Dynamical Properties of Eccentric Nuclear Disks: Stability, Longevity, and Implications for Tidal Disruption Rates in Post-Merger Galaxies
Ann-Marie Madigan, Andrew Halle, Mackenzie Moody, Mike McCourt, Chris Nixon, Heather Werkne
DDynamical Properties of Eccentric Nuclear Disks: Stability, Longevity,and Implications for Tidal Disruption Rates in Post-Merger Galaxies
Ann-Marie Madigan, Andrew Halle, Mackenzie Moody, Michael McCourt, Chris Nixon, & Heather Wernke February 7, 2018
Abstract
In some galaxies, the stars orbiting the supermassive black hole take the form of an eccentric nuclear disk, in which everystar is on a coherent, apsidally-aligned orbit. The most famous example of an eccentric nuclear disk is the double nucleus ofAndromeda, and there is strong evidence for many more in the local universe. Despite their apparent ubiquity however,a dynamical explanation for their longevity has remained a mystery: differential precession should wipe out large-scaleapsidal-alignment on a short timescale.Here we identify a new dynamical mechanism which stabilizes eccentric nuclear disks, and explain for first time thenegative eccentricity gradient seen in the Andromeda nucleus. The stabilizing mechanism drives oscillations of the eccentricityvectors of individual orbits, both in direction (about the mean body of the disk) and in magnitude. Combined with thenegative eccentricity gradient, the eccentricity oscillations push some stars near the inner edge of the disk extremely close tothe black hole, potentially leading to tidal disruption events.Order of magnitude calculations predict extremely high rates in recently-formed eccentric nuclear disks ( ∼ − − gal − ). Unless the stellar disks are replenished, these rates should decrease with time as the disk depletes in mass. Ifeccentric nuclear disks form during gas-rich major mergers, this may explain the preferential occurrence of tidal disruptionevents in recently-merged and post-merger (E + A / K + A) galaxies.
The tidal gravity of supermassive black holes (SMBHs) givesrise to some of the most energetic phenomena in the uni-verse: tidal disruption events, hyper-velocity stars, and grav-itational wave inspirals of compact stellar remnants. Fun-damentally, the rates of these events are determined by thedynamics of stars in the vicinity of SMBHs. It is commonlyassumed that nuclear star clusters are spherically symmetric,and that gravitational two-body scattering is the dominantprocess determining the rates of these events. However, thedistribution of stars around SMBHs is often observed to behighly asymmetric, and coherent torques between stellarorbits may strongly dominate over two-body interactions.For a historical perspective, we look to our most massive Astrophysical and Planetary Sciences & JILA,CU Boulder, CO; [email protected] UC Berkeley, CA Princeton University, NJ UC Santa Barbara, CA; NASA Hubble Postdoctoral Fellow Theoretical Astrophysics Group, Department of Physics & Astronomy,University of Leicester galactic neighbor.The asymmetric nucleus of the Andromeda galaxy (M31)has been a puzzle since its discovery by balloon-borne exper-iments (Light et al., 1974). It took nearly two decades fora telescope of sufficient resolution (
Hubble Space Telescope ;HST) to resolve the nucleus into two distinct components: afaint peak (P2) lying approximately at the bulge center, anda brighter component (P1) offset by 0. (cid:48)(cid:48) ≈ a r X i v : . [ a s t r o - ph . GA ] F e b nd a fainter one at periapsis (P2), where orbits pinch to-gether and create an enhancement in surface brightness.The periapsis peak is faint, however, and is only visible fora narrow range of nearly edge-on orientations. Though theM31 disk at first seems exotic, seeing an improbable arrange-ment in a nearby galaxy suggests that these disks shouldin fact be common. Indeed, while no systematic search foreccentric nuclear stellar disks has been conducted, thereis growing evidence that they are unexceptional. In theLauer et al. (2005) sample of 65 non-dust-obscured early-type galaxies, ∼
20% show features consistent with eccentricnuclear stellar disks seen from different angles (offset nu-clei, nuclei with central minima and double nuclei). This isdespite the observational challenges of detecting such sig-natures; stable eccentric nuclear disks live within the radiusof influence of the SMBH, a scale difficult to resolve exceptfor the closest and / or most massive galaxies (see § 5).Hopkins & Quataert (2010a,b) show that eccentric nu-clear stellar disks may originate in gas-rich galaxy mergers:they form when gas that is funneled to the center of the po-tential fragments and forms stars on aligned eccentric orbits.The presence and prevalence of these disks may thereforeencode a wealth of information about galaxy formationand merger histories. Despite the importance and apparentubiquity of eccentric nuclear disks however, the stability ofsuch disks has remained a mystery. Individual orbits withinthe disk should precess at different rates, smearing out thecoherent structure on a differential precession timescale of ∼ Myr, far shorter than the (cid:166)
Gyr ages of the stars in the M31disk. This has led many (e.g., Merritt, 2013) to concludethat the eccentric disk we see in M31 is a transient feature.In this paper we identify a new dynamical mechanismwhich stabilizes eccentric nuclear stellar disks against smear-ing by differential precession, thus explaining the longevityof these structures. We show that in an isolated system, asmall fraction of outermost orbits break away to form theirown oppositely-precessing structure, for which there may beevidence in the M31 nucleus (Menezes et al., 2013; Brown& Magorrian, 2013). We also show that the mechanism nat-urally reproduces the negative eccentricity gradient of theM31 disk (Peiris & Tremaine, 2003). An important corollaryof our disk stabilizing mechanism is that the stars undergooscillations in orbital eccentricity. During these oscillations,some fraction of the stars approach plunging orbits and aresusceptible to tidal disruption by the SMBH near periap-sis. Thus, we propose that eccentric nuclear disks, via theirstabilizing mechanism, fundamentally change the rates atwhich stars interact with SMBHs. Such interactions may sig- nificantly enhance the production rates of tidal disruptionevents, hyper-velocity stars and binaries, and gravitationalwave inspirals of compact objects.We present the paper as follows: in § 2 we describe a newdynamical