Merging Compact Binaries Near a Rotating Supermassive Black Hole: Eccentricity Excitation due to Apsidal Precession Resonance
MMerging Compact Binaries Near a Rotating Supermassive Black Hole:Eccentricity Excitation due to Apsidal Precession Resonance
Bin Liu , Dong Lai Cornell Center for Astrophysics and Planetary Science, Cornell University, Ithaca, NY 14853, USA
We study the dynamics of merging compact binaries near a rotating supermassive black hole(SMBH) in a hierarchical triple configuration. We include various general relativistic effects thatcouple the inner orbit, the outer orbit and the spin of the SMBH. During the orbital decay dueto gravitational radiation, the inner binary can encounter an “apsidal precession resonance” andexperience eccentricity excitation. This resonance occurs when the apsidal precession rate of theinner binary matches that of the outer binary, with the precessions driven by both Newtonianinteractions and various post-Newtonian effects. The eccentricity excitation requires the outer orbitto have a finite eccentricity, and is most effective for triples with small mutual inclinations, in contrastto the well-studied Lidov-Kozai effect. The resonance and the associated eccentricity growth mayoccur while the binary emits gravitational waves in the low-frequency band, and may be detectableby future space-based gravitational wave detectors.
I. INTRODUCTION
The detections of gravitational waves from mergingbinary black holes (BHs) [1–5] have motivated manyrecent studies on their formation channels. These in-clude the traditional isolated binary evolution [e.g., 6–13] and chemically homogeneous evolution [e.g., 14, 15],gas-assisted mergers [e.g., 16], and various flavors of dy-namical channels that involve either strong gravitationalscatterings in dense clusters [e.g., 17–27] or more gen-tle “tertiary-induced mergers” [e.g., 28–38]. Many recentstudies have shown that merging BH and neutron star(NS) binaries can be formed efficiently (via Lidov-Kozai(LK) oscillations [39–41]) with the aid of a tertiary bodythat moves on an inclined (outer) orbit relative to theorbit of the inner binary. Furthermore, the efficiency ofthe merger can be enhanced when the triple is part of aquadruple system [e.g., 42–45], or more generally, whenthe outer orbit experiences quasi-periodic external forc-ing [e.g., 46–48].In the standard LK-induced merger scenario, the lead-ing order of post-Newtonian (PN) effect of general rel-ativity (GR) gives rise to apsidal precession of the in-ner binary, and this tends to suppress LK oscillations orlimit the maximum eccentricity [e.g., 49–51]. However,several numerical studies [e.g., 52–54] based on seculartriple equations (see [55–57] for recent, more systematicstudy of the secular triple equations in PN theory) alsofound evidence that with small mutual inclinations (noLK oscillations are allowed to occur), significant eccen-tricity excitation might still be achieved under some con-ditions (e.g., the clearest example of this phenomenonis displayed in Figure 14 of [52]; [54] added some other(generally non-essential) PN terms and called this “GR-induced eccentricity excitation”). The physical expla-nation of the eccentricity growth at low inclinations interms of “apsidal procession resonance” was provided in[58] in the context of merging compact binaries with ter-tiary (low-mass) companions. It was shown that a sec-ular resonance occurs when the total apsidal precession of the inner binary (driven by GR and the outer binary)matches the precession rate of the outer binary (drivenby the inner binary), and this resonance allows efficient“transfer” of eccentricity from the outer binary to theinner binary.In this paper, we are interested in stellar-mass BHbinary (BHB) mergers induced by a supermassive BH(SMBH). Such BHBs may exist in abundance in the nu-clear cluster (with a central SMBH) due to various dy-namical processes, i.e., scatterings and mass segregation[59, 60]. Importantly, our recent study [61] shows thatseveral GR effects (including some of the “cross terms”studied in [55]) introduced by a rotating SMBH can gen-erate extra precessions on the BH orbits, significantly in-creasing the merger fraction (as well as the merger rate).This opens the question of how these GR effects modifythe eccentricity growth mechanism due to the “apsidalprecession resonance”. We address this issue systemat-ically in this paper. We focus our attention to isolatedBHB-SMBH triple systems, and do not consider otherprocesses related to scatterings and relaxation with sur-rounding stars in the cluster [47, 62, 63], which may alsochange the character of SMBH-induced mergers of stellarBHBs.Our paper is organized as follows. In Section II, wereview the essential GR effects (including the “crossterms”) relevant to BHB-SMBH triples and present thesecular equations in PN theory. In Section III, we presentsome numerical examples to illustrate how secular apsidalresonance influences the orbital decay of BHB in triples.In Section IV, we perform analytical calculations for theeccentricity excitation for coplanar systems residing nearthe resonance, and explore the parameter space whichcan lead to the eccentricity excitation. In Section V, weextend our calculations to systems with slightly inclinedouter binary and spin orientations. We summarize ourmain results in Section VI. a r X i v : . [ a s t r o - ph . H E ] M a y II. EVOLUTION OF BHB NEAR A SMBHA. Equations for Standard LK-Induced Merger
