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%astro-ph/0609046
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\newcommand{\browna}{\citet{2005ApJ...622L..33B}}
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\newcommand{\browns}{\citet{2006ApJ...640L..35B}}
\newcommand{\brownsp}{\citep{2006ApJ...640L..35B}}
\newcommand{\genzel}{\citet{2003ApJ...594..812G}}
\newcommand{\yutremaine}{\citet{2003ApJ...599.1129Y}}
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%\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. This work was supported in
part by Harvard University grants.
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