SUPERNOVAE IN THE CENTRAL PARSEC: A MECHANISM FOR PRODUCING SPATIALLY ANISOTROPIC HYPERVELOCITY STARS

Figure 1 shows the expected number of SNe within 10⁸ yr of a star formation event per 10⁶ M_☉ of initial stellar mass and its dependence on the initial mass function (IMF) power-law index Γ (where dN(M)/dM∝M^−Γ) and the stellar mass limits. The number of SNe is assumed to be equal to the number of stars more massive than 8 M_☉, the SN limit (by 100 Myr all of these stars will have undergone SNe; Woosley et al. 2002). Observations (Kollmeier et al. 2009) and theory (Nayakshin 2006) constrain the IMF of stars close to Sgr A* to be normal or top-heavy (i.e., Γ > −2.3), so that we expect ∼1–2 × 10⁴ SNe per 10⁶ M_☉ of stars formed in the GC 100 Myr ago.

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The star formation history of the GC is poorly constrained, but observations (Blum et al. 2003) indicate that ≳ 3 × 10⁵ M_☉ of stars formed within r = 2.5 pc of Sgr A* in the last 10⁸ yr and a similar or higher mass formed between 1 Gyr and 100 Myr ago. We investigate initial stellar masses in the range ∼3–10 × 10⁵ M_☉ with a high binary fraction f_bin = 0.7–1 (e.g., Kobulnicky & Fryer 2007), giving GC SN rates ∼2 × 10⁻⁵–2 × 10⁻⁴ yr⁻¹. We use a fiducial value of 10⁻⁴ yr⁻¹ when presenting results.

Observations of isolated radio pulsars (Hobbs et al. 2005; Arzoumanian et al. 2002) suggest that many core-collapse SNe produce remnants with kick velocities

$$\begin{equation} 30 \: {\rm km \: s^{-1}} \lesssim v_{\rm k} \lesssim 1200 \: {\rm km \: s^{-1}} \end{equation} \tag{ 1 }$$

(corresponding to 1σ deviations from the two peaks in the distribution found by Arzoumanian et al. 2002). For binaries with equal mass, 10 M_☉ components and typical orbital periods 1–300 days (Raguzova & Popov 2005), the orbital velocities of the members around their common center of mass (the "internal" orbit) range between

$$\begin{equation} 50 \: {\rm km \: s^{-1}} \lesssim v_{\rm int} \lesssim 320 \: {\rm km \: s^{-1}}. \end{equation} \tag{ 2 }$$

Clearly, the kick velocities can be sufficient to unbind massive binaries in this period range.

The velocities of binaries in circular orbits around Sgr A* (the "external" orbit) follow

$$\begin{equation} v_{\rm ext} \simeq 130 R_{\rm pc}^{-1/2} \: {\rm km \: s^{-1}}, \end{equation} \tag{ 3 }$$

where R_pc is the radius of the external orbit in parsecs. These external orbital velocities are of comparable magnitude to the internal orbital velocities and their vector sum determines the subsequent orbits of ejected binary members. If the internal and external orbital velocity vectors of a secondary star are aligned when the binary is disrupted, then the secondary may be ejected from the GC as an HVS. The natal kick helps unbind the binary, but it is the combination of the internal and external orbital velocities that produces HVSs. This alignment requirement naturally results in an anisotropic distribution of HVSs, with a higher density of objects in the plane of the stellar disk. Since the disk alignment is unknown (the present-day disks are not aligned with any larger structures in the Galaxy), the observed plane in the HVS distribution could represent a fossil imprint of the orientation of a ≲ 10⁶ M_☉ stellar disk in the GC ∼ 100 Myr ago.

We use Monte Carlo simulations of binary SNe in the GC to constrain HVS production rates and possible orientations of the proposed stellar disk.

Binaries are initially distributed in a thin disk (external orbits at z = 0) extending from r_in = 0.04 pc to r_out = 2.5 pc from a 4 × 10⁶ M_☉ point mass. The radial stellar density profile is a power law with Σ∝r^−0.5. Calculations assume circular, Keplerian external orbits, appropriate for r_out ≲ 3 pc. Primary masses range between M_{p, min} = 8 and M_{p, max} = 100 M_☉ and follow a Salpeter IMF. Secondary masses are selected from M_{s, min} = 1 M_☉ − M_p. Binary mass ratios are uniformly distributed.

