The Effect of General Relativistic Precession on Tidal Disruption Events from Eccentric Nuclear Disks
Abstract
An eccentric nuclear disk consists of stars moving on apsidallyaligned orbits around a central black hole. The secular gravitational torques that dynamically stabilize these disks can also produce tidal disruption events (TDEs) at very high rates in Newtonian gravity. General relativity, however, is known to quench secular torques via rapid apsidal precession.
Here we show that for a disk to black hole mass ratio of , the system is in the full loss cone regime. The magnitude of the torque per orbital period acting on a stellar orbit means that general relativistic precession does not have a major effect on the dynamics.
Thus we find that TDE rates from eccentric nuclear disks are not affected by general relativistic precession. Furthermore, we show that orbital elements between successive TDEs from eccentric nuclear disks are correlated, potentially resulting in unique observational signatures.
JILA and Department of Astrophysical and Planetary Sciences, CU Boulder, Boulder, CO 80309, USA ^{†}^{†}heather.wernke@colorado.edu
JILA and Department of Astrophysical and Planetary Sciences, CU Boulder, Boulder, CO 80309, USA
1 Introduction
A tidal disruption event (TDE) occurs when a star is violently ripped apart by a black hole’s tidal forces (Hil75b). When a star is tidally disrupted, roughly half of the stellar debris remains bound to the black hole while the other half of the debris escapes. The gravitationally bound debris forms an accretion disk which feeds the black hole, producing a flare (Ree88). The current detection rate of flares from TDEs is about two per year (vanVelzen2018) and this is expected to increase with new surveys such as the Large Synoptic Survey Telescope (LSST) (vanVelzen2011).
TDE flares can provide insight into the mysteries of many areas of astrophysics. They illuminate central black holes in otherwise quiescent galaxies (Maksym2013; Macleod2014). We can use their observations to test theories of accretion physics and relativistic jets (Zauderer2011; Bloom2011; vanVelzen2016; Ale17). Tidal disruptions of white dwarfs should even produce gravitational waves detectable by the Light Interferometer Space Antenna (LISA) (Macleod2014). Additionally, we can test our understanding of gravitational stellar dynamics near supermassive black holes by comparing theoretical TDE rates with observations.
1.1 Loss cone dynamics
The rate of TDEs due to stellar twobody relaxation has been studied extensively (Fra76; Lig77; Sha78; Coh78; Stone2016a). Twobody relaxation is the diffusive process by which stars exchange energy and angular momentum amongst themselves, sometimes scattering a star onto a tidally disrupting orbit. It is faster to reach such an orbit by diffusion in angular momentum than in energy (Fra76).
In order for a star in these systems to get close enough to the supermassive black hole (SMBH) to tidally disrupt, it must enter the loss cone. The loss cone defines the region containing orbits with pericenters inside the tidal disruption radius of the black hole. The tidal disruption radius is:
(1) 
where is the mass of the black hole, is the mass of the star, and is the radius of the star (Ree88). Orbits within the loss cone have angular momenta less than the angular momentum of an orbit with a pericenter equal to the tidal radius,
(2) 
There are two loss cone regimes, defined by the parameter q,
(3) 
where is the change in angular momentum per orbital period (Lig77). If , stars take multiple orbital periods to enter the loss cone. This is known as the empty loss cone regime or the diffusion limit because the time for a star to enter the loss cone is greater than the time for the star to be destroyed. If , stars can jump into and out of the loss cone within one orbital period. This is known as the full loss cone regime or the pinhole limit, because the loss cone is continuously populated by stars. The division between the two loss cone regimes for a spherical nuclear star cluster lies close to the radius of influence of the black hole (Lig77). The TDE rate, in this case, is also dominated by stars coming from this region.
