Evolution of Retrograde Orbiters in an AGN Disk

AGN disks have been proposed as promising locations for the mergers of stellar mass black hole binaries (BBHs). Much recent work has been done on this merger channel, but the majority focuses on stellar mass black holes (BHs) orbiting in the prograde direction. Little work has been done to examine the impact of retrograde orbiters (ROs) on the formation and mergers of BBHs in AGN disks. Quantifying the retrograde contribution is important, since roughly half of all orbiters should initially be on retrograde orbits when the disk forms. We perform an analytic calculation of the evolution of ROs in an AGN disk. Because this evolution could cause the orbits of ROs to cross those of prograde BBHs, we derive the collision rate between a given RO and a given BBH orbiting in the prograde direction. ROs experience a rapid decrease in the semi-major axis of their orbits while also becoming highly eccentric in less than a million years. This rapid orbital evolution leads to very small collision rates between retrograde BHs and prograde BBHs, meaning that they are unlikely to break apart or ionize existing BBHs in AGN disks. The rapid orbital evolution of ROs could instead lead to extreme mass ratio inspirals and gravitationally lensed BBH inspirals. These could be detected by the Laser Interferometer Space Antenna (LISA), and even cause disruption of the inner disk, which may produce electromagnetic signatures.


INTRODUCTION
Active galactic nucleus (AGN) disks are promising locations (McKernan et al. 2012(McKernan et al. , 2014Bellovary et al. 2016; Bartos et al. 2017;Stone et al. 2016;McKernan et al. 2018;Leigh et al. 2018;Secunda et al. 2019Secunda et al. , 2020Yang et al. 2019a,b;Tagawa et al. 2020;McKernan et al. 2019;Gröbner et al. 2020;; The LIGO Scientific Collaboration & the Virgo Collaboration 2020) for producing the stellar mass black hole binary (BBH) mergers detected by the Advanced Laser In-terferometer Gravitational Wave Observatory (aLIGO) and Advanced Virgo (Acernese et al. 2014;Aasi et al. 2015;Abbott et al. 2019). An AGN disk is a favorable location for BBH mergers detectable by aLIGO because the gas disks will act to decrease the inclination of intersecting orbiters and harden existing BBHs (McKernan et al. 2014;Yang et al. 2019a). Additionally, stellar mass black holes (BHs) on prograde orbits will exchange energy and angular momentum with the gas disk, causing migration in both the inward and outward radial directions (Bellovary et al. 2016;Secunda et al. 2019Secunda et al. , 2020. In particular these orbiters will migrate towards regions of the disk where positive and negative torques cancel out, known as migration traps. As these prograde orbiters (POs) migrate towards the migration traps, they will encounter each other at small relative velocities. Consequently, BBHs form that could merge on timescales of 10 − 500 years (Baruteau et al. 2011;McKernan et al. 2012McKernan et al. , 2018Leigh et al. 2018; Baruteau & Lin 2010).
Despite an abundance of recent publications on BHs in AGN disks, thus far studies have largely ignored the impact of retrograde orbiters (ROs) in an AGN disk. We could expect that since bulges have little net rotation, perhaps nuclei lack net rotation as well. Consequently, roughly half of the initial BH population of a nuclear star cluster should be on retrograde orbits when the gas disk forms. While MacLeod & Lin (2020) found that orbiters with high initial inclinations will flip from prograde to retrograde orbits as they are ground down into alignment with the disk, the population of orbiters initially aligned with the disk or on slightly inclined orbits should be roughly half retrograde. These ROs will be impacted by the disk in a significantly different way from POs due to their larger velocities relative to the gas disk. Additionally, ROs will encounter POs in the disk with large relative velocities, meaning they are less likely to form BBHs with POs and more likely to ionize binaries in the disk (Leigh et al. 2018). Therefore ROs could have a significant affect on the number of BBHs and mergers in AGN disks.
We aim to calculate the evolution of BHs initially orbiting in the retrograde direction when the gas disk appears, and predict whether these ROs interact with BHs and/or BBHs orbiting in the prograde direction. In §2 we derive equations for the time evolution of the energy, angular momentum, and eccentricity of a RO in a Sirko & Goodman (2003) AGN disk. In §3 we derive the collision rate between a RO and a BBH orbiting in the prograde direction as a function of the semi-major axis and eccentricity of the RO. We use these derivations to give three fiducial examples in §4. Finally, in §5 we discuss the implications of the results of our model for gravitational and electromagnetic wave detections of BHs and BBHs in AGN disks.

