The Uncertain Masses of Progenitors of Core Collapse Supernovae and
  Direct Collapse Black Holes
  We show that it is not possible to determine the final mass $M_{\rm fin}$ of
a red supergiant (RSG) at the pre-supernova (SN) stage from its luminosity $L$
and effective temperature $T_{\rm eff}$ alone. Using a grid of stellar models,
we demonstrate that for a given value of $L$ and $T_{\rm eff}$, a RSG can have
a range of $M_{\rm fin}$ as wide as 3 to $45~\mathrm{M}_{\odot}$. While the
probability distribution within these limits is not flat, any individual
determination of $M_{\rm fin}$ for a RSG will be degenerate. This makes it
difficult to determine its evolutionary history and to map $M_{\rm fin}$ to an
initial mass. Single stars produce a narrower range that is difficult to
accurately determine without making strong assumptions about mass loss,
convection, and rotation. Binaries would produce a wider range of RSG $M_{\rm
fin}$. However, the final Helium core mass M$_{\rm He-core}$ is well determined
by the final luminosity and we find $\log (\mathrm{M}_{\rm He-core}/M_{\odot})
= 0.659 \log (L/\mathrm{L}_{\odot}) -2.630$ Using this relationship, we derive
M$_{\rm He-core}$ for directly imaged SN progenitors and one failed SN
candidate. The value of $M_{\rm fin}$ for stripped star progenitors of SNe IIb
is better constrained by $L$ and $T_{\rm eff}$ due to the dependence of $T_{\rm
eff}$ on the envelope mass $M_{\rm env}$ for $M_{\rm env} \lesssim
1~$M$_{\odot}$. Given the initial mass function, our results apply to the
majority of progenitors of core collapse SNe, failed SNe and direct collapse
black holes.
