Analytical Solution to a simplified dark matter profile

Coordinates transformations and definitions. GC:= Galactic centre, GW:= Galactic West, GE:= Galactic East, SGP:= South Galactic Pole, NGP:= North Galactic Pole (Image credit:M. Hütten, C. Combet and D. Maurin).

We want to make a coordinates transformation from heliocentric coordinates to Galactocentric coordinates. Assuming that the GC is located at distance R_sun we can write

The astrophysical factor in the (theta,Psi) direction is obtained by integrating over the line of sight the dark matter density squared profile

We can use a simplified form for the dark matter profile to obtain an analytical solution to the line-of-sight integration like shown below: