Insights from Analytical Theory of Eccentric Circumbinary Disks

Eccentric cavities in circumbinary disks precess on timescales much longer than the binary orbital period. These long-lived steady states can be understood as trapped modes in an effective potential primarily determined by the binary quadrupole and the inner-disk pressure, with associated frequencies ω Q and ω P . Within this framework, we show that the ratio ω P / ω Q is the main parameter determining the mode spectrum, and obtain a thorough understanding of it by systematically solving this problem with various degrees of sophistication. We first find analytical solutions for truncated power-law disks and use this insight in disks with smooth central cavities. Our main findings are as follows: (i) The number of modes increases for thinner disks and more-equal-mass binaries. (ii) For 2D disks, the normalized ground-mode frequency, ω 0 /( ω Q + ω P ), decreases monotonically with the ratio ω P / ω Q . (iii) For thin disks, ω P ≪ ω Q , the ground-mode frequency coincides with the maximum of the effective potential, which tracks the gravitational quadrupole frequency inside the inner-disk cavity, and is thus rather sensitive to the density profile of the cavity, where these modes are localized. (iv) For thick disks, ω P ≫ ω Q , increasing the pressure anchors the peak of the effective potential at the inner cavity radius as the ground-mode extends farther out and its frequency decreases. (v) In agreement with numerical simulations, with ω P / ω Q ≃ 0.1, we find that disk precession is rather insensitive to the density profile, and ground-mode frequencies for 3D disks are about half the value of those for 2D disks.