1.2 Status of observations of TDEs
In deriving theoretical TDE rates, we typically assume that stars come from an isotropic, spherical distribution around the black hole and are driven to the black hole through twobody relaxation (Wan04; Stone2016a). Theoretical TDE rates in these spherical nuclear star clusters have been calculated to be yr gal (Wan04), and more recently yr gal (Stone2016a). In observations, however, TDEs are preferentially found in postmerger or poststarburst galaxies (K+A/E+A galaxies) at much higher rates. K+A/E+A galaxies are a relatively rare subtype of elliptical galaxy that underwent a major starburst about 11.5 Gyr ago (Couch87; Pog04). K+A/E+A galaxies make up of the galaxies in the local universe, and yet, the observed TDE rates in these K+A/E+A galaxies are yr gal, which pushes the observed TDE rate of ‘normal’ galaxies down to yr gal (French2016). There is even (tentative) evidence that the TDE rate could be as high as yr gal in ultraluminous infrared galaxies (ULIRGs), which are typically in the process of merging (Tadhunter2017; Dou2017). We learn from these observations that merging galaxies and postmerger galaxies tend to have elevated TDE rates.
Several dozen TDE candidates have been identified in the last two decades, from UV/optical to Xray. TDE candidates are generally identified as flaring events, inconsistent with supernovae, at the centers of galaxies. Candidates are typically excluded if the host galaxy shows signs of AGN activity. There have been a number of alternative ideas to explain these flaring events at galactic centers. Proposed TDE impostors include supernovae in AGN disks and black hole accretion disk instabilities (Sax16). One distinguishing feature that can be used to discriminate between real TDEs and impostors is the critical black hole mass beyond which a TDE will not be observable, known as the Hills mass (Hil75b). The Hills mass results from the fact that the tidal radius and Schwarzschild radius of a black hole scale differently with mass of the black hole. The Schwarzschild radius is given by
(4) 
where is the gravitational constant and is the speed of light. Equating the tidal radius to the Schwarzschild radius yields a Hills mass of for a solartype star. Above this limit, the star plunges into the black hole without emitting a flare. A rapidly spinning black hole can raise this limit to (Kesden2012). Recently, vanVelzen2018 presented the black hole mass function of optical/UVselected TDE candidates and showed a sharp decrease in the number of candidates above . This is consistent with the direct capture of stars when the black hole is above the Hills mass and provides strong evidence that we are seeing TDEs rather than impostors.
1.3 Secular dynamics and eccentric nuclear disks
Twobody relaxation is not the only form of relaxation present in galactic nuclei. Resonant relaxation^{2}^{2}2Note that this is a secular (orbitaveraged) effect; ‘resonant’ here refers to the resonance between the azimuthal and radial frequency of a Kepler orbit. arises in nearKeplerian potentials (Rauch1996). A particle on a nearKeplerian orbit traces out the same path repeatedly. On a timescale less than the precession timescale, the orbits remain fixed, and exert mutual gravitational torques on each other. Thus, the angular momentum relaxation can be greatly enhanced, while the energy relaxation is unaffected (Rauch1996).
Resonant relaxation is most effective for stars orbiting close to the central supermassive black hole (in the absence of general relativity). This means that in an isotropic, spherical stellar distribution, where TDEs come most often from near the radius of influence, resonant relaxation will not greatly increase the rate or number of TDEs (Rau98). Not all galactic nuclei, however, are spherical like our galactic center. The nucleus of our nearest galactic neighbor, Andromeda (M31), has a very different configuration.
The Andromeda Galaxy (M31) has an elongated nucleus that resolves into two distinct brightness peaks. The doublenucleus can be explained by a thick, apsidallyaligned eccentric nuclear disk of Keplerian orbits around a SMBH (Tremaine1995). The two brightness peaks correspond to apoapsis and periapsis of the eccentric nuclear disk.
While it may seem like the central disk in M31 is an unusual and unlikely arrangement, the fact that we see it in our closest major galaxy suggests that it is a common configuration. In fact, despite observational challenges, Lauer2005 found that about 20% of nearby, earlytype galaxies have features consistent with eccentric nuclear disks seen from different angles on the sky.
1.4 TDEs from eccentric nuclear disks
The stability of eccentric nuclear disks has long been a mystery. One would expect that the apsidal precession of individual orbits would spread out the disk into an axisymmetric structure on a timescale much shorter than the age of the stars. In a recent paper (Madigan2018), we proposed that the same secular mechanism that stabilizes eccentric nuclear disks is responsible for producing high rates of TDEs.
