DYNAMICAL AND EVOLUTIONARY CONSTRAINTS ON THE NATURE AND ORIGIN OF HYPERVELOCITY STARS

In the IMBH inspiral scenario (Hansen & Milosavljević 2003; Yu & Tremaine 2003), an IMBH inspirals to the GC through dynamical interactions with stars. In the late stages of the inspiral, when the IMBH is already close to the MBH in the GC (<0.01 pc or even less, depending on the IMBH mass), it may closely interact with stars and scatter them at very high velocities, thus producing HVSs (Levin 2006; Baumgardt et al. 2006; Löckmann & Baumgardt 2007; Sesana et al. 2008). Consequently, the population of HVSs ejected by an IMBH in the GC should be strongly correlated with the stellar population in the central 0.01 pc of the GC. ¹ In this scenario, the stellar type of the stars (and hence their mass) only negligibly affect the possibility of their ejection as HVSs. Therefore, any star in the close region of the MBH may become an HVS in this case.

Results of N-body simulations (Baumgardt et al. 2006; Löckmann & Baumgardt 2007; Sesana et al. 2008) show that only a fraction of the stars are ejected as HVSs, and most are ejected at lower velocities. Sesana et al. (2008) find a total number of 500 (2500) HVSs ejected for an IMBH of mass ∼5 × 10³ M_☉ (∼1.5 × 10⁴ M_☉). They find the total number of stars ejected during the inspiral to be about 10⁴ (3.5 × 10⁴). These numbers indicate that a fraction of f_HVS ∼ 0.05 (0.07) of all ejected stars during the IMBH inspiral are HVSs. Current observations infer ∼100 unbound B-type HVSs exist in the galaxy and, therefore, ∼100/f_HVS ∼ 2000 (1400) B-type stars must have existed at a distance of 0.001 (0.01) pc from the MBH during the short time of IMBH inspiral. Current observations show ∼1 (30) B-type stars at 0.001 (0.01) pc from the MBH (Eisenhauer et al. 2005; Ghez et al. 2005); less luminous B-type stars might be missed so this could get a factor of 3–8 times larger. ² One may suggest that the stellar population in the GC at the time of inspiral (up to ∼10⁸ yr ago) is significantly different than currently observed; however, the possibility of so many B-type stars existing in this small region of the GC is highly unlikely, and would require a yet unknown process for producing such overabundance of B-type stars. Given our theoretical understanding of this process, the current observations of the GC, and the inferred number of HVSs, the IMBH inspiral scenario is unlikely to be the main origin for the ejection of currently observed HVSs.

In the SBH kicks scenario (Miralda-Escudé & Gould 2000; O'Leary & Loeb 2007), SBHs in the close environment of the MBH (mostly less than 0.01 pc or even less than 0.001 pc; O'Leary & Loeb 2007) strongly interact with stars and scatter them at high velocities thus producing HVSs. Consequently, and similar to the IMBH scenario, the population of HVSs ejected by an IMBH in the GC should be strongly correlated with the stellar population in the central 0.01 pc of the GC. In this scenario, the stellar type of the stars (and hence their mass) affects the possibility of their ejection as HVSs, but not strongly for the B MS stars of 3–4 M_☉ currently observed.

Results of analytical calculations by O'Leary & Loeb (2007) indicate that only a fraction of the stars are ejected as HVSs, and most are ejected at lower velocities. They find that the ejection rate of stars at velocities lower than required for HVSs (≲800 km s⁻¹) is given by (v_ej/800 km s⁻¹)^−2.5. For the 100 unbound HVSs inferred from observations, about ∼100 × (100/800)^−2.5 ≃ 1.8 × 10⁴ B-type stars have been ejected from the central 0.01 pc with velocities greater than 100 km s⁻¹, i.e., for the lifetime of such 3–4 M_☉ stars, a star formation rate of at least ∼(2 × 10⁴)/(2 × 10⁸) = 10⁻⁴ B stars per year is required. Given that most of the stars are not scattered outside the GC region, this is only a lower limit on the required star-formation rate, and at least as many stars should have been left over in this region.

