In the weak-field and slow-motion approximation of general relativity, the rotation of a body discriminates between the opposite directions of motion of a pair of counter-revolving tests particles orbiting it along geometrically identical trajectories: it is the so-called gravitomagnetic clock effect. In this paper, we analytically calculate the gravitomagnetic corrections to both the draconitic and to the anomalistic periods of arbitrarily inclined, quasi-circular orbits for a generic orientation of the spin axis of the primary. While the anomalistic period is left unchanged, the draconitic one experiences a generally non-vanishing correction which, to zero order in the eccentricity, gains a minus sign if the velocity of the test particle is reversed. As a result, a gravitomagnetic draconitic clock effect arises since a generally non-zero difference of the draconitic periods of a pair of counter-orbiting test particles arises. Remarkably, it is independent of their initial conditions, with some advantages from an experimental point of view because, in principle, the data could be collected independently of each other and at different epochs. Numerically integrations of the equations of motion confirm our analytical results. The relativistic time difference turns out to be up to about $0.5$ ms and $0.1$ ms around Jupiter and the Sun, respectively, while it is as little as $0.6$ $\mu$s around the Earth.
Mon, 21 Jul 14
Comments: LaTex2e, 22 pages, 1 figure, no tables, 59 references