We consider a hierarchical triple system, composedof an inner BH binary of masses m , m and a dis-tant companion (the SMBH) of mass m that movesaround the center of mass of the inner bodies. The re-duced mass for the inner binary is µ in ≡ m m /m ,with m ≡ m + m . Similarly, the outer binaryhas µ out ≡ ( m m ) /m with m ≡ m + m .The semi-major axes and eccentricities are denoted by a in , a out and e in , e out , respectively. Therefore, theorbital angular momenta of two orbits are given by L in = L in ˆ L in = µ in (cid:112) Gm a in (1 − e ) ˆ L in and L out =L out ˆ L out = µ out (cid:112) Gm a out (1 − e ) ˆ L out . We definethe mutual inclination between L in and L out (inner andouter orbits) as I . Throughout the paper, for convenienceof notation, we will frequently omit the subscript “in” forthe inner orbit.To study the evolution of the merging inner BH binaryunder the influence of the tertiary companion, we usethe double-averaged (DA; averaging over both the innerand outer orbital periods) secular equations of motion(see section 2.1 of [38] for discussion of double vs singleaveraged equations). For the inner binary, the dynamicsof the angular momentum L and eccentricity e vectorsare given by d L dt = d L dt (cid:12)(cid:12)(cid:12)(cid:12) LK + d L dt (cid:12)(cid:12)(cid:12)(cid:12) GW , (1) d e dt = d e dt (cid:12)(cid:12)(cid:12)(cid:12) LK + d e dt (cid:12)(cid:12)(cid:12)(cid:12) GR + d e dt (cid:12)(cid:12)(cid:12)(cid:12) GW , (2)where we include the contributions from the Newtonianpotential of the external companion (the first terms inEquations (1)-(2); following the notation of [38], we labelthese with the subscript “LK” since they can generate LKoscillations for sufficiently inclined orbits – although inthis paper we focus on non-LK regime), the leading orderPN correction, and the dissipation due to gravitationalwaves (GW) emission.The explicit DA equations of d L /dt | LK and d e /dt | LK ,are provided in [50]. The Newtonian (LK) terms induceprecession of eccentricity vectors on the timescale T LK = 1 ω LK = 1 n m m (cid:18) a out , eff a (cid:19) , (3)where n = ( Gm /a ) / is the mean motion of the innerbinary, and a out , eff ≡ a out (cid:112) − e is the effective outerbinary separation. Again, we label this T LK , because forsufficiently inclined orbits, this is the LK timescale foroscillations in eccentricity and orbital inclination.GR (1-PN correction) introduces pericenter precessionof the inner binary, d e dt (cid:12)(cid:12)(cid:12)(cid:12) GR = ω GR , in ˆ L × e , (4) with the precession rate ω GR , in = 3 G / m / c a / (1 − e ) . (5)Similar equations apply to the outer binary, with d e out dt (cid:12)(cid:12)(cid:12)(cid:12) GR = ω GR , out ˆ L out × e out , (6) ω GR , out = 3 G / m / c a / (1 − e ) . (7)During the LK oscillations, the short-range effect cap-tured in Equation (4) plays a crucial role in determiningthe maximum eccentricity e max of the inner binary [e.g.,49], that can be evaluated analytically [e.g., 50, 51].Gravitational radiation draws energy and angular mo-mentum from the BH orbit. The rates of change of L and e are given by [64] d L dt (cid:12)(cid:12)(cid:12)(cid:12) GW = − G / c µ m / a / e / − e ) ˆ L , (8) d e dt (cid:12)(cid:12)(cid:12)(cid:12) GW = − G c µm a (1 − e ) / (cid:18) e (cid:19) e . (9)For reference, the merger time due to GW radiation ofan isolated binary with the initial semi-major axis a andeccentricity e is approximately given by T m = T m , (1 − e ) / = 5 c a G m µ (1 − e ) / (10) (cid:39) (cid:18) M (cid:12) m (cid:19) (cid:18) M (cid:12) µ (cid:19)(cid:18) a . (cid:19) (1 − e ) / yrs . Equations (1)-(2), as well as the similar equations ofmotion of the outer binary (without GW emission), com-pletely determine the evolution of the triple system. The
Standard LK-Induced Merger mechanism (as studied inmost previous works) considers sufficiently large mutualinclinations, and includes the apsidal precession due toGR (Equation (4)), but neglects the GR effects associ-ated with the rotating tertiary companion. This is ade-quate when the tertiary mass m is not much larger thanthe masses of the inner BHB. However, for BHB-SMBHtriples, with m (cid:29) m , m , several GR effects involvingthe SMBH can qualitatively change the efficiency andoutcomes of LK-induced mergers [61]. B. Additional GR effects due to rotating SMBH
For a rotating SMBH, the spin angular momentum isgiven by S = χ Gm /c , where χ (cid:54) χ = 1 in the numerical examples pre-sented this paper). In GR, the vectors L , L out , S , e and e out are coupled to each other, inducing time evolu-tion of these vectors. In a systematical post-Newtonianframework of triple dynamics [55–57], there are numerousterms. We summarize the most essential effects below(also the leading-order effects). The related equationsare either from the classical work on binaries with spin-ning bodies [65] (see also [66] and references therein), orcan be derived (or extended to include eccentricity) “byanalogy”, i.e., by recognizing that the inner binary’s or-bital angular momentum L behaves like a “spin”. Aswe see below, the vector forms of the equations we useare much more transparent than equations based on or-bital elements (see [55–57]), especially we are deal withmisaligned L , L out and S . Effect I: the coupling between L out and S . In theBHB-SMBH system, the angular momentum of the outerbinary L out and the spin angular momentum S of m precesses around