Sustaining a Warm Corona in Active Galactic Nuclei Accretion Discs
  Warm coronae, thick ($\tau_{\mathrm{T}}\approx 10$-$20$, where
$\tau_{\mathrm{T}}$ is the Thomson depth) Comptonizing regions with
temperatures of $\sim 1$ keV, are proposed to exist at the surfaces of
accretion discs in active galactic nuclei (AGNs). By combining with the
reflection spectrum, warm coronae may be responsible for producing the smooth
soft excess seen in AGN X-ray spectra. This paper studies how a warm corona
must adjust in order to sustain the soft excess through large changes in the
AGN flux. Spectra from one-dimensional constant density and hydrostatic warm
coronae models are calculated assuming the illuminating hard X-ray power-law,
gas density, Thomson depth and coronal heating strength vary in response to
changes in the accretion rate. We identify models that produce warm coronae
with temperatures between $0.3$ and $1.1$ keV, and measure the photon indices
and emitted fluxes in the $0.5$-$2$ keV and $2$-$10$ keV bands. Correlations
and anti-correlations between these quantities depend on the evolution and
structure of the warm corona. Tracing the path that an AGN follows through
these correlations will constrain how warm coronae are heated and connected to
the accretion disc. Variations in the density structure and coronal heating
strength of warm coronae will lead to a variety of soft excess strengths and
shapes in AGNs. A larger accretion rate will, on average, lead to a warm corona
that produces a stronger soft excess, consistent with observations of local
Seyfert galaxies.
