Observable signature of magnetic tidal coupling in hierarchical triple systems

TL;DR

This work addresses the resonant dynamics of a compact-object binary orbiting a supermassive black hole, focusing on strong-field magnetic tidal interactions. By extending the Hamiltonian treatment to $0.5$PN order and incorporating quadrupolar magnetic tides, the authors identify a new class of precession resonances with $q=1$ that induce additional eccentricity excitations and accelerate mergers. They develop an analytical framework based on averaged Hamiltonians and fundamental outer-orbit frequencies, and validate it with numerical solutions of the Lagrange planetary equations that include gravitational-wave radiation reaction. The results imply observable imprints in the gravitational-wave signal, potentially detectable by LISA, and establish magnetic tidal coupling as a distinctive strong-gravity effect in b-EMRI systems.

Abstract

We study hierarchical triple systems formed by a compact binary orbiting a supermassive black hole (SMBH), focusing on the role of relativistic magnetic tidal interactions. Extending previous analyses of precession resonances to 0.5 post-Newtonian order, we incorporate quadrupolar magnetic tidal moments, which have no Newtonian counterpart. We find that magnetic tides introduce new resonances absent at lower order, leading to additional eccentricity excitations and significantly modifying the binary's long-term evolution. Numerical solutions of the Lagrange Planetary Equations confirm these analytical predictions and reveal how resonance strength depends on orbital eccentricity and inclination. The resulting dynamics accelerates the binary merger and imprints distinctive signatures on gravitational waves, potentially observable by LISA. Our findings identify magnetic tidal coupling as a novel strong-gravity effect and establish its importance for the resonant dynamics of compact-object binaries near SMBHs.

Figures (5)

