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

摘要

We develop a general approach to analytically calculate the perturbations (Delta delta au _ ext {p}) of the orbital component of the change (delta au _ ext {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 au _ ext {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 au _ ext {p}^ ext {GE} ight| lesssim 10^3~ ext {s},~left| Delta delta au _ ext {p}^ ext {LT} ight| lesssim 0.6~ ext {s},left| Delta delta au _ ext {p}^{Q_2} ight| lesssim 0.04~ ext {s}). Faster (left( P_mathrm{b}= 5~ ext {years} ight) ) and more eccentric (left( e=0.97 ight) ) orbits would imply net shifts per revolution as large as (left| leftlangle Delta delta au _ ext {p}^ ext {GE} ight angle ight| lesssim 10~ ext {Ms},~left| leftlangle Delta delta au _ ext {p}^ ext {LT} ight angle ight| lesssim 400~ ext {s},left| leftlangle Delta delta au _ ext {p}^{Q_2} ight angle ight| lesssim 10^3~ ext {s}), depending on the other orbital parameters and the initial epoch. For the propagation delays, we have (left| delta au _ ext {prop}^ ext {LT} ight| lesssim 0.02~ ext {s},~left| delta au _ ext {prop}^{Q_2} ight| lesssim 1~upmu ext {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 ext {s}), or, perhaps, even 1–10?(upmu ext {s}). Our method is, in principle, neither limited just to some particular orbital configuration nor to the dynamical effects considered in the present study.