High-accuracy analytical solutions for the projected mass (counts) and
  surface density (brightness) of Einasto profiles
  The Einasto profile has been successful in describing the density profiles of
dark matter haloes in $\Lambda$CDM N-body simulations. It has also been able to
describe multiple components in the surface brightness profiles of galaxies.
However, analytically projecting it to calculate quantities under projection is
challenging. In this paper, we will see the development of a highly accurate
analytical approximation for the mass (or counts) enclosed in an infinitely
long cylindrical column for Einasto profiles--also known as the projected mass
(or counts)--using a novel methodology. We will then develop a self-consistent
high-accuracy model for the surface density from the expression for the
projected mass. Both models are quite accurate for a broad family of functions,
with a shape parameter $\alpha$ varying by a factor of 100 in the range $0.05
\lesssim \alpha \lesssim 5.0$, with fractional errors $\sim 10^{-6}$ for
$\alpha \lesssim 0.4$. Profiles with $\alpha \lesssim 0.4$ have been shown to
fit the density profiles of dark matter haloes in N-body simulations as well as
the luminosity profiles of the outer components of massive galaxies. Since the
projected mass and the surface density are used in gravitational lensing, I
will illustrate how these models facilitate (for the first time) analytical
computation of several quantities of interest in lensing due to Einasto
profiles. The models, however, are not limited to lensing and apply to similar
quantities under projection, such as the projected luminosity, the projected
(columnar) number counts and the projected density or the surface brightness.
