The Short-period S-stars S4711, S62, S4714 and the Lense–Thirring Effect due to the Spin of Sgr A*

Recently, some S-stars (S4711, S62, S4714) orbiting the supermassive black hole (SMBH) in Sgr A^* with short orbital periods (7.6 yr ≤ P _b ≤ 12 yr) were discovered. It was suggested that they may be used to measure the general relativistic Lense–Thirring (LT) precessions of their longitudes of ascending node Ω induced by the SMBH's angular momentum J _•. In fact, the proposed numerical estimates hold only in the particular case of a perfect alignment of J _• with the line of sight, which does not seem to be the case. Moreover, the inclination I and the argument of perinigricon ω also undergo LT precessions for an arbitrary orientation of J _• in space. We explicitly show the analytical expressions of ${\dot{I}}^{LT},{\dot{\Omega }}^{LT},{\omega }^{LT}$ in terms of the SMBH's spin polar angles i ^•, ε ^•. It turns out that the LT precessions, in arcseconds per year $\left({}^{\prime \prime }{yr}^{-1}\right)$, range within $\left|{\dot{I}}^{LT}\right|\lesssim 7^{\prime \prime }{yr}^{-1}$, $\left|{\dot{\Omega }}^{LT}\right|\lesssim 9^{\prime \prime }{yr}^{-1}$, and $-{13}^{\prime \prime }{yr}^{-1}\lesssim {\dot{\omega }}^{LT}\lesssim {14}^{\prime \prime }{yr}^{-1}$ for S4714, $\left|{\dot{I}}^{LT}\right|\lesssim 5^{\prime \prime }{yr}^{-1}$, $\left|{\dot{\Omega }}^{LT}\right|\lesssim 5^{\prime \prime }{yr}^{-1}$, and $\left|{\dot{\omega }}^{LT}\right|\lesssim {10}^{\prime \prime }{yr}^{-1}$ for S62, and $\left|{\dot{I}}^{LT}\right|\lesssim {0.3}^{\prime \prime }{yr}^{-1}$, $\left|{\dot{\Omega }}^{LT}\right|\lesssim {0.3}^{\prime \prime }{yr}^{-1}$, and $\left|{\dot{\omega }}^{LT}\right|\lesssim {0.7}^{\prime \prime }{yr}^{-1}$ for S4711. For each star, the corresponding values of $i_{max}^{\bullet },{\epsilon }_{max}^{\bullet }$, and $i_{min}^{\bullet },{\epsilon }_{min}^{\bullet }$ are determined as well, along with those $i_0^{\bullet },{\epsilon }_0^{\bullet }$ that cancel the LT precessions. The LT perinigricon precessions ${\dot{\omega }}^{LT}$ are overwhelmed by the systematic uncertainties in the Schwarzschild ones due to the current errors in the stars' orbital parameters and the mass of Sgr A^* itself.