Lense-Thirring precession of neutron-star accretion flows: Relativistic versus classical precession

Problem
The vertical (Lense-Thirring) precession of innermost accretion flows is considered a sensitive indicator of neutron star (NS) rotation and equation of state (EoS), particularly because this precession vanishes for non-rotating stars. However, observational data on low-frequency quasi-periodic oscillations (LF QPOs) in NS low-mass X-ray binaries (LMXBs) have failed to show a strong correlation with the NS spin frequency. Previous studies utilizing the Lense-Thirring (LT) metric, which is linear in the NS angular momentum, suggest a monotonic relationship between spin and precession frequency. This paper investigates whether the widely used LT metric is sufficient to describe precession behavior across the astrophysically relevant range of NS angular momentum, or if the interplay between relativistic frame-dragging and classical precession effects (arising from the NS quadrupole moment) leads to more complex, non-monotonic behaviors that could explain the observational discrepancies.

Methodology
The authors employ the Hartle-Thorne (HT) spacetime metric, which accounts for the NS mass ($M$), angular momentum ($J$), and quadrupole moment ($Q$), rather than the linear LT metric. They utilize dimensionless parameters for angular momentum ($j = J/M^2$) and quadrupole moment ($q = Q/M^3$), introducing a quadrupole parameter $\tilde{q} = q/j^2$ which depends on the EoS but not on the rotational frequency.

The study calculates precession frequencies for two distinct types of motion:

1. Geodesic motion: Free test particles and slender tori.
2. Non-geodesic fluid flows: Thick accretion tori with constant specific angular momentum and a polytropic equation of state, incorporating pressure corrections.

The analysis covers a range of NS masses ($1.4M_\odot$ to $2.5M_\odot$) and rotational frequencies corresponding to millisecond pulsars. The authors examine specific radii within the innermost accretion region, including the innermost stable circular orbit (ISCO), and radii defined by resonant ratios of epicyclic frequencies ($r_{3:1}, r_{2:1}, r_{3:2}$). To ensure physical realism, the study incorporates surface restrictions imposed by specific EoSs (XMLSLZ, VGBCMR, and MIT bag for quark stars), determining whether the maximum precession frequency occurs at the ISCO, the stellar surface, or an intermediate local maximum. All calculations are reproducible via an associated Wolfram Mathematica notebook.

Key Contributions

• Generalization of Precession Models: The work extends previous findings on the effect of the NS quadrupole moment by providing a full, general description of precession behavior for both geodesic and non-geodesic flows using the HT metric.
• Demonstration of Non-Monotonicity: The paper demonstrates that the LT metric is insufficient for calculating precession frequencies across a broad range of angular momenta. It shows that even for moderately oblate NSs, the dependence of precession frequency on angular momentum is not monotonic.
• Identification of Frequency Maxima: The authors identify that precession frequencies within the innermost accretion region exhibit local maxima at relatively low values of NS angular momentum.
• Equation of State Dependence: The study provides specific examples of precession behavior for different EoSs (nuclear and quark matter), showing how the quadrupole parameter influences the location and magnitude of frequency maxima.

Results

• Competition of Effects: The results highlight a competition between relativistic precession (driven by frame-dragging) and classical precession (driven by the spatial distribution of mass/quadrupole moment). While relativistic effects dominate at low spins, the classical contribution increases with spin, eventually causing the total precession frequency to decrease or reverse orientation.
• Existence of Maxima: For any given quadrupole parameter $\tilde{q} \gtrsim 1.5$, the precession frequency at the ISCO exhibits a maximum ($\nu_{max}^{ISCO}$) at a specific angular momentum $j_{max}^{ISCO} \approx 0.95 \tilde{q}^{0.94}$. Similar maxima are found at radii $r_{3:1}, r_{2:1}$, and $r_{3:2}$.
• Fluid vs. Geodesic: While fluid flows (thick tori) introduce pressure corrections that can shift frequencies, the qualitative behavior regarding the existence of maxima remains similar to that of geodesic motion.
• Surface Effects: When accounting for the NS surface, the maximum observable frequency is determined by the ISCO, the surface, or a local radial maximum, depending on the spin and EoS. However, the non-monotonic trend persists regardless of whether the surface is inside or outside the ISCO.
• Divergence from LT Metric: Frequencies predicted by the linear LT metric are substantially higher than the maxima found in the HT spacetime and fail to capture the decrease in frequency at higher spins.

Significance
The paper concludes that the non-monotonic relationship between NS spin and precession frequency implies that very different groups of accreting NSs—specifically slow rotators and fast rotators—can display identical precession frequencies. This finding offers a potential explanation for the lack of observed correlation between LF QPO frequencies and NS spin in previous studies. By demonstrating that the LT metric is insufficient for capturing the full behavior of precession in realistic, oblate NS spacetimes, the work suggests that the interplay between relativistic and classical precession must be considered to accurately model and interpret QPO observations. The authors provide a reproducible framework for these calculations, allowing for further investigation into specific NS and quark star models.