Gravitational Radiation-Driven Chaotic Tide in a White Dwarf-Massive Black Hole Binary as a Source of Repeating X-ray Transients

TL;DR

This work demonstrates that gravitational-wave backreaction can drive chaotic growth of the WD’s dynamical tide in WD–MBH binaries over a wide range of pericenter distances, causing the tidal energy to grow roughly linearly with the number of orbits and potentially reach the stellar binding energy. When wave breaking occurs, partial tidal disruption or mass transfer to the MBH can produce repeating X-ray transients (pTDEs) with recurrence times of years, while dynamical tidal stripping near $r_p/R_t \approx 2.7$ can generate quasi-periodic eruptions with shorter recurrence times. The authors develop an efficient iterative map coupling the $(\ell,m)=(2,2)$ f-mode to the orbital evolution under 2.5PN GW backreaction and a parametrized non-conservative mass-transfer model, outlining three observational regimes—LISA EMRIs, pTDEs, and QPEs—organized by the pericenter distance and mass-transfer parameters $\sigma_1,\sigma_2,\sigma_3$. The study provides a framework for linking WD–MBH dynamics to multi-messenger signals, while acknowledging the need for detailed hydrodynamic modeling to solidify the mass-loss physics and refine predictions for LISA and X-ray observatories.

Abstract

Eccentric white dwarf-massive black hole binaries can potentially source some extreme X-ray transients, including the recently observed quasi-periodic X-ray eruptions and tidal disruption events at the galactic nuclei. Meanwhile, they are one of the target gravitational wave sources with extreme mass ratios for future millihertz gravitational wave missions. In this work, we focus on the tidal evolution and orbital dynamics of such binaries under the influence of the tidal backreaction and the dissipative effect from gravitational wave emission. We find that the latter can cause the dynamical tide to evolve chaotically after more than one orbital harmonics encounter the mode resonance through orbital decay. Different from the tidal-driven chaos, which only occurs for pericenter distances within around two times the tidal radius ($R_t$), this gravitational wave-driven chaos can happen at much larger pericenter distances. The growth scales similarly to a diffusive process, in which the tidal energy grows linearly with time on average over a long duration. If the tidal energy eventually approaches the stellar binding energy, achievable when the pericenter distance is less than $4R_t$, it can cause mass ejection as the wave breaks due to nonlinear effects. We show that this can potentially lead to the repeating partial tidal disruptions observed at galactic centers. That means these disruptions can occur at a much larger pericenter distance than previous analytical estimates. Furthermore, if the system can evolve to a close pericenter distance of about 2.7$R_t$, the white dwarf can further lose mass via tidal stripping at each pericenter passage. This provides a mechanism for producing the quasi-periodic eruptions.

Figures (8)

• Figure 1: The various behavior of the highly eccentric WD-MBH binary in the parameter space formed by the initial $r_p$ and the fractional mass loss of the WD at wave breaking. From top to bottom, the regions correspond to where the system remains as a coherent GW source for LISA, a source for pTDEs, and a potential QPE source due to tidal stripping. There is also a region where the WD becomes unbound from the binary before reaching the tidal stripping stage due to mass transfer.
• Figure 2: Left panel: The maximum mode energy (in units of the initial orbital energy $E_\text{orb, 0}$) over 200 orbits with initial $P_\text{orb} = 9$ hr and different $r_p$, obtained with the iterative method in Sec. \ref{['sec:formulation']}. The blue dots are the case that only includes tidal backreaction, and the red dots include both tidal and GW backreaction. A blue dashed line indicates the initial $r_p$ where the change in mode phase at the first pericenter crossing equals unity. A red dashed line indicates where the GW orbital decay effect causes one resonance at the 200th orbit. Keep in mind that its position depends on the number of orbits. A cyan dashed line shows that the averaged mode energy roughly follows $N_\text{orb} \Delta E_{a, 0}$. The initial orbital energy and the stellar binding energy are shown with the horizontal lines. Right panel: The mode energy at different numbers of orbits, $N_\text{orb}$, with initial $r_p$ at $4.4 R_t$. The effects of tidal back reaction and GW backreaction are included. The vertical blue dotted lines correspond to $\omega_a P_k$ being integer multiples of $2\pi$. The inset shows the large $N_\text{orb}$ evolution of the mode energy, and the green line illustrates the linear growth of its averaged value. The horizontal black dashed line gives the SPA of the first resonance energy \ref{['eq:dq_res']}.
• Figure 3: The mode energy evolution normalized by the WD stellar binding energy. The initial $r_p/R_t$ is $2.9$, and the fractional mass ejected by the WD per orbit, $\sigma_1$, is chosen as $5\times 10^{-4}$. The purple region corresponds to $r_p/R_t \leq 2.7$, where we expect the mass loss via dynamical tidal stripping further alters the orbital dynamics. The horizontal dashed line is used to indicate the threshold for wave breaking, i.e., $E_a = 10\% E_*$.
• Figure 4: The eccentricity evolution under the effects of tide, GW backreaction, and wave breaking. The black curve corresponds to the same system in Fig. \ref{['fig:Emode_MT']}. A vertical black dotted line shows where $r_p/R_t = 2.7$ for the case with $\sigma_1 = 5\times 10^{-4}$. For $r_p/R_t$ smaller than that, we expect dynamical tidal stripping. The blue curve has the same initial pericenter distance and period, but a different $\sigma_1$. The red dashed line indicates the $e = 1$ limit where the orbit unbinds.
• Figure 5: The mode energy (left panel) and orbital period (right panel) at different numbers of orbits during dynamical tidal stripping. The effects of tidal and GW backreaction are included. We choose the initial values $r_p/R_t = 2.7$, $q_a = 0$, and $\sigma_1=2\times10^{-4}$. We end the evolution when the binary unbinds, which is indicated by the vertical dashed lines. When mode energy is above 10%$E_*$, we assume wave breaking to happen. We use the parameters $\sigma_1 = 5\times 10^{-4}$ and $\sigma_3 = M_2/M_1$ for the mass transfer during wave breaking.