ORBITAL EVOLUTION
For a BH orbiting in an AGN disk in the retrograde direction the relative velocity (v rel = v − v disk ) between the orbiter and the disk is highly supersonic, with Mach number v rel /c s ∼ (h/r) −1 1, where h/r is the disk aspect ratio. The gas drag force on a BH of mass m can be approximated as dynamical friction (Binney 1987;Ostriker 1999), where ρ is the local mass density of the disk, and Λ ∼ hv 2 rel /Gm, where h is the scale height of the disk, m the mass of the RO, and G is the gravitational constant. We assume m is small enough that Λ 1. The additional contribution to the drag from Bondi-Hoyle-Lyttleton accretion onto the BH will be smaller by a factor of ∼ (ln Λ) −1 , and so we neglect it.
The orbital energy of the BH is, where M is the mass of the SMBH and a is the semimajor axis of the orbit. Neglecting accretion onto the BH, and where v is the velocity of the RO. Defining the angular momentum of the AGN disk as positive, the angular momentum for a RO becomes where e is the eccentricity. The torque on the orbiter is whereê φ is a unit vector in the azimuthal direction and r is the radial distance of the RO from the central SMBH.
Using Equations 2 and 5, for small changes da, de 2 in a and e 2 we get and Using equation 7 to put equation 8 in terms of dE instead of da and substituting in the mean motion, n = − GM/a 3 , gives the change in eccentricity in terms of the change in energy and angular momentum, Using the fact that E = m(v 2 /2 − GM/r), L = mrv φ = m GM a(1 − e 2 ) and v disk =ê φ GM/r, we obtain and which allows us to eliminate the velocities in equations 4 and 6 in favor of r. r, as a function of the azimuthal angle φ, is where φ p is the angle at pericenter. We set φ p = 0, because our disk is axisymmetric. By Kepler's Second Law, the time interval dt corresponding to the angular interval dφ are related by, where P is the orbital period. We can use Equation 14 to write the average the change in energy, angular momentum, and eccentricity over one orbital in terms of dφ. Apart from the velocities, dE/dt and dL/dt depend on r through the midplane density ρ(r) and Λ. In a Sirko & Goodman (2003) AGN disk, we have ρ(r) ∝ r γ , with γ = 3/2 for a radiation-pressure-dominated disk with constant opacity. We ignore the slight variation in ln Λ along the orbit.
We define which has the same dimensions as dE/dt and ndL/dt. We can now write and where I E and I L are the dimensionless integrals, and We integrate these equations numerically to solve for the eccentricity and semi-major axis of a RO as a function of time. We discuss the orbital evolution of ROs for three different fiducial initial eccentricities in §4.

COLLISION RATES
In this section we estimate the collision rate of a RO (body 1) and a prograde BBH (body 2). We assume that the apsidal precession rate due to both relativistic effects and disk self-gravity is rapid compared to the interaction rate, such that the probability of finding an orbiter in a given area element rdrdφ is independent of azimuth, φ. Therefore, the collision probability is proportional to the fraction (dt/P ) i of the orbit of body i spent between r and r + dr, whereÊ,Φ, andL, are the total energy, potential energy, and angular momentum per unit mass. The factor of 2 occurs in the numerator because the orbit crosses a given radius r twice per orbit, provided that a(1 − e) < r < a(1 + e). The second line follows from plugging in the relations P = 2π a 3 /GM , Orbiters in an AGN disk will be excited onto slightly inclined orbits by turbulent motions in the disk, but the inclination will also be damped by drag forces from the gas. Without a specific model for turbulence, we assume for simplicity that the probability of finding an orbiter at height z to z + dz is gaussian, with a scale height h BH estimated as follows.