It is unlikely that regular star formation could have formed such stars so close to the MBH, given the tidal forces in this region (that would disrupt a progenitor molecular cloud). These stars might have formed continuously through a fragmentation of a gaseous disk, although so close to the MBH even such star formation would most likely be prohibited or be inefficient given the required Toomre criteria in this region (Levin 2007). Nevertheless, these stars could have formed at some larger distance such as the young stars observed at less than 0.5 pc scale stellar disk in the GC (Paumard et al. 2006) and continuously migrated close to the MBH (Levin 2007). Such a scenario would require the star formation rate in this region (most likely below 0.5 pc) to be greater than 10⁻⁴ yr⁻¹, and would also require an efficient mechanism for transferring a large fraction of these formed stars to the central 0.01 pc within short times. The relaxation time in the GC is much longer than the lifetime of such stars and, therefore, some other migration mechanism would be required (e.g., migration in a gaseous disk; Levin 2007). The central region of the GC probably contains ≲5 × 10² B-type stars similar to the observed HVSs (F. Eisenhauer 2006, private communication) and, therefore, implies a star formation rate of less than 5 × 10²/2 × 10⁸ = 2.5 × 10⁻⁶ or so B stars per year. This rate is much smaller than the minimally required rate of ∼10⁻⁴ yr⁻¹ we found above for explaining the inferred number of HVSs in this scenario. More simply put we would have expected to observe ∼10⁴ B stars in the GC region where only ∼500 are observed (or even less in the central 0.01 pc), suggesting that this scenario is unlikely to be the main origin for the ejection of currently observed HVSs, although it could explain a fraction of them.

In the binary disruption scenario (Hills 1988), binaries are disrupted by the MBH in the GC if they come closer than the tidal radius. One star is captured by the MBH where the other is ejected at high velocity thus producing HVSs. The fraction of ejected stars with velocities lower than those of HVSs is strongly dependent on the semimajor axis distribution of the binaries (where higher velocity stars are ejected from disruption of closer binaries; Hills 1991; Bromley et al. 2006), which is unknown in the GC. The velocity of an ejected star was found in numerical simulations (Hills 1988; Bromley et al. 2006) to scale as 1

$$\begin{equation} v_{\mathrm{BH}} = 892\;\mathrm{km\; s^{-1}} \left(\frac{a}{1\;\mathrm{AU}}\right)^{-1/2}\left(\frac{M_{\rm bin}}{8\,M_{\odot }}\right)^{1/3}\!\left(\frac{M_{BH}}{3.7 \times 10^{6}\;}\right)^{1/6}. \end{equation} \tag{ 1 }$$

To reproduce the high HVS velocities we consider binaries with a < 0.95 AU. These are tidally disrupted at r_t < 3.8 × 10⁻⁴ pc and eject an unbound HVS with v_BH ≳ 920 km s⁻¹, which could be observed as an HVS with velocity greater than 450 km s⁻¹ at 55 kpc from the GC, given estimated Galactic potential difference between the center and 55 kpc of v₅₅ ∼ 800 km s⁻¹ (Carlberg & Innanen 1987).

For the semimajor axis distribution of massive binary stars, which is strongly biased toward close binaries, a large fraction of all binaries (f_bin ∼ 0.3–0.9) have semimajor axis short enough (≲ AU; Garmany et al. 1980; Abt 1983; Kobulnicky & Fryer 2008) such that the binary disruption by the MBH would lead to a ejection of HVS. Therefore, given the ∼100 HVSs inferred from observations one would require ∼330/(f_bin/0.3) binaries to be disrupted. This does not constrain the stellar population from which the binaries originate (most originate from the central 10 pc of the GC where ∼10⁴ such binaries exist), but may constrain the number of captured stars (Perets et al. 2007). In each binary disruption, the companions to the ejected stars are captured by the MBH. The capture semimajor axis distance to the MBH is linearly dependent on the semimajor axis of the original stellar binary (Hills 1991), which is ≲0.02 pc for the companion of a HVS and, therefore, 100–300 such stars should be captured near the MBH during the last ∼10⁸ yr in this region. This is generally consistent with current observations of ∼100 massive B stars at less than 0.04 pc from the MBH, where the number of less massive (less luminous) B stars may be a few times larger (see Footnote ²).