The forces that cause precession in eccentricity vectors also result in a buildup of gravitational torques between orbits. These torques change the eccentricities of individual orbits as they are perturbed ahead of, or behind, the disk. Differential precession driven by these eccentricity changes holds the disk together. The orbits in an eccentric nuclear disk undergo oscillations in eccentricity. During the high eccentricity phase of an oscillation, a star can be tidally disrupted as it moves through pericenter. The gravitational torques due to secular dynamics are much more efficient at refilling the loss cone than twobody relaxation, which has typically been used to determine TDE rates. We proposed that secular torques in eccentric nuclear disks can produce the observed high rate of TDEs in K+A/E+A galaxies (Madigan2018). Hop10a; Hop10b show that eccentric nuclear disks can form via the merging of gasrich galaxies, meaning that it would be likely to find eccentric nuclear disks in postmerger, K+A/E+A galaxies. TDE rates in eccentric nuclear disks could be as high as yr gal at early times in the life of the disk (Madigan2018).
Several other mechanisms have been theorized to explain the enhanced TDE rates in K+A/E+A galaxies. One of these theories is an enhanced rate due to SMBH binaries after the starburst. Chen2011 show that the TDE rate should scale weakly with the SMBH mass ratio. This would indicate that TDEs would be seen primarily after minor mergers, which are more common. TDEs are preferentially observed, however, in mergers with a more equal SMBH mass ratio, indicating that the TDE rate is not driven by SMBH binaries (French2017). Another theory involves more dense spherical star clusters resulting in enhanced twobody relaxation (Stone2016b; Stone2017).
1.5 This work
In Madigan2018, we evolved eccentric nuclear disks with body simulations in Newtonian gravity. Rapid apsidal precession due to general relativity, however, can quench secular dynamical mechanisms; a wellknown example of this is the KozaiLidov effect (Ford2000; Blaes2002; Naoz2013b). Resonant relaxation in a spherical cluster also gets quenched at low semimajor axes by general relativistic precession as the orbits move too rapidly to allow torques to build up coherently (Rauch1996; Madigan2011).
Similarly, one might expect general relativistic precession to disrupt the secular torques of the eccentric nuclear disk, greatly decreasing the TDE rate. As eccentricity increases due to secular torques, the general relativistic precession rate also increases (Equation 5). One would therefore expect eccentric orbits to precess ahead of the disk, escaping completely until joining back up on the other side and recircularizing. In this case, we should see fewer TDEs with general relativity than without it.
The goal of this work is to explore the effects of general relativity on TDEs occurring in eccentric nuclear disks, and to quantify the distribution of orbital elements of TDEs that originate in eccentric nuclear disks. We do this using body simulations with and without general relativity. We present the paper in the following manner: in Section 2 we describe the initial conditions and parameters for our simulations, and compare the number of TDEs that occur with and without general relativity. We track the orbital elements of a single tidally disrupted star in order to show how quickly the orbit is torqued to an extreme eccentricity. In Section 3 we explore the unique orbital elements of tidally disrupted stars from eccentric nuclear disks, including the penetration factor, inclination distribution, and change in eccentricity vector between TDEs. In Section 4 we summarize and discuss our results.
2 body Simulations of Eccentric Nuclear Disks with General Relativistic Precession
We run body simulations of eccentric nuclear disks with REBOUND (Rein2012) and the IAS15 integrator (Rein2015). We implement general relativity as a postNewtonian approximation with REBOUNDX^{3}^{3}3https://github.com/dtamayo/reboundx. In this paper, we show results from simulations with the following parameters: N=100 stars^{4}^{4}4In Madigan2018, we used a range of stars. Our results were qualitatively the same for the different . We use stars in order to reduce computing time., each with an initial eccentricity of 0.8, a range of semimajor axes () with a surface density of , Rayleigh distributed inclinations with mean , and a disk mass of . We want to qualitatively understand the effects of general relativistic precession rather than obtain an exact number for the TDE rate.
In each of these simulations, we examine the effect that general relativity has on the number of tidal disruption events. The orbitaveraged precession rate due to general relativity is given by
(5) 
Equation 5 is a first order postNewtonian approximation in general relativity yielding corrections to Newtonian accelerations of (Ein16). We track the general relativistic precession rate in our simulations by calculating the change in the orientation of the eccentricity vector at each time step.
A star is considered tidally disrupted if at any point in the simulation its radius . We treat stars as point masses and do not extract them from our simulation after they are disrupted, but they are counted only once as a TDE.
