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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 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 . (2.17) R = > 0 , (2.13) ≤ ≤ ≤ dr2 The potential problems posed by the turning points can for the extremum to be stable. This implies that the orbit will be marginally stable if R′′ = 0. Thus, the be eliminated by reparametrizing θ. Following Ref. [5], ′ ′′ we use three equations R = R = R = 0 can be numerically solved for a given ιLSO to yield r, E, Lz and Q at the 2 2 LSO. z = cos θ = z− cos χ , (2.18)

where C. The constants in the transition regime 2 2 2 2 2 Q + Lz + a (µ E ) β(z z+)(z z−) = βz z − We need a model of the phase space trajectory, − − − µ2 [E(t),L (t),Q(t)] near the LSO in order to compute the Q z + , (2.19) world line of the compact object as it transitions from µ2 inspiral to plunge. To this end, we Taylor expand about the LSO to obtain and β = a2(µ2 E2)/µ2. The θ-equation of motion now − E(t) ELSO + (t tLSO)E˙ LSO , (2.14) becomes ≃ − Lz(t) Lz,LSO + (t tLSO)L˙ z,LSO , (2.15) ≃ − dχ β(z+ z) Q(t) QLSO + (t tLSO)(Q˙ LSO + δQ˙ ) = 2 − , (2.20) ≃ − dt γ p+ a Ez(χ)/µ +δQ , (2.16) which are natural generalizations of equations (3.4) and where (3.5) of Ref. [2]. The overdot denotes differentiation with E (r2 + a2)2 2MraL respect to t. We will later see that our initial condition γ = a2 z . (2.21) µ ∆ − − ∆µ for t amounts to choosing tLSO, the instant at which the   compact object crosses the LSO. This choice is consistent with the procedure in Ref. [2] — Eq. (3.14) of Ref. [2] Equation (2.20) can now be integrated without turning implies a choice of tLSO = 0. points because χ varies from 0 to π to 2π as θ varies from The constant terms in Eq. (2.16), δQ and δQ˙ , are θmin to θmax and back to θmin. needed to guarantee that the trajectory remains circu- lar as we enter the transition. As the notation suggests, these constants are small compared to QLSO and Q˙ LSO. E. The prescription They are discussed in more detail when we discuss initial conditions for the transition in Sec. II F. The expressions (2.14), (2.15) and (2.16) do not include In keeping with our main objective of obtaining the conservative effects of the self force. Pound and Poisson world line [r(t),θ(t), φ(t)] through the transition regime, [10] have demonstrated that this omission will lead to we eliminate τ by dividing equation (2.1) by (2.4) and observationally significant changes. Inclusion of these ef- squaring the result to obtain fects would effectively alter the potentials, Eq. (2.5) - 2 Eq. (2.8) leading to slight deviations of (E,Lz,Q)LSO dr R(r, χ) and rLSO (for a given ιLSO) from their geodesic values. = 2 F. (2.22) dt Vt(r, χ) ≡ The exact impact of these effects will not be known until   we know what the corrections are. We will later see that One more time derivative gives the acceleration: our results posses all the expected qualitative features despite this handicap. Moreover, the prescription in [2] d2r 1 ∂ R ∂ R dχ/dt and its generalization presented here can easily incorpo- = + . (2.23) dt2 2 ∂r V 2 ∂χ V 2 dr/dt rate these effects once they are known.   t   t   The fluxes at the LSO remain a parameter in our code. We use the code developed in [5] to provide us the di- Ideally, Eq. (2.23) must have other additive terms pro- 2 2 3 portional to non-zero powers of µ. This is analogous to mensionless fluxes, (M/µ) E˙ , (M/µ )L˙ z and (1/µ )Q˙ at the LSO. Equivalently, we can use the expressions in [13] Eq. (3.10) of [2]. Excluding this term amounts to ignor- (with zero eccentricity) for the dimensionless fluxes. ing the conservative self force. D. Reparametrization of the θ-equation Since the transition phase is in the proximity of the LSO, we can Taylor expand F about rLSO, ELSO, Lz,LSO and QLSO to obtain Numerical integration of the θ-equation warrants some care. The issue arises because dθ/dt vanishes at the turn- 4

