Gravitational attraction and potential of spherical shell with radially dependent density

Solutions to the direct problem in gravimetric interpretation are well-known for wide class of source bodies with constant density contrast. On the other hand, sources with non-uniform density can lead to relatively complicated formalisms. This is probably why analytical solutions for this type of sources are rather rare although utilization of these bodies can sometimes be very effective in gravity modeling. I demonstrate an analytical solution to that problem for a spherical shell with radial polynomial density distribution, and illustrate this result when applied to a special case of 5th degree polynomial. As a practical example, attraction of the normal atmosphere is calculated.