On the virialization of disk winds: Implications for the black hole mass estimates in active galactic nuclei

Estimating the mass of a supermassive black hole in an active galactic nucleus usually relies on the assumption that the broad line region (BLR) is virialized. However, this assumption seems to be invalid in BLR models that consist of an accretion disk and its wind. The disk is likely Keplerian and therefore virialized. However, beyond a certain point, the wind material must be dominated by an outward force that is stronger than gravity. Here, we analyze hydrodynamic simulations of four different disk winds: an isothermal wind, a thermal wind from an X-ray-heated disk, and two line-driven winds, one with and the other without X-ray heating and cooling. For each model, we determine whether gravity governs the flow properties by computing and analyzing the volume-integrated quantities that appear in the virial theorem: internal, kinetic, and gravitational energies. We find that in the first two models, the winds are non-virialized, whereas the two line-driven disk winds are virialized up to a relatively large distance. The line-driven winds are virialized because they accelerate slowly so that the rotational velocity is dominant and the wind base is very dense. For the two virialized winds, the so-called projected virial factor scales with inclination angle as 1/sin ² i. Finally, we demonstrate that an outflow from a Keplerian disk becomes unvirialized more slowly when it conserves the gas specific angular momentum, as in the models considered here, than when it conserves the angular velocity, as in the so-called magneto-centrifugal winds.