each other due to spin-orbit coupling ifthe two vectors are misaligned (1.5 PN effect) [65, 67]: d L out dt (cid:12)(cid:12)(cid:12)(cid:12) L out S = ω L out S ˆ S × L out , (11) d e out dt (cid:12)(cid:12)(cid:12)(cid:12) L out S = ω L out S ˆ S × e out − ω L out S (ˆ L out · ˆ S )ˆ L out × e out , (12) d ˆ S dt (cid:12)(cid:12)(cid:12)(cid:12) S L out = ω S L out ˆ L out × ˆ S , (13)where the orbit-averaged precession rates are ω L out S = GS (4 + 3 m /m )2 c a (1 − e ) / = ω S L out S L out . (14)Since in our case, S can be easily larger than L out , thede-Sitter precession (Equation (13)) is negligible. Effect II: the coupling between L and L out . In addi-tion to the Newtonian precession (driven by the tidalpotential of m ), L experiences an additional de-Sitterlike (geodesic) precession in the gravitational field of m introduced by GR. This is a 1.5 PN spin-orbit couplingeffect, with L behaving like a “spin”. We have d L dt (cid:12)(cid:12)(cid:12)(cid:12) L in L out = ω (GR)L in L out ˆ L out × L , (15) d e dt (cid:12)(cid:12)(cid:12)(cid:12) L in L out = ω (GR)L in L out ˆ L out × e , (16)and the feedback from ˆ L , e on L out and e out are givenby (see [65]) d L out dt (cid:12)(cid:12)(cid:12)(cid:12) L out L in = ω (GR)L out L in ˆ L × L out , (17) d e out dt (cid:12)(cid:12)(cid:12)(cid:12) L out L in = ω (GR)L out L in ˆ L × e out (18) − ω (GR)L out L in (ˆ L out · ˆ L )ˆ L out × e out , with ω (GR)L in L out = 32 G ( m + µ out / n out c a out (1 − e ) = ω (GR)L out L in L out L , (19) where n out = ( Gm tot /a ) / . Note the similarity be-tween Equations (17)-(18) and Equations (11)-(12).Note that both Equations (15)-(16) are required tokeep L · e = 0. Equations (15)-(16) can also be repro-duced through the “cross terms” in the PN equations ofmotion of hierarchical triple systems [55, 56]. Effects III: the coupling between L and S . Since thesemimajor axis of the inner orbit ( a ) is much smaller thanthe outer orbit ( a out ), the inner binary can be treatedas a single body approximately. Therefore, the angularmomentum ˆ L is coupled to the spin angular momentum S of m , and experiences Lens-Thirring precession. Thisis a 2 PN spin-spin coupling effect, with L behaving likea “spin”. We have d L dt (cid:12)(cid:12)(cid:12)(cid:12) L in S = ω L in S ˆ S × L − ω L in S (ˆ L out · ˆ S )ˆ L out × L , (20) d e dt (cid:12)(cid:12)(cid:12)(cid:12) L in S = ω L in S ˆ S × e − ω L in S (ˆ L out · ˆ S )ˆ L out × e . (21)Note that Equation (21) is required to keep L · e = 0. Theback-reaction on the outer binary is gives (see Equations64,65,70 of [65]) d L out dt (cid:12)(cid:12)(cid:12)(cid:12) S L in = − ω S L in (cid:104) (ˆ L out · ˆ L )ˆ S + (ˆ L out · ˆ S )ˆ L (cid:105) × L out , (22) d e out dt (cid:12)(cid:12)(cid:12)(cid:12) S L in = − ω S L in (cid:26) (ˆ L out · ˆ L )ˆ S + (ˆ L out · ˆ S )ˆ L + (cid:104) (ˆ L · ˆ S ) − L out · ˆ L )(ˆ L out · ˆ S ) (cid:105) ˆ L out (cid:27) × e out . (23)In the above, the orbit-averaged precession rates are ω L in S = GS c a (1 − e ) / = ω S L in L out L . (24)Note ω L in S /ω (GR)L in L out ∼ ( V out /c ) χ (where V out is theorbital velocity of the outer binary and χ is the dimen-sionless spin parameter of the SMBH). Thus Effect III isimportant only when the outer binary is relativistic. III. MERGING BHB WITH A COPLANARSMBH: NUMERICAL EXAMPLES
We begin with coplanar systems (ˆ L = ˆ L out ) withthe SMBH spin either aligned (ˆ S = ˆ L ) or anti-aligned(ˆ S = − ˆ L ) with respect to the orbit. Figure 1 showstwo examples. All Newtonian (up to the octupole order)and GR effects discussed in Section II are included in thecalculation. In the left panels (with ˆ L = ˆ L out = ˆ S ),the eccentricity of the inner binary is negligible initially, FIG. 1: The evolution of an inner merging BH-BH binary near a rotating SMBH in a coplanar orbital configuration, with theSMBH spin either parallel (left) or anti-parallel (right) to the orbits.. The masses of BHs are m = 30 M (cid:12) , m = 20 M (cid:12) , and m = 10 M (cid:12) , respectively. The initial semimajor axes of the inner and outer binaries are a = 0 . a = 0 . a out = 90AU, respectively. The eccentricities and longitudes of the periapsis of two orbits are initialized as e = 10 − , e out , = 0 . (cid:36) = (cid:36) out , = 0, respectively. The blue lines are obtained by the numerical integrations of the timeevolution equations. The red-dashed lines on the second panels depict the analytical maximum eccentricity of the inner binaryat the corresponding semi-major axis (see Section IV B). The subfigures in the third and bottom panels show the difference ofouter eccentricity ( e out − e out , ) and zoom-in of the GW peak frequency near the apsidal precession resonance. but undergoes small-amplitude oscillations at the earlyphase, due to the Newtonian perturbation from the outerbinary. During the orbital decay, the eccentricity gets ex-cited twice and achieves a value of ∼ .
27. This is theresult of the “apsidal precession resonance”, which allowsthe inner binary to efficiently “gain” some eccentricityfrom the outer binary (the eccentricity of the outer bi-nary is initially 0 .
