Two-body relaxation in the EMRI-TDE disk model for Quasi Periodic Eruptions

Abstract

Quasi Periodic Eruptions (QPEs) are luminous bursts of soft X-rays recently discovered in galactic nuclei. They repeat on timescales of hours to weeks, superimposed to an otherwise stable quiescent X-ray level, consistent with emission from a radiatively efficient accretion flow around relatively low-mass MBHs. Although their physical origin is still debated, their quasi-periodicity naturally arises within the 'impact model', in which the X-ray bursts are generated by the interaction between an sBH or a star in a close orbit around the central MBH and the accretion disk formed by a tidal disruption event (TDE). While this model is consistent with the phenomenology of QPEs, it remains unclear whether such specific physical configurations are sufficiently commonto explain the observed QPE number density. We present the first end-to-end quantitative calculation of the expected QPE rate and abundance within the framework of the impact model. To this purpose, we combine the rates of TDEs and extreme mass-ratio inspirals (EMRIs) around MBHs spanning a range of masses masses. We employ the public code \textsc{PhaseFlow} to simulate seven systems with MBH masses between $10^5 M_\odot$ and $10^8 M_\odot$, each sourronded by a three-component population: one composed of $1M_\odot$ stars, and two consisting of sBHs with masses of $10M_\odot$ and $40 M_\odot$. Based on the emission constraints available in the literature, we restrict to sBH EMRIs on prograde orbit with eccentricity $e<0.5$ and inclination $ι<20^{\circ}$ with respect to the accretion disk. For stellar EMRIs the constraints instead arise from the requirement that the star avoid tidal disruption. We find that the predicted QPE number density spans the range $10^{-12} \rm Mpc^{-3}$ to $10^{-6} \rm Mpc^{-3}$, depending on the assumed orbital period interval and on the adopted eccentricity and inclination thresholds.

Figures (11)

• Figure 1: Schematic representation of the QPE disk-impact model. The central big black dot represents the MBH, the small gray dot represents the compact orbiter, either a sBH or a star, the yellow area represents the TDE accretion disc, and the dashed line represents the EMRI trajectory. The angle between the disk and the EMRI orbital plane is the inclination $i$. The violet bubbles represent the stripped mass that expands both above and below the disk after the impact. As soon as the bubbles become optically thin, they emit electromagnetic radiation ($L_{\mathrm{QPE}}$)
• Figure 2: Trajectories of EMRIs dominated by GW emission. In the blue region the gravitational-wave timescale $t_\mathrm{GW}$ is shorter than the relaxation timescale $t_\mathrm{rlx}$; the orange area encloses plunging orbits; the white area encloses orbits dominated by two-body relaxation. The shaded area encloses observable QPEs, that are determined by two parameters in our work: the maximum semi major axis $a_\mathrm{QPE}^\mathrm{max}$ (horizontal, dashed) and the maximum eccentricity $e_\mathrm{QPE}^\mathrm{max}$ (vertical, dashed). The black curves show the trajectory of sBHs that form EMRIs from Point A' and A" and become observable QPEs. In this case, $M_\bullet = 3\times10^6M_\odot$, $M_\mathrm{CO} = 40M_\odot$.
• Figure 3: Black Hole Mass Function $\Phi$ used in this work 2024Izquierdo-Villalba.
• Figure 4: Differential rate of TDEs, shown through the related function $\mathrm{d}\dot N / \mathrm{d log} \epsilon = \epsilon \mathcal{F}_\mathrm{TDE}$ as a function of the absolute value of the orbital energy $\epsilon=-E$ for a star with $M_\mathrm{CO} = 1M_\odot$, $M_\bullet = 3\times 10^6M_\odot$, at $t=8\times 10^6 yr$. The peak of $\mathrm{d}\dot N / \mathrm{d log} \epsilon$ is located at $r_c(E) =$0.884 pc
• Figure 5: Differential rate of EMRIs $\mathcal{F}_\mathrm{EMRI}$ and direct plunges $\mathcal{F}_\mathrm{sBH}$ as functions of energy $\epsilon$ and eccentricity of the loss cone orbit $e$. We consider sBHs of $M_\mathrm{CO} = 40 M_\odot$ in a system with $M_\bullet = 3\times 10^6 M_\odot$ that has relaxed for $1$ Gyr. Our estimate of the EMRI rate is given by the light blue colour-filled area. The eccentricity at the peak of the differential rate is $1-e=3.35\times10^{-7}$.