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

Fabio Antonini; James C. Lombardi; David Merritt

doi:10.1088/0004-637X/731/2/128

1. INTRODUCTION

Tidal breakup of binary stars by the supermassive black hole (SMBH) at the Galactic center (GC) has been invoked to explain a number of otherwise puzzling discoveries, including the hypervelocity stars (HVSs) that are observed 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 the SMBH (Eisenhauer et al. 2005; Gillessen et al. 2009). As first pointed out by J. Hills, close passage of a binary star near an SMBH can result in an exchange interaction, such that one component of the binary is ejected with greater than escape velocity while the other star is scattered onto a tightly bound orbit (Hills 1988; Yu & Tremaine 2003). The predictions of this model are broadly consistent with the observed properties of both the HVSs (Bromley et al. 2006) and the S-stars (Perets et al. 2009; Perets & Gualandris 2010).

The origin of the binary progenitors of the HVSs is not clear. One possibility is that the binaries originated at distances of a few parsecs from the GC and were subsequently scattered inward by "massive perturbers" (Perets et al. 2007). In this scenario, most of the binaries will lie on either unbound or weakly bound orbits with respect to the SMBH, and they will encounter it only once before being scattered onto different orbits. Alternatively, the binaries may form closer to the SMBH, perhaps in the young (or an older) stellar disk that is observed between ∼0.04 pc and ∼0.5 pc from the SMBH (Nayakshin & Cuadra 2005, 2007; Paumard et al. 2006; Levin 2007). Also, Perets (2009b) suggested that binaries could be left near the GC through a triple disruption by the SMBH. In these latter cases, the binaries would be bound to the SMBH and would encounter it many times before being disrupted.

If the finite sizes of stars are taken into account, a number of outcomes are possible in addition to simple binary disruption. The two stars can collide, resulting in a merger if the relative velocity is less than stellar escape velocities (Ginsburg & Loeb 2007). Since the radius of tidal disruption of single stars by the SMBH is comparable to the binary disruption radius, stars can also be tidally disrupted by the SMBH, either before or after their close interaction with each other.

In Paper I (Antonini et al. 2010), we presented the results of a large number of N-body integrations of point-mass binary stars on eccentric orbits around the GC SMBH. In many cases, the trajectories of the two stars were found to imply a physical collision, assuming that the unperturbed stars had radii similar to those of normal main-sequence stars of the same mass. The probability of physical encounters was found to increase significantly if the binaries were allowed to complete many orbits about the SMBH. In some cases, one or both stars also passed close enough to the SMBH that gravitational tides would be expected to significantly affect their internal structure.

In this paper, we use smoothed-particle hydrodynamics (SPH) simulations to study the binaries from Paper I that approached closely enough to physically interact. The N-body simulations of Paper I were first used to identify initial conditions that resulted in close interactions between the two stars. The point-mass stars were then realized as macroscopic, fluid-dynamical models and integrated forward in the gravitational field of the SMBH using an SPH algorithm. As in Paper I, we followed the trajectories for multiple orbits around the SMBH, allowing us, for the first time, to investigate the consequences of repeated tidal interactions with the SMBH (movies can be found at http://astrophysics.rit.edu/fantonini/tbbs2/).

In Section 2, we briefly discuss timescales for binary disruption at the GC. Our initial conditions and numerical methods are described in Section 3 and the results in Section 4. Some observable consequences are presented in Section 5. Section 6 sums up.

2. THE SURVIVAL TIME OF BINARIES AT THE GALACTIC CENTER

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

$\begin{equation} |E|/({M_{\rm b} \sigma ^2}) \lesssim 1, \end{equation} \tag{ 1 }$

with E the internal orbital energy of the binary, M_b the binary mass, and σ the one-dimensional velocity dispersion of the stellar background. In principle, because most of the binaries at the GC are expected to be "soft," |E| ≲ M_bσ², and because the binary evaporation time t_ev is a function of the distance from the SMBH, the variation of t_ev with galactocentric radius can be used to constrain the origin of the HVSs (Perets 2009a). If the evaporation time at some radius is shorter than the lifetime of a typical main-sequence star, the stellar population in this region would be dominated by isolated (i.e., single) stars.

Here we show that the survival time of binaries at galactocentric distances r < 0.1 pc is likely to be comparable to the typical main-sequence lifetimes of most stars in this region. We also show that, within a radius of r ∼ 0.3 pc, t_ev becomes essentially independent of radius.

Beyond ∼1 pc from Sgr A*, the mass density determined from the stellar kinematics follows ρ ∼ r^−β, 1.5 ≲ β ≲ 2 (e.g., Oh et al. 2009). At smaller radii, number counts of the dominant (old) stellar population near the GC suggest a space density that is weakly rising, or falling, toward the SMBH, inside a core of radius ∼0.5 pc (Buchholz et al. 2009; Do et al. 2009; Bartko et al. 2010). Approximating the mass density as a broken power law,

$\begin{equation} \rho (r)=\rho _0 \left(\frac{r}{r_0} \right)^{-\gamma } \left[ 1+\left(\frac{r}{r_0} \right)^2\right]^{(\gamma -\beta)/{2}} \end{equation} \tag{ 2 }$

with r₀ = 0.3 pc and β = 1.8, and setting ρ₀ = 1.3 × 10⁶ M_☉ pc⁻³ gives a good fit to the space density outside the core (e.g., Merritt 2010). At smaller radii, the uncertainties in ρ are represented by the poorly determined value of γ.

The evaporation time is given by (Binney & Tremaine 1987)

$\begin{equation} t_{\rm ev}=\frac{M_{\rm b}\sigma }{M 16\sqrt{\pi } \rho a_0 \ln \Lambda }, \end{equation} \tag{ 3 }$

where ln Λ is the Coulomb logarithm, M is the mass of the field stars, a₀ is the binary semimajor axis, and σ is calculated from the Jeans equation,

$\begin{equation} \rho (r)\sigma (r)^2 = G\int _r^{\infty } dr^{\prime } r^{\prime -2} [M_{\bullet }+M_\star ({<}r^{\prime })]\rho (r^{\prime }), \end{equation} \tag{ 4 }$

with M_⋆(< r) the total mass in stars within r and M_• the mass of the central black hole. Hereafter, we adopt M_• = 4 × 10⁶ M_☉ (Ghez et al. 2008; Gillessen et al. 2009). In Figure 1, we plot the evaporation time of binaries in the density model of Equation (2) as a function of galactocentric radius, assuming M_b = 2M, ln Λ = 15, and two different values of the internal slope: γ = 0.5 which is representative of the observed distribution and γ = 1.8 which corresponds approximately to a relaxed system around an SMBH (Bahcall & Wolf 1976).

**Figure 1.** Evaporation time of binaries vs. galactocentric radius for different values of the binary semimajor axis a₀. Solid curves show the evaporation time for the density model of Equation (2) with γ = 0.5 while the dashed curves correspond to the coreless model with slope γ = 1.8. The filled gray region gives the ages of the S-stars (Eisenhauer et al. 2005).
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From the figure, it is clear that the survival time of binaries would be greatly increased if the distribution of stars at the GC has a low-density core (for comparison, see also Figure 1 in Perets 2009a). The figure also shows that, at any radius, for a₀ ≲ 1 au, this time is larger than the typical lifetime of the S-stars. Therefore, it cannot be excluded that the S-stars were initially part of binary systems originating at galactocentric distances of few tens of milliparsecs.

We finally note that, in the context of this paper, it might be more appropriate to compare t_ev with the timescale required to drive the eccentricities of the stars away from their initial values to produce quasi-radial orbits. At the GC, this time can be of the order of a few Myr (Löckmann et al. 2008; Madigan et al. 2009; Merritt et al. 2009; Perets et al. 2009; Fujii et al. 2010), which is much shorter than the typical evaporation timescale of binaries with a₀ ⩽ 0.2 au.

3. INITIAL CONDITIONS AND NUMERICAL METHOD

3.1. Orbital Initial Conditions

In Paper I, we used the high-accuracy numerical integrator ARCHAIN (Mikkola & Merritt 2008, 2006) to study the dynamics of main-sequence binary stars on highly elliptical orbits whose periapsides lay close to the SMBH. We determined the final orbital properties of both ejected and bound stars. Initial conditions consisted of equal-mass binaries on circular relative orbits with random orientations, initial separations a₀ in the range 0.05–0.2 au, and individual stellar masses M of 3–6 M_☉. The binaries were given a tangential initial velocity in the range 4–85 km s⁻¹ and initial 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 dimension to the particles and investigate the probability of stellar collisions and mergers. In detail, we defined the minimum impact parameter for a collision as 2R, and the two stars were assumed to coalesce when their relative velocity upon collision was lower than the escape velocity from their surface. The binary separations adopted in this work were close to the extremes of the interval within which HVSs can be produced: small semimajor axes, a₀ ≲ 0.05 au, result in contact binaries, while for a₀ > 0.2 au few stars would be ejected with velocities sufficient 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 additional set of ∼2000 integrations of binaries with component masses M₂ = 1–3 M_☉ and M₁ = 6 M_☉. Initial conditions and results of these new runs are summarized in the Appendix. In the following, for the case of unequal-mass binaries, we distinguish between the primary and secondary stars' quantities using the subscripts 1 and 2, respectively.