7, and decreases only slightly as thesystem passes through the resonance) (see more detailsin Section IV C). The bottom panel shows the time evo-lution of the peak frequency of GW, given by [30] f GW = (1 + e ) . π (cid:115) G ( m ) a (1 − e ) . (25)We see that the peak frequency is above 10 − Hz at res-onance, and thus the system is in the LISA sensitivityband. After the resonances, the gravitational radiationreduces e gradually, circularizing the inner binary beforethe final merger.For reference, the right panel of Figure 1 shows theevolution for a system with ˆ L = ˆ L out = − ˆ S . A simi-lar resonance feature occurs, although at different loca-tion (semi-major axis), and the inner binary eccentricitybuilds up to as high as 0 . IV. APSIDAL PRECESSION RESONANCE:ANALYTICAL RESULTS FOR COPLANARSYSTEMS
For coplanar (ˆ L = ˆ L out ), non-dissipative (with nogravitational radiation) systems, the secular dynamicscan be understood analytically. We first consider thecase of small eccentricities, before studying the finite ec-centricity case. A. Linear (low- e ) Systems For systems with e , e out (cid:28)
1, the evolution of e and e out is governed by the linear Laplace-Lagrange equations[68], with proper inclusion of the related GR precessionterms [58]. Define the complex eccentricity variables E in = e in exp( i(cid:36) in ) , E out = e out exp( i(cid:36) out ) , (26)where (cid:36) in , (cid:36) out are the longitude of pericenter of theinner and outer orbits. The evolution equations are ddt (cid:18) E in E out (cid:19) = i (cid:18) ω in ν in , out ν out , in ω out (cid:19) (cid:18) E in E out (cid:19) , (27)where ω in = 34 n m m (cid:18) aa out (cid:19) (28)+ ω GR , in + ω (GR)L in L out ∓ ω L in S ,ω out = 34 n out m m m (cid:18) aa out (cid:19) (29)+ ω GR , out ∓ ω L out S − ω (GR)L out L in ± ω S L in ,ν in , out = − n (cid:18) aa out (cid:19) m ( m − m ) m , (30) ν out , in = − n out (cid:18) aa out (cid:19) m m ( m − m ) m . (31)where in Equations (28)-(29) the upper sign denotes thecase of ˆ L = ˆ L out = ˆ S , and the lower sign the ˆ L = ˆ L out = − ˆ S case.Starting with e = 0, e out = e out , at t = 0, Equation(27) can be solved to determine the time evolution of e ( t )[see 58]. We find that e ( t ) oscillates between e and e max ,given by e max = 2 e out , | ν in , out | (cid:112) ( ω in − ω out ) + 4 ν in , out ν out , in . (32)Clearly, e max attains the peak value when ω in = ω out , atwhich e peak = e out , (cid:12)(cid:12)(cid:12)(cid:12) ν in , out ν out , in (cid:12)(cid:12)(cid:12)(cid:12) / = e out , (cid:12)(cid:12)(cid:12)(cid:12) L (cid:48) out L (cid:48) in (cid:12)(cid:12)(cid:12)(cid:12) / = e out , m / m / m / ( m m ) / (cid:18) a out a (cid:19) / , (33)where L (cid:48) in = L in ( e = 0) and L (cid:48) out = L out ( e out = 0). Wecall this “apsidal precession resonance”.The linear theory is valid only for e (cid:28) e out (cid:28) L (cid:48) out (cid:29) L (cid:48) in , as in thecases studied in this paper ( m (cid:29) m , m ), even a small e out , may lead to unphysically larger e peak . In prac-tice, Equation (33) is useful in the sense that wheneverit predicts e peak of order unity of larger, we can expectthe inner binary to attain significant eccentricity, but theprecise value of e can only be obtained using nonlinearcalculations, as we discuss next. B. Finite Eccentricities
For finite eccentricities, Equation (27) breaks down,but the eccentricity evolution of a coplanar triple canstill be understood using energy and angular momentumconservations, without the need of numerical integrationsof the equations of motion.The total energy of the triple can be written as Φ tot =Φ N + Φ extra , whereΦ N = µ Φ (cid:20) − − e − ε oct e (4 + 3 e ) cos ∆ (cid:36) (cid:21) (34) is the Newtonian potential energy between the inner andouter orbits in the octupole order [52, 58, 69], and Φ extra is the effective energy associated to the GR effects. InEquation (34), ∆ (cid:36) ≡ (cid:36) in − (cid:36) out , with (cid:36) in , (cid:36) out thelongitude of pericenters, andΦ ≡ Gm a a (1 − e ) / , ε oct ≡ m − m m aa out e out − e . (35)Various GR effects introduce extra apsidal precession d e dt (cid:12)(cid:12)(cid:12)(cid:12) extra = ˙ ω extra ˆ L × e . (36)In terms of the effective potential, ˙ ω extra is given by (see[49] and Appendix A)˙ ω extra = − √ − e e µ √ Gm a ∂ Φ extra ∂e . (37)The effective Φ extra can be obtained fromΦ extra = − (cid:90) ˙ ω extra eµ √ Gm a √ − e de. (38)Equation (37) is the canonical relations between Delau-nay variables. Equations (36)-(38) can also apply to theouter binary. Consequently, the potential energy associ-ated with the periastron advance in the inner and outerorbits are given by [49]Φ GR , in = − G m m m c a √ − e , (39)Φ GR , out = − G m m m c a (cid:112) − e . (40)Similarly, the potentials associated with the GR EffectsI-III (Equations (12), (16) and (21)) due to the rotatingSMBH areΦ L out S = ± G m (3 m + 4 m ) S c n out a (1 − e ) , (41)Φ L in L out = G m m m (4 m + 3 m ) √ − e c nn out aa (1 − e ) , (42)Φ L in S = ∓ G m m S √ − e c naa (1 − e ) / . (43)In Equations (41) and (43), the upper (lower) sign refersto the ˆ S = ˆ L (ˆ S = − ˆ L ) case. The detailed derivationof the potentials is presented in Appendix A.In the absence of dissipation (no GW), the total energyΦ tot (the sum of Equation (34) and (39)-(43)) is con-served. This energy conservation, together with angularmomentum conservation, L tot = L + L out = constant,completely determine the secular evolution of the triplesystem.Suppose e = 0 and e out = e out , at t = 0, we can usethe conservation of L tot and Φ tot to determine e max , the FIG. 2: The maximum eccentricity e max of the inner binaryas a function of the semimajor axis a for systems startingwith e = 0 and e out , = 0 .
