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We have performed numerical calculations of a binary interacting with a gas disk, using 11 different numerical methods and a standard binary−disk setup. The goal of this study is to determine whether all codes agree on a numerically converged solution and to determine the necessary resolution for convergence and the number of binary orbits that must be computed to reach an agreed-upon relaxed state of the binary−disk system. We find that all codes can agree on a converged solution (depending on the diagnostic being measured). The zone spacing required for most codes to reach a converged measurement of the torques applied to the binary by the disk is roughly 1% of the binary separation in the vicinity of the binary components. For our disk model to reach a relaxed state, codes must be run for at least 200 binary orbits, corresponding to about a viscous time for our parameters, 0.2( a ^2 Ω _B / ν ) binary orbits, where ν is the kinematic viscosity. The largest discrepancies between codes resulted from the dimensionality of the setup (3D vs. 2D disks). We find good agreement in the total torque on the binary between codes, although the partition of this torque between the gravitational torque, orbital accretion torque, and spin accretion torque depends sensitively on the sink prescriptions employed. In agreement with previous studies, we find a modest difference in torques and accretion variability between 2D and 3D disk models. We find cavity precession rates to be appreciably faster in 3D than in 2D.

Quasi-periodic eruptions (QPEs) are luminous X-ray outbursts recurring on hour timescales, observed from the nuclei of a growing handful of nearby low-mass galaxies. Their physical origin is still debated, and usually modeled as (a) accretion disk instabilities or (b) interaction of a supermassive black hole (SMBH) with a lower mass companion in an extreme mass-ratio inspiral (EMRI). EMRI models can be tested with several predictions related to the short- and long-term behavior of QPEs. In this study, we report on the ongoing 3.5 yr NICER and XMM-Newton monitoring campaign of eRO-QPE1, which is known to exhibit erratic QPEs that have been challenging for the simplest EMRI models to explain. We report (1) complex, non-monotonic evolution in the long-term trends of QPE energy output and inferred emitting area; (2) the disappearance of the QPEs (within NICER detectability) in 2023 October, and then the reappearance by 2024 January at a luminosity of ∼100× fainter (and temperature of ∼3× cooler) than the initial discovery; (3) radio non-detections with MeerKAT and Very Large Array observations partly contemporaneous with our NICER campaign (though not during outbursts); and (4) the presence of a possible ∼6 day modulation of the QPE timing residuals, which aligns with the expected nodal precession timescale of the underlying accretion disk. Our results tentatively support EMRI–disk collision models powering the QPEs, and we demonstrate that the timing modulation of QPEs may be used to jointly constrain the SMBH spin and disk density profile.

The manner in which warps in accretion disks evolve depends on the magnitude of the viscosity. For small viscosity (α < H/R), the warp evolves in a wave-like manner; for large viscosity, H/R < α ≪ 1, it evolves diffusively. Here, α is the viscosity parameter and H/R is the disk aspect ratio. Currently there is no simple set of equations that describes the evolution in both regimes. In this paper, we describe a possible solution to this problem and introduce a set of one-dimensional equations that describe the evolution of a warped disk that are applicable in both high- and low-viscosity regimes for arbitrary tilts, but small warps. [ABSTRACT FROM AUTHOR]