Importance of relativistic pericenter precession in identifying the presence of a third body near eccentric binaries

TL;DR

This work develops an analytic timing model for highly eccentric binaries in hierarchical triples, incorporating conservative $1\mathrm{PN}$ corrections that generate relativistic pericenter precession and significantly alter GW burst timing. Using a perturbative osculating-orbit framework, it adds $1\mathrm{PN}$ and $2.5\mathrm{PN}$ terms alongside Newtonian third-body perturbations, deriving coupled evolution equations for the inner-binary parameters and burst timings. The study shows that $1\mathrm{PN}$ precession markedly changes timing signals and induces modulations in semilatus rectum and eccentricity, while neglecting these effects biases estimates of the tertiary’s mass and separation by orders of magnitude in some regimes. Detectability analyses indicate that with $3\mathrm{G}$ detectors, the third body can be identified within finite separations, and the timing signature becomes more robust to initial-phase variations when $1\mathrm{PN}$ corrections are included, highlighting the practical importance of relativistic precession in environmental inference for eccentric binaries.

Abstract

Many astrophysical processes can produce gravitational wave (GW) sources with significant orbital eccentricity. These binaries emit bursts of gravitational radiation during each pericenter passage. In isolated systems, the intrinsic timing of these bursts is solely determined by the properties of the binary. The presence of a nearby third body perturbs the system and alters the burst timing. Accurately modeling such perturbations therefore offers a novel approach to detecting the presence of a nearby companion. Existing timing models account for Newtonian dynamics and leading-order radiation reaction effects but neglect the higher order post-Newtonian (PN) contributions to the inner binary. In this paper, we present an improved timing model that incorporates conservative PN corrections that lead to the precession of the binary's pericenter. We find that these PN corrections significantly impact the binary's orbital evolution and the timing of the GW burst. In particular, 1PN precession gives rise to distinctive modulation features in the binary's semilatus rectum and eccentricity. These modulations encode valuable information about the presence and properties of the third body, including its mass and distance. Furthermore, unmodeled 1PN effects significantly bias the tertiary's mass and distance. Finally we assess the detectability of GW bursts from such perturbed systems and demonstrate that the inclusion of PN corrections is crucial for accurately capturing the orbital dynamics of hierarchical triples.

Figures (9)

• Figure 1: Geometric representation of a triple system. The inner binary follows an eccentric orbit with reduced mass $\mu$ orbiting the total mass $m$ at a separation $r$. The third body of mass $m_3$ is located at a much larger distance $R \gg r$ and moves on a circular outer orbit. The angle $V$ denotes the true anomaly of the inner orbit, and $\omega$ is the longitude of pericenter. The true anomaly of the outer orbit is $V_3$. Both orbits are assumed to be coplanar. The inclination angle $\iota_3$ describes the orientation of the system relative to the detector.
• Figure 2: Allowed values of tertiary distance $R$ and mass $m_3$ for various physical constraints discussed in Sec. \ref{['sec:parameter space']}. Different colors represent different physical conditions. Solid lines denote $p=30 m$ and dashed lines show $p=200m$. We choose an equal mass binary with $m=30M_{\odot}$ and $e =0.99$. The horizontal line represents the radius of the innermost stable circular orbit for a maximally spinning black hole $(R_{\rm min} = m_3)$. The arrows denote the region of allowed parameters.
• Figure 3: Difference in GW burst timings with 1PN ($t$) and without 1PN ($t_{\rm no, 1PN}$) corrections as a function of number of bursts. Each dot represents the difference in timing recorded at each pericenter passage. The black and red dots denote the unperturbed and perturbed binary, respectively. The horizontal dotted line shows the typical detection threshold ($0.1$ sec) for the Einstein Telescope. The initial binary parameters are $m=30 M_{\odot}, e_0=0.99, p_0=200 m, \omega_0=0, V_{3,0} =\pi/3$ perturbed by a third body of mass $m_3=10^6 m$ located at $R=20 m_3$. For both the unperturbed binary and perturbed binary, the 1PN corrections significantly impact the GW burst timings.
• Figure 4: Evolution of the semi-latus rectum, eccentricity, and longitude of pericenter of the inner binary. The properties of the system are the same as those in Fig. \ref{['fig:timing']}. Different colors represent different physical scenarios. The perturbed binary with 1PN precession shows distinctive modulation features in $p$ and $e$. The longitude of the pericenter increases rapidly with time when 1PN precession is included.
• Figure 5: Left: fractional errors on $m_3/R^3$ due to neglecting 1PN corrections as a function of $m_3/R^3$ and for different values of $\omega_0$. Other binary parameters are the same as those in Fig. \ref{['fig:timing']}. Right: errors as a function of $e_0$ for different values of $p_0$ and $\omega_0 = 0$. The unmodeled 1PN corrections in the timing model lead to significantly large errors in the tertiary mass and distance.