aa r X i v : . [ a s t r o - ph ] A ug Stellar Relaxation Processes Near the GalacticMassive Black Hole † By Tal Alexander , Faculty of Physics, Weizmann Institute of Science, PO box 26, Rehovot 76100, Israel William Z. and Eda Bess Novick Career Development ChairThe massive black hole (MBH) in the Galactic Center and the stars around it form a unique stellar dynamicslaboratory for studying how relaxation processes affect the distribution of stars and compact remnants andlead to close interactions between them and the MBH. Recent theoretical studies suggest that processesbeyond “minimal” two-body relaxation may operate and even dominate relaxation and its consequencesin the Galactic Center. I describe loss-cone refilling by massive perturbers, strong mass segregation andresonant relaxation; review observational evidence that these processes play a role in the Galactic Center;and discuss some cosmic implications for the rates of gravitational wave emission events from compactremnants inspiraling into MBHs, and the coalescence timescales of binary MBHs.
1. Introduction
The M • ∼ × M ⊙ massive black hole (MBH) in the Galactic Center (GC) (Eisenhauer et al. et al. et al. et al. et al. et al. et al. † Invited talk. To appear in “2007 STScI spring symposium: Black Holes”, eds, M. Livio & A. M.Koekemoer, Cambridge University Press, in press.
Relaxation Processes Near the Galactic MBH
Milky Way is the archetype of the subset of galaxies with low-mass MBHs that are key targetsfor planned space-borne gravitational wave detectors, such as the Laser Interferometer SpaceAntenna (LISA). GC studies may help understand the effect of such relaxation processes on theopen questions of the cosmic EMRI event rate and the EMRI orbital characteristics.Before turning to a discussion of the non-standard relaxation processes that are expected tooperate in the GC, it is useful to briefly review the dynamics leading to close interactions with aMBH (loss-cone theory) and the dynamical conditions in the GC.1.1.
Infall and inspiral into a MBH
Stars can fall into the MBH either by losing orbital energy, so that the orbit shrinks down tothe size of the last stable circular orbit ( r LSCO = 3 r s for a non-rotating MBH, where the eventhorizon is at the Schwarzschild radius r s = 2 GM • /c ), or by losing orbital angular momentumso that the orbit becomes nearly radial and unstable (periapse r p < r s for a star with zero orbitalenergy falling into a non-rotating MBH) † . The timescale to lose energy by 2-body scattering, T E ≡ | E/ ˙ E | is of the order of the relaxation time, T E ∼ T R ∼ ( M • /M ⋆ ) τ dyn ( r ) /N ⋆ ( < r ) log N ⋆ ( < r ) , (1.1)where N ⋆ ( < r ) is the number of stars inside r , τ dyn ( r ) ∼ p r /GM • is the dynamical timeand spherical symmetry and a Keplerian velocity dispersion are assumed, σ ∼ GM • /r . Themaximal angular momentum available for an orbit with energy E is that of a circular orbit, J c ( E ) = GM • / √ E (using here the stellar dynamical sign convention E ≡ − v / − φ ( r ) > ).The timescale for losing angular momentum, T J ≡ | J/ ˙ J | , can be much shorter than T E when J < J c , since T J = [ J/J c ( E )] T E . (1.2)As a consequence, almost all stars that reach the MBH, and are ultimately destroyed by a closeinteraction with it, do so by being scattered to low- J “loss-cone” orbits (near radial orbits with J < J lc ≃ √ GM • q , where q is the maximal periapse required for the close interaction of interestto occur. Frank & Rees 1976; Lightman & Shapiro 1977). The rate of close interaction events, Γ lc , is set by the replenishment rate of stars into the loss-cone. When the replenishment mecha-nism is diffusion in phase space by 2-body scattering, Γ lc ∝ T − R , which is typically a very lowrate. Close to the MBH, at high- E , where the relative size of the loss-cone in phase-space is large( J lc /J c ∝ √ qE ), relaxation is too slow to replenish the lost stars, and the loss-cone is on averageempty. Farther out, at low E , where the loss-cone is small, relaxation can replenish the lost stars,the loss-cone is full (isotropic distribution of stars) and the local replenishment rate is maximal.Nevertheless, the contribution to the total replenishment rate from the low- E , full loss-cone re-gions of phase-space, where the timescales are longer and the stellar densities lower, remainssmall compared to that from the empty loss-cone regions at high- E (Lightman & Shapiro 1977).The observational and theoretical interest in such close interactions motivated numerous inves-tigations of alternative efficient loss-cone replenishment mechanism, such as 2-body relaxation innon-spherically symmetric potentials (Magorrian & Tremaine 1999; Berczik et al. et al. et al. a ; Levin 2007), or perturbations by a massive accretion disk or a secondary IMBH (Polnarev & Rees1994; Levin et al. † If the stars are tidally disrupted before falling in the MBH, the relevant distance scale is the tidaldisruption radius r t ∼ R ⋆ ( M • /M ⋆ ) / > r s rather than the event horizon r s . . Alexander: Relaxation Processes Near the Galactic MBH ‡ and so the inspiral time scales with the number of periapse passages, and hence with the initialorbital period.An infall or inspiral event can occur only if the star, once deflected into the loss-cone, avoidsbeing re-scattered out of it (and in the case of inspiral, also avoids being scattered directly into theMBH). Because inspiral processes are slow, stars can avoid re-scattering, complete the inspiraland decay to an interesting, very short period orbit with high emitted dissipative power, only ifthey are deflected into the loss cone from an initially short period orbit, with E > E crit . Figure (1)shows a schematic description of the phase-space evolution of infalling and inspiraling stars, andthe emergence of a critical energy scale. For inspiral by GW emission into a M • ∼ O (10 M ⊙ ) MBH, E crit corresponds to an initial distance scale of r crit ∼ . pc (the ansatz r ↔ E = GM • / a , is assumed here, where a is the Keplerian semi-major axis). The EMRI event rate isthen approximately (Hopman & Alexander 2005) Γ lc ∼ N GW ( < r crit ) /T R ( r crit ) ∝ N GW ( < r crit ) N ⋆ ( < r crit ) /τ dyn ( r crit ) , (1.3)where N GW ( < r ) is the number of potential GW sources (compact remnants) within distance r of the MBH. A critical energy can be similarly defined for infall processes. Because infall ismuch faster than inspiral, E crit is much lower ( r crit much larger). For example, the critical radiusfor tidal disruption in the GC is r crit ∼ few pc (Lightman & Shapiro 1977; Syer & Ulmer 1999;Magorrian & Tremaine 1999). Most of the stars that infall or inspiral originate near r crit .Equation (1.3) shows that the degree of central concentration of compact remnants stronglyaffects the EMRI event rate. Mass segregation therefore substantially increases the predictedEMRI event rate from inspiraling O (10 M ⊙ ) stellar black holes (SBHs), which are the mostmassive, long-lived objects in the population (Hopman & Alexander 2006 b ; §3). Similarly, thecapture of compact remnants very near the MBH by 3-body exchanges between the MBH andbinaries (§2) can also strongly affect the EMRI rate (Perets, Hopman & Alexander, 2007, inprep.). The dependence of Γ lc on T R is not trivial, since r crit itself depends on T R : the shorterthe relaxation time, the faster stars are scattered into the loss-cone, but also out of it. Detailedanalysis shows that the two effects cancel out for n ⋆ ∝ r − / stellar cusps. Since in most galacticnuclei the logarithmic slope of the density profile is not much different from − / , the EMRIrate is expected to be roughly independent of the relaxation time (Hopman & Alexander 2005).It should be emphasized that this result applies only to 2-body relaxation, and needs to be re-examined if other loss-cone replenishment mechanisms dominate the dynamics.1.2. The dynamical state of the stellar system around the Galactic MBH
The stellar system around the Galactic MBH is expected to be in a state of dynamical relaxationin a high density cusp. This is a direct consequence of the low mass of the Galactic MBH andof the M • /σ relation, the tight observed correlation between the mass of central MBHs andthe typical velocity dispersion in the bulges of their host galaxies, M • ∝ σ β , where . β . (Ferrarese & Merritt 2000; Gebhardt et al. β = 4 is assumed here for simplicity; theconclusions below are reinforced if β > . ‡ E.g. the GW energy emitted per orbit scales as ∆ E ∝ ( M ⋆ c /M • )( r p /r s ) − / (Peters 1964). T. Alexander:
Relaxation Processes Near the Galactic MBH
J = J (E) clsco log E log J lc log E log J
1D J−scattering"Spherical background"
J = J (E) c log E lsco log E crit log J lc log E log J TransitionphaseE−dissipationphase
Detectable GW
J−scattering phase F IGURE
1. A schematic representation of the phase-space (log E, log J ) trajectories leading a star to theMBH. Each segment of the random-walk trajectory represents the change in the phase coordinates oversome fixed time step ∆ t . The shaded areas on top ( E > E
LSCO ) and on the left (
J < J lc ) are regions of phasespace where no stable orbits exist. The diagonal boundary on the right is s the maximal angular momentum J c ( E ) . Left: Infall without dissipation. A star with initially high J is scattered with roughly equal relativemagnitude in E and J . Eventually a random kick will send it to a low- J orbit, where J -scattering is muchfaster than E -scattering, making it plunge directly into the MBH. Right: Inspiral with dissipation. Energydissipation by the emission of GW can lead to very rapid orbital decay on low- J orbits, faster than the meantime between scattering events, thus enabling the star to reach a short-period orbit with detectable GWemission (narrow horizontal shaded strip on top). Statistically, nearly all stars with initial energy E > E crit will ultimately inspiral into the MBH, while nearly all stars with
E < E crit will ultimately plunge into theMBH, following a trajectory similar to the one depicted in the left panel.