3 2 1 ∂ F 3 ∂ F F (r, L ,E,χ,ι) (r rLSO) + (L L LSO)(r rLSO) z ≃ 6 ∂r3 − ∂r∂L z − z, − LSO z LSO 2 2 ∂ F ∂ F + (E ELSO)(r rLSO )+ (Q QLSO)(r rLSO) . (2.24) ∂r∂E − − ∂r∂Q − − LSO LSO

1 Thus, the acceleration now becomes :

2 3 2 2 2 d r 1 1 ∂ F 2 ∂ F ∂ F ∂ F = (r rLSO) + (L L LSO)+ (E ELSO)+ (Q QLSO) dt2 2 2 ∂r3 − ∂r∂L z − z, ∂r∂E − ∂r∂Q −  LSO z LSO LSO LSO ∂F dχ/dt + . (2.25) ∂χ dr/dt 

We have not expanded the second term in Eq. (2.23) where because we do not know the value of χ at r = rLSO a priori. Similarly, the φ-equation takes the form 1 ∂3 R α = , (2.30) −4 ∂r˜3 Σ2  LSO 2 ˙ 2 dφ Vφ(r, χ) 1 ∂ R E˜ ∂ R = . (2.26) β = 2 + 2 dt Vt(r, χ) 2 ∂L˜ ∂r˜ Σ ˜˙ ∂E∂˜ r˜ Σ " z   Lz   Q˜˙ ∂2 R + (2.31) The trajectory in the transition phase can now be com- ˜˙ ∂Q∂˜ r˜ Σ2 puted by integrating equations (2.25), (2.26) and (2.20) Lz  #LSO from some starting point outside the LSO to some ending 1 dL˜ dL˜ /dt˜ κ(t) = z = z , (2.32) point inside the LSO, for a given ιLSO, with time varying −µ/M dτ˜ −(µ/M)(dτ/dt) E, L and Q. z κ0 = κ LSO , (2.33) | −1/5 τ0 = (αβκ0) , (2.34)

withr ˜ = r/M, t˜ = t/M, E˜ = E/µ, L˜z = Lz/(µM) and ˜ 2 F. Initial conditions Q = Q/(µM) . These definitions reduce to those presented in Ref. [2] when ι = 0. It is useful to observe that κ does not scale The angles, φ and χ can be set to zero without loss with µ. We evaluate dτ/dt, α, β and κ0 at θ = π/2 ιLSO − of generality. Setting χ = 0 corresponds to starting the because we do not know θLSO a priori. Notice that X and inspiral at θ = θmin. T are dimensionless. The choice of initial radius depends explicitly on µ. In The smoothness of the transition implies that there is Ref. [2], the authors define parameters, α, β, κ, τ0 and no fixed instant at which the transition starts or ends. Motivated by the choices in Ref. [2], we set T 1 at R0. These are used to scale out the perturbing mass from ≃ − the equation of motion and initial conditions. Although t = 0 and stop the numerical integrator when X Xe = 5. ≤ we prefer to retain dimensions in the equations of motion, − we specify initial conditions in a dimensionless form, in- In summary, our initial conditions are T = 1, φ = 0 2 − dependent of µ. This will be useful be in interpreting our and χ =0at t =0 . Setting T = 1 at t = 0 allows us to − results and making comparisons with Ref. [2]. Following calculate tLSO and hence E(0) and Lz(0) from equations Ref. [2], we define (2.14) and (2.15). We then solve R(E,Lz,Q,r) = 0 and dR/dr = 0 to obtain r(0) and Q(0). This is analogous to Sec. IIIC of Ref. [2] where they enforce X = √ T to 2 5 − µ / r rLSO determine X at t = 0. X = − , (2.27) M R0  2/5 −3/5 R0 = (βκ0) α , (2.28) 1 5 µ / t˜ t˜LSO dτ 2 T = − , (2.29) It is important to keep |T | small enough that our Taylor expan- M τ0 dt sion about the LSO remains a valid approximation. 1 LSO 2 Note that Eq. (2.24) ignores terms of order (µ/M) and higher.