model which explains the stability of eccentricnuclear disks. In § 3 we compare predictions of our modelwith N -body results, showing that an eccentric disk developsa negative eccentricity gradient, and that individual orbitsundergo large-amplitude oscillations in eccentricity and thatsome may even flip in inclination. In § 4 we describe how ourdynamical model can explain the enhanced tidal disruptionevents seen in recently-merged and post-merger galaxies.In § 5 we summarize our findings and discuss our results. The orbit of a star near a SMBH of mass M • can be describedby two vectors: the (specific) angular momentum vector j = r × v , which defines the orbital plane, and the eccentricityvector e = ( v × j ) / ( GM • ) − ˆ r , which points toward periapsisand whose magnitude equals the eccentricity of the orbit.In a perfectly Keplerian potential, the vectors j and e areconstant. If we add a small non-Keplerian force f , however,the orbit vectors evolve in time according to j (cid:48) = r × f = τ (1a) e (cid:48) = f × j GM • + v × τ GM • , (1b)where τ ≡ r × f is the specific torque produced by thenon-Keplerian force.An eccentric disk is characterized by apsidally-alignedorbits, i.e., orbits with aligned e -vectors. Non-Keplerianforces induce precession of the e -vectors (equation 1b) atrates that depend on the semi-major axes of the orbits (or-bits near the inner edge of the disk have smaller angularmomentum and experience larger forces than orbits at theouter, less dense edge of the disk). Hence we would ex-pect the e -vectors to spread out on a differential precessiontimescale, and that the apsidally-aligned orbits observed inthe M31 nucleus and in so many other galaxies should be ashort-lived, transient phenomenon.The forces that drive differential precession in e -vectors,however, also result in coherent gravitational torques (equa-tion 1a) acting between stellar orbits (Rauch & Tremaine,1996). In the inner arcsecond ( ≈ ∼
15% of the mass of theSMBH ( M • ≈ − × M (cid:12) ; Bender et al., 2005). Hence the2 x (cid:228) x (cid:228) x (cid:228) x (cid:228) x (cid:228) x (cid:228) x (cid:228) x (cid:228) x (cid:228) x precessionOrbit Leads Disk: e j precessionOrbit Lags Behind Disk: e j Figure 1:
Physics of orbital oscillations within a stable eccentric disk.
The entire disk (black) precesses in the prograde direction (counter-clockwise in this figure). If an orbit (red) moves ahead of the disk(left panel), it feels a gravitational pull towards the bulk of the disk. This torques the orbit, decreasing its angular momentum and thusincreasing its orbital eccentricity ( j ∝ ( − e ) ). This lowers the orbit’s precession rate, allowing the bulk of the disk to catch up withit. The reverse happens for an orbit which lags behind the bulk of the disk (right panel). The overall effect is to stabilize the disk: anyorbit which is perturbed off the disk is driven back toward it by torques and differential precession. The mechanism inducing thisstability leads to oscillations in eccentricity. secular (orbit-averaged) gravitational torques exerted bythe disk orbits on each other are strong and can effectivelycounteract e -vector differential precession.The stability of a disk depends entirely on the direction ofits e -vector precession. In Madigan et al. (2009), we focusedon the case in which an eccentric stellar disk is embeddedin a more massive ∼ symmetric nuclear star cluster (such asin the Milky Way Galactic center; Feldmeier et al., 2014).This additional gravitational potential leads to retrogradeprecession of the orbits ( e (cid:48) · v p < v p is the velocityat periapsis). Retrograde precession, combined with mutualgravitational torques, results in an ‘eccentric disk instability’which propels the orbits apart.Here we focus on the case in which the mass of the asym-metric eccentric disk is much greater than the backgroundstellar potential (as is true for the M31 nucleus; Kormendy& Bender, 1999), such that the direction of precession isreversed. Prograde precession ( e (cid:48) · v p >
0) leads insteadto stability: orbits which precess ahead of the disk feel a An intuitive explanation for this retrograde precession can be seenfrom equation 1b. In Gauss’s approximation, we spread the mass of the starover its orbit, with the density inversely proportional to the instantaneousvelocity. Hence most of the mass of the orbit is located at apoapsis. Forsufficiently eccentric orbits and for typical nuclear star cluster densityprofiles, the forces experienced at this location dominate over those atperiapsis. A spherical gravitational potential results in an inward radialforce at apoapsis; the first term in equation 1b gives retrograde precession. gravitational pull toward the disk behind it. We show thisin the left panel of figure 1. This gravitational force createsa torque τ z < j = GM • a ( − e ) (2a) E = GM • a . (2b)The torque does not affect the energy, or equivalently thesemi-major axis, of the orbit. Hence the torque raises theorbital eccentricity of the orbit. Increasing the eccentric-ity slows its angular precession rate ( ∝ f × j / e ), stallingthe orbit until it is reabsorbed by the mean body of thedisk. A similar analysis shows that orbits which lag behindthe disk decrease in eccentricity, precess more rapidly andare driven back towards the bulk of the disk (right panelof figure 1). This stability mechanism implies both that acoherent precessing eccentric disk maintains its shape inresponse to perturbations, and that perturbed orbits un-dergo oscillations in eccentricity and in orientation aboutthe mean body of the disk. A similar analysis shows that suf-ficiently massive eccentric disks are stable to perturbationsin inclination resulting from out-of-plane forces. Stronglyperturbed and / or extremely eccentric orbits can flip theirorientation however; see § 3.4.3 N -body Simulations of Eccentric Disks We can use these results to study stable models of eccentricdisks using N -body simulations. We use the REBOUND codewith the IAS15 integrator (Rein & Liu, 2012; Rein & Spiegel,2015) with no gravitational softening. We chose generalinitial conditions for disk orbits, initializing N = − a = − Σ ∝ a − . The diskto SMBH mass ratio is M disk / M • = e = Our model posits that eccentric orbits in an asymmetric diskare stable so long as they apsidally precess in a progradedirection. We verify this with N -body simulations. Figure 2shows results for an eccentric disk with initial orbital eccen-tricities e = ( e x , e y ) . As expected, the bulk ofthe eccentric nuclear disk stably precesses with progrademotion (counterclockwise; blue points). The outermost or-bits (5 −
