------------------------------------------------------------------------ p.tex MNRAS, Sept. 2006, submitted Date: Tue, 5 Sep 2006 09:07:36 -0400 X-Mailer: Apple Mail (2.752.2) X-Junkmail-Status: score=10/50, host=mr02.lnh.mail.rcn.net X-Junkmail-SD-Raw: score=unknown, refid=str=0001.0A0B0203.44FD752F.0048,ss=1,fgs=0, ip=207.172.4.11, so=2006-05-09 23:27:51, dmn=5.2.113/2006-07-26 X-MailScanner-Information: Please contact postmaster@aoc.nrao.edu for more information X-MailScanner: Found to be clean X-MailScanner-SpamCheck: not spam, SpamAssassin (score=1.68, required 5, autolearn=disabled, OBSCURED_EMAIL 1.68) X-MailScanner-SpamScore: s X-MailScanner-From: roleary@cfa.harvard.edu X-Spam-Status: No %astro-ph/0609046 \documentclass[useAMS,usenatbib, usegraphicx]{mn2e} \usepackage{epsfig} %\usepackage{lscape} %\usepackage{graphicx} %\usepackage{color} \usepackage{amsmath} \usepackage{amssymb} %\usepackage{amsfonts} %\usepackage{amstext} %\usepackage{amsbsy} \usepackage{natbib} %\loadbold \bibliographystyle{mn2e} %\newcommand{\tbfrac}[2]{\genfrac{}{}{0pt}{0}{#1\strut}{#2\strut}} \newcommand{\kms}{\ensuremath{{\rm km\,s}^{-1}}} \newcommand{\msun}{\ensuremath{M_{\odot}}} \newcommand{\hills}{\citet{1988Natur.331..687H}} \newcommand{\hillsp}{\citep{1988Natur.331..687H}} \newcommand{\browna}{\citet{2005ApJ...622L..33B}} \newcommand{\brownap}{\citep{2005ApJ...622L..33B}} \newcommand{\browns}{\citet{2006ApJ...640L..35B}} \newcommand{\brownsp}{\citep{2006ApJ...640L..35B}} \newcommand{\genzel}{\citet{2003ApJ...594..812G}} \newcommand{\yutremaine}{\citet{2003ApJ...599.1129Y}} \newcommand{\perets}{2006astro.ph..6443P} \newcommand{\infinity}{{\infty}} \newcommand{\apj}{ApJ} \newcommand{\apjl}{ApJ} \newcommand{\mnras}{MNRAS} \newcommand{\aj}{AJ} \newcommand{\apjs}{ApJS} \newcommand{\nat}{Nat} \newcommand{\pasj}{PASJ} \newcommand{\aap}{A\&A} \newcommand{\letter}{{paper}} \newcommand{\sag}{Sgr~A*} \newcommand{\rmd}{{\rm d}} \newcommand{\physrep}{Physics Reports} \newcommand{\aaps}{A\&AS} %\begin{document} \title[Hypervelocity Stars from the Galactic Centre]{Production of Hypervelocity Stars through Encounters with Stellar-Mass Black Holes in the Galactic Centre} \author[O'Leary \& Loeb]{Ryan M.\ O'Leary\thanks{E-mail:roleary@cfa.harvard.edu} and Abraham Loeb\thanks{E-mail:aloeb@cfa.harvard.edu}\\ Harvard-Smithsonian Center for Astrophysics, 60 Garden St., MS 10, Cambridge, MA 02138, USA\\} \begin{document} \maketitle \begin{abstract} %Newly-formed as well as old stars Stars within 0.1\,pc of the supermassive black hole \sag\ at the Galactic centre are expected to encounter a cluster of stellar-mass black holes (BHs) that have segregated to that region. Some of these stars will scatter off an orbiting BH and be kicked out of the Galactic centre with velocities up to $\sim\,2000\,$\kms. We calculate the resulting ejection rate of hypervelocity stars (HVSs) by this process under a variety of assumptions, and find it to be comparable to the tidal disruption rate of binary stars by \sag, first discussed by \hills. Under some conditions, this novel process is sufficient to account for all the B-type HVSs observed in the halo, and to dominate the production rate of all HVSs with lifetimes much less than the relaxation time-scale at a distance $\sim 2\,$pc from \sag\ ($\gtrsim 2\,$Gyr). Since HVSs are produced by at least two unavoidable processes, the statistics of HVSs could reveal bimodal velocity and mass distributions, and can constrain the distribution of BHs and stars in the innermost $0.1\,$pc around \sag. \end{abstract} \begin{keywords} Galaxy:centre--Galaxy:kinematics and dynamics--stellar dynamics \end{keywords} \section{Introduction} %%\subsection{Motivation} Hypervelocity stars (HVSs) have velocities so great ($\sim\,1000\,$\kms) that they are gravitationally unbound to the Milky Way galaxy. Since the discovery of the first HVS by \browna, six additional HVSs have been found \citep{2005Apj...634L.181E, 2005A&A...444L..61H, 2006ApJ...640L..35B, 2006ApJ...647..303B}. The age and radial velocities of all but one HVS are consistent with them originating from the Galactic centre\footnote{In the case of \citet{2005Apj...634L.181E}, the star appears to originate from the Large Magellanic Cloud.