The effects of orbital motion on LISA time delay interferometry

In an effort to eliminate laser phase noise in laser interferometer spaceborne gravitational wave detectors, several combinations of signals have been found that allow the laser noise to be cancelled out while gravitational wave signals remain. This process is called time delay interferometry (TDI). In the papers that defined the TDI variables, their performance was evaluated in the limit that the gravitational wave detector is fixed in space. However, the performance depends on certain symmetries in the armlengths that are available if the detector is fixed in space, but that will be broken in the actual rotating and flexing configuration produced by the LISA orbits. In this paper we investigate the performance of these TDI variables for the real LISA orbits. First, addressing the effects of rotation, we verify Daniel Shaddock's result that the Sagnac variables α (t), β (t) and γ (t) will not cancel out the laser phase noise, and we also find the same result for the symmetric Sagnac variable ζ (t). The loss of the latter variable would be particularly unfortunate since this variable also cancels out gravitational wave signal, allowing instrument noise in the detector to be isolated and measured. Fortunately, we have found a set of more complicated TDI variables, which we call Δ Sagnac variables, one of which accomplishes the same goal as ζ (t) to good accuracy. Finally, however, as we investigate the effects of the flexing of the detector arms due to non-circular orbital motion, we show that all variables, including the interferometer variables, X(t), Y(t) and Z(t), which survive the rotation-induced loss of direction symmetry, will not completely cancel laser phase noise when the armlengths are changing with time. This unavoidable problem will place a stringent requirement on laser stability of ∼5 Hz Hz-1/2