At ∼ 500 R s in a Sirko & Goodman (2003) disk with a 10 8 M SMBH, Σ ≈ 7 × 10 5 g cm −2 and h ≈ 1.28 × 10 14 . The eddy turnover speed will be v edd α 1/2 c s , where α is the ShakuraSunyaev (Shakura & Sunyaev 1973) viscosity parameter (10 −2 for a Sirko & Goodman (2003) disk) and c s is the isothermal sound speed at the midplane. The turnover time of eddies of size l edd is If we limit τ edd to τ edd Ω −1 , where Ω = (GM/r 3 ) −1/2 = c s /h is the orbital frequency, l edd /h v edd /c s = α 1/2 . Therefore the eddy mass is The eddy mass m edd at 500 R s in our AGN disk is roughly 3 × 10 −3 M . Assuming equipartition of vertical kinetic energies, For a 10 M BH in our AGN disk h BH 1.7 × 10 −3 h 2.17 × 10 11 cm. For simplicity we assume that h BH is independent of radius.
Since the area of the annulus is 2πrdr and the distribution over height is given by eq. (23), the probability per unit volume dV = rdrdφdz of finding the body near a given point (r, φ, z) is We will assume e 2 , the eccentricity of the BBH orbiting in the prograde direction, is ∼ 0, because the disk acts to circularize POs (Tanaka & Ward 2004). This assumption gives 2π 2 a 2 , if e 2 1.
(29) If the annuli of the two bodies overlap, the expected interaction rate between them becomes The interaction cross section of the BBH and the RO is, Here s bin is the semi-major axis of the binary itself, which we take to be the mutual Hill radius of the two BHs in the binary f is the Safronov number, a dimensionless "gravitational focusing" factor, where m 2 is the total mass of the BBH and v 2 rel = (v φ,1 − v φ,2 ) 2 + (v r,1 − v r,2 ) 2 + (v z,1 − v z,2 ) 2 is the relative velocity between the BBH and the RO. σ is taken in the limit that f 1.
Because v rel will be very large, the encounters will be fast and gravitational focusing will not be important.
Next, we assume that the z components of the velocities, v z,i , are negligible and that e 2 ∼ 0. Since e 2 ∼ 0, (34) Substituting into eq. (30) gives, with v rel given by eq. (34) and σ(v rel ) by eq. (31). The two terms in parentheses in the denominator of the second term define the limits where a collision is possible given our assumptions, since both terms must be positive. That is, it is not possible for a collision to take place if a 2 is greater than the apocenter of the RO or less than the pericenter of the RO. We discuss the collision rate for three different fiducial eccentricities in §4.

RESULTS
The solid lines in Figure 1 show the evolution of the semi-major axis (top) and eccentricity (bottom) for ROs orbiting a 10 8 M SMBH with initial eccentricities e 0 =0.1, 0.5, and 0.7, calculated through numerical integration of Equations 7, 16, 17, and 18. All orbiters were initiated with a = 500 R s and integrated over time until they reached e = 0.999. ROs at all e 0 see a dramatic increase in their eccentricity and decrease in their semi-major axis within 10 5 years. All orbiters reach an eccentricity of 0.99 in under a Myr. ROs with greater e 0 become highly eccentric on shorter timescales. For example, orbiters with e 0 0.5 reach e = 0.999 within 100 kyr.
For the special case of a BH on a circular retrograde orbit, v = −v disk , where v disk is the velocity of the disk ( GM/r), and the relative velocity between the orbiter  (Peters 1964) in our integration. All orbiters begin with a semi-major axis of 500 Rs and are evolved until they reach an eccentricity of 0.999 for the integration that does not include GW circularization, or merge with the SMBH for the integration that does. and the disk is v rel = 2v disk . Equations 3 and 4 from §2 can be used to calculate the evolution of the semi-major axis for a 10 M BH on this circular, retrograde orbit around a 10 8 M SMBH in a Sirko & Goodman (2003) AGN disk. If the BH is initially at a radius of ∼ 10 3 R s , Λ ∼ (4M/m)(h/r) ∼ 10 5 and ρ ∼ 10 −7 g cm −3 , which (36) The dashed black lines in Figure 1 show the evolution of ROs with the same initial conditions as the colored lines, when accounting for gravitational wave (GW) circularization (Peters 1964). We evolve e and a by the rates in Peters (1964) at the values we find for e and a after evolving them with Equations 7, 16, 17 and 18 from §2. These rates are integrated over time until a = 0. GW circularization becomes more rapid as the eccentricity of the orbiter increases, slowing the eccentricity driving once a high eccentricity is reached. The maximum eccentricity reached is now 0.982, 0.997, and 0.998, and the eccentricity at the time of merger is 0.932, 0.984, and 0.983 for ROs with e 0 =0.1, 0.5, and 0.7, respectively. Figure 2 shows the collision rate per orbit, τ −1 coll , as a function of time calculated with Equation 35 for the three example e 0 . The mass of the RO is 10 M and the total mass of the binary m 2 = 20 M . For simplicity we take the semi-major axis of the BBH, a 2 , to be constant at 330 R s , i.e. the location of the migration trap in a Sirko & Goodman (2003) AGN disk (Bellovary et al. 2016). a 1 and e 1 evolve over time as calculated above, with GW circularization and a 1 =500 R s , initially. If a 2 is greater than the apocenter distance or less than the pericenter distance of the RO for a given orbit we take τ −1 coll = 0.