Although less likely, the initial semimajor axis distribution of B MS binaries in the GC environment may behave like that of lower mass stars (Duquennoy & Mayor 1991). Most binaries with large semimajor axis would not survive for long in the central regions of the GC (<10 pc; from which most disrupted binaries originate; see Figure 1), and so only the closer binaries (about half of the primordial population) survive. In this case, the fraction of disrupted binaries that lead to ejection of HVSs (binaries closer than ≲ AU) is ∼0.16. Therefore, we obtain a total number of ∼100/0.16 = 625 stars captured by the MBH during the last ∼10⁸ yr. However, many of these captured stars would be captured at much larger distances than the companions of HVSs and would be distributed up to distances of ∼pc from the MBH; i.e., the constrain we have is of ∼625 B-type stars (more likely a few times more, if both stars are captured, and also taking into account the larger impact parameter for wider binaries) in the central pc. This is still marginally consistent with current observations in this region, but might be excluded in future observations. We conclude that the scenario of HVS ejection from binary disruption is consistent with current observations of B stars in the GC given a binary distribution of B stars binaries similar to that observed in the solar neighborhood, and marginally consistent if the binary distribution is similar to that of lower mass stars in the solar neighborhood.

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Binaries may survive for a Hubble time unless destroyed due to stellar evolutionary processes (e.g., merger or disruption due to mass transfer or mass loss) or subjected to dynamical interactions. In dense environments, the latter possibility may play an important role in the evolution of binary systems. In such environments binaries (soft binaries; Heggie 1975) may gradually evaporate due to perturbations from encounters with other single stars if

$$\begin{equation} |E|/m_{\rm bin}\sigma ^{2}<1, \end{equation} \tag{ 2 }$$

where E = −Gm₁m₂/2a is the orbital energy of a binary with component masses m₁ and m₂ and separation a, m_bin = m₁ + m₂ is the binary mass, and σ is the velocity dispersion of stars in the system. Due to the high-velocity dispersions in the GC, all but the closest (contact) binaries are soft binaries. Hard, close binaries can become harder due to interactions with other stars, i.e., become even closer. However the hardening changes the orbital energy of these binaries at a rate of less than ∼20% per relaxation time for marginally hard binaries, and even less for harder binaries (see, e.g., Equations (8)–(113) in Binney & Tremaine 1987). This would only slightly change the distribution of periods of close binaries. Given the uncertainties in the distribution of binaries in the GC this effect is negligible and does not contribute much to the processes of binary evolution in the GC. Most of the binary population in the GC is in soft binaries. The evaporation time of such binaries is given by (Binney & Tremaine 1987)

$$\begin{equation} t_{\rm evap}=\frac{m_{12}}{m}\frac{\sigma }{16\sqrt{\pi }\rho a\ln \Lambda }, \end{equation} \tag{ 3 }$$

where ρ is the stellar density, m is the typical mass of a star in this region, and ln Λ is the Coulomb logarithm. In the GC, σ is dependent on r; $\sigma \sim \sqrt{GM(<r)/r}$, where M(>r) is the enclosed mass up to distance r from the MBH. Figure 1 shows the evaporation time for binaries with different semimajor axis (10⁻²–10² AU) in the central regions in the GC, taking ρ(r) = ρ₀(r/r₀)^−α, where r₀ = 0.4 pc, ρ₀ = 1.2 × 10⁶ M_☉ pc⁻³, α = 1.4 for r < r₀, and α = 2 for r > r₀ (Genzel et al. 2003). The binary mass ratio is assumed to be 1 (m₁₂ = 2m).

