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We have performed numerical calculations of a binary interacting with a gas disk, using eleven 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, $0.2 (a^2 \Omega_B /\nu)$ binary orbits, where $\nu$ is the kinematic viscosity. We did not investigate dependence on binary mass ratio, eccentricity, disk temperature, or disk viscosity; therefore, these benchmarks may act as guides towards expanding converged solutions to the wider parameter space but might need to be updated in a future study that investigates dependence on system parameters. We find the most major discrepancies between codes resulted from the dimensionality of the setup (3D vs 2D disks). Beyond this, 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., Comment: ApJ Accepted

Binary black holes embedded within the accretion disks that fuel active galactic nuclei (AGN) are promising progenitors for the source of gravitational wave events detected by LIGO/VIRGO. Several recent studies have shown that when these binaries form they should be highly eccentric and retrograde. However, many uncertainties remain concerning the orbital evolution of these binaries as they either inspiral towards merger or disassociate. Previous hydrodynamical simulations exploring their orbital evolution have been predominantly two-dimensional, or have been restricted to binaries on nearly circular orbits. We present the first high-resolution, three-dimensional local shearing-box simulations of both prograde and retrograde eccentric binary black holes embedded in AGN disks. We find that retrograde binaries shrink several times faster than their prograde counterparts and exhibit significant orbital eccentricity growth, the rate of which monotonically increases with binary eccentricity. Our results suggest that retrograde binaries may experience runaway orbital eccentricity growth, which may bring them close enough together at pericenter for gravitational wave emission to drive them to coalescence. Although their eccentricity is damped, prograde binaries shrink much faster than their orbital eccentricity decays, suggesting they should remain modestly eccentric as they contract towards merger. Finally, binary precession driven by the AGN disk may dominate over precession induced by the supermassive black hole depending on the binary accretion rate and its location in the AGN disk, which can subdue the evection resonance and von Ziepel-Lidov-Kozai cycles., Comment: Corrected figures 11, 13

The accretion disks that fuel active galactic nuclei (AGN) may house numerous stars and compact objects, formed in situ or captured from nearby star clusters. Embedded neutron stars and black holes may form binaries and eventually merge, emitting gravitational waves detectable by LIGO/VIRGO. AGN disks are a particularly promising environment for the production of high-mass gravitational wave events involving black holes in the pair instability mass gap, and may facilitate electromagnetic counterparts to black hole binary mergers. However, many orders of magnitude separate the typical length scales of binary formation and those on which gravitational waves can drive binary inspirals, making binary mergers inside the disk uncertain. Previous hydrodynamical simulations of binaries have either been restricted to two dimensions entirely, or focused on binaries aligned with the midplane of the disk. Herein we present the first three-dimensional, high-resolution, local shearing-box hydrodynamical simulations of disk-embedded binaries over a range of orbital inclinations. We find that retrograde binaries can shrink up to four times as quickly as prograde binaries, and that all binaries not perfectly aligned (or anti-aligned) with the AGN disk are driven into alignment. An important consequence of this is that initially retrograde binaries will traverse the inclinations where von Zeipel-Lidov-Kozai oscillations can drive binary eccentricities to large values, potentially facilitating mergers. We also find that interactions with the AGN disk may excite eccentricities in retrograde binaries and cause the orbits of embedded binaries to precess., Comment: 22 pages, 15 figures. Comments welcome

We study the close encounters between two single black holes (BHs) embedded in an AGN disk using a series of global 2D hydrodynamics simulations. We find that when the disk density is sufficiently high, bound BH binaries can be formed by the collision of their circum-single disks. Our analysis demonstrates that, after a BH pair passes the pericenter of their relative trajectory, a gas post-collision drag may slow down the BHs, possibly forcing the two BHs to stay tightly bound. A binary formed by a close encounter can have a compact semi-major axis, large eccentricity, and retrograde orbital angular momentum. We provide a fitting formula that can accurately predict whether a close encounter can form a binary based on the gas mass and the incoming energy of the encounter. This fitting formula can be easily implemented in other long-term simulations that study the dynamical evolution of BHs in AGN disks., Comment: 10 pages, 5 figures

