Post-Keplerian effects on radial velocity in binary systems and the possibility of measuring General Relativity with the S2 star in 2018

Abstract

One of the directly measured quantities which are used in monitoring the orbital motions of many of the S stars revolving around the Supermassive Black Hole (SMBH) in the Galactic Center (GC) is their radial velocity (RV) $V$ obtained with near-infrared spectroscopy. Here, we devise a general approach to calculate both the instantaneous variations $\Delta V\left(t\right)$ and the net shifts per revolution $⟨\Delta V⟩$ induced on such an observable by some post-Keplerian (pK) accelerations. In particular, we look at the general relativistic Schwarzschild (gravitoelectric) and Lense-Thirring (gravitomagnetic frame-dragging) effects, and the mass quadrupole. It turns out that we may be on the verge of measuring the Schwarzschild-type 1pN static component of the SMBH's field with the S2 star for which RV measurements accurate to about $\simeq 30-50{\text{km s}}^{-1}$ dating back to $t_0=2003.271$ are currently available, and whose orbital period amounts to $P_{\text{b}}=16$ yr. Indeed, while its expected general relativistic RV net shift per orbit amounts to just $⟨\Delta V^{\text{GE}}⟩=-11.6{\text{km s}}^{-1}$, it should reach a peak value as large as $\Delta V_{\text{max}}^{\text{GE}}\left(t_{\text{max}}\right)=551{\text{km s}}^{-1}$ at $t_{\text{max}}=2018.35$. The periastron shift $\Delta {\omega }^{\text{GE}}$ of S2 over the same time span will not be larger than $0.2$ deg, while the current accuracy in estimating such an orbital element for this star is of the order of $0.6$ deg. The frame-dragging and quadrupole-induced RV shifts are far smaller for S2, amounting to, at most, $0.19{\text{km s}}^{-1},0.0039{\text{km s}}^{-1}$, respectively. Further studies should be dedicated to the impact on the RV of possible diffused mass distribution in the GC and of other individual stars inside and outside the orbit of S2.

LaTex2e, 27 pages, 2 tables, 5 figures. Version matching the one at press in Monthly Notices of the Royal Astronomical Society (MNRAS)

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• On the nature of the fast moving star S2 in the Galactic Center
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• APSIS - an Artificial Planetary System in Space to probe extra-dimensional gravity and MOND

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• Seeing Black Holes : from the Computer to the Telescope
• Multi-scale Constraints on Scalar-Field couplings to Matter: The Geodetic and Frame-Dragging Effects
• Post-Newtonian effects on some characteristic timescales of transiting exoplanets
• Cosmic censorship conjecture in a general Kerr-Newman black hole
• Relativistic effects in orbital motion of the S-stars at the Galactic Center
• A stellar fly-by close to the Galactic centre: Can we detect stars on highly-relativistic orbits?
• The GWs from the S-stars revolving around the SMBH at Sgr A*
• Precession shift in curvature based Extended Theories of Gravity and Quintessence fields
• Impact of Lorentz violation models on exoplanets dynamics