• Figure 1: Evolution of the eccentricity $e$ of the inner binary as a function of the normalized time $\hat{t}/T_{\rm RR}$, comparing the case with only the $0$PN tidal coupling (black curve) to that including both the $0$PN and $0.5$PN tidal couplings (blue curve), for a quasi-circular outer orbit. The left panel corresponds to a region closer to the ISO of the SMBH, while the right panel shows a region farther from it. In the former, the first precession resonance encountered by the inner binary arises solely from the $0$PN term in the Hamiltonian \ref{['Hmarckaveinner_plusprecession']}; in the latter, the first resonance is due exclusively to the magnetic $0.5$PN contribution. The parameters and initial conditions are: $M_*=4\times 10^6\, \rm M_{\odot}$, $m_1=25\rm M_{\odot}$, $m_2=1.5\, \rm M_{\odot}$, $\hat{e}=0.08$, $\hat{a}=7.5 GM_*/c^2 \sim 0.3\,\text{AU}$ (left panel), $\hat{a}=10 G M_*/c^2 \sim 0.4 \text{AU}$ (right panel), $T_{\rm RR}\sim 6\,\text{yrs}$, GW peak frequency $f_{\mathrm{GW}}\sim0.06\, \text{Hz}$, $a_0\sim 0.0003\, \text{AU}$ (close to the first resonance), $e_0=0.01$, $\gamma_0=0^{\circ}$, $I_0=20^{\circ}$, $\vartheta_0=0^{\circ}$, $\hat{\Psi}_0=0^{\circ}$. Numbers in parentheses indicate the $(k,l)$ pairs corresponding to each resonance peak, which satisfy the resonance condition \ref{['eq:res_condition_q1']}.
• Figure 2: Evolution of the first precession resonance encountered by the inner binary as a function of the outer semi-major axis $\hat{a}$, expressed in units of $GM_*/c^2$. The solid black curve corresponds to the $\dot{\gamma} = \tfrac{1}{2}\Omega_{\hat{r}}$ resonance, i.e., $(k, l) = \left(\tfrac{1}{2}, 0\right)$, which arises solely from the $0$PN electric coupling in the Hamiltonian \ref{['Hmarckaveinner_plusprecession']}. The dashed black curve represents the $\dot{\gamma} = -\Omega_{\hat{r}} + \Omega_{\hat{\Psi}}$ resonance, i.e., $(k, l) = (-1, 1)$, which originates from the $0.5$PN magnetic coupling. In the left panel, we set $\hat{e} = 0.08$, while in the right panel $\hat{e} = 0.5$. In both panels, the blue solid line marks the semi-major axis of the ISO, defined in Eq. \ref{['eq:a_iso']}, for a SMBH of mass $M_* = 4 \times 10^6\,M_\odot$. Notice that, for a fixed outer semi-major axis $\hat{a}$, the inner binary evolves upward along the y-axis.
• Figure 3: Evolution of the eccentricity $e$ of the inner binary as a function of $\hat{t}/T_{\rm RR}$, including only the $0$PN tidal coupling (black curve) and both the $0$PN and $0.5$PN tidal couplings (blue curve), for a quasi-circular outer orbit. In the left panel, the initial mutual inclination between the inner and outer orbits is $I_0 \simeq 0^{\circ}$ (spin-aligned configuration), where precession resonances driven by the $0.5$PN magnetic coupling are enhanced. In contrast, in the right panel we set $I_0 = 60^{\circ}$, where all $0.5$PN magnetic contributions to precession resonances are suppressed. For both panels, the following parameters and initial conditions are used: $M_*=4\times 10^6\, \rm M_{\odot}$, $m_1=25\rm M_{\odot}$, $m_2=1.5\, \rm M_{\odot}$, $\hat{e}=0.05$, $\hat{a}=10 G M_*/c^2 \sim 0.4 \text{AU}$, $T_{\rm RR}\sim 6\,\text{yrs}$, GW peak frequency $f_{\mathrm{GW}}\sim 0.06\, \text{Hz}$, $a_0\sim 0.0003\, \text{AU}$ (close to first resonance), $e_0=0.01$,$\gamma_0=0^{\circ}$, $I_0=0.00001^{\circ}\sim0^{\circ}$ (left panel), $I_0=60^{\circ}$ (right panel), $\vartheta_0=0^{\circ}$, $\hat{\Psi}_0=0^{\circ}$.
• Figure 4: Evolution of the eccentricity $e$ of the inner binary as a function of $\hat{t}/T_{\rm RR}$, including only the $0$PN tidal coupling (black curve) and both the $0$PN and $0.5$PN tidal couplings (blue curve). In the left panel, even for a moderate outer eccentricity of $\hat{e} = 0.2$, the $0$PN electric coupling in the Hamiltonian \ref{['Hmarckaveinner_plusprecession']} begins to contribute to precession resonances that, in the quasi-circular limit, arise solely from the $0.5$PN magnetic term. In the right panel, for a higher outer eccentricity ($\hat{e} = 0.5$), both the $0$PN electric and $0.5$PN magnetic couplings contribute to all precession resonances, making the two curves nearly indistinguishable. The only notable difference is that, with the $0.5$PN magnetic term included, the eccentricity jumps are slightly larger, leading to a faster inspiral of the inner binary. For both panels, the following parameters and initial conditions are used: $M_*=4\times 10^6\, \rm M_{\odot}$, $m_1=25\rm M_{\odot}$, $m_2=1.5\, \rm M_{\odot}$, $\hat{e}=0.2$ (left panel), $\hat{e}=0.5$ (right panel), $\hat{a}=10 G M_*/c^2 \sim 0.4 \text{AU}$ (left panel), $\hat{a}=15 G M_*/c^2 \sim 0.6 \text{AU}$ (right panel), $T_{\rm RR}\sim 6\,\text{yrs}$ (left panel), $T_{\rm RR}\sim 32\,\text{yrs}$ (right panel), GW peak frequency $f_{\mathrm{GW}}\sim 0.06\, \text{Hz}$ (left panel) , GW peak frequency $f_{\mathrm{GW}}\sim 0.03\, \text{Hz}$ (right panel) , $a_0\sim 0.0003\, \text{AU}$ (left panel) (close to first resonance), $a_0\sim 0.0005\, \text{AU}$ (right panel) (close to first resonance), $e_0=0.06$, $\gamma_0=0^{\circ}$, $I_0=20^{\circ}$, $\vartheta=0^{\circ}$, $\hat{\Psi}=0^{\circ}$.
• Figure 5: Evolution of the eccentricity $e$ of the inner binary for different values of the parameter $\alpha$. The following parameters and initial conditions are used: $M_*=4\times 10^6\, \rm M_{\odot}$, $m_1=25\rm M_{\odot}$, $m_2=1.5\, \rm M_{\odot}$, $\hat{e}=0.08$, $\hat{a}=10 G M_*/c^2 \sim 0.4 \text{AU}$, RR timescale as defined in Eq. \ref{['T_RR_approx']}$T_{\rm RR}\sim 6\,\text{yrs}$, GW peak frequency $f_{\mathrm{GW}}\sim 0.06\, \text{Hz}$, $a_0\sim 0.0003\, \text{AU}$ (close to first resonance), $e_0=0.01$, $\gamma_0=0^{\circ}$, $I_0=15^{\circ}$, $\vartheta_0=0^{\circ}$, $\hat{\Psi}_0=0^{\circ}$.