10% by mass) are pushed via two-body relaxationbeyond the initial outer edge of the disk, and precess moreslowly with retrograde motion (clockwise; red points). Thedirection of precession of orbits at intermediate semi-majoraxes is time-dependent – these orbits undergo both progradeand retrograde precession as secular torques change theireccentricities. Dissipation decreases the mean precessionrate of the disks. This is also due to two-body relaxationwhich increases the semi-major axis range of the stars asthey scatter off one another.In figure 3, we show the transition between theprograde-precessing eccentric nuclear disk and the outer-most retrograde-precessing orbits. The orange line showsthe mean longitude of periapsis (cid:36) as a function of time forthe innermost 76% of the disk orbits (in semi-major axis).The dark blue line shows the same for the outermost 6%of the disk orbits. Three things stand out from this plot.The first is that the transition between the inner prograding-precessing disk and the outer retrograde precessing orbitsis not abrupt. Second, the outer orbits precess at abouthalf the rate of the prograde-precessing disk. Third, theprecession rate of the outer orbits slows when the orbitsoverlap with the main disk. This last point is importantobservationally: the two-oppositely precessing structures will not necessarily appear as separate components. Theslow-down in precession is a direct result of angular mo-mentum exchange. As the outer orbits approach the innerprograde-precessing disk, they loses angular momentum,increase in mean orbital eccentricity, and slow down in pre-cession. As they precesses past the disk, they are torqued tohigher angular momenta, decrease in mean orbital eccentric-ity, and speed up in precession. Hence they appear to racearound in precession towards the inner disk again. As theprograde-precessing disk is far more massive, it experiencessmaller eccentricity variations.Our idealized, low- N simulations can not provide a di-rect comparison to the M31 nucleus. Due to the artificiallyhigh two-body relaxation rate, the stars diffuse rapidly insemi-major axes thus decreasing both the orbital precessionrates and the strength of the torques exerted between or-bits. Both of these affect the eccentricity structure of thedisk which feeds back into the precession rates. A carefulconvergence study will therefore be required to make aconcrete comparison with the M31 nucleus. We must alsoadd the M31 bulge potential and general relativistic pre-cession, explore a smaller mass ratio between the SMBHand the disk, and expand the semi-major axis range of thedisk. Nevertheless, across a broad range of parameter space,our simulations generically show that the outermost orbits( ≤
10% in mass) break away into an oppositely-precessingstructure, and there may be observational evidence of thisin the M31 nucleus.Brown & Magorrian (2013) explore non-parametric mod-els of the M31 nuclear disk. They find that some sections ofthe disk are not aligned with others. This is consistent withour model. In detail, however, they find that orbits are anti-aligned with respect to the main disk both inside r < (cid:48)(cid:48) r > (cid:48)(cid:48) .2. This is not what we predict, but thisis likely an artifact of biaxial symmetry in their model. Theinner structure that they describe behaves similarly to theinner disk in our simulations: eccentricities, inclinations,and their standard deviations decrease with semi-major axis.There are differences in the main disk structure that theyfind however. The inclinations of orbits increase with radiuswhich is the opposite of what we find. We note howeverthat our simulations do not include a grainy ( N -body) back-ground potential; this has been shown to be important inthe growth of orbital inclinations of disk stars (Löckmannet al., 2009).Menezes et al. (2013) recently reported the discoveryof an eccentric H α -emitting disk in the inner 0. (cid:48)(cid:48) igure 2: Time evolution of an eccentric disk
Eccentricity vectors of stars in an N -body simulation (projected into the disk plane), plotted every secular timescale ( ∼
100 orbits)(left to right, top to bottom). The dashed grey circle indicates values of e =
1. Stars are initialized with eccentricities e = = e -vectors spread out due to orbits oscillating back and forthacross the mean body of the disk and high eccentricity orbits gaining significant inclinations. The very outermost orbits precess withretrograde motion (clockwise). They exchange angular momentum via coherent gravitational torques with the prograde-precessingdisk when they encounter it at low relative azimuthal angles. t [ t sec ] h ̟ i ( d e g ) Figure 3:
Mean precession of disk orbits
Mean longitude of periapsis, (cid:36) , as a function of time in units ofthe secular timescale, t sec ≡ M • / M disk P . The various colored linesshow different sections of the disk binned in semi-major axis tohighlight the reversal in the direction of precession of the outerorbits. The orange line shows 〈 (cid:36) 〉 for the innermost 76% of thedisk orbits (in semi-major axis). The red (76% - 82%), green (82%- 88%), and light blue (88% - 94%) lines show the precession forsections of the disk increasing in semi-major axis. The dark blueline shows 〈 (cid:36) 〉 for the outermost 6% of the disk orbits which havea net precession in the opposite, retrograde, direction. or gaseous in origin – is not yet known. Using a simpleorbit model, they derive Kepler elements that show it tobe related to, but distinct from, the P1-P2 stellar disk. Itseccentricity, inclination and position angle of ascendingnodes ( e = i = ◦ , P. A. = ◦ ) are very similar tothat derived for the stellar P1-P2 disk ( e = i = ◦ , P.A. = ◦ ). Noticeably however, the argument of periapsis ω = ◦ is very different from that of the stellar P1-P2disk, ω = ◦ . This translates to a longitude of periapsis difference of ∆ (cid:36) ∼ ◦ between the two structures.Though it is not yet clear how to map the Brown & Magor-rian models and Menezes et al. H α -observations to oursimulations or even to each other, a stable eccentric nucleardisk is likely to be more complex than a single monolithicstructure. Orbits of the same eccentricity with low semi-major axeswill precess faster than those with high semi-major axes(from equation 1b; e (cid:48) ∝ f × j ∝ ( − e ) / a − / taking f ∝ a − ). For the inner disk to stably precess as one body,the orbits at low semi-major axes must increase their ec-centricities with respect to the mean, while the orbits at (cid:36) is the angle between the eccentricity vector and a reference direc-tion. Figure 4:
