Lindblad resonance torques in relativistic discs – II. Computation of resonance strengths

Mon. Not. R. Astron. Soc. 414, 3212–3230 (2011) doi:10.1111/j.1365-2966.2011.18619.x Lindblad resonance torques in relativistic discs – II. Computation of resonance strengths Christopher M. Hirata Caltech M/C 350-17, Pasadena, CA 91125, USA Accepted 2011 February 28. Received 2011 February 24; in original form 2010 October 10 ABSTRACT Key words: accretion, accretion discs – black hole physics – relativistic processes. 1 I N T RO D U C T I O N This is the second in a series of two papers devoted to a relativistic computation of torques from an external perturber on a thin disc due to interactions at the Lindblad resonances, that is, locations in the disc where the orbital frequency  and the radial epicyclic frequency κ satisfy κ = ±m( − s ), where s is the pattern speed of the perturbation. Such resonances have been extensively studied in the non-relativistic case (e.g. Lynden-Bell & Kalnajs 1972; Goldreich & Tremaine 1978, 1979, 1980; Lin & Papaloizou 1979). In the first paper (Hirata 2011, hereinabter Paper I), we performed this computation for a general time-stationary, axisymmetric space– time with an equatorial plane of symmetry and a metric perturbation hαβ that respects the equatorial symmetry. This paper (Paper II) completes the evaluation of the Lindblad torque in the case of most interest: the perturbation of the accretion disc surrounding a Schwarzschild or Kerr black hole by a small secondary also orbiting in the equatorial plane. Such computations of the Lindblad resonant strengths may be relevant in the context of electromagnetic counterparts to binary black hole mergers, particularly if an inner disc is involved (Chang et al. 2010). (The more complicated case of perturbations outside the equatorial plane – as may occur in the case  E-mail: of a merger where the primary hole is rotating and the secondary is in an inclined orbit – is left to future work.) The resonant torque formula in Paper I depended on the geodesic properties in the unperturbed space–time as well as being proportional to the square of the absolute value of the resonant amplitude S (m) , which was a function of the eimφ Fourier component of the metric perturbation hαβ and its spatial derivative hαβ,r . The construction of these perturbations generally depends on the solution for the Weyl tensor component ψ 4 , which may be solved using a separable wave equation with a source given by the stress-energy tensor associated with the perturber (Teukolsky 1973); then hαβ may be obtained by applying a second-order differential operator to a master potential (Chrzanowski 1975), which may be derived from ψ 4 (Wald 1978). Fortunately, for our computations, there is a way to circumvent the Chrzanowski (1975) procedure: Paper I showed that the particular combination of metric perturbations we require is related to P (m) , the power delivered to a test particle in a slightly eccentric orbit by the eimφ component of the perturbation. By replacing the perturber with an equivalent gravitational wave source – either incoming from past null infinity in the case of an inner Lindblad resonance (ILR) or emerging from the past horizon in the case of an outer Lindblad resonance (OLR) – we may equate P (m) with the power absorbed from the gravitational wave. However, energy is conserved on a time-independent background metric and thus P (m) can be related to the interference between the equivalent gravitational wave representing the perturbation and the  C 2011 The Author C 2011 RAS Monthly Notices of the Royal Astronomical Society  We present a fully relativistic computation of the torques due to Lindblad resonances from perturbers on circular, equatorial orbits on discs around Schwarzschild and Kerr black holes. The computation proceeds by establishing a relation between the Lindblad torques and the gravitational waveforms emitted by the perturber and a test particle in a slightly eccentric orbit at the radius of the Lindblad resonance. We show that our result reduces to the usual formula when taking the non-relativistic limit. Discs around a black hole possess an m = 1 inner Lindblad resonance (ILR) with no Newtonian–Keplerian analogue; however, its strength is very weak even in the moderately relativistic regime (r/M ∼ few tens), which is in part due to the partial cancellation of the two leading contributions to the resonant amplitude (the gravitoelectric octupole and gravitomagnetic quadrupole). For equatorial orbits around Kerr black holes, we find that the m = 1 ILR strength is enhanced for retrograde spins and suppressed for prograde spins. We also find that the torque associated with the m ≥ 2 ILRs is enhanced relative to the non-relativistic case; the enhancement is a factor of 2 for the Schwarzschild hole even when the perturber is at a radius of 25M. Lindblad resonances – II which satisfy ρ ρ̄ = −1 . The Weyl scalar ψ 4 used to describe the emitted gravitational waveform is ψ4 = −Cαβγ δ nα m̄β nγ m̄δ = −Rαβγ δ nα m̄β nγ m̄δ , where Cαβγ δ is the Weyl tensor and the equivalence to the component formed from the Riemann tensor Rαβγ δ is due to the Newman– Penrose basis conditions. The horizons of the black hole are at the radial coordinate  (6) rh± = M ± M 2 − a 2 . Particles very close to the horizon (r − rh+ → 0+ ) rotate at a pattern speed of the hole’s angular velocity:  1 − 1 − a2 a a . (7) = 2 = H = 2Mrh+ 2a M rh+ + a 2 Note that for real coordinates, m̄ = m∗ and ρ̄ = ρ ∗ , where denotes the complex conjugate; however, we will occasionally analytically continue r to complex values, in which case the barred quantities are not the complex conjugates of the unbarred quantities: ρ̄(r, θ ) = ρ ∗ (r ∗ , θ ∗ ) = ρ ∗ (r, θ ). Finally, we define ∗ K ≡ ω(r 2 + a 2 ) − am 2 K E R R M E T R I C A N D N OTAT I O N (5) (8) and use the angular operator 2.1 The metric and null tetrad IL†n ≡ ∂θ − m csc θ + aω sin θ + n cot θ. We parametrize the Kerr black hole sequence with the gravitational mass M and the specific angular momentum a. We use relativistic units where the Newtonian gravitational constant and the speed of light are equal to unity. The dimensionless angular momentum is a ≡ a/M. The Kerr metric in Boyer–Lindquist coordinates (Boyer & Lindquist 1967) is   2Mr 4Mar 2 sin2 θ dtdφ dt 2 − ds = − 1 − + + (r 2 + a 2 )2 − dr 2 + a 2 sin2 θ sin2 θ dφ 2 dθ 2 , (1) where ≡ r2 − 2Mr + a2 and ≡ r2 + a2 cos 2 θ . The contravariant metric coefficients are (r 2 + a 2 )2 − a 2 sin2 θ , g tt = − g tφ = − g rr = 2aMr , g φφ = and g θθ = − a 2 sin2 θ , sin2 θ −1 . The standard Newman–Penrose basis is a r 2 + a2 ∂t + ∂φ + ∂r , l =   2 r + a2 a ∂t + ∂φ − ∂r , n = 2 ia sin θ ∂t + ∂θ + i csc θ ∂φ √ and m= 2 (r + ia cos θ ) −ia sin θ ∂t + ∂θ − i csc θ ∂φ √ . m̄ = 2 (r − ia cos θ) Expressions involving m can be simplified if we use −1 −1 ρ= and ρ̄ = , r − ia cos θ r + ia cos θ  C 2011 The Author, MNRAS 414, 3212–3230 C 2011 RAS Monthly Notices of the Royal Astronomical S (...truncated)