}, the most natural site for producing them (Hills 1988). As more HVSs are found, they can be used to constrain many properties of both the Galactic centre as well as the Milky-Way galaxy as a whole, providing information about the Galactic potential \citep{2005ApJ...634..344G,2006astro.ph..8159B}, the merger history of the \sag\ \citep{2005astro.ph..8193L,2006astro.ph..7455B}, as well as the mass distribution of stars near \sag. Nearly 17 years before the initial discovery by \browna, \hills\ first proposed that HVSs should populate the galaxy and provide indirect evidence for the existence of a supermassive black hole (SMBH) in the Galactic centre. \hills\ showed that when a tight stellar binary (with a separation $<0.1\,$AU) gets sufficiently close to a SMBH, it will be disrupted by its strong tidal field. Consequently, one member of the binary could be ejected from the Galactic centre with sufficient energy to escape the gravitational potential of the entire galaxy \citep{1991AJ....102..704H,2005MNRAS.363..223G}. The other binary member is expected to remain in a highly eccentric orbit around the SMBH \citep[see, e.g.,][]{2003ApJ...592..935G,2006MNRAS.368..221G}. \yutremaine\ analysed the production rate of HVSs in more detail, specifically for stars originating near the SMBH in our galaxy, \sag. The authors also corrected the initial calculation of \hills\ and accounted for the diffusion of hard binaries into the ``loss--cone'', finding the production rate of HVSs to be $\sim\,10^{-5}\,$yr$^{-1}$, nearly three orders of magnitude below the previous estimate. \yutremaine\ also looked at two additional mechanisms for producing HVSs. They found that the rate could easily be higher ($\sim\,10^{-4}\,$yr$^{-1}$) if \sag\ had a massive binary companion \citep[see also][]{2005astro.ph..8193L,2006astro.ph..7455B,2006astro.ph..4299S}, and also determined that star-star scattering near \sag\ resulted in a nearly undetectable rate, due to physical collisions among the two stars. In examining the rate estimate from \yutremaine\, one may adopt a simple mass function (MF) for stars, $\rmd n/\rmd m \propto m^{-\beta}$ to estimate the total number of B--type HVSs in the Galactic halo. Using the Salpeter MF slope $\beta = 2.35$ and a lower mass limit of $0.5\,\msun $ one expects there to be about $\sim\,80$ HVSs with masses between $3$ and $5\,\msun$, consistent with the observed rate for such stars \brownsp. However this is an overestimate. The Salpeter MF overestimates the total number of stars with mass $>\,3\,\msun$, since star formation occurs continuously over time near the Galactic centre and massive stars have short lifetimes \citep{\perets}. Accounting for continuous star formation as well as a more realistic distribution of binary parameters, \cite{\perets} found a lower total HVS rate of $5\times 10^{-7}\,$yr$^ {-1}$ yielding $\sim 1$ HVS in the entire galaxy with mass between $3$ and $5\,\msun$. The discrepancy between this calculation and the number of observed HVSs may be overcome by massive perturbers in the Galactic centre, which reduce the relaxation time and increase the rate to that observed \citep{\perets}. Nevertheless, this last estimate still assumes that all stars are initially relaxed and therefore fill the entire energy- momentum space. The lifetimes of the stars observed by \browns\ are still smaller than the enhanced relaxation time, except in the most optimistic case of \citet{\perets}. We argue then that the total number of B-type HVSs given by \cite{\perets} may still be an overestimate, and an additional source of HVSs may be required. In this \letter, we propose a novel source of HVSs in the Milky Way: in the dense stellar cusp near \sag\, {\it stars scatter off stellar-mass black holes} (BHs) that are segregated there \citep{1993ApJ...408..496M} and recoil out of the Galactic centre. The existence of stars on radial orbits originating from the Galactic centre was first proposed by \cite{2000ApJ...545..847M} as evidence for a segregated BH cluster. Here, we show that there should exist a high-velocity tail of stars, significant enough in number to account for some if not all of the HVSs observed by \browns. This \letter\ is organised as follows. In \S \ref{stars} we describe the distribution of stars and BHs around \sag. We describe in detail our assumptions and calculations in \S \ref{theory}, and present our results in \S \ref{results}. Finally, in \S \ref{discussion}, we discuss the observational implications and prospects for future work. \section{The cusp of stars and BHs around \sag} \label{stars} Using near-infrared adaptive optics imaging, \genzel\ showed that the density of stars in the innermost $\sim 2\,$pc of the Galactic centre is well-fit by a broken power-law radial profile, $\rho(R) = 1.2 \times 10^6 (R/0.4\,{\rm pc})^{-\alpha}\,\msun\,$pc$^{-3}$, where $\alpha \approx 1.4$ for $R < 0.4\,$pc and $\alpha \approx 1.8$ for $R > 0.4\,$pc, assuming that \sag\ is at a distance of $8\,$kpc. Indeed, for a dynamically relaxed stellar system, stars near a SMBH are expected to be in a cusp profile with $1.5 < \alpha < 1.75$, depending on the mass distribution of the stars (\citealp{1976ApJ...209..214B,1977ApJ...216..883B}; but see also \citealp{2006ApJ...645L.133H}). \genzel\ normalised the density profile by assuming that the total mass of stars within $1.9\,$pc of \sag\ is $ \approx 3.2\times 10^6\,\msun$, and that the mass distribution follows the same density profile as the observed stars with a constant mass--to--light ratio. Invariably, stellar evolution and mass segregation should alter the mass--to--light ratio at different radii \citep{2006ApJ...645L.133H}, and cause the actual density profile to deviate from the observed number counts of stars. Besides the observed cusp of stars, there are two additional structures found near \sag\ that suggest ongoing star formation very close to the SMBH. Within 1 arcsec ($\approx\,.04\,$pc) of \sag\ is a nearly isotropic cluster of massive B--type stars, the so-called ``S-- stars'' \citep{2005ApJ...628..246E,2005ApJ...620..744G}, whose origin remains a mystery (see \citealp{2005PhR...419...65A} for a review). Outside this cluster, at 1--3 arcsec (0.04--0.1\,pc), is at least one disk of young stars, with a stellar population distinctly younger than the S--stars \citep{2003ApJ...594..812G,2006ApJ...643.1011P}. The spectral identification of $\gtrsim 30$ post main sequence blue supergiants and Wolf-Rayet stars constrains the age to $\lesssim 8\,$Myr \citep{2006ApJ...643.1011P}. The stars have top-heavy initial mass function with $\beta \approx 1.35$ (\citealp{2006MNRAS.366.1410N,2006ApJ...643.1011P}; see also \citealp{2005MNRAS.364L..23N}). In the same observations of the disk, \citet{2006ApJ...643.1011P} found the density of OB--type stars to fall off more steeply than the observed cusp of stars, with no positive detections outside of $0.5\,$pc. Dispersed throughout the stars there should be a group of stellar- mass BHs brought there through mass segregation \citep{1993ApJ...408..496M,2000ApJ...545..847M}. Assuming that all the stars in the stellar cusp around \sag\ formed $10\,$Gyr ago, \citet{2000ApJ...545..847M} predicted that there should be $\sim 2.5 \times 10^4$ BHs within the central pc of the Milky Way. In their analysis, \citet{2000ApJ...545..847M} estimated that $1.6\%$ of the stellar mass is in BHs given an approximately Salpeter initial mass function. All BHs within $5\,$pc relax to the central pc through mass segregation, where they form an $\alpha = 1.75- 2.0$ cusp. More detailed studies support this argument. \citet{2006ApJ...645L.133H} numerically solved the time--dependant Boltzmann equations for a three mass model near \sag\ with $10\,\msun$ BHs, $1.4\,\msun$ neutron stars and $1\,\msun$ stars. They found that there should be $\approx 1.8\times 10^3$ BHs and $\approx 3\times 10^4\,\msun$ of stars within $0.1\,$pc of \sag\ in an $\alpha_ {\rm BH} \approx 2.0$ , $\alpha_{*} \approx 1.4$ cusp respectively. In addition, large $N$ Monte-Carlo simulations of relaxation and stellar evolution in the Galactic centre found the formation of a similar cusp of BHs \citep{2006astro.ph..3280F}. However, in these simulations the density of BHs at $0.1\,$pc was found to be ten times higher and the density of stars to be about fives times less than in \citet{2006ApJ...645L.133H}. \section{Theory and assumptions} \label{theory} To calculate the rate at which stars are ejected from the Galactic centre through their scattering off BHs there, we generalise the analysis of \yutremaine\ to multi--mass systems \citep[in a form similar to that of][]{1969A&A.....2..151H}. In our calculations we assume that \sag\ has a mass $M_{\rm smbh} = 