7. The system parameters arethe same as in Figure 1. The black and blue lines are theresults from numerical integration with and without the GWemission. The red dashed lines are achieved by the analyticalformula according to the conservation laws.FIG. 3: The apsidal precession rates of the inner and outerbinaries (Equations 45 and 46) as a function of a . The systemparameters are the same as Figure 1. The black dots representthe resonance location where ω in = ω out . maximum eccentricity attained by the inner binary. Notethat since L out (cid:29) L , conservation of L tot implies e out (cid:39) e out , + 1 − e , e out , L − L L out , , (44)where L and L out , are the initial values of L and L out ,respectively. Solving the energy and angular momentumconservation laws (Equation (44)) yields e as a functionof ∆ (cid:36) . The maximum eccentricity e max is achieved at either ∆ (cid:36) = 0 or π , depending on the initial value of∆ (cid:36) , and whether ∆ (cid:36) librates or circulates.Figure 2 shows e max obtained by different methods asa function of a where the parameters are the same as inFigure 1. Here, the black solid lines are from the numer-ical integrations including the GW emission, while thethe blue solid lines are achieved by integrating the equa-tions without the GW radiation. We evolve the systemfor about ∼ yrs for a given semimajor axis a andrecord e max during the evolution. The red dashed linesare the analytical prediction using the two conservationlaws.The numerical result (without the GW) are in a goodagreement with the analytical calculation. We see thatthere are two evident peaks of e max in the blue and reddashed lines, indicating the resonance arises twice (seealso Figure 3). However, in the “real” evolution of thesystem (i.e. with GW), the eccentricity preserves thememory of the excitation: Once the growth in e hap-pens, the orbit can only be circularized by GW emissiongradually. C. Eccentricity Excitation in coplanar System
Our analysis in Section IV A for linear (low- e ) systemsshows that the inner binary attains a peak eccentricityat the resonance ω in = ω out . For finite eccentricities, theresonance is not precise, but we expect a similar peakeccentricity occurs when ω in = ω out , with ω in = ω LK , in + ω GR , in + ω (GR)L in L out ∓ ω L in S , (45) ω out = ω GR , out ∓ ω L out S − ω (GR)L out L in ± ω S L in , (46)where the various frequencies include the dependence offinite e out (cid:39) e out , . Note that in Equation (46), the lasttwo terms are much smaller than the corresponding termsin Equation (45) since L in /L out (cid:28) ω in and ω out as a function of a forthe examples depicted in Figure 1. When a is relativelylarge, the Newtonian interaction is strong and ω LK , in isthe dominated source in ω in . So we have ω in (cid:29) ω out .On the other hand, near the merger of the inner binary( a is small), the GR effect becomes important and dom-inate the precession, leading to ω in (cid:28) ω out . In between,we see that ω in crosses ω out twice, creating two “apsidalprecession resonances” during the orbital decay. D. Parameter Space
For a given set of system parameters, the criterion ofapsidal precession resonance ( ω in = ω out ) provides a goodestimation on the resonance radius (resonance occurs atthe location a = a in , Res for a given a out ; or a out = a out , Res for a given a ). Combining the analytical analysis in Sec-tion IV B, we can predict the maximum eccentricity ofthe inner binary at resonance. FIG. 4: Parameter space in the a − a out plane, as well as the orbital period of the inner binary P in - a out plane, where the apsidalprecession resonance occurs. The parameters are m = 30 M (cid:12) , m = 20 M (cid:12) , and m = 10 M (cid:12) . The outer binary is set upwith two different eccentricities, e out = 0 . e out = 0 . R g = ( Gm ) /c . The color-coded dots represent the values of the maximum eccentricity of the inner binary(with negligible initial eccentricity) due to the resonance, as calculated by energy and angular momentum conservations. Here,the inner and outer curves are from the triple systems with aligned and anti-aligned ˆ S , respectively.FIG. 5: Same as Figure 4, but in the m − a out plane, for a = 0 .
02 AU.
Figure 4 illustrates the level of eccentricity excitationin the a ( P in ) − a out plane for the given m and e out . Thegreen region corresponds to the space where the triplesystem is dynamically unstable, where the dot-dashedline is the instability limit according to [70]. The dottedlines indicate the innermost stable circular orbits (ISCO)for the outer binary, where R g = ( Gm ) /c (the ISCOranges from R g to 9 R g depending on the spin magnitudeand orientation relative to the orbit). The color-codeddots denote the locations ( a in , Res ) at resonance and thevalues of e max . The inner part apply to the systems withˆ L = ˆ L out = ˆ S and outer part ˆ L = ˆ L out = − ˆ S .We see that in the left panel of Figure 4, for the reso-nance to occur, the BHB must be close to the SMBH. In the case of aligned ˆ S , two locations of a in , Res tend to con-verge as a out increases. The maximum e max is achievedat the largest a out . Furthermore, the eccentricity of theouter binary is crucial to produce significant e max (asshown in the right panel).Figure 5 shows the similar results but in the m − a out plane (with a fixed at 0 .
02 AU). The upper andlower color-coded dots are from the case of aligned andanti-aligned ˆ S , respectively. We find that the apsidalprecession resonance is expected for a wide range of m .In particular, small m leads to larger values of e max fora given a and e out .Figure 6 displays the resonance locations for four typesof inner binaries (as labeled) in P in − a out plane, with four FIG. 6: Similar to Figure 4, but for the coplanar triple systems with aligned ˆ S . Here, we only show the systems satisfying thestability criterion and outside the ISCO. The outer eccentricity e out is set to be 0 . different SMBH masses. Note that we only restrict to theconfiguration where ˆ S is aligned with ˆ L out , and haveconsidered the criteria of stability [70] and ISCO. We seethat, in terms of the occurrence of resonances, the smallmass SMBH (top left panel) favors the low mass binary,especially the BH-NS binary and NS-NS binary. To pro-duce resonance, these binaries must be concentrated inthe inner-most region ( ∼ AU) around the SMBH. Sincethe BHB emits GW in the low frequency range beforechirping to the LIGO band, if the time evolution on thebinary eccentricity could be measured (e.g. sources nearthe Galactic center), it would provide a useful test for thebinary formation channels. As the mass of the SMBH in-creases, the apsidal precession resonance can emerge forthe high mass binaries, and the regions of interest are lo-cated farther from the SMBH, as shown in other panelsof Figure 6.
V. RESONANCE IN THE INCLINED SYSTEMS
In this section, we extend our exploration to the gen-eral cases of mutually inclined inner/outer binaries, andmisaligned spin of SMBH with respect to ˆ L out . Since nosimple analytical result can be derived, we sample a to determine the resonance location numerically for givenouter binary parameters.We first consider the same example as in Figure 1 (withaligned ˆ S ), but increase the initial inclination to thehigher values. We integrate the secular equations of mo-tion with no GW radiation, evolving the system for ∼ yrs, and record e max during the evolution. The results areshown in the upper panel of Figure 7. Compared to thefiducial example (black lines; the same as in Figure 2),we see that the resonance locations change and spread forlarger inclinations. As I approaches 40 ◦ , the dynamics islargely determined by the LK oscillations when a (cid:38) . a ’s, LK oscillations are suppressed), andthe eccentricity excitation due to resonance tends to beerased.We can see several examples of the eccentricity evo-lution in the upper panel of Figure 8, where a is fixedto be 0 .
