsis separation at the first contact is r0/(2R) = 0.86.After the first periapsis passage, as consequence of theSMBH perturbation, the binary star becomes very ec-centric. As the stars move through the circum-binaryenvelope formed during the previous encounters, the or-bit gradually circularizes and shrinks. By t ≈ 60 days,after approximately 30 collisions, the two stellar nucleimerge. The final product has an oblate shape which is acommon characteristic of all the merger remnants formedin our simulations.

As another example, Figure 8 displays column densityplots of simulation M13 where a merger occurs betweena 6 and a 1M⊙ stars. The stars collide after 1.48 daysfrom the moment of the closest approach to the SMBH,and subsequently merge in the following ∼ 1 day. Duringthe merger, the high density core of the lower mass starrapidly sinks to the center of the companion star. Thetail-like feature observed at t = 1.86 days in the figure, ismostly material coming from the the secondary star thatloses part of its outermost envelope while sinking to thecenter of the merger remnant.

Figure 9 shows chemical composition profiles of themerger products for runs M4 and M13, after one periap-sis passage. In simulation M4, the remnant has a massof roughly 5.9 M⊙, its composition profile is very similarto that of the parent stars (see Figure 2). Based on howlong it would take a normal star of that mass to evolve tothat central hydrogen abundance using the TWIN stellarevolution code, we estimate that a “normal” 5.9M⊙ starwould reach a core helium abundance Y = 0.35 (assum-ing Z = 0.02) after ∼ 2Myr. By colliding two 50 Myrold 3M⊙ stars, we have effectively made a more massive(M ∼ 6M⊙), younger (age ∼ 2 Myr) star. The mergerremnant of run M13 shows a peculiar composition pro-file when compared to a normal star, but quite normalfor merger products. In M13 (and in M12 as well), thelow mass star drops to the center of the merger productbringing its fresh hydrogen fuel along and significantly re-juvenating the core. As a result, the maximum He doesnot occur at the center of the merger product. The mainreason of the negligible amount of hydrodynamic mixingis that the cores of the initial stars are very dense anddifficult to break even in a head-on collision (Lombardi etal. 1995, 1996). However, we cannot exclude that otherprocesses occurring on a thermal timescale (as opposedto the hydrodynamic timescale) can produce a signifi-cant degree of mixing in the stars as they evolve towardthermal equilibrium (Sills et al. 1997).

Another interesting result, plotted in Figure 9, is that alarge fraction of the Lithium/Beryllium/Boron from theparent stars gets ejected, indicating that a significant gapin the abundances of these elements in the S-star pop-ulation might be observational evidence of rejuvenationthrough merger. Reduced atmospheric Lithium abun-dances are for instance observed in blue stragglers andcan also be a strong indicator of mixing (Hobbs & Math-ieu 1991; Pritchet & Glaspey 1991).

We finally note that as the stars keep orbiting theSMBH, their chemical profile and their spinning con-figuration will change in time and therefore the statesdisplayed in Figures 7, 8 and 9 should be intended asnot permanent. The evolution will typically lead toward


Fig. 5.— Column density plots for simulation C5 on the X −Z plane. In this case the binary has an internal semimajor-axis a0 = 0.1auand its components have masses M1 = 6M⊙ and M2 = 3 M⊙. Time t = 0 corresponds to the periapsis passage of the binary externalorbit. Between t = 0 and t = 0.9 days the panels are centered on the binary center of mass. The two bottom-right panels are centered onthe center of mass of either the captured (left) or ejected (right) star. The black hole is outside the images. At t = 0.13 days the starscollide. Subsequently, the 6M⊙ member is ejected at hypervelocity while the secondary star remains bound to the SMBH.

Fig. 6.— Stars ejected in our simulations after a collision with the companion star. The stars show, typically, an oblate envelopesurrounding a high density spherical nucleus. Only in simulation C4, where the impact is more “grazing,” the collisional product isspherically symmetric even in its outermost envelope.


smaller spins and a lower Lithium/Beryllium/Boronabundances in the stars. We will come back to this pointbelow.

4.3. Clean Ejection of Hypervelocity StarsEven when the stars do not collide, tidal torque and

mass-loss can occur if the periapsis distance of the binarycenter of mass initially lies within the Roche limit of itsmember stars. Similarly to what is done in the previoustwo subsections, we analyze here the first binary-SMBHinteraction, while we discuss the following evolution ofthe bound stars in the next subsection. Some of the re-sults of our SPH simulations, for which there is not adirect collision between the stars, are listed in Table 3.As expected, for ζ & 1 there is no mass-loss, and thestars maintain their initial configuration essentially un-altered (runs H2 and H3). Conspicuous mass-loss insteadoccurs for runs H6 and H8 in which at least one of thestars crosses its tidal radius. Interesting, although theusual condition for tidal disruption is well satisfied (i.e.,λ < 1), in both runs the stars are not fully disrupted bythe SMBH’s tidal gravity at the first periapsis passage.In the table we also give the final values of the spin pa-rameter J that, in general, are found to be a very smallfraction of the breakup value. Figure 10 shows the tem-poral evolution of J for the cases of Table 3 that havethe highest value of the final spin. The interaction withthe SMBH induces a strong rotation only in the starsof runs H6 and H8. We conclude that, unless the starspenetrate deeply their tidal disruption radius, it seemsunlikely that the SMBH tides at periapsis alone can pro-duce a significant spin-up of the stars.

As an example, Figure 11 gives column density plots ofrun H8. In this simulation, the primary and secondarystars have masses 1 M⊙ and 6 M⊙ respectively. The pe-riapsis of the external orbit (∼ 0.8au) is initially insidethe tidal radius of the 6M⊙ member but it is still outsidethe tidal radius of the 1M⊙ star. At periapsis, the starsare squeezed by the SMBH’s tidal gravity. In the processthe primary star losses a large fraction of its initial mass(∼ 3.4M⊙), while the secondary loses only ∼ 0.03M⊙.After the interaction with the SMBH the binary is bro-ken apart and the lightest member becomes an HVS. Be-cause of tidal heating during the periapsis passage, thestars are perturbed from their thermal equilibrium stateand their radii are somewhat enlarged with respect to anormal main-sequence star of the same mass.

