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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.

With hydrodynamical simulations we examine the evolution of a highly misaligned circumbinary disc around a black hole binary including the effects of general relativity. We show that a disc mass of just a few per cent of the binary mass can significantly increase the binary eccentricity through von-Zeipel–Kozai–Lidov (ZKL) like oscillations provided that the disc lifetime is longer than the ZKL oscillation time-scale. The disc begins as a relatively narrow ring of material far from the binary and spreads radially. When the binary becomes highly eccentric, disc breaking forms an inner disc ring that quickly aligns to polar. The polar ring drives fast retrograde apsidal precession of the binary that weakens the ZKL effect. This allows the binary eccentricity to remain at a high level and may significantly shorten the black hole merger time. The mechanism requires the initial disc inclination relative to the binary to be closer to retrograde than to prograde.

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.

While giant planet occurrence rates increase with stellar mass, occurrence rates of close-in super-Earths decrease. This is in contradiction to the expectation that the total mass of the planets in a system scale with the protoplanetary disc mass and hence the stellar mass. Since the snow line plays an important role in the planet formation process, we examine differences in the temperature structure of protoplanetary gas discs around stars of different mass. Protoplanetary discs likely contain a dead zone at the mid-plane that is sufficiently cold and dense for the magneto-rotational instability to be suppressed. As material builds up, the outer parts of the dead zone may be heated by self-gravity. The temperature in the disc can be below the snow line temperature far from the star and in the inner parts of a dead zone. The inner icy region has a larger radial extent around smaller mass stars. The increased mass of solid icy material may allow for the in situ formation of larger and more numerous planets close to a low-mass star. Super-Earths that form in the inner icy region may have a composition that includes a significant fraction of volatiles.