Following Sana et al. (2012), binary internal orbital periods are selected from the range 1–300 days, following a (logP/days)^−0.55 distribution. Internal orbital planes are randomly orientated and uncorrelated with the disk plane. We also consider the effects of binary hardening (evolution to shorter internal orbital periods), which may happen in a few tens of external orbits as binaries migrate through a gaseous disk (Baruteau et al. 2011). We examine the cases of no hardening, weak hardening (binary separation reduced by 2), and strong hardening (separation reduced by 10), with circular orbits assumed in all cases. In each case, we reject binary systems which, after hardening, would have separation smaller than the sum of the two stellar radii. The latter are calculated using Equation (2) in Gvaramadze et al. (2009).

Pre-SN evolution is governed by mass loss from the primary. We adopt the pre-explosion–zero-age main-sequence mass relation of Limongi & Chieffi (2009). SN mass loss is assumed to be instantaneous and remnant masses are taken from Limongi & Chieffi (2009). Kick velocities are drawn from the bimodal distribution of Arzoumanian et al. (2002) and follow an isotropic spatial distribution. These are added to the remnant orbital velocities. We do not consider the effects of the SN ejecta on the secondary.

For disrupted binaries, we calculate the velocities of the secondaries after they escape the remnant. For those secondaries with velocities greater than the escape velocity from the SMBH, we calculate their terminal velocities. (Real velocities would be somewhat smaller since we do not consider the extended background potential.) If the terminal velocity is greater than 500 km s⁻¹, then we label the star as an HVS.

Each calculation samples N_tot = 10⁷ SNe for each case of binary hardening. The results were divided into 20 subsamples to examine statistical variations. Statistical convergence was tested by varying N_tot, with N_tot = 10⁷ being well converged.

The probability that an SN disrupts a binary is p_disr ≳ 93%. This result is higher than earlier models (e.g., Hills 1983, where p_disr ∼ 70%–80%) because we consider SN progenitors with masses down to 8 M_☉. Low-mass progenitors lose a larger fraction of their mass during the explosion and so are more easily unbound. Hence, top-heavy IMFs give lower binary disruption rates. We ran model H0.5 with an IMF slope of 1.3 and found p_disr = 90%, with a negligible effect on the other probabilities (see below). Disruption rates are independent of external orbital radii.

The high disruption probabilities yield a fiducial SN-induced binary disruption rate for the GC Γ_disr ∼ 10⁻⁴ yr⁻¹, with uncertainties giving a range 2 × 10⁻⁵–2 × 10⁻⁴ yr⁻¹. This is comparable to the tidal disruption rates of binaries as found by Perets et al. (2007). In that model, tidal disruption rates are enhanced by rare encounters with massive perturbers, such as GMCs. We find similar disruption rates as a natural result of SNe in massive binaries without recourse to external encounters.

Although most secondary stars remain bound to the SMBH even after binary disruption, a fraction attains a large enough velocity to escape from the sphere of influence. We give the secondary star ejection probability per SN explosion in the fourth column of Table 1; it ranges from ∼4% for unhardened binaries to ∼25% for strong hardening. For moderate hardening, we find a stellar ejection rate of ∼8 × 10⁻⁶ yr⁻¹.

The production rate of unbound stars is ∼3–4 times higher for binaries with external orbital radii R ∼ 0.1–0.5 pc, where v_ext ∼ v_int, than elsewhere. Closer to the SMBH, v_ext ≫ v_int and the central potential is too deep for stars to easily escape. Further out, external velocities are too low to significantly boost the terminal velocities. Hence, our results are insensitive to the inner and outer disk boundaries.