5

The trajectory is adiabatic before the start of the stant, transition. At t = 0, we must impose the condition [14, 15, 16] that circular orbits remain circular even un- ∆T 3.3 3.4 , (2.37) ≃ − der adiabatic radiation reaction. Thus, requiring that ˙ ˙ ′ 2 when X = 5 for all values of a and ι. Again, this is R = dR/dt = 0 and R = d R/drdt = 0 leads to expres- e − sions (3.5) and (3.6) of [5] forr ˙(0) and Q˙ (0) respectively. a consistent generalization of the result in Ref. [2] where they find ∆T 3.3 for all circular, equatorial orbits. We can now substitute Q(0) and Q˙ (0) in Eq. (2.16) to ≃ obtain two independent equations, Radial trajectory in the transition regime 3.4 Q(0) = QLSO tLSO(Q˙ LSO + δQ˙ )+ δQ and(2.35) − Q˙ (0) = Q˙ LSO + δQ,˙ (2.36) 3.35 which can be used to evaluate δQ and δQ˙ .

3.3

G. Code algorithm and numerical results r/M 3.25 The previous sections developed the steps required to calculate the compact body’s trajectory as it transitions 3.2 from inspiral to plunge. We now summarize the algo- rithm that was actually used to implement this prescrip- tion: 0 50 100 150 200 250 300 350 400 450 (1) Take ιLSO as input. t/M (2) Compute E and Lz at the LSO. ˙ ˙ (3) Obtain E and Lz at the LSO from the code developed FIG. 1: Radial trajectory during the transition (black line) in [5]. We may also use the expressions in [13] (which re- from inspiral to plunge for a compact object of mass µ = − duce to the results in [5] for circular orbits), which will 10 5M in a nearly circular orbit around a black hole with be particularly useful when we generalize to eccentric or- spin a = 0.8M. The compact object crosses the LSO at time tLSO = 137.5M. The inclination of the orbit at tLSO is ιLSO = bits. ◦ (4) Choose initial conditions T 1, φ = 0 and χ = 0 37 . The red line is a plunging geodesic matched to the end at t = 0. ≃ − of the transition. (5) Calculate E(0) and Lz(0) from equations (2.14) and (2.15). (6) Solve for r(0) and Q(0) by imposing R = 0 and Angular trajectory in the transition regime dR/dr =0 at t = 0. 2.5 (7) Computer ˙(0) and Q˙ (0) from equations (3.5) and (3.6) of [5]. 2 (8) Substitute Q(0) and Q˙ (0) in Eq. (2.16) to evaluate (rad) 1.5 δQ and δQ˙ . θ (9) Use a Runge-Kutta integrator on (2.25), (2.26) and 1

(2.20) to compute the coordinates at the next step. A 0 50 100 150 200 250 300 350 400 450 time step of δt 0.05M works well. t/M ≃ (10) Update the “constants”, Ei+1 = Ei + Eδt˙ , Lz,i+1 = 6 Lz,i + L˙ zδt and Qi+1 = Qi + (Q˙ + δQ˙ )δt. The subscript i denotes a discrete time instant. 4 (11) Repeat steps (9)-(11) until X(t) 5. (rad)

≃− φ The primary objective of this calculation is to compute 2 the world line of the compact object during the transi- tion. Figures 1 and 2 illustrate r, θ and φ motions of 0 0 50 100 150 200 250 300 350 400 450 the compact object for a typical set of parameters. We t/M also show a plunging geodesic matched to the end of the transition. FIG. 2: Angular motion during the transition for a compact Table I shows the parameters and transit times for a object around a spinning black hole with identical parameters range of inclination angles. In general, we find that the as in Fig. 1. transit time increases with inclination. However, the di- mensionless transit time ∆T remains approximately con- 6

