  The transient, ultra-luminous X-ray source CXOM31 J004253.1+411422 in the
Andromeda galaxy is most likely a 10 solar mass black hole, with
super-Eddington luminosity at its peak. The XMM-Newton spectra taken during the
decline then trace luminosities of 0.86-0.27 L_Edd. These spectra are all
dominated by a hot disc component, which roughly follows a constant inner
radius track in luminosity and temperature as the source declines. At the
highest luminosity the disc structure should change due to advection of
radiation through the disc. This advected flux can be partly released at lower
radii thus modifying the spectral shape. To study the effect of advection at
luminosities close to Eddington we employ a fully relativistic slim disc model,
SLIMBH, that includes advective cooling and full radiative transfer through the
photosphere based on TLUSTY. The model also incorporates relativistic photon
ray-tracing from the proper location of the disc photosphere rather than the
mid-plane as the slim disc is no longer geometrically thin. We find that these
new models differ only slightly from the non-advective (standard) BHSPEC models
even at the highest luminosities considered here. While both discs can fit the
highest luminosity data, neither is a very good fit to the *lower*
luminosities. This could indicate a missing physical process that acts in low
luminosity discs and subsides as the disc luminosity approaches the Eddington
limit.

High Energy Astrophysical Phenomena