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

## TL;DR

This paper develops a method to detect post-Keplerian relativistic effects on the radial velocity of stars orbiting the Galactic Center's SMBH, suggesting the Schwarzschild effect may be measurable with current data on the S2 star.

## Contribution

It introduces a general approach to quantify relativistic effects on radial velocities in binary systems, specifically applied to the S2 star near the SMBH, highlighting the potential to measure general relativistic effects.

## Key findings

- Relativistic RV shift per orbit for S2 is about -11.6 km/s.
- Peak relativistic RV effect could reach 551 km/s in 2018.
- Current RV measurement accuracy is sufficient to detect the Schwarzschild effect.

## 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 $\left\langle\Delta V\right\rangle$ 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 $\simeq 30-50~\textrm{km s}^{-1}$ dating back to $t_0 = 2003.271$ are currently available, and whose orbital period amounts to $P_\textrm{b} = 16$ yr. Indeed, while its expected general relativistic RV net shift per orbit amounts to just $\left\langle\Delta V^\textrm{GE}\right\rangle = -11.6~\textrm{km s}^{-1}$, it should reach a peak value as large as $\Delta V_\textrm{max}^\textrm{GE}\left(t_\textrm{max}\right) = 551~\textrm{km s}^{-1}$ at $t_\textrm{max} = 2018.35$. The periastron shift $\Delta\omega^\textrm{GE}$ 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, $0.19~\textrm{km s}^{-1},0.0039~\textrm{km s}^{-1}$, 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.