Inclination Evolution of Protoplanetary Disks Around Eccentric Binaries

It is usually thought that viscous torque works to align a circumbinary disk with the binary's orbital plane. However, recent numerical simulations suggest that the disk may evolve to a configuration perpendicular to the binary orbit ("polar alignment") if the binary is eccentric and the initial disk-binary inclination is sufficiently large. We carry out a theoretical study on the long-term evolution of inclined disks around eccentric binaries, calculating the disk warp profile and dissipative torque acting on the disk. For disks with aspect ratio $H/r$ larger than the viscosity parameter $\alpha$, bending wave propagation effectively makes the disk precess as a quasi-rigid body, while viscosity acts on the disk warp and twist to drive secular evolution of the disk-binary inclination. We derive a simple analytic criterion (in terms of the binary eccentricity and initial disk orientation) for the disk to evolve toward polar alignment with the eccentric binary. When the disk has a non-negligible angular momentum compared to the binary, the final "polar alignment" inclination angle is reduced from $90^\circ$. For typical protoplanetary disk parameters, the timescale of the inclination evolution is shorter than the disk lifetime, suggesting that highly-inclined disks and planets may exist orbiting eccentric binaries.