4.4. The Bound PopulationAfter the initial encounter between the binary and the

SMBH, one star remains in a bound orbit around theSMBH in all of our simulations. Like the orbit of the ini-tial binary about the SMBH, the orbit of such a boundstar is highly eccentric: 0.96 < e < 1. In those casesin which the bound star is a merger product (runs M1through M13), the semimajor-axis a very nearly equalsthe semimajor-axis of the initial binary about the SMBH:a ≈ 1000au, corresponding to an orbital period of about16 years. We note that these orbital periods are compa-rable to those of the S-Stars in the GC. In those casesin which a HVS star is ejected (runs C1 through C5 andruns H1 through H8), the ejection energy comes at theexpense of the orbital energy of the bound star, whichconsequently has a somewhat smaller semimajor-axis:

100au . a . 700au, corresponding to orbital periodsof 0.5 to 9 years.

4.4.1. Tidal StrippingWe find that the periapsis separation of a bound star

remains remarkably constant from one orbit to the next,even when there is significant mass loss due to Rochelobe overflow at periapse. As an example, consider therun C1 in which the initial stars have the dimensionlessRoche lobe parameter ζ = 0.67. As expected for ζ < 1,the bound star does indeed lose mass each time it sweepspast the black hole. The gradual decrease of the massM of the bound star can be seen in the top panels ofFigure 12. The mass δM lost per orbit, shown by thestar symbols in the top panel, increases with each or-bit, until after 96 periapsis passages the star has beencompletely disrupted. We note from the bottom panelthat the periapsis separation rper = a(1 − e) is nearlyunchanged during this entire process: in this and othercases, we find the bound star returns to the nearly samerelative separation from the black hole regardless of massloss. The apoapsis separation d = a(1 + e) is also some-what constant, although it decreases at late times whenthe mass loss is greatest and strong tidal effects removeenergy from the orbit.

As another example, the lower panels of Figure 12 givethe evolution of the merger remnant formed in run M5.In this case most of the mass-loss occurs during the firstperiapsis passages where very high entropy material is re-moved from the outer layers of the star that responds byreducing its radius. In the following evolution, mass-lossessentially ceases. We note that merger products have avery non-uniform density profile characterized by a ex-tended low density envelope and a dense central region.Subsequent passages of the star by the SMBH will there-fore cause the depletion of the outermost stellar region,unveiling its hot central core. An example of this phe-nomenon is shown in Figure 13, where we plot columndensity plots for the merger remnant of run M7. Afterabout twenty orbits mass loss stops and the envelope hasbeen completely removed.

We find that the dimensionless parameter ζ is stronglycorrelated with whether and how quickly a bound starloses mass through Roche lobe overflow. For example inrun H2, the bound star has ζ > 1 and does not experiencea collision or merger that would change ζ: it consequentlycontinues to orbit the SMBH without ever suffering anymass loss. In all the cases with ζ < 1, the bound staris ultimately destroyed after repeated episodes of Rochelobe overflow, with smaller values of ζ generally corre-sponding to fewer orbits before disruption. Figure 14shows that the number Np of periapsis passages beforedisruption grows exponentially with the initial ζ. In ad-dition, this number of passages depends only weakly onwhether the interaction type is a collision (red crosses),merger (black triangles), or clean ejection of a HVS (bluecircles).

The rightmost data point in Figure 14, correspondingto run C2, deserves some discussion. In this case, ζ1 =ζ2 = 1.19 > 1, so that neither binary component wouldlose mass if it were not for the collision induced on thefirst periapsis passage. This collision both increases theradius and slightly decreases the mass of the bound star,effectively decreasing its ζ parameter to a value below 1.


Fig. 7.— Column density plots for simulation M4 on the X − Z plane. The binary has an internal semimajor-axis a0 = 0.2au and itscomponents have masses M = 3M⊙. Time t = 0 corresponds to the periapsis passage of the binary external orbit.

Fig. 8.— Column density plots for simulation M13 on the X − Y plane. The binary has an internal semimajor-axis a0 = 0.1au and itscomponents have masses M1 = 6 M⊙ and M2 = 1M⊙. Time t = 0 corresponds to the periapsis passage of the binary external orbit.


Fig. 9.— Upper panels: composition profile of the merger rem-nant formed after one periapsis passage in run M4 (see also Figure7). The merger product of two equal mass stars has a compositionprofile very similar to that of its parent stars. For a ∼ 6 M⊙ star,a final He abundance in the core of Y = 0.35 corresponds to aneffective age of ∼ 2 Myr.Lower panels: composition profile of the merger remnant formedafter one periapsis passage in run M13 (see also Figure 8). In thiscase, the merger product has a peculiar profile if compared to a“normal” star. Its core is strongly hydrogen-enriched as a conse-quence of the low-He fluid transported by the low mass star alongwith it to the center.

Thus, on the subsequent passage past the black hole, thestar loses more mass, now due to Roche lobe overflow.The response of this particular star to mass loss is that itsradius remains roughly constant. From equation (7), theζ parameter then stays below 1 and slowly decreases asthe mass ratio q decreases with each successive passage.Ultimately, after nearly 600 periapsis passages, the staris completely pulled apart.

Fig. 10.— Temporal evolution of the dimensionless spin param-eter J for some of the simulations in which there is not a stellarcollision, at times near the first periapsis passage. Dashed linesgive J for the stars captured by the SMBH, while the solid curvescorrespond to the ejected stars. Time is scaled in order to havet = 0 at the moment of the closest approach of the binary with theSMBH.