Typical low mass (<3 M_☉) binaries have a log normal distribution of semimajor axis centered around ∼30 AU (Duquennoy & Mayor 1991). As can be clearly seen in Figure 1 most such binaries cannot survive for long close to the MBH. Many of the peculiar properties of stars evolved in binaries are due to their long-term evolution in such systems (Abt 1983). Since binaries close to the MBH are disrupted in very short time scales, the component stars in these binaries would become single stars, and effectively evolve as isolated stars. Consequently, peculiar stellar populations that require long-term evolution in binaries are not expected to form in these regions. As discussed earlier, the scenarios of HVS ejection by SBHs or by an IMBH are most efficient at close distances of ∼0.001–0.01 pc from the MBH and, therefore, most HVSs are ejected from these regions in these scenarios (O'Leary & Loeb 2007; Sesana et al. 2008). At such distances from the MBH the velocity dispersion is of the order of a few 10²–10³ km s⁻¹, and even the closest binaries are soft and would be disrupted in less than 10⁷ yr (see Figure 1), i.e., shorter than the MS lifetimes of most stars. Consequently, the hypervelocity binaries that were suggested as a possible signature of the IMBH inspiral scenario (Lu et al. 2007) and peculiar stellar populations evolved and observed mainly in binaries, are not expected to be ejected as HVSs in these scenarios (see also Löckmann & Baumgardt 2007). Other stellar populations include, for example, subdwarf B (sdB) stars (Maxted et al. 2001; Han et al. 2003), Am stars (see also Hansen 2007), and BY Dra stars (see Abt 1983 for a review).

In the binary disruption scenario for ejection of HVSs a different picture arises. In this case, most binaries disrupted by the MBH come from much larger distances from the MBH (≳2 pc; Perets et al. 2007) than HVSs ejected in the SBH kick or IMBH inspiral scenarios. At these distances binaries could survive longer (Figure 1). However, a HVS is ejected following the disruption of the binary, destroying the possible progenitor of any binary-evolved peculiar star. Consequently, only stars that already evolved in a binary to become peculiar prior to the disruption of the binary could be ejected as peculiar-type HVSs. Unfortunately the lifetime of many peculiar stars at this phase are usually much shorter than their lifetime on the MS (e.g., the lifetime of sdB stars is of the order of 10⁸ to 2 × 10⁸ yr; Dorman et al. 1993; whereas their progenitor MS lifetimes could be a few Gyr) and, therefore, fine tuning would be required for the ejection of HVSs in this case (i.e., they need to be ejected within the short time after they become peculiar, and before they end their life at this phase) and they would be rare. Nevertheless, if observed, they are expected to be single stars, which would be a strong signature of their binary disruption origin, since such stellar populations are expected to be and usually observed as binary stars.

Some stellar populations do not exist in binaries, or exist only in long-period binaries. In the binary disruption scenario, such stellar populations are not (or rarely) expected to be ejected as HVSs. For example, Be-type stars and A4-F2–type stars are usually observed with large semimajor axis, and their binary fraction at smaller semimajor axis (<∼ AU) is low (Abt 1983; Abt & Cardona 1984). Statistics of this type of stars in HVSs observations (or of stars very close to the MBH in the GC, where they could have been captured in the binary disruption mechanism; see, e.g., Gould & Quillen 2003; Perets et al. 2007), could give a measure of this possible signature for the HVSs origin from binary disruptions.

