OLD-POPULATION HYPERVELOCITY STARS FROM THE GALACTIC CENTER: LIMITS FROM THE SLOAN DIGITAL SKY SURVEY

The search for hypervelocity stars (HVS) in the stellar halo was originally proposed in order to indirectly probe the Galactic center (GC) at optical wavelengths (Hills 1988). In his seminal paper, Hills first suggested that these objects would constitute a "minor impurity of metal-rich stars" in the otherwise metal-poor galactic halos that host supermassive black holes (SMBHs) at their centers. These stars, having been ejected from the GC via binary-exchange collisions with a putative SMBH at speeds in excess of Galactic escape would have provided the dynamical evidence for such an object at a time when direct detection seemed remote. Indeed, while Hills advocated proper-motion surveys at that time, he was also the first to recognize that these objects could be discovered by their velocities noting that their radial velocities (RVs) would be "hard to rationalize away" (Hills 1988). Nearly two decades after this prediction, HVS were first discovered serendipitously in spectroscopic surveys of blue stars in the Galactic halo owing to their very large RVs (Brown et al. 2005; Edelmann et al. 2005; Hirsch et al. 2005). Efficient color-selection methods for the spectroscopic targeting of blue stars in the halo have proven extremely effective to increase samples of young HVS (Brown et al. 2007a, 2007b, 2009). To date, however, no low-mass (F/G) HVS have been detected, perhaps owing to their lack of existence or perhaps simply due to the difficulty in detecting these objects in the old stellar halo.

Despite the tremendous advances in infrared instrumentation since Hills' prediction (and the associated confirmation of the SMBH at the GC), it remains extraordinarily difficult to image any but the brightest, youngest stars at the center of the Galaxy—the "S-stars" (e.g., Genzel et al. 1997; Ghez et al. 1998, 2000; Schődel 2002). The existence of these young stars is a major mystery since one would naively expect that the conditions near an SMBH are sufficiently hostile to prevent in situ star formation of any kind. A key question is whether or not these stars represent the "tip of the iceberg" in an otherwise normal star-forming region or whether star formation is fundamentally different at the GC. Gould & Quillen (2003) first suggested that blue stars were being ejected from the GC as a natural consequence of the process (presumed to be binary disruption) that bound the blue S stars to SgrA*. Because HVS are relatively rare, it was previously not feasible to amass sufficiently large numbers of spectra of old halo stars to use HVS to go beyond the Gould & Quillen (2003) prediction that there should be young, blue stars in the halo. In the era of deep, wide-field spectroscopic surveys, it is now conceivable to transform Hills' original idea to find the "impurity of metal-rich" HVS in the Galactic halo to obtain, among other things, detailed information about star formation at the GC, which is otherwise obscured from direct view, to at least a factor of 2.5 lower in mass than the young population currently being probed (Brown et al. 2005, 2007a, 2007b, 2009).

The Sloan Digital Sky Survey (SDSS) provides an exceptional database to search for HVS that would otherwise be missed from targeted surveys. The SDSS has obtained over 300,000 stellar spectra since it began operations (York et al. 2000; Gunn et al. 1998, 2006). The data are reduced by an automated pipeline that produces reliable spectral classifications and RVs. With this enormous database of stellar spectra it is already possible to detect HVS among a broad class of stars. In particular, the size of the SDSS stellar sample makes SDSS sensitive to F and G type HVS, which require a substantially different search strategy than their O and B counterparts (Kollmeier & Gould 2007). However, in contrast to the photometric survey, the SDSS RV catalog is not complete in any dimension. In this work, we examine the colors and RVs of a large sample of halo stars in order to place limits on the ejection of metal-rich F/G stars from the GC. As we will describe, we do not explicitly cut on metallicity until the final step, although our search criteria (which are based on the photometric and kinematic properties of our sample stars) are specifically formulated based on the assumption that we are searching for metal-rich stars. This means that interesting high-velocity metal-poor stars can survive the penultimate step. We take note of these even though they are ultimately excluded from our well defined final metal-rich sample.

