LAUSR

A warm corona at the surface of an accretion disc has been proposed as a potential location for producing the soft excess commonly observed in the X-ray spectra of active galactic nuclei (AGNs). In order to fit the observed data the gas must be at temperatures of $\sim 1$ keV and have an optical depth of $\tau_{\mathrm{T}}\approx 10$--$20$. We present one-dimensional calculations of the physical conditions and emitted spectra of a $\tau_{\mathrm{T}}=10$ or $20$ gas layer subject to illumination from an X-ray power-law (from above), a blackbody (from below) and a variable amount of internal heating. The models show that a warm corona with $kT \sim 1$ keV can develop, producing a strong Comptonized soft excess, but only if the internal heating flux is within a relatively narrow range. Similarly, if the gas density of the layer is too large then efficient cooling will stop a warm corona from forming. The radiation from the hard X-ray power-law is crucial in producing a warm corona, indicating that a warm and hot corona may co-exist in AGN accretion discs, and their combined effect leads to the observed soft excess. Intense heating of a warm corona leads to steep X-ray spectra with ionised Fe K$\alpha$ lines, similar to those seen in some narrow-line Seyfert 1 galaxies.