Structure of a stable eccentric nuclear disk
Top : eccentricity gradient as a function of semi-major axis after ∼
200 orbits. Dots indicate the mean values of disk stars. Thegray region shows one-sigma standard deviation. Stars at theinnermost edge of the disk reach e ∼
1, and are susceptible todisruption. At t = e = Bottom :inclination gradient as a function of semi-major axis. Dots indicatethe median values of disk stars. The vertical dashed lines indicatethe division between the stable prograde-precessing disk and theouter retrograde-precessing orbits. high semi-major axes must become more circular. They dothis naturally via the coherent torques as described in § 2.Hence our model predicts that a stable equilibrium eccen-tric stellar disk will have a negative eccentricity gradient:orbital eccentricities decrease as a function of semi-majoraxis.Our N -body simulations verify this prediction. As anexample we show results from a simulation with initialdisk eccentricity e = de / da < 〈 e 〉 ≈ 〈 e 〉 ≈ a (cid:166) di / da <
0. The highest values at low semi-major axes are drivenby the ‘flipping’ of extreme eccentricity orbits to i ∼ ◦ (see § 3.4). In this high eccentricity disk, secular (orbit-averaged) torques dominate the dynamical evolution of the6 igure 5: Oscillations of orbits
Eccentricity and longitude of periapsis e − (cid:36) tracks for two starsin a simulation. Data are plotted every orbit over ∼
300 orbitalperiods. The inner (orange) orbit oscillates about the main bodyof the disk and varies in eccentricity as predicted from our simplemodel. We subtract the mean longitude of periapsis of the innerdisk to highlight this. The outer (blue) orbit demonstrates theexchange of angular momentum between the inner and outer diskas they precess past each other. system. Orbits with high eccentricities have less angularmomentum and therefore require less force to be torquedaround, and so a negative inclination gradient is a naturalconsequence of the negative eccentricity gradient.One might expect from the inclination distribution thatthe inner disk is more geometrically thick than the outerdisk. This is not the case however, as pointed out by Haas& Šubr (2016): the high eccentricity orbits have ω valuessuch that they are relatively embedded in the disk. In otherwords, inclinations grow via orbits rolling over their majoraxes. Since the orbits are so eccentric, their high inclinationsdo not lift them far above the disk in 3D space. In termsof the azimuthal distribution of orbits, the stable eccentricnuclear disk is a coherent, prograde-precessing structurewith a relatively small deviation of (cid:36) values, whereas theoutermost orbits form a less coherent structure. The fullthree-dimensional structure of eccentric disks, which varieswith semi-major axis and also with azimuth, will greatly af-fect the observational signatures of disks seen from differentangles. As discussed in § 2, the stabilizing forces which hold thedisk together against differential precession also drive oscil-lations in the individual orbits eccentricity vectors. Theseoscillations are driven by gravitational torques – which ad- just orbital eccentricities and limit the differential precessionrate – and are damped by two-body relaxation. In figure 5we show longitude of periapsis (cid:36) tracks for two stars over ∼
300 orbital periods, subtracting the mean longitude ofperiapsis of the main disk to highlight orbital oscillations.The inner (orange) orbit oscillates with counter-clockwisemotion about the main body of the disk as described in§ 2. When ahead of the disk ( (cid:36) − 〈 (cid:36) 〉 > (cid:36) − 〈 (cid:36) 〉 < i e ≡ arctan e y / e x (Madigan & McCourt, 2016).We note that this angle is equivalent to the longitude ofperiapsis (cid:36) (= Ω + ω ) for orbits of zero inclination, i e = Ω − arctan [ cos ω , cos i sin ω ] + π/
2. The precession rate,that is the time rate of change of i e , is di (cid:48) e = − j f r GM • e cos ψ + τ || v r GM • e (cid:149) e + cos ψ (cid:152) , (3a) ∼ ( e − − ) / π t − (3b)where ψ is the true anomaly of the star on its orbit, τ || ≡ τ · ˆ j is the perpendicular component of the torque and v r is theradial velocity of the star at ψ (Madigan et al., 2017). Thefirst term in equation 3a describes precession due to a radial,non-Keplerian specific force f r . For simplicity we evaluatethis at apoapsis (cos ψ = −
1) and ignore the second termwhich results from fitting Kepler elements to a slightly non-Keplerian orbit. We approximate the magnitude of the radial force experienced at apoapsis as f r ∼ GM disk / a and definethe secular dynamical timescale t sec ≡ (cid:129) M • M disk (cid:139) P (4)We denote the angular difference between the e -vectorof a test orbit and the mean e -vector of the disk by δ i e . Tofirst order, differences in eccentricity drive spreading of thedisk according to δ i (cid:48) e = di (cid:48) e de δ e (5) Note that this approximation is valid only for high eccentricity orbits.Furthermore, the rough approximation of the radial force, f r , which wealso made in § 3.2, does not account for the radial spread in orbits in thedisk. t [ t sec ] i ( d e g ) . . . . . . . . . . . e t sec i ( d e g ) . . . . . . . . . . . e t sec − − i e − h i e i ( d e g ) i ( d e g ) Figure 6:
