Ionised Accretion Discs in Active Galactic Nuclei: The Effects of a
  Lamppost with a Variable Height
  The X-ray emitting corona irradiates and ionises the surface of the inner
accretion disc in Active Galactic Nuclei (AGNs). The ionisation parameter of
the inner disc at a radius $r$ from the black hole, $\xi(r)$, can be used to
infer information about the location of the corona. Here, a new formula is
derived that predicts $\xi(r,h)$ for a disc irradiated by a X-ray source at a
height $h$ above the black hole symmetry axis (i.e., a lamppost geometry). The
equation is independent of the black hole mass and the X-ray spectrum, and
accounts for the effects of gravitational light-bending on the ionisation state
and a variable coronal dissipation factor. We predict a strong ionisation
gradient across the surface of the inner disc that depends on the black hole
spin and lamppost height. For a fixed $h$, the ionisation parameter is also
expected to increase as $\lambda^3$, where $\lambda$ is the observed bolometric
Eddington ratio of the AGN. Comparing this formula to the observed
$\xi$-$\lambda$ relationship for Mrk 335 yields $h \propto \lambda^{0.5-0.6}$,
supporting the view of a dynamic X-ray corona in AGNs.