The MBH radius of dynamical influence is conventionally defined as r h ∼ GM • /σ ∝ M / • .The mass in stars within the radius of influence is of the order of the mass of the MBH, so theirnumber is N h ∼ M • /M ⋆ , where M ⋆ is the mean stellar mass, and the average stellar densitywithin r h is ¯ n h ∼ N h /r h . The two-body relaxation time at r h is T R ∼ ( M • /M ⋆ ) τ h /N h . It thenfollows that T R ∝ M / • and ¯ n h ∝ M − / • . Evaluated for the Galactic MBH, T R ∼ O (1 Gyr)
2. Massive perturbers
Massive perturbers and the loss-cone
The relaxation time (Eq. 1.1) is proportional to ( M ⋆ n ⋆ ) − . This can be readily understood byconsidering the “ Γ ∼ nv Σ ” collision rate between stars of mass M ⋆ and mean space densityin volume V, n ⋆ = N ⋆ /V , where the cross-section Σ ∼ πr c is evaluated for collisions at thecapture radius r c = 2 GM ⋆ /v , the minimal radius for a soft encounter with a typical velocity v .The rate of scattering by stars is then Γ ⋆ ∼ n ⋆ M ⋆ /v ∼ T − R (integration over all collision radii. Alexander: Relaxation Processes Near the Galactic MBH M p ≫ M ⋆ and space density n p = N p /V ≪ n ⋆ will scatter stars at the capture radius r c = G ( M ⋆ + M p ) /v at a rate of Γ p ∼ n p ( M ⋆ + M p ) /v ∼ n p M p /v . MPs could well dominate the relaxation even if they are very rare, as long as µ ≡ M p N p /M ⋆ N ⋆ > . (2.1)Efficient relaxation by MPs was first suggested by Spitzer & Schwarzschild (1951, 1953) toexplain stellar velocities in the Galactic disk. Its relevance for replenishing the loss-cone wassubsequently investigated in the context of Solar system dynamics for the scattering of Oortcloud comets to the Sun (Hills 1981; Bailey 1983), and more recently as a mechanism for estab-lishing the M • /σ correlation by fast accretion of stars and dark matter (Zhao et al. et al. et al. r crit relative to the spatial distribution of the MPs( r crit increases with the loss-cone size, and in the case of inspiral also with the efficiency of thedissipative process). MPs are extended objects, which cannot survive in the strong tidal field ofthe MBH (IMBHs could be the one exception). Generally, MPs in galactic centers could alsobe affected by an intense central radiation field, whether the AGN’s or the stars’, or by outflowsassociated with accretion on a MBH. These processes introduce an inner cutoff r MP to the MPdistribution. A plausible estimate is r MP & O ( r h ) . This is the case in the GC, where the clumpycircumnuclear gas ring lies outside the central . pc, on a scale comparable to r h . The eventrates of processes such as tidal disruption of single stars ( r crit ∼ r h ) or GW EMRI ( r crit ≪ r h ),where stellar relaxation by itself efficiently fills the loss-cone at r crit < r < r MP , will not be muchenhanced by additional relaxation due to MPs (the stellar distribution function (DF) cannot bemore random than isotropic). In contrast, the event rates of processes whose loss-cone is large,and which would have remained empty beyond r MP in the absence of MPs, can be increased byorders of magnitude by the presence of MPs. Most of the enhancement is due to MPs near r MP (Perets et al. M and semi-major axis a that interact withthe MBH at a distance r p < r t ∼ a ( M • /M ) / , leading to the capture of one star around theMBH and the ejection of the other as a HVS (Hills 1988), and the orbital decay of a binary MBHof total mass M , mass ratio M /M = Q < and semi-major axis a by interactions with starsat a distance r p . O ( a ) (the “slingshot effect”) (Begelman et al. Massive perturbers in the Galactic Center
MPs in the GC include GMCs, stellar clusters and possibly IMBHs, if these exist. Direct ob-servational evidence (Fig. 2) indicates that the dominant MPs on the r ∼ – pc scale are O (100) GMCs in the mass range – M ⊙ , with rms mass of ∼ M ⊙ and a typical size of R p ∼ pc (the quoted range includes an order of magnitude uncertainty in the mass determina- T. Alexander: Relaxation Processes Near the Galactic MBH C u m u l a t i v e D i s t. MP Mass (M o ) F IGURE
2. The observed MPs in the GC and their effect on the relaxation time. Left: The observed massfunction of molecular cloud massive perturbers in the GC (adapted from Perets et al. (2007) with permissionfrom the
Astrophysical Journal ). Lower ( ◦ ) and upper (virial) ( (cid:3) ) mass estimates for the molecular clumpsin the inner ∼ pc, based on the molecular line observations of Christopher et al. (2005), and lower ( △ ) andupper (virial) ( ▽ ) mass estimates for the GMCs in the inner ∼ pc of the Galaxy, based on the molecularline observations of Oka et al. (2001). Right: The relaxation time as function of distance from the GalacticMBH due to stars alone, the upper (GMC1) and lower (GMC2) mass estimates of the molecular clumpsand GMCs and due to upper (Clusters1) and lower (Clusters2) estimates on the number and masses ofstellar clusters. The sharp transitions at r = 1 . and pc are artifacts of the non-continuous MP distributionassumed here. GMCs dominate the relaxation in the GC. tion), (Oka et al. r ∼ . - pc scale, O (10) molecularclumps † with masses in the range – M ⊙ , with rms mass of ∼ M ⊙ and a typical sizeof R p ∼ . pc (Christopher et al. ∼ observed stellar clusters (Figer et al. et al. ∼ × ∼ M ⊙ stars in the central pc (Figer et al. µ ∼ – × (Eq. 2.1). Figure (2) showsa more detailed estimate of the local relaxation time for the various molecular cloud models,taking into account, among other considerations, the Coulomb factors. The relaxation time isindeed substantially decreased, by factors of − relative to that by stars alone, dependingon distance from the center, and on the GMC mass estimates. If IMBHs do exist, then the effectsof accelerated relaxation will be even stronger than predicted here, and probably extend all theway to the center. 2.3. Galactic and cosmic implications
With stellar relaxation alone, the empty loss-cone region of MBH-binary interactions is large( r t ∝ a ) and extends out to > pc. However, the MPs that exist in the Galaxy on thatscale accelerate relaxation, efficiently fill the loss-cone, and thus increase the binary disrup-tion rate by several orders of magnitude, making binary disruptions dynamically and obser-vationally relevant (Perets et al. † The division of a quasi-continuous medium into individual clouds is somewhat arbitrary, since severalsub-clumps can be identified as a single cloud, depending on the spatial resolution of the observations andthe adopted definition of a cloud. For a fixed total MP mass, M = M p N p within a region of size r , therelaxation time scales with N p as T R ∝ ( M p N p ) − = M − N p ; the more massive and less numerousthe clouds, the shorter T R . The value of T R thus depends on the way clouds are counted. Obviously, thestatistical treatment of relaxation is valid only for N p ≫ and R p ≪ r . . Alexander: Relaxation Processes Near the Galactic MBH −2 −1 0 1 2 3 4 5 6 7 8 900.020.040.060.080.10.120.140.160.180.2 log(P/days) N o r m a li s e d P D F InitialEvolved at100 pc (MPs)Evolved at10 pc (stars) −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 000.20.40.60.811.2 log(r/pc) N o r m a li z e d P D F Stars MPs F IGURE