−6 TABLE I: Fluxes and transit times for different inclinations. We set a = 0.5M, µ = 10 M , M = 1, T (0) = −1 and Xe = −5. ◦ 2 2 3 ιLSO rLSO/M (M/µ) E˙ LSO (M/µ )L˙ z,LSO (1/µ )Q˙ z,LSO α β R0 κ0 τ0 t/M ∆T − 10 3 4.23 −0.00457 −0.0422 −0.000572 0.00311 0.0327 0.0699 0.603 2.80 944.9 3.36 10 4.26 −0.00446 −0.0409 −0.00684 0.00304 0.0327 0.0677 0.604 2.81 952.4 3.36 20 4.32 −0.00415 −0.0375 −0.0241 0.00284 0.0329 0.0615 0.610 2.82 974.9 3.36 30 4.43 −0.00368 −0.0323 −0.0481 0.00254 0.0333 0.0523 0.618 2.84 1012.6 3.36 40 4.59 −0.00314 −0.0262 −0.0733 0.00219 0.0342 0.0416 0.630 2.86 1065.9 3.35 50 4.78 −0.002594 −0.0198 −0.0946 0.00184 0.0363 0.0309 0.643 2.88 1134.4 3.35 60 5.01 −0.00208 −0.0139 −0.108 0.00152 0.0403 0.0211 0.657 2.90 1217.9 3.35

R(r) to solve dR/dr = 0 and d2R/(drdt) = 0 for r(0) TABLE II: Variation of transit time with perturbing mass, ◦ andr ˙(0) respectively. In contrast, we solve the equations µ/M. We set a = 0.9M, ιLSO = 0.001 , M = 1, Ts = −1 and exactly. This leads to differences of less than 1%. Xe = −5. Note that rLSO = 2.32M. µ/M t/M ∆T −3 10 118.9 3.449 III. ECCENTRIC ORBITS − 10 4 185.6 3.397 10−5 292.2 3.375 The methods developed thus far only discussed circular − 10 6 461.9 3.367 orbits. We now extend this technique to include non-zero − 10 7 731.3 3.363 eccentricity. In the absence of radiation reaction, the − 10 8 1158.6 3.362 geodesic equations admit bound eccentric orbits. These orbits are conventionally parametrized by the semi-latus rectum, p, and the eccentricity, e. The radial coordinate H. Comparison with Ref. [2] can now be expressed as p r(t)= . (3.1) The results in Ref. [2] provide an important sanity 1+ e cos ψ(t) check for the case of circular, equatorial orbits. However, we have to account for the minor differences between the The angle ψ(t) is analogous to the eccentric anomaly and two approaches. Ref. [2] makes the approximations can be solved for numerically. The geodesic has turning points at ψ = 0,π. Deep in the adiabatic inspiral, the dφ dφ and (2.38) compact object’s trajectory is well approximated by a dt ≃ dt ISCO sequence of orbits with slowly varying p(t) and e(t).

dτ dτ Geodesics beyond the LSO do not have turning points , (2.39) dt ≃ dt (where dr/dt = 0). This changes the situation consider- ISCO ably because the parameters, p and e are not well-defined which lead to anymore. Thus, the trajectory ceases to have turning points somewhere during the transition from inspiral to dL˜z/dt˜ κ = , plunge. We will later show that this feature is naturally (µ/M)(dτ/dt) − buried in our model of the transition. dL˜ /dt˜ ISCO z | , (2.40) ≃ −(µ/M)(dτ/dt)ISCO A. The last stable orbit which is a dimensionless constant. In our prescription, dτ/dt varies with time. This time dependence has to be As with circular orbits, the last stable bound geodesic enforced because dτ/dt is a function of θ, whose value at is an important reference in our procedure. The inner the LSO is not known a priori. The circular, equatorial and outer turning points (rmin and rmax) of the LSO are case in Ref. [2] does not suffer from this pathology be- related to eLSO and pLSO through cause θ = π/2 at all times. Thus, we treat κ as a slowly varying function of time. Table II shows the transit times pLSO rmin = and (3.2) for a nearly equatorial orbit (ιLSO =0.001) and a range 1+ eLSO of mass ratios. As the becomes smaller, the pLSO rmax = . (3.3) variation in κ decreases, and the dimensionless transit 1 eLSO time converges to the limit where κ is constant. − Our initial conditions differ slightly from those used Our goal is to determine pLSO and the constants in Ref. [2]. Effectively, they use the Taylor expansion of (E,Lz,Q) at the LSO for a given ιLSO and eLSO. This 7 can be achieved by requiring that Q(0) and Q˙ (0) ; E(t), Lz(t) and Q(t) are independent. As discussed in Sec. II C, equations (3.6), (3.7) and dR = 0at r = rmin , (3.4) (3.8) do not include conservative effects of the self force. dr Just as the circular case, this will lead to a slight shift of min max R = 0at r = r and r = r . (3.5) (E,Lz,Q)LSO and pLSO (for a given eLSO and ιLSO) with respect to their geodesic values. Again, our motivation to Recall that the function R is given by Eq. (2.5) and ι is stick with this approximation stems from the facts that: defined by Eq. (2.9). We require Eq. (3.4) to be satisfied (a) These effects can be incorporated into our prescrip- because the inner most turning point corresponds to a tion once they are known, and (b) Our results show the local maximum of ( R). Equation (3.5) enforces the generally expected behavior, at least qualitatively. compact object’s velocity− to vanish at the turning points. Numerical methods to calculate the change in the Equations (3.4), (3.5) and (2.9) can be solved numerically Carter constant due to gravitational-wave backreaction for p and (E,L ,Q) at the LSO. Appendix A describes z have recently become available [17, 18]. Work is in the details of this numerical procedure. progress implementing that result in the code we use to compute the rate of change of orbital constants [19]. For ˙ B. The constants during the transition now, we use the approximate expressions for Q described in [13]; it will be a simple matter to update our code ˙ As with the circular case, our initial conditions are such when more accurate Q results are available. that we effectively choose the LSO crossing to occur at t = tLSO. This allows us to expand the constants about the LSO to obtain C. The prescription for eccentric orbits

