Is there still something left that Gravity Probe B can measure?

We perform a full analytical and numerical treatment, to the first post-Newtonian (1pN) order, of the general relativistic long-term spin precession of an orbiting gyroscope due to the mass quadrupole moment $J_2$ of its primary without any restriction on either the gyro's orbital configuration and the orientation in space of the symmetry axis $\boldsymbol{\hat{k}}$ of the central body. We apply our results to the past spaceborne Gravity Probe B (GP-B) mission by finding a secular rate of its spin's declination $δ$ which may be as large as $\lesssim 30-40\,\mathrm{milliarcseconds\,per\,year\,(\mathrm{mas\,yr}^{-1}})$, depending on the initial orbital phase $f_0$. Both our analytical calculation and our simultaneous integration of the equations for the parallel transport of the spin 4-vector and of the geodesic equations of motion of the gyroscope confirm such a finding. For GP-B, the reported mean error in measuring the spin's declination rate amounts to $σ^\mathrm{GP-B}_{\dotδ}=18.3\,\mathrm{mas\,yr}^{-1}$. We also calculate the general analytical expressions of the gravitomagnetic spin precession induced by the primary's angular momentum $\boldsymbol J$. In view of their generality, our results can be extended also to other astronomical and astrophysical scenarios of interest like, e.g., stars orbiting galactic supermassive black holes, exoplanets close to their parent stars, tight binaries hosting compact stellar corpses.

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