3. A schematic representation of the mapping of the initial binary period distribution to the semi–major axis of the tidally captured star. Left: The bimodal initial period distribution for old white dwarf /main sequence binaries (adapted from Willems & Kolb 2004), and its subsequent evolution due to GW co-alescence (for the shortest periods) and to slow evaporation by field stars at 100 pc and faster evaporationat 10 pc. Right: the resulting semi-major axis distribution of the captured stars, due to scattering by stars,which occurs on the O (10 pc) scale, and due to scattering by MPs, which occurs on the O (100 pc) scale. et al. et al. ‡ ( v &
500 km s − ), observed tens of kpc away from the GC (Hirsch et al. et al. et al. et al. a ), and the origin of the puz-zling “S-stars” (Gould & Quillen 2003; Ginsburg & Loeb 2006), a cluster of ∼ – main-sequence B-stars ( M ⊙ . M ⋆ . M ⊙ , main sequence lifespan t ⋆ ∼ few × – few × yr)on random tight orbits around the MBH in the central few × . pc (Eisenhauer et al. et al. et al. ∼ . of MBH-binary encoun-ters lead to capture, and that the mean semi-major axis of the captured star is related to that ofthe original binary by (Hills 1988, 1991) h a i ∼ ( M • /M ) / a , (2.2)which implies a very high initial eccentricity, − e = r t / h a i = ( M /M • ) / ∼ O (0 . . Thetidal capture process can be viewed as a mapping between the properties of field binaries far fromthe MBH, and the orbital properties of the captured stars: wide binaries result in wide capturedorbits, and vice versa (Fig. 3). The mean velocity of the ejected star at infinity (neglecting thegalactic potential) is (cid:10) v ∞ (cid:11) ∼ √ GM / M / • /a . (2.3)This translates, for example, to v ∞ ∼ − for a × M ⊙ B-star binary with a = 0 . AU, well above the escape velocity from the Galaxy.Of particular interest is the connection between the HVSs and the S-stars that is implied by thebinary tidal disruption scenario. The stellar binary mass ratio distribution is peaked around ∼ (Duquennoy & Mayor 1991; Kobulnicky et al. ‡ HVS candidates are chosen for spectroscopy by color, to maximize the contrast against the halo popu-lation, and so are pre-selected to have B-type spectra (e.g. Brown et al. b ). T. Alexander:
Relaxation Processes Near the Galactic MBH the number of tidally captured S-stars for different MP populations (Perets et al. a ∼ . +0 . − . AU) and their fractionamong young massive stars in the field ( f ∼ . ) (Kobulnicky et al. ∼ pc of the Galaxy (isothermal, normalizedby the observations of Genzel et al. et al. × . M ⊙ binary (main sequence B-stars with a lifespan of t ⋆ ∼ × yr). Dynami-cal evaporation is negligible for such short-lived binaries. The steady state number of capturedS-stars is then h N ⋆ i = Γ lc t ⋆ . Figure (4) shows that with stellar relaxation alone, tidal capture can-not explain the S-star population. However, relaxation by GMCs is consistent with the observednumber of S-stars, as well as with the spatial extent of the cluster of ∼ . pc, which reflectsthe hardness of young massive binaries in the field (Eq. 2.2). It is also consistent with the factthat the S-cluster does not include any star earlier then O8V/B0V. Such short-lived binaries arevery rare in the field, and their mean number in the S-cluster is predicted to be h N ⋆ i < .The MP-induced binary tidal disruption scenario also predicts that there should be – hyper-velocity ∼ M ⊙ B-stars at distances between and kpc from the GC. This is consis-tent with the total number of ± extrapolated by Brown et al. (2006 a ), based on the HVSs de-tected at these distances in their field of search. The tidal disruption scenario predicts an isotropicdistribution of HVSs around the GC, and a random ejection history, in contrast to models wherethe ejection is related to a discrete binary MBH merger event (Yu & Tremaine 2003; Haardt et al. et al. et al. a ) and thussupport the tidal disruption scenario.The tidal disruption scenario can naturally explain many of the properties of the S-stars andHVSs, but it has two potential flaws. (1) The predicted high eccentricities of the captured stars arelarger than those observed for a few of the S-stars ( e ∼ . , Eisenhauer et al. t ⋆ ∼ × yr) is shorterby a factor . than the MP-accelerated relaxation time in the inner ∼ pc (Fig. 2), where asubstantial fraction of the binaries are scattered from. Thus if a binary in those regions starts ona near-circular orbit, MP-induced relaxation is not fast enough to scatter it to a J < J lc orbit (Eq.1.2) within its lifetime. However, as the timescale discrepancy is not large, and as it affects onlythe most massive binaries in the central few pc, where the determination of T R is ambiguous(see footnote on page 6), this does not appear to be a fatal flaw of this scenario. It does howeverhighlight the importance of observationally quantifying the relaxation time in the GC and thedistribution and properties of the field binaries.Low-mass binaries are also deflected to the MBH by MPs and tidally disrupted at rates as highas ∼ − yr − (Perets et al. et al. Relaxation Processes Near the Galactic MBH −2 −1 Semi Major Axis (pc) N o . o f M S B − S t a r s GMC1GMC2Clusters1StarsObserved Clusters2 -F IGURE
4. A comparison between the cumulative number of S-stars (main sequence B stars) observed or-biting the Galactic MBH on randomly oriented orbits (vertical bar), and the predicted number captured by3-body tidal interactions of the MBH with binaries deflected to the center by massive perturbers, for differ-ent massive perturbers models (Perets et al. to the MBH and therefore the rate of GW bursts increases significantly. In particular, the rate ofGW bursts from white dwarfs increases from ∼ . − (Hopman et al. ∼ − (Perets et al. 2007, in prep.).Binary MBHs form in the aftermath of galactic mergers, when the two MBHs sink by dy-namical friction to the center of the merged galaxy. Once the binary hardens, the orbital decaycontinues by 3-body interactions with stars that are deflected to the center and extract energyfrom the binary, until the orbit becomes tight enough for efficient GW emission, which rapidlyleads to coalescence. Simulations show that when the loss-cone is replenished by stellar relax-ation alone, the interaction rate is too slow for the binary MBH to coalesce within a Hubble time(e.g. Berczik et al. et al. † for resolving the stalling problemis by accelerated MP-induced loss-cone replenishment (Perets & Alexander 2007)Figure (5) shows the time to coalescence, as function of the binary MBH mass, for differentmerger and MP scenarios, based on a combination of extrapolation of the Galactic MP populationto early type galaxies, on extra-galactic observations of molecular gas in galactic centers, and onresults from galactic merger simulations. The results show that MPs allow binary MBHs in gas-rich galaxies to coalesce within a Hubble time over nearly the entire range of M . The situationwith respect to gas-poor galaxies is less clear, since it is harder to model reliably the MPs there † Other possible routes are by interactions with gas in “wet mergers” (Ivanov et al. et al. et al. et al. et al. et al.
