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

📝 Abstract

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 nonrelativistic limit. Discs around a black hole possess an m=1 inner Lindblad resonance 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 inner Lindblad resonances is enhanced relative to the nonrelativistic case; the enhancement is a factor of 2 for the Schwarzschild hole even when the perturber is at a radius of 25M.

💡 Analysis

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 nonrelativistic limit. Discs around a black hole possess an m=1 inner Lindblad resonance 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 inner Lindblad resonances is enhanced relative to the nonrelativistic case; the enhancement is a factor of 2 for the Schwarzschild hole even when the perturber is at a radius of 25M.

📄 Content

arXiv:1010.0759v2 [astro-ph.HE] 1 Mar 2011 Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 30 October 2018 (MN LATEX style file v2.2) Lindblad resonance torques in relativistic discs: II. Computation of resonance strengths Christopher M. Hirata Caltech M/C 350-17, Pasadena, California 91125, USA 22 February 2011 ABSTRACT 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 nonrelativistic 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 rela- tivistic 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 nonrelativistic case; the enhancement is a factor of 2 for the Schwarzschild hole even when the perturber is at a radius of 25M. Key words: accretion, accretion discs – relativistic processes – black hole physics. 1 INTRODUCTION This is the second in a series of two papers devoted to a relativistic computation of torques from an external per- turber on a thin disc due to interactions at the Lind- blad resonances, i.e. locations in the disc where the orbital frequency Ωand the radial epicyclic frequency κ satisfy κ = ±m(Ω−Ωs), where Ωs is the pattern speed of the per- turbation. Such resonances have been extensively studied in the nonrelativistic case (e.g. Lynden-Bell & Kalnajs 1972; Goldreich & Tremaine 1978, 1979, 1980; Lin & Papaloizou 1979). In the first paper (“Paper I”), we performed this com- putation for a general time-stationary, axisymmetric, space- time with an equatorial plane of symmetry and a metric per- turbation hαβ that respects the equatorial symmetry. This paper (“Paper II”) completes the evaluation of the Lind- blad 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 equa- torial 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 com- plicated case of perturbations outside of the equatorial plane – as may occur in the case 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 spacetime 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 perturba- tions generally depends on the solution for the Weyl tensor component ψ4, which may be solved using a separable wave equation with a source given by the stress-energy tensor associated with the perturber (Teukolsky 1973); and then hαβ may be obtained by applying a second-order differential operator to a master potential (Chrzanowski 1975), which may be derived from ψ4 (Wald 1978). Fortunately, for our computations there is a way to circumvent the Chrzanowski (1975) procedure: Paper I showed that the particular com- bination 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 c⃝0000 RAS 2 Hirata time-independent background metric, and thus P(m) can be related to the interference between the equivalent gravita- tional wave representing the perturbation and the gravita- tional wave emitted by the test particle. This allows us to ex- press the resonant amplitude and hence the resonant torque in terms of the waveforms emitted by the perturber and the test particle (both to future null infinity and into the fu- ture horizon), so that standard methods t

This content is AI-processed based on ArXiv data.