Similarly to the stars from run C2, the 6M⊙ star in runM13 has ζ = 1.22 > 1. This star indeed makes the firstperiapsis passage without immediately losing any mass;however, while the binary recedes away from the SMBH,the merger causes mass ejection. The resulting 6.84M⊙merger product is large enough that ζ drops below 1,and, on the subsequent periapsis passages, mass is lostthrough Roche lobe overflow. As a result of sheddingits high entropy outer layers, the merger product shrinkssufficiently that ζ is pushed back toward ζ ≈ 1. Bycomparing runs C2 and M13, we conclude that the fateof binary stars with ζ & 1 depends not simply on theinitial values of ζ but also on the type of their interactionand the response of the bound star to mass loss.

In several of our simulations, merger products formedfrom stars with ζ > 1 are large enough that ζ drops be-low 1 and at least some mass is lost due to Roche lobeoverflow on the second and later periapsis passages (runsM1, M3, M4, M5, M7, M8, M9, M11, M12, and M13).Because shock heating is preferentially distributed to theouter layers of a merger product (Lombardi et al. 2002),this Roche lobe overflow always strips away very highentropy material and the product responds by decreas-ing its radius. In this way, the ζ parameter graduallyapproaches a value ≈ 1, corresponding to an eccentricsemidetached binary consisting of the bound star andthe SMBH. In our SPH simulations of such cases, wetypically follow the dynamics for several hundred orbits,without seeing an appreciable decrease in the mass of thebound star: indeed at late times the mass loss typicallyfluctuates between 0 and 2 SPH particles per periapsispassage. Such situations necessarily challenge the massresolution limit of our simulations, and it is difficult tosay whether such small levels of mass loss are physically


Fig. 11.— Column density plots for simulation H8 on the X − Z plane. The binary has an internal semimajor-axis a0 = 0.1 au and itscomponents have masses M1 = 6 M⊙ and M2 = 1M⊙. Time t = 0 corresponds to the periapsis passage of the binary external orbit. Thefirst four panels are centered to center of mass of the binary, while in the two bottom-right panels the origin is the center of mass of eitherthe captured (left) or ejected star (right).

meaningful or simply a numerical artifact. In any case,in nature, thermal relaxation in the outermost layers ofsuch a merger product would tend to retract it inside ofits Roche lobe and stabilize the star against further massloss.

4.4.2. Internal Structure

Figure 15 shows the composition profiles as a func-tion of enclosed mass fraction for the bound star in casesM4 and M13. The dotted curves show the profiles aftertwo periapse passages, while the solid curves show thesame profiles once the bound star has effectively reacheda steady state. Mass loss experienced during multiplepassages removes the outer layers of the star, so that thecomposition profiles shift slightly to larger enclosed massfractions m/M . We note that the helium profile for M13is qualitatively similar to that of the case G merger prod-uct in Sills et al. (1997) (see Fig. 2 in that paper): bothhave a maximum helium abundance at an intermediateradius inside the star. In both case G and our M13, thestrange Y profile is caused by a low mass star sinkingto the center of the collision product and displacing thelarge Y fluid outward. The stellar track for the case Gproduct is shown in Figures 4 and 6 of Sills et al. (1997).In Figure 6, we see that, on the main sequence, the caseG product is somewhat bluer and brighter than a nor-mal main sequence star of the same mass. In Figure4, we see that the case G product is a little bluer andbrighter than a different collision product (case J) withbasically the same mass but without the dense hydrogencore. It is the increased helium content in the stellar in-terior that makes the opacity lower (compared to othermain sequence stars of the same mass) and thus bluer

and brighter. So, by analogy, our M13 merger productwould be a little bluer and brighter than a normal main-sequence star of the same mass.

The elements Li, Be, and B are potentially interestingobservational indicators of the history of a star. These el-ements burn at temperatures of about 2.5×106, 3.5×106,and 5× 106 K, respectively, and therefore can exist onlyin thin outer layers of the parent stars. During a dynam-ical interaction, these elements can be removed either byejecting the outer layers of the star or by redistributingthem to an environment too hot for their long term ex-istence. Although Be, B, and Li can exist after the firstperiapsis passage (see Fig. 9), the effect of multiple pas-sages is typically to remove these elements completely.Although there are cases where B still exists in the finalbound star, it is always severely depleted. For example,in run M13 the B level at the surface of the final productis only ∼3% of the surface value in the 6 M⊙ parent starfrom which it originated.

In Table 5, we summarize some properties of the boundstars that survive in our SPH simulations (i.e., that arenot ultimately disrupted by the SMBH). The “numberof passages” represents the number of periapse passagesbefore the mass loss effectively shuts off, which we de-fined as having 2 or fewer SPH particles ejected. Thecentral hydrogen abundance is given by Xc, which al-ways equals the central hydrogen abundance of the low-est mass parent star. We also list the effective age ofthe bound star based on its mass and central hydrogenabundance. We evaluate this effective age, based on howlong it would take a normal star of that mass to evolve tothat central hydrogen abundance using the TWIN stel-lar evolution code. It is known that the contraction of


Fig. 12.— Evolution versus time (and number of orbits) in theruns C1 and M5. From the top to the bottom panel: mass δM•gained per orbit by the SMBH (filled circles) and mass δM lost perorbit by the stars (star symbols), cumulative mass ∆M• bound tothe SMBH, mass M of the bound star, apoapsis d of the boundstar, periapsis rper of the bound star.

a merger product to the main-sequence is very similarto the contraction of a pre-main-sequence star to themain-sequence. In the latter case, the most importantvariable is the mass (for a given hydrogen abundance).In the former case, the two important variables are es-sentially mass and central hydrogen abundance (Sills &Lombardi 1997). In runs C5 and H2 the bound star isonly a slightly perturbed version of one of the binarycomponents, and the ages in these cases is the same 50Myr as that component. For mergers of two 3 M⊙ stars,the effective age of the merger product is in the range of14 to 22 Myr. For mergers of two 6 M⊙ stars, the effec-

tive age is in the range of 6 to 9 Myr. Mergers of unequalmass stars (M12 and M13) also significantly rejuvenatea star: for example in M13, the sinking of the 1 solarmass star to the center of the merger product essentiallyresets the nuclear clock to only 0.3Myr after the ZAMS.