Relaxation Processes Near the Galactic MBH BMBH Mass (M o ) C o a l esce n ce T i m e ( y r s ) Stars Majorgas−poorMinorQ=0.05 Majorgas−rich −3 −2 −1 Time (yrs) S e m i − M a j o r A x i s ( p cs ) F IGURE
5. Accelerated binary MBH mergers in the presence of MPs (Perets & Alexander 2007). Left: Thetime to coalescence as function of binary MBH mass, for different merger scenarios distinguished by themass ratio Q between the two MBHs and the MP contents of host galaxies. The age of the universe isindicated by the dotted horizontal line. Stellar relaxation alone cannot supply a high enough rate of starsfor the slingshot mechanism to complete the merger within a Hubble time. However, in minor mergers( Q = 0 . ) major gas-rich mergers ( Q = 1 ) with MPs merger is possible within a Hubble time for all butthe most massive MBHs. Right: The evolution of the binary MBH semi-major axis as function of time formajor mergers ( Q = 1 ) in the presence of MPs (solid line) and stellar relaxation alone (dashed line), forbinary MBH masses of , , and M ⊙ (from bottom up). (probably clusters rather than GMCs). However, even for such galaxies, MPs allow coalescencewithin a Hubble time up to masses of M . M ⊙ .Efficient binary MBH coalescence by MPs has various implications. It increases the cos-mic rate of GW events from MBH-MBH mergers, it increase the “mass deficit” in the galac-tic core (the stellar mass ejected from the core by the slingshot effect) (Milosavljevi´c et al. et al. et al. et al. et al. et al.
3. Strong mass segregation
The Bahcall-Wolf solution of moderate mass-segregation
The 2-body relaxation timescale around the Galactic MBH, T R ∼ O (1 Gyr) , is short enoughfor the old stellar population there to relax to a universal steady-state configuration, indepen-dently of the initial conditions. This configuration was investigated by Bahcall & Wolf (1976,1977). The Bahcall-Wolf solution predicts that in the Keplerian potential near a MBH, starsof mass M ⋆ in a multi-mass population , M < M ⋆ < M , have a DF that is approximately apower-law of the specific orbital energy ǫ , f M ( ǫ ) ∝ ǫ p M , where p M ∝ M ⋆ with a proportionalityconstant p M /M ⋆ ≃ / (4 M ) . In a Keplerian potential, this DF corresponds to a density cusp n M ( r ) ∝ r − α M , where α M = 3 / p M . Elementary considerations show that α = 7 / ( p = 1 / )for a single mass population (e.g. Binney & Tremaine 1987, § d E ( r ) / d t ∼ N ⋆ ( < r ) E ⋆ ( r ) /T R ( r ) ∼ r / − α = const (using the relations N ⋆ ( < r ) ∝ r − α ,. Alexander: Relaxation Processes Near the Galactic MBH M ean m a ss [ M o ] K [mag] dlog(N)/dKOld star fractionAll stars Old stars (t>10 yr) GC population synthesis model S u r f a c e den s i t y [ s t a r s / a r cs e c ] Projected distance [arcsec] x2RC/HBSchoedel et al 200714.75 6. Left: A theoretical population model for the central few pc of the GC (Alexander & Sternberg1999; Alexander 2005), assuming continuous star formation over the past 10 Gyr (Figer et al. > Gyr) in the population, as function of the K -band magnitude in the GC (for DM + A K = 17 . mag, Eisenhauer et al. K -band luminosity function. Top panel:The mean mass of all stars and of the old stars only, as function of the K -band magnitude. The concentrationof old Red Clump / horizontal branch giants around K ∼ . is clearly seen as an excess in the luminos-ity function, as an increase the fraction of old stars and as a decrease in the mean stellar mass relative tostars both immediately brighter and fainter (Schödel et al. Σ( R ) ( R = 1” corresponds to ∼ . pc in the GC) in 3 adjacent K -magnitude bins, centered around thebin associated with the Red Clump giants ( . < K < . ), with broken power-law fits Σ ∝ R − Γ (Schödel et al. Astronomy and Astrophysics ). Top: . < K < . (density multiplied by for display purposes). Middle: . < K < . (the Red Clump / horizontalbranch range). Bottom: . < K < . . E ⋆ ∼ r − and T R ∝ r α − / , §1.2). The Bahcall-Wolf solution reproduces this result for a singlemass population, and predicts that it should apply also to the heaviest stars in a multi-mass pop-ulation. The Bahcall-Wolf solution thus implies that at most ∆ α = 1 / between the lightest andheaviest stars in the population. The predicted degree of segregation is moderate.Theoretical considerations, results from dynamical simulations and GC observations, hint thatthe moderate segregation solution should not and does not always hold, even in relaxed systems.As formulated, the solution depends only on the stellar masses, but not on the mass function.However, this cannot apply generally, since in the limit where the massive objects are very rare,they are expected to sink efficiently to the center by dynamical friction, and create a cusp muchsteeper than α = 7 / . As shown below (§3.2), models of the present-day mass function in the cen-tral few pc of the GC suggest that the massive objects are relatively rare. Dynamical simulationsof mass segregation in the GC based on such a mass function (Fig. 8) indeed show steep cusps( α > ) for the heaviest masses. Finally, the observed surface density distribution of GC stars inthe magnitude bin . < K < . , which corresponds to the low-mass ( . . M ⋆ . M ⊙ )Red Clump / horizontal branch giants (Fig. 6), is substantially flatter than that of the higher-massgiants ( M ⋆ ∼ M ⊙ ) that populate the adjacent bins of brighter and fainter magnitudes (Fig. 6;Schödel et al. ∆ α & (Levi 2006). While none of thesehints for strong mass-segregation is decisive in itself, and other explanations are possible, takentogether they motivate a re-examination of the mass-segregation solution in a relaxed system.2 T. Alexander: Relaxation Processes Near the Galactic MBH The relaxational self-coupling parameter Assume for simplicity a stellar system with a two-mass population of light stars of mass M L ,total initial number N L and local density n L ( r ) and heavy stars of mass M H , total initial number N H and local density n H ( r ) . The self interaction rate is then Γ LL ∝ n L M L /v for the light starsand Γ HH ∝ n H M H /v for the heavy stars (§2). In the limit where the heavy stars are testparticles ( n H /n L ≪ M L /M H , or equivalently Γ HH / Γ LL ≪ ), the heavy stars interact mostlywith the light ones, lose energy and sink to the center by dynamical friction. Conversely, in thelimit Γ HH / Γ LL ≫ , the heavy stars interact mostly with each other, effectively decouple fromthe light stars and establish an α = 7 / cusp typical of a single mass population. This suggeststhat the global relaxational self-coupling parameter (cf Eq. 2.1), defined as µ ≡ N H M H /N L M L , (3.1)can be used to determine whether the system settles into the moderate (Bahcall-Wolf) mass-segregation solution ( µ > ) or the strong mass-segregation solution ( µ < ). This hypothesisis borne out by the numerical results presented below † (Alexander & Hopman 2007, in prep.;§3.3). For a continuous mass distribution, µ can be generalized to µ ≡ Z M M M ⋆ (d N/ d M ⋆ )d M ⋆ , Z M M M ⋆ (d N/ d M ⋆ )d M ⋆ , (3.2)the ratio between the 2nd moments of the mass distribution of the heavy ( M ⋆ > M ) and light( M ⋆ < M ) stars, for some suitable choice of the light/heavy boundary mass M .The value of µ depends on the population’s present-day mass function. So-called universalinitial mass functions (IMFs), which extend all the way from the brown