------------------------------------------------------------------------ ms.tex submitted to ApJ Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit X-ESAFE-STATUS: Mail clean X-ESAFE-DETAILS: Clean 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=2.68, required 5, autolearn=disabled, OBSCURED_EMAIL 1.68, SARE_OEM_OBFU 1.00) X-MailScanner-SpamScore: ss X-MailScanner-From: hagai.perets@weizmann.ac.il X-Spam-Status: No %astro-ph/0606443 %% LyX 1.4.1 created this file. For more info, see http://www.lyx.org/. %% Do not edit unless you really know what you are doing. \documentclass[english,twocolumn]{emulateapj} %\documentclass[12pt,preprint,english]{aastex} \usepackage[T1]{fontenc} \usepackage[latin1]{inputenc} \setcounter{tocdepth}{3} \usepackage{color} \usepackage{graphicx} \usepackage{amssymb} \usepackage{amsmath} \usepackage{times} \usepackage{natbib} \makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands. %% Because html converters don't know tabularnewline \providecommand{\tabularnewline}{\\} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands. %\newcommand{\Ms}{M_{\star}} %\newcommand{\Rs}{R_{\star}} %\newcommand{\Ls}{L_{\star}} \newcommand{\ns}{n_{\star}} \newcommand{\Es}{E_{\star}} %\newcommand{\Mbh}{M_{\bullet}} %\newcommand{\Mo}{M_{\odot}} %\newcommand{\Ro}{R_{\odot}} %\newcommand{\Lo}{L_{\odot}} \newcommand{\peryr}{\mathrm{yr}^{-1}} \newcommand{\yr}{\mathrm{yr}} \newcommand{\pc}{\mathrm{pc}} \newcommand{\perpc}{\mathrm{pc}^{-1}} \newcommand{\nbh}{n_{\bullet}} \newcommand{\Md}{M_{\mathrm{disk}}} \newcommand{\half}{{1\over 2}} \newcommand{\scr}{s_{\mathrm{crit}}} \shorttitle{Massive Perturber-driven interactions} \shortauthors{Perets, Hopman and Alexander} \usepackage{babel} \makeatother \begin{document} \newcommand{\Ms}{M_{\star}} \newcommand{\Rs}{R_{\star}} \newcommand{\Ls}{L_{\star}} \newcommand{\Mo}{M_{\odot}} \newcommand{\Ro}{R_{\odot}} \newcommand{\Lo}{L_{\odot}} \newcommand{\Mbh}{M_{\bullet}} \newcommand{\np}{n_{p}} \newcommand{\Np}{N_{P}} \newcommand{\Mp}{M_{p}} \newcommand{\Ns}{N_{\star}} \newcommand{\SgrA}{SgrA^{\star}} \newcommand{\rMP}{r_{\mathrm{MP}}} %\centerline{\rule{\columnwidth}{1pt}} \title{Massive perturber-driven interactions of stars with a massive black hole } \author{Hagai B. Perets, Clovis Hopman and Tal Alexander\altaffilmark{1}} \begin{abstract} We study the role of massive perturbers (MPs) in deflecting stars and binaries to almost radial ({}``loss-cone'') orbits, where they pass near the central massive black hole (MBH), interact with it at periapse $q$, and are ultimately destroyed. MPs dominate dynamical relaxation when the ratio of the 2nd moments of the MP and star mass distributions, $\mu_{2}\!\equiv\!\left.N_{p}\left\langle M_{p}^{2}\right\rangle \right/N_{\star}\left\langle M_{\star}^{2}\right\rangle $, satisfies $\mu_{2}\!\gg\!1$. The observed MPs in the nucleus of the Galaxy (giant molecular clouds and stellar clusters), and plausibly in late type galaxies generally, have $10^{2}\!\lesssim\!\mu_{2}\!\lesssim\!10^{5}$. MPs thus shorten the relaxation timescale by $10^{2-5}$ relative to 2-body relaxation by stars alone. We show this increases by $10^{1-3}$ the rate of large-$q$ interactions with the MBH, where loss-cone refilling by stellar 2-body relaxation is inefficient. We extend the Fokker-Planck loss-cone formalism to approximately account for relaxation by rare encounters with MPs. We show that binary--MBH exchanges driven by MPs may explain the origin of the young main sequence B stars that are observed very near the Galactic MBH, and may increase by orders of magnitude the ejection