2.1 Effects of General Relativity
We find that the TDE rate with general relativistic precession is the same as in Newtonian gravity. About 12% of disk stars are tidally disrupted^{5}^{5}5This percentage is smaller than in Madigan2018 because we have a more rigorous TDE criterion and exclude partial disruptions from our analysis. for a black hole during a time of 1000 orbital periods, where each orbital period is roughly 1000 years. We have compiled results from 45 simulations with general relativistic precession and 100 simulations without general relativistic precession. The mean percent and standard deviation of tidally disrupted disk stars is shown in Figure 2.1. The number of TDEs is approximately equal for both general relativistic simulations and Newtonian simulations. This means that general relativistic precession does not quench secular torques in eccentric nuclear disks.
In order to understand this, we track the orbital elements of a single star (with general relativistic precession) which suffers a tidal disruption event in Figure 2.1. We see a star that develops an eccentricity such that its orbital angular momentum is less than the loss cone angular momentum. The star also passes through pericenter while it is at a high eccentricity, meaning that the star is close enough to the black hole to be tidally disrupted. We also see that the star’s orbital inclination flips by as it reaches extreme eccentricity (see discussion in Section 3.2). Panel 4 shows the general relativistic precession rate, which we track by calculating the change of in each time step. tracks the orientation of the eccentricity vector in the plane of the disk and is given by
(6) 
(Madigan2016). Here and are the x and y components of the eccentricity vector. We use instead of the argument of periapsis, , or the longitude of periapsis, , to avoid effects of changing inclination. As an orbit rolls over its major axis, the eccentricity vector remains close to the plane, even though the inclination grows. and , however, will change with the flipping inclination. We see that the rate of change of is very small until the star reaches pericenter at an extreme eccentricity where there is a large jump due to general relativity. This jump in precession rate is only present for a fraction of an orbital period. The final panel of Figure 2.1 shows the torque acting on the orbit in units of the circular angular momentum which we explore in the next section.
2.2 Magnitude of Torque from Disk
Here we calculate the magnitude of the torque exerted on a typical orbit by the disk. The orbit is described by its specific angular momentum and energy
(7a)  
(7b) 
For an eccentric orbit, the specific torque is given by
(8) 
is the orbital radius and is the specific gravitational force felt by an orbit due to the rest of the disk. This force is defined by
(9) 
where is the mass of the eccentric nuclear disk. Approximating by the semimajor axis yields a torque
(10) 
Normalizing the torque by the circular angular momentum yields
(11) 
where is the orbital period. Hence, in our body simulations, in which , the normalized torque per orbital period should be on the order of . The final panel in Figure 2.1 shows that indeed our example star experiences a torque of . This magnitude of torque can change an orbit’s eccentricity from to within one orbital period. That is, the change in angular momentum required to produce a TDE can occur within one orbital period, suggesting that our system is in the full loss cone regime. By assuming a black hole and a solartype star, we find from Equation 3, that in our simulations, putting the system well within the full loss cone or pinhole regime. This explains why general relativity is ineffective at shutting down the TDE production. In the full loss cone regime, a stellar orbit can be propelled from outside the loss cone to inside in less than an orbital period. General relativistic precession only acts strongly when the star approaches pericenter, at which point it is too late to avoid disruption.
Not all eccentric nuclear disks will be in the full loss cone regime. The transition from full loss cone to empty loss cone occurs when such that
(12) 
For a SMBH of , solartype stars, and a disk inner edge of pc, we find that disks with are in the full loss cone regime.
3 Unique Orbital Elements
Twobody relaxation predicts that the time between individual TDEs ( years) is much greater than the time it takes for a TDE disk to accrete onto the black hole. If stars come from eccentric nuclear disks however, the typical timescale between individual TDEs can be much shorter (110 yr; Madigan2018), and TDE disks could potentially overlap with one another. This could have interesting observational consequences especially if the orbital parameters of TDEs are correlated.
3.1 Penetration Factor
The strength of a tidal disruption may be quantified by the dimensionless penetration factor,
(13) 
where is the tidal radius and is the pericenter of the star’s orbit (Pre77). In Figure 3.1 we show the distribution of penetration factors in our simulations.
