Gravitational Redshift for Rapidly Rotating Neutron Stars

TL;DR

The paper addresses errors in the gravitational redshift treatment within the Oblate Schwarzschild approximation for rapidly rotating neutron stars. It introduces an empirical ZAM redshift correction, fits it across a large EOS library, and integrates it into an Improved OS framework that uses Schwarzschild ray-tracing with a corrected redshift factor. The main result is that OS-induced flux errors are typically below 1% for NICER-like pulsars, but can grow to a few percent at spin frequencies near 600 Hz, justifying the adoption of the corrected method in future high-precision pulse-profile analyses. This work enhances the reliability of NS mass-radius inferences from X-ray timing and spectra, particularly for faster rotators and upcoming missions with higher precision.

Abstract

The Oblate Schwarzschild (OS) approximation is a method often used to compute the flux of X-rays emitted from a rapidly rotating neutron star. In this approximation, the oblate shape of the rotating star is embedded in the Schwarzschild metric, which is used to compute the redshift of photon energies as they propagate from the star to the telescope. In this paper, we demonstrate that there are small errors introduced by the standard treatment of photon redshift in the OS approximation and provide a simple method to correct these errors. These errors are constant in phase, so this results in a constant absolute reduction in the flux. For PSR J0740+6620, the most rapidly spinning of the pulsars observed by NICER, we estimate the flux errors are less than 1\%, which is an order of magnitude smaller than the uncertainty in the distance, so this does not affect the mass and radius constraints found for this pulsar. The errors for the other pulsars observed by NICER are even smaller. However, this correction should be included when analyzing data for more rapidly rotating X-ray pulsars with spin frequencies near 600 Hz.

Figures (3)

• Figure 1: Redshift of ZAM photons as a function of latitude on the oblate stellar surface for the EOS DS(CMF)-1 star (upper panel) and the EOS A star (lower panel) whose properties are given in Table \ref{['tab:fiducial']}. The star's equator corresponds to $\cos\theta=0$ and the spin poles are at $\cos\theta=1$. The solid black line is the exact value of the ZAM redshift computed using rns and Equation (\ref{['eq:zam-z']}). The dashed yellow line corresponds to the approximation described in Section \ref{['sec:ZOS']}. The solid blue and green curves correspond to the Schwarzschild and Kerr metric approximations respectively. Quasi-Kerr approximations are shown with dotted curves. The green dotted curve shows the quasi-Kerr approximation when the correct values of $a$ and $q$ are used. The lower panel shows the quasi-Kerr approximation when overly large values of $a$ and $q$ are used.
• Figure 2: ZAM redshift correction, $\chi$, (upper panel) and the residual fractional errors (lower panel) in $z_0$ as functions of the dimensionless parameters $M/R$ and $\overline{\Omega}$. The surface represents the function defined by Equation \ref{['eq:fit']} with $n=3$ and $m=2$. Colored points correspond to the individual calculations for each stellar model in the library.
• Figure 3: Lightcurves (upper panels) for monochromatic light emitted from $\theta=41^\circ$ and observed at $\zeta=20^\circ$, for stars constructed with EOS L, with $M=1.4 M_\odot$ and spin frequencies of 300 Hz (left) and 600 Hz (right). The exact light curves obtained from raytracing on the correct metric 2007ApJ...654..458Cadeau are shown with solid lines. The original and corrected OS approximations are shown with dotted and dash-dotted curves. The SD approximation is shown with a dash-dotted curve with 3 dots for comparison. The percent differences between the two OS approximations with respect to the exact lightcurve is shown in the lower panel. Due to the very large errors in the SD approximation, the percent error for SD is not shown for 600 Hz.