Effect of an arbitrary spin orientation on the quadrupolar structure of an extended body in a Schwarzschild spacetime

The influence of an arbitrary spin orientation on the quadrupolar structure of an extended body moving in a Schwarzschild spacetime is investigated. The body dynamics is described by the Mathisson-Papapetrou-Dixon model, without any restriction on the motion or simplifying assumption on the associated spin vector and quadrupole tensor, generalizing previous works. The equations of motion are solved analytically in the limit of small values of the characteristic length scales associated with the spin and quadrupole variables with respect to the characteristic length of the background curvature. The solution provides all corrections to the circular geodesic on the equatorial plane taken as the reference trajectory due to both the dipolar and quadrupolar structures of the body as well as the conditions which the nonvanishing components of the quadrupole tensor must fulfill in order for the problem to be self-consistent.