In the last three columns of Table 5 we list centraltemperature Tc, internal energy U and thermal timescaletthermal of surviving bound stars. The central tempera-ture is defined as the temperature in the star where thedensity is highest and therefore is not always the tem-perature of the highest temperature SPH particle. Thecentral temperature Tc of all the stars is large enough tosustain nuclear burning in the core. The global thermaltime scale can be estimated as

tthermal = U/⟨L⟩, (10)

where ⟨L⟩ is a mass weighted average of the luminosityL throughout the entire star. To calculate L, we takeadvantage of the fact that the parent stars are massiveenough to be fully radiative. In addition, shock heatingprevents any convective zones from existing in a newlyformed merger product. Thus, we obtain the luminosityL from the equation of radiative transport,

L =163

π a c r2T 3∇T/(κρ) , (11)

where c the speed of light, and the opacity κ is calculatedfrom the OPAL opacity tables.

As example, in Figure 16, we show a more detailedlook at the interior of the merger products of runs M4and M5. From top to bottom, we give the luminosityL, temperature T , and radius r as a function of enclosedmass m. Our merger products typically achieve a surfaceluminosity comparable to the Eddington luminosity

Ledd =4πGc

κM ∼ 3.8 × 104L⊙

M

M⊙, (12)

although such a high luminosity would diminish rapidlyas the star contracts to the main-sequence in a timetthermal usually of order ∼ 0.1Myr.

5. DISCUSSIONThe observed rotation rates of HVSs may give impor-

tant clues to their formational history. Hansen (2007) hasproposed that, as a consequence of tidal locking in closebinaries, HVSs ejected by the Hills mechanism shouldrotate systematically slower than field stars of similarspectral type. López-Morales & Bonanos (2008) foundthat the late B-type star HVS 8 has a rotational velocityof ∼ 260 km s−1, more typical of single B-type stars andtherefore seemingly contrary to the hypothesis of a bi-nary origin for this star. In order to explain the observa-tions, other ejection mechanisms have been invoked, suchas ejection by a close encounter with a massive black holebinary or with a stellar black hole orbiting the galacticcenter SMBH (Yu & Tremaine 2003; Levin 2006; Sesanaet al. 2006; Löckmann, & Baumgardt 2008). However, inthis paper we have shown that there are two other poten-tially important ways with which the stars can somewhatincrease their rotation even in the binary disruption sce-nario: tidal torque by the SMBH at periapsis (if the starsenter within their tidal disruption radius) and/or a col-lision between the two binary members.


Fig. 13.— Evolution of the merger remnant formed in run M7 after each periapsis passage when the star sets to hydrostatic equilibriumand until mass loss ceases. Time increases from left to right and form top to bottom. Initially the merger remnant has a large low-densityenvelope that is completely removed after several obits around the SMBH.

TABLE 5

Some properties of the bound stars that survive in our SPH simulations. “Number of Passages” gives the number of SMBH-star encounters beforemass-loss ceases. Mass, orbital eccentricity and semimajor-axis of the star are given by M , e and a respectively. The value of J is the finaldimensionless spin parameter, while Xc is the central hydrogen abundance. The corresponding effective age is also listed.In the last three columns we give central temperature Tc ( the temperature where the density is highest.), internal energy U and thermal timescaletthermal.

Run Number of Passages M e a J Xc Effective Age Tc U tthermal( M⊙) (au) (Myr) ×106 (K) ×1050 (erg) (Myr)

C5 8 2.998 0.974 212 0.009 0.63 50 22 0.141 0.4

H2 2 3.000 0.978 316 0.000 0.63 50 23 0.148 1

M1 30 4.83 0.995 1010 0.009 0.63 16 31 0.326 0.1

M3 30 4.94 0.996 996 0.010 0.63 16 31 0.321 0.2

M4 18 4.27 0.996 1000 0.009 0.63 22 32 0.292 0.2

M5 22 5.15 0.962 1020 0.014 0.63 14 32 0.309 0.05

M7 25 8.39 0.997 973 0.016 0.56 9 39 0.710 0.1

M8 90 10.7 0.996 1030 0.014 0.56 6 39 0.944 0.07

M9 27 8.74 0.995 1010 0.010 0.56 9 39 0.736 0.06

M11 14 10.1 0.980 1010 0.005 0.56 7 36 0.793 0.03

M12 27 7.27 0.992 1020 0.009 0.63 7 32 0.269 0.03

M13 79 6.32 0.997 1060 0.017 0.70 0.3 18 0.391 0.1

With the help of simplifying approximations, we areable to relate the rotational parameter J calculated inour simulations to the observable rotational velocity vafter that star has thermally relaxed back to the mainsequence. In particular, we approximate that the starrotates rigidly, that its rotation does not drastically af-fect its structure, and that the rotational parameter J isconserved during relaxation. Using L = Iv/R = c1MRv

and E = −c2GM2/R in equation (9), we obtain J =c1c

1/22 v(R/(GM))

1/2. Clearly c1 and c2 are simply nu-merical coefficients related to the moment of inertia Iand total energy E of a star, respectively. For B-typemain sequence stars, we find c1c

1/22 ∼ 0.04 to 0.05 and

R/M ∼0.5 to 0.8 R⊙/M⊙ using models from the TWINstellar evolution code. Solving for v in terms of J , we


Fig. 14.— The number of periapsis passages past the black holeneeded to disrupt the bound star versus the initial dimensionlessRoche lobe parameter ζ1. The different data points represent thescenarios in which the bound star suffers a collision (red crosses:runs C1, C2, C3, and C4), is formed in a merger (black triangles:runs M2, M6, and M10), or is cleanly separated from the HVS(blue circles: runs H1, H3, H4, H5, H6, H7, and H8).

findv ∼ 1.2 × 104J km s−1, (13)

accurate to within ∼ 30% for most B-type main sequencestars. Given the J values of ejected stars in our simula-tions (see Fig. 4 and Table 4), we estimate from equation(13) that the post-relaxation rotational velocity v canbe as large as ∼400 or 500 km s−1 for HVS stars (con-sider runs C3 and H6). We therefore conclude that therotation of the star HVS 8, for example, is completelyconsistent with a binary origin.

