As part of the Panchromatic Hubble Andromeda Treasury (PHAT) multi-cycle program, we observed a 12 times 6.5 area of the bulge of M31 with the WFC3/UVIS filters F275W and F336W. From these data we have assembled a sample of sim4000 UV-bright, old stars, vastly larger than previously available. We use updated Padova stellar evolutionary tracks to classify these hot stars into three classes: Post-AGB stars (P-AGB), Post-Early AGB (PE-AGB) stars and AGB-manque stars. P-AGB stars are the end result of the asymptotic giant branch (AGB) phase and are expected in a wide range of stellar populations, whereas PE-AGB and AGB-manque (together referred to as the hot post-horizontal branch; HP-HB) stars are the result of insufficient envelope masses to allow a full AGB phase, and are expected to be particularly prominent at high helium or {alpha} abundances when the mass loss on the RGB is high. Our data support previous claims that most UV-bright sources in the bulge are likely hot (extreme) horizontal branch stars (EHB) and their progeny. We construct the first radial profiles of these stellar populations, and show that they are highly centrally concentrated, even more so than the integrated UV or optical light. However, we find that this UV-bright population does not dominate the total UV luminosity at any radius, as we are detecting only the progeny of the EHB stars that are the likely source of the UVX. We calculate that only a few percent of MS stars in the central bulge can have gone through the HP-HB phase and that this percentage decreases strongly with distance from the center. We also find that the surface density of hot UV-bright stars has the same radial variation as that of low-mass X-ray binaries. We discuss age, metallicity, and abundance variations as possible explanations for the observed radial variation in the UV-bright population.
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