We choose these initial parameters to resemble the most common conditions in Secunda et al. (2019) and Secunda et al. (2020), who find that BHs migrate towards the migration trap at ∼ 330 R s in a Sirko & Goodman (2003) AGN disk, where they start forming BBHs on timescales similar to the orbital evolution of ROs (∼ 10 4 − 10 5 years) and remain for the lifetime of the disk. This over-dense population of BBHs is a prime target for a RO to interact with. However, future work should look at a wider range of BBH orbital parameters, AGN disk parameters, and actively migrating prograde BBHs to determine collision rates for a wider range of initial conditions (see e.g. McKernan et al. 2019;McKernan et al. 2020;Tagawa et al. 2020, on the prevalence of BBH mergers away from a trap). Preliminary tests show that changing the location of the BBH and having BBHs migrate within the inner 500 R s does not have a significant affect on τ −1 coll . τ −1 coll ∼ O(10 −10 ) yr −1 initially for ROs with e 0 =0.5,0.7 and then increases to ∼ O(10 −8 ) yr −1 as GW circularization becomes important. At first, τ −1 coll = 0 for the RO with e 0 =0.1, since its orbit will not cross the orbit of the BBH until its semi-major axis has decreased and its eccentricity has increased. Once the orbits do cross τ −1 coll starts out relatively high, around O(10 −7 ). Then, while the eccentricity is still low, τ −1 coll decreases as the semi-major axis decreases reaching a minimum of around 6 × 10 −10 . Next, as the eccentricity increases, the decrease in semi-major axis causes τ −1 coll to increase to ∼ 10 −6 yr −1 . Finally, the semi-major axis becomes too small for the RO to cross the orbit of the BBH, and τ −1 coll = 0. The total probability of an encounter summed over all orbits before the RO reaches a = 0 is 5.0 × 10 −4 , 1.6 × 10 −5 , and 4.1 × 10 −6 for our fiducial runs with e 0 =0.1, 0.5, and 0.7, respectively. Our calculations suggest that an interaction between a RO and a BBH orbiting in the prograde direction is most likely to occur for ROs with smaller e 0 , because their orbits have more time to evolve to smaller semi-major axes before they are driven to high eccentricities. However, our calculated probability of interaction is still very small for these orbiters, suggesting that the likelihood of an interaction between a prograde BBH and a RO is small.

DISCUSSION
ROs migrate to the inner disk on timescales of tens to hundreds of kiloyears, depending on their initial eccentricity. They also experience a rapid increase in their eccentricity, reaching e 0.999 or e 0.98 in less than a megayear, without and with GW circularization, respectively. The collision rate per orbit between ROs and prograde orbiting 20 M BBHs in the migration trap of a Sirko & Goodman (2003) AGN disk is small. Therefore, the probability of a RO interacting with these prograde BBHs before it reaches a high eccentricity in our fiducial examples is low. GW circularization only has a minimal affect on our 10 M ROs until they reach e 0.98. Afterwards, GWs quickly lead to coalescence with the SMBH before the orbits of the retrograde BHs can become much more circular. However, as the mass of the RO increases, the rate of GW circularization increases while the rate of eccentricity driving from the gas decreases. As a result, more massive ROs take longer to reach their maximum eccentricities and never become as eccentric as the fiducial examples shown here. In some cases, such as a 50 M RO with e 0 =0.1, ROs will circularize after they reach their maximum eccentricity before merging with the SMBH.
