Color-luminosity relations for the resolved hot stellar populations in the centers of M31 and M32

We present Faint Object Camera (FOC) ultraviolet images of the central 14 " x 14 " of Messier 31 and Messier 32. The hot stellar populations detected in the composite UV spectra of these nearby galaxies are partially resolved into individual stars, and their individual colors and apparent magnitudes are measured. We detect 433 stars in M31 and 138 stars in M32, down to detection limits m(F275W) = 25.5 mag and m(F175W) = 24.5 mag. We investigate the luminosity functions of the sources, their spatial distribution, their color-magnitude diagrams, and their total integrated far-UV flux. Comparison to IUE and Hopkins Ultraviolet Telescope (HUT) spectrophotometry and WFPC2 stellar photometry indicates consistency at the 0.3 mag level, with possible systematic offsets in the FOC photometry at a level less than this. Further calibrations or observations with the Space Telescope Imaging Spectrograph (STIS) will be necessary to resolve the discrepancies. Our interpretation rests on the assumption that the published FOC on-orbit calibration is correct. Although M32 has a weaker UV upturn than M31, the luminosity functions and color-magnitude diagrams (CMDs) of M31 and M32 are surprisingly similar and are inconsistent with a majority contribution from any of the following: post-asymptotic giant branch (PAGB) stars more massive than 0.56 M. (with or without associated planetary nebulae), main sequence stars, and blue stragglers. Both the luminosity functions and color-magnitude diagrams are consistent with a dominant population of stars that have evolved from the extreme horizontal branch (EHB) along tracks with masses between 0.47 and 0.53 M.. These stars are well below the detection limits of our images while on the zero-age EHB but become detectable while in the more luminous (but shorter) AGB-manque and post-early asymptotic giant branch (PEAGB) phases. The FOC observations require that only very small fractions of the main-sequence populations (2% in M31 and 0.5% in M32) in these two galaxies evolve though the EHB and post-EHB phases, with the remainder evolving through bright PAGE evolution that is so rapid that few if any stars are expected in the small held of view covered by the FOG. A model with a flat EHB star mass distribution reproduces the HUT and IUE spectra of these two galaxies reasonably well, although there is some indication that an additional population of very hot (T-eff > 25,000 K) EHB stars may be needed to reproduce the HUT spectrum of M31 near the Lyman limit and to bring integrated far-UV fluxes of M31 and M32 into agreement with IUE. In addition to the post-EHB population detected in the FOG, we find a minority population (similar to 10%) of brighter stars that populate a region of the CMD that cannot be explained by canonical post-MB evolutionary tracks. The nature of these stars remains open to interpretation. The spatial distributions of the resolved UV-bright stars in both galaxies are more centrally concentrated than the underlying diffuse emission, implying that stellar populations of different ages and/or metallicities might be responsible for each component.