We present the cumulative distribution of ejected star terminal velocities in Figure 2. In the top panel, we show the distributions as fractions of all ejected stars. This shows that models with stronger hardening not only eject more stars, but also eject them with higher average velocities. In the bottom panel, the distributions are convolved with a SN rate of 10⁻⁴ yr⁻¹ and the probabilities of stellar ejection. We also show the HVS production rates predicted by tidal disruption models: Yu & Tremaine (2003), who considered binary injection into low angular momentum orbits due to stellar relaxation only (dashed line labeled "YT03"); an equivalent model by Perets et al. (2007), who used a better-constrained binary distribution to find a production rate of a factor 10³ lower ("P+07, only stars"); and two models from Perets et al. (2007) involving relaxation accelerated by gravitational interactions with massive perturbers such as GMCs ("P+07, highest rate" refers to their model GMC1, which gives the highest binary disruption rate and largest number of HVSs, while "P+07, S star match" refers to model GMC2, which provides the number of S stars in the central ∼0.04 pc around Sgr A* consistent with observations). The two vertical dashed lines represent the velocity cut at 275 km s⁻¹ used by Brown et al. (2012b) to identify HVSs and the 400 km s⁻¹ Galactic escape velocity used by the same authors.³ We also give the production probabilities per SN explosion for stars with velocities higher than the two thresholds in Table 1.

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In 10⁸ yr, our models H1, H0.5, and H0.1 predict 37, 75, and 512 stars ejected with terminal velocities v_term > 275 km s⁻¹, respectively. These values are somewhat optimistic, because we do not consider deceleration while escaping the Galactic gravitational potential. The latter can be crudely approximated as ΔΦ ∼ σ²ln (R/R_infl), with σ = 100 km s⁻¹ the velocity dispersion in the Galaxy and R_infl ≃ 2 pc the radius of influence of Sgr A*. Taking R = R_Solar = 8 kpc, we find that only stars with v_term ≳ 400 km s⁻¹ have radial velocities above the cut at the solar circle. For these, our models predict 6, 19, and 109 such stars. Therefore, a binary orbit hardening by slightly more than a factor of two is enough to produce HVS numbers consistent with observations. In addition, the ratio of unbound to bound HVSs is ∼0.3 in all our models, slightly lower than the observationally derived ∼0.5 (Brown et al. 2012b).

The highest velocities we expect to find in 10⁸ yr are ∼500, ∼600, and ∼700 km s⁻¹ for models H1, H0.5, and H0.1, respectively. Accounting for the deceleration while escaping the galaxy, these velocities are reduced to ∼400, ∼530, and ∼640 km s⁻¹, respectively. Such velocities cannot explain the fastest known HVSs, for example, US708 with v_rf ≃ 750 km s⁻¹ (Hirsch et al. 2005), but the vast majority of currently known ones are moving slower than these maximal radial velocities.

Brown et al. (2012b) present observational evidence for anisotropy in the spatial distribution of HVSs (see their Figures 4 and 5). They find a strong excess of HVSs in the northern galactic hemisphere at longitudes 240° < l < 300° and a mild excess at latitudes 45° < b < 60°. Overall, their result provides a 3σ confirmation that the HVS distribution is not spherically symmetric.

We produce plots equivalent to Figure 5 from Brown et al. (2012b) in Figure 3 for models H1 and H0.5. We apply a similar a priori cut as in Brown et al. (2012b): we only consider stars with |b| > 20° and reduce the statistical weight of stars with b < 0 by a factor three. We only take into account stars that have v_term > 400 km s⁻¹; in Section 4.2 above, we showed that these stars would have v_radial ∼ 275 km s⁻¹ at the solar circle, equivalent to the velocity cut Brown et al. (2012b) used to define HVSs. We compare the HVS distribution obtained from an thin disk with an inclination i = 45° to the Galactic plane and line-of-nodes direction Ω pointing toward Galactic longitude l = 120° with an initially spherically symmetric distribution in the left panels. We find a clear anisotropy caused by the disk. The middle panels show the effects of disk inclination. Angles i ≲ 60° yield a longitude distribution similar to that observed, while i ≃ 45° produces a comparable latitude distribution. The effects of the line-of-nodes direction Ω are shown in the right panels and the longitude distribution can be seen to have a strong peak at Ω + π/2 and a weaker one at Ω − π/2. The anisotropy is weaker in the H0.5 model than in H1 because hardened binaries have larger internal orbital velocities, weakening the effect of correlated external motions (i.e., the disk). Nevertheless, anisotropy is still present in H0.5. The anisotropy would be amplified if the internal orbits aligned with the disk due to the same torques that cause binary hardening.