rate of hyper-velocity stars. We suggest that MP-driven relaxation plays an important role in the capture of stars on very tight orbits around the MBH, leading to their tidal orbital decay and disruption or to inspiral through the emission of gravitational waves from zero-eccentricity orbits. We show that loss-cone refilling by MPs leads to rapid orbital decay and coalescence of binary MBHs, thereby solving the stalling problem. \end{abstract} \keywords{black hole physics --- clusters --- galaxies: nuclei --- stars: kinematics --- giant molecular clouds } \section{Introduction} \label{s:intro} There is compelling evidence that massive black holes (MBHs) lie in the centers of all galaxies \citep{Fer+00,Geb+03,Shi+03}, including in the center of our Galaxy \citep{Eis+05,Ghe+05}. The MBH affects the dynamics and evolution of the galaxy's center as a whole (e.g. \citealt{Bah+76}) and it also strongly affects individual stars or binaries that approach it. Such close encounters, which may be extremely energetic, or involve non-gravitational interactions, or post-Newtonian effects, have been the focus of many studies (see review by \citealt{Ale05}). These processes include the destruction of stars by the MBH, either by falling whole through the event horizon, or by being first tidally disrupted and then accreted (e.g. \citealt{Ree88}); tidal scattering of stars on the MBH \citep{Ale+01b}; the capture and gradual inspiral of stars into the MBH, accompanied by the emission of gravitational waves or by tidal heating (e.g. \citealt{Ale+03b,Ale+03a}); or dynamical exchange interactions in which incoming stars or binaries energetically eject a star tightly bound to the MBH and are captured in its place very near the MBH (e.g. \citealt{Ale+04,Gou+03}). The interest in such processes is driven by their possible implications for the growth of MBHs, for the orbital decay of a MBH binary, for the detection of MBHs, for gravitational wave (GW) astronomy, as well as by observations of unusual stellar phenomena in our Galaxy, e.g. the puzzling young population of B-star very near the Galactic MBH \citep{Eis+05}, or the hyper-velocity B stars at the edge of the Galaxy \citep{Bro+05,Fue+06,Bro+06}, possibly ejected by 3-body interactions of a binaries with the MBH. Here we focus on close encounters with the MBH whose ultimate outcome ({}``event'') is the elimination of the incoming object from the system, whether on the short infall (dynamical) time, if the event is prompt (e.g. tidal disruption or 3-body exchange between a binary and the MBH), or on the longer inspiral time, if the event progresses via orbital decay (e.g. GW emission or tidal capture and heating). Such processes are effective only when the incoming object follows an almost zero angular momentum ({}``loss-cone'') orbit with periapse closer to the MBH than some small distance $q$. To reach the MBH, or to decay to a short period orbit, both the infall and inspiral times must be much shorter than the system's relaxation time $t_{r}$ \citep{Ale+03b}. The fraction of stars initially on loss-cone orbits is very small and they are rapidly eliminated. Subsequently, the close encounter event rate is set by the dynamical processes that refill the loss-cone. The loss-cone formalism used for estimating the event rate (\citealt{Fra+76,Lig+77,Coh+78}) usually assumes that the system is isolated and that the refilling process is 2-body relaxation. This typically leads to a low event rate, set by the long 2-body relaxation time. Two-body relaxation, which is inherent to stellar systems, ensures a minimal loss-cone refilling rate. Other, more efficient but less general refilling mechanisms were also studied with the aim of explaining various open questions (e.g. the stalling problem of MBH binary coalescence, \citealt{Ber+05,Mer+05}, or MBH feeding, \citealt{Zha+02,Mir+05}) or in the hope that they may lead to significantly higher event rates for close encounter processes. These mechanisms include chaotic orbits in triaxial potentials \citep{Nor+83,Ger+85,Mer+04b} (the presence of a MBH may however destroy the triaxiality near the center; \citealt{Mer+98,Hol+02,Sel02}); increased fraction of low angular momentum orbits in non-spherical potentials \citep{Mag+99}; accelerated resonant relaxation of angular momentum near the MBH where the orbits are Keplerian \citep{Rau+96,Rau+98,Hop+06a,Lev06}; perturbations by a massive accretion disk or an intermediate mass black hole (IMBH) companion \citep{Pol+94,Zha+02,Lev+05}. Most of these mechanisms require special circumstances to work (e.g. specific asymmetries in the potential), or are short-lived (e.g. the IMBH will eventually coalesce with the MBH). Here we explore another possibility, which is more likely to apply generally: accelerated relaxation and enhanced rates of close encounters driven by massive perturbers (MPs). Efficient relaxation by MPs were first suggested in this context by \citet{Zha+02} as a mechanism for establishing the $M_{\bullet}/\sigma$ relation \citep{Fer+00,Geb+00} by fast accretion of stars and dark matter. \citet{Zha+02} also noted the possibility of increased tidal disruption flares and accelerated MBH binary coalescence due to MPs. In this study we investigate in detail the dynamical implications of relaxation by MPs. We evaluate its effects on the different modes of close interactions with the MBH, in particular 3-body exchanges, which were not considered by \citet{Zha+02}, and apply our results to the Galactic Center (GC), where observations indicate that dynamical relaxation is very likely dominated by MPs. This paper is organized as follows. In \S\ref{s:outline} we present the main concepts and procedures of our calculations. The observational data and theoretical predictions about MPs in the inner $\sim\!100$ pc of the GC are reviewed in \S\ref{s:GC_MPs}. In section \S\ref{s:interact} we explore the implications of relaxation by MPs for various types of interactions with the MBH. We summarize our results in \S\ref{s:summary}. \section{Loss-cone refilling } \label{s:outline} In addition to stars, galaxies contain persistent dense structures% \footnote{Structures that persist at least as long as the local galactic dynamical time and are substantially denser than the ambient stellar mass distribution.% } such as molecular clouds, open clusters and globular clusters with masses up to $10^{4}$--$10^{7}\,\Mo$. Such structures can perturb stellar orbits around the MBH much faster than 2-body stellar relaxation (hereafter {}``stellar relaxation''), provided they are numerous enough. This condition can be quantified by considering a test star randomly scattered by perturbers with masses in the interval ($M_{p},M_{p}\!+\!\mathrm{d}M_{p}$) and number density $(\mathrm{d}N_{p}/\mathrm{d}M_{p})\mathrm{d}M_{p}$, approaching it with relative velocity $v$ on orbits with impact parameters in the interval ($b,b\!+\!\mathrm{d}b$).$ $ The minimal impact parameter still consistent with a small angle deflection is $b_{\min}\!=\! GM_{p}/v^{2}$ (the capture radius), where $v$ is of the order of the local velocity dispersion $\sigma$. Defining $B\!\equiv\! b/b_{\min}\!