Gravitomagnetic effects for polar circular geodesic orbits around a central rotating body by Lorenzo Iorio | Papers by Lorenzo

In this paper we are interested in the general relativistic motion
around a central rotating body of mass M. In particular, we wish to elucidate the
gravitomagnetic effect on the motion of a test particle following a polar circular
geodesic orbit in a plane containing the proper angular momentum J of the central
gravitating source. The first general relativistic correction of order O(c−2) to the
Keplerian period is proportional to J2/pM5. Such correction, which turns out not
to be a mere coordinate effect, is insensitive to the sense of motion of the test particle
on its orbit, contrary to the well known case of a circular, equatorial orbit yielding
the gravitomagnetic clock effect for a couple of counter–orbiting test particles. For a
LAGEOS like satellite in the gravitational field of the Earth it is of order of 10−9 s.
Unfortunately, due to the uncertainty in the Earth’s GM, the error in the classical
Keplerian period is of the order of 10−5 s, so that the obtained gravitomagnetic
effect is, at present, undetectable.