046 AU. The maximum eccentricity varies. Theirregular behavior in the case of I = 40 ◦ arises due tothe combined influences of Effects I-III.Alternatively, we vary the spin orientation of theSMBH, choosing the initial misalignment angle betweenˆ S and ˆ L out (i.e., θ L out ) to be 0 ◦ , 10 ◦ and 60 ◦ , whilekeeping the mutual binary inclination angle to zero. Theresults are shown in the lower panel of Figure 7. Some-what surprisingly, in this case, e max is sensitive to θ L out FIG. 7: The maximum eccentricity e max as a function of theinner binary semimajor axis a , obtained by integrating theoctupole equations of motion and including the GR effects(but no GW emission). The upper panel shows the resultsfrom various initial inclinations (as labeled) and aligned ˆ S (parallel to ˆ L out ). The lower panel shows the results fromdifferent initial ˆ S − ˆ L out misalignment angles (as labeled)and coplanar tripla. The other parameters are the same as inFigure 1.FIG. 8: Evolution examples of the orbital eccentricity in theinner binary. The parameters are picked from the cases inFigure 7, and a = 0 . a = 0 . and it can grow to larger values, approaching the unity.This is because Effect I plays a crucial role, and LK os-cillations can be triggered (especially for θ L out = 60 ◦ )due to an inclination resonance [e.g., 61]. To illustratethe e evolution, in the lower panel of Figure 8, we see thegrowth of e becomes chaotic for the fixed a = 0 . S , the system can still en-counter the apsidal precession resonance. Figure 9 showsan example of a merging BHB with the inclined SMBH( I = 10 ◦ ). The initial misalignment angles between ˆ L out and ˆ S are 10 ◦ (left panel) and 170 ◦ (right panel), respec-tively. We find that the behaviors of e and e out , in par-ticular the excitation of the inner binary eccentricity, aremore significant than the case of coplanar triple. Whenthe resonance is encountered during the orbital decay, e increases while e out decreases. The systems have themaximum eccentricities in excess of 0 . I and θ L out , the evolution may become chaotic, and theeccentricity of the inner binary can easily to grow to closeto unity [e.g., 61]. Such “GR-enhanced” channel mayplay an important role in BHB mergers. A comprehen-sive parameter space study is beyond the scope of thispaper and we leave it to a future work. VI. DISCUSSION AND CONCLUSION
In this paper, we have studied the dynamics of com-pact BH-BH binaries under the influence of a nearby ro-tating SMBH in a hierarchical triple configuration. Wehave presented the general secular equations of motionthat govern the evolution of the (BH-BH)-SMBH triplesystem, including various general relativistic (GR) effectsthat couple the inner and outer orbits and the spin of theSMBH (Section II). These post-Newtonian equations ofmotion are derived and extended from previous work onbinaries with spinning bodies [65]. In our recent work[61], we have shown that several of these GR effects cansignificantly influence the rate of tertiary induced binarymergers via Lidov-Kozai mechanism. In this paper, wefocus on systems with small mutual inclinations such thatLidov-Kozai oscillation does not happen. We show thatcompact binaries near a SMBH can experience an “ap-sidal precession resonance”, where the pericenter preces-sion rate of the inner binary matches that of the outerbinary. Both precessions are driven by the combined ef-fects of Newtonian gravitational interaction and generalrelativity. The resonance results in an efficient “transfer”of eccentricity from the outer binary to the inner binary,leading to eccentricity growth of the inner binary. Anexample of the eccentricity growth due to apsidal pre-cession resonance during binary merger near a SMBH isshown in in Figure 1.We provide analytical analysis for coplanar systemswith small eccentricities (linear theory; Section IV A)and finite eccentricities (non-linear theory; Section IV B).The linear theory gives a useful criterion for the res-onance (Equations (32), (45) and (46)), but the non-linear theory is needed to accurately predict the valueof maximum eccentricity excitation (see Figure 2). Thegrowth of the eccentricity in the inner binary at res-0
FIG. 9: Similar to Figure 1, but for triple systems with finite initial binary inclinations and ˆ S − ˆ L out misalignment angles: I = 10 ◦ , θ L out = 10 ◦ (left) and θ L out = 170 ◦ (right). The other parameters are the same as in Figure 1. The qualitativebehavior of the inclined systems is similar to the coplanar systems, but the maximum eccentricities due to resonance are larger. onance can be understood as “angular momentum ex-change” (i.e., eccentricity exchange) between the innerand outer binaries. For the systems studied in this paper( m SMBH = m (cid:29) m , m ), even a weakly eccentric outerorbit (SMBH’s orbit) can excite appreciable eccentricityin the inner BH-BH binary, and the peak eccentricity in-creases as the outer binary becomes more eccentric (seeFigure 4).The eccentricity growth due to apsidal resonance gen-erally operates for triple systems with small mutual in-clinations (in contrast to Lidov-Kozai oscillations, whichrequire high mutual inclinations) and allows for generallymisaligned spin