3.5\times 10^6\,\msun$ and is at a distance of $8\,$kpc. \subsection{Two--body scattering} We would like to identify the conditions under which a scattering between a BH and a star would result in the ejection of a HVS. In each scattering, a star of mass $m$ and velocity $\bmath v$ undergoes a change in velocity (\citealp{1987gady.book.....B}, Eqs. 7-10a - 10b) \begin{eqnarray} \label{deltav} {\bmath \delta v} = \frac{-2m'bw^3}{G(m'+m)^2} \left(1+\frac{b^2w^4} {G^2(m'+m)^2}\right)^{-1} \hat{\bmath{w}} \nonumber\\ + \frac{2m'bw}{m'+m} \left(1+\frac{b^2w^4}{G^2(m'+m)^2}\right)^{-1} \hat{\bmath{w}}_{\perp}, \end{eqnarray} where $b$ is the impact parameter of the encounter, $w = |{\bmath w}| = |{\bmath v}-{\bmath v}'|$ is the relative velocity, $m'$ is the mass of the BH, and ${\bmath v}'$ is the BH's velocity. The direction of ${\bmath \delta v}$ is determined by $\hat{\bmath w}$ and $\hat{\bmath w}_ {\perp}$, the unit vectors along and perpendicular to $\bmath{w}$, respectively. We are seeking circumstances under which the star's velocity at infinity is greater than some threshold ejection speed $v_{\rm ej}$, \begin{equation} \label{vinf} v_{\infinity}^2 = |\bmath{v}+\bmath{\delta v}|^2 - \frac{2 G M_{\rm smbh}}{r} \geq v_{\rm ej}^2, \end{equation} where $r$ is the distance from \sag\ at which the encounter occurred. At $55\,$kpc, the distance at which many HVSs have been observed, the star would have a velocity $v_{\rm 55} \approx (v_\infinity^2-(800\,\kms)^2)^{1/2}$ \citep{1987AJ.....94..666C}. The star and BH will not physically collide so long as the impact parameter of the encounter satisfies \citep{ 2003ApJ...599.1129Y} \begin{equation} \label{collisions} b > R_{*} \sqrt{1+\frac{2G(m+m')}{R_{*}w^2}}, \end{equation} where $R_{*}$ is the radius of the star. For our calculations we determine $R_{*}$ by fitting a broken power-law to the solar metallicity stellar models of \citet{1992A&AS...96..269S}. Encounters in which the BH passes through the star at a speed much larger than the star's escape speed $(Gm/R_*)^{1/2}\sim 500~{\rm km~s^{-1}}$ do not lead to coalescence and could also result in HVSs, but in our conservative estimates we omit them from our ejection statistics. \subsection{Total rate} The population of stars near \sag\ can be described by the seven--dimensional distribution function (DF) $f_*({\bmath r}, {\bmath v}, m)$, where $m$ is the star's mass, ${\bmath v}$ is the star's velocity, and ${\bmath r}$ is the star's position relative to \sag\ which is located at ${\bmath r} = \bmath{0}$. Similarly, we describe the DF of BHs by $f_ {\rm BH}({\bmath r}', {\bmath v}', m')$. In our analysis we restrict our attention to stars scattering off BHs only, since in star--star scattering, physical collisions between the stars limit the total production rate of HVSs (Yu \& Tremaine 2003). This can be seen by looking at the ejection speed of a minimally bound star after an encounter with another star, $v_{\infinity} \approx \sqrt{2 v \delta v} \approx \sqrt{4 v G 2 m/(2 R_{*}v)} \approx 800\,\kms$, which is the minimum velocity required to get to 55\,kpc at rest. However, for a star-BH encounter $v_{\infinity} \approx \sqrt{4 v G m'/(R_{*}v)} \approx 2700\,\kms$, where $m'$ is now the mass of the BH. The probability for a test star to encounter a BH at an impact parameter $b$ within an infinitesimal time interval $\rmd t$ is \citep{1960AnAp...23..467H,2003ApJ...599.1129Y} \begin{equation} \Gamma({\bmath r}, {\bmath v}) \rmd t= \rmd t \int b\,\rmd b \int w\, \rmd^3{\bmath v}' \int \rmd{\Psi} \int \rmd m'\, f_{\rm BH}({\bmath r}, {\bmath v}', m'), \end{equation} where $m'$ is the BH mass, and $\Psi$ is the angle between the $({\bmath v}, {\bmath v}')$-plane and $({\bmath v} - {\bmath v'}, {\bmath \delta v})$-plane. The integrated rate at which all stars undergo such encounters in a small volume, $\rmd^3{\bmath r}$, around position ${\bmath r}$ is \begin{equation} \mathcal{R}({\bmath r}) = \int \rmd^3 {\bmath v}\int \rmd m\,f_* (\bmath r, \bmath v, m)\,\Gamma({\bmath r}, {\bmath v}). \end{equation} Finally, the total rate of ejecting stars with velocity $\geq v_{\rm ej}$ is \begin{equation} \label{finalrate} \frac{\rmd N_{\rm ej}}{\rmd t} = \int \rmd^3{\bmath r} \mathcal{R} ({\bmath r}), \end{equation} where we limit the integration over $b$, $w$, and $\Psi$ such that $v_{\infinity} > v_{\rm ej}$, as in Equation~(\ref{vinf}), and exclude collisions (Eq.