The transition from adiabatic inspiral to geodesic plunge for a compact object around a massive Kerr black hole: Generic orbits Pranesh A Sundararajan Department of Physics and MIT Kavli Institute, MIT, 77 Massachusetts Ave., Cambridge, MA 02139 (Dated: November 4, 2018) The inspiral of a stellar mass compact object falling into a massive Kerr black hole can be broken into three different regimes: An adiabatic inspiral phase, where the inspiral timescale is much larger than the orbital period; a late-time radial infall, which can be approximated as a plunging geodesic; and a regime where the body transitions from the inspiral to plunge. In earlier work, Ori and Thorne have outlined a method to compute the trajectory during this transition for a compact object in a circular, equatorial orbit. We generalize this technique to include inclination and eccentricity. PACS numbers: 04.25.-g, 04.80.Nn, 97.60.Lf I. INTRODUCTION AND MOTIVATION Teukolsky equation in the time-domain. The world line of the compact object serves as an input to this code. While the world line in the adiabatic phase can be calcu- Extreme mass ratio inspirals (EMRIs), in which stel- lated from a frequency-domain based Teukolsky equation lar mass compact objects radiate gravitational energy solver [5, 6], the trajectory in the transition regime for and fall into their massive black hole companions, are completely generic obits remains unknown. This calcu- promising sources of gravitational waves. LISA [1], the lation will provide the missing link needed to generate a proposed space based gravitational wave detector should complete inspiral trajectory. detect waves from the last stages of such inspirals. A clear theoretical understanding of the dynamics of EM- A number of researchers are working towards solving RIs is vital to the detection of these gravitational waves. the self-force problem exactly [7, 8]. Such an exact solu- tion can be separated (at least qualitatively) into time- The small mass ratios, which typically lie in the range reversal symmetric and asymmetric components. The µ/M = 10−5 10−8, allow EMRIs to be treated within − symmetric component (the “conservative self-force”) con- the framework of perturbation theory. The trajectory serves the integrals of motion. On the other hand, of the compact object can be roughly broken into three the asymmetric component (the “dissipative self-force”) regimes: (a) An adiabatic inspiral phase, during which leads to non-zero time derivatives of the integrals of mo- the dominant inspiral mechanism arises from the radia- tion. Recent advances demonstrate that we are mak- tion reaction force on the smaller object. In this stage, ing steady progress on this problem. For example, the the time scale over which the characteristic radial sep- self force is now essentially understood for circular orbits aration (between the compact object and its black hole around Schwarzschild black holes [8]. Although approxi- companion) changes is large compared to the orbital pe- mate, the results in this paper may serve as an indepen- riod. This allows us to approximate the trajectory as a dent check for these solutions. It is worth noting that if sequence of bound geodesics. (b) A plunge phase, during it becomes possible to include the conservative force in which stable geodesics do not exist. It has been shown a simple way, we should be able to build its impact into [2] that the effect of radiation reaction is negligible dur- the formalism developed here. This work may also be of arXiv:0803.4482v3 [gr-qc] 8 May 2008 ing the plunge and that this phase can be modeled as interest for numerical relativity — a perturbative inspiral a geodesic infall. (c) A regime where the spiraling com- constructed by the techniques discussed here may be an pact object transitions from adiabatic inspiral to geodesic accurate point of comparison for full numerical inspirals plunge. The course of motion at this juncture shows as- for small ratios (and may even be useful, if not so accu- pects of both, the self-force from radiation reaction and rate, for mass ratios that are not strictly perturbative). the effects of unstable geodesics. Ref. [9] discusses the transition when the compact ob- In [2], Ori and Thorne introduce a method to predict ject is in an eccentric, equatorial orbit. However, the the motion when the object is constrained to an approxi- focus of that paper is to calculate the transit time and mately circular, equatorial orbit. We generalize this pro- estimate the probability for LISA to observe such a tran- cedure to