Inclination and eccentricity of a single star in the stably-precessing inner disk. ( Top ): Inclination ‘flips’ (green) correspond to high eccentricity peaks (orange). Evolution is shown as a function of time in units of thesecular timescale t sec ≡ M • / M disk P . ( Bottom ): Zoom-in on inclination flip (green circles, plotted every orbital period), with eccentricity(left; orange diamonds) and relative i e ( ≡ arctan e y / e x ) value with respect to the main body of the inner disk (right; blue pentagons). From equation 3b, di (cid:48) e de = − (cid:18) e − (cid:112) e − − (cid:19) (cid:129) π t sec (cid:139) (6)Next we model the azimuthal in-plane force felt by atest orbit by the disk. The eccentric disk has some meanazimuthal spread (angular spread of the eccentricity vectorsin the disk plane) which we denote as φ disk . A test orbit lyingoutside this range experiences a force f ∼ − GM disk / ( a δ i e ) .Inside the disk, however, the force should not tend to infinity.In fact opposing forces ∼ cancel near the middle of the disk.Here we approximate the force as f ∼ − αδ i e , where α = GM disk / ( a φ ) . The time rate of change of eccentricity of a test orbit within the disk is given by e (cid:48) = E ( j · τ )( GM • ) e + j ( v · f )( GM • ) e (7a) = (cid:112) e − − φ δ i e (cid:129) π t sec (cid:139) (7b)where we have ignored the second half of the equation 7aas it oscillates over an orbital period, and used τ ∼ r f cos ( δ i e ) ∼ r f .Differentiating equation 5, and equating δ e (cid:48) = e (cid:48) , gives δ i (cid:48)(cid:48) e = di (cid:48) e de e (cid:48) (8a) = − ( e φ disk ) − δ i e (cid:129) π t sec (cid:139) (8b)This yields the equation for simple harmonic motion, δ i (cid:48)(cid:48) e = ω δ i e , with the oscillation frequency ω osc = ( e φ disk ) − / (cid:129) π t sec (cid:139) . (9)The oscillation period is t osc = πω osc = ( e φ disk ) / t sec (10)Taking representative values of e ∼ φ disk ∼ O (100 P). This isthe order of magnitude we see for oscillations in figures 5,6, and 8. Since stars with very high orbital eccentricity have low an-gular momentum, they are easily torqued around and mayundergo orbital ‘flips’. In figure 6 we focus on one such star.The top plot shows its eccentricity (orange; right y -axis)and inclination (green; left y -axis) evolution over the en-tire simulation. High eccentricity peaks often correspondto a rapid reversal of orbital inclination. The bottom twoplots show a zoom-in at a time when the orbital inclination(green circles, plotted every orbital period) flips from pro-grade to retrograde and back in tandem with eccentricityoscillations (orange diamonds). In the bottom right plot,we show i e ( ≡ arctan e y / e x ) relative to the mean i e of theinner disk (blue pentagons), essentially quantifying the dif-ference in eccentricity vectors in the plane of the disk. Thedouble-peaked eccentricity profile and corresponding dou-ble inclination flip can be explained using simple Newtonianmechanics as follows:As the orbit precesses ahead of the main body of the disk( t ∼ t ∼ i ∼ ◦ and the torque is now acting in the samedirection as the angular momentum vector of the orbit andso starts to circularize it. As the disk overtakes the orbit( t ∼ ? observesimilar behavior in simulations of nearly-coplanar, counter-rotating orbits, using a softened Gauss code which secularlyevolves systems of gravitationally interacting Kepler ellipses. Eccentricities oscillate from near-circular to near-radial andback, as inclinations flip between prograde and retrogradenear the peaks of the eccentricity oscillation. These flipshave also been detected in N -body simulations of mutually-interacting eccentric orbits and disks (Löckmann et al., 2009;Kazandjian & Touma, 2013; Haas & Šubr, 2016; Šubr &Haas, 2016).Li et al. (2014) study similar near-coplanar flips in the con-text of hierarchical three-body systems using Hamiltoniandynamics. They identify the dynamics with the octupoleeccentric Kozai-Lidov mechanism (third order in semi-majoraxis ratio between inner and outer binary). If we relatethe outer perturber in their models with the coherent ec-centric disk in ours, it is unsurprising that we find similardynamics. We note however, that the high eccentricities andsimilar semi-major axes of eccentric nuclear disk stars failthe stability criterion for hierarchical three-body systems ε = a a e − e < a ≈ a , e (cid:166) ε (cid:166) N -body simulations. General relativistic precessionacts in a prograde direction at a rate which increases withorbital eccentricity. This increases the differential preces-sion rate between perturbed, high-eccentricity orbits andthe disk. Preliminary results suggest however that the rateof inclination flips remain constant; secular torques fromthe eccentric nuclear disk are strong enough to push or-bits through zero angular momentum faster than generalrelativistic precession can respond (Wernke et al, in prep.). A Tidal Disruption Event (TDE) occurs when a star is de-stroyed by the tidal force of a supermassive black hole (Hills,1975). A significant fraction of the disrupted stellar gas re-mains bound to the supermassive black hole (Rees, 1988),and its subsequent accretion produces a luminous flare.Dozens of TDE candidates have been observed to date inX-ray, UV and optical wavelengths (for review see Komossa,2015). Here we propose that eccentric nuclear disks maybe one of the dominant sources of TDEs in the universe.9e have shown that a stable eccentric nuclear disk devel-ops a negative eccentricity gradient. Consequently, a star’sdistance of closest approach to the black hole, p = a ( − e ) ,strongly decreases with decreasing distance to the blackhole. The eccentricity oscillations which occur on top of thisnegative eccentricity gradient means that low semi-majoraxis stars at the peak of their eccentricity oscillation arevulnerable to the tidal effects of the SMBH at periapsis.Stars will disrupt if, during their orbital oscillations, theireccentricities approach values of unity ( e + δ e → δ e − e ∼ (cid:118)(cid:116) e ( + e )( − e ) φ − / δ i e (12)where, as before, e is the initial eccentricity of the orbit, δ e is the magnitude of the eccentricity oscillation, and φ disk quantifies the angular “width” of the disk in radians, compa-rable to the amplitude of oscillation ( δ i e ). The disk rapidlyproduces TDEs whenever the TDE threshold δ e / ( − e ) (cid:166) e in figure 7,taking a range of values for φ disk and δ i e : 0.5 − e (cid:166) e (cid:166) N -bodysimulation with initial disk eccentricity e = ∼ two precession periods of the inner disk, all five stars aretorqued to ( − e ) < − .Since the mechanism stabilizing the disk as a whole setsup a large-scale eccentricity gradient, simply disruptingthe innermost stars will not deplete the supply: they willbe dynamically repopulated by differential precession onapproximately an oscillation timescale. This is the secular-dynamical analogue of “refilling the loss cone”, and it is much faster than the diffusive processes typically consideredwhen computing predicted TDE rates (e.g., Wang & Merritt,2004; Stone & Metzger, 2016). However, if we consideran eccentric nuclear disk in isolation, we see that such anelevated rate of TDEs cannot persist indefinitely. Since eachTDE removes mass but not angular momentum from thedisk, it lowers the mean eccentricity of the disk. Eventuallythe disk will drop below the TDE threshold δ e / ( − e ) =