The paper is organized as follows. In Section 2, we summarize the stellar spectra and the sample selection for stars that will be used for this analysis. In Section 3, we present limits on the ejection of old-population HVS. We compare this limit with young-population HVS in Section 4. In Section 5, we discuss candidate metal-poor HVS uncovered in our study. We present our discussion and conclusions in Section 6. Detailed derivations are put in the Appendix for the interested reader.

The sample used here is part of an ongoing study to find high-velocity stars within the SDSS (G. K. Knapp et al. 2009, in preparation). All RVs for stars acquired by the SDSS as of 2007 January 15 were extracted from the SDSS database. Selected objects were required to have RV errors not exceeding 100  km s⁻¹, as determined from the automated stellar template fitting procedure, and required to be free of pipeline flags indicating likely problems in the automated velocity determination. This yielded a sample of 291,111 objects. Velocities were converted to Galactocentric velocity assuming that the local rotation speed is 220  km s⁻¹ and that the Sun moves at (+9,+12,+7)  km s⁻¹ relative to the local standard of rest in the direction of the GC, Galactic rotation, and the north Galactic pole, respectively. All spectra with Galactocentric RVs in excess of |V_G| ⩾ 350  km s⁻¹ were examined to ensure the sample was free of catastrophic velocity errors. In Figure 1, we show the distribution of stars from this sample that match our photometric criteria (which are discussed below). We show the distributions of color, magnitude, Galactocentric RV, and velocity error to give a sense of the overall properties of the sample. The blue points in the figure will be discussed in Section 5.

Zoom In Zoom Out Reset image size

Kollmeier & Gould (2007) argued that large RV samples, such as SDSS, could probe an as yet undiscovered old population of stars ejected from the GC and would be most sensitive to stars near the main-sequence turnoff within this population. Moreover, they argued that such surveys would be more sensitive to the bound rather than unbound members of this population, simply because the bound stars accumulate over the lifetime of the Galaxy and can be seen still orbiting, while the unbound stars can only be viewed during their exit. In this section, we analyze the sensitivity of SDSS to both classes of ejectees, and we tabulate the candidates derived from this database. We present detailed derivations of our sensitivity calculation in Appendix for the interested reader.

3.1. Selection Criteria for Bound Ejectees

We begin by asking what the sensitivity of the SDSS survey is to bound turnoff stars and whether any such candidates are in the sample. Of course, this requires that we establish selection criteria that remove the vast majority of contaminants but still retain significant sensitivity to bound ejectees. We adopt the following two criteria:

$$\begin{equation} v_{r,G}> v_{\rm cut}= 400\,\, {\rm km\, s}^{-1},\qquad v_{\odot \rm -circle} < v_{\rm esc}, \end{equation} \tag{ 1 }$$

where v_r,G is the observed RV converted to the Galactic frame, $v_{\odot \rm -circle}$ is the velocity that the star has when it crosses the solar circle, and v_esc is the Galactic escape velocity, again measured at the solar circle.

The first criterion is purely observational (apart from the implied assumption of solar motion) and is self-contained. The threshold v_cut is established empirically from the observed velocity distribution (G. K. Knapp et al. 2009, in preparation). The second criterion requires that we specify both a model of the Galaxy (to determine the value of v_esc) and a model of the photometric properties of the target population (metal-rich ejectees), so that we can estimate $v_{\odot \rm -circle}$ from its observed v_r,G and from the observed fluxes in several bands.