The orbital initial conditions that we adopt in the SPH simulations correspond to binaries that enter well within their tidal disruption radius r_bt, approximated by the expression (Miller et al. 2005):

$\begin{equation} r_{\rm bt} \sim \left(\frac{M_{\bullet }}{M_{\rm b}} \right)^{1/3}a_0, \end{equation} \tag{ 5 }$

with M_b = M₁ + M₂ the total mass of the binary. Therefore, the binaries in all of our simulations are strongly perturbed at the first periapse passage. Moreover, we focus on cases where the tidal perturbations from the SMBH on the single stars are expected to be significant. This corresponds to binaries with periapsides that lie close to the tidal disruption radius of a single star, or

$\begin{equation} r_{{\rm t}} \sim \left(\frac{M_{\bullet }}{M} \right)^{1/3}R. \end{equation} \tag{ 6 }$

We selected three types of initial conditions that can be classified according to the final outcome of the SPH simulations.

1.
Stellar collision (without merger). In some cases, gravitational perturbations from the SMBH can lead to a physical collision between the two stars. If the relative velocity at collision is sufficiently high, the stars can survive the interaction and avoid a merger. Soon after the collision, one star can be ejected at high velocity.
2.
Stellar merger. If the two stars collide with a relative velocity at impact smaller than approximately the escape velocity from their surfaces, coalescence occurs, resulting in the formation of a new, more massive star. The merger remnant will remain bound to the SMBH unless extreme mass loss occurs during coalescence.
3.
(Clean) ejection of an HVS. The tidal breakup of the binary by the SMBH results in the ejection of a star at very high velocity. The former companion of the ejected star loses energy in the process and is deposited onto a tight orbit around the SMBH. Here, "clean" means that the member stars of the binary do not collide with each other during the process of ejection.

For the sake of simplicity, in all the SPH simulations we chose the initial apoapsis of the external binary orbit to be d = 2000 au ≈0.01 pc (except for case H8 which has d = 1700 au; see Table 1). However, starting the SPH simulations from this distance would greatly increase the required computational time. The stars were therefore initially placed at a point of the orbit corresponding to a much smaller distance from the SMBH: r₀ = 5r_bt. This choice for r₀ was motivated by the fact that at these distances, the tidal forces from the SMBH are still too weak to significantly influence the internal structure of the stars. In addition, since r₀ is considerably larger than r_bt, at the initial time the internal binary eccentricity is still close to zero (e ≲ 10⁻²). As in Paper I, the plane of the binary's orbit about the SMBH is the x–z plane.

Table 1. Summary of the SPH Simulations

Run	M₁ (M_☉)	M₂ (M_☉)	a₀ (au)	r_per (au)	λ₁ (λ₂)	ζ₁(ζ₂)	SPH	N-body
C1	3	3	0.05	1.50	1.36	0.668	Collision+HVS	Collision+HVS
C2	3	3	0.2	2.67	2.43	1.19	Collision+HVS	Collision+HVS
C3	6	6	0.05	1.50	1.07	0.526	Collision+HVS	Merger
C4	6	6	0.2	2.04	1.46	0.716	Collision+HVS	Collision+HVS
C5	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	Merger
M2	3	3	0.05	1.50	1.36	0.668	Merger	Merger
M3	3	3	0.05	4.18	3.79	1.86	Merger	Collision+HVS
M4	3	3	0.2	4.18	3.79	1.86	Merger	Merger
M5	3	3	0.2	38.2	34.7	17.0	Merger	Merger
M6	6	6	0.05	2.67	1.91	0.935	Merger	Collision+HVS
M7	6	6	0.05	2.67	1.91	0.935	Merger	Merger
M8	6	6	0.05	4.18	2.99	1.46	Merger	Merger
M9	6	6	0.05	5.05	3.62	1.77	Merger	Merger
M10	6	6	0.2	2.04	1.46	0.716	Merger	Merger
M11	6	6	0.2	20.4	14.6	7.13	Merger	Merger
M12	6	3	0.1	8.08	5.78 (7.34)	2.83 (3.59)	Merger	Merger
M13	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	HVS
H2	3	3	0.05	7.07	6.42	3.14	HVS	HVS
H3	3	3	0.2	2.04	1.86	0.909	HVS	HVS
H4	6	6	0.05	1.50	1.07	0.526	HVS	HVS
H5	6	6	0.2	2.04	1.46	0.716	HVS	HVS
H6	6	6	0.2	0.60	0.429	0.210	HVS	HVS
H7	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)

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3.2. SPH Numerical Techniques and Initial Conditions

SPH is a Lagrangian method in which the fluid is represented by a finite number of fluid elements or "particles." Associated with each particle i are, for example, its position r_i, velocity v_i, and mass m_i. Each particle also carries a purely numerical smoothing length h_i that determines the local spatial resolution and is used in the calculation of fluid properties such as acceleration and density. For a recent review of SPH, see Rosswog (2009). The SPH code used in this work is presented in Gaburov et al. (2010), with the augmentation that the analytic solution to the Kepler two-body problem can be used to advance a star bound to the SMBH through those portions of the orbit when hydrodynamic effects are negligible (see Section 4.4). The equation of state is ideal gas plus radiation pressure and radiative cooling and heating is neglected. To calculate the gravitational accelerations and potentials, we use direct summation on NVIDIA graphics cards, softening with the usual SPH kernel as in Hernquist & Katz (1989). Thus, gravity is softened only in interactions between neighbors. The use of such a softening with finite extent (as opposed, for example, to Plummer softening) increases the accuracy and stability of SPH models, consistent with the studies of Athanassoula et al. (2000) and Dehnen (2001).

In our simulations, the SMBH is a compact object particle that interacts gravitationally, but not hydrodynamically, with the rest of the system. The gravity of the SMBH is softened according to a density profile defined by the standard SPH cubic spline kernel with a constant smoothing length h_• = 20 R_☉. This approach has the advantage that the treatment of gravity is unsoftened for separations r > 2h_•. We note that h_• is small compared to the periapsis separation r_per in all cases, so our code is able to follow bound stars around the black hole for many orbits without introducing spurious secular effects from gravitational softening. The SMBH is allowed to move in response to gravitational pulls. However, because the 4 × 10⁶ M_☉ SMBH is much more massive than any of the binaries being considered, the SMBH always remains very near the center of mass of the system, which we take to be the origin.

Before simulating the interaction of a binary with the SMBH, we must first prepare an SPH model for each binary component in isolation. To compute stellar structure and composition profiles, we use the TWIN stellar evolution code (Eggleton 1971; Glebbeek & Pols 2008; Glebbeek 2008) from the MUSE software environment (Portegies Zwart et al. 2009). We evolve main-sequence stars with initial helium abundance Y = 0.28 and metallicity Z = 0.02. The 3 M_☉ star is evolved to an age of 50 Myr, yielding the same 2.15 R_☉ (0.01 au) radius as in the corresponding models of Paper I. The 1 and 6 M_☉ stars are each evolved to 18.2 Myr, yielding stellar radii of 0.891 and 3.44 R_☉, respectively. Figure 2 shows the resulting composition profiles for our 1, 3, and 6 M_☉ stars, colored red, green, and blue, respectively.

**Figure 2.** Fractional chemical abundances (by mass) vs. enclosed mass fraction m/M for our M = 1 M_☉ (red curves), 3 M_☉ (green curves), and 6 M_☉ (blue curves) stars, as calculated by the TWIN stellar evolution code.
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Initially, we place the SPH particles on a hexagonal close packed lattice, with particles extending out to a distance only a few smoothing lengths less than the full stellar radius. After the initial particle parameters have been assigned according to the desired profiles from TWIN, we allow the SPH fluid to evolve into hydrostatic equilibrium. During the relaxation calculation, the drag force we include is the normal artificial viscosity but in the acceleration equation only.

