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 support, with associated frequencies $\omega_Q$ and $\omega_P$. Within this framework, we show that the ratio $\omega_P/\omega_Q$ 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: (i) The number of modes increases for thinner disks and more-equal-mass binaries. (ii) For 2D disks, the normalized ground-mode frequency, $\omega_0/(\omega_Q+\omega_P)$, decreases monotonically with the ratio $\omega_P/\omega_Q$. (iii) For thin disks, $\omega_P\ll\omega_Q$, 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, $\omega_P\gg\omega_Q$, increasing pressure support 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 $\omega_P/\omega_Q \simeq 0.1$, we find that disk precession is rather insensitive to the density profile and ground-mode frequencies for 3D disks are about half the value for 2D disks.