Post-Keplerian perturbations of the orbital time shift in binary pulsars: an analytical formulation with applications to the galactic center

Abstract

We develop a general approach to analytically calculate the perturbations Delta delta tau _text {p} of the orbital component of the change delta tau _text {p} of the times of arrival of the pulses emitted by a binary pulsar p induced by the post-Keplerian accelerations due to the mass quadrupole Q_2, and the post-Newtonian gravitoelectric (GE) and Lense–Thirring (LT) fields. We apply our results to the so-far still hypothetical scenario involving a pulsar orbiting the supermassive black hole in the galactic center at Sgr A^*. We also evaluate the gravitomagnetic and quadrupolar Shapiro-like propagation delays delta tau _text {prop}. By assuming the orbit of the existing main sequence star S2 and a time span as long as its orbital period P_mathrm{b}, we obtain left| Delta delta tau _text {p}^text {GE}right| lesssim 10^3~text {s},~left| Delta delta tau _text {p}^text {LT}right| lesssim 0.6~text {s},left| Delta delta tau _text {p}^{Q_2}right| lesssim 0.04~text {s}. Faster left( P_mathrm{b}= 5~text {years}right) and more eccentric left( e=0.97right) orbits would imply net shifts per revolution as large as left| leftlangle Delta delta tau _text {p}^text {GE}rightrangle right| lesssim 10~text {Ms},~left| leftlangle Delta delta tau _text {p}^text {LT}rightrangle right| lesssim 400~text {s},left| leftlangle Delta delta tau _text {p}^{Q_2}rightrangle right| lesssim 10^3~text {s}, depending on the other orbital parameters and the initial epoch. For the propagation delays, we have left| delta tau _text {prop}^text {LT}right| lesssim 0.02~text {s},~left| delta tau _text {prop}^{Q_2}right| lesssim 1~upmu text {s}. The results for the mass quadrupole and the Lense–Thirring field depend, among other things, on the spatial orientation of the spin axis of the Black Hole. The expected precision in pulsar timing in Sgr A^* is of the order of 100~upmu text {s}, or, perhaps, even 1–10 upmu text {s}. Our method is, in principle, neither limited just to some particular orbital configuration nor to the dynamical effects considered in the present study.