LIGO/Virgo has detected several binary black hole (BBH) merger events that may have originated in the accretion disks of Active Galactic Nuclei (AGN). These events require individual black hole masses that fall within the pair instability supernova mass gap, and therefore these black holes may have been grown from hierarchical mergers. AGN disks are a prime environment for hierarchical mergers, and thus a potential location for the progenitors of BBH gravitational wave events. Understanding how a BBH embedded in an AGN disk interacts with the surrounding environment is thus crucial for determining if this interaction can lead to its merger. However, there are few high fidelity simulations of this process, and almost all are two-dimensional. We present the results from 3D, high-resolution, local shearing-box simulations of an embedded BBH interacting with an AGN disk. In these first simulations of their kind, we focus on determining the mass accretion rate and the orbital evolution rate at different BBH separations. We find that circular, equal-mass BBHs with separations greater than 10% of their Hill radius contract while accreting at a super-Eddington rate. At smaller separations, however, our 3D simulations find that BBHs expand their orbits. This result suggests that it may be difficult for an AGN disk to push a BBH to merger, but we discuss several mechanisms, including MHD turbulence and radiative and mechanical feedback, that could alleviate this difficulty., Comment: 25 pages, 16 figures, accepted to ApJ

Binary black hole (BBH) mergers, particularly those with component masses in the pair-instability gap, may be produced by hierarchical mergers in the disks surrounding Active Galactic Nuclei (AGN). While the interaction of an embedded BBH with an AGN disk is typically assumed to facilitate a merger, recent high-resolution hydrodynamical simulations challenge this assumption. However, these simulations often have simplified treatments for the gas thermodynamics. In this work, we model the possible consequence of various feedback from an embedded BBH with a simple model that maintains an enhanced temperature profile around each binary component. We show that when the minidisks around each BH become hotter than the background by a factor of three, the BBH orbital evolution switches from expansion to contraction. By analyzing the gravitational torque profile, we find that this change in direction is driven by a weakening of the minidisk spirals and their positive torque on the binary. Our results highlight the important role of thermodynamics around BBHs and its effect on their orbital evolution, suggesting that AGN disks could be efficient factories for BBH mergers., Comment: 11 pages, 5 figures, ApJL in press

Recent simulations show that giant planets of about one Jupiter mass migrate inward at a rate that differs from the Type II prediction. Here we show that at higher masses, planets migrate outward. Our result differs from previous ones because of our longer simulation times, lower viscosity, and our boundary conditions that allow the disk to reach viscous steady state. We show that, for planets on circular orbits, the transition from inward to outward migration coincides with the known transition from circular to eccentric disks that occurs for planets more massive than a few Jupiters. In an eccentric disk, the torque on the outer disk weakens due to two effects: the planet launches weaker waves, and those waves travel further before damping. As a result, the torque on the inner disk dominates, and the planet pushes itself outward. Our results suggest that the many super-Jupiters observed by direct-imaging at large distances from the star may have gotten there by outward migration., Comment: 11 pages, 5 figures. ApJL accepted

We find that, under certain conditions, protoplanetary disks may spontaneously generate multiple, concentric gas rings without an embedded planet through an eccentric cooling instability. Using both linear theory and non-linear hydrodynamics simulations, we show that a variety of background states may trap a slowly precessing, one-armed spiral mode that becomes unstable when a gravitationally-stable disk rapidly cools. The angular momentum required to excite this spiral comes at the expense of non-uniform mass transport that generically results in multiple rings. For example, one long-term hydrodynamics simulation exhibits four long-lived, axisymmetric gas rings. We verify the instability evolution and ring formation mechanism from first principles with our linear theory, which shows remarkable agreement with the simulation results. Dust trapped in these rings may produce observable features consistent with observed disks. Additionally, direct detection of the eccentric gas motions may be possible when the instability saturates, and any residual eccentricity leftover in the rings at later times may also provide direct observational evidence of this mechanism., Comment: 18 pages, 12 figures, accepted to ApJ