Figure 3 shows energies versus time both during and after the relaxation process. From the internal energy U and potential energy W curves, it is apparent that the star oscillates on a hydrodynamical timescale, specifically with a fundamental period of about 0.08 days. During relaxation, these oscillations are damped by the drag force, which does negative work on the system and decreases the total energy E toward that of a minimum energy equilibrium state. At a time of 1.84 days (=100 G^−1/2 M^−1/2_☉ R_☉^3/2), the drag force is removed and the star is allowed, as a test of stability, to evolve dynamically in isolation. During this dynamical evolution, the internal energy U and gravitational energy W each remain nearly constant. By t = 9 days, the kinetic energy T has diminished to nearly 10 orders of magnitude less than the total energy E in magnitude, corresponding to an exceedingly small amount of noise in an otherwise static model. The overall level of energy conservation is excellent: extrapolating forward the drift in total energy E, which is linear in time, we find it would take about 2.4 × 10⁵ days (660 years or 3 × 10⁶ oscillation periods) of hydrodynamical evolution to reach a 1% error in total energy.

Our approach allows the parent stars to be modeled very accurately. As an example, Figure 4 plots both desired profiles and SPH particle data for the 6 M_☉ star. The structure and composition profiles of the SPH model closely follow the desired TWIN profiles. Our relaxed models remain static and stable when left to evolve dynamically in isolation: indeed, the particle data shown in the lower panels of Figure 4 are nearly identical to those in the upper panels, demonstrating that there are no significant changes in the model even after more than 4000 days (or equivalently 5 × 10⁴ oscillation periods) of hydrodynamical evolution.

The binaries in our simulations are then created simply by shifting two stellar models, each taken from the end of a relaxation calculation, to the appropriate initial position and velocity provided by the N-body code. We begin with binary components irrotational in the inertial frame, which allows us to more easily study any rotation imparted during the subsequent interaction.

Unless otherwise noted, our simulations employ N ≈ 4 × 10⁴ SPH particles: such a particle number provides an appropriate balance between resolution and the need sometimes to follow the hydrodynamics for time intervals exceeding 10⁵ dynamical timescales (corresponding to hundreds of orbits around the SMBH).

3.3. Timescale Considerations and Orbital Advancement

Because of shock heating in collisions and mergers, in addition to tidal heating during the periapse passage, the bound stars are out of thermal equilibrium and larger than a normal main-sequence star of the same mass. The global thermal readjustment of the bound stars proceeds on a thermal timescale t_thermal ≈ U/L, where U is the total internal energy in the star and L is its luminosity. The SPH simulations confirm that the internal energy U of the bound star after one periapsis passage is comparable to the total internal energy of the star(s) from which the bound star came and typically U ≈ (2–7) × 10⁴⁹ erg, with the larger values generally corresponding to more massive stars. For weak collisions and clean ejections of HVSs, the luminosity of the bound star will be comparable to the value it had in the initial binary: the thermal timescale in such cases is then roughly 10⁵–10⁷ years. Guided by calculations of blue stragglers (Sills et al. 1997), we estimate that the luminosity of a bound star produced by a merger or strong collision may be up to ∼100 times larger than that of a main-sequence star of the same mass. Thus, the luminosity of our most massive merger products could be briefly as large as ∼10⁶ L_☉ ≈ 4 × 10³⁹ erg s⁻¹, so that the global thermal timescale t_thermal ≳ 600 years (although the local thermal timescale in the outer layers of the star could be less). We conclude that thermal adjustment over an orbital period is small and often completely negligible, and we therefore do not attempt to model the thermal relaxation here.

Although the orbital period is small compared to the thermal timescale, it is large compared to the hydrodynamical timescale, which is about an hour. Following the full hydrodynamics of a multiple orbit encounter would therefore not be practical. What we do instead is wait for the star(s) to move sufficiently far away from the black hole and then advance any bound star around most of its Kepler two-body orbit. At the same time, we remove from the simulation any HVS, any ejecta, and any gas that has become bound to the SMBH. As long as the periapse passages are treated hydrodynamically, our results are not sensitive to precisely which portion of the orbit is treated in the two-body approximation. In practice, we wait at least 8 days after periapse, and at least 8 days after the merger or ionization of a binary, before measuring the orbital elements and implementing the two-body analytic solution. The orbital advancement is performed such that the distance from the black hole to the bound star is unchanged but that the objects are now approaching one another. We preserve the orientation of the orbit and spin of the bound star during the orbital advancement.

In order to test the reliability of the method, we run a portion of the first orbit for one of our simulations (C1 in Table 1) without any orbital advancement. We found that, at about 20 days after we would have applied the advancement, their masses had decreased by an additional 0.002 M_☉. We note that even though we retain too much mass, this "extra" mass is far from the stars, so it does not significantly participate in the hydrodynamics and it will get ejected in the next passage. We conclude that neglecting hydrodynamics far from periapsis is a reasonable approximation.

4. RESULTS

The first important result of the SPH simulations performed in this work is that their qualitative outcome agrees very well with that of the N-body integrations devised in Paper I. Among a total of 26 simulations, only in 3 cases did the N-body approach fail to match the results of the hydrodynamic calculations.

Table 1 reports the chosen initial conditions as well as a qualitative 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 parameterized by the dimensionless quantity: λ = r_per/r_t. In general, stars on orbits meeting the condition λ < 1 are tidally disrupted (Luminet & Carter 1986; Evans & Kochanek 1989). However, even when tidal disruption does not occur, we expect that mass will be stripped from the outer regions of any star passing within its Roche limit (Paczynski 1971). In the table, ζ gives the Roche lobe radius (evaluated at periapsis) in units of the stellar radius:

$\begin{equation} \zeta =\frac{0.49q^{2/3}}{0.6q^{2/3}+{\rm ln}(1+q^{1/3})}\times \frac{r_{\rm per}}{R}, \end{equation} \tag{ 7 }$

with q = M/M_• (Eggleton 1983). Although this formula was derived under the assumption of circular orbit, it has been shown to work reasonably well even for eccentric binaries, if used at periapsis (Regös et al. 2005). Note that because q ≪ 1 in the present work, the approximate relation ζ ≈ 0.49λ exists between λ and ζ.

The results of our SPH simulations are presented in what follows. We first describe the product of one binary–SMBH interaction (Sections 4.1–4.3), and then we successively follow the evolution of the bound stars as they perform several revolutions around the SMBH (Section 4.4).

4.1. Stellar Collisions

Ginsburg & Loeb (2007) noted that the tidal breakup of stellar binaries interacting with the SMBH can lead, under some circumstances, to a physical collision between the two member stars. In this section, we investigate the cases in which the two stars collide with a relative impact speed large enough that they do not merge upon impact. Some results of the SPH simulations are shown in Table 2, where we also compare the asymptotic ejection velocity of the HVSs (v_ej), the semimajor axis (a), and eccentricity (e) of the captured stars with the same quantities obtained in the N-body simulations.

Table 2. Ejection Velocity v_ej of the HVS, Orbital Semimajor Axis a and Eccentricity e of the Captured Star, and Distance of Closest Approach Between the Two Stars (r₀) in the SPH (N-body) Calculations

Run	v_ej (km s⁻¹)	a (au)	e	r₀/(R₁ + R₂)	ΔM_b/M_b	ΔM_•/M_b
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

Notes. The quantity ΔM_b/M_b gives the fraction of mass lost from the binary, while ΔM_•/M_b is the fraction of the mass lost from the binary that remains bound to the SMBH. All quantities are evaluated after the first periapsis passage, once the stars have retreated far from the SMBH.

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The agreement between the two methods is remarkably good. However, the SPH simulations systematically produce smaller values of v_ej and larger a. This is a consequence of the efficient energy and angular momentum transfer occurring between the stars: the ejected star slows down and the captured star gains speed during the collision.

From Table 2, it is clear that the mass loss from the binary is smaller than a few percent of the total mass, and always below ∼0.1 M_☉. This is consistent with previous simulations of stellar collisions that often found a small fractional mass loss (Benz & Hills 1987, 1992; Freitag & Benz 2005). Our calculations indicate that, after a collision, the mass ejected from the SMBH–stars system is comparable to, although smaller than, the mass that is ejected from the binary but remains bound to the SMBH; this mass loss has a small but measurable impact on the subsequent evolution of the stars' orbits as demonstrated by comparing the SPH and the N-body quantities in the table. Debris will eventually settle into a torus-like structure about the SMBH that will subsequently evolve due to viscosity, mass inflow, radiative cooling, and winds.

Spin-up is expected to be one of the main signatures of either a (off-axis) collision (Alexander & Kumar 2001) or a tidal encounter with a massive black hole (Evans & Kochanek 1989). In our simulations, the close stellar encounter as well as the SMBH tides at periapsis lead therefore to some degree of rotation in the stars with angular frequency:

$\begin{equation} \Omega _{\rm tot} \approx \sqrt{\Omega _*^2 + \Omega _\bullet ^2} \approx \sqrt{\frac{G(M_1+M_2)}{r_{\rm 0}^3}+ \frac{{\it GM}_{\bullet }}{r_{\rm per}^3}}, \end{equation} \tag{ 8 }$

where Ω_* and Ω_• are, respectively, the angular velocity induced by the interaction with the companion star and that imparted by the SMBH tides; r₀ is the distance of closest approach between the stars. The ratio Ω_*/Ω_• in the cases considered here is always greater than 1 and varies from a maximum of ∼20 (simulation C5) to a minimum of ∼2 (simulation C4).