dwarf boundary M ∼ . M ⊙ to M & M ⊙ (e.g. the Salpeter (1955) IMF, and its subsequent refinements, theMiller & Scalo (1979) and Kroupa (2001) IMFs), result in evolved populations, old star-bursts orcontinuously star forming populations, that naturally separate into two mass scales, the O (1 M ⊙ ) scale of low-mass main-sequence dwarfs, white dwarfs and neutrons stars, and the O (10 M ⊙ ) scale of stellar black holes, and typically have µ < (Fig. 7). Such evolved populations are thuswell-approximated by the simple 2-mass population model. In particular, the volume-averagedstellar population in the central few pc of the GC is reasonably well approximated by a 10 Gyrold, continuously star-forming population with a universal IMF ‡ (Alexander & Sternberg 1999;Fig. 6). Generally, 10 Gyr old, continuously star-forming populations with a power-law IMF, d N/ d M ⋆ ∝ M − γ⋆ , have µ < for γ & , and µ > for γ . . Since the critical value γ = 2 isclose to the generic Salpeter index γ = 2 . , it is quite possible that both the moderate and strongsegregation solutions are realized around galactic MBHs, depending on the system-to-systemscatter in the IMF (and perhaps also realized in clusters around IMBHs, if such exist).3.3. Solutions of the Fokker-Planck energy equation The steady state configuration of stars around a MBH can be described in terms of the diffusionof stars in phase space, from an infinite reservoir of unbound stars with a given mass function(the host galaxy, far from the MBH), to an absorbing boundary at high energy where stars aredestroyed (the MBH event horizon, tidal disruption radius, or collisional destruction radius). † In the limit M H /M L ≫ , it may be necessary to take explicitly into account the the dynamical frictiontimescale in order to obtain a more accurate segregation criterion. Here µ is adopted for its simplicity. ‡ Note that recent analysis of late type giants in the GC suggests that the IMF in the inner ∼ pc ofthe GC could typically be a flat γ ∼ . power-law (Maness et al. µ ≫ in theinner ∼ pc, possibly a volume-averaged µ > in the inner few pc (the “collection basin” for stellar BHs,Miralda-Escudé & Gould 2000), and hence moderate segregation. . Alexander: Relaxation Processes Near the Galactic MBH d N / d M M [M ] µ = 1.50 µ = 0.13 µ = 0.15 Miller-Scalo IMF (x100)Continuous SF t=10 GyrOld burst t=10 Gyr (x0.01) F IGURE 7. The predicted values of the global relaxational self-coupling parameter µ for a “universal”Miller & Scalo (1979) IMF (top line, shifted by × for display purposes), an evolved mass functionassuming continuous stars formation over 10 Gyr (middle line), and an evolved star formation burst 10 Gyrold (bottom line, shifted by × . for display purposes) (Alexander & Hopman 2007, in prep.). The massfunctions of the old populations develop excesses in the ∼ . – . M ⊙ range due to the accumulation ofwhite dwarfs and neutron stars, and in the ∼ M ⊙ range due to the accumulation of stellar black holes(here represented by a simplified discrete mass spectrum, see Alexander 2005, table 2.1). Bahcall & Wolf (1976, 1977) simplified the full Fokker-Planck treatment in ( E, J ) phase spaceby integrating over J so as to reduce it to E only, by assuming a Keplerian potential, and byrecasting it in the form of a particle conservation equation. In dimensionless form these can bewritten as (Hopman & Alexander 2006 b ) ∂∂τ g M ( x, τ ) = − x / ∂∂x Q M ( x, τ ) − R M ( x, τ ) , (3.3)where M , x and τ are the dimensionless mass, energy, and time, respectively, g M is the di-mensionless DF, Q M is the flow integral, which expresses the diffusion rate of stars by 2-bodyscattering to energies above x , and R M ∝ g M /T R is the J -averaged effective loss-cone term. Q M and R M are non-linear functions of the set of DFs { g M } . The equations are solved for { g M } byfinite difference methods starting from an arbitrary initial DF and integrating forward in time un-til steady state is reached, subject to the boundary conditions that no stars exist at energies abovesome destruction energy x D , g M ( x > x D ) = 0 , and that the unbound stars are drawn from anisothermal distribution with a given mass function, g M ( x < 0) = N M exp ( M x ) . Bahcall & Wolf(1977) showed that the stellar space density distribution, n M ( r ) ∝ Z r/r h −∞ g M ( x ) p r/r h − x d x , (3.4)does not depend strongly on the exact form of the loss-cone term, and proceeded to use in theirmass-segregation calculations a simplified version of Eq. (3.3) by setting R M = 0 . This approx-imation can be justified by noting that while the existence of a loss-cone drastically increases4 T. Alexander: Relaxation Processes Near the Galactic MBH 100 1000 10000 100000 1e+06 1e+07 1e+08 1e+09 0.001 0.01 0.1 1 10 M / p c [ p c ] r [pc]M. Freitag 2005 priv. comm. α WD =1.5 α BH =0.5 α BH =1.75 α BH =2.75 α BH =2.2 r - α fitBHWD l og n ( p c - ) log r (pc)N MS :N WD :N NS :N BH = 10 :10 -1 :10 -2 :10 -3 MS (1.0 M o )WD (0.6 M o )NS (1.4 M o )BH (10. M o )r -2.1 r -1.5 F IGURE 8. Numerical models of the mass distribution in the GC showing strong segregation. Right: Thedensity distribution of M ⊙ stellar BHs and . M ⊙ white dwarfs at 10 Gyr in an approximate model ofthe GC, with the evolved universal IMF of Fig. (5), derived by M. Freitag (priv. comm., reproduced here withpermission. See also Freitag et al. α L ≃ . power-law cusp. The distribution of massivestellar BHs can be approximated by a piece-wise broken power-law, with α H ∼ . at r ∼ . pc. Left:The space density of a simplified 4-component population model for the GC, as given by the solutionof the Fokker-Planck equations with a loss-cone term (adapted from Hopman & Alexander 2006 b , withpermission from the Astrophysical Journal ). The logarithmic slope of the density cusp of the stellar BHs at r = 0 . pc is α ≃ . , as compared to α ≃ . for the lighter species. the flow rate of stars into the MBH, it typically affects only a small volume in phase spacenear J ∼ . This translates to small changes only in the J -integrated DF g M ( x ) , mostly for x → x D , and even smaller changes in n M ( r ) due to the smoothing effect of the g M ( x ) → n M ( r ) transformation (Eq. 3.4). Here we adopt this approximation to allow direct comparison with theBahcall & Wolf (1977) results, after verifying that the inclusion of the loss-cone term indeeddoes not significantly change the derived stellar cusps (cf Fig. 8 and 9).We calculated a suite of such Fokker-Planck mass-segregation models for 2-mass popula-tions with different mass ratios M H /M L and mass functions N H /N L , spanning a very widerange of the global relaxational self-coupling parameter values † , − < µ < (Alexan-der & Hopman 2007, in prep.). The DFs are not exact power-laws, and the logarithmic slopes p M ( x ) = d log g M / d log x depend somewhat on energy, especially near the boundaries. However,analysis of the results is considerably simplified by the fact that the values of p M ( x ) vary mono-tonically with µ at all x , and so the order ranking of p M for different models does not dependon the choice of x . Figure (9) shows p L and p H at x = 10 , which corresponds to r ∼ . pcin the GC. This choice samples g M ( x ) in a representative region, far from either boundaries at x = 0 and x D = 10 , and translates to an observationally relevant region in the GC, which is closeenough