~[\ref{collisions}]). In order to find the total rate of HVS we integrate Equation~(\ref{finalrate}) from some minimum radius $r_{\rm min}$ out to $0.1\,$pc using a multidimensional Monte-Carlo numerical integrator. Because the DF of the stars, $f_{\rm *}$, is not continuous, other integration routines are unsuitable for our calculations. We integrate all of our results until they converge to a residual statistical error of $< 10\,\%$. \subsection{Distribution functions} Unfortunately, there are no observations to constrain the expected number or distribution of BHs near \sag. For simplicity, we assume that all BHs have a mass of $10\,\msun$, are distributed isotropically, and follow an $\alpha_{\rm BH}$ cusp density profile so that \begin{equation} \label{bhdist} f_{\rm BH} \propto E^{\alpha_{\rm BH} - 1.5} \delta(m' - 10\,\msun), \end{equation} where $E= (G M_{\rm smbh} / r - v'^2/2)$ is the negative specific energy of a BH and $\alpha_{\rm BH} = 2$. Equation~(\ref{bhdist}) is normalised so that there are $N_{\rm BHs} = 1800$ BHs within $0.1\,$pc of \sag, consistent with the calculations of \cite{2000ApJ...545..847M} and \cite{2006ApJ...645L.133H}. The stellar DF, on the other hand, should be much shallower than the BH's within $\sim\,0.1\,$pc of \sag. Observations as well as the calculations of \cite{2006ApJ...645L.133H} suggest $\alpha_{\rm stars} = 1.4$ in this region (see \S \ref{stars}). This gives \begin{equation} \label{stardist} f_{*} \propto E^{-0.1} f(r,m) \frac{\rmd n}{\rmd m} , \end{equation} where $\rmd n / \rmd m \propto m^{-\beta}$ is the current mass function of stars, and $f(r,m)$ is a function used to account for the effects of relaxation. We truncate Equation~(\ref{stardist}) so that $f_{*} = 0 $ for $E > G m'/R_{*}$. Stars would need to undergo physical collisions with the BHs, which are rare, in order to relax to a higher specific energy \citep{1976ApJ...209..214B}. In our calculations we normalise Equation~(\ref{stardist}) by requiring that the total mass in stars for $r < 0.1\,$pc be equal to $6\times10^4\,\msun$ \citep{2003ApJ...594..812G}, where we set $f(r,m)=1$. We look at three different MFs in order to determine how the number of HVSs may depend on the mass distribution in the cusp. In our `conservative' model, we set $\beta =2.35$ and $m_{\rm min} = 0.5\,\msun$. We also look at two top-heavy MFs, `heavy' ($\beta = 2.35$, $m_{\rm min}= 2\,\msun$) and `flat' ($\beta= 1.35$, $m_{\rm min}=2\,\msun$). In our model of the stellar DF, we set \begin{equation} \label{frm} f(r,m) = \begin{cases} 1 & r > 0.04\,{\rm pc},\\ 1 & t_{\rm MS}(m) > t_{\rm r}, \\ \frac{t_{\rm MS}(m)}{t_{\rm r}} e^{t_{\rm MS}(m)/t_{\rm r}-1} & {\rm otherwise,} \end{cases} \end{equation} where $t_{\rm r}=800\,$Myr is the non-resonant relaxation time-scale at $r = 0.04\,$pc (assuming that it is dominated by BHs), and $t_{\rm MS}(m)$ is the main sequence lifetime of a star of mass $m$. We fit a broken power-law for $t_{\rm MS}(m)$ based on the solar metallicity stellar models of \citet{1992A&AS...96..269S}. Equation~(\ref{frm}) is based on both observations and theory. Since we observe young stars down to $\sim 0.04\,$pc of \sag, we simply assume that $f(r,m) = 1$ for $r > 0.04\,$pc \citep{2006ApJ...643.1011P}. We note that although many of these stars appear to be in a disk, this disk reflects just the latest episode of star formation in the long history of the Galactic centre. Presumably, similar episodes have happened in the past, forming disks around \sag\ with no preferred orientation (such as in the possible second counter- rotating disk near \sag; \citealp{2006ApJ...643.1011P}). For $r < 0.04\,$pc, only relaxation and strong encounters can bring the stars to smaller radii. Therefore, we compare the lifetime of a star $t_{\rm MS}(m)$ to the relaxation time at $0.04\,$pc, $t_{\rm r} \approx 800\,$Myr \citep[see, e.g.,][]{2006ApJ...645.1152H,2006ApJ...645L.133H}. In order to determine the distribution of unrelaxed stars, we interpret the calculation of \citet{2006ApJ...645.1152H}. Their Figure~(1) suggests that the density distribution of an unrelaxed system is proportional to the final expected cusp profile. In addition, the density of stars seems to grow exponentially, which