E(t) ELSO + (t tLSO)E˙ LSO , (3.6) ≃ − Our next task is to derive equations of motion to map L (t) L LSO + (t tLSO)L˙ LSO , (3.7) z ≃ z, − z, the phase space trajectory to an actual world line. The Q(t) QLSO + (t tLSO)Q˙ LSO. (3.8) angular equations, Eq. (2.20) and Eq. (2.26), remain un- ≃ − affected. Our strategy for the radial equation is to ex- Notice that we no longer need the corrections, δQ and δQ˙ pand the geodesic equation about (ELSO,Lz,LSO,QLSO). because there are no additional symmetries to constrain This leaves us with

d2r 1 ∂F ∂F dχ/dt = + , (3.9) dt2 2 ∂r ∂χ dr/dt   ∂F ∂2F ∂2F ∂2F (E ELSO)+ (L L )+ (Q QLSO) ∂r ≃ ∂r∂E − ∂r∂L z − z,LSO ∂r∂Q −  LSO z LSO LSO

∂F + (r, χ; ELSO,L LSO,QLSO) . (3.10) ∂r z, 

Note that we only expand about the constants, not the This amounts to choosing tLSO. We can now determine r-coordinate, because there is no unique r at the LSO. [E(0),Lz(0),Q(0)], which can be mapped to (p,e,ι) at In the absence of the first three terms in Eq. (3.10), t = 0. This mapping is allowed because the trajectory is the equation of motion is simply a geodesic at the LSO. adiabatic before t = 0. The coordinates at any point on This is consistent with our intuitive notion of “expanding the geodesic defined by [E(0),Lz(0),Q(0)] can serve as about the LSO”. The existence of turning points presents our initial conditions. For simplicity, we choose a complication while integrating Eq. (3.10) numerically. We present a method to tackle this in Appendix B. p r(0) = , (3.11) D. Initial conditions 1+ e dr (0) = 0 , (3.12) We need initial conditions for r and dr/dt before we dt start the numerical integrator. Motivated by the initial φ(0) = 0 , (3.13) conditions for circular orbits, we set T 1 at t = 0. χ(0) = 0 . (3.14) ≃ − 8

The equations of motion can now be easily integrated Radial trajectory in the transition regime across the LSO. 11 10