Figure 5 plots the temporal evolution of the dimensionless spin parameter (Peebles 1971):

$\begin{equation} J=\frac{L|E|^{1/2}}{{\it GM}^{5/2}}, \end{equation} \tag{ 9 }$

where L is the spin angular momentum of the star and E is its binding energy. In all cases, there is a sharp increase in the stars' spins during the periapsis passage followed by a second gradual decrease toward the final relaxed (spinning) configuration. Note that the captured stars have values of the final spin slightly larger than that of the ejected stars. This finding is consistent with the results of Paper I (but see Sari et al. 2010 as well), where it is shown that the captured member is always the star with the smallest value of the closest approach distance to the SMBH implying, as we expect from Equation (8), a larger tidal torque at periapsis.

Figure 5 also shows the temporal evolution of the stellar masses near the first periapsis passage. Each star typically loses about 1% of its mass, with captured stars (again, those that pass closer to the SMBH) losing slightly more mass than their ejected counterparts. In the cases with an equal-mass binary, both stars lose a comparable amount of mass. The captured star in simulation C5 actually gains mass that had been lost from its binary companion, as discussed in more detail below. In all cases, the stellar masses stabilize to an essentially constant value by the time the orbital advancement technique is implemented, which is where the curves terminate.

Figure 6 presents column density snapshots for simulation C5. The simulation models a stellar binary with a₀ = 0.1 au and with components of masses M₁ = 6 M_☉ and M₂ = 3 M_☉ (see Table 1). The time indicated on the panels is shifted in order to have t = 0 at the moment of the closest approach of the binary with the central black hole. Between t = 0 and t = 0.9 days the reference frame is the center of mass of the binary while in the lower right panels, we switch to the frame in which the center of mass of either the captured (left) or ejected (right) star is at the origin.

The first contact between the stars occurs ∼0.1 days after the periapsis passage. The interaction leads to an episode of mass transfer between the stars (observed at ∼0.35 days in the figure). The smaller star gains mass (∼0.02 M_☉) in the collision, while the larger star loses ∼0.04 M_☉. Only 6 × 10⁻³ M_☉ becomes ejecta from the entire system (i.e., stars plus SMBH) after the first periapsis passage, while the remainder of the gas lost from the binary remains bound to the black hole. The mass loss from the binary upon impact is therefore of order ∼10⁻² M_☉. Note that, in this simulation, the penetration factor λ of both stars is large enough that the mass loss from the binary can be completely attributed to the stellar impact rather than to the tidal perturbations from the SMBH.

The two bottom right panels give column density plots of the captured (left) and ejected (right) stars, both showing a low-density, oblate envelope surrounding a compact spherical nucleus with a central density almost unaltered with respect to that of the parent stars. This particular configuration is common to almost all the other collisional products as shown in Figure 7 which gives column density plots of the ejected stars. Rotation as well as asymmetric shape can have a fundamental role in the future evolution of the stars and their observable characteristics; a star with a rapidly rotating nucleus can have its main-sequence lifetime considerably extended with respect to their non-rotating counterpart (Clement 1994). Only in run C4 is the HVS ejected after the collision slowly spinning and spherical even in its outermost envelope. In this simulation, the stars experience a more grazing collision which leads to some envelope ejection but leaves the stars' structure essentially unchanged.

**Figure 7.** Stars ejected in our simulations after a collision with the companion star. The stars show, typically, an oblate envelope surrounding a high-density spherical nucleus. Only in simulation C4, where the impact is more "grazing," the collisional product is spherically symmetric even in its outermost envelope.
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4.2. Mergers

Stellar collisions due to either binary evolution or dynamical interactions are thought to be the main formation channel of blue stragglers in star clusters (Collier & Jenkins 1984; Leonard 1989; Mateo et al. 1990). Similar processes have been proposed in the past to explain the puzzling presence of the young massive stars observed at galactocentric distances of few mpc, where star formation is thought to be strongly inhibited by the SMBH tides (Genzel et al. 2003; Eisenhauer et al. 2005). In Paper I, we showed that gravitational encounters involving stellar binaries and the SMBH lead, for a wide range of orbital parameters, to a stellar collision and that among the collisional products, stellar coalescence occurs in more than 80% of the cases. In this section, we study this latter outcome and investigate the properties of the resulting stars to clarify whether they would be expected to posses features commonly associated with the S-star population.

Table 3 gives the orbital parameters (i.e., eccentricity and semimajor axis) of the merger products in our simulations as well as the mass ejected from the binary and the fraction of mass captured by the SMBH after the first periapsis passage. The table shows that the merger remnants lie on an orbit very close to the initial orbit of the center of mass of the binary around the black hole, implying only a small effect of the mass loss on the dynamical evolution of the stars. The mass ejected from the binary after the first periapsis passage is typically larger than that found in collisions that do not end up with a merger (see Table 2) and is of order ∼10⁻² times the initial mass of the binary. In many cases, most of the mass ejected from the binary during the merger remained bound to the SMBH. This is a quite different situation with respect to that found in Section 4.1, where approximately half of the mass ejected from the binary remained unbound to the black hole; in these previous runs one of the two stars is always found on an escaping trajectory and, consequently, the debris associated with such a star will also tend to escape the SMBH. The last column in the table gives the dimensionless spin parameter defined in Equation (9) of the final merger products that show very large spins, some of them close to the "breakup" value (i.e., J = 1).

Table 3. Same as Table 2 but for Stellar Mergers After the First Periapsis Passage

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

Note. Here the quantities in parentheses refer to the initial orbit of the binary center of mass.

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In principle, it is possible that, as a consequence of the mass loss occurring near periapsis, the resulting merger product gains orbital energy and escapes the SMBH (see, for instance, Faber et al. 2005). Although we do not exclude this outcome for a different set of initial conditions, in our simulations this mechanism does not produce HVSs, and in all cases the merger remnant is still on a bound orbit around the SMBH. Another more important consideration is that, subsequent to the merger, the tidal heating results in some degree of expansion and a weakly bound configuration for the merger product. Because the tidal radius of the newly formed star is much larger than that of its progenitors, the star will successively lose more mass with each periapsis passage and eventually be torn apart by the SMBH tides. And in fact, as it is shown below, this is the final outcome of some of our simulations.

An example of merger is displayed in Figure 8, which involves a binary with a₀ = 0.2 au and equal-mass components of masses M = 3 M_☉ (run M4). In this run, the stars collide for the first time after ∼4 days from the time corresponding to the periapsis passage; the internal periapsis separation at the first contact is r₀/(2R) = 0.86. After the first periapsis passage, as consequence of the SMBH perturbation, the binary star becomes very eccentric. As the stars move through the circum-binary envelope formed during the previous encounters, the orbit gradually circularizes and shrinks. By t ≈ 60 days, after approximately 30 collisions, the two stellar nuclei merge. The final product has an oblate shape which is a common characteristic of all the merger remnants formed in our simulations.

As another example, Figure 9 displays column density plots of simulation M13 where a merger occurs between 6 and 1 M_☉ stars. The stars collide after 1.48 days from the moment of the closest approach to the SMBH, and subsequently merge in the following ∼1 day. During the merger, the high-density core of the lower mass star rapidly sinks to the center of the companion star. The tail-like feature observed at t = 1.86 days in the figure, is mostly material coming from the secondary star that loses part of its outermost envelope while sinking to the center of the merger remnant.

Figure 10 shows chemical composition profiles of the merger products for runs M4 and M13, after one periapsis passage. In simulation M4, the remnant has a mass of roughly 5.9 M_☉, its composition profile is very similar to that of the parent stars (see Figure 2). Based on how long it would take a normal star of that mass to evolve to that central hydrogen abundance using the TWIN stellar evolution code, we estimate that a "normal" 5.9 M_☉ star would reach a core helium abundance Y = 0.35 (assuming Z = 0.02) after ∼2 Myr. By colliding two 50 Myr old 3 M_☉ stars, we have effectively made a more massive (M ∼ 6 M_☉), younger (age ∼2 Myr) star. The merger remnant of run M13 shows a peculiar composition profile when compared to a normal star, but quite normal for merger products. In M13 (and in M12 as well), the low-mass star drops to the center of the merger product bringing its fresh hydrogen fuel along and significantly rejuvenating the core. As a result, the maximum He does not occur at the center of the merger product. The main reason of the negligible amount of hydrodynamic mixing is that the cores of the initial stars are very dense and difficult to break even in a head-on collision (Lombardi et al. 1995, 1996). However, we cannot exclude the possibility that other processes occurring on a thermal timescale (as opposed to the hydrodynamic timescale) can produce a significant degree of mixing in the stars as they evolve toward thermal equilibrium (Sills et al. 1997).