1, causing the TDE rate to dwindle. This will also ceaseevolution of the disk, so we predict most eccentric disks tohave inner edges with e ∼ − N -body simulations and in figure 7, we observe thatthe inner edge of the disk needs to drop below e ∼ N -body simulations are overly-simplistic. Nevertheless,the fact that they agree with one another is encouraging.Given the negative eccentricity gradient, an inner edge of e ∼ 〈 e 〉 ∼ 〈 e 〉 ∼ N -body calculations. As the two-body relaxation rate scalesas ∼ N − , dissipation in our low- N simulations is artificiallyhigh. Convergence studies with high- N simulations arecomputationally demanding and beyond the scope of thiswork. However, we expect the effect of dissipation is tolower the amplitude of the oscillations and thus to artificiallyreduce the TDE rates in our simulations. Additionally, wewill need to improve our calculations with the addition ofbulge gravitational potentials and apsidal precession due togeneral relativity (although general relativistic precessiondoes not appear to lower TDE rates; Wernke et al, in prep.).Though we cannot yet use N -body simulations to cal-culate the TDE rate, we can make an order of magnitudeestimate for the TDE soon after disk formation. We take arepresentative SMBH mass of M • = M (cid:12) and an SMBH-disk mass ratio similar to what is observed in M31 ( ∼ M disk = M (cid:12) . We estimate that ∼
10% of stars are disrupted soon after disk formation . Wemake the simplest approximation and convert this directlyto 10 stars. The TDEs won’t happen instantaneously; weestimate that they occur over ten oscillation times, whichyields 10 stars being disrupted over 10 − orbital pe-riods (equation 10). The stars are preferentially disruptedat the inner edge of the disk. From observations of theM31 nuclear eccentric disk and the (non-apsidally aligned)nuclear disk in the Galactic center (Lu et al., 2009), weestimate the inner disk radius to be ∼ − r H ≈ In simulations with mean initial disk eccentricity of e = e = ( − e ) < − at some point duringthe first 2000 orbital periods. igure 7: TDE Threshold
A star is in danger of being disrupted by the SMBH when its equi-librium eccentricity plus its perturbed eccentricity tends to one, δ e / ( − e ) → e (cid:166) δ e / ( − e ) =
1, and the disk no longer produces TDEs. where r H is the radius of influence of the SMBH. The orbitalperiod at this radius is ∼
300 yr. Hence, we arrive at a rateof 0.3 − − gal − . We emphasize that this is an extremelycrude estimate. Nevertheless it illustrates the ease at whichhigh TDE rates can be generated by an eccentric nucleardisk. There is essentially no classical angular momentumbarrier, since self-gravity of the disk actively forces stars intothe black hole.Our model anticipates anomalously high TDE rates fromgaseous, recently-merged galaxies (Hopkins & Quataert,2010a,b). TDEs at this stage may be difficult to distinguishfrom quasar or AGN activity. Certainly, overlapping eventswill obscure the characteristic TDE luminosity signature( L ∼ t − / ; Phinney, 1989). TDEs interacting with a pre-existing magnetized accretion disk, built up from previousTDEs, may account for observed jetted TDEs (Tchekhovskoyet al., 2014; Kelley et al., 2014). Radiation pressure in hy-peraccreting TDE envelopes may also produce jets in TDEs(Coughlin & Begelman, 2014), and significant variabilitycan be induced in the fallback rates through gravitationalfragmentation of the tidal debris stream (Coughlin & Nixon,2015). TDEs involving high-metallicity stars (Batra & Bald-win, 2014; Kochanek, 2016) may help explain observedabundance anomalies in quasars.Tadhunter et al. (2017) recently discovered the first TDE candidate in a nearby ultra-luminous infrared galaxy(ULIRG). ULIRGs represent the peak of major, gas-richgalaxy mergers. Unlike in most ULIRGs, the central re-gion of this galaxy is not heavily enshrouded in dust andso presents us with a rare view of its star-forming, activenucleus. Given this unusually clear sightline and the smallsample of ULIRGs from which the event was discovered, theauthors suggest that the TDE rate in such galaxies could be ∼ − yr − . Interestingly, the TDE flare was unusuallyprolonged compared with typical TDEs, with the light curveflattening at late times rather than following the typical L ∼ t − / decline predicted for isolated events.Since the eccentric disk loses stellar mass but not angu-lar momentum through TDEs, its mean eccentricity willdecrease with time. Lowering the eccentricity at the inneredge of the disk reduces the probability of TDEs, as shownin figure 7. Hence the TDE rate will decrease with time.Eventually, individual events will be detectable on top of thebaseline AGN activity. This may in some cases explain theso-called “changing-look quasars” (MacLeod et al., 2016),in which transient broad H α emission lines, possibly due toTDE flares, are seen superimposed on quasar spectra.When the time between TDEs is longer than the ac-cretion time of the stellar material onto the SMBH ( ∼ − gal − yr − ), TDEs will be individually identifiable.Thus we predict that TDEs occur at a lower rate in post-merger galaxies (with the rate continuing to decrease withtime since merger) than in ULIRGs, but that events shouldbehave more like typical TDEs following the theoretical L ∼ t − / decline. There may already be evidence to sup-port this: Arcavi et al. (2014) and French et al. (2016)have discovered that TDEs frequently occur in K + A galaxies.K + A (or E + A) galaxies are so called due to Balmer absorp-tion features in their spectra (characteristic of an A star)which appear superimposed on an old K star or (E)arly-typegalaxy population (Dressler & Gunn, 1983). The Balmerabsorption in K + A galaxies points to a significant starburstpopulation with ages ∼ − years, while low H α in-dicates a lack of ongoing star formation. There is strongevidence that galaxy-galaxy interactions and / or mergerstrigger the starburst (Yang et al., 2004), and the most lu-minous K + A galaxies appear to be successors of ULIRGs.There is even a connection to the formation of eccentricnuclear disks: the population of A-stars in K + A galaxiesis often very centrally concentrated, a consequence of gasdriven to the center during the merger (e.g., Yang et al.,2008), which hydrodynamic simulations of galaxy forma-tion suggest leads to eccentric disks (Hopkins & Quataert,11 t [ t sec ] − − − − − − − − e Figure 8:
Eccentricity evolution of stars in the inner prograde-precessing disk.
