Star formation in self-gravitating disks in active galactic nuclei. I.
  Metallicity gradients in broad line regions
  It has been suggested that the high metallicity generally observed in active
galactic nuclei (AGNs) and quasars originates from ongoing star formation in
the self-gravitating part of accretion disks around the supermassive black
holes. We designate this region as the star forming (SF) disk, in which metals
are produced from supernova explosions (SNexp) while at the same time inflows
are driven by SNexp-excited turbulent viscosity to accrete onto the SMBHs. In
this paper, an equation of metallicity governed by SNexp and radial advection
is established to describe the metal distribution and evolution in the SF disk.
We find that the metal abundance is enriched at different rates at different
positions in the disk, and that a metallicity gradient is set up that evolves
for steady-state AGNs. Metallicity as an integrated physical parameter can be
used as a probe of the SF disk age during one episode of SMBH activity. In the
SF disk, evaporation of molecular clouds heated by SNexp blast waves
unavoidably forms hot gas. This heating is eventually balanced by the cooling
of the hot gas, but we show that the hot gas will escape from the SF disk
before being cooled, and diffuse into the BLRs forming with a typical rate of
$\sim 1\sunmyr$. The diffusion of hot gas from a SF disk depends on ongoing
star formation, leading to the metallicity gradients in BLR observed in AGNs.
We discuss this and other observable consequences of this scenario.