Another interesting result, plotted in Figure 10, is that a large fraction of the lithium/beryllium/boron from the parent stars gets ejected, indicating that a significant gap in the abundances of these elements in the S-star population might be observational evidence of rejuvenation through merger. Reduced atmospheric lithium abundances are, for instance, observed in blue stragglers and can also be a strong indicator of mixing (Hobbs & Mathieu 1991; Pritchet & Glaspey 1991).

We finally note that as the stars keep orbiting the SMBH, their chemical profile and their spinning configuration will change in time and therefore the states displayed in Figures 8, 9, and 10 should be intended as not permanent. The evolution will typically lead toward smaller spins and a lower lithium/beryllium/boron abundances in the stars. We will come back to this point below.

4.3. Clean Ejection of Hypervelocity Stars

Even when the stars do not collide, tidal torque and mass loss can occur if the periapsis distance of the binary center of mass initially lies within the Roche limit of its member stars. Similarly to what is done in the previous two subsections, we analyze here the first binary–SMBH interaction, while we discuss the following evolution of the bound stars in the next subsection. Some of the results of our SPH simulations, for which there is not a direct collision between the stars, are listed in Table 4. As expected, for ζ ≳ 1 there is no mass loss, and the stars maintain their initial configuration essentially unaltered (runs H2 and H3). Conspicuous mass loss instead occurs for runs H6 and H8 in which at least one of the stars crosses its tidal radius. Interestingly, although the usual condition for tidal disruption is well satisfied (i.e., λ < 1), in both runs the stars are not fully disrupted by the SMBH's tidal gravity at the first periapsis passage. In the table, we also give the final values of the spin parameter J that, in general, are found to be a very small fraction of the breakup value. Figure 11 shows the temporal evolution of J and the stellar masses for the cases of Table 4 that have the highest value of the final spin. The interaction with the SMBH induces a strong rotation only in the stars of runs H6 and H8. We conclude that, unless the stars penetrate deeply their tidal disruption radius, it seems unlikely that the SMBH tides at periapsis alone can produce a significant spin-up of the stars.

**Figure 11.** Same as Figure 5, but for some of the simulations in which there is neither a stellar collision nor a merger.
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Table 4. Same as Table 2 but for Simulations in which the Stars do not Collide and One Member Becomes an HVS

Run	v_ej (km s⁻¹)	a (au)	e	ΔM_b/M_b	ΔM_•/M_b	J_captured(J_ejected)
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)

Note. J_captured gives the spin of the stars that remain bound to the SMBH, while J_ejected refers to the ejected stars.

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As an example, Figure 12 gives column density plots of run H8. In this simulation, the primary and secondary stars have masses 1 M_☉ and 6 M_☉, respectively. The periapsis of the external orbit (∼0.8 au) is initially inside the tidal radius of the 6 M_☉ member but it is still outside the tidal radius of the 1 M_☉ star. At periapsis, the stars are squeezed by the SMBH's tidal gravity. In the process, the primary star loses a large fraction of its initial mass (∼3.4 M_☉), while the secondary loses only ∼0.03 M_☉. After the interaction with the SMBH the binary is broken apart and the lightest member becomes an HVS. Because of tidal heating during the periapsis passage, the stars are perturbed from their thermal equilibrium state and their radii are somewhat enlarged with respect to a normal main-sequence star of the same mass.

4.4. The Bound Population

After the initial encounter between the binary and the SMBH, one star remains in a bound orbit around the SMBH in all of our simulations. Like the orbit of the initial binary about the SMBH, the orbit of such a bound star is highly eccentric: 0.96 < e < 1. In those cases in which the bound star is a merger product (runs M1 through M13), the semimajor axis a very nearly equals the semimajor axis of the initial binary about the SMBH: a ≈ 1000 au, corresponding to an orbital period of about 16 years. We note that these orbital periods are comparable to those of the S-Stars in the GC. In those cases in which an HVS star is ejected (runs C1 through C5 and runs H1 through H8), the ejection energy comes at the expense of the orbital energy of the bound star, which consequently has a somewhat smaller semimajor axis: 100 au ≲ a ≲ 700 au, corresponding to orbital periods of 0.5–9 years.

4.4.1. Tidal Stripping

The most significant hydrodynamic effects occur near the periapsis, where induced collisions and mergers are most likely to occur and where tidal stripping is at its greatest.

We 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 Roche lobe overflow at periapse. As an example, consider the run C1 in which the initial stars have the dimensionless Roche lobe parameter ζ = 0.67. As expected for ζ < 1, the bound star does indeed lose mass each time it sweeps past the black hole. The gradual decrease of the mass M of the bound star can be seen in the top panels of Figure 13. The mass δM lost per orbit, shown by the star symbols in the top panel, increases with each orbit, until after 96 periapsis passages the star has been completely disrupted. We note from the bottom panel that the periapsis separation r_per = a(1 − e) is nearly unchanged during this entire process: in this and other cases, we find the bound star returns to the nearly same relative separation from the black hole regardless of mass loss. The apoapsis separation d = a(1 + e) is also somewhat constant, although it decreases at late times when the mass loss is greatest and strong tidal effects remove energy from the orbit.

As another example, the lower panels of Figure 13 give the evolution of the merger remnant formed in run M5. In this case, most of the mass loss occurs during the first periapsis passages where very high entropy material is removed from the outer layers of the star that responds by reducing its radius. In the following evolution, mass loss essentially ceases. We note that merger products have a very non-uniform density profile characterized by a extended low-density envelope and a dense central region. Subsequent passages of the star by the SMBH will therefore cause the depletion of the outermost stellar region, unveiling its hot central core. An example of this phenomenon is shown in Figure 14, where we plot column density plots for the merger remnant of run M7. After about 20 orbits mass loss stops and the envelope has been completely removed. Similar mechanisms, involving tidal stripping suffered by late-type giants during close passages around a intermediate massive black hole, have been invoked in the past (Miocchi 2007) to explain the extreme horizontal-branch stars observed in some globular clusters (Rich et al. 1997).

**Figure 14.** Evolution of the merger remnant formed in run M7 after each periapsis passage when the star sets to hydrostatic equilibrium and until mass loss ceases. Time increases from left to right and form top to bottom. Initially, the merger remnant has a large low-density envelope that is completely removed after several obits around the SMBH.
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We find that the dimensionless parameter ζ is strongly correlated with whether and how quickly a bound star loses mass through Roche lobe overflow. For example in run H2, the bound star has ζ > 1 and does not experience a collision or merger that would change ζ: it consequently continues to orbit the SMBH without ever suffering any mass loss. In all the cases with ζ < 1, the bound star is ultimately destroyed after repeated episodes of Roche lobe overflow, with smaller values of ζ generally corresponding to fewer orbits before disruption. Figure 15 shows that the number N_p of periapsis passages before disruption grows exponentially with the initial ζ. In addition, this number of passages depends only weakly on whether the interaction type is a collision (red crosses), merger (black triangles), or clean ejection of an HVS (blue circles).

The rightmost data point in Figure 15, corresponding to run C2, deserves some discussion. In this case, ζ₁ = ζ₂ = 1.19 > 1, so that neither binary component would lose mass if it were not for the collision induced on the first periapsis passage. This collision both increases the radius and slightly decreases the mass of the bound star, effectively decreasing its ζ parameter to a value below 1. Thus, on the subsequent passage past the black hole, the star loses more mass, now due to Roche lobe overflow. The response of this particular star to mass loss is that its radius remains roughly constant. From Equation (7), the ζ parameter then stays below 1 and slowly decreases as the mass ratio q decreases with each successive passage. Ultimately, after nearly 600 periapsis passages, the star is completely pulled apart.

Similarly to the stars from run C2, the 6 M_☉ star in run M13 has ζ = 1.22 > 1. This star indeed makes the first periapsis passage without immediately losing any mass; however, while the binary recedes away from the SMBH, the merger causes mass ejection. The resulting 6.84 M_☉ merger product is large enough that ζ drops below 1, and, on the subsequent periapsis passages, mass is lost through Roche lobe overflow. As a result of shedding its high entropy outer layers, the merger product shrinks sufficiently that ζ is pushed back toward ζ ≈ 1. By comparing runs C2 and M13, we conclude that the fate of binary stars with ζ ≳ 1 depends not simply on the initial values of ζ but also on the type of their interaction and the response of the bound star to mass loss.