In the fiducial calculation in §3 and §4 we have chosen a total mass of 20 M for the BBH, but Secunda et al. (2020) found that BBHs near the migration trap often grow as massive as 100 M , and occasionally even 1000 M . The former BBH mass would increase the probability of interaction for ROs with e 0 =0.1 to about 4.3%. A 1000 M BBH would be almost certain to collide with a RO with e 0 =0.1, and has a probability of interaction of ∼11% with a RO with e 0 =0.5. However, in our fiducial examples ROs take under a megayear to merge with the SMBH, and in (Secunda et al. 2020) these 1000 M BBHs take several megayears to form. ROs from further out in the disk or that are ground down from inclined orbits into the disk could perhaps replenish the supply of ROs at later times, although MacLeod & Lin (2020) find that ROs on inclined orbits will flip to prograde orbits as they align with the disk.
If a RO were to interact with a PO the two could potentially merge or form a BBH. This interaction outcome would be most likely to occur at apocenter of the retrograde BH's orbit where the relative velocities of the POs and ROs would be smallest. A merger would also be more likely if the orbiters are very far out from the central SMBH, where both of their orbital velocities will be lower. However, due to the high relative velocities of POs and ROs, the total interaction energy is likely to be positive. As a result, ROs would most likely act to ionize existing prograde BBHs (e.g. Leigh et al. 2016Leigh et al. , 2018. For example, the hard-soft boundary describes the binary separation over which a BBH will tend to be disrupted or ionized when it encounters a tertiary. The hard-soft boundary for a RO with e 0 =0.1 interacting with a 100 M BBH in our fiducial model, would be at most 5.6 × 10 −4 AU, depending on when the RO and BBH interact. A BBH this compact would likely merge rapidly due to GW emission and not survive long enough to undergo a collision. Whether ROs will form BBHs with each other is uncertain. ROs' large eccentricities may lead to large relative velocities among them, preventing them from becoming bound. However, if ROs after experiencing orbital decay did undergo a GW inspiral in the innermost disk, they would have a higher probability of being gravitationally lensed by the SMBH, which could be detected by the Laser Interferometer Space Antenna (LISA) (Amaro-Seoane et al. 2017;Nakamura 1998;Takahashi & Nakamura 2003;Kocsis 2013;D'Orazio & Loeb 2019;Chen et al. 2019). This population of orbiters in the innermost disk could also perturb the inner disk, which may be detectable by electromagnetic observations (McKernan et al. 2013;McKernan et al. 2014;Blanchard et al. 2017;Ross et al. 2018;Ricci et al. 2020).
Finally, the rapid orbital decay of these retrograde BHs would likely lead to extreme mass ratio inspirals (EMRIs), from coalescence of these BHs with the SMBH. LISA is most sensitive to EMRIs where the SMBH is 10 5 − 10 6 M (Babak 2017), i.e. 2-3 orders of magnitude less massive than the SMBH mass used here. Nonetheless, EMRIs involving a 10 8 M SMBH could be detectable at low redshift by LISA. ROs in most cases will still be on eccentric orbits when they merge. As a result they will produce exotic waveforms, that would identify them as ROs. LISA could also potentially localize its detections to only a few candidate AGN (Babak 2017).
The examples studied here provide evidence that ROs will not change previously predicted BH merger rates, because we find the probability of interaction between a RO and a BBH to be low. However, wider parameter studies including initially inclined orbits,(e.g., Just et al. 2012;Kennedy et al. 2016;Panamarev et al. 2018;MacLeod & Lin 2020;Fabj et al. 2020), lower mass SMBH, higher mass BBH, varying disk density and scale height profiles, and POs away from the migration trap should investigate where RO-BBH interactions may become important. Instead of interacting with BBHs, we find that ROs are likely to become EMRIs. Because these EMRIs are often on highly eccentric orbits at the time of merger, their LISA-observable waveforms will be extremely distinctive and may even allow for measurement of gas effects (Derdzinski et al. 2019(Derdzinski et al. , 2020. Such observations will provide critical insights for our models of both nuclear star clusters and AGN disks.