orientations of the SMBH. (see Figure 7and 8). The GR effects (especially Effect I; see Equa-tions (11)-(14)) play an important role, and can makethe orbital evolution of the BH binary chaotic. The ec-centricity of the merging binary can attain a larger valuecompared to the corresponding coplanar case (see Figure9).The apsidal precession resonance can occur while the binary is emitting gravitational waves in the low-frequency band, thus potentially detectable by futuregravitational wave detectors operating at low frequen-cies, such as LISA, DECIGO [71] and TianQin [72]. Bi-nary mergers near the Galactic Center would be of greatinterest, particularly if the eccentricity evolution can betracked. On the other hand, the apsidal precession reso-nance may play a role in the scenario of BH binary mergerin the Active Galactic Nuclei disk [e.g., 16]. In this case,a BH binary aligns its orbit with the disk, and movesto the migration trap close to the central SMBH. Thefinal configuration of (BH-BH)-SMBH system may thensatisfy the criterion of resonance studied in this paper. VII. ACKNOWLEDGMENTS
This work is supported in part by the NSF grant AST-1715246 and NASA grant NNX14AP31G. [1] B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese,K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Ad-hikari, V. B. Adya, et al., Phys. Rev. X , 031040 (2019).[2] B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese,K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Ad-hikari, V. B. Adya, et al., arXiv:1811.12940. [3] B. Zackay, T. Venumadhav, L. Dai, J. Roulet, and M.Zaldarriaga, Phys. Rev. D , 023007 (2019).[4] T. Venumadhav, B. Zackay, J. Roulet, L. Dai, and M.Zaldarriaga, arXiv:1904.07214.[5] B. Zackay, L. Dai, T. Venumadhav, J. Roulet, and M.Zaldarriaga, arXiv:1910.09528. [6] V. M. Lipunov, K. A. Postnov, and M. E. Prokhorov,Astron. Lett. , 492 (1997).[7] V. M. Lipunov, V. Kornilov, E. Gorbovskoy, D. A. H.Buckley, N. Tiurina, P. Balanutsa, A. Kuznetsov, J.Greiner, V. Vladimirov, D. Vlasenko et al., Mon. Not.R. Astron. Soc. , 3656 (2017).[8] P. Podsiadlowski, S. Rappaport, and Z. Han, Mon. Not.R. Astron. Soc. , 385 (2003).[9] K. Belczynski, M. Dominik, T. Bulik, R. O’Shaughnessy,C. Fryer, and D. E. Holz, Astrophys. J. Lett. , L138(2010).[10] K. Belczynski, D. E. Holz, T. Bulik, and R.O’Shaughnessy, Nature (London) , 512 (2016).[11] M. Dominik, K. Belczynski, C. Fryer, D. E. Holz, E.Berti, T. Bulik, I. Mandel, and R. O’Shaughnessy, As-trophys. J. , 52 (2012).[12] M. Dominik, K. Belczynski, C. Fryer, D. E. Holz, E.Berti, T. Bulik, I. Mandel, and R. O’Shaughnessy, As-trophys. J. , 72 (2013).[13] M. Dominik, E. Berti, R. O’Shaughnessy, I. Mandel, K.Belczynski, C. Fryer, D. E. Holz, T. Bulik, and F. Pan-narale, Astrophys. J. , 263 (2015).[14] I. Mandel and S. E. De Mink, Mon. Not. R. Astron. Soc. , 2634 (2016).[15] P. Marchant, N. Langer, P. Podsiadlowski, T. M. Tauris,and T. J. Moriya, Astron. Astrophys. , A50 (2016).[16] I. Bartos, B. Kocsis, Z. Haiman, and S. Mrka, Astrophys.J. , 165 (2017).[17] S. F. P. Zwart and S. L. W. McMillan, Astrophys. J. Lett. , L17 (2000).[18] R. M. O’Leary, F. A. Rasio, J. M. Fregeau, N. Ivanova,and R. OShaughnessy, Astrophys. J. , 937 (2006).[19] M. C. Miller and V. Lauburg, Astrophys. J. , 917(2009).[20] S. Banerjee, H. Baumgardt, and P. Kroupa, Mon. Not.R. Astron. Soc. , 371 (2010).[21] J. M. B. Downing, M. J. Benacquista, M. Giersz, and R.Spurzem, Mon. Not. R. Astron. Soc. , 1946 (2010).[22] B. M. Ziosi, M. Mapelli, M. Branchesi, and G. Tormen,Mon. Not. R. Astron. Soc. , 3703 (2014).[23] C. L. Rodriguez, M. Morscher, B. Pattabiraman, S. Chat-terjee, C.-J. Haster, and F. A. Rasio, Phys. Rev. Lett. , 051101 (2015).[24] J. Samsing and E. Ramirez-Ruiz, Astrophys. J. Lett. , L14 (2017).[25] J. Samsing and D. J. D’Orazio, Mon. Not. R. Astron.Soc. , 5445 (2018).[26] C. L. Rodriguez, P. Amaro-Seoane, S. Chatterjee, and F.A. Rasio, Phys. Rev. Lett. , 151101 (2018).[27] L. Gond´an, B. Kocsis, P. Raffai, and Z. Frei, Astrophys.J. , 5 (2018).[28] O. Blaes, M. H. Lee, and A. Socrates, Astrophys. J. ,775 (2002).[29] M. C. Miller and D. P. Hamilton, Astrophys. J. , 894(2002).[30] L. Wen, Astrophys. J. , 419 (2003).[31] F. Antonini and H. B. Perets, Astrophys. J. , 27(2012).[32] F. Antonini, S. Toonen, and A. S. Hamers, Astrophys. J. , 77 (2017).[33] K. Silsbee and S. Tremaine, Astrophys. J. , 39 (2017).[34] B. Liu, and D. Lai, Astrophys. J. Lett. , L11 (2017).[35] B. Liu, and D. Lai, Astrophys. J. , 68 (2018).[36] L. Randall and Z.-Z. Xianyu, Astrophys. J. , 93 (2018).[37] B.-M. Hoang, S. Naoz, B. Kocsis, F. A. Rasio, and F.Dosopoulou, Astrophys. J. , 140 (2018).[38] B. Liu, D. Lai, and Y.-H. Wang, Astrophys. J. , 41(2019).[39] M. L. Lidov, Planetary and Space Science , 719 (1962).[40] Y. Kozai, Astron. J. , 591 (1962).[41] S. Naoz, Annu. Rev. Astron. Astrophys. , 441 (2016).[42] X. Fang, T. A. Thompson, and C. M. Hirata, Mon. Not.R. Astron. Soc. , 4234 (2018).[43] B. Liu and D. Lai, Mon. Not. R. Astron. Soc. , 4060(2019).[44] G. Fragione and B. Kocsis, Mon. Not. R. Astron. Soc. , 4781 (2019).[45] M. Zevin, J. Samsing, C. Rodriguez, C.-J. Haster, andE. Ramirez-Ruiz, Astrophys. J. , 91 (2019).[46] A. S. Hamers and D. Lai, Mon. Not. Roy. Astron. Soc. , 1657 (2017).