If the penetration factor is greater than or equal to one, the star will be tidally disrupted. If the penetration factor is less than, but close to one, the star may have its outer layers stripped, with a stellar core remaining intact (Iva01; Guillochon2013; Bog14; Mai17). If the penetration factor is too large however, the star will fall straight into the black hole without emitting an electromagnetic flare. This occurs when the . For a nonspinning, black hole and solartype stars this occurs at .
We see in Figure 3.1 that the probability distribution function, , is fully consistent with the full loss cone or pinhole regime (Coughlin2017). This is significant because the critical radius (where ) is typically found near the radius of influence of the black hole. We find that eccentric nuclear disks bring the critical radius orders of magnitude within the radius of influence, to a radius smaller than the inner edge of the disk.
3.2 Inclination Distribution
In a spherical, isotropic stellar system dominated by twobody relaxation, there should be no correlation between the orbital angular momentum vectors of consecutive TDEs, and so we would expect to see an isotropic distribution of TDE inclinations. This is quite different in the case that stars are originating in an eccentric nuclear disk.
We find that whenever a star reaches a high eccentricity in our simulations, it undergoes an inclination flip of . Figure 3.2 is an example of a double peak in eccentricity corresponding to a double flip in inclination. As the orbit is negatively torqued by the disk to extreme eccentricity (in blue), its angular momentum vector decreases until it passes through zero. At this point, the inclination (in green) flips and the orbit switches from a prograde orientation (with respect to the disk) to a retrograde orientation. The orbit now feels a positive torque causing it to circularize and precess quickly back towards the disk. On the other side of the disk, the angular momentum will again decrease, pass through zero and change direction. The orbit is prograde again after the second flip. These double peaks of inclination were also seen in our Newtonian simulations (Madigan2018).
As the orbits flip from prograde to retrograde and back, the percentage of stars on retrograde orbits fluctuates throughout a given simulation. Figure 3.2 is an example of the percentage of retrograde orbits in a disk for a single simulation. Most of these retrograde orbits lie near the inner edge of the disk. This may lead to interesting observational signatures in the velocity moments of eccentric nuclear disks.
The flipping of orbits in inclination results in an anisotropic distribution of TDE inclinations (see Figure 3.2).
Stars preferentially tidally disrupt at orbital inclinations of and with respect to the disk midplane. More disruptions occur at . This is because the stars get the first opportunity to disrupt at an inclination of , while their orbit is ahead of the disk. The probability for a star to disrupt in one orbital period is
(14)  
We estimate for the inner edge of the disk, where is the radius of influence of the black hole, based on the disks in M31 and the Galactic center (Madigan2018). We take to be 5pc from observations of the Galactic center (Lu2009). We find that . Out of 100 stars vulnerable to disruption, 15 will tidally disrupt at an inclination of . 85 stars will then flip inclinations and have a 15% chance (1213) of tidally disrupting at an inclination of . Therefore, we find that the number of TDEs at is 85% the number at , or in general,
(15) 
This explains the height difference that we see in the inclination distribution in Figure 3.2.
For a spinning Kerr black hole, the tidal and capture crosssections shift towards negative angular momenta (Bel92). The asymmetric crosssections make it easier for stars on retrograde orbits to be captured, meaning that prograde TDEs will be preferentially observed. If the spin angular momentum vector of the black hole is aligned with the orbital angular momentum vector of the disk, then the preference for TDEs from eccentric nuclear disks to have orbital inclination puts them at the perfect orientation to be visibly disrupted by a Kerr black hole.
3.3 Eccentricity Vector
We plot the eccentricity vectors of tidally disrupted stars at the time of TDE in the top panel of Figure 3.3. The eccentricity vectors precess together in a prograde direction while remaining in the plane. This means that when a stellar orbit flips in inclination, it flips over its major axis. The bottom panel of Figure 3.3 shows the angular momentum vectors of the same tidally disrupted stars at the time of TDE. The spread in angular momentum vectors confirms that the orbits roll over their major axes.
