Tidal contributions to neutron star orbital precession around Sgr $A^*$

Technical Summary: Tidal contributions to neutron star orbital precession around Sgr A∗

Problem Statement
The Galactic Centre, hosting the supermassive black hole Sgr A* (mass $\approx 4.3 \times 10^6 M_\odot$), serves as a laboratory for testing gravity in strong-field regimes. While the population of main-sequence S-stars is well-documented, recent discoveries of millisecond pulsars near the Galactic Centre suggest a significant population of neutron stars (NS) may also exist in this environment. Neutron stars are unique probes of dense matter physics, with their internal structure governed by the Equation of State (EOS) of cold, strongly interacting matter. A key structural parameter is the tidal Love number ($k_2$), which quantifies the star's static quadrupolar deformation in response to an external tidal field. This paper investigates the magnitude of the tidal contribution to the orbital precession of neutron stars orbiting Sgr A* and compares it against the dominant General Relativistic (GR) orbital precession. The central question is whether the tidal deformation of a neutron star induces a measurable shift in its orbital precession, potentially offering a new probe of NS structure.

Methodology
The study employs the Tolman–Oppenheimer–Volkoff (TOV) equations to model static, spherically symmetric neutron star configurations.

• Stellar Modeling: The authors generate a sequence of equilibrium configurations using a polytropic EOS ($P = K\rho^\Gamma$) with $K=200$ and $\Gamma=2$. By integrating the TOV equations, they determine the stellar mass ($M$), radius ($R$), and compactness ($C = GM/c^2R$).
• Tidal Love Number Calculation: The quadrupolar tidal Love number ($k_2$) is derived by solving the linearized Einstein equations for an $\ell=2$ static, even-parity perturbation on the TOV background. The interior solution is matched to the exterior Schwarzschild spacetime at the stellar surface to obtain $k_2$ as a function of compactness.
• Precession Estimation: The tidal angular shift per orbit ($\Delta\phi_{\text{tidal}}$) is calculated using the analytical expression for a neutron star orbiting a companion of mass $M_{bh}$ (Sgr A*) with orbital eccentricity $e$ and pericentre distance $r_p$. The formula used is:
  $$\Delta\phi_{\text{tidal}} = \frac{30\pi k_2}{(1+e)^5} \left(1 + \frac{3}{2}e^2 + \frac{1}{8}e^4\right) \frac{M_{bh}}{M} \left(\frac{R}{r_p}\right)^5$$
  The study assumes a highly eccentric orbit ($e=0.9$) and evaluates the precession for various pericentre distances ($r_p$), including values corresponding to the innermost stable circular orbit (4.5 au) and observed S-stars (e.g., S-2 at 12 au).

Key Contributions and Results
The paper provides a quantitative assessment of tidal precession effects for neutron stars in the Sgr A* environment, yielding the following specific findings:

• Dependence on Parameters: The tidal precession exhibits a strong dependence on the ratio of the stellar radius to the pericentre distance, scaling as $(R/r_p)^5$. Consequently, the effect decreases sharply as the pericentre distance increases.
• Compactness and $k_2$: For the stellar configurations considered, the calculated tidal angular shift decreases with increasing stellar compactness. Conversely, larger Love numbers ($k_2$) enhance the tidal contribution. The study finds that $k_2$ varies within a narrow range ($\approx 0.2259–0.2291$) for the adopted model.
• Magnitude of the Effect: Despite the enhancement expected during closer encounters (small $r_p$) and high eccentricity, the predicted tidal shifts are extremely small. Within the displayed parameter range, the maximum tidal shift reaches approximately $3.5 \times 10^{-18}$ microarcseconds ($\mu$as).
• Comparison with GR Precession: The ratio of tidal precession to General Relativistic precession ($\Delta\phi_{\text{tidal}}/\Delta\phi_{\text{GR}}$) spans the range of approximately $10^{-33}$ to $10^{-28}$. This indicates that GR precession overwhelmingly dominates the orbital dynamics.

Significance and Claims
The authors conclude that while the theoretical framework for calculating tidal contributions to orbital precession is established, the practical detectability of this effect is severely limited by the physical scales involved.

• Dominance of GR: The study asserts that relativistic precession dominates the orbital dynamics by many orders of magnitude, rendering the tidal contribution negligible in the context of current or near-future observational capabilities.
• Detection Challenges: The paper explicitly states that the direct detection of the neutron-star tidal contribution through orbital precession around Sgr A* would be "exceptionally challenging" under the assumptions of the adopted TOV model.
• Role of the Fifth-Power Dependence: The authors attribute the negligible signal primarily to the $(R/r_p)^5$ suppression factor. For neutron stars, the ratio of the stellar radius to the pericentre distance is sufficiently small that this suppression outweighs the enhancement provided by the large black-hole-to-stellar mass ratio.

In summary, the paper establishes that within the investigated parameter range and using the TOV stellar model, tidal corrections play a negligible role in the orbital precession of neutron stars around Sgr A*, and their direct measurement via this method is not feasible.