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 in the Galactic Center at Sgr A${}^{\ast }$. We also evaluate the gravitomagnetic and quadrupolar Shapiro-like propagation delays $\delta {\tau }_{\text{prop}}$. By assuming the orbit of the existing S2 main sequence star and a time span as long as its orbital period $P_{\text{b}}$, 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 $\left(P_{\text{b}}=5\text{yr}\right)$ and more eccentric $\left(e=0.97\right)$ orbits would imply net shifts per revolution as large as $\left|⟨\Delta \delta {\tau }_{\text{p}}^{\text{GE}}⟩\right|\lesssim 10\text{Ms},\left|⟨\Delta \delta {\tau }_{\text{p}}^{\text{LT}}⟩\right|\lesssim 400\text{s},\left|⟨\Delta \delta {\tau }_{\text{p}}^{Q_2}⟩\right|\lesssim 10^3\text{s}$, depending on the other orbital parameters and the initial epoch. For the propagation delays, we have $\left|\delta {\tau }_{\text{prop}}^{\text{LT}}\right|\lesssim 0.02\text{s},\left|\delta {\tau }_{\text{prop}}^{Q_2}\right|\lesssim 1\mu \text{s}$. The expected precision in pulsar timing in Sgr A${}^{\ast }$ is of the order of $100\mu \text{s}$, or, perhaps, even $1-10\mu \text{s}$.

LaTex2e, 2 tables, 4 figures, 36 pages. Entirely rewritten version which corrects previous erroneous results concerning the orbital time delays

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