we incorporate %model as $\propto t/t_{\rm r}e^{t/t_{\rm r} - 1}$ as in Equation~(\ref{frm}). This approximation is, again, conservative. It is based on the relaxation of the entire density cusp, which spans distances of over four orders of magnitudes, whereas we are interested in only about one order of magnitude of migration (see \S~\ref{results}). In addition, it is consistent with the expected number of stars with mass $m > 3.5\,\msun$ in the central $1$ arcsec of \sag\ \citep[][see their Table~2]{2003ApJ...594..812G} \section{Results} \label{results} \begin{figure} \begin{center} \includegraphics[width=\columnwidth]{f1.ps} \end{center} \caption{\label{velocity} The total rate of stars with ejected velocities $v_{\infinity} > v_{\rm ej}$ versus $v_{\rm ej}$ for $r_{\rm min} = .001\,$pc. The coloured curves from top to bottom are for our conservative (black), heavy (blue) and flat (red) MFs. The dashed curves are for the same MFs but include all encounters that result in a collision. The total rate of HVSs with masses between $3$ and $5\,\msun$ is also plotted including (`dash-dotted') and excluding (`dotted') collisions. In this case we plot only one MF (blue) to avoid confusion. The top axis is labelled to show the corresponding velocity at a galactocentric radius of $55\,$kpc, $v_{\rm 55}$.} \end{figure} \begin{figure} \begin{center} \includegraphics[width=\columnwidth]{f2.ps} \end{center} \caption{\label{massdep} The total rate of HVSs ($v_{\rm ej} = 1000\,\kms$) in $0.5\,\msun$ intervals. As in Figure~\ref{velocity}, the black solid curve is for our conservative MF with $\beta = 2.35$ and $m_{\rm min} = 0.5\,\msun$ and the red curve is for our flat MF with $\beta = 1.35$ and $m_{\rm min} = 2\,\msun$. The `dot-dashed' curves are for the same MFs but assuming that the cusp is fully relaxed (i.e., $f(r,m)=1$). The top and bottom black `dotted' curves are for a power-law with slope -3.3, and -7 respectively for comparison.} \end{figure} In our results, we consider a star to be a HVS when $v_{\infinity} > 1000\,\kms$ ($v_{\rm ej} = 1000\,\kms$). In Figure~\ref{velocity}, we show how the total ejection rate depends on the minimum ejection velocity, $v_{\rm ej}$, with $r_{\rm min} = 0.001\,$pc. For small ejection velocities ($v_{\rm ej} \sim 100\,\kms$) the rate decreases roughly as a power-law $\propto v_{\rm ej}^{-2.6}$ independent of the MF. However, for velocities $v_{\rm ej} > 800\,\kms$ physical collisions between stars and BHs begin to suppress the rate of HVSs. In our conservative model ($\beta = 2.35$, $m_{\rm min} = 0.5\,\msun$), the ejection rate decreases rapidly from $1.4\times 10^{-6}\,$yr$^{-1}$ for $v_{\rm ej} = 1000\,\kms$ (with an observed velocity of $v_{\rm 55} > 600\,\kms$ at 55\,kpc from \sag) down to $2.7\times 10^{-8}\,$yr$^{-1}$ for $v_{\rm ej} = 2000\,\kms$ ($v_{\rm 55} > 1800\,\kms)$. Thus the observed distribution of HVSs produced from this mechanism should be truncated sharply around $2000~{\rm km~s^{-1}}$, in contrast to tidally disrupted binaries which easily eject HVSs with velocities exceeding $4000\,\kms$ \citep{2006MNRAS.368..221G,2006astro.ph..8159B}. The total number of HVSs strongly depends on the number of low mass stars in the inner cusp. In our conservative model, the total rate of HVSs is $1.4\times 10^{-6}\,$yr$^{-1}$, over two orders-of-magnitude greater than in the flat MF model ($5.6\times 10^{-9}\,$yr$^{-1}$). There are three causes for this: {\it (i)} Equation~(\ref{stardist}) is normalised based on the total amount of mass within $0.1\,$pc whereas the rate depends on the number density, {\it (ii)} the more massive stars are not entirely relaxed, and {\it (iii)} more massive stars get a smaller kick from encounters with BHs. Nevertheless, even though the total rate of HVSs can vary greatly due to the mass distribution of stars, the rate of higher-mass HVSs is not as strongly dependent on the MF. In Figure~\ref{massdep}, the rate of HVSs is plotted in $0.5\,\msun$ intervals, so that the total rate is the sum over all bins. Despite the different MFs ($\Delta \beta = 1$), the rate of HVSs with mass between ($m$) and ($m+ 0.5\,\msun$) drops off rapidly, and can be approximated as a power-law with slope $-7$. This slope is steeper than the number of intermediate mass stars in the inner $.01\,$pc ($\propto