For the Galactic model, we adopt an isothermal sphere characterized by a rotation speed v_c = 220  km s⁻¹, truncated in density at a Galactocentric radius R_b. For this model R_b = R₀exp [(v_esc/v_c)²/2 − 1] where R₀ = 8 kpc is the distance to the GC. We will adopt v_esc = 550  km s⁻¹, which implies R_b = 8.37 R₀, as a default, but will also consider other values. This model agrees well with more complex models in the literature at the solar circle. However, at larger radii, triaxiality can significantly affect the trajectories of stars and hence their velocities. Because we do not know the true potential of the Milky Way, we perform our calculations on an exceedingly simple model. We will consider two classes of metal-rich stars, "F stars," defined observationally as 0.3 < (g − i)₀ ⩽ 0.5, and "G stars," defined as 0.5 < (g − i)₀ ⩽ 0.75. We adopt for these absolute magnitudes M_g = 3.5 and M_g = 5.5, respectively, which holds for a metal-rich population. It is important to keep in mind that these assumptions apply only to the putative metal-rich ejectee population: the photometric properties of non-metal-rich-ejectee stars (including metal-poor ejectees and nonejectees), which obviously dominate the SDSS sample overall, are completely irrelevant.

From the observed extinction corrected magnitude g₀ of the star (determined from the Schlegel et al. 1998 extinction maps), we can infer its distance, $r=10^{(g_o-M_g)/5 - 2}$ kpc, and hence (from its position on the sky), the angle ϕ between the Sun and the GC, as seen from the star. Then, since the ejectee is assumed to be on a Galactocentric radial orbit,⁵ its full three-dimensional velocity is $v=|v_{r,G}{\rm{sec}}\, \phi |$, and hence its velocity at the solar circle (assuming, as is always the case for F stars in SDSS, that R < R_b) is

$$\begin{equation} v_{\odot \rm -circle} = \sqrt{(v_{r,G}{\rm{sec}}\,\phi)^2 + 2v_c^2\ln (R/R_0)}. \end{equation} \tag{ 2 }$$

It is immediately obvious that this equation places strong constraints on the observational parameter space in which the search can be conducted. For instance, if the star is fairly bright, so that r ≪ R₀, then cos ϕ>v_cut/v_esc. Since, for this geometry, cos ϕ ∼ −cos lcos b, this constraint eliminates the roughly v_cut/v_esc = 73% of the sky that is not sufficiently close to the Galactocentric (or anticentric) directions. At the opposite extreme, a bound star at R>R₀exp [(v_esc/v_cut)²/2] ∼ 2.57 R₀ cannot be probed in any direction because its velocity (and hence RV) will fall below the threshold. At intermediate distances, these two effects combine with different relative weights.

3.2. Candidate Bound Metal-rich Ejectees

3.3. Bound-ejectee Sensitivity of SDSS

To calculate the sensitivity of the SDSS spectroscopic survey, we first evaluate n_i, the number density of stars per steradian, per magnitude satisfying our color criteria, for our two classes of stars as a function of magnitude and position on the sky using the SDSS photometric catalog. This was done using 20 individual patches of sky, each with area 1 deg², and interpolating these values to obtain n_i for other parts of the sky. The fraction of ejected stars probed by a single observation of the ith star is given by (see Appendix for derivation)

$$\begin{equation} f_i= {\ln 10\over 2.5}{ r_i^3\over 4\pi R_i^2 v P_i n_i}. \end{equation} \tag{ 3 }$$

We sum Equation (3) over all stars in the SDSS spectroscopic catalog that meet our color criteria. Figure 2 shows the result for several cases. The three bold curves (blue) are for F stars under the assumption of v_esc = 550  km s⁻¹ and for three different thresholds, v_cut = 400, 410, and 420. The solid (red) curve is for F stars with v_cut = 400  km s⁻¹, v_esc = 520  km s⁻¹, while the dashed (green) curve is for G stars with v_cut = 400  km s⁻¹, v_esc = 550  km s⁻¹. The two other curves will be explained later.