We perform a series of high-resolution 2D hydrodynamical simulations of equal-mass binary black holes (BBHs) embedded in active galactic nucleus (AGN) accretion disks to study whether these binaries can be driven to merger by the surrounding gas. We find that the gravitational softening adopted for the BBH has a profound impact on this result. When the softening is less than ten percent of the binary separation, we show that, in agreement with recent simulations of isolated equal-mass binaries, prograde BBHs expand in time rather than contract. Eventually, however, the binary separation becomes large enough that the tidal force of the central AGN disrupts them. Only when the softening is relatively large do we find that prograde BBHs harden. We determine through detailed analysis of the binary torque, that this dichotomy is due to a loss of spiral structure in the circum-single disks orbiting each BH when the softening is a significant fraction of the binary separation. Properly resolving these spirals -- both with high resolution and small softening -- results in a significant source of binary angular momentum. Only for retrograde BBHs do we find consistent hardening, regardless of softening, as these BBHs lack the important spiral structure in their circum-single disks. This suggests that the gas-driven inspiral of retrograde binaries can produce a population of compact BBHs in the gravitational-wave-emitting regime in AGN disks, which may contribute a large fraction to the observed BBH mergers., Comment: accepted by ApJ, a new Table 2 is added

In simulations of viscously evolving accretion disks, the inner boundary condition is particularly important. If treated incorrectly, it induces incorrect behavior very quickly, because the viscous time is shortest near the inner boundary. Recent work has determined the correct inner boundary in Eulerian simulations. But in quasi-Lagrangian simulations (e.g., SPH, moving mesh, and mesh-less), where the inner boundary is modeled by removing mass within a finite zone, the inner density profile typically becomes anomalously depleted. Here we show how the boundary condition should be applied in such codes, via a simple modification of the usual approach: when one removes mass, one must speed up the remaining material so that the disk's angular momentum is unchanged. We show with both 1D and 2D moving-mesh (AREPO) simulations that this scheme works as desired in viscously evolving disks. It produces no spurious density depletions and is independent of the mass removal rate, provided that the disk is adequately resolved and that the mass removal rate is not so extreme as to trigger instabilities. This "torque-free" mass removal technique permits the use of quasi-Lagrangian codes to simulate viscously evolving disks, while including a variety of additional effects. As an example, we apply our scheme to a 2D simulation of an accretion disk perturbed by a very massive planet, in which the disk is evolved to viscous steady state., Comment: 10 pages, 5 figures, Accepted to ApJL

We investigate how planets interact with viscous accretion disks, in the limit that the disk is sufficiently low mass that the planet migrates more slowly than the disk material. In that case, the disk's surface density profile is determined by the disk being in viscous steady state (VSS), while overflowing the planet's orbit. We compute the VSS profiles with 2D hydrodynamical simulations, and show that disk material piles up behind the planet, with the planet effectively acting as a leaky dam. Previous 2D hydrodynamical simulations missed the pileup effect because of incorrect boundary conditions, while previous 1D models greatly overpredicted the pileup due to the neglect of non-local deposition. Our simulations quantify the magnitude of the pileup for a variety of planet masses and disk viscosities. We also calculate theoretically the magnitude of the pileup for moderately deep gaps, showing good agreement with simulations. For very deep gaps, current theory is inadequate, and we show why and what must be understood better. The pileup is important for two reasons. First, it is observable in directly imaged protoplanetary disks, and hence can be used to diagnose the mass of a planet that causes it or the viscosity within the disk. And second, it determines the planet's migration rate. Our simulations determine a new Type-II migration rate (valid for low mass disks), and show how it connects continuously with the well-verified Type-I rate., Comment: 27 pages, 17 figures, accepted to ApJ