\ge\!1$, the encounter rate is then \begin{align} \left(\frac{\mathrm{d}^{2}\Gamma}{\mathrm{d}M_{p}db}\right)\mathrm{d}M_{p}\mathrm{d}b\, & \sim\left(\frac{\mathrm{d}N_{p}}{\mathrm{d}M_{p}}\right)\mathrm{d}M_{p}vb_{\min}^{2}2\pi B\mathrm{d}B\nonumber \\ & =\frac{G^{2}}{v^{3}}\left[\left(\frac{\mathrm{d}N_{p}}{\mathrm{d}M_{p}}\right)M_{p}^{2}\right]\mathrm{d}M_{p}2\pi B\mathrm{d}B\,.\end{align} The total rate is obtained by integrating over all MP masses and over all impact parameters between $b_{\min}$ and $b_{\max}$. Here we are interested in perturbations in the specific angular momentum $J$ of a star relative to the central MBH, and so $b_{\max}\!\sim\! r$, the radial distance of the star from the center. MPs with substantially larger impact parameters are much less efficient because their effect on the MBH-star pair is tidal rather then direct. The relaxation rate due to all MPs at all impact parameters is then \begin{align} t_{r}^{-1} & =\int_{b_{\min}}^{b_{\max}}\mathrm{d}b\mathrm{\int}\mathrm{d}M_{p}\left(\frac{\mathrm{d}^{2}\Gamma}{\mathrm{d}M_{p}db}\right)\nonumber \\ & \sim\log\Lambda\frac{G^{2}}{v^{3}}\int\mathrm{d}M_{p}\left(\frac{\mathrm{d}N_{p}}{\mathrm{d}M_{p}}\right)M_{p}^{2}\,,\end{align} where $\log\Lambda\!=\!\log(b_{\max}/b_{\min}$) is the Coulomb logarithm (here the dependence of $\log\Lambda$ and $v$ on $M_{p}$ is assumed to be negligible). Typically $\log$$\Lambda\!\gtrsim\!10$; the omission of large angle scattering by encounters with $b\!<\! b_{\min}$ is thus justified because it introduces only a small logarithmic correction. This formulation of the relaxation time is equivalent to its conventional definition (\citealt{Spi87}) as the time for a change of order unity in $v^{2}$ by diffusion in phase space due to scattering, $t_{r}\!\sim\! v^{2}/D(v^{2})$, where $D(v^{2})$ is the diffusion coefficient. If the stars and MPs have distinct mass scales with typical number densities \textcolor{black}{$N_{\star}$ and $N_{p}$ and rms masses} $\left\langle M_{\star}^{2}\right\rangle ^{1/2}$ and $\left\langle M_{p}^{2}\right\rangle ^{1/2}$ (\textcolor{black}{$\left\langle M^{2}\right\rangle \!\equiv\!\int M^{2}(\mathrm{d}N/\mathrm{d}M)\mathrm{d}M/N$})\textcolor{black}{, then MPs dominate if} the ratio of the 2nd moments of the MP and star mass distributions, $\mu_{2}\!\equiv\!\left.N_{p}\left\langle M_{p}^{2}\right\rangle \right/N_{\star}\left\langle M_{\star}^{2}\right\rangle $, satisfies $\mu_{2}\!\gg\!1$ (note that for a continuous mass spectrum, this condition is equivalent to $-\mathrm{d}\log N/\mathrm{d}\log M\!<\!2$). \textcolor{black}{} As discussed in detail in \S\ref{s:GC_MPs}, the central $\sim\!100\,\mathrm{pc}$ of the Galactic Center (GC) contain $10^{8}-10^{9}$ solar masses in stars, and about $10^{6}-10^{8}$ solar masses in MPs such as open clusters and GMCs of masses $10^{3}-10^{7}\,\Mo$ (\citealt{Oka+01,Fig+02,Vol+03,Gus+04,Bor+05}). An order of magnitude estimate indicates that MPs in the GC can reduce the relaxation time by several orders of magnitude, \begin{align} \frac{t_{r,\star}}{t_{r,\mathrm{MP}}} & =\mu_{2}\sim\frac{(N_{p}M_{p})M_{p}}{(N_{\star}M_{\star})M_{\star}}\nonumber \\ & =10^{3}\left[\frac{(N_{\star}\Ms/N_{p}M_{p})}{10^{2}}\right]^{-1}\left[\frac{(M_{p}/\Ms)}{10^{5}}\right]\,.\end{align} This estimate is borne by more detailed calculations (Fig. \ref{f:tr} and table \ref{t:models}), using the formal definition $t_{r}\!