Orbital eccentricities of stars in an N -body simulation with initial disk eccentricity of e = t sec ≡ M • / M disk P ( = P ). The widths of the eccentricity peaks are often narrow, just a few orbits wide. Though we selectthese stars for their high eccentricities, we note that 36% of stars in this simulation reach 1 − e < − . ∼ − yr − per K + A galaxy, and only a few × − yr − per galaxy in ‘normal’ star-forming or ellipticalgalaxies.In our model, when the inner edge of the disk drops to e (cid:174) e ∼
1. Tidal interactions are greatly reduced and sothe eccentric disk structure ‘fossilizes’. The double nucleusin M31 is an example of such a remnant disk.
This paper focuses on the dynamics of eccentric nuclear disks ,in which the stars orbiting the super-massive black hole inthe center of a galaxy all have eccentric, apsidally-alignedorbits. In an eccentric nuclear disk, the individual stellarorbits nearly overlap, such that the entire star cluster takesthe form of an elliptical disk with the black hole at onefocus. While such a configuration at first seems exceedinglyimprobable, an eccentric nuclear disk has been observed inthe nearby galaxy M31, and quite possibly in many othergalaxies as well. Eccentric nuclear disks are only discerniblein very nearby galaxies, and double nuclei appear only fornearly edge-on configurations; the low probability of detect-ing eccentric nuclear disks therefore suggests they may bequite common in the local universe. It is surprising that suchan apparently finely-tuned structure should be seen in somany galaxies, and this result suggests that some unknownprocess actively creates and stabilizes eccentric disks. While self-gravity seems like a perfect candidate for sta-bilizing eccentric disks, ostensibly one would expect self-gravity to instead de stabilize these disks since it drives differ-ential precession. Stellar orbits precess in the non-keplerianpotential caused by the disk’s gravity; since stars near theinside and outside of the disk feel different gravitationalpotentials, they should precess at different rates, smearingout the angular extent of the disk.We have shown that such smearing by differential preces-sion does not in fact occur; the same non-keplerian forceswhich drive precession also torque the orbits and changetheir eccentricities. Since the precession rate also dependson eccentricity ( ∝ f × j / e ), we find the system reaches astable equilibrium in which the orbital energies and angularmomenta of individual stars balance such that each starhas a nearly identical precession rate, independent of itslocation within the disk. We sketch the dynamics of thismechanism in figure 1, and we show that the system is sta-ble in the sense that stars undergo oscillations about theirequilibrium configuration (see figure 5). Our model ex-plains the (cid:166) Gyr longevity of the M31 disk, and its negativeeccentricity gradient.In addition, our model makes a number of predictions,which may be testable in M31 and other nearby galaxies:1. We predict the presence of a less massive ( ≤ N -body12imulations, indicates that the inner and outer disk canreproduce M31’s double nucleus structure.2. In addition to the negative eccentricity gradient in M31,we also predict a negative inclination gradient (§ 3.2).3. Our model predicts a small population of counter-orbiting stars at low semi-major axes due to rapid in-clination flips (§ 3.4).4. Our stability model predicts that eccentric disks maybe quite common in nearby galaxies, especially post-merger galaxies. These disks will only manifest asdouble-nuclei over a narrow range of parameter space,but may appear as offset nuclei or as a central “hole”in the stellar distribution (Lauer et al., 2005). Using cosmological simulations, Hopkins & Quataert(2010a,b), show that tidal torques can drive gas into galac-tic nuclei during galaxy mergers. If enough gas enters thegalactic nucleus, it forms a lopsided, eccentric disk orbitingthe supermassive black hole. When this gas fragments andforms stars, it may produce an eccentric stellar disk verysimilar to the ones we study here.Not all eccentric disks are stable; reversing the logic infigure 1, one can see that if the orbits precess in a retro-grade manner, the disk becomes unstable. Retrograde pre-cession would naturally result from a massive symmetricbackground stellar population, such as the one found in theMilky Way Galactic Center. This may explain why we do notobserve one in our own Galaxy, despite the observationalevidence for a recent star formation event in a disk (Levin &Beloborodov, 2003). However, we note that during a galaxymerger, in-spiraling SMBHs are expected to scour out a corein stars in any prior nuclear star distribution leading to theconditions necessary for stability of a stable eccentric disk.Once a stable eccentric disk forms, it should persist untileither the background potential changes, or until the galaxyundergoes another merger. In either event, eccentric nucleardisks may last for many Gyr, serving as persistent markersof gas-rich galaxy mergers. The population of nearby eccen-tric disks may therefore encode valuable information aboutgalaxy mergers and evolution in the local universe.