Accretion disks can be eccentric: they support $m=1$ modes that are global and slowly precessing. But whether the modes remain trapped in the disk---and hence are long-lived---depends on conditions at the outer edge of the disk. Here we show that in disks with realistic boundaries, in which the surface density drops rapidly beyond a given radius, eccentric modes are trapped and hence long-lived. We focus on pressure-only disks around a central mass, and show how this result can be understood with the help of a simple second-order WKB theory. We show that the longest lived mode is the zero-node mode in which all of the disk's elliptical streamlines are aligned, and that this mode decays coherently on the viscous timescale of the disk. Hence such a mode, once excited, will live for the lifetime of the disk. It may be responsible for asymmetries seen in recent images of protoplanetary disks., Comment: Submitted to ApJ. Comments are welcomed

Accretion disks around stars, or other central massive bodies, can support long-lived, slowly precessing $m=1$ disturbances in which the fluid motion is nearly Keplerian with non-zero eccentricity. We study such `slow modes' in disks that are subject to both pressure and self-gravity forces. We derive a second-order WKB dispersion relation that describes the dynamics quite accurately, and show that the apparently complicated nature of the various modes can be understood in a simple way with the help of a graphical method. We also solve the linearized fluid equations numerically, and show that the results agree with the theory. We find that when self-gravity is weak ($Q\gtrsim 1/h$, where $Q$ is Toomre's parameter, and $h$ is the disk aspect ratio) the modes are pressure dominated. But when self-gravity is strong ($1

We run a suite of dissipative N-body simulations to determine which regions of phase space for smooth disk migration are consistent with the GJ876 system, an M-dwarf hosting three planets orbiting in a chaotic 4:2:1 Laplace resonance. We adopt adaptive mesh refinement (AMR) methods which are commonly used in hydrodynamical simulations to efficiently explore the parameter space defined by the semi-major axis and eccentricity damping timescales. We find that there is a large region of phase space which produces systems in the chaotic Laplace resonance and a smaller region consistent with the observed eccentricities and libration amplitudes for the resonant angles. Under the assumptions of Type I migration for the outer planet, we translate these damping timescales into constraints on the protoplanetary disk surface density and thickness. When we strongly (weakly) damp the eccentricities of the inner two Laplace planets, these timescales correspond to disk surface densities around ten thousand (a few hundred) grams per square centimeter and disk aspect ratios between 1-10%. Additionally, smooth migration produces systems with a range of chaotic timescales, from decades and centuries to upwards of thousands of years. In agreement with previous studies, the less chaotic regions of phase space coincide with the system being in a low energy double apsidal corotation resonance. Our detailed modeling of multi-planetary systems coupled with our AMR exploration method enhances our ability to map out the parameter space of planet formation models, and is well suited to study other resonant chain systems such as Trappist-1, Kepler-60, and others., Comment: 12 pages, 10 figures. Accepted to ApJ

We study thermal convection in a rotating fluid in order to better understand the properties of convection zones in rotating stars and planets. We first derive mixing-length theory for rapidly-rotating convection, arriving at the results of Stevenson (1979) via simple physical arguments. The theory predicts the properties of convection as a function of the imposed heat flux and rotation rate, independent of microscopic diffusivities. In particular, it predicts the mean temperature gradient; the rms velocity and temperature fluctuations; and the size of the eddies that dominate heat transport. We test all of these predictions with high resolution three-dimensional hydrodynamical simulations of Boussinesq convection in a Cartesian box. The results agree remarkably well with the theory across more than two orders of magnitude in rotation rate. For example, the temperature gradient is predicted to scale as the rotation rate to the 4/5th power at fixed flux, and the simulations yield $0.75\pm 0.06$. We conclude that the mixing length theory is a solid foundation for understanding the properties of convection zones in rotating stars and planets., Comment: 9 pages, 6 figures, to be published in ApJ