Zoom In Zoom Out Reset image size

Several features are apparent from Figure 2. First, the sensitivity is peaked quite close to the escape velocity. This may seem surprising since the fraction of ejectees probed in this regime declines exponentially with the square of the velocity for R_* < R_b, and then even more steeply (see Equation (A10) in the  Appendix). However, at low v, distant (i.e., faint) stars do not contribute because their declining speeds at large R bring them below the first selection cut in Equation (1). Most distant stars gain by r³, at least for r < R₀. Second, the sensitivity falls quite steeply with increasing v_cut, approximately a factor of 2 for each 10  km s⁻¹. This is basically a consequence of the same volume effect just analyzed. The same effect also accounts for the dramatic falloff in sensitivity if the escape velocity proves lower than our default v_esc = 550  km s⁻¹. Finally, sensitivity to bound G-star ejectees is about 1/3 that of F stars.

We have found zero metal-rich bound ejectees. Let us initially adopt the extremely simple model that the number of bound F-star ejectees is uniform as a function of $v_{\odot -\rm circle}$. Integrating under the entire (upper-blue) curve in Figure 2 yields $\int d v\Gamma _{\rm thresh}^{-1} = 4.0\;\rm Myr$  km s⁻¹. However, from Figure 2, it is clear that about 75% of this sensitivity comes from only a 35  km s⁻¹ interval. The fact that there are zero detections implies that at 1 − e⁻ⁿn⁰/n! → 95% confidence, there are fewer than n = 3 expected detections. This implies an upper limit on the ejection rate

$$\begin{eqnarray} &&\fl \Gamma < {3\over \langle \Gamma _{\rm thresh}^{-1}\rangle } = 3{35\,{\rm \, {\rm km\, s}^{-1}}\over 0.75 \times 4\;{\rm Myr} \, \, {\rm km\, s}^{-1}}< 35\;{\rm Myr}^{-1} \nonumber\\&&[-6pt]\\&&[-6pt] \fl (505\,\, {\rm km\, s}^{-1}< v_{\odot \rm -circle}<540\,\, {\rm km\, s}^{-1}).\hphantom{aaaaaaaaa}\hspace{3pt}\nonumber \end{eqnarray} \tag{ 4 }$$

Since the peak of the G-star curve in Figure 2 is lower than the F-star peak by a factor of ∼2.5, the limit on G-star ejectees is weaker by about the same factor.

3.4. Unbound Ejectees

We select candidate unbound-ejected stars with exactly the criteria as bound stars (Equation (1)), except reversing the sign on the second condition, $v_{\odot \rm -circle}\ge v_{\rm esc}$, and of course demanding that they be exiting rather than entering the Galaxy. Then, following a very similar derivation to the one given for the bound case (see Equations (A3) and (A9) in Appendix for derivations of these), we immediately derive their analogs for the unbound case,

$$\begin{equation} \Gamma _{\rm thresh}^{-1} = \sum _i t_i,\qquad t_i= {\ln 10\over 5}{ r_i^3\over 4\pi R_i^2 v n_i}. \end{equation} \tag{ 5 }$$

This quantity is shown in Figure 2 as a function of $v_{\odot \rm -circle}$ for F stars (bold-dashed, magenta) and G stars (dotted, cyan) for the case v_esc = 550  km s⁻¹, v_cut = 400  km s⁻¹. Nothing prevents us from evaluating Equation (5) even below v_esc, and we display the extensions of these curves in Figure 2. Their physical meaning is that it is possible to detect bound ejectees during their first orbit, and for orbital times of order or longer than τ_MW, this is actually a more accurate representation of the detection rate than averaging over full orbits. Thus, the bound rate is really the maximum of the bound curve and extension of the unbound curve.

Again, we find no metal-rich unbound ejectees of either F or G type. Since Γ_thresh is essentially constant in the unbound regime at (0.05 Myr)⁻¹ for F stars and (0.01 Myr)⁻¹ for G stars, this lack of detections places an upper limit on the ejection of metal-rich unbound F and G stars of

$$\begin{equation} \Gamma ^F_{\rm unbound} < 60\;\rm Myr^{-1},\qquad \Gamma ^G_{\rm unbound} < 300\;\rm Myr^{-1}, \end{equation} \tag{ 6 }$$

at 95% confidence. Equation (6) is consistent with, but less stringent than, the limit derived in Equation (4).

