Is it possible to measure the Lense-Thirring orbital shifts of the short-period S-star S4716 orbiting Sgr A$^\ast$?

Abstract

The maximal values of the general relativistic Lense-Thirring (LT) orbital shifts $\Delta I^\mathrm{LT},\,\Delta\Omega^\mathrm{LT}$ and $\Delta\omega^\mathrm{LT}$ of the inclination $I$, the longitude of the ascending node $\Omega$ and the perinigricon $\omega$ of the recently discovered star S4716, which has the shortest orbital period $\left(P_\mathrm{b}=4.02\,\mathrm{yr}\right)$ of all the S-stars that orbit the supermassive black hole (SMBH) in Sgr A$^\ast$, are of the order of $\simeq 5-16$ arcseconds per revolution $\left(^{\prime\prime}\,\mathrm{rev}^{-1}\right)$. Given the current error $\sigma_\omega = 0.02^\circ$ in determining $\omega$, which is the most accurate orbital parameter of S4716 among all those affected by the SMBH's gravitomagnetic field through its angular momentum ${\boldsymbol{J}}_\bullet$, about 48 yr would be needed to reduce $\sigma_\omega$ to $\simeq 10\%$ of the cumulative LT perinigricon shift over the same time span. Measuring $\Delta I^\mathrm{LT}$ and $\Delta\Omega^\mathrm{LT}$ to the same level of accuracy would take even much longer. Instead, after just 16 yr, a per cent measurement of the larger gravitoelectric (GE) Schwarzschild-like perinigricon shift $\Delta\omega^\mathrm{GE}$, which depends only on the SMBH's mass $M_\bullet$, would be possible. On the other hand, the uncertainties in the physical and orbital parameters entering $\Delta\omega^\mathrm{GE}$ would cause a huge systematic bias of $\Delta\omega^\mathrm{LT}$ itself. The SMBH's quadrupole mass moment $Q_2^\bullet$ induces orbital shifts as little as $\simeq 0.01-0.05\,^{\prime\prime}\,\mathrm{rev}^{-1}$.