A Bias-corrected Luminosity Function for Red Supergiant Supernova Progenitor Stars

The apparent tension between the luminosity functions of red supergiant (RSG) stars and of RSG progenitors of Type II supernovae (SNe) is often referred to as the RSG problem and it motivated some to suggest that many RSGs end their life without an SN explosion. However, the luminosity functions of RSG SN progenitors presented so far were biased to high luminosities, because the sensitivity of the search was not considered. Here, we use limiting magnitudes to calculate a bias-corrected RSG progenitor luminosity function. We find that only (36 +/- 11)% of all RSG progenitors are brighter than a bolometric magnitude of -7 mag, a significantly smaller fraction than (56 +/- 5)% quoted by Davies & Beasor. The larger uncertainty is due to the relatively small progenitor sample, while uncertainties on measured quantities such as magnitudes, bolometric corrections, extinction, or SN distances, only have a minor impact, as long as they fluctuate randomly for different objects in the sample. The bias-corrected luminosity functions of RSG SN progenitors and Type M supergiants in the Large Magellanic Cloud are consistent with each other, as also found by Davies & Beasor for the uncorrected luminosity function. The RSG progenitor luminosity function, hence, does not imply the existence of failed SNe. The presented statistical method is not limited to progenitor searches, but applies to any situation in which a measurement is done for a sample of detected objects, but the probed quantity or property can only be determined for part of the sample.