In several of our simulations, merger products formed from stars with ζ > 1 are large enough that ζ drops below 1 and at least some mass is lost due to Roche lobe overflow on the second and later periapsis passages (runs M1, M3, M4, M5, M7, M8, M9, M11, M12, and M13). Because shock heating is preferentially distributed to the outer layers of a merger product (Lombardi et al. 2002), this Roche lobe overflow always strips away very high entropy material and the product responds by decreasing its radius. In this way, the ζ parameter gradually approaches a value ≈1, corresponding to an eccentric semidetached binary consisting of the bound star and the SMBH. In our SPH simulations of such cases, we typically follow the dynamics for several hundred orbits, without seeing an appreciable decrease in the mass of the bound star: indeed at late times the mass loss typically fluctuates between 0 and 2 SPH particles per periapsis passage. Such situations necessarily challenge the mass resolution limit of our simulations, and it is difficult to say whether such small levels of mass loss are physically meaningful or simply a numerical artifact. In any case, in nature, thermal relaxation in the outermost layers of such a merger product would tend to retract it inside of its Roche lobe and stabilize the star against further mass loss.

To better understand the effects of the numerical resolution, we vary the number of particles used to model several of the scenarios. The results for scenario M7 are summarized in Table 5, where we list the mass, eccentricity, and semimajor axis of the bound star after 25 orbits. We find a good agreement of results at all resolutions tested and a convergence of these results as the number of particles N is increased up to the value used in this paper (≈4 × 10⁴). In particular, the final orbital data have converged to within ∼0.02%.

Table 5. Resolution Study for Scenario M7

N	M (M_☉)	e	a (au)
4,957	0.9972	1147.1	8.468
9,889	0.9973	1146.7	8.410
19,933	0.9974	1146.2	8.392
39,877	0.9973	1146.2	8.390

Notes. The particle number is given by N, while the mass, orbital eccentricity, and semimajor axis of the bound star after 25 orbits around the SMBH are given by M, e, and a, respectively.

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We also extend our resolution study to cases in which the bound star is ultimately disrupted. We find that, for various particle numbers from N ≈ 5 × 10³ up to 8 × 10⁴, the simulations of the same initial conditions all behave very similarly for at least the first ∼100 orbits around the SMBH. The top three frames of Figure 16, for example, demonstrate this consistency for the mass M, eccentricity e, and semimajor axis a of the bound star after 50 orbits. For situations in which the star orbits the SMBH more than ∼100 times, the simulations at various resolutions diverge at late times, with higher resolutions simulations in which there is a collision or merger requiring more periapsis passages to disrupt the bound star (see C2 and M6 the bottom frame of Figure 16).

In addition to the scenarios shown in Figure 16, we also study resolution effects in simulations of several other cases in which the bound star is ultimately disrupted (specifically C3, H1, H3, H4, H6, and M2), with the particle number N varying from 5 × 10³ up to 4 × 10⁴.

As with the Figure 16 data, the consistency of the results is again very good for N_p ≲ 100. For example, for case C4, each of four such simulations predict that it would take somewhere in the range of 89–91 periastron passages to completely disrupt the bound star. Furthermore, for case C3 all simulations predict that it takes 8 periastron passages to disrupt the bound star, in case H4 all simulations predict that it takes 10 passages, and in case H6 all simulations predict that it takes 2 passages. We conclude that the particle number employed in this paper is sufficient to model accurately the evolution for at least ∼100 orbits around the SMBH.

Finally, as an illustrative example, Figure 17 gives column density plots for run M6 after 50 periapsis passages and for simulations with different numbers of particles.

4.4.2. Internal Structure

Figure 18 shows the composition profiles as a function of enclosed mass fraction m/M for the bound star in cases M4 and M13. Here, m is the mass enclosed within an isodensity surface and M is the total bound mass. The dotted curves show the profiles after two periapse passages, while the solid curves show the same profiles once the bound star has effectively reached a steady state. Mass loss experienced during multiple passages removes the outer layers of the star, decreasing the bound mass M and causing the composition profiles to shift slightly to larger enclosed mass fractions m/M.

We note that the helium profile for M13 is qualitatively similar to that of the case G merger product in Sills et al. (1997, see their Figure 2): both have a maximum helium abundance at an intermediate radius inside the star. In both case G and our M13, the strange helium profile is caused by a low-mass star sinking to the center of the collision product and displacing the helium-rich fluid outward. The stellar track for the case G product is shown in Figures 4 and 6 of Sills et al. (1997). In Figure 6, we see that, on the main sequence, the case G product is somewhat bluer and brighter than a normal main-sequence star of the same mass. In Figure 4, we see that the case G product is a little bluer and brighter than a different collision product (case J) with basically the same mass but without the dense hydrogen core. It is the increased helium content in the stellar interior that makes the opacity lower (compared to other main-sequence stars of the same mass) and thus bluer and brighter. So, by analogy, our M13 merger product would be a little bluer and brighter than a normal main-sequence star of the same mass.

The elements Li, Be, and B are potentially interesting observational indicators of the history of a star. These elements burn at temperatures of about 2.5 × 10⁶, 3.5 × 10⁶, and 5 × 10⁶ K, respectively, and therefore can exist only in thin outer layers of the parent stars. During a dynamical interaction, these elements can be removed either by ejecting the outer layers of the star or by redistributing them to an environment too hot for their long term existence. Although Be, B, and Li can exist after the first periapsis passage (see Figure 10), the effect of multiple passages is typically to remove these elements completely. Although there are cases where B still exists in the final bound star, it is always severely depleted. For example, in run M13 the B level at the surface of the final product is only ∼3% of the surface value in the 6 M_☉ parent star from which it originated.

In Table 6, we summarize some properties of the bound stars that survive in our SPH simulations (i.e., that are not ultimately disrupted by the SMBH). The "number of passages" represents the number of periapse passages before the mass loss effectively shuts off, which we defined as having two or fewer SPH particles ejected. The central hydrogen abundance is given by X_c, which always equals the central hydrogen abundance of the lowest mass parent star. We also list the effective age of the bound star based on its mass and central hydrogen abundance. We evaluate this effective age, based on how long it would take a normal star of that mass to evolve to that central hydrogen abundance using the TWIN stellar evolution code. It is known that the contraction of a merger product to the main sequence is very similar to the contraction of a pre-main-sequence star to the main sequence. In the latter case, the most important variable is the mass (for a given hydrogen abundance). In the former case, the two important variables are essentially mass and central hydrogen abundance (Sills & Lombardi 1997). In runs C5 and H2, the bound star is only a slightly perturbed version of one of the binary components, and the ages in these cases are the same 50 Myr as that component. For mergers of two 3 M_☉ stars, the effective age of the merger product is in the range of 14–22 Myr. For mergers of two 6 M_☉ stars, the effective age is in the range of 6–9 Myr. Mergers of unequal-mass stars (M12 and M13) also significantly rejuvenate a star: for example, in M13, the sinking of the 1 solar mass star to the center of the merger product essentially resets the nuclear clock to only 0.3 Myr after the zero-age main sequence (ZAMS).

Table 6. Some Properties of the Bound Stars that Survived in Our SPH Simulations

Run	Number of Passages	M	e	a	J	X_c	Effective Age	T_c	U	t_thermal
		(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

Notes. "Number of passages" gives the number of SMBH–star encounters before mass loss ceases. The mass, orbital eccentricity, and semimajor axis of the star are given by M, e, and a, respectively. The value of J is the final dimensionless spin parameter, while X_c is the central hydrogen abundance. The corresponding effective age is also listed. In the last three columns, we give the central temperature T_c (the temperature where the density is highest), internal energy U, and thermal timescale t_thermal.

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In the last three columns of Table 6, we list the central temperature T_c, internal energy U, and thermal timescale t_thermal of surviving bound stars. The central temperature is defined as the temperature in the star where the density is highest and therefore is not always the temperature of the highest temperature SPH particle. The central temperature T_c of all the stars is large enough to sustain nuclear burning in the core. The global thermal timescale can be estimated as

$\begin{equation} t_{\rm thermal}=U/\langle L \rangle, \end{equation} \tag{ 10 }$

where 〈L〉 is a mass-weighted average of the luminosity L throughout the entire star.