=\! v^{2}/D(v_{||}^{2})$ with $M_{\star}\rho_{\star}\rightarrow\int(\mathrm{d}N_{p}/\mathrm{d}M_{p})M_{p}^{2}\mathrm{d}M_{p}$ (e.g. \citealt{Bin+87}, Eqs. 8-69 to 8-70). A similar result is indicated by simulations of spatial diffusion of stars in the central 100 pc \citep{Kim+01}. \subsection{Non-coherent loss-cone refilling} \label{ss:noncoherent} The Fokker-Planck approach to the loss-cone problem (e.g. \citealt{Coh+78}) assumes that the effects of multiple small perturbations on the orbit of a test star dominate over the rarer strong close encounters ($b_{\max}/b_{\min}\!\gg\!1)$, and that the cumulative effect can be described as diffusion in phase space. The change in the angular momentum of the test star then grows non-coherently, $\Delta J\!\propto\!\sqrt{t}$. The change over one orbital period $P$ is $J_{D}\!=\! J_{c}(E)\sqrt{P/t_{r}}$, where $J_{c}\!=\!\sqrt{2(\psi-E})r$ is the maximal (circular) angular momentum for a stellar orbit of specific relative energy $E\!=\!-v^{2}/2+\psi(r)$, and $\psi\!\equiv\!-\phi$ is the negative of the gravitational potential, so that $E\!>\!0$ for bound orbits. The magnitude of $J_{D}$ relative to the $J$-magnitude of the loss-cone, \begin{equation} J_{lc}\!\simeq\!\sqrt{2G\Mbh q}\,,\end{equation} determines the mode of loss-cone refilling. The relative volume of phase space occupied by the loss-cone, $J_{lc}^{2}/J_{c}^{2}(E)$, increases with $E$ (decreases with $r$) while $P$ decreases. Near the MBH (high $E$) $J_{D}\!\ll\! J_{lc}$, stars diffuse slowly into the loss-cone, and are promptly destroyed over an orbital period, leaving the loss-cone always nearly empty. In this empty loss-cone regime, the loss-cone is relatively large, but the refilling rate is set by the long relaxation timescale (e.g. \citealt{Lig+77}), \begin{eqnarray} \left(\frac{\mathrm{d}\Gamma}{\mathrm{d}E}\right)_{\mathrm{empty}} & \simeq & \frac{N_{\star}(E)}{\log(J_{c}/J_{lc})t_{r}}\nonumber \\ & = & \frac{J_{D}^{2}(E)}{J_{c}^{2}(E)}\frac{N_{\star}(E)}{\log[J_{c}(E)/J_{lc}]P(E)}\,,\label{e:Gempty}\end{eqnarray} where $N_{\star}(E)$ is the stellar number density per energy interval. Far from the MBH (low $E$) $J_{D}\!\gg\! J_{lc}$, stars diffuse across the loss-cone many times over one orbit, and the loss cone is always nearly full. In this full loss-cone regime the refilling rate is set by the short orbital time, but the loss cone is relatively small, \begin{equation} \left(\frac{\mathrm{d}\Gamma}{\mathrm{d}E}\right)_{\mathrm{full}}\simeq\frac{J_{lc}^{2}}{J_{c}^{2}(E)}\frac{N_{\star}(E)}{P(E)}\,.\label{e:Gfull}\end{equation} Note that here and elsewhere we make the simplifying approximation that the period is a function of energy only, which is true only for motion in a Keplerian potential. The total contribution to loss-cone refilling is dominated by stars with energies near the critical energy $E_{c}$ (equivalently, critical typical radius $r_{c}$) separating the two regimes (\citealt{Lig+77}; see \S\ref{s:interact}). Within $r_{c}$ ($E\!>\! E_{c}$), an object, once deflected into the loss cone, can avoid being scattered out of it before reaching the MBH% \footnote{In the case of inspiral, $E_{c}$ is determined by the condition $J_{D}\!=\! J_{lc}$ over the inspiral time, rather than the much shorter orbital period, which results in a much smaller $r_{c}$ than for direct infall. Inspiraling stars with $E\!>\! E_{c}$ can avoid being scattered directly into the MBH before completing the orbital decay. There is no contribution to inspiral events from regions outside $r_{c}$ ($E\!