The J values for unmerged stars in our simulations thatare bound to the SMBH after the first periapsis passageindicate, via equation (13), that the post-relaxation ro-tational velocity v would be typically . 400km s−1 butcould be as large as ∼ 1000km s−1. These large rotationvelocities, however, correspond to stars that penetratedeeply within their tidal disruption radius (e.g. runs H6and H8) and therefore are eventually destroyed after sev-eral orbits. Merger products obtain even larger spins af-ter the first periapsis passage (see Table 3). However,as the merger product keeps orbiting the SMBH, J de-creases as mass gets pulled off the outside of the star,where the specific angular momentum is greatest. Thebound stars that survive to orbit the SMBH end our sim-ulations with J . 0.017 (see Table 5), corresponding topost-relaxation rotational velocities v . 200km s−1. Asour simulations began with irrotational stars, the actualfinal rotational velocity of a bound star could be largerif the parent stars had significant spin.

Deep near-IR observations of the GC show that the S-stars are B0-B9 main-sequence stars with rotational ve-locities similar to those of field stars of the same spectraltype (Alexander 2005). Our final bound stars therefore

Fig. 15.— Upper panels: composition profile of the mergerproduct in run M4 after two (dotted curves) and eighteen (solidcurves) periapsis passages, corresponding to masses M = 4.75M⊙and M = 4.27M⊙, respectively. Lower panels: composition profileof the merger product in run M13 after two (dotted curve) and76 (solid curves) periapsis passages, corresponding to masses M =6.46M⊙ and M = 6.32M⊙, respectively.

have properties very similar to those of the S-stars: theirmasses qualify them as spectral type B main-sequencestars, and their post-relaxation rotational speeds are ofthe correct general magnitude. For example, the rota-tional speeds of our fastest rotators are consistent withthe 220±40 km s−1 value for the S-star SO-2 (Ghez et al.2003). However, we note that the orbital eccentricitiesof our bound stars (0.96 < e < 1) are larger than thatof SO-2 (e ≈ 0.87), as similarly found in simulations byGinsburg & Loeb (2006) and Hansen (2007).

For tidal torque from the black hole to have a signifi-cant effect on stellar rotation, the stars should enter deep


Fig. 16.— From top to bottom: luminosity L, temperature T ,and radius r as a function of enclosed mass m for the final mergerproducts of runs M4 (upper panel) and M5 (lower panel). L, r andm are in solar units, T is in Kelvin.

into their disruption zone (i.e., rper < rt).When no collision occurs between the components of

a binary, the distance of closest approach of the twostars to the SMBH typically changes little due to theencounter. In such cases, a necessary condition for sig-nificant spin-up is that the binary itself be on an orbitthat passes within ∼ rt of the SMBH. This implies inturn that the fractional change in orbital angular mo-mentum with respect to the SMBH, per orbit, be of or-der unity. This condition is satisfied in the so-called “fullloss cone” regime, which, in a galaxy like the Milky Way,extends inward to ∼ 0.2 times the SMBH influence ra-dius, or to r ≈ 0.5pc (e.g. Wang & Merritt 2004). 1

1 Assuming a ρ ∼ r−2 density cusp.

Inside this region, which is the region of interest for thecurrent study, evolution onto loss-cone orbits is diffu-sive, and most binaries would be tidally disrupted beforefinding themselves on orbits that intersect ∼ rt. Wenote however that in the “massive-perturber scenario”the apoapsis distance of the binary is of the order of aparsec (i.e., > 0.5pc) and therefore the fractional changein orbital angular momentum with respect to the SMBH,per orbit, can be of sufficient to put the binaries on atrajectory that passes within rt of the SMBH. Even in-side 0.5pc, other dynamical processes like resonant re-laxation (Rauch & Tremaine 1996), scattering from aintermediate mass black hole (Merritt et al. 2009) or per-haps eccentric instability in a disc (Madigan et al. 2008)can produce larger changes in orbital angular momentumthan in the case of two-body relaxation alone. On theother hand tidal spin-up is not expected to be very effi-cient because it is important only for the narrow rangeof periapses: 12rt . r . rt (for r .

12rt the star is fully

disrupted; for r > rt tidal torque is small). As a con-sequence of the previous condition, the ejected star willlose a large fraction of its mass. An observational indi-cator of the history of the star would be, even in thiscase, a deficit in the abundances of light elements (suchas Lithium) that can exist only in thin outer layers of theparent star and that are typically removed by the tidalinteraction with the SMBH.

One of the main arguments against rejuvenation ofthe S-stars through merger is the apparent “normality”of their spectra (Figer 2008). We note, however, thatthe envelope of our merger products will not look sig-nificantly different than that of normal stars (comparethe right edge of the plots in Figure 15 with the rightedge of Figure 2). In fact, if the parent stars are of equalmass, the merger product will have very normal profilesthroughout the star. If the parent stars are of signifi-cantly different mass, then the profiles are more peculiarand one should worry about how this affects the stellarevolution. The main effect would probably be to changethe opacity and therefore shift slightly the color and lu-minosity. But, unless significant mixing is induced, thechemical composition of the outermost envelope will re-main similar to that of the higher mass star (with thepossible exception of Li, Be, and B levels: see §4.2).

The tidal disruption of a star passing close enough bya SMBH to enter its tidal radius, produces a luminousUV/X-ray flare of radiation as the bound stellar gas fallsback onto the black hole and is accreted (Rees 1988).Tidal flares are of great interest because they can probethe presence of SMBHs in galaxies with otherwise no ev-idence of an active nucleus and can be used to measurethe mass and spin of the central black hole (Komossa etal. 2004; Gezari et al. 2008, 2009). Computations of thetidal disruption of stars have been performed by severalstudies in the past, with the aim of understanding the ob-servational signatures of these events (Ulmer 1999; Bog-danović 2004; Lodato et al. 2009; Strabbe & Quataert2009; Kasen & Ramirez-Ruiz 2010). We note that thereare many important scenarios in our paper that havebeen so far almost completely ignored: (i) multiple pas-sages by the same star; (ii) merger of two stars resultingin disruption due to the increased size; (iii) partial tidaldisruptions.


