Phys. Rev. D 106, 084058 (2022) - Dynamical perturbations around an extreme mass ratio inspiral near resonance

Dynamical perturbations around an extreme mass ratio inspiral near resonance

Makana Silva and Christopher Hirata

Phys. Rev. D 106, 084058 – Published 28 October 2022

Abstract

Extreme mass ratio inspirals (EMRIs)—systems with a compact object orbiting a much more massive (e.g., Galactic Center) black hole—are of interest as a new probe of the environments of galactic nuclei, and their waveforms are a precision test of the Kerr metric. This work focuses on the effects of an external perturbation due to a third body around an EMRI system. This perturbation will affect the orbit most significantly when the inner body crosses a resonance with the outer body, and results in a change of the conserved quantities (energy, angular momentum, and Carter constant) or equivalently of the actions, which results in a subsequent phase shift of the waveform that builds up over time. We present a general method for calculating the changes in action during a resonance crossing, valid for generic orbits in the Kerr spacetime. We show that these changes are related to the gravitational waveforms emitted by the two bodies (quantified by the amplitudes of the Weyl scalar $ψ_{4}$ at the horizon and at $\infty$ ) at the frequency corresponding to the resonance. This allows us to compute changes in the action variables for each body, without directly computing the explicit metric perturbations, and therefore we can carry out the computation by calling an existing black hole perturbation theory code. We show that our calculation can probe resonant interactions in both the static and dynamical limit. We plan to use this technique for future investigations of third-body effects in EMRIs and their potential impact on waveforms for LISA.

Received 30 July 2022
Accepted 26 September 2022

DOI:https://doi.org/10.1103/PhysRevD.106.084058

Physics Subject Headings (PhySH)

General relativity Gravitational waves

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Makana Silva ^1,2,* and Christopher Hirata ^1,2,3,†

¹Center for Cosmology and Astroparticle Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus OH, 43210, USA
²Department of Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus OH, 43210, USA
³Department of Astronomy, The Ohio State University, 140 West 18th Avenue, Columbus OH, 43210, USA

^*silva.179@osu.edu
^†hirata.10@osu.edu

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Vol. 106, Iss. 8 — 15 October 2022

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Figure 1
A model of our three-body system. The test particle (purple) can take any generic geodesic while the external perturber (blue) is assumed to be on an equatorial, circular orbit. The test particle will follow some inspiral trajectory due to the emission of gravitational radiation. The presence of the external perturber will affect the emitted radiation from the test particle by its own gravitational field. The power of the emitted radiation can be computed in the regions between the particle and the BH ( $r < r_{1}$ ) and the particle infinity ( $r > r_{1}$ ), where $r_{1}$ is the location of the perturbing body. The $ψ_{4}$ scalar for both regions will have contributions from the pure gravitational radiation due to the perturber.
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Figure 2
The setup for the precession resonance problem (Sec. 4a). The pericenter (P) and apocenter (A) of the inner body precess at approximately the same rate as the outer body orbits, so that the angle $Θ_{F} + δ$ between the inner apsidal line (P–A) and the outer perturber is roughly constant. The shaded region shows the part of the inner body orbit that is “above” the plane of the page ( $z_{inertial} > 0$ ). The angle $δ$ is a small correction that oscillates around zero due to the argument of pericenter $ω_{K}$ not being in the same plane as the other angles. The figure is not to scale.
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Figure 3
The plots shows the tidal torque ( $⟨ {\dot{\tilde{L}}}_{z} ⟩$ ) on an EMRI due to an effectively static perturber on a circular orbit of radius $r_{outer} = 500 M$ . The left (right) plot is for an EMRI orbit with eccentricity $e = 0.63$ (0.942), pericenter $r_{p} = 30 M$ ( $10 M$ ), and inclination $θ_{inc} = π / 4$ . We compare the Keplerian (dashed) and our torques and find about an 8% difference in amplitude.
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Figure 4
This shows the tidal torque on the inner body at different tidal resonance conditions, ${\vec{N}}_{inner} = (3, 0, - 2)$ and $(- 3, 0, 2)$ , and orbit parameters (shown in each panel). The radius of the outer perturber is $r_{outer} = 500 M$ , effectively making the perturber “static.”.
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Figure 5
The tidal torques per unit mass of the inner and outer bodies with the respective $1 / r_{outer}^{3}$ scaled out for the mode vector ${\vec{N}}_{inner} = (1, 2, - 2)$ and for a pericenter, inclination, and black hole mass and spin of $3 M$ , $π / 4$ , $M = 1$ , and $a_{*} = 0.9$ , respectively The apocenter changes according to the outer orbit radius, decreasing as the outer radius decreases. Each curve represents the scaled torques on the EMRI system due to a perturbing orbit at a specific radius. Regardless of the outer orbit radius, we can see a similar behavior with the tidal torques on the EMRI.
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Physical Review D

covering particles, fields, gravitation, and cosmology

Dynamical perturbations around an extreme mass ratio inspiral near resonance

Makana Silva and Christopher Hirata

Phys. Rev. D 106, 084058 – Published 28 October 2022

Abstract

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Physical Review D

covering particles, fields, gravitation, and cosmology

Dynamical perturbations around an extreme mass ratio inspiral near resonance

Makana Silva and Christopher Hirata

Phys. Rev. D 106, 084058 – Published 28 October 2022

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