Eccentric nuclear disks are stable in the sense that theyundergo stable oscillations about their equilibrium config-uration. Over the course of these oscillations, individual stars swing in both eccentricity and in orientation. Since theequilibrium configuration of the disk places the innermoststars at the highest eccentricity, these oscillations may takethe innermost stars to an eccentricity of one if the overalldisk eccentricity is high enough. Such stars are on plungingorbits and are therefore susceptible to tidal disruption bythe supermassive black hole. This leads to the corollary thateccentric nuclear disks may be remarkably prolific sourcesof TDEs: the self-gravity of the disk actively funnels starsinto the black hole, with essentially no analogue of the losscone. This process should continue until the stellar diskpartially circularizes, likely after losing a significant fractionof its mass to TDEs.While we cannot yet precisely quantify either the totalnumber of TDEs from a single eccentric nuclear disk, or pre-cisely how the TDE rate evolves with time, we can describethe expected behavior in a broad qualitative manner: • At early times the TDE rate should be very high (0.3 − − gal − ) and nuclei will be difficult to distinguishfrom an AGN. Along many lines of sight, TDEs will beblocked by the eccentric disk itself. • As stars are destroyed through tidal forces, the diskloses mass though not angular momentum and soshould become less eccentric over time. Since a lowerinitial eccentricity decreases the number of stars suscep-tible to disruption, the TDE rate will decrease with time,unless the mass of the disk is somehow replenished. • When the rate has dropped sufficiently such that theaverage TDE duration is shorter than time period be-tween TDE events ( ∼ yr − gal − ), individual TDEswill be easier to identify. At this point a galaxy may bein the post-merger phase (E + A / K + A). • When the inner edge of the disk drops to e (cid:174) e ∼
1. Tidal interactions are greatly reduced, thedisk ceases to evolve, and the eccentric disk ‘fossilizes.’This state may persist for many Gyr.
If our hypothesis is correct, and eccentric nuclear disks area dominant source of TDEs, gas-rich mergers and K + A / E + Agalaxies should host eccentric stellar disks within the radiusof influence of their SMBHs. Unfortunately both of thesegalaxy types are rare and typically too distant to resolvenuclear scales. When unresolved, an eccentric nuclear diskwill appear as an asymmetric nucleus, as in the case of M3113efore the era of H ST. Stone & van Velzen (2016) recentlydemonstrated using HST photometry that one of our near-est E + A galaxies, NGC 3156 (22 Mpc), has a significantlyasymmetric nucleus.Our model predicts a high rate of TDEs originating froma recently-formed eccentric nuclear disk. Therefore weexpect the age of stars being tidally disrupted to increaseas a function of time since merger. Recent major, gas-richmergers or (U)LIRGs should host TDEs of the youngest stars,with K + A / E + A galaxies hosting an older ( ∼ − M • (cid:174) M (cid:12) ). Eccentric nuclear disks orbiting moremassive SMBHs will still produce high eccentricity orbitsand in-falling stars will add to the growth of the SMBH, butthey will not result in observable TDEs.The mechanism driving stars to high eccentricities worksvia secular gravitational torques, not two-body relaxation.Therefore we might expect some stars to repeatedly grazethe loss cone before fully disrupting. Partial TDEs and TDEswith stripped outer envelopes should be commonly observed.The latter should lack hydrogen in their spectra (e. g. PS1-10jh; Strubbe & Murray, 2015). Multiple stellar systems, ifthey prevail in eccentric nuclear disks, will be more easilydisrupted than individual stars. This could result in hyper-velocity stars, binaries and tightly-bound high eccentricitystars close to the SMBH, detracting somewhat from theoverall TDE rate. Other likely outcomes include doubletidal disruptions and disruptions of highly magnetized stars(Mandel & Levin, 2015; Bradnick et al., 2017). We will study the structure of eccentric nuclear disks asa function of initial disk parameters such as eccentricityand surface density. We will also explore how the diskresponds to background stellar potentials and losing stel-lar mass via TDEs. Ultimately we would like to calculatethe time-dependent TDE rate due to evolving eccentric nu-clear disks. We will include the effects of general relativity,background gravitational potentials, and stellar mass segre-gation.With a suite of N -body simulations of differing initialconditions, we will be able to calculate the photometricand spectroscopic signatures of eccentric nuclear disks atdifferent resolutions. These may be used to determine theprevalence of eccentric nuclear disks in the local universe,and directly compared with high-resolution studies of the nuclear environments of ULIRGS (e.g., Medling et al., 2014)and E + A / K + A galaxies. Since eccentric nuclear disks arelong-lived, it’s also worth considering their observationalimplications; in particular, how their presence may biasSMBH mass measurements.We will additionally study the effect that the eccentricnuclear disk has on the background stellar population sur-rounding the SMBH. Torques from the non-axisymmetricdisk potential, analogous to the eccentric Kozai-Lidov mech-anism (Li et al., 2015; Naoz, 2016), should produce higheccentricity orbits in this older stellar population. This willresult in an older TDE population along with hyper-velocitystars and hyper-velocity binaries. We are interested to ex-plore whether the latter can account for the currently unex-plained distribution of unusual Ca-rich supernovae observedoffset from their host galaxies (Perets et al., 2010; Kasli-wal et al., 2012; Lyman et al., 2014; Foley, 2015). Finally,torques due to eccentric nuclear disks are an exciting possi-ble channel for the production of extreme-mass ratio inspi-rals (EMRIs) of compact stellar remnants into supermassiveblack holes, a prime target for low-frequency, space-based,gravitational wave observatories such as
LISA . Acknowledgments
AM thanks Mariska Kriek, Marcel van Daalen, and Freekevan de Voort for their encouragement at UC Berkeley. Shethanks Decker French, Ann Zabludoff and Nadia Zakam-ska for enlightening conversations. AM was supportedby UC Berkeley’s Theoretical Astrophysics Center. MkMcCwas supported by NASA grant NNX15AK81G and HST-HF2-51376.001-A, under NASA contract NAS5-26555. CN wassupported by the Science and Technology Facilities Coun-cil (STFC) (grant number ST / M005917 / References
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