<\! E_{c}$), since the probability of an object to remain on its low-$J$ trajectory over the many orbital periods required to complete the inspiral, is vanishingly small. % }. The empty and full loss-cone regimes of infall processes can be interpolated to give a general approximate expression for the differential event rate for these non-coherent encounters (e.g. \citealt{You77}),\begin{equation} \frac{\mathrm{d}\Gamma}{\mathrm{d}E}\simeq\frac{j^{2}(E)}{J_{c}^{2}(E)}\frac{N_{\star}(E)}{P(E)}\,,\label{e:dGdE}\end{equation} with \begin{equation} j^{2}(E)\equiv\min\left[\frac{J_{D}^{2}(E)}{\log(J_{c}(E)/J_{lc})},J_{lc}^{2}\right]\,,\end{equation} where $j$ is the loss-cone limited angular momentum change per orbit, which expresses the fact that the loss-cone can at most be completely filled during one orbit. \subsection{Coherent loss-cone refilling} The loss-cone formalism can be generalized to deal with MPs in an approximate manner with only few modifications. Formally, $\log\Lambda\!\sim\!4$ for MPs ($b_{\min,\mathrm{MP}}/b_{\min,\star}\!\propto\! M_{p}/\Ms$), much less than for relaxation by stars. Nevertheless, the error introduced by neglecting encounters with $b\!<\! b_{\min,\mathrm{MP}}$ is not very large because the finite size of the MPs is also $R_{p}\!\sim\! b_{\min,\mathrm{MP}}$ (\S\ref{s:GC_MPs}), and penetrating encounters are much less efficient. However, the assumption of multiple non-coherent encounters with MPs over one orbital period is not necessarily justified because of their small number density. To address this, we modify the treatment of the empty loss-cone regime (the contribution to the event rate from regions where the loss-cone is already filled by stellar relaxation can not be increased by MPs, see \S\ref{s:interact}). We define rare encounters as those with impact parameters $b\!\le\! b_{1}$, where $b_{1}$ is defined by \begin{equation} P\int_{b_{\min}}^{b_{1}}\mathrm{d}b(\mathrm{d}\Gamma/\mathrm{d}b)\!=\!1\,,\label{e:b1}\end{equation} the differential rate is estimated simply by $(\mathrm{d}\Gamma/\mathrm{d}b)\!=\! N_{p}v2\pi b$, with \begin{equation} b_{\min}\!=\!\min(0.1R_{p},GM_{p}/v^{2})\,,\label{e:bmin}\end{equation} and $v^{2}\!=\! GM(<\! r)/r$. When $P\int_{b_{1}}^{b_{\max}}\mathrm{d}b(\mathrm{d}\Gamma/\mathrm{d}b)\!>\!1$, with $b_{max}\!=\! r$, all encounters with $b>b_{1}$ are defined as frequent encounters that occur more than once per orbit, and add non-coherently% \footnote{In the marginal cases of $P\int_{b_{\min}}^{b_{\max}}\mathrm{d}b(\mathrm{d}\Gamma/\mathrm{d}b)\!<\!1$ or $P\int_{b_{1}}^{b_{\max}}\mathrm{d}b(\mathrm{d}\Gamma/\mathrm{d}b)\!<\!1$, all encounters are considered rare. % }. Note that even when $P\int_{b_{1}}^{b_{\max}}\mathrm{d}b(\mathrm{d}\Gamma/\mathrm{d}b)\!>\!1$ for all MPs, perturbations by rare, very massive MPs may still occur less than once per orbit. Our treatment is approximate. A complete statistical treatment of this situation lies beyond the scope of this study. When the typical number of encounters per orbit is less than one, the fractional contributions from different individual encounters, $\delta J$, should be averaged coherently ($\Delta J\!\propto\! t$), subject to the limit that each encounter can at most fill the loss-cone. The loss-cone limited change in angular momentum per orbit due to rare encounters is therefore \begin{equation} j_{R}^{2}(E)=\left[P\!\int_{b_{\min}}^{b_{1}}\mathrm{d}b\frac{\mathrm{d}\Gamma}{\mathrm{d}b}\min\left(\delta J,J_{lc}\right)\right]^{2}\,.\label{e:jR}\end{equation} In contrast, frequent uncorrelated collisions add up non-coherently ($\Delta J\!\propto\!\sqrt{t}$), and it is only their final value that is limited by the loss-cone (individual steps $\delta J$ may exceed $J_{lc}$, but can then be partially cancelled by opposite steps during the same orbit). The loss-cone limited change in angular momentum per orbit due to frequent encounters is therefore \begin{equation} j_{F}^{2}(E)=\min\left[\frac{1}{\log(J_{c}/J_{lc})}P\!