t_{ms} m^{-2.35}\propto m^{-4.35}$). Quite clearly, if the inner $.01\,$pc is not relaxed, as assumed here, any HVSs with mass $\gtrsim 10\,\msun$ could not have come from this mechanism, even under the most optimistic conditions. However, if the centre is assumed to be relaxed, then the slope of the power-law is $\approx -3.3$. The ejection rate depends very strongly on the extent to which stars populate the inner edge of the observed disk, as shown in Figure~\ref{minrad}. For our conservative MF with $\beta = 2.35$ and $m_{\rm min}= .5\,\msun$ the HVS ejection rate is $\approx 1\times10^{-6}\,$yr$^{-1}$ for $r_{\rm min} = 0.001\,$pc, and $2\times10^{-7}\,$yr$^{-1}$ for $r_{\rm min} = 0.01\,$pc. For $r_{\rm min} \lesssim .001\,$pc the total rate increases only logarithmically with inner radius, doubling by $10^{-4}\,$pc. However, interior to $.001\,$pc, stellar collisions may flatten the cusp of stars and cause the rate to increase even more slowly \citep{2006astro.ph..3280F}. Since \browns\ initiated a targeted search for HVSs in the Galactic halo, it is useful to forecast the number of observable HVSs that would originate from encounters with stellar-mass BHs. Current observations indicate that there are $\sim 33 \pm 17$ hypervelocity B-stars with masses between $3$--$5\,\msun$ (W. Brown, private communication). {\it Can scattered stars alone explain the abundance of the observed HVSs?} The answer is positive but not under the most conservative conditions assumed thus far, which predict only $\approx 1-2 (N_{\rm BH}/1800)$ such HVSs in the galaxy depending on the MF. However, if we assume that the B-stars are in fact fully relaxed down to 0.001\,pc ($10^{-4}\,$pc) then we get $\sim 8$ (20) HVSs, and $\sim 30$ (50) HVSs if we ignore all collisional effects. Instead, if we set $v_{\rm ej} = 900\,\kms$, the ejection velocity of HVS7 \citep{1987AJ.....94..666C,2006ApJ...647..303B}, we get $\sim 3$ HVSs with up to $\sim 40$ HVSs when we ignore collisions and assume that they are fully relaxed. Assuming that the $3$--$5\,\msun$ stars are fully relaxed is consistent with both theory and observations. If the BH density was ten times larger and the number density of stars ten times smaller (as suggested by the simulations of \citealp{2006astro.ph..3280F}), then the relaxation time-scale would be about $80\,$Myr and the expected total number of stars within 1 arcsec of \sag\ would be within constraints. Ignoring collisions is also not unreasonable. The relative speed of the stars and BHs $|{\bmath w}| \gg 500\,\kms$, is typically much larger than the surface escape velocity of the star. Since any such collision would be brief and not very luminous (unless coalescence follows at low impact speeds), it would be difficult to identify star-BH collision events in external Galactic nuclei. \begin{figure} \begin{center} \includegraphics[width=\columnwidth]{f3.ps} \end{center} \caption{\label{minrad} The HVS ejection rate ($v_{\rm ej} = 1000\,\kms$) versus the minimum radius of integration $r_{\rm min}$. The lines are coloured as in Fig.~\ref{velocity}.} \end{figure} \section{Summary and discussion} \label{discussion} We constructed a variety of models for the distribution of stars and BHs in the innermost $0.1\,$pc of the Galactic centre. Assuming that the cusp and disk of stars we observe today represent a steady state over the past $\sim\,100\,$Myr, we calculated the rate at which stars will scatter off BHs and populate the Milky Way halo. We showed that the total ejection rate of HVSs can be comparable to that produced by the tidal disruption of binaries, and may exceed it for intermediate mass stars. Our results are consistent with the total number of ejected stars from Fokker-Plank simulations by \citet{2006astro.ph..3280F}. Using the same optimistic assumptions of \citet{\perets}, i.e., assuming that the B-stars are fully relaxed, we can account for most, if not all, of the stars observed by \browns. We demonstrated that the velocity distribution as well as the mass distribution of HVSs should be truncated at high values for the BH scattering process compared to binary disruption events (see Figs. 1 and 2). Better statistics of HVS detections could therefore determine the relative significance of these two plausible channels. In our study, the ejected stars originate from the