Dust Extinction Law in Nearby Star-Resolved Galaxies. I. M31 Traced by
  Supergiants
  The dust extinction laws and dust properties in M31 are explored with a
sample of reddened O-type and B-type supergiants obtained from the LGGS. The
observed spectral energy distributions (SEDs) for each tracer are constructed
with multiband photometry from the LGGS, PS1 Survey, UKIRT, PHAT Survey,
Swift/UVOT and XMM-SUSS. We model the SED for each tracer in combination with
the intrinsic spectrum obtained from the stellar model atmosphere extinguished
by the model extinction curves. Instead of mathematically parameterizing the
extinction functions, the model extinction curves in this work are directly
derived from the silicate-graphite dust model with a dust size distribution of
$dn/da \sim a^{-\alpha}{\rm exp}(-a/0.25),~0.005 < a < 5~\mu {\rm m}$. The
extinction tracers are distributed along the arms in M31, with the derived
MW-type extinction curves covering a wide range of $R_V$ ($\approx 2 - 6$),
indicating the complexity of the interstellar environment and the inhomogeneous
distribution of interstellar dust in M31. The average extinction curve with
$R_V \approx 3.51$ and dust size distribution $dn/da \sim a^{-3.35}{\rm
exp}(-a/0.25)$ is similar to those of the MW but rises slightly less steeply in
the far-UV bands, implying that the overall interstellar environment in M31
resembles the diffuse region in the MW. The extinction in the $V$ band of M31
is up to 3 mag, with a median value of $ A_V \approx 1$ mag. The multiband
extinction values from the UV to IR bands are also predicted for M31, which
will provide a general extinction correction for future works.