The Brown et al. (2007a, 2007b) survey in essence covered an entire volume of the Galaxy (defined by magnitude limits) rather than a scattering of targets within the Galaxy, such as SDSS. Thus, to calculate Γ_thresh one should perform a volume integral. An argument similar to the one given in Appendix leads to the equation

$$\begin{eqnarray} &&\fl \Gamma _{\rm thresh}^{-1} = \int _{\rm area-covered} {d\sin b d l\over 4\pi }\hphantom{aaaaaaaaaaaaaaaa} \nonumber\\ &&\fl \times\int _{r_{\rm min}}^{r_{\rm max}} {d r\,r^2\over [R(r,l,b)]^2\left[v^2 - 2v_c^2\ln (R/R_0)\right]^{1/2}}, \end{eqnarray} \tag{ 7 }$$

where we have for convenience assumed the form of the Galactic potential within R < R_b. We estimate this expression for the Brown et al. (2007a, 2007b) survey by noting that it was 80% complete over an area $\Delta \Omega =5000\;\rm deg^{2}$ and that most of the sensitivity was in regions where R ∼ r, by adopting an average value for the velocity in the denominator of 〈v〉 = 600  km s⁻¹, and approximating Δr = r_max − r_min = 40 kpc. We then obtain $\Gamma _{\rm thresh}^{-1} = 80\% \times (\Delta \Omega /4\pi)(\Delta r/\langle v\rangle) = 6.4\;\rm Myr$. Hence, the four detections imply $\Gamma ^B_{\rm unbound} = 4\Gamma _{\rm thresh}= 0.625\;\rm Myr^{-1}$.

Thus, the upper limit given by Equation (6) shows that F-star ejectees are no more than 100 times more common than B-star ejectees. Since F stars themselves are about 100 times more common than B stars in the Galaxy as a whole, this result is already of some interest. Nevertheless, it would certainly be better to achieve at least a few times better sensitivity.

While our photometric criteria are tuned for metal-rich stars, they are sufficiently broad that they do not exclude metal-poor stars. We therefore find some metal-poor candidate ejectees in our sample, including five metal-poor F star ejectees and one metal-poor G star ejectee. Because they are contaminants of our selection and not targets, they do not enter into our derived limits. However, they are of interest and should be followed up, so we present them here.

5.1. Bound Metal-poor Ejectees

5.2. Unbound Metal-poor Ejectees

There is one metal-poor F star candidate that survives our selection criteria and one metal-poor G star. The proper motions for both of these stars are inconsistent with a Galactocentric origin, as we now show. Were it ejected from the GC, based on its current position, a star should exhibit a proper motion in the direction

$$\begin{equation} {\rm tan}(\theta) = {\rm tan}(l){\rm sin}(b), \end{equation} \tag{ 8 }$$

where θ is measured from Galactic north through east. The F star candidate has a measured proper motion from USNO-B ∩ SDSS (Monet et al. 2003; Pier et al. 2003; Munn et al. 2004) of 34 mas yr^-1 at 55° (with small errors)—significantly in conflict with the predicted value θ = 330°.

Unlike the F star, the G star candidate cannot be discarded based on the its proper-motion direction alone. The measured proper motion of 6 mas yr^-1 is sufficiently small that the measured direction of 183° is poorly determined, given the errors. We therefore compare the predicted and observed vector proper motions, which (unlike the case of the F star) require a distance estimate. From its color, magnitude, and metallicity, we estimate a distance of 5.0  kpc, and it is therefore expected to be moving at 41 mas yr^-1 at 244°. Since the typical Munn et al. (2004) errors are ∼4 mas yr^-1, it is extremely unlikely that this star is an unbound ejectee.

