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We study the effects of general relativity (GR) on the evolution and alignment of circumbinary disks around binaries on all scales. We implement relativistic apsidal precession of the binary into the hydrodynamics codephantom. We find that the effects of GR can suppress the stable polar alignment of a circumbinary disk, depending on how the relativistic binary apsidal precession timescale compares to the disk nodal precession timescale. Studies of circumbinary disk evolution typically ignore the effects of GR, which is an appropriate simplification for low-mass or widely separated binary systems. In this case, polar alignment occurs, provided that the disks initial misalignment is sufficiently large. However, systems with a very short relativistic precession timescale cannot polar align and instead move toward coplanar alignment. In the intermediate regime where the timescales are similar, the outcome depends upon the properties of the disk. Polar alignment is more likely in the wavelike disk regime (where the disk viscosity parameter is less than the aspect ratio,α<H/r), since the disk is in good radial communication. In the viscous disk regime, disk breaking is more likely. Multiple rings can destructively interact with one another, resulting in short disk lifetimes and the disk moving toward coplanar alignment. Around main-sequence star or stellar mass black hole binaries, polar alignment may be suppressed far from the binary, but in general, the inner parts of the disk can align to polar. Polar alignment may be completely suppressed for disks around supermassive black holes for close binary separations.

We examine the geometry of the post–asymptotic giant branch (AGB) star binary AC Her and its circumbinary disk. We show that the observations describe a binary orbit that is perpendicular to the disk with an angular momentum vector that is within 9° of the binary eccentricity vector, meaning that the disk is close to a stable polar alignment. The most likely explanation for the very large inner radius of the dust is a planet within the circumbinary disk. This is therefore both the first reported detection of a polar circumbinary disk around a post-AGB binary and the first evidence of a polar circumbinary planet. We consider the dynamical constraints on the circumbinary disk size and mass. The polar circumbinary disk feeds circumstellar disks with gas on orbits that are highly inclined with respect to the binary orbit plane. The resulting circumstellar disk inclination could be anywhere from coplanar to polar depending upon the competition between the mass accretion and binary torques.

We use numerical simulations of circumplanetary disks to determine the boundary between disks that are radially truncated by the tidal potential and those where gas escapes the Hill sphere. We consider a model problem, in which a coplanar circumplanetary disk is resupplied with gas at an injection radius smaller than the Hill radius. We evolve the disk using thePhantomsmoothed particle hydrodynamics code until a steady state is reached. We find that the most significant dependence of the truncation boundary is on the disk aspect ratioH/R. Circumplanetary disks are efficiently truncated forH/R≲ 0.2. ForH/R≃ 0.3, up to about half of the injected mass, depending on the injection radius, flows outward through the decretion disk and escapes. As expected from analytic arguments, the conditions (H/Rand Shakura–Sunyaevα) required for tidal truncation are independent of planet mass. A simulation with largerα= 0.1 shows stronger outflow than one withα= 0.01, but the dependence on transport efficiency is less important than variations ofH/R. Our results suggest two distinct classes of circumplanetary disks: tidally truncated thin disks with dust-poor outer regions, and thicker actively decreting disks with enhanced dust-to-gas ratios. Applying our results to the PDS 70 c system, we predict a largely truncated circumplanetary disk, but it is possible that enough mass escapes to support an outward flow of dust that could explain the observed disk size.

We report the analysis of circumbinary (CB) discs formed in a radiation hydrodynamical simulation of star cluster formation. We consider both pure binary stars and pairs within triple and quadruple systems. The protostellar systems are all young (ages < 105 yrs). We find that the systems that host a CB disc have a median separation of ≈11 au, and the median characteristic radius of the discs is ≈64 au. We find that 89 per cent of pure binaries with semimajor axes a < 1 au have a CB disc, and the occurrence rate of CB discs is bimodal with log-separation in pure binaries with a second peak at a ≈ 50 au. Systems with a > 100 au almost never have a CB disc. The median size of a CB disc is between ≈5 and 6 a depending on the order of the system, with higher order systems having larger discs relative to binary separation. We find the underlying distributions of mutual inclination between CB discs and binary orbits from the simulation are in good agreement with those of observed CB discs.

We analyse the eccentric response of a low-mass coplanar circumbinary disc to secular tidal forcing by a Keplerian eccentric orbit central binary. The disc acquires a forced eccentricity whose magnitude depends on the properties of the binary and disc. The largest eccentricities occur when there is a global apsidal resonance in the disc. The driving frequency by the binary is its apsidal frequency, which is equal to zero. A global resonance occurs when the disc properties permit the existence of a zero apsidal frequency free eccentric mode. Resonances occur for different free eccentric modes, which differ in the number of radial nodes. For a disc not at resonance, the eccentricity distribution has somewhat similar form to the eccentricity distributions in discs at resonance that have the closest matching disc aspect ratios. For higher disc aspect ratios, the forced eccentricity distribution in a 2D disc is similar to that of the fundamental free mode. The forced eccentricity distribution in a 3D disc is similar to that of higher order free modes, not the fundamental mode, unless the disc is very cool. For parameters close to resonance, large phase shifts occur between the disc and binary eccentricities that are locked in phase. Forced eccentricity may play an important role in the evolution of circumbinary discs and their central binaries.

