Dynamical Measurement of Supermassive Black Hole Masses: QPE Timing Method

TL;DR

The paper introduces the QPE timing method to measure SMBH masses and spins by dynamically tracing a stellar-mass object as it collides with an accretion disk in Kerr spacetime. It builds a comprehensive Bayesian framework that jointly models forced EMRI trajectories and disk motion, enabling inference of $M_\bullet$ and $a$ from QPE timing data, with a focus on CATCH-era monitoring. Through application to GSN 069 and extensive mock-data experiments, it demonstrates robust mass determination and shows how observation strategy, disk precession, and Lense–Thirring precession affect parameter recoverability, highlighting the importance of long, uninterrupted monitoring. The work underscores the potential of QPE timing to extend precise SMBH parameter measurements into the low-mass regime ($M_\bullet \lesssim 10^7 M_\odot$) and informs practical planning for future X-ray facilities.

Abstract

Quasi-periodic eruptions (QPEs) are intense repeating soft X-ray bursts with recurrence times about a few hours to a few weeks from galactic nuclei. More and more analyses show that (at least a fraction of) QPEs are the result of collisions between a stellar mass object (SMO, a stellar mass black hole or a main sequence star) and an accretion disk around a supermassive black hole (SMBH) in galactic nuclei. Previous studies have shown the possibility of reconstructing the SMO trajectory from QPE timing data, consequently measuring the SMBH mass from tracing a single SMO. In this paper, we construct a comprehensive Bayesian framework for implementing the QPE timing method, explore the optimal QPE observation strategy for measuring SMBH masses, and forecast the measurement precision expected in the era of multi-target X-ray telescope, Chasing All Transients Constellation Hunters (CATCH). Simulations of CATCH observations of GSN 069 and eRO-QPE2 like QPEs confirm the possible applications of the QPE timing method in precise measurement of SMBH masses (and spins), especially in the lower mass end ($\lesssim 10^7 M_\odot$) where QPEs prevail and relevant dynamical timescales are reasonably short to be measured.

Figures (14)

• Figure 1: GSN 069 QPE light curves from 4 observations during Dec. 2018-May 2019 Miniutti2019Miniutti2023b, where the recurrence times show an alternating long-short pattern $T_{\rm long, short}$, with both $T_{\rm long}$ and $T_{\rm short}$ showing clear variations while the sum of two consecutive recurrence times $T_{\rm long} + T_{\rm short}$ remains approximately a constant Zhou2024a.
• Figure 2: Schematic picture of the EMRI+disk model, where the EMRI collides the disk at a different location each time due to apsidal precession, and the EMRI orbital plane precesses on an even longer Lense-Thirring precession timescale if the central SMBH is spinning.
• Figure 3: Top panel: light curve data of GSN 069 along with the best-fit EMRI trajectories, where the vertical bands are the inferred starting times $t_0^{(k)}\pm \sigma(t_0^{(k)})$ of the QPEs. Middle panel: distance to the disk midplane $z_{\rm disk}(t)$ of the best-fit orbits for the disk precession and alignment hypothesis ($\mathcal{H}_1$), where the orange horizontal lines denote the disk surface $z=H$ and the verticals bands are the inferred starting times $t_0^{(k)}\pm \tilde{\sigma}(t_0^{(k)})$, with $\tilde{\sigma}(t_0^{(k)})=\sqrt{(\sigma(t_0^{(k)}))^2+\sigma_{\rm sys}^2}$. Bottom panel: same to the midle panel but for the vanilla hypothesis ($\mathcal{H}_0$).
• Figure 4: Best-fit EMRI trajectories obtained with three different strategies, where $z_{\rm disk}(t)$ is the $z$-component. The vertical bands indicate the simulated data $t_0^{(k)}\pm\sigma(t_0^{(k)})$. The orange horizontal line marks the disk surface. Blue: strategy A. Green: strategy B. Red: strategy C.
• Figure 5: Probability distributions $f(T_{\rm aps})\Delta T_{\rm aps}$ of the apsidal precession period $T_{\rm aps}$ obtained from three different observation strategies, where $\Delta T_{\rm aps}$ is the bin size.