The macroscopic precession model: describing quasi-periodic oscillations including internal structures of test bodies

The relativistic precession model (RPM) is widely-considered as a benchmark framework to interpret quasi-periodic oscillations (QPOs), albeit several observational inconsistencies suggest that the model remains incomplete. The RPM ensures \emph{structureless test particles} and attributes precession to geodesic motion alone. Here, we refine the RPM by incorporating the internal structure of rotating test bodies, while preserving the test particle approximation (TPA), and propose a \emph{macroscopic precession model} (MPM) by means of the Mathisson-Papapetrou-Dixon (MPD) equations, applied to a Schwarzschild background, which introduces 1) a shift in the Keplerian frequency and 2) an \emph{effective spin correction} to the radial epicyclic frequency that, once the spin tensor is modeled, reproduces a quasi-Schwarzschild-de Sitter (SdS) correction. We apply the MPM to eight neutron star low mass X-ray binaries (NS-LMXBs), performing Markov chain Monte Carlo (MCMC) fits to twin kHz QPOs and find observational and statistical evidence in favor of precise power law spin reconstructions. Further, our model accurately predicts the $3:2$ frequency clustering, the disk boundaries and the NS masses. From the MPM model, we thus conclude that complexity of QPOs can be fully-described including the test particle internal structure.

💡 Research Summary

The relativistic precession model (RPM) has long served as a benchmark for interpreting the twin kilohertz quasi‑periodic oscillations (kHz QPOs) observed in neutron‑star low‑mass X‑ray binaries (NS‑LMXBs). RPM treats the accreting matter as a collection of structure‑less test particles moving on geodesics, identifying the lower QPO frequency with the periastron‑precession frequency (ν_L = (Ω_φ – Ω_r)/2π) and the upper QPO frequency with the Keplerian (azimuthal) frequency (ν_U = Ω_φ/2π). While successful in many respects, several observational facts—most notably the ubiquitous 3:2 frequency clustering, the need for an effective cosmological‑constant‑like term, and systematic tensions when fitting data with pure Schwarzschild or Kerr metrics—indicate that RPM is incomplete.

In this work the authors propose a “macroscopic precession model” (MPM) that retains the test‑particle approximation (TPA) but endows each particle with an intrinsic spin, thereby accounting for its internal angular momentum. The dynamics are derived from the Mathisson‑Papapetrou‑Dixon (MPD) equations:  Dp^μ/Dτ = –½ u^π S^{ρσ} R^μ_{ πρσ},  DS^{μν}/Dτ = p^μ u^ν – p^ν u^μ, with the Tulczyjew‑Dixon supplementary condition S_{λν} p^ν = 0. By exploiting the Killing vectors of a static, spherically symmetric Schwarzschild spacetime, the authors obtain modified conserved energy and angular momentum (E + ΔE, L – ΔL) that contain spin‑dependent corrections proportional to the spin tensor components S^{tr} and S^{ϕr}. The spin tensor is reduced to a single radial function S^{tr}=C_n r^n, where n is a macroscopic index describing the spatial distribution of the internal spin (n=1 filament‑like, n=2 disk‑like, n=3 spherical‑like) and C_n is a small amplitude.

Solving the MPD equations perturbatively in the dimensionless spin parameter κ = |S_0|/(mr) ≪ 1, the authors find that the azimuthal frequency receives a first‑order correction:  Ω_φ = ±√(M/r³) ∓ s κ_0 (2n+1) r^{–6}(n–1) /