\int_{b_{1}}^{b_{\max}}\!\mathrm{d}b\frac{\mathrm{d}\Gamma}{\mathrm{d}b}\delta J^{2},J_{lc}^{2}\right]\,.\label{e:jF}\end{equation} The total loss-cone limited angular momentum change per orbit is then approximated by \begin{equation} j^{2}=\min\left(j_{R}^{2}+j_{F}^{2},J_{lc}^{2}\right)\,,\label{e:jtot}\end{equation} and the differential event rate is calculated by Eq. (\ref{e:dGdE}), $\mathrm{d}\Gamma/\mathrm{d}E\!=\!\left[j^{2}(E)/J_{c}^{2}(E)\right]N_{\star}(E)/P(E)$. The contribution of rare encounters is evaluated in the impulse approximation by setting $\delta J\!\sim\! GM_{p}r/bv$ in Eq. (\ref{e:jR}). We find that the this contribution by GC MPs (\S\ref{s:GC_MPs}) is generally small. Frequent encounters are the regime usually assumed in the Fokker-Planck treatment of the loss-cone problem (e.g. \citealt{Lig+77}). To evaluate the contribution of frequent encounters, we do not calculate $\delta J$ directly, but instead calculate the sub-expression $I=P\int_{b_{1}}^{b_{\max}}\mathrm{d}b(\mathrm{d}\Gamma/\mathrm{d}b)\delta J^{2}$ in Eq. (\ref{e:jF}) in terms of the $b$-averaged diffusion coefficient $D(v_{t}^{2})$, after averaging over the orbit between the periapse $r_{p}$ and apoapse $r_{a}$ and averaging over the perturber mass function (this is essentially equivalent to the definition of $J_{D}$ in terms of $t_{r}$, \S\ref{ss:noncoherent}), \begin{eqnarray} I & = & \int\mathrm{d}M_{p}\left(2\int_{r_{p}}^{r_{a}}\frac{r^{2}D(\Delta v_{t}^{2})}{v_{r}}\mathrm{d}r\right)\nonumber \\ & \simeq & \int\mathrm{d}M_{p}\left(2\int_{0}^{2r}\frac{r^{2}D(\Delta v_{\perp}^{2})}{v}\mathrm{d}r\right)\,.\label{e:IF}\end{eqnarray} The assumptions involved in the last approximate term (\citealt{Mag+99}) are that the star is on a nearly radial orbit ($v_{r}\!\rightarrow\! v$, $r_{p}\!\rightarrow\!0$, $r_{a}\!\rightarrow\!2r$) and that $D(v_{t}^{2}$) (the diffusion coefficient of the transverse velocity relative to the MBH) can be approximated by $D(\Delta v_{\perp}^{2})$ (the diffusion coefficient of the transverse velocity relative to the stellar velocity $\mathbf{v}$), given explicitly by (\citet{Bin+87}, Eq. 8-68) \begin{equation} D(\Delta v_{\perp}^{2})=\frac{8\pi G^{2}(\mathrm{d}N_{p}/\mathrm{d}M_{p})M_{p}^{2}\ln\Lambda}{v}K\left(\frac{v}{\sqrt{2}\sigma}\right)\,,\label{e:delta_v}\end{equation} where $K(x)\!\equiv\!\mathrm{erf}(x)(1-1\left/2x^{2}\right.)+\exp(-x^{2})\left/\sqrt{\pi}x\right.$ and where a spatially homogeneous distribution of MPs with a Maxwellian velocity distribution of rms 1D velocity $\sigma$ was assumed. % \begin{figure} %\includegraphics[clip,width=1\columnwidth,keepaspectratio]{f1} \plotone{f1.eps} \caption{\label{f:tr}Relaxation time as function of distance from the MBH, for stars (solid line) and for each of the 3 MP models separately, as listed in table \ref{t:models}: clusters (dash-dot line), GMC1 and GMC2 (dashed line). The discontinuities are artifacts of the assumed sharp spatial cutoffs on the MP distributions. 2-body stellar processes dominate close to the MBH, where no MPs are observed to exist. However, at larger distances massive clumps (at $1.5\!0.4\,\mathrm{pc}$ and $\Ms\!=\!1\,\Mo$) in terms of $r$, using $N_{\star}$ to derive the appropriate density of perturbed objects (single stars \S\ref{ss:single} or binaries \S\ref{ss:binary}). At each $r$ we calculate $b_{\min}$ (Eq. \ref{e:bmin}), $b_{1}$ (Eq. \ref{e:b1}), $j_{R}$ (Eq. \ref{e:jR}) and $j_{F}$ (Eq. \ref{e:jF}). The integral $I$ (Eq. \ref{e:IF}) is evaluated by taking $v^{2}\!\rightarrow\! GM(