Generic gravito-magnetic clock effects

General relativity predicts that two counter-orbiting clocks around a spinning mass differ in the time required to complete the same orbit. The difference in these two values for the orbital period is generally referred to as the gravito-magnetic (GM) clock effect. It has been proposed to measure the GM clock effect using atomic clocks carried by satellites in prograde and retrograde orbits around the Earth. The precision and stability required for satellites to accurately perform this measurement remains a challenge for current instrumentation. One of the most accurate clocks in the Universe is a millisecond pulsar, which emits periodic radio pulses with high stability. Timing of the pulsed signals from millisecond pulsars has proven to be very successful in testing predictions of general relativity and the GM clock effect is potentially measurable in binary systems. In this work we derive the generic GM clock effect by considering a slowly-spinning binary system on an elliptical orbit, with both arbitrary mass ratio and arbitrary spin orientations. The spin-orbit interaction introduces a perturbation to the orbit, causing the orbital plane to precess and nutate. We identify several different contributions to the clock effects: the choice of spin supplementary condition and the observer-dependent definition of a full revolution and "nearly-identical" orbits. We discuss the impact of these subtle definitions on the formula for GM clock effects and show that most of the existing formulae in the literature can be recovered under appropriate assumptions.