to the MBH to be nearly Keplerian, but still contains a large number of observed stars toallow meaningful statistics (cf Fig. 6).Figure (9) shows that for µ > , the Fokker-Planck calculations recover the Bahcall-Wolfsolution: p H ≃ / irrespective of the mass ratio, and p L ≃ (1 / M L /M H ) . However, for µ < there is a marked qualitative change in the nature of the solutions, as anticipated by the analysis in§3.2. The more the light stars dominate the population (the smaller µ ), the more they approachthe single population solution p L = 1 / (§3.1). The heavy stars settle to a much steeper cusp with p H > / . Figure (9) also shows the grid of models explored by Bahcall & Wolf (1977), which,while large, covers only the µ > range. This explains why the strong segregation branch of the † It is unlikely that real stellar system will have relaxational self-coupling parameters µ ≪ . . How-ever, the study of such models is useful for understanding the mathematical properties of the solution. . Alexander: Relaxation Processes Near the Galactic MBH µ ≃ . has a low mass ratio M H /M L = 1 . ,where the two solution branches are not very different).The µ < models explored here follow the p M ∝ M ⋆ relation noted by Bahcall & Wolf(1977) for the approximate mass-segregation solutions without a loss-cone term. Therefore, inthose models where the limit p L → / is reached (for M H /M L = 1 . , ), the heavy stars reachthe asymptotic value p H → (1 / M H /M L ) . It remains to be seen whether this result also holdsfor higher mass ratios, and for the full Fokker-Planck equation (Eq. 3.3) with the loss-cone term.A realistic evolved stellar system, such as the GC, is expected to have a maximal mass ratioof at least M H /M L = 10 and µ ≃ . (Fig. 7). The mass-segregation calculations indicatethat for these parameters the stellar BHs are expected to form an α H ≃ . – . cusp (Figs. 8,9), significantly steeper than the α H = 1 . predicted by the Bahcall-Wolf solution of moderatesegregation. It is encouraging that this logarithmic slope is very close to that found in numericalsimulations (Fig. 8) and that it is broadly consistent with what is needed to explain the observedtrend in the stellar surface density distributions in the GC in terms of mass segregation (Fig. 6). Inother systems the moderate segregation solution may apply. For example, if the globular clusterM15 contains an IMBH (e.g. Gerssen et al. et al. µ ∼ and a relatively shallow α = 7 / cusp of stellar BHs. Full-scale numeric simulations thatare free of the restrictive assumptions of the analytic approach adopted here (Keplerian potential,fixed boundary conditions, approximate treatment of the loss-cone and a fixed 2-mass stellarpopulation) are needed to verify and test these predictions in more detail.Strong segregation will affect the cosmic rates of EMRI events. A detailed analysis of theanticipated change relative to the various discrepant published rate estimates depends on theirspecific assumptions (e.g. the assumed mass function, slope of the cusp, normalization of thestellar number density), and is outside the scope of this review. 4. Resonant relaxation Resonant relaxation dynamics The effect of 2-body relaxation on a test star is incoherent: the star experiences randomly ori-ented, uncorrelated perturbations from the ambient stars, and as a result its orbit deviates in arandom-walk fashion from its original phase-space coordinates (in a stationary spherical smoothedpotential where E and J would have been conserved in the continuum limit, ∆ E ∝ √ t and ∆ J ∝ √ t due to 2-body interactions). Resonant relaxation (RR) (Rauch & Tremaine 1996;Rauch & Ingalls 1998) is a form of accelerated relaxation of the orbital angular momentum,which occurs when approximate symmetries in the potential restrict the orbital evolution of theperturbing stars. This happens in the almost Keplerian potential near a MBH, where the orbitsare approximately fixed ellipses (the potential of the enclosed stellar mass far from the MBH, orGeneral Relativistic (GR) precession near the MBH, eventually leads to deviations from pure el-liptical orbits), or in a non-Keplerian, but nearly spherically symmetric potential around a MBH,where the orbits approximately conserve their angular momentum and move on rosette-like pla-nar orbits (the fluctuations in the potential due to stellar motions eventually lead to deviationsfrom strictly planar orbits). As long as the symmetry is approximately conserved, on times shorterthan the coherence timescale t ω , the orbit of a test star with semi-major axis a experiences cor-related (coherent) perturbations † , which can be described as a constant residual torque exertedby the superposed potentials of the N ⋆ ( < a ) randomly oriented elliptical “mass wires” (in aKeplerian potential ) or “mass annuli” (in a non-Keplerian spherical potential) that represent the † RR is better described as “coherent relaxation”. The term “resonant” refers to the equality of the radialand azimuthal orbital periods in a Keplerian potential, which results in closed ellipse orbits. Relaxation Processes Near the Galactic MBH p L , p H a t r ~ . p c [ n (r) ∝ r - / - p ] Global µ = N H M H2 / N L M L2 Bahcall & Wolf 1977 M H stars common Alexander & Hopman 2007 M H stars rare GC (UIMF) r -2.2 M15 (Murphy et al 1997) r -7/4 p H p L BW77 modelsM H / M L = 10.3.01.5 F IGURE 9. Fokker-Planck mass-segregation results. The logarithmic slopes p H and p L of the DFs of theheavy stars (thick lines) and light stars (narrow lines) , evaluated at ( r ∼ . pc in the GC), as function ofthe global relaxational self-coupling parameter µ , for mass ratios of M H /M L = 1 . , , (Alexander &Hopman 2007, in prep.). The logarithmic slope of the stellar density of massive stars in the GC, assuming auniversal IMF and continuous star formation history ( µ ∼ . , Fig. 7, α H = 3 / p H ≃ . ) is indicatedby a cross on the left, and for globular cluster M15 (assuming it harbors an IMBH), on the right(estimatedat µ ∼ , based on the mass function model of Murphy et al. (1997), α H ≃ . ). The results for themodels calculated by Bahcall & Wolf (1977) are indicated by circles. orbitally-averaged mass distribution of individual perturbing stars. The magnitude of the residualtorque is then ˙ J ∼ N / ⋆ ( < a ) GM ⋆ /a and the change in the angular momentum of the test starincreases linearly with time, ∆ J ∼ ˙ Jt (for t < t w ). The orbital energy, on the other hand, remainsunchanged, since the potential is constant.RR in a Keplerian potential is called scalar RR since it changes both the magnitude and direc-tion of J . Scalar RR can therefore change a circular orbit into an almost radial, MBH-approachingone. In contrast, RR in a non-Keplerian spherical potential is called vector RR since, for reasonsof symmetry, it changes only the direction of J , but not its magnitude (Fig. 10). Vector RR canrandomize the orbital orientations, but does not play a role in supplying stars to the loss-cone.On timescales longer than the coherence time, the orbital orientations of the perturbing starsdrift, and coherence is lost. the maximal change in angular momentum during the linear coher-ence time, ∆ J ω ∼ ˙ Jt ω then becomes the “mean free path” for a random walk in J -space, whosetime-step is t ω . On timescales longer than the coherence time, the angular momentum changesincoherently ∝ √ t , but much faster than it would have in the absence of RR. The energy isunaffected by RR and always evolves incoherently ∝ √ t on the long non-resonant relaxationtimescale (Fig. 11). The RR timescale T RR is defined, like the incoherent 2-body relaxationtimescale, as the time to change J by order J c (Eqs. 1.1, 1.2), T RR ∼ ( J c / ∆ J ω ) t ω , which can. Alexander: Relaxation Processes Near the Galactic MBH Stationary ellipses in point mass potentialPlanar rosettes inspherical potential Effect on perturbed starPerturbing stars Scalar resonant relaxationVector resonant relaxation F IGURE 10. A sketch comparing the symmetries leading to scalar and vector RR. Top: The torques by fixedelliptical “mass wires” in a Keplerian potential lead to rapid changes in both the direction and magnitude ofthe orbital angular momentum of a test star. Bottom: The torques by fixed “mass annuli” in a non-Keplerianspherical potential lead to rapid changes in the direction, but not in the magnitude of the orbital angularmomentum of a test star. be expressed as (Hopman & Alexander 2006 a ) T RR = A ωRR N ⋆ ( < a ) µ ( < a ) P ( a ) t ω ≃ A ωRR N ⋆ ( < a ) (cid:18) M • M ⋆ (cid:19) P ( a ) t ω , (4.1)where µ is the relative enclosed enclosed stellar mass, µ = N ⋆ M ⋆ / ( M • + N ⋆ M ⋆ ) , P is theradial orbital period, and the last approximate equality holds in the Keplerian regime. Here andbelow, the constants A ωRR are numerical factors of order unity that depend on the specifics of thecoherence-limiting process, on the orbital characteristics of the test star, and probably also on theparameters of the stellar distribution. These constants are not well-determined at this time.The coherence time depends on the symmetry assumed and on the process that breaks it. Fora non-relativistic near-Keplerian potential, the limiting process is precession due to the potentialof the distributed stellar mass † , t ω = t M ∼ M • N ⋆ ( < a ) M ⋆ P ( a ) . (4.2)Remarkably, the resulting RR timescale does not depend on N ⋆ . Close to the MBH is muchshorter than the non-coherent 2-body relaxation timescale (here denoted for emphasis as T NR ), T MRR = A MRR M • M ⋆ P ( a ) ∼ N ⋆ ( < a ) M ⋆ M • T NR . (4.3)Yet closer to the MBH, it is GR precession that limits the coherence, t ω = t GR = 83 (cid:18) JJ LSO (cid:19) P ( a ) , (4.4)where J LSO = 4 GM • /c is the last stable orbit for ǫ ≪ c . The GR precession is prograde, while † The enclosed stellar mass N ⋆ M ⋆ changes the Keplerian period P ∝ M − / • by ∆ P/P = N ⋆ M ⋆ / M • = ∆ ϕ/ π . Identifying de-coherence with a phase drift ∆ ϕ = π then implies t M ∼ ( π/ ∆ ϕ ) P . Relaxation Processes Near the Galactic MBH R M S c hange τ ∆ E/E ∝ τ ∆ |J|/J c (E) → ∝ τ | ∆ J|/J c (E) → ∝ τ F IGURE 11. A correlation analysis of N -body simulations ( N = 200 ) showing the relaxation of energy andof scalar and vector angular momentum around a MBH in the Keplerian limit ( M ⋆ /M • = 3 × − ) for athermal population of stars, as function of the elapsed time-lag (Eilon, Kupi & Alexander, 2007, in prep.).The change in ∆ E/E , ∆ J/J c and | ∆ J | /J c is plotted as function of the normalized time-lag τ = ∆ t/P in the range ≤ τ ≤ . The mass precession coherence time of the system is τ M = t M /P ∼ . × ,and the potential fluctuation coherence time is τ φ = t φ /P ∼ . × , so both scalar and vector RR areexpected to grow linearly over the plotted range. More detailed analysis shows that the ∆ J/J c ( E ) is afunction of both energy and angular momentum, which for τ → scales as √ τ , and for ≪ τ ≪ τ w scalesas τ , and that | ∆ J | /J c is simply proportional to ∆ J/J c . The correlation analysis is an efficient method forquantifying relaxation in N -body results (cf Rauch & Tremaine 1996, Fig. 1). The theoretical predictionsfor ∆ J/J c and | ∆ J | /J c fit the numeric results very well. As expected, ∆ E/E ∝ √ τ at all time-lags. that due to the distributed mass is retrograde, and so they may partially cancel each other. Theircombined effect on the scalar RR timescale is T sRR ≃ A sRR N ⋆ ( < a ) (cid:18) M • M ⋆ (cid:19) P ( a ) (cid:12)(cid:12)(cid:12)(cid:12) t M − t GR (cid:12)(cid:12)(cid:12)(cid:12) . (4.5)Since t M increases with r , while t GR decreases with r , scalar RR is fastest at some finite distancefrom the MBH, which typically coincides with ∼ r crit / for LISA EMRI targets (Fig. 13).Precession does not affect vector RR. The coherence in a non-Keplerian spherical potential islimited by the change in the total gravitational potential φ = φ • + φ ⋆ caused by the fluctuationsin the stellar potential, φ ⋆ , due to the realignment of the stars as they rotate by π on their orbits, t ω = t φ = φ ˙ φ ⋆ ∼ N / ⋆ ( < a )2 µ P ( a ) ≃ M • N / ⋆ ( < a ) M ⋆ P ( a ) , (4.6)where the last approximate equality holds in the Keplerian regime. The vector RR timescale isobtained by substituting t φ in Eq. (4.1), T vRR = 2 A vRR N / ⋆ ( < a ) µ ( < a ) P ( a ) ≃ A vRR (cid:18) M • M ⋆ (cid:19) P ( a ) N / ⋆ ( < a ) . (4.7)Vector RR is much faster than scalar RR (Fig. 13).. Alexander: Relaxation Processes Near the Galactic MBH Resonant relaxation and EMRI rates The efficiency of scalar RR quickly decreases with distance from the MBH, since the coher-ence time falls as M ⋆ ( < r ) /M • grows. At r h , where M ⋆ ( < r h ) /M • ∼ O (1) , scalar RR isalmost completely quenched. Because r crit ∼ r h for tidal disruption (Lightman & Shapiro 1977;§1.1), RR does not significantly enhance the tidal disruption rate (Rauch & Tremaine 1996). Incontrast, r crit ∼ . pc for EMRI events, where M ⋆ ( < r crit ) /M • ≪ and T sRR is near itsminimum. Scalar RR therefore dominates the dynamics of the loss-cone for GW EMRI events(Hopman & Alexander 2006 a ). Scalar RR accelerates the flow of stars in phase-space from large- J orbits to low- J orbits that approach the MBH and can lose orbital energy and angular momen-tum by the emission of GWs. However, if unchecked, RR would continue to rapidly drive thestars to plunging orbits that fall directly into the MBH. This is where GR precession is pre-dicted to play an important role (Hopman & Alexander 2006 a ). Orbits with very small periapse, r p ∼ few × r s , where GW emission becomes appreciable, are also orbits where the GR precessionrate becomes large enough ( t GR becomes short enough) to quench RR, and allow the EMRI in-spiral to proceed undisturbed. This subtle cancellation, which is critical for the observability ofEMRI events, still has to be verified by direct simulations.The effect of scalar RR can be included in an approximate way in the Fokker-Planck equa-tion (Eq. 3.3) as an additional loss-cone term R RR ∝ χg/T sRR , where the efficiency factor χ parametrizes the uncertainties that enter through the various order-unity factors A ωRR (Eq. 4.1).Such calculations show that the poorly determined value of the efficiency can strongly affect thepredicted EMRI rates (Fig. 12). As the efficiency rises, the EMRI rate first increases becausestars are supplied faster to the loss-cone, but when the efficiency continues to rise the stars aredrained so rapidly into the MBH, that the EMRI rates are strongly suppressed.Rauch & Tremaine (1996) explored the efficiency of RR by a few small-scale N -body simu-lations, and noted a large variance around the derived mean efficiency. Here we use their meanefficiency as the reference point ( χ = 1 ), but consider also