Predicting the radiative effects of multiple passages ofa star by a SMBH is outside the scope of the presentwork. But, it seems likely that the light curve result-ing from these repeated tidal events might show a se-ries of small peaks, separated roughly by the orbital pe-riod, before finally producing the large peak that is ob-served as the “tidal disruption.” Assuming the star on aparabolic trajectory, after the star-SMBH encounter, themost bound material moves on a orbit with semi-majoraxis a = 12r

2per/R and returns to periapsis after a time

t0 =2πr3per

(GM•)1/2 (2R)3/2

= 0.22

×(

rperrt

)3 (R

R⊙

)3/2 (M

M⊙

)−1 (M•

4 × 106 M⊙

)1/2yr ,(14)

that it is also the time when the flare starts, while thepeak return rate occurs at t ∼ 1.5t0 (Evans & Kochanek1989; Li et al. 2002).

As previously discussed, merger products are verylarge, so it is easy to strip off lots of mass during the earlyperiapsis passages, while at later time the mass loss of-ten ceases. The light curve resulting from these repeatedtidal events will eventually show a series of small peaksof declining intensity (see Figure 12). The result of therepeated (partial-)tidal disruption of a star with a largeenvelope (e.g.., late-type giants) will show a similar lightcurve. In collisions without mergers, there is some ex-pansion in size but it is not as dramatic as in a merger.So it is not until late times that there is significant massoverflowing the Roche lobe. The light curve will showpeaks of increasing intensity until the last brightest flareproduced by the full tidal disruption of the star.

In future work, one could model the material that be-comes bound to the black hole more carefully. In par-ticular, it would be interesting to identify possible sig-natures of interactions between the bound star and theaccretion torus left behind from previous periapse pas-sages. Quasi-periodic emission may be detected if X-rayflares arise every time the star crosses the torus plane(Dai et al. 2010).

6. SUMMARYIn this paper, we carried out hydrodynamic simulations

of binary stars in orbit about the supermassive black hole(SMBH) at the Galactic center. In the N -body simula-tions of Paper I, we assigned physical sizes to stars basedon a simple mass-radius relation and predicted which bi-naries would merge, i.e., undergo a collision with relativevelocity less than escape velocity. The fluid simulationspresented in this paper were found to be quite consistentwith the N -body simulations, in the sense that when thelatter predicted a stellar merger, the fluid stars typicallymerged as well. The merger rates presented in that paperare therefore confirmed by the present work. Additionalresults from the fluid simulations presented in this paperare summarized below.

1 The central temperature of the merged stars in allour simulations is large enough that there wouldstill be nuclear burning in the stellar core. Mergerremnants have thermal time scales . 0.1Myr andsurface luminosities that are predicted to be a large

fraction of the Eddington luminosity (Table 5 andFigure 16).

2 Assuming an age of 50Myr for our 3 M⊙ stars and18.2Myr for our 6 M⊙ stars, we found that mergerproducts of two 3 M⊙ stars have effective ages inthe range 14−22 Myr, i.e., this is how long it wouldtake a normal star of the same total mass to evolveto have the same central hydrogen abundance. Formergers of two 6 M⊙ stars, the effective age is in therange 6−9 Myr. Mergers of unequal mass stars alsosignificantly “rejuvenate” a star.

3 Mass loss due to a collision or merger is alwaysless than a few percent of the initial mass of thebinary. After a collision without merger, the massassociated with the captured star remains boundto the SMBH, while the unbound debris originatesmostly from the ejected star. In mergers, becausethere is no escaping star, most of the ejecta remainbound to the SMBH.

4 Much greater fractional mass loss can occur whenone or both stars is tidally perturbed by the SMBH.Even if the initial periapsis of the binary lies be-yond the tidal disruption radius of a single star,tidal stripping can still occur, if one star is scat-tered onto a tighter orbit around the SMBH or ifthe two stars merge, forming a temporarily moreextended object.

5 We performed the first simulations of repeated tidalencounters of a star with a SMBH. In all caseswhere the star or merger product has a distanceof closest approach to the SMBH smaller than theRoche limit (ζ < 1; eq. 6), disruption occurs afterrepeated periapse passages, with smaller values ofζ corresponding to fewer orbits before disruption.Figure 14 shows that the number Np of periapsepassages before complete disruption grows expo-nentially with ζ. Moreover, the critical number ofpassages depends weakly on whether the interac-tion type is a collision, merger, or clean ejection ofa HVS.

6 Tidal spin-up of a star that does not suffer a colli-sion is significant only in a narrow range of periapsedistances, rt/2 . rper . rt, with rt the tidal dis-ruption radius of a single star with respect to theSMBH. For r . rt/2, the stars are fully disruptedat the first encounter with the SMBH, while theytypically survive for rper ≈ rt. If the high rota-tional velocity of, for instance, HVS 8 is the resultof such an encounter, the star would have neededto lose a large fraction of its mass before ejectionfrom the Galactic center. When the stars collideor merge, more efficient spin-up can occur, result-ing in an increased rotational velocity for both theejected and captured stars.

7 In stellar mergers, elements that can exist only inthe outermost envelope of the stars such as Li,Be and B are severely depleted, partially due toenvelope ejection during the merger but mostlydue to tidal truncation by the SMBH in subse-quent periapse passages. However, the envelopes


of the merger products do not otherwise differ sig-nificantly from those of the parent stars. In thecase of equal mass stars, composition profiles fol-lowing a merger are essentially “normal”, while forunequal mass binaries the lower mass star sinks tothe center of the merger remnant effectively reset-ting the nuclear clock of the star to the ZAMS.