[47] C. Petrovich and F. Antonini, Astrophys. J. , 146(2017).[48] G. Fragione, E. Grishin, N. W. C. Leigh, H. B. Perets,and R. Perna, Mon. Not. R. Astron. Soc. , 47 (2019).[49] D. Fabrycky and S. Tremaine, Astrophys. J. , 1298(2007).[50] B. Liu, D. J. Mu˜noz, and D. Lai, Mon. Not. Roy. Astron.Soc. , 747 (2015).[51] K. R. Anderson, D. Lai, and N. I. Storch, Mon. Not. Roy.Astron. Soc. , 3066 (2017).[52] E. B. Ford, B. Kozinsky, and F. A. Rasio, Astrophys. J. , 385 (2000).[53] R. A. Mardling, Mon. Not. R. Astron. Soc. , 1768(2007).[54] S. Naoz, B. Kocsis, A. Loeb, and N. Yunes, Astrophys.J. , 187 (2013).[55] C. M. Will, Phys. Rev. D , 044043 (2014).[56] C. M. Will, Phys. Rev. Lett. , 191101 (2018).[57] H. Lim, and C. L. Rodriguez, arXiv:2001.03654.[58] B. Liu, D. Lai, and Y.-F. Yuan, Phys. Rev. D , 124048(2015).[59] R. M. O’Leary, B. Kocsis, and A. Loeb, Mon. Not. Roy.Astron. Soc. , 2127 (2009).[60] N. W. C. Leigh, A. M. Geller, B. McKernan, K. E. S.Ford, M.-M. Mac Low, J. Bellovary, Z. Haiman, W. Lyra,J. Samsing, M. O’Dowd, B. Kocsis, and S. Endlich, Mon.Not. R. Astron. Soc. , 5672 (2018).[61] B. Liu, D. Lai, and Y.-H. Wang, Astrophys. J. Lett. ,L7 (2019).[62] J. H. VanLandingham, M. C. Miller, D. P. Hamilton, andD. C. Richardson, Astrophys. J. , 77 (2016).[63] A. S. Hamers, B. Bar-Or, C. Petrovich, and F. Antonini,Astrophys. J. , 2 (2018).[64] P. C. Peters, Phys. Rev. , B1224 (1964).[65] B. M. Barker and R. F. O’Connell, Phys. Rev. D , 329(1975).[66] ´E. Racine, Phys. Rev. D , 044021 (2008)[67] Y. Fang and Q. G. Huang, Phys. Rev. D , 103005(2019).[68] C. D. Murray, and S. F. Dermott, Solar System Dynam-ics, Cambridge U. Press, NY (1999).[69] S. Naoz, W. M. Farr, Y. Lithwick, and F. A. Rasio, Mon.Not. R. Astron. Soc. , 2155 (2013).[70] L. G. Kiseleva, S. J. Aarseth, P. P. Eggleton, and R. deLa Fuente Marcos, in ASP Conf. Ser. 90, The Origins, Evolution, and Destinies of Binary Stars in Clusters, ed.E. F. Milone and J.-C. Mermilliod (San Francisco, CA:ASP), 433 (1996).[71] S. Kawamura et al., Class. Quant. Grav. , 094011(2011).[72] J. Luo et al. (TianQin Collaboration), Class. Quant.Grav. , 035010 (2016). Appendix A: The Effective Potentials of GR Effects
In the hierarchical coplanar triple system, the secularHamiltonian is given by H = H + H + Φ= − Gm m a − Gm m a out + Φ N + Φ extra , (A1)where Φ N is given by Equation (34), and Φ extra isthe effective potential due to various GR effects. Forthe coplanar systems (ˆ L = ˆ L out ), we have H = H ( (cid:36) in , L in , (cid:36) out , L out ), where (cid:36) in and (cid:36) out are the lon-gitude of pericenter of the inner and outer binaries,respectively, and L in = µ (cid:112) Gm a (1 − e ), L out = µ out (cid:112) Gm a out (1 − e ) are the respective canonicalmomenta. The canonical equations of motion for the in-ner and outer orbits are˙ (cid:36) in = ∂ H ∂L in , ˙ L in = − ∂ H ∂(cid:36) in , (A2)˙ (cid:36) out = ∂ H ∂L out , ˙ L out = − ∂ H ∂(cid:36) out . (A3)Since H depends on (cid:36) in and (cid:36) out through ( (cid:36) in − (cid:36) out )(see Equation (34)), the second equations in (A2) and(A3) yield angular momentum conservation L in + L out = constant . (A4)In vector form, the first equations in (A2) and (A3) implythat the apsidal precession due to Φ extra is given by d e dt (cid:12)(cid:12)(cid:12)(cid:12) extra = ˙ (cid:36) (in)extra ˆ L × e (A5) d e out dt (cid:12)(cid:12)(cid:12)(cid:12) extra = ˙ (cid:36) (out)extra ˆ L out × e out , (A6)where ˙ (cid:36) (in)extra = ∂ Φ extra ∂L in , ˙ (cid:36) (out)extra = ∂ Φ extra ∂L out . (A7)Thus, for a given apsidal precession rate, the “common”extra potential can be evaluated byΦ extra = (cid:90) ˙ (cid:36) (in)extra dL in = (cid:90) ˙ (cid:36) (out)extra dL out . (A8) Consider the various GR effects discussed in Section II.For Effect I, Equation (12) reduces to d e out dt (cid:12)(cid:12)(cid:12)(cid:12) L out S = ( ∓ (cid:36) L out S )ˆ L out × e out . (A9)The upper (lower) sign refers to the ˆ S = ˆ L (ˆ S = − ˆ L )case. Using (14), the effective potential can be obtained:Φ L out S = (cid:90) ( ∓ (cid:36) L out S ) µ out √ Gm a out d (cid:113) − e = ± G m (3 m + 4 m ) S c n out a (1 − e ) . (A10)For Effect II, Equations (16) and (18) reduce to d e dt (cid:12)(cid:12)(cid:12)(cid:12) L in L out = (cid:36) (GR)L in L out ˆ L out × e , (A11) d e out dt (cid:12)(cid:12)(cid:12)(cid:12) L out L in = − (cid:36) (GR)L out L in ˆ L out × e out . (A12)Uding Equation (19), the common effective potential canbe obtained:Φ L in L out = (cid:90) (cid:36) (GR)L in L out µ √ Gm a d (cid:112) − e = (cid:90) ( − (cid:36) (GR)L out L in ) µ out √ Gm a out d (cid:113) − e = G m m m (4 m + 3 m ) √ − e c nn out aa (1 − e ) . (A13)For Effect III, Equations (21) and (23) reduce to d e dt (cid:12)(cid:12)(cid:12)(cid:12) L in S = ( ∓ (cid:36) L in S )ˆ L out × e , (A14) d e out dt (cid:12)(cid:12)(cid:12)(cid:12) S L in = ( ± (cid:36) S L in )ˆ L out × e out . (A15)Again, the upper (lower) sign refers to the ˆ S = ˆ L (ˆ S = − ˆ L ) case. Using (24), the effective potential is given byΦ L in S = (cid:90) ( ∓ (cid:36) L in S ) µ √ Gm a d (cid:112) − e = (cid:90) ( ± (cid:36) S L in ) µ out √ Gm a out d (cid:113) − e = ∓ G m m S √ − e c naa (1 − e ) / ..