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Brown et al. (2012a) recently put constraints on the difference between the stellar age and the travel time from the GC for five HVSs (the "arrival time"). In our model, the arrival time is approximately equal to the main-sequence lifetime of the primary star. We plot the fraction of HVSs as a function of the lifetime of the primary star in Figure 4. The main-sequence lifetime is given by t_MS ≃ 1.2 × 10¹⁰(M/M_☉)^−2.5 yr and is a very steep function of mass. Stars with M ≃ 8 M_☉ survive for ∼70 Myr and will have been the companions of HVSs with the longest arrival times.

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We find ∼5%–10% of HVSs have companions with main-sequence lifetimes t_MS > 50 Myr, within 3σ of the arrival time results of Brown et al. (2012a). It is also worth noting that massive stars tend to have companions of similar masses (Halbwachs et al. 2003), such that the companions of the most massive stars are more likely to have undergone SNe explosions themselves, reducing the fraction of observable HVSs with short arrival times.

Baruteau et al. (2011) present an analysis of binary evolution and migration through gaseous disks. One of the findings is that the disk exerts a torque on the migrating binaries, which hardens them. Typical hardening timescales are a few tens of external orbital times. In our case, the external orbital timescale is $t_{\rm orb} \sim 7500 R_{\rm pc}^{3/2}$ yr and binaries are expected to harden on a timescale t_harden ∼ 10⁵ yr, much shorter than the lifetime of even the most massive stars. This validates our assumption of binary orbits remaining circular after hardening.

Since the hardening torque acts perpendicular to the disk plane, it also works to align the binary internal orbit with its external orbit within the disk. The effect is unlikely to be strong, as the torque direction is essentially random on length scales much smaller than the disk scale height, H ∼ 200R_pc AU for H/R = 10⁻³, but may lead to a weak correlation between internal and external orbital velocity planes. Our calculations show that even a perfect correlation does not increase the stellar ejection rates significantly, however, the spatial anisotropy of ejected star directions does become more pronounced.

Another potential binary hardening mechanism is the common envelope phase at the end of the primary's life. This phase of binary star evolution may occur when the binary separation is of the same order as the primary star's radius so that the secondary star is engulfed by the envelope of the primary. The binary orbit is then hardened as (internal) orbital energy is used in unbinding the primary's envelope. The efficiency with which orbital energy is extracted by this process is very uncertain (see, e.g., Ivanova et al. 2013) and we do not include its effects in our model. However, we note that common envelope evolution would be more frequent in a binary population with orbits hardened via migration through a disk and may play an important role in HVS production.

In our calculations, we have considered only core-collapse SN explosions. White dwarfs in binary systems can explode as Type Ia SNe; since there is no remnant in that case, the binary disappears and the secondary star can also be ejected from the GC. It is difficult to estimate the rate of these SNe, because it depends on many parameters of the system, so we cannot constrain the rate of ejection of low-mass stars. Their dynamics, however, are very similar to those presented in this paper. Although the binary components have lower masses, the orbital separations can also be lower due to one binary component being compact, therefore the internal orbital velocities should be comparable. Furthermore, since the secondary star does not need to escape the gravity of the SN remnant, it loses less orbital energy while leaving the system. Pan et al. (2013) recently interpreted the HVS US708 (Hirsch et al. 2005) as a possible former companion of a Type Ia SN progenitor.

Undisrupted binaries may remain visible as pulsars or X-ray binaries (XRBs) until the secondary star dies. Using an approximate 5% non-disruption probability (see Table 1), we estimate an XRB creation rate Γ_puls < 10⁻⁵ yr. Typical XRB lifetimes ≲ 10⁷ yr (King et al. 2001) lead us to predict a population of <100 GC XRBs at any time. This number may be significantly reduced as undisrupted binaries are likely to have eccentric orbits and possibly be prone to tidal disruption. The fate of XRBs and the remnants in general is complicated by stellar dynamics within the GC, binary evolution, and accretion physics, and we intend to examine these issues in future work.