Deprojecting Sersic Profiles for Arbitrary Triaxial Shapes: Robust
  Measures of Intrinsic and Projected Galaxy Sizes
  We present the analytical framework for converting projected light
distributions with a S\'ersic profile into three-dimensional light
distributions for stellar systems of arbitrary triaxial shape. The main
practical result is the definition of a simple yet robust measure of intrinsic
galaxy size: the median radius $r_\mathrm{med}$, defined as the radius of a
sphere that contains 50% of the total luminosity or mass, that is, the median
distance of a star to the galaxy center. We examine how $r_\mathrm{med}$
depends on projected size measurements as a function of S\'ersic index and
intrinsic axis ratios, and demonstrate its relative independence of these
parameters. As an application we show that the projected semi-major axis length
of the ellipse enclosing 50% of the light is an unbiased proxy for
$r_\mathrm{med}$, with small galaxy-to-galaxy scatter of $\sim$10% (1$\sigma$),
under the condition that the variation in triaxiality within the population is
small. For galaxy populations with unknown or a large range in triaxiality an
unbiased proxy for $r_\mathrm{med}$ is $1.3\times R_{e}$, where $R_{e}$ is the
circularized half-light radius, with galaxy-to-galaxy scatter of 20-30%
(1$\sigma$). We also describe how inclinations can be estimated for individual
galaxies based on the measured projected shape and prior knowledge of the
intrinsic shape distribution of the corresponding galaxy population. We make
the numerical implementation of our calculations available.