9

E. Code implementation and numerical results 8

7 r/M Taking eccentricity into account changes our algorithm 6 slightly. We summarize the code’s algorithm as follows: (1) Take ιLSO and eLSO as input. 5 (2) Compute E, Lz and Q at the LSO. 4 (3) Obtain E˙ , L˙ z and Q˙ at the LSO from the expressions in Ref. [13]. 3 (4) Choose initial conditions T 1, φ = 0 and χ = 0 ≃ − 2 at t = 0. 0 50 100 150 200 250 300 350 400 t/M (5) Calculate E(0), Lz(0) and Q(0) from equations (3.6), (3.7) and (3.8). (6) Map [E(0),Lz(0),Q(0)] to (p,e,ι). FIG. 3: Radial trajectory during the transition (black line) from inspiral to plunge for a compact object of mass µ = (7) Set r = p/(1 + e) and dr/dt =0 at t = 0. −6 (8) Use a Runge-Kutta integrator on (3.9), (2.26) and 10 M in an eccentric orbit around a black hole with spin (2.20) to compute the coordinates at the next step. A a = 0.8M. The compact object crosses the LSO at time tLSO = 196.7M. The inclination and eccentricity of the orbit time step of δt 0.05M works well. ◦ ≃ ˙ at tLSO are ιLSO = 45 and eLSO = 0.6 respectively. The (10) Update the “constants”, Ei+1 = Ei + ELSOδt, red line is an unstable geodesic matched to the end of the Lz,i+1 = Lz,i + L˙ z,LSOδt and Qi+1 = Qi + Q˙ LSOδt. The transition. subscript i refers to a discrete time instant. (11) Repeat steps (9)-(11) until X 5. ≃− Radial trajectory in the transition regime Recall that the local minimum of the potential R is 3.2 less than zero for bound orbits and is greater than zero for a plunging geodesic. The minimum is exactly zero at 3.1 the LSO. These conditions can be used as sanity checks 3 while performing the numerical integration. Figures 3, 4 and 5 show a typical trajectory during the 2.9 transition from inspiral to plunge. The compact object 2.8 starts at the minimum of the last bound geodesic be- r/M fore the plunge. The radial coordinate increases until it 2.7 reaches a maximum where R = dr/dt = 0. Subsequently, 2.6 it turns around and heads toward the minimum. After executing a number of “whirls” near the minimum, the 2.5 trajectory becomes unstable, and thus plunges into the central black hole. The whirls are evident from the angu- 2.4 lar trajectory plotted in Fig. 5. We also show a plunging 2.3 geodesic matched to the end of the transition. Notice 280 300 320 340 360 380 t/M that the plunge spends quite a bit of time at r 2.8M — much more time than the transition trajectory.∼ This FIG. 4: Same as Fig. 3, but zooming in on the final “whirls”. is because the radiation emission built into the transition trajectory’s construction pushes it off this marginally sta- ble orbit rather quickly. Table III shows the various parameters and transit F. Comparison with Ref. [9] times for a range of eccentricities. Note that the pa- rameters α, β, R0, κ0 and τ0 (which are defined in Sec. As mentioned in the introduction, there are differences IIF) are evaluated at pLSO. In general, we find that the between our generalized prescription and the method de- transit time is proportional to α. This is not surprising veloped in Ref. [9], which only models the transition when because α is the first term in the Taylor expansion of the the compact object is in an eccentric, equatorial orbit. potential, R. We also observe some degree of correlation First, we set our initial conditions at the start of the between the transit time and τ0, the parameter used to LSO, whereas Ref. [9] sets the initial conditions at the define the dimensionless time. end of the LSO. This educated choice allows Ref. [9] to 9

−5 ◦ TABLE III: Fluxes and transit times for different eccentricities. We set a = 0.8M, µ = 10 , ιLSO = 45 , M = 1, Ts = −1 and Xe = −5. 2 2 3 eLSO pLSO/M (M/µ) E˙ LSO (M/µ )L˙ z,LSO (1/µ )Q˙ z,LSO α β R0 κ0 τ0 t/M ∆T 10−4 3.58 −0.00974 −0.0619 −0.153 0.00517 0.0530 3.04 0.113 7.98 486.3 3.34 0.1 3.70 −0.00857 −0.0545 −0.136 0.00351 0.0506 3.54 0.0969 8.97 448.2 2.81 0.2 3.84 −0.00795 −0.0479 −0.120 0.00220 0.0484 4.33 0.0832 10.2 373.5 2.10 0.3 3.96 −0.00751 −0.0419 −0.105 0.00117 0.0463 5.83 0.0714 12.1 341.1 1.66 0.4 4.09 −0.00693 −0.0361 −0.0900 0.000365 0.0442 10.8 0.0604 15.9 332.7 1.25 0.5 4.22 −0.00607 −0.0300 −0.0745 −0.000280 0.0420 11.5 0.0496 17.7 331.6 1.14 0.6 4.35 −0.00450 −0.0236 −0.0582 −0.000801 0.0401 5.41 0.0385 15.2 338.8 1.37 0.7 4.49 −0.00351 −0.0168 −0.0413 −0.00123 0.0381 3.57 0.0272 15.1 381.9 1.56 0.8 4.62 −0.00206 −0.0100 −0.0245 −0.00159 0.0362 2.44 0.0162 16.1 507.2 1.95

