Discovery of a red supergiant donor star in SN2010da/NGC 300 ULX-1
  SN2010da/NGC 300 ULX-1 was first detected as a supernova impostor in May 2010
and was recently discovered to be a pulsating ultraluminous X-ray source. In
this letter, we present VLT/X-shooter spectra of this source obtained in
October 2018, covering the wavelength range 350-2300 nm. The $J$- and $H$-bands
clearly show the presence of a red supergiant donor star that is best matched
by a MARCS stellar atmosphere with $T_{\rm eff} = 3650 - 3900$ K and
$\log(L_{\rm bol}/L_{\odot}) = 4.25\pm0.10$, which yields a stellar radius $R =
310 \pm 70 R_{\odot}$. To fit the full spectrum, two additional components are
required: a blue excess that can be fitted either by a hot blackbody (T
$\gtrsim 20,000$ K) or a power law (spectral index $\alpha \approx 4$) and is
likely due to X-ray emission reprocessed in the outer accretion disk or the
donor star; and a red excess that is well fitted by a blackbody with a
temperature of $\sim 1100$ K, and is likely due to warm dust in the vicinity of
SN2010da. The presence of a red supergiant in this system implies an orbital
period of at least 0.8-2.1 years, assuming Roche lobe overflow. Given the large
donor-to-compact object mass ratio, orbital modulations of the radial velocity
of the red supergiant are likely undetectable. However, the radial velocity
amplitude of the neutron star is large enough (up to 40-60 km s$^{-1}$) to
potentially be measured in the future, unless the system is viewed at a very
unfavorable inclination.