We describe the first grid-based simulations of the polar alignment of a circumbinary disc. We simulate the evolution of an inclined disc around an eccentric binary using the grid-based code athena++ . The use of a grid-based numerical code allows us to explore lower disc viscosities than have been examined in previous studies. We find that the disc aligns to a polar orientation when the α viscosity is high, while discs with lower viscosity nodally precess with little alignment over 1000 binary orbital periods. The time-scales for polar alignment and disc precession are compared as a function of disc viscosity, and are found to be in agreement with previous studies. At very low disc viscosities (e.g. α = 10−5), anticyclonic vortices are observed along the inner edge of the disc. These vortices can persist for thousands of binary orbits, creating azimuthally localized overdensities and multiple pairs of spiral arms. The vortex is formed at ∼3–4 times the binary semimajor axis, close to the inner edge of the disc, and orbits at roughly the local Keplerian speed. The presence of a vortex in the disc may play an important role in the evolution of circumbinary systems, such as driving episodic accretion and accelerating the formation of polar circumbinary planets.

Mutually misaligned circumbinary planets may form in a warped or broken gas disk or from later planet–planet interactions. With numerical simulations and analytic estimates we explore the dynamics of two circumbinary planets with a large mutual inclination. A coplanar inner planet causes prograde apsidal precession of the binary and the stationary inclination for the outer planet is higher for larger outer planet orbital radius. In this case a coplanar outer planet always remains coplanar. On the other hand, a polar inner planet causes retrograde apsidal precession of the binary orbit and the stationary inclination is smaller for larger outer planet orbital radius. For a range of outer planet semimajor axes, an initially coplanar orbit is librating meaning that the outer planet undergoes large tilt oscillations. Circumbinary planets that are highly inclined to the binary are difficult to detect—it is unlikely for a planet to have an inclination below the transit detection limit in the presence of a polar inner planet. These results suggest that there could be a population of circumbinary planets that are undergoing large tilt oscillations.

A test particle orbit around an eccentric binary has two stationary states in which there is no nodal precession: coplanar and polar. Nodal precession of a misaligned test particle orbit centres on one of these stationary states. A low-mass circumbinary disc undergoes the same precession and moves towards one of these states through dissipation within the disc. For a massive particle orbit, the stationary polar alignment occurs at an inclination less than 90°, which is the prograde-polar stationary inclination. A sufficiently high angular momentum particle has an additional higher inclination stationary state, the retrograde-polar stationary inclination. Misaligned particle orbits close to the retrograde-polar stationary inclination are not nested like the orbits close to the other stationary points. We investigate the evolution of a gas disc that begins close to the retrograde-polar stationary inclination. With hydrodynamical disc simulations, we find that the disc moves through the unnested crescent shape precession orbits and eventually moves towards the prograde-polar stationary inclination, thus increasing the parameter space over which circumbinary discs move towards polar alignment. If protoplanetary discs form with an isotropic orientation relative to the binary orbit, then polar discs may be more common than coplanar discs around eccentric binaries, even for massive discs. This has implications for the alignment of circumbinary planets.

We present hydrodynamical simulations to model the accretion flow from a polar circumbinary disc on to a high eccentricity (e = 0.78) binary star system with near unity mass ratio (q = 0.83), as a model for binary HD 98800 BaBb. We compare the polar circumbinary disc accretion flow with the previously studied coplanar case. In the coplanar case, the circumbinary disc becomes eccentric and the accretion alternates from being dominant on to one binary member to the other. For the polar disc case involving a highly eccentric binary, we find that the circumbinary disc retains its initially low eccentricity and that the primary star accretion rate is always about the same as the secondary star accretion rate. Recent observations of the binary HD 98800 BaBb, which has a polar circumbinary disc, have been used to determine the value of the $\rm H\,\alpha$ flux from the brighter component. From this value, we infer that the accretion rate is much lower than for typical T Tauri stars. The eccentric orbit of the outer companion HD 98800 A increases the accretion rate on to HD 98800 B by ∼20 per cent after each periastron passage. Our hydrodynamical simulations are unable to explain such a low accretion rate unless the disc viscosity parameter is very small, α < 10−5. Additional observations of this system would be useful to check on this low accretion rate.