Angular trajectory in the transition regime Radial trajectory in the transition regime 2.4 2.5

2 2.35

1.5 (rad)

θ 2.3 1 2.25 0.5 0 50 100 150 200 250 300 350 t/M 2.2 r/M

6 2.15

4 2.1 (rad) φ 2 2.05

0 2 0 50 100 150 200 250 300 350 250 260 270 280 290 300 310 t/M t/M

FIG. 5: Angular trajectory during the transition for the same FIG. 6: Comparison of our trajectory with approximate an- set of parameters as in Fig. 3. alytic results from Ref. [9]. The compact object is in an ec- centric, equatorial trajectory with parameters eLSO = 0.6 and − − µ = 10 6M. Its mass is µ = 10 6M and is around a black hole with spin a = 0.8M. The black line shows our trajectory; derive an analytic form for the trajectory. Second, we the blue line is obtained from Ref. [9]. The observed deviation differ in the choice of final conditions. 3 is because the approximation in Ref. [9] is somewhat more re- In attempting to make comparisons with Ref. [9], we strictive than ours. found a number of typographical errors. Thus, we extract the essence of the calculation in Ref. [9] and present it in a form that (hopefully) makes the errors obvious. We start by expressing the radial geodesic equation as

2 The orbit is unstable if the local maximum of V (r) is dr + V (r) = 0 , (3.15) negative. Define dτ   where R I = Max V (r) = V (rmax) . (3.17) V (r) = . (3.16) − { } − −Σ2

′ ′′ Note that this implies V (rmax) = 0 and V (rmax) < 0. 3 See Sec. IID3 and Ref. [20] of Ref. [9] for a description their We Taylor expand Eq. (3.15) about the maximum of V (r) choice of final conditions. corresponding to some (E,Lz,Q) just beyond the LSO to 10 get the transit time is correlated with α, the coefficient of the first term in the Taylor expansion of the radial potential. 2 d(δr) ′ The code developed in [3] and [4] solves the Teukolsky + V (rmax)+ δrV (rmax) dτ equation in the time-domain and thus computes gravita-   tional waveforms for almost any given trajectory of the 1 2 ′′ + δr V (rmax) = 0 ,(3.18) compact object. We intend to generate waveforms by 2 2 feeding the world lines calculated using this prescription 2 d(δr) 1 2 ′′ γ + δr V (rmax)= I to the time-domain Teukolsky equation-solver. The re- ⇒ dt 2   sulting waveforms will be useful for LISA data analysis 2 2 routines. It is also possible to use these waveforms to es- 2 d(δr) δr γ = I (3.19) timate recoil velocities from mergers of compact objects ⇒ dt − τ 2   s with black holes. where

