Stability of Gas Clouds in Galactic Nuclei: An Extended Virial Theorem
  Cold gas entering the central $1$ to $10^2$ pc of a galaxy fragments and
condenses into clouds. The stability of the clouds determines whether they will
be turned into stars or can be delivered to the central supermassive black hole
(SMBH) to turn on an active galactic nucleus (AGN). The conventional criteria
to assess the stability of these clouds, such as the Jeans criterion and Roche
(or tidal) limit, are insufficient here, because they assume the dominance of
self-gravity in binding a cloud, and neglect external agents, such as pressure
and tidal forces, which are common in galactic nuclei. We formulate a new
scheme for judging this stability. We first revisit the conventional Virial
theorem, taking into account an external pressure, to identify the correct
range of masses that lead to stable clouds. We then extend the theorem to
include an external tidal field, crucial for the stability in the region of
interest -- in dense star clusters, around SMBHs. We apply our extended Virial
theorem to find the correct solutions to practical problems that until now were
controversial, namely, the stability of the gas clumps in AGN tori, the
circum-nuclear disk in the Galactic Center, and the central molecular zone of
the Milky Way. The masses we derive for these structures are orders of
magnitude smaller than the commonly-used Virial masses (equivalent to the Jeans
mass). Moreover, we prove that these clumps are stable, contrary to what one
would naively deduce from the Roche (tidal) limit.
