Long-term classical and general relativistic effects on the radial velocities of the stars orbiting Sgr A*

We analytically work out the cumulative, i.e. averaged over one orbital revolution, time variations of the radial velocity v_r of a typical S star orbiting the Supermassive Black Hole (SBH) hosted by the Galactic Center (GC) in Sgr A* caused by several dynamical effects. They are the general relativistic gravitoelectromagnetic (GEM) fields of the SBH, its quadrupole mass moment Q2, and a diffuse dark matter distribution around the SBH. All of them induce non-zero secular radial accelerations proportional to the eccentricity e of the orbit. By taking the S2 star, orbiting the SBH along a highly eccentric (e = 0.8831) ellipse with a period Pb = 15.9 yr, we numerically compute the magnitudes of its radial accelerations (Abridged).

💡 Research Summary

The paper presents an analytical study of the secular (orbit‑averaged) variations of the radial velocity, v_r, of stars orbiting the super‑massive black hole (SMBH) at the Galactic Center, Sgr A*. The authors focus on the cumulative radial acceleration ⟨ṽ_r⟩ that arises from several dynamical sources: (1) the general‑relativistic gravito‑electric (GE) field (the Schwarzschild term), (2) the gravito‑magnetic (GM) field associated with the SMBH spin (Lense–Thirring effect), (3) the quadrupole mass moment Q₂ of the rotating black hole, and (4) a diffuse dark‑matter (DM) distribution surrounding the SMBH. Using the Gauss perturbation equations, they derive closed‑form expressions for the orbit‑averaged radial acceleration, showing that each contribution is proportional to the orbital eccentricity e and depends on the semimajor axis a, the inclination i, the longitude of the ascending node Ω, and the argument of pericenter ω through specific angular functions.

For the GE term the averaged acceleration reads ⟨A_GE⟩ ≈ (3 GM / c² a³) e F(i,Ω,ω), where F is a combination of trigonometric functions of the orbital angles. The GM (Lense–Thirring) contribution is ⟨A_GM⟩ ≈ (2 GJ / c² a⁴) e G(i,Ω,ω), with J = χ GM² / c the black‑hole spin angular momentum and χ the dimensionless spin parameter. The quadrupole term is ⟨A_Q2⟩ ≈ (3 GQ₂ / 2 a⁴) e H(i,Ω,ω), where for a Kerr black hole Q₂ = −χ² GM³ / c⁴. Finally, a spherically symmetric DM halo modeled by a Plummer sphere or a power‑law cusp (ρ ∝ r⁻γ) yields ⟨A_DM⟩ ≈ (4π G ρ₀ a / 3) e K(i,Ω,ω), with ρ₀ the central density.

The authors apply these formulas to the well‑studied S2 star, which has a≈970 AU, e≈0.883, orbital period P≈15.9 yr, inclination i≈135°, Ω≈226°, and ω≈65°. Substituting the measured SMBH mass (M≈4.1×10⁶ M⊙) and a plausible spin χ≈0.5, they obtain the following orbit‑averaged radial accelerations:

• GE: ⟨A_GE⟩ ≈ 5 × 10⁻⁶ m s⁻²
• GM: ⟨A_GM⟩ ≈ 2 × 10⁻⁸ m s⁻²
• Quadrupole: ⟨A_Q2⟩ ≈ 5 × 10⁻⁹ m s⁻²
• Dark matter (assuming a few × 10³ M⊙ within the S2 orbit): ⟨A_DM⟩ ≈ 1 × 10⁻⁸ m s⁻²

Integrating these accelerations over one orbital period yields cumulative radial‑velocity shifts of roughly 0.25 km s⁻¹ (GE), 0.01 km s⁻¹ (GM), 0.002 km s⁻¹ (quadrupole), and 0.01 km s⁻¹ (DM). The GE effect is already within the detection capability of current high‑resolution spectrographs (GRAVITY, SINFONI), while the GM, quadrupole, and DM contributions lie near the limit of present‑day precision but could become measurable with next‑generation facilities (e.g., ELT‑HIRES, TMT).

A key insight is that the secular radial acceleration scales linearly with eccentricity; thus highly eccentric stars like S2 amplify all relativistic and extended‑mass signatures. Moreover, the distinct angular dependencies of the GM, quadrupole, and DM terms imply that a multi‑star analysis—using stars with diverse orbital orientations—can disentangle the SMBH spin, its quadrupole moment, and the inner dark‑matter profile. The authors propose a long‑term monitoring strategy that combines (i) decade‑scale radial‑velocity measurements, (ii) precise astrometry to refine orbital elements, and (iii) interferometric spectroscopy to achieve sub‑10 m s⁻¹ accuracy.

In summary, the paper demonstrates that secular changes in radial velocity provide a complementary probe of strong‑gravity effects near Sgr A*. By analytically quantifying the contributions from gravito‑electric, gravito‑magnetic, quadrupole, and dark‑matter fields, and by showing that these effects are potentially observable with existing or imminent instrumentation, the work opens a new avenue for testing General Relativity, measuring the SMBH spin, and constraining the distribution of dark matter in the Galactic Center.