To calculate L, we take advantage of the fact that the parent stars are massive enough to be fully radiative. In addition, shock heating prevents any convective zones from existing in a newly formed merger product. Thus, we obtain the luminosity L exiting a closed surface by the integral L = ∮F · da, where da is an area element on the surface and the diffusive radiative flux F = −4acT³∇T/(3κρ). Here, a is the radiation constant, c is the speed of light, and κ is the opacity. The surface integral is easily converted to a volume integral by the divergence theorem. The result, L = ∫∇ · FdV, is straightforward to estimate in SPH:

$\begin{equation} L=\sum _i \frac{m_i}{\rho _i}({\bf \nabla }\,{\cdot}\, {\bf F})_i, \end{equation} \tag{ 11 }$

where the sum is over only those particles positioned inside the surface under consideration. Because SPH calculations cannot properly resolve the photosphere, Equation (11) cannot be used to give a reliable total luminosity. However, Equation (11) does allow us to study the luminosity profile throughout the bulk of the system.

As an example, in Figure 19, we show a more detailed look at the interior of the final bound stars in runs M4 and C5. From top to bottom, we give the luminosity L, temperature T, and radius r as a function of enclosed mass m. To evaluate the luminosity profile, we use Equation (11) on each SPH particle, summing over particles of larger density and calculating the opacity κ from the OPAL tables. Our merger and collision products typically achieve a maximum luminosity in their outermost layers that is comparable to the Eddington luminosity

$\begin{equation} L_{\rm edd}=\frac{4\pi G c}{\kappa } M \sim 3.8\times 10^4\, L_{\rm \odot } \frac{M}{\,M_\odot }, \end{equation} \tag{ 12 }$

although such a high luminosity would diminish rapidly as the star contracts to the main sequence in a time t_thermal usually of order ∼0.1 Myr.

**Figure 19.** From top to bottom: luminosity L, temperature T, and radius r as a function of enclosed mass m for the final bound stars of runs M4 (upper panels) and C5 (lower panels). L, r, and m are in solar units, while T is in Kelvin.
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5. DISCUSSION

The observed rotation rates of HVSs may give important clues to their formational history. Hansen (2007) has proposed that, as a consequence of tidal locking in close binaries, HVSs ejected by the Hills mechanism should rotate systematically slower than field stars of similar spectral type. López-Morales & Bonanos (2008) found that the late B-type star HVS 8 has a rotational velocity of ∼260 km s⁻¹, more typical of single B-type stars and therefore seemingly contrary to the hypothesis of a binary origin for this star. In order to explain the observations, other ejection mechanisms have been invoked, such as ejection by a close encounter with a massive black hole binary or with a stellar black hole orbiting the GC SMBH (Yu & Tremaine 2003; Levin 2006; Sesana et al. 2006; Löckmann & Baumgardt 2008). However, it has been noted that a larger statistic would be certainly required in order for the rotation to be used as a signature for the origin of HVSs and/or S-stars (Perets 2009a). Furthermore, in this paper we have shown that there are two other potentially important ways with which the stars can somewhat increase their rotation even in the binary disruption scenario: tidal torque by the SMBH at periapsis (if the stars enter within their tidal disruption radius) and/or a collision between the two binary members.

With the help of simplifying approximations, we are able to relate the rotational parameter J calculated in our simulations to the observable rotational velocity v after that star has thermally relaxed back to the main sequence. In particular, we approximate that the star rotates rigidly, that its rotation does not drastically affect its structure, and that the rotational parameter J is conserved during relaxation. Using L = Iv/R = c₁MRv and E = −c₂GM²/R in Equation (9), we obtain J = c₁c^1/2₂v(R/(GM))^1/2. Clearly, c₁ and c₂ are simply numerical coefficients related to the moment of inertia I and total energy E of a star, respectively. For B-type main-sequence stars, we find c₁c^1/2₂ ∼ 0.04–0.05 and R/M ∼ 0.5–0.8 R_☉/M_☉ using models from the TWIN stellar evolution code. Solving for v in terms of J, we find

$\begin{equation} v\sim 1.2\times 10^4 J\;{\rm km\;s}^{-1}, \end{equation} \tag{ 13 }$

accurate to within ∼30% for most B-type main-sequence stars. Given the J values of ejected stars in our simulations (see Figure 5 and Table 4), we estimate from Equation (13) that the post-relaxation rotational velocity v can be as large as ∼400 or 500 km s⁻¹ for HVS stars (consider runs C3 and H6). We therefore conclude that the rotation of the star HVS 8, for example, is completely consistent with a binary origin.

The J values for unmerged stars in our simulations that are bound to the SMBH after the first periapsis passage indicate, via Equation (13), that the post-relaxation rotational velocity v would be typically ≲ 400 km s⁻¹ but could be as large as ∼1000 km s⁻¹. These large rotation velocities, however, correspond to stars that penetrate deeply within their tidal disruption radius (e.g., runs H6 and H8) and therefore are eventually destroyed after several orbits. Merger products obtain even larger spins after the first periapsis passage (see Table 3). However, as the merger product keeps orbiting the SMBH, J decreases as mass gets pulled off the outside of the star, where the specific angular momentum is greatest. The bound stars that survive to orbit the SMBH end our simulations with J ≲ 0.017 (see Table 6), corresponding to post-relaxation rotational velocities v ≲ 200 km s⁻¹. As our simulations began with irrotational stars, the actual final rotational velocity of a (bound) star could be larger or smaller if the parent stars had significant spin, depending on the orientation of the spin axis with respect to the external orbital plane and/or the plane on which the collision occurs. If the spin axis is aligned with the angular momentum of the external orbit, the initial spin will sum up with that acquired due to the tidal torque from the SMBH. A larger spin will also result from a collision, if the angular momentum of the inner binary is initially aligned with the spin axis of the stars. We stress here that, in general, the effect of an initial spin on the final rotation of the stars can be complicated, and for this reason we decided to ignore initial rotation and begin with binary components irrotational in the inertial frame, which allows us to more easily measure any rotation imparted during the subsequent interaction.

Deep near-IR observations of the GC show that the S-stars are B0-B9 main-sequence stars with rotational velocities similar to those of field stars of the same spectral type (Alexander 2005). Our final bound stars therefore have properties very similar to those of the S-stars: their masses qualify them as spectral type B main-sequence stars, and their post-relaxation rotational speeds are of the correct general magnitude. For example, the rotational speeds of our fastest rotators are consistent with the 220 ± 40 km s⁻¹ value for the S-star SO-2 (Ghez et al. 2003). However, we note that the orbital eccentricities of our bound stars (0.96 < e < 1) are larger than that of SO-2 (e ≈ 0.87), as similarly found in simulations by Ginsburg & Loeb (2006) and Hansen (2007).

For tidal torque from the black hole to have a significant effect on stellar rotation, the stars should enter deep into their disruption zone (i.e., r_per < r_t).

When no collision occurs between the components of a binary, the distance of closest approach of the two stars to the SMBH typically changes little due to the encounter. In such cases, a necessary condition for significant spin-up is that the binary itself be on an orbit that passes within ∼r_t of the SMBH. This implies in turn that the fractional change in orbital angular momentum with respect to the SMBH, per orbit, be of order unity. This condition is satisfied in the so-called full loss cone regime, which, in a galaxy like the Milky Way, extends inward to ∼0.2 times the SMBH influence radius, or to r ≈ 0.5 pc (e.g., Wang & Merritt 2004).³ Inside this region, which is the region of interest for the current study, evolution onto loss-cone orbits is diffusive, and most binaries would be tidally disrupted before finding themselves on orbits that intersect ∼r_t. We note however that in the "massive-perturber scenario" the apoapsis distance of the binary is of the order of a parsec (i.e., >0.5 pc) and therefore the fractional change in orbital angular momentum with respect to the SMBH, per orbit, can be of sufficient to put the binaries on a trajectory that passes within r_t of the SMBH. Even inside 0.5 pc, other dynamical processes like resonant relaxation (Rauch & Tremaine 1996), scattering from an intermediate mass black hole (Merritt et al. 2009), or perhaps eccentric instability in a disk (Madigan et al. 2009) can produce larger changes in orbital angular momentum than in the case of two-body relaxation alone. On the other hand, tidal spin-up is not expected to be very efficient because it is important only for the narrow range of periapses: $\frac{1}{2}r_{{\rm t}} \lesssim r \lesssim r_{{\rm t}}$ (for $r \lesssim \frac{1}{2}r_{{\rm t}}$ the star is fully disrupted; for r > r_t tidal torque is small). As a consequence of the previous condition, the ejected star will lose a large fraction of its mass. An observational indicator of the history of the star would be, even in this case, a deficit in the abundances of light elements (such as lithium) that can exist only in thin outer layers of the parent star and that are typically removed by the tidal interaction with the SMBH.