With TDEs preferentially occurring in the plane with inclinations of or , the debris streams from two sequential TDEs could cross and produce unique observational signatures. Bonnerot2018preprint derive the conditions necessary for a tidal stream crossing to occur, which depend on disk properties, tidal stream widths, and the time between consecutive TDEs. If the pericenter shift between tidally disrupted stars is positive, the time delay is small, and the inclination offset is less than the width of the tidal streams, crossing of the tidal streams could occur. While our simulations do not allow for the calculation of accurate time delays (due to the low nature of our simulations) or stream width, we calculate the pericenter shift between two TDEs with an angle, . This tracks the orientation of the orbit of the first TDE with respect to the orientation of the orbit of the second TDE. We track the orientation of the orbits in our simulation with , defined in Equation 6. We, therefore, calculate between TDEs as
(16) 
We show, in Figure 3.3, a distribution of the between pairs of TDEs in our simulations. We see that about 20% of consecutive TDEs occur with a small (), positive , satisfying one of the conditions for tidal streams crossing. We expect that this condition would be met due to prograde precession of the eccentric nuclear disk.
4 Discussion
This paper focuses on the dynamics of eccentric nuclear disks with general relativistic precession. In Madigan2018, we showed that the same secular mechanism that keeps eccentric nuclear disks stable results in extremely high TDE rates. This work did not include general relativity, however, which is known to quench secular torques via rapid apsidal precession. In this paper, we show that secular gravitational torques push the orbits of stars to extremely high eccentricities within one orbital period (full loss cone regime). This does not allow general relativistic precession enough time to suppress the TDE rate. The geometry of eccentric nuclear disks is key: the torques acting on an orbit from the rest of the disk stars are coherent. Our results point to the following conclusions and implications:

General relativistic precession does not affect the TDE rate from eccentric nuclear disks as stars at the inner edge of the disk are in the full loss cone regime. TDEs occur in simulations with general relativity as often as they occur in Newtonian simulations.

TDEs from eccentric nuclear disks do not follow an isotropic distribution of inclinations; they preferentially disrupt at inclinations of and with respect to the midplane of the disk. Overlapping TDE disks may have similar (or opposing) angular momenta that can build up (or cancel each other out).

The probability of disrupting stars on prograde orbits is higher than the probability of disrupting stars on retrograde orbits for a nonspinning, Schwarzschild black hole in an eccentric nuclear disk. Similarly, the number of prograde captured stars (within the Schwarzschild radius so that a flare will not be observed), will also be greater than the number of retrograde captured stars for a nonspinning, Schwarzschild black hole.

If an eccentric nuclear disk forms during a gasrich major merger, it is likely that the central gaseous accretion event that produces the disk aligns the disk angular momentum with that of the central SMBH. Spinning, Kerr black holes have asymmetric tidal and capture crosssections (Bel92). For black holes with mass greater than the Hills mass, the only observable TDEs are those on prograde orbits aligned with the black hole spin. The preference for TDEs from eccentric nuclear disks to have orbital inclination puts them in the perfect orientation to be observably disrupted by such black holes. There may be evidence of TDEs by extremely massive black holes already. ASSASN15lh is a TDE candidate found in a galaxy with a central SMBH much more massive than the Schwarzschild Hills mass () (Leloudas2016).

In steadystate, eccentric nuclear disks have a nonnegligible fraction of retrograde orbiting stars (10%). Most of these will lie at the inner edge of the disk. This will lead to interesting observational signatures in the velocity moments of eccentric nuclear disks.
Finally, we look back to our nearest neighbor, Andromeda (M31). To date, no TDEs have been observed from its center. This may be due to the fact that the eccentric nuclear disk in M31 is very old, on the order of Gyr (Sil98).
Unless continuously replenished, an eccentric nuclear disk loses mass due to stars being destroyed by tidal forces, but it does not lose significant angular momentum. The disk, therefore, becomes less eccentric with time, causing the TDE rate to decrease (Madigan2018). At , the mass of the M31 black hole is also greater than the Hills mass (Bender2005). We should not expect to observe TDEs, unless the black hole is spinning.
We gratefully acknowledge support from NASA Astrophysics Theory Program under grant NNX17AK44G. Simulations in this paper made use of the REBOUND code which can be downloaded freely at https://github.com/hannorein/rebound. This work utilized the RMACC Summit supercomputer, which is supported by the National Science Foundation (awards ACI1532235 and ACI1532236), the University of Colorado Boulder, and Colorado State University. The Summit supercomputer is a joint effort of the University of Colorado Boulder and Colorado State University.
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