8 If a bound star has a large low density envelope(as in our merger products), conspicuous mass lossof decreasing amplitude characterizes the first fewperiapse passages, eventually becoming negligible.The envelope is thereby removed but the densernucleus survives. More compact stars that enterthe tidal disruption sphere lose some fraction oftheir mass during each periapse passage, with fulldisruption occurring only after many orbits. Re-peated tidal flares, of either increasing or decreas-ing intensity and separated roughly by the orbitalperiod of the star, are the predicted observationalsignature of such events.

We thank E. Gaburov for his development of the GPUlibrary used to calculate gravitational forces and ener-gies in this paper, and S. Mikkola who wrote the AR-CHAIN code. We also thank H. Perets, W. R. Brown,J. Faber, A. Gualandris, N. Ivanova, S. J. Kenyon, S. Ko-mossa, S. Portegies Zwart, and A. Robinson for helpfuldiscussions. Column density plots in this paper wereproduced using the publicly available visualisation toolSPLASH (Price 2007). This work was supported bygrants NNX10AF84G and NNX07AH15G from NASAand by grants AST-0807910 and AST-0821141 from theNSF.

APPENDIX

UNEQUAL MASS BINARIES: N-BODY SIMULATIONS

TABLE A1HVSs (vej > 1000 km s

−1), Collision and Merger frequency(%)

M2(M⊙) d(pc) HVSs (total) HVSs (primary) HVSs (secondary) Collisions Mergers1 0.01 33. 6 1.29 32.3 7.74 7.103 0.01 52.9 25.5 27.4 9.35 8.711 0.1 62.5 31.3 31.2 6.77 6.453 0.1 59.3 29.3 30.0 11.6 11.3

In the following, we briefly present the results of new N -body simulations of unequal-mass binaries passing by SgrA*. The integrations are performed by using the high accuracy N -body code ARCHAIN (Mikkola & Merritt 2008;Mikkola & Merritt 2006). The binaries are initially placed at a distance d = 0.1 − 0.01pc from the SMBH with apurely tangential velocity corresponding to periapses between the tidal disruption radius of the secondary star andrbt. The primary star has mass M1 = 6 M⊙, while the mass of the secondary is either M2 = 1M⊙ or 3M⊙. We adopta0 = 0.1au for the internal semimajor-axis of the binaries. In all the simulations the final integration time is fixed toone orbital period of the binary orbit around the SMBH. In total we perform 1200 simulations.

As expected, we find a systematically larger ejection velocity for the less massive star (Yu & Tremaine 2003). Forbinaries with M2 = 1M⊙(3M⊙) and initial distance d = 0.01pc, the mean asymptotic ejection velocity of the primaryand secondary stars are respectively: v1 ∼ 1000km s−1 (1600km s−1) and v2 ∼ 2800km s−1 (2700km s−1). When theinitial apoapsis of the binary is increased to d = 0.1pc we found: v1 ∼ 1350km s−1 (2180km s−1) and v2 ∼ 3500km s−1(3200km s−1). Table A1 gives the fraction of collisions, mergers, and HVSs (vej > 1000km s−1) in the simulations.Note that in this table any merger is also counted as a collision. In the table we report the probability of ejection,distinguishing between the two components of the binary. It is clear that the initial distance of the binary from thecentral black hole plays a fundamental role in determining which member is ejected. For large initial distances (i.e.,d = 0.1pc) the ejection probability is almost independent on the stellar mass, while for d = 0.01pc, the lighter star ispreferentially ejected. These results are consistent with the findings of Sari et al. (2010) that used an approximatedmethod to study the dynamical evolution of binaries on parabolic orbits. Our simulations suggest that, in the limitthat the external orbital energy of the binary goes to zero, the ejection probability becomes an independent functionof the stellar mass.

REFERENCES

Alexander, T. 2005, PhR, 419, 65Alexander, T., & Kumar, P., 2001, ApJ, 549, 948Antonini, F., Faber, J., Gualandris, A., & Merritt, D. 2010, ApJ,

713, 90, (Paper I)Bahcall, J. N., & Wolf, R. A. 1976, ApJ, 209, 214Bartko et al. 2010, ApJ, 708, 834

Benz, W., & Hills, J. G. 1987, ApJ, 323, 614Benz, W., & Hills, J. G. 1992, ApJ, 389, 546Binney, J. & Tremaine, S. 1987, Galactic Dynamics (Princeton,

NJ:Princeton University Press)Bogdanović, T., Eracleous, M., Mahadevan, S., Sigurdsson, S., &

Laguna, P. 2004, ApJ, 610, 707


Bromley, B. C., Kenyon, S. J., Geller, M. J., Barcikowski, E.,Brown, W. R., & Kurtz, M. J. 2006, ApJ, 653, 1194

Brown, W. R., Geller, M. J., Kenyon, S. J., & Kurtz, M. J. 2005,ApJ, 622, L33

Brown, W. R., Geller, M. J., Kenyon, S. J., & Kurtz, M. J. 2006,ApJ, 647, 303

Brown, W. R., Geller, M. J., Kenyon, S. J., Kurtz, M. J., &Bromley, B. C. 2007, ApJ, 671, 1708

Brown, W. R., Geller, M. J., & Kenyon, S. J. 2009, ApJ, 690, 1639Buchholz, R. M., Schödel, R., & Eckart, A. 2009, A&A, 499, 483Cement, M. J., 1994, ApJ, 420, 797Collier, A. C., & Jenkins, C. R. 1984, MNRAS, 211, 391Dai, L. J., Fuerst, S. V., & Blandford, R. 2010, MNRAS, 402, 1614Do, T., Ghez, A. M., Morris, M. R., Lu, J. R., Matthews, K., Yelda,

S., & Larkin, J. 2009, ApJ, 703, 1323Eggleton P. P., 1971, MNRAS, 151, 351Eggleton, P. P. 1983 ApJ, 268, 368Eisenhauer, F. et al. 2005, ApJ, 628, 246Evans, C. R., & Kochanek, C. S. 1989 ApJ, 346, 13Faber, J. A., Rasio, F, A., & Willems, B. 2005 Icar., 175, 248Figer, D. F. 2008, in Massive Stars: From PopIII

and GRBs to the Milky Way (in press); preprint athttp://arxiv.org/abs/0803.1619

Freitag, M., & Benz, W. 2005, MNRAS, 358, 1133Fujii, M., Iwasawa, M., Funato, Y., & Makino, J. 2010, ApJ, 716,