Recently, it was suggested that the rotational velocity of HVSs can serve as a signature for their origin (Hansen 2007). Observations show that field A- and B-type MS stars that evolve in binaries have lower average rotational velocities than isolated stars (Abt et al. 2002; Abt & Boonyarak 2004). If HVSs origin is from the binary disruption scenario, they are expected to form in binaries and, therefore, be slower rotators on average. Lower rotational velocities have been observed even for relatively young MS stars in binaries, suggesting that the low rotations are related to their formation in a binary and are not a consequence of their later evolution in that system (Abt & Boonyarak 2004). Consequently, stars formed in binaries should show this signature even if their binaries have been disrupted in a short time.

We point out that the rotational velocity distribution of stars both isolated and in binaries is very wide spread (see Figure 2) and, therefore, some statistics are required to test this signature and the rotational velocity of a single star, which cannot pinpoint to its origin as a single or a binary star (many of the stars formed in isolation are quite slow rotators, Abt & Boonyarak 2004, where as some of the binary evolved stars are very fast rotators). Using a Kolmogorov–Smirnov test we find out that ≳25 B MS HVSs are required, on average, to be able to differentiate between these distributions with a ⩾95% confidence level, if all these HVSs are taken from the same distribution (either all evolved in binaries or all evolved in isolation).

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Binaries formed close to the MBH in the GC are soft binaries and would shortly after be disrupted due to perturbing encounters with other stars. Consequently, the binary components, now single stars, should also have lower rotational velocities, on average, similar to other stars formed in binaries. Since the binary fractions of stars are high (e.g., greater than 70% for B stars in young clusters; Kobulnicky & Fryer 2008; Kouwenhoven et al. 2007), many, probably most, of the A and B MS stars in the GC are expected to have formed in binaries as slow rotators, and later on become single stars. If HVSs were ejected due to the SBH or IMBH kick scenarios, most of them are therefore expected to be relatively slower rotators. Still, a non-negligible fraction of the stars are formed as isolated stars and will possibly be faster rotators.

We can construct the rotational velocity distribution of the currently single-star populations close to the MBH (<0.01 pc). The stellar population in this region is composed of single stars formed in isolation and of single stars originally formed in binaries that have evaporated (with the appropriate fractions). The constructed rotational velocities distribution is the combination of the single stars' and binary stars' rotational velocities with the appropriate weights that depend on the binary fraction in the population. To be conservative we take a lower limit on the initial binary fraction of ∼35% (Abt 1983) where one should recall that each evaporated binary contributes two stars to the combined single-star population. In other words, any star chosen from our constructed rotational velocities distribution in the GC has a 35 · 2/(35 · 2 + 65) = 0.51 probability to originally form in a binary and, therefore, have a rotational velocity chosen from the binary stars distribution, although it is currently a single star. Again using a Kolmogorov–Smirnov test, we find out that ≳100 B MS HVSs are required, on average, to be able to differentiate between these distributions with a ⩾95% confidence level, if all these HVSs are taken from the same distribution (either all from the constructed distribution for the stellar population in the close environment of the MBH, or all evolved in binaries). Given the small number of HVSs observed and inferred to exist, such a signature for the HVSs origin is unfortunately quite weak (even weaker if a higher binary fraction is assumed).

Recently, Strom et al. (2005) and Wolff et al. (2007) have shown that the rotational velocity distribution in denser environments lack the cohort of slow rotators, thus showing very different rotational velocity distribution than field stars. Given these observations and our poor knowledge of the star formation environments in the GC (both close to the MBH and further out), it would be difficult to use the rotational velocities of HVSs as a tracer for their ejection scenario. We conclude that the rotational velocities of A and B MS HVSs are strongly dependent on the formation environment of these stars, but are most likely not good tracers for the ejection scenario of HVSs. Data on the rotational velocity distribution of stars close to the MBH and further away, may be an important clue for our understanding of their ejection mechanism, but even in that case too large statistics may be required for them to be used as a signature for the HVS ejection scenario.

