Surface mass density of the Einasto family of dark matter haloes: Are
  they Sersic-like?
  Recent advances in N-body simulations of dark matter halos have shown that
three-parameter models, in particular the Einasto profile characterized by d ln
{\rho}(r)/d ln r / r with a shape parameter {\alpha} < 0.3, are able to produce
better fits to the 3D spatial density profiles than two-parameter models like
the Navarro, Frenk and White (NFW), and Moore et al. profiles.
  In this paper, we present for the first time an analytically motivated form
for the 2D surface mass density of the Einasto family of dark matter haloes, in
terms of the 3D spatial density parameters for a wide range of the shape
parameter 0.1 < {\alpha} < 1. Our model describes a projected (2D) Einasto
profile remarkably well between 0 and (3 - 5) r_{200}, with errors less than
0.3 per cent for {\alpha} < 0.3 and less than 2 per cent for {\alpha} as large
as 1. This model (in 2D) can thus be used to fit strong and weak lensing
observations of galaxies and clusters whose total spatial (3D) density
distributions are believed to be Einasto-like. Further, given the dependence of
our model on the 3D parameters, one can reliably estimate structural parameters
of the spatial (3D) density from 2D observations. We also consider a
Sersic-like parametrization for the above family of projected Einasto profiles
and observe that fits with a Sersic profile are sensitive to whether one fits
the projected density in linear scale or logarithmic scale and yield widely
varying results. Structural parameters of Einasto-like systems, inferred from
fits with a Sersic profile, should be used with caution.
