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The Einasto profile has been successful in describing the density profiles ofdark matter haloes in $\Lambda$CDM N-body simulations. It has also been able todescribe multiple components in the surface brightness profiles of galaxies.However, analytically projecting it to calculate quantities under projection ischallenging. In this paper, we will see the development of a highly accurateanalytical approximation for the mass (or counts) enclosed in an infinitelylong cylindrical column for Einasto profiles--also known as the projected mass(or counts)--using a novel methodology. We will then develop a self-consistenthigh-accuracy model for the surface density from the expression for theprojected 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 tofit the density profiles of dark matter haloes in N-body simulations as well asthe luminosity profiles of the outer components of massive galaxies. Since theprojected mass and the surface density are used in gravitational lensing, Iwill illustrate how these models facilitate (for the first time) analyticalcomputation of several quantities of interest in lensing due to Einastoprofiles. The models, however, are not limited to lensing and apply to similarquantities under projection, such as the projected luminosity, the projected(columnar) number counts and the projected density or the surface brightness.

High-accuracy analytical solutions for the projected mass (counts) and surface density (brightness) of Einasto profiles