include inclined and eccentric trajectories. A sition. Our intent is to generate the world line during few modifications to the prescription in [2] are introduced the transition. We also choose our initial conditions dif- to handle such generic orbits. The results from our gen- ferently than they are chosen in Ref. [9]; we discuss these eralized prescription are in excellent agreement with [2]. differences in more detail in Sec. III. The simple calculation described in this paper is meant The rest of the paper is organized as follows: Section II to serve as a stopgap for many other open and important discusses circular orbits with arbitrary inclination. Sec. problems. There has been recent progress ([3, 4] and ref- III generalizes the formalism developed in Sec. II to in- erences therein) in the development of a code to solve the clude eccentricity. Finally, we summarize our results in 2 Sec. IV. The potentials can be expressed as: 1 2 R = E a2 + r2 aL µ2 − z − ∆ + (L aE)2 + r2µ2 + Q , (2.5) µ2 z − II. THE TRANSITION TRAJECTORY FOR 1 CIRCULAR ORBITS V = [Q θ µ2 − cos2 θ a2 µ2 E2 + L2/ sin2 θ , (2.6) − z Up to initial conditions, a set of three constants, the en- 1 ergy, E, the component of the angular momentum along V = L /sin2 θ aE φ µ z − the spin axis, Lz, and the Carter constant, Q define a a 2 2 geodesic. The Carter constant has an approximate in- + E r + a Lza , (2.7) terpretation of being the square of the component of µ∆ − angular momentum perpendicular to the spin axis. As 1 2 Vt = a Lz aE sin θ + the compact object radiates, the “constants” that de- µ − fine its geodesic will gradually evolve. (We will refer to r2+a2 E r2 + a2 L a . (2.8) [E(t),Lz(t),Q(t)] as the “constants”, although they are µ∆ − z slowly evolving.) A common approach to model the adi- abatic regime consists of treating the motion as the se- The parameters (r,θ,φ,t) are the Boyer-Lindquist coor- quence of geodesics [5, 6] defined by these evolving con- dinates, M is the black hole mass, µ is the perturbing stants. As pointed out in [10], this limit amounts to a “ra- mass, Σ = r2 + a2 cos2 θ,∆= r2 2Mr + a2 and a diative” or “dissipative” approximation. A true adiabatic is the spin parameter of the black hole.− The constants approximation would be a sequence of orbits in which (E,Lz,Q) represent the actual energy, momentum and each orbit included conservative self corrections. Since Carter constant (in units of M, M 2 and M 4 respectively), we currently use purely geodesic orbits as our background not the dimensionless versions of them. By introduc- motion (in lieu of a self-force enhanced description), we ing the perturbing mass explicitly, our notation deviates will refer to a sequence of geodesics as an “adiabatic in- from previous literature. We do this in order to show the spiral” throughout this paper. Thus, within the adiabatic dependence of the transition phase on the mass of the approximation, the world line of a particle is computed perturbing object. We also set G = c = 1 everywhere. by mapping [E(t),Lz(t),Q(t)] to [r(t),θ(t), φ(t)]. The symbols r, θ and φ are the usual Boyer-Lindquist coor- dinates. B. The last stable orbit In contrast, the plunge can be treated as a single unsta- A standard but not unique definition of the “inclina- ble geodesic with almost constant E, Lz and Q. Thus, the passage from adiabatic inspiral to geodesic plunge tion” of a Kerr geodesic is given by must contain both these features — slowly evolving “con- L cos ι = z , (2.9) stants” and marginal stability. 2 Lz + Q L2 Q = p z . (2.10) ⇒ cos2 ι L2 − z It is possible to use ι to eliminate the Carter constant. A. Kerr Geodesics Thus, any circular orbit can be parametrized by its radius (r) and inclination (ι). The last stable orbit (LSO) serves as an important ref- The following system of first order equations describes erence point — the inspiral is adiabatic well before the geodesics in a Kerr [11, 12] geometry: compact object crosses the LSO and is approximately a plunge well after the crossing. Since the transition occurs in the vicinity of the LSO, a preliminary step in our com- dr Σ = √R, (2.1) putation is to determine r and (E,Lz,Q) at the LSO for dτ ± a given inclination at the LSO, ιLSO. Note that ι changes dθ with time because it is a function of [E(t),L (t),Q(t)]. Σ = Vθ , (2.2) z dτ ± Circular orbits satisfy dφ p Σ = V , (2.3) dτ φ R =0and (2.11) dt ′ dR Σ = V . (2.4) R = =0 . (2.12) dτ t dr 3 ′ We must have R = 0 because the LSO lies at an ex- ing points, θmax and θmin, where tremum of R. We also require that 2 ′′ d R 0 θmin θmax π .