values smaller and larger. Figure(12) shows the GW inspiral rate and direct plunge rate as function of the unknown efficiency χ , relative to the no-RR case ( χ = 0 ), derived from Fokker-Planck calculations of a single masspopulation. The EMRI rate rises to ∼ times more than is expected without RR, peaking at . χ . , but then falls rapidly to zero at χ & . The strong χ -dependence of the EMRI ratesprovides strong motivation to determine the RR efficiency and its dependence on the parame-ters of the system both numerically (Gürkan & Hopman 2007; Eilon, Kupi & Alexander 2007,in prep; Fig 11), and by direct observations of the only accessible system at present where RReffects may play a role—the stars around the Galactic MBH.4.3. Resonant relaxation and stellar populations in the GC The stellar population in the GC includes both young and old stars, and is composed of distinctsub-populations, each with its own kinematical properties (see Alexander 2005 for a review). Asshown below, RR can naturally explain some of the systematic differences between the variousdynamical components in the GC. Conversely, GC observations of these populations can thentest the various assumptions and approximations that enter into analytic treatment of RR, and inparticular constrain the poorly determined RR efficiency.Figure (13) summarizes the typical distance scales and ages associated with these populations,and compares them with the various relaxation timescales. The calculation of the relaxationtimescales are approximate since they assume a single mass population. The non-resonant 2-body relaxation timescale (Eq. 1.1) is roughly independent of radius in the GC, T NR ∼ few × yr (assuming a mean stellar mass of M ⋆ = 1 M ⊙ ; in a multi-mass system it is expected to decreaseto T NR ∼ yr in the inner 0.001 pc due to mass segregation, Hopman & Alexander 2006 b ).Because neither the RR efficiency, nor the mass function is known with confidence, the scalarRR timescale, T sRR , is shown for two different assumptions; χM ⋆ = 1 M ⊙ and χM ⋆ = 10 M ⊙ .0 T. Alexander: Relaxation Processes Near the Galactic MBH −2 −1 −1 χ Q ( χ ) / Q N R GW inspiralPrompt infall Inspiral Plunge NR + RR r a t e / NR r a t e Resonant relaxation efficiency Rauch & Tremaine 1996 F IGURE 12. The relative rates of GW EMRI events and direct infall (plunge) events, as function of theunknown efficiency of RR, χ , normalized to χ = 1 for the values derived by Rauch & Tremaine (1996). As discussed in §4.1, beyond r ∼ . pc, T sRR > T NR due to mass precession, and the loss-cone replenishment is dominated by non-coherent relaxation. T sRR decreases toward the MBH,until it reaches a minimum, where it starts increasing again due to GR precession. The distancescale where T sRR is shortest happens to coincide with the volume r . r crit , where most GWEMRI sources are expected to lie and where T sRR ≪ T NR , so RR dominates EMRI loss-conedynamics (§4.2). In contrast to scalar RR, the vector RR timescale T vRR (shown here for anassumed χM ⋆ = 1 M ⊙ ) decreases unquenched toward the MBH.Dynamical populations and structures whose estimated age exceed these relaxation timescalesmust be relaxed. Those whose age cannot be determined, but whose lifespan exceeds the relax-ation timescales may be affected, unless we are observing them at an atypical time soon afterthey were created. The youngest dynamical structure observed in the GC is the stellar disk (orpossibly two non-aligned disks) (Levin & Beloborodov 2003; Genzel et al. et al. ∼ young massive OB stars with an age of t ⋆ ∼ ± Myr, onco-planar, co-rotating orbits that extend between ∼ . – . pc. The inner edge of the disk issharply defined and it coincides with the outer boundary of the S-stars cluster (§2.2). Figure (13)shows that the vector RR timescale equals the age of the stellar disk at its inner edge, and so isconsistent with the spatial extent of the disk. Even if the S-stars were initially the inner part ofthe disk (this does not appear likely given that they are systematically lighter than the disk stars),vector RR would have efficiently randomized their orbital inclination. However, their measuredhigh eccentricities (Eisenhauer et al. et al. M ⋆ ∼ – M ⊙ ,Genzel et al. r & . pc appear dynamically relaxed (Genzel et al. Relaxation Processes Near the Galactic MBH −3 −2 −1 1a (pc)10 t ( M y r) T NR T _ s RR T v RR Star disksS- starsGW sources Relaxed giants F IGURE 13. Evidence for resonant relaxation in the GC in the age .vs. distance from the MBH plane.The spatial extent and estimated age of the various dynamical sub-populations in the GC (shaded ar-eas) is compared with the non-resonant 2-body relaxation timescale (top line, for assumed mean massof M ⋆ = 1 M ⊙ ) and with the scalar RR timescale (two curved lines, top one for χM ⋆ = 1 M ⊙ , bottom onefor χM ⋆ = 10 M ⊙ ) and vector RR timescale (bottom line, for χM ⋆ = 1 M ⊙ ). The populations include theyoung stellar rings in the GC (filled rectangle in the bottom right); the S-stars, if they were born with thedisks (open rectangle in the bottom left); the maximal lifespan of the S-stars (filled rectangle in the middleleft); the dynamically relaxed old red giants (filled rectangle in the top right); and the reservoir of GWinspiral sources, where the age is roughly estimated by the progenitor’s age or the time to sink to the center(open rectangle in the top left). Stellar components that are older than the various relaxation times must berandomized. (Hopman & Alexander 2006 a , reproduced with permission from the Astrophysical Journal ). It should be noted that the effect of RR on the stellar density distribution is not expected to belarge even quite close to the MBH ( r . . pc), unless the efficiency χ is very high, because theRR-induced depletion of the DF at high energies is smoothed by the transformation from the DFto n ⋆ ( r ) (Eq. 3.4) and by the contribution of unbound stars to the central density. 5. Summary Relaxation processes play an important role in the GC, where the 2-body relaxation time isshorter than the age of the system and the stellar density is high. The scaling laws that followfrom the M • /σ relation imply that the same must hold for all galaxies with M • . few × M ⊙ .Relaxation processes affect the distribution of stars and compact remnants, lead to close interac-tions between them and the MBH, and may be related to the unusual stellar populations that areobserved in the GC. These are of relevance because of the very high quality stellar data comingfrom the GC, and because galactic nuclei with low-mass MBHs like the GC are expected to beimportant GW EMRI targets for the next generation of space borne GW detectors. In addition,efficient relaxation mechanisms that operate and can be studied in the GC may play a role evenin galactic nuclei with high-mass MBHs, where 2-body relaxation is unimportant.Three processes beyond minimal two-body relaxation were discussed here: accelerated loss-cone replenishment by MPs, strong mass-segregation in evolved populations, and rapid RR. Ev-idence was presented that these processes operate and may even dominate relaxation and itsconsequences in the GC: The S-stars and HVSs are consistent with relaxation by GMCs; thereare hints for strong mass segregation in the central density suppression of the low-mass Red2 T. 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