Self-gravitating warped discs around supermassive black holes

We consider warped equilibrium configurations for stellar and gaseous disks in the Keplerian force-field of a supermassive black hole, assuming that the self-gravity of the disk provides the only acting torques. Modeling the disk as a collection of concentric circular rings, and computing the torques in the non-linear regime, we show that stable, strongly warped precessing equilibria are possible. These solutions exist for a wide range of disk-to-black hole mass ratios $M_d\mathrm{/}{M}_{bh}$, can span large warp angles of up to $\pm \sim 120\mathrm{deg}$, have inner and outer boundaries, and extend over a radial range of a factor of typically two to four. These equilibrium configurations obey a scaling relation such that in good approximation $\text{phidot}\mathrm{/}\mathrm{\Omega }\propto M_d\mathrm{/}{M}_{bh}$ where $\text{phidot}$ is the (retrograde) precession frequency and $\mathrm{\Omega }$ is a characteristic orbital frequency in the disk. Stability was determined using linear perturbation theory and, in a few cases, confirmed by numerical integration of the equations of motion. Most of the precessing equilibria are found to be stable, but some are unstable. The main result of this study is that highly warped disks near black holes can persist for long times without any persistent forcing other than by their self-gravity. The possible relevance of this to galactic nuclei is briefly discussed.