inner $\sim 0.1\,$pc near \sag, whereas the tidally disrupted binaries that produce HVSs originally come from $\gtrsim\,2\,$pc \citep{2003ApJ...599.1129Y,2006astro.ph..6443P}. The observations of the young disk of stars \citep{2006ApJ...643.1011P}, as well as the cluster of S-stars, suggest that the population of stars 0.1\,pc from \sag\ should be younger and more massive than at $2\,$pc. This is further supported by the steep drop in B-stars outside of 0.5\,pc \citep{2006ApJ...643.1011P}. In addition, binaries with B-stars ($t_{\rm ms} \lesssim 300\,$Myr) may also not be fully relaxed as has been assumed in many previous calculations, and therefore the diffusion rate of the most massive binaries into the loss-cone may be lower by orders of magnitudes. In the survey of \citet{2006ApJ...647..303B}, the observed HVSs have the same colour as blue horizontal branch stars (BHBs) and have yet to be distinguished from B-stars with high resolution spectroscopy. As \citet{2006ApJ...647..303B} pointed out, until now, no effective mechanism for ejecting such low mass stars was known since it is unlikely for them to be in a tight binary. However, the calculations shown here suggest that low mass stars, such as BHBs and their progenitors, can become HVSs with high efficiency if they are in the inner 0.1\,pc of \sag. The spectroscopic identification of one hypervelocity BHB star would be strong evidence in support of the mechanism presented here. However, the expected rate of hypervelocity BHBs depends on uncertain details of stellar evolution and goes beyond the scope of this work. There is a curious link between the number of HVSs observed by \citet{2006ApJ...647..303B} and the S-stars observed orbiting \sag. Some S-stars may be the former companions to the HVSs from tidally disrupted binaries \citep{2006MNRAS.368..221G}. Interestingly, there exists a similar connection between HVSs scattered off of the BHs and the S-stars as well, although perhaps not one--to--one. For every strong encounter that produces a HVS, there is likely another encounter which can bring the star closer to \sag, perhaps kicking out the BH instead, similar to the scenario proposed by \citet{2004ApJ...606L..21A}. In this scenario, the scattering of a small fraction of young stars from many previous disks into the inner $0.04\,$pc may result in a population similar to the S-stars (O'Leary, R. \& Loeb, A. 2006, in preparation). There is obviously considerable uncertainty in our model. The rate of diffusion of stars both close to and far from \sag\ into eccentric orbits is important to understanding both the mechanisms as well as the source of the observed B-type HVSs in the Galactic halo. In both cases, it is most likely that the stars formed on relatively circular orbits, whether in a disk around \sag\ or in an inspiralling cluster. Large scale simulations similar to those already done by \citet{2006astro.ph..3280F} can help resolve the uncertainties of relaxation, and with modifications, may be able to account for the conditions of continuous star formation over long periods of time. In our discussion we have not considered the effects of resonant relaxation \citep{1996NewA....1..149R,2006ApJ...645.1152H} near \sag, which may have two counteracting effects on our rate calculation. Resonant relaxation may flatten the cusp and deplete the number density of stars and BHs in the innermost 0.01\,pc; at the same time, it may also drive more massive stars into the same region producing more B-type HVSs \citep{2006ApJ...645.1152H}. In our analysis, we also neglected the migration of massive objects near \sag. If, as suggested by \citet{2006ApJ...641..319P}, many intermediate mass BHs (with masses $ \gtrsim 10^3\,\msun$) populate the inner pc of the Galactic centre and merge with \sag\ every $\sim 10^7$--$10^8\,$yr, then the cusp of stellar-mass BHs and stars would not regenerate fast enough to produce HVSs through BH-star encounters \citep{2006astro.ph..7455B} but instead could produce them through IMBH-star encounters \citep{2005astro.ph..8193L}. \section*{Acknowledgements} We would like to thank Reinhard Genzel for discussing his most recent results on the stars near \sag, as well as Warren Brown for helpful discussions and comments on our manuscript. 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