L80Gaburov E., Lombardi J. C., Jr., Portegies Zwart S., 2010,

MNRAS, 402, 105Genzel, R., Schödel, R., Ott, T., Eisenhauer, F., Hofmann, R., &

Lehnert, M. et al. 2003, ApJ, 594, 812Gezari, S., et al. 2008, ApJ, 676, 944Gezari, S., et al. 2009, ApJ, 698, 1367Ghez, A. M., et al. 2003, ApJ, 586, L127Ghez, A. M., et al. 2008, ApJ, 689, 1044Ginsburg, I., & Loeb, A. 2006 MNRAS, 368, 221Ginsburg, I., & Loeb, A. 2007 MNRAS, 376, 492Gillessen, S., Eisenhauer, F., Trippe, S., Alexander, T., Genzel, R.,

Martins, F., & Ott, T. 2009, ApJ, 692, 1075Glebbeek E., 2008, PhD thesis, Utrecht UniversityGlebbeek E., Pols O. R., 2008, A&A, 488, 1017Gualandris, A., Portegies Zwart, S., & Sipior, M. S. 2005, MNRAS,

363, 223Hansen, C. J., Kawaler, S. D., & Trimble, W. 2004, Stellar Interiors,

Springer-Verlag New York, Inc.Hansen, B. M. S., 2007, ApJ, 671, L133Hills, J. G. 1988, MNRAS, 368, 221Hobbs, L. M.& Mathieu, R. D. 1991, PASP, 103, 431Kasen, D., & Ramirez-Ruiz, E. 2010, ApJ, 714, 155Komossa, S., Halpern, J., Schartel, N., Hasinger, G, Santos-Lleo,

M., Predehl, P. 2004, ApJ, 603, L17Levin, Y. 2006, ApJ, 653, 1203Levin, Y. 2007, MNRAS, 374, L515Leonard, P. J. T. 1989, AJ, 98, 217

Li, L., Narayan, R., & Menou, K. 2002, ApJ, 576, L753Löckmann, U. & Baumgardt, H. 2008, MNRAS, 384, 323Löckmann, U., Baumgardt, H., & Kroupa, P. 2008, ApJ, 683, L151Lodato, G., King, A. R., & Pringle, J. E. 2009, MNRAS, 392, 332Lombardi Jr. J. C., Rasio F. A., & Shapiro S. L. 1995, ApJL, 445,

L117Lombardi, J. C., Jr., Rasio, F. A., & Shapiro, S. L. 1996, ApJ, 468,

797Lombardi J. C., Jr., Warren J. S., Rasio F. A., Sills A., & Warren

A. R. 2002, ApJ, 568, 939López-Morales, M., Bonanos, A. Z., 2008, ApJ, 685, L47Luminet, J. P., & Carter, B. 1986, ApJS, 61, 219Madigan, A., Levin, Y., & Hopman, C. 2009, ApJ, 697, L44Mateo, M., Harris, H. C., Nemec, J., & Olszewski, E. W. 1990, AJ,

100, 469Merritt, D., Gualandris, A., & Mikkola, S. 2009, ApJ, 693L, 35Merritt, D. 2010, ApJ, 718, 739Mikkola, S., & Merritt, D. 2008, AJ, 135, 2398Mikkola, S., & Merritt, D. 2006, MNRAS, 372, 219Miller, M. C., Freitag, M., Hamilton, D. P., & Lauburg, V. M.

2005, ApJ, 631, 117Nayakshin, S. & Cuadra, J. 2005, A&A, 437, 437Nayakshin, S. & Cuadra, J. 2007, MNRAS, 379, 21Oh, S., Kim, S. S., & Figer, D. F. 2009, Journal of Korean

Astronomical Society, 42, 17Paczynski, H.B. 1971, ARA&A, 9, 183Paumard, T. et al. 2006, ApJ, 643, 1011Peebles, P. J. E. 1971, A&A, 11, 377Perets, H. B. 2009, ApJ, 690, 795Perets, H. B., Hopman, C., & Alexander, T. 2007, ApJ, 656, 709Perets, H. B., Gualandris, A., Kupi, G., Merritt, D., & Alexander,

T. 2009, ApJ, 702, 884Perets, H. B. &Gualandris, A. 2010, ApJ, 719, 220Portegies Zwart S., McMillan S., Harfst S. e. a., 2009, New

Astronomy, 14, 369

Price D. J. 2007, Publications of the Astronomical Society ofAustralia, 24, 159

Pritchet, C. J., & Glaspey, J. W. 1991, ApJ, 373, 105Rauch, K. P., & Tremaine, S. 1996, New Astronomy, 1, 149Rees, M. J. 1988, Nature, 333, 523Regös, E., Bailey, V. C., & Mardling, R. 2005, MNRAS, 358, 544Rosswog S., 2009, NewAR, 53, 78Sesana, A., Haardt, F., & Madau, P. 2006, ApJ, 651, 392Sari, R., Kobayashi, S., & Rossi, E. M. 2010, ApJ, 708, 605Sills, A., & Lombardi, J. C., Jr. 1997, ApJ, 484, L51Sills, A., Lombardi, J. C., Jr., Bailyn, C. D., Demarque, P., Rasio,

F. A., & Shapiro, S. L., 1997, ApJ, 487, 290Strabbe, L. E., & Quataert, E. 2009, MNRAS, 400, 2070Ulmer, P. 1999, ApJ, 514, 180Wang, J., & Merritt, D. 2004, ApJ, 600, 149Yu, Q., & Tremaine, S. 2003, ApJ, 599, 1129

Documents

TIDAL BREAKUP OF BINARY STARS AT THE GALACTIC CENTER. II.astrophysics.rit.edu/fantonini/tbbs2/ms.pdf · Tidal breakup of binary stars at the Galactic Center-II 3 that, at any radius,