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

TL;DR

This paper addresses how Lense-Thirring precession in the innermost NS accretion flows depends on rotation and quadrupole deformation. By employing the Hartle-Thorne spacetime and defining the quadrupole parameter tilde{q}=q/j^2, the authors analyze both geodesic and non-geodesic (fluid torus) precession frequencies, revealing a non-monotonic dependence on the spin $j$ with maxima at moderate values. The key result is that relativistic frame-dragging and classical quadrupole effects compete, producing local maxima in the precession frequency and enabling slow and fast rotators to exhibit similar observable frequencies, which helps explain the weak correlation between QPO frequencies and NS spin. The findings are illustrated with EOS-specific examples (including MIT bag) and are supported by a reproducible Mathematica notebook, making them directly applicable to modeling NS QPOs and constraining NS EoS.

Abstract

The vertical (Lense-Thirring) precession of the innermost accretion flows has been discussed as a sensitive indicator of the rotational properties of neutron stars (NSs) and their equation of state because it vanishes for a non-rotating star. In this work, we apply the Hartle-Thorne spacetimes to study the frequencies of the precession for both geodesic and non-geodesic (fluid) flows. We build on previous findings on the effect of the NS quadrupole moment, which revealed the importance of the interplay between the relativistic and classical precession. Because of this interplay, the widely used Lense-Thirring metric, linear in the NS angular momentum, is insufficient to calculate the behaviour of the precession frequency across an astrophysically relevant range of NS angular momentum values. We find that even for a moderately oblate NSs, the dependencies of the precession frequency on the NS angular momentum at radii within the innermost accretion region have maxima that occur at relatively low values of the NS angular momentum. We conclude that very different groups of accreting NSs -- slow and fast rotators -- can display the same precession frequencies. This may explain the lack of evidence for a correlation between the frequencies of the observed low-frequency quasiperiodic oscillations and the NS spin. In our work, we provide a full, general description of precession behaviour, and also examples that assume specific NS and quark star (MIT bag) equation of state. Our calculations are reproducible using the associated Wolfram Mathematica notebook.

Figures (5)

• Figure 1: Frequencies of orbital motion around rotating NSs. a) NS spacetime approximated using the LT metric. b) Spacetime of extremely compact NSs approximated using the Kerr limit of the HT metric. c) Spacetime of highly oblate NSs approximated by the HT metric. The upper panels show the Keplerian and (axisymmetric) epicyclic frequencies. The bottom panels show the precession frequencies. The full lines correspond to the frequencies of a test particle motion and slender tori. The colour-shaded regions indicate frequencies relevant to the non-geodesic flows of non-slender tori. The dashed lines in panel b indicate the frequencies of a test particle motion in the Kerr spacetime calculated using the appropriate coordinate transformation.
• Figure 2: Change in the orientation of the LT precession and behaviour of local frequency maxima shown for $j=0.35$. As in Figure \ref{['fig:freq2']}, the solid lines correspond to the frequencies of test particle motion and slender tori, while the colour-shaded regions indicate the frequencies relevant to non-geodesic flows. The full black circles indicate local maxima of the geodesic precession frequency, $\nu_{\mathrm{max}}^0$.
• Figure 3: LT precession frequency in LT and HT spacetimes. a) At the ISCO. b) Close to the ISCO. c) Close to the location of the maximum of the geodesic radial epicyclic frequency. d) At a larger radius, outside of the location of the maximum of the geodesic radial epicyclic frequency. The red colour corresponds to the LT metric. Lines in shades of blue correspond to the HT metric with varying $q$. The displayed full lines show the frequencies relevant to the test particle motion and slender tori. The colour-shaded regions indicate frequencies relevant to non-geodetic flows and cover all allowable torus thicknesses. The dotted line in panel a denotes the maximum frequency at the ISCO given by equation (\ref{['equation:jmax']}).
• Figure 4: Maximum and average value of the LT frequency. Left panel: Maximum of the frequency, $\nu_{\mathrm{MAX}}$, as a function of the angular momentum of the star. Middle panel: Frequency value averaged from $r_\mathrm{ISCO}$ to $r_\mathrm{2:1}$ as a function of the star's angular momentum. Right panel: Frequency averaged from $r_\mathrm{ISCO}$ to $r_\mathrm{3:2}$ as a function of the star's angular momentum. As in Figure \ref{['fig:lt']}, the red lines correspond to the LT metric, while the lines in shades of blue correspond to the HT metric and varying $q$. The colour-shaded regions then indicate the frequencies relevant to non-geodetic flows and cover the full range of allowable torus thicknesses.
• Figure 5: Maximum LT precession frequency as a function of angular momentum of the star based on specific EoS and a fixed gravitational mass, $M$ (top panels), or a fixed non-rotating mass, $M_0$ (bottom panels). Left panels: XMLSLZ(NL3) EoS. Middle panels: VGBCMR(D1M) EoS. Right panels: MIT bag EoS. The coloured solid lines show the values of $\nu_{\mathrm{MAX}}$ when the star's surface is below the ISCO, while the coloured dashed lines show the values of $\nu_{\mathrm{MAX}}$ when the surface is above the ISCO. The circles on the individual lines enable us to determine the radius at which the maximum displayed frequency is reached. For values of $j$ below that indicated by the black filled circle (or when there is no circle), the frequency corresponds to the edge of the relevant radial interval: the ISCO (solid lines) or the NS surface (dotted lines). For higher values of $j$, below the value indicated by the open circle, the frequency corresponds to the maximum above the edge. Above this value, it again corresponds to the ISCO for the solid lines and the NS surface for the dotted lines. The grey lines then indicate the value of the corresponding local geodesic maximum, $\nu_{\mathrm{max}}^0$.