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