One of the main arguments against rejuvenation of the S-stars through merger is the apparent "normality" of their spectra (Figer 2008). We note, however, that the envelope of our merger products will not look significantly different than that of normal stars (compare the right edge of the plots in Figure 15 with the right edge of Figure 2). In fact, if the parent stars are of equal masses, the merger product will have very normal profiles throughout the star. If the parent stars are of significantly different masses, then the profiles are more peculiar and one should worry about how this affects the stellar evolution. The main effect would probably be to change the opacity and therefore shift slightly the color and luminosity. But, unless significant mixing is induced, the chemical composition of the outermost envelope will remain similar to that of the higher mass star (with the possible exception of Li, Be, and B levels: see Section 4.2).

The tidal disruption of a star passing close enough by an SMBH to enter its tidal radius produces a luminous UV/X-ray flare of radiation as the bound stellar gas falls back onto the black hole and is accreted (Rees 1988). Tidal flares are of great interest because they can probe the presence of SMBHs in galaxies with otherwise no evidence of an active nucleus and can be used to measure the mass and spin of the central black hole (Komossa et al. 2004; Gezari et al. 2008, 2009). Computations of the tidal disruption of stars have been performed by several studies in the past, with the aim of understanding the observational signatures of these events (Ulmer 1999; Bogdanović 2004; Gomboc & Čadež 2005; Lodato et al. 2009; Strabbe & Quataert 2009; Guillochon et al. 2009; Kasen & Ramirez-Ruiz 2010). We note that there are many important scenarios in our paper that have been so far almost completely ignored: (1) multiple passages by the same star, (2) merger of two stars resulting in disruption due to the increased size, and (3) partial tidal disruptions.

Predicting the radiative effects of multiple passages of a star by an SMBH is outside the scope of the present work. But, it seems likely that the light curve resulting from these repeated tidal events might show a series of small peaks, separated roughly by the orbital period, before finally producing the large peak that is observed as the "tidal disruption." Assuming the star is on a parabolic trajectory, after the star–SMBH encounter, the most bound material moves on an orbit with semimajor axis $a=\frac{1}{2}r^2_{\rm per}/R$ and returns to periapsis after a time

$\begin{eqnarray} t_0&=&\frac{2\pi r_{\rm per}^3}{\left({\it GM}_{\bullet } \right)^{1/2} (2R)^{3/2} }= 0.22 \left(\frac{r_{\rm per}}{r_{\rm t} } \right)^3 \left(\frac{R}{R_{\odot }} \right)^{3/2} \left(\frac{M}{\,M_\odot } \right)^{-1}\nonumber \\ &&\times \left(\frac{M_{\bullet }}{4 \times 10^6 \,M_\odot }\right)^{1/2} {\rm yr}, \end{eqnarray} \tag{ 14 }$

which is also the time when the flare starts, while the peak return rate occurs at t ∼ 1.5t₀ (Evans & Kochanek 1989; Li et al. 2002).

As previously discussed, merger products are very large, so it is easy to strip off lots of mass during the early periapsis passages, while at later time the mass loss often ceases. The light curve resulting from these repeated tidal events will eventually show a series of small peaks of declining intensity (see Figure 12). The result of the repeated (partial-)tidal disruption of a star with a large envelope (e.g., late-type giants) will show a similar light curve. In collisions without mergers, there is some expansion in size but it is not as dramatic as in a merger. So it is not until late times that there is significant mass overflowing the Roche lobe. The light curve will show peaks of increasing intensity until the last brightest flare produced by the full tidal disruption of the star.

In future work, one could model the material that becomes bound to the black hole more carefully. In particular, it would be interesting to identify possible signatures of interactions between the bound star and the accretion torus left behind from previous periapse passages. Quasi-periodic emission may be detected if X-ray flares arise every time the star crosses the torus plane (Dai et al. 2010).

6. SUMMARY

In this paper, we carried out hydrodynamic simulations of binary stars in orbit about the SMBH at the GC. In the N-body simulations of Paper I, we assigned physical sizes to stars based on a simple mass–radius relation and predicted which binaries would merge, i.e., undergo a collision with relative velocity less than escape velocity. The fluid simulations presented in this paper were found to be quite consistent with the N-body simulations, in the sense that when the latter predicted a stellar merger, the fluid stars typically merged as well. The merger rates presented in that paper are therefore confirmed by the present work. The principal, new findings of our work are summarized here.

1.
The central temperature of the merged stars in all our simulations is large enough that there would still be nuclear burning in the stellar core. However, mergers tend to "rejuvenate" stars, in the sense that the lower mass star sinks to the center of the merger remnant, effectively resetting the nuclear clock of the merger product to the ZAMS. Even the products of equal-mass mergers exhibit significant rejuvenation.
2.
Mass loss during collisions is generally small, but large fractional mass loss can occur when one or both stars is tidally perturbed by the SMBH; if the two stars merge, a temporarily more extended object is formed, further enhancing the mass-loss rate. When the merger product has a distance of closest approach to the SMBH smaller than its Roche limit, total disruption always occurs after repeated periapse passages. Repeated tidal flares, separated by roughly the orbital period, are predicted to precede the disruption.
3.
In stellar mergers, elements that can exist only in the outermost envelope of the stars such as Li, Be, and B are severely depleted, due primarily to tidal truncation by the SMBH in subsequent periapse passages. However, the envelopes of the merger products do not otherwise differ significantly from those of the parent stars.

We finally stress that SPH calculations neglect radiative and heat transport, and therefore can follow the system only over hydrodynamical timescales that are typically of the order of a few hours. For these reasons, in this paper, we were able to discuss the relaxed structure of merger products only qualitatively and the relaxation time only in order of magnitude. In a subsequent paper, we plan to present the results of stellar evolution calculations that can follow the evolution of the SPH merger products over much longer thermal and nuclear timescales and determine their track in a color–magnitude diagram. These calculations will allow us to compare the observable properties of our models with the properties of stars observed at the GC.

We thank E. Gaburov for his development of the GPU library used to calculate gravitational forces and energies in this paper, and S. Mikkola who wrote the ARCHAIN code. We also thank H. Perets, W. R. Brown, J. Faber, A. Gualandris, N. Ivanova, S. J. Kenyon, S. Komossa, S. Portegies Zwart, and A. Robinson for helpful discussions. Column density plots in this paper were produced using the publicly available visualization tool SPLASH (Price 2007). This work was supported by grants NNX10AF84G and NNX07AH15G from NASA and by grants AST-0807910 and AST-0821141 from the NSF.

APPENDIX: UNEQUAL-MASS BINARIES: N-BODY SIMULATIONS

In the following, we briefly present the results of new N-body simulations of unequal-mass binaries passing by Sgr A*. The integrations are performed by using the high-accuracy N-body code ARCHAIN (Mikkola & Merritt 2008, 2006). The binaries are initially placed at a distance d = 0.1–0.01 pc from the SMBH with a purely tangential velocity corresponding to periapses between the tidal disruption radius of the secondary star and r_bt. The primary star has mass M₁ = 6 M_☉, while the mass of the secondary is either M₂ = 1 M_☉ or 3 M_☉. We adopt a₀ = 0.1 au for the internal semimajor axis of the binaries. In all the simulations, the final integration time is fixed to one 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). For binaries with M₂ = 1 M_☉(3 M_☉) and initial distance d = 0.01 pc, the mean asymptotic ejection velocities of the primary and secondary stars are, respectively, v₁ ∼ 1000 km s⁻¹ (1600 km s⁻¹) and v₂ ∼ 2800 km s⁻¹ (2700 km s⁻¹). When the initial apoapsis of the binary is increased to d = 0.1 pc we found: v₁ ∼ 1350 km s⁻¹ (2180 km s⁻¹) and v₂ ∼ 3500 km s⁻¹ (3200 km s⁻¹). Table 7 gives the fraction of collisions, mergers, and HVSs (v_ej > 1000 km 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 the central black hole plays a fundamental role in determining which member is ejected. For large initial distances (i.e., d = 0.1 pc), the ejection probability is almost independent on the stellar mass, while for d = 0.01 pc, the lighter star is preferentially ejected. These results are consistent with the findings of Sari et al. (2010) that used an approximated method to study the dynamical evolution of binaries on parabolic orbits. Our simulations suggest that, in the limit that the external orbital energy of the binary goes to zero, the ejection probability becomes an independent function of the stellar mass.

Table 7. HVSs (v_ej > 1000 km s⁻¹), Collision, and Merger Frequency (%)

M₂ (M_☉)	d (pc)	HVSs (Total)	HVSs (Primary)	HVSs (Secondary)	Collisions	Mergers
1	0.01	33.6	1.29	32.3	7.74	7.10
3	0.01	52.9	25.5	27.4	9.35	8.71
1	0.1	62.5	31.3	31.2	6.77	6.45
3	0.1	59.3	29.3	30.0	11.6	11.3

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

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ABSTRACT

1. INTRODUCTION

2. THE SURVIVAL TIME OF BINARIES AT THE GALACTIC CENTER