Current observation of HVSs detect B-type stars of limited magnitude. Such HVSs could either be MS B stars (3–4 M_☉; possibly blue stragglers) or hot BHB stars. The velocity distribution of HVSs shows a marked asymmetry between HVSs, with many more HVSs with positive Galactocentric radial velocities than HVSs with negative ones (Brown et al. 2007b). This was suggested to infer that the observed HVSs have short lifetimes and, therefore, bound HVSs are too short lived to be observed returning with negative radial velocities (Brown et al. 2007b; Kollmeier & Gould 2007; Svensson et al. 2008; Perets et al. 2008).

If HVSs are ejected continuously, such as in the ejection scenarios of the binary disruption by a MBH or scattering by SBHs, then bound HVSs ejected at earlier times could now be observed returning with negative radial velocities. In this cases, no asymmetry in the HVSs velocity distribution should be observed (up to the escape velocity from the galaxy, above which no returning stars are expected at any time). Consequently, the observations of asymmetry may raise a grave problem for these scenarios, unless there is a special physical reason for ejecting stars with short lifetimes. One explanation could be related to the survival probability of binaries in the GC.

Hot BHB stars have been suggested to form through evolution in binaries and may have high binary fraction, similar to sdB stars (Peterson & Green 2002; Peterson et al. 2002). If this is the case, then the fast evaporation of binaries close to the MBH in the GC would exclude the formation of such stars (as well as sdB stars) and no such BHB HVS would be ejected in the SBH kick or IMBH inspiral scenarios (see the discussion in Section 3.1). In this case, hot BHB stars would be very rare in the population of HVSs, and therefore all or most of the observed HVSs will be B MS stars that naturally have short lifetimes consistent with the asymmetric velocity distribution of observed HVSs. Alternatively, even if some of the HVSs are hot BHB stars (e.g., from the binary disruption scenario, see Section 3.1), they had to be ejected only after they evolved to this stage, and therefore their propagation time as HVSs is limited to their lifetime at this phase, which is short (a few 10⁸ yr) and comparable to that of MS B stars. In both cases, an asymmetric velocity distribution of the HVSs would be expected.

HVSs could also be blue straggler stars (which would possibly give them longer propagation times, and therefore different observable velocity asymmetry). In this case, the same arguments could be introduced as for the hot BHB stars. Since the evolution of blue stragglers is also through mass transfer in binaries (or stellar collisions, however, this would not happen for an ejected star), and most if not all of the field blue stragglers are in binaries (Carney et al. 2001; note, however, that these refer to lower mass blue stragglers), we may expect to see only blue straggler HVSs that have been already ejected after they evolved to this phase. Such stars are practically indistinguishable from regular MS stars, and their lifetime at this stage is as short.

Another possibility for explaining HVSs velocity asymmetry is the case of a limited time span for the ejection of HVSs, such as expected during an IMBH inspiral in the GC; i.e., a short-lived discrete event, and not long-lived continuous process such as discussed above. In this case, stars are expected to be ejected only during the limited and relatively short timescale at which the IMBH could eject HVSs (unless several such inspiral events happened). Such a timescale could be as large as 10⁸ yr (Löckmann & Baumgardt 2007), which could marginally fit the observed ejection time span of the unbound HVSs (Figure 8 in Brown et al. 2007b) ³.

Recently, the possibility of rare massive binary encounter in dense young clusters (Gvaramadze et al. 2008) has been suggested for ejection of HVSs. Such a process is also a continuous process which should have similarly lead to a symmetric velocity distribution. However, in this scenario mostly massive stars (and hence short lifetimes) are expected to be ejected, which could explain the lack or high velocity of returning stars. However, it is not at all clear whether the necessary conditions in such young clusters exist, and whether the frequency of such rare strong encounters could, to begin with, explain the observed population of HVSs (and especially unbound HVSs; see Perets 2008 for a short discussion on this).

We conclude that the currently observed B-type HVSs are most likely MS B stars, and suggest that hot BHB HVSs could only be produced in the binary disruption scenario. However, even in the latter scenario these are not expected to be frequent.