We have presented the first limit on old-population ejectees from the GC based on data obtained with the SDSS. We have shown that the ejection of F/G stars is no more than 100 times more common than O/B star ejectees. We note that this limit is obtained simply from analyzing the stellar spectra obtained by SDSS as part of its main projects, not via a survey designed to preferentially select these objects. Already, this limit suggests that the mass function at the GC is not weighted toward low mass star formation—a fact impossible to determine by direct imaging of the GC alone. It is also possible that the ejection mechanism is dramatically less effective for low-mass stars as suggested by, e.g., Perets et al. (2007). We additionally find a population of candidate ejectees that are metal-poor (see Table 1). While they do not figure into our limits, these objects may represent a population of metal-poor stars at the GC. These stars may represent a low-metallicity population at the GC. Another possibility is that our candidates are low-metallicity runaway stars ejected from the disk. Proper-motion measurements would determine whether they are indeed ejectees from the GC. Ongoing surveys, such as the second epoch of the Sloan Extension for Galactic Understanding and Exploration (SEGUE-2; Yanny et al. 2009), will determine conclusively if star formation at the GC is indeed weighted heavily toward young, massive stars.

We thank Warren Brown and Hagai Perets for early comments on this manuscript. We thank the referee whose careful report improved the paper. J.A.K. was supported by Hubble Fellowship HF-01197 and by a Carnegie-Princeton Fellowship during this work. A.G. was supported by NSF grant AST-0757888. J.A.K. and A.G. acknowledge the support of the Kavli Institute for Theoretical Physics in Santa Barbara during the completion of this work. T.C.B. acknowledges partial funding of this work from grants PHY 02-16783 and PHY 08-22648: Physics Frontiers Center/Joint Institute for Nuclear Astrophysics (JINA), awarded by the U.S. National Science Foundation. This research has made use of NASA's Astrophysics Data System, of the SIMBAD database operated at CDS, Strasbourg, France, and of the plotting and analysis package SM, written by Robert Lupton and Patricia Monger. Funding for the creation and distribution of the SDSS-I and SDSS-II Archives has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council of the United Kingdom. The SDSS Web site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are The American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, Cambridge University, Case Western Reserve University, The University of Chicago, The Chinese Academy of Sciences (LAMOST), Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, The Kavli Institute for Particle Physics and Cosmology, the Korean Scientist Group, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, The Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.

Initially, consider a spectroscopic survey that obtains RVs of all stars in a small angular area Ω over a narrow magnitude range g₀ ± Δg₀/2, and over a specified narrow range of colors. Consider a star in this sample that is ejected from the GC, with current Galactocentric RV v_r, and assume that it has an absolute magnitude (estimated from its color) M_g. The star then has distance from us $r= 10^{(g_0-M_g)/5 - 2}$ kpc, so the volume of such a survey is

$$\begin{equation} \Delta V = {\ln 10\over 5}\Delta g_0\Omega r^3. \end{equation} \tag{ A1 }$$

It then has a three-dimensional Galactocentric velocity $v = v_r{\rm{sec}}\,\phi$, where ϕ is the angle between us and the GC, as seen from the star.

As seen from the GC, the probed volume covers an area ΔA and has a thickness Δw. Of course, ΔV = ΔAΔw. A fraction ΔA/4πR² of all ejected stars will pass through this volume at some time (assuming isotropic ejection), where R is the star's Galactocentric distance, and it will spend a fraction 2Δw/vP of its time in this volume, where P is the orbital period, and where the factor of 2 arises from ingoing+outgoing radial orbits. Hence, observations of this volume will probe a fraction

$$\begin{equation} f_{\rm vol}={\Delta A\over 4\pi R^2}{2\Delta w\over vP} = {\ln 10\over 2.5}{\Delta g_0\Omega r^3\over 4\pi R^2 vP} \end{equation} \tag{ A2 }$$

of all ejected stars. Now consider that we do not observe all stars in this volume, but only one. Clearly, the fraction of ejected stars that we probe falls by a factor of N_vol, where N_vol is the number of stars that satisfy our detection criteria. That is, N_vol = nΩΔg₀, where n is the number density of stars per steradian, per magnitude, and satisfying the color criteria. Hence, the fraction of all ejected stars probed by this single observation of the ith star is

$$\begin{equation} f_i= {\ln 10\over 2.5}{ r_i^3\over 4\pi R_i^2 v P_i n_i}. \end{equation} \tag{ A3 }$$