δr = r(t) rmax , (3.20) − Acknowledgments dt V γ = = t , (3.21) dτ Σ rmax rmax The author is very grateful to Scott A. Hughes for 2 ′′ invaluable guidance throughout the development of this τs = 2/ V (rmax) . (3.22) | | paper. The author also thanks Gaurav Khanna for help- The solution of Eq. (3.19) in the regime of interest is ful discussions. The author also thanks Adrian Liu for spotting an embarassing error in an earlier version of the t tc paper. This work was supported by NASA Grant No. r(t) = rmax √Iτ sinh − , (3.23) − s γτ NNG05G105G.  s  where tc is an integration constant. We can compare our numerical solution with Eq. (3.23) by letting the two APPENDIX A: THE LSO FOR ECCENTRIC trajectories intersect at some arbitrary instant. This free- ORBITS dom is equivalent to choosing initial conditions. For ex- ample, Fig. 6 shows the two trajectories near rmax for The following set of equations need to be solved in which tc is chosen such that they intersect at t = 300M. The compact object has mass µ = 10−6M and is in order to compute p and (E,Lz,Q) at the LSO for a given inclination (ι) and eccentricity, (e): an eccentric orbit with eLSO = 0.6 around a black hole with spin a = 0.8M. Notice that Eq. (3.23) is valid only in the immediate vicinity of rmax because (dt/dτ) R(r,E,Lz) = 0 , (A1) and (E,Lz,Q) are assumed constant. Inclusion of the 1+ e R r ,E,L = 0 and (A2) time-dependence of (dt/dτ) is crucial because it leads to 1 e z  −  time varying γ, which alters the natural timescale in Eq. dR (3.23). This explains the observed deviation at large val- (r,E,Lz) = 0 . (A3) ues of δr . dr | | Recall that R is given by Eq. (2.5). The carter constant, Q can be eliminated using Eq. (2.9). Applying an it- IV. SUMMARY AND FUTURE WORK erative technique to solve the above equations directly can lead to problems because the terms that do not con- The primary focus of this paper is to provide an ap- tain r are identical in equations (A1) and (A2). We can proximate model for the trajectory of a compact object skirt around this problem by solving the equivalent set as it transitions from an adiabatic inspiral to a geodesic of equations, plunge. We have presented a generalization of the pro- cedure in Ref. [2], where circular, equatorial orbits are R1(r,E,Lz) = R(r,E,Lz)=0 , (A4) treated. We derive approximate equations of motion [Eq. 1+ e (2.25) and Eq. (3.10)] by Taylor expanding the geodesic R2(r,E,L ) = R r ,E,L R1(r,E,L ) z 1 e z − z equations about the LSO and subjecting them to evolv-  −  ing E, Lz and Q. We can now readily integrate these = 0 and (A5) equations numerically. Figures 1 and 2 show the radial dR R3(r,E,L ) = (r,E,L )=0 , (A6) and angular trajectories for a typical inclined, circular z dr z orbit. We also plot the plunging geodesic that it tran- sitions to. Figures 3 and 5 are analogous plots for an using the standard Newton-Raphson method described eccentric orbit. Our numerical experiments suggest that in [20]. This iterative procedure takes an initial guess for 11

−7 the solution as input. We use when Bi < x, where x 10 . The method outlined here works| | well for a large≃ fraction of parameter space. r0 = 6.1(1 a/2) , (A7) − 1 2qv3 + q2v4 Lz,0 = r0v cos ι − and (A8) 1 3v2 +2qv3 − APPENDIX B: NUMERICAL INTEGRATION 1 2v2 + qv3 ACROSS TURNING POINTS E0 = − p . (A9) 1 3v2 +2qv3 − As mentioned in Sec. IIIC, Eq. (3.10) passes through p T where q = a/M, r = M/r and S0 = (r0,Lz,0, E0) is turning points. The numerical integrator can accumu- our initial guess for S = (r, L , E)T . Let S denote the late error when dr/dt 0. This section describes our z i → solution at any givenp iteration. The algorithm consists algorithm to resolve the issue. of incrementing Si as follows: Let tp denote the instant at which dr/dt = 0. The ra- dial motion is highly symmetric about the turning point. Si+1 = Si + λ δSi , (A10) Thus, we must have, × (A11) dr dr = , (B1) where dt − dt − tp+ǫ tp ǫ −1 δSi = J Bi , (A12) i where ǫ is an infinitesimal duration of time. When the ∂R1/∂r ∂R1/∂E ∂R1/∂Lz radial velocity becomes very small, we exploit this sym- Ji =  ∂R2/∂r ∂R2/∂E ∂R2/∂Lz  , (A13) metry and set ∂R3/∂r ∂R3/∂E ∂R3/∂Lz  i  T  dr dr Bi = ( R1,i, R2,i, R3,i) , (A14) = , (B2) − − − dt − dt − tp+δt tp δt and λ 0.1. The subscript “i” denotes that the expres- ≃ sions are evaluated at (ri,Lz,i, Ei). We stop iterating which is the discretized version of Eq. (B1).

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