To determine the period, we apply our adopted Galactic model, first finding that the apocenter of the orbit R_m will be

$$\begin{eqnarray} &&\fl R_m = R_*\quad (R_*\le R_b);\hphantom{aaaaaaaaaaaaaaaaa}\nonumber\\&&[-1pt]\\&&[-12pt] \fl R_m = {R_b\over 1 + \ln (R_b/R) - (v/v_c)^2/2} \quad (R_*>R_b),\nonumber \end{eqnarray} \tag{ A4 }$$

where

$$\begin{equation} R_* \equiv R\exp {(v/v_c)^2\over 2}. \end{equation} \tag{ A5 }$$

The velocity v' at any point in the orbit R' will be given by

$$\begin{eqnarray} &&\fl {1\over 2}\biggl ({v^{\prime }\over v_c}\biggr)^2 = \ln {R_*\over R^{\prime }} \quad (R^{\prime }\le R_b);\hphantom{aaa}\hspace{-1pt}\nonumber\\&&[-6pt]\\&&[-6pt] \fl {1\over 2}\biggl ({v^{\prime }\over v_c}\biggr)^2 = {R_b\over R^{\prime }}- {R_b\over R_m} \quad (R^{\prime }> R_b).\nonumber \end{eqnarray} \tag{ A6 }$$

Integrating the equation dt = dR'/v'(R') over one full period yields

$$\begin{equation} P = \sqrt{2\pi }{R_*\over v_c} \quad (R_*\le R_b), \end{equation} \tag{ A7 }$$

$$\begin{eqnarray} &&\fl P = \sqrt{2\pi }{R_*\over v_c}{\rm erfc}\,[\sqrt{\ln (R_*/R_b)}]\hphantom{aaaaaaaaaaaaaaaaa}\nonumber\\&&[-6pt]\\&&[-6pt] \fl +\,\sqrt{2}{R_m\over v_c} \biggl [\sqrt{1-x} + {\arccos \sqrt{x}\over \sqrt{x}} \biggr ] \quad (R_*>R_b),\nonumber \end{eqnarray} \tag{ A8 }$$

where x ≡ R_b/R_m.

The total sensitivity of the survey is then simply the sum f_tot ≡ ∑_if_i over all stars. If there are N_tot bound stars orbiting, then the expected number of detections will simply be N_exp = f_totN_tot. It will prove useful to express the same result as

$$\begin{equation} \Gamma _{\rm thresh}^{-1} =\tau _{\rm MW}\sum _i f_i, \end{equation} \tag{ A9 }$$

where τ_MW = 10 Gyr is the lifetime of the Milky Way. In this formulation, N_exp = Γ/Γ_thresh, where Γ is the mean bound-ejection rate averaged over the lifetime of the Galaxy.

To understand the basic properties of f_i, let us initially restrict attention to the first case, R_* ⩽ R_b. Then

$$\begin{equation} f_i= {\ln 10\over 5}(2\pi)^{-3/2} \biggr ({r_i\over R_i}\biggr)^3{\exp (-z^2/2)\over z n_i}, \qquad z\equiv {v\over v_c}. \end{equation} \tag{ A10 }$$

If we consider some typical values for relatively faint turnoff stars in SDSS (g₀ ∼ 18.5), r ∼ 10 kpc, R ∼ 16 kpc, z = 2, and $n=100\;\rm deg^{-2}$, then f_i ∼ 1.5 × 10⁻⁹. Hence, by obtaining RVs for about 10⁴ stars, one could probe a bound population of F stars ejected from the GC provided it had at least 10⁵ members. This corresponds to $\Gamma _{\rm thresh}= 10\;\rm Myr^{-1}$.