Fundamental modes of rotating neutron stars with various degrees of differential rotation in dynamical spacetimes

TL;DR

This study addresses the frequencies of fundamental oscillation modes in rotating neutron stars with differential rotation using dynamical spacetime. It employs 2D axisymmetric GR hydrodynamics with a polytropic EOS and a $j$-constant rotation law across five differential-rotation degrees $\tilde{A}$, exciting $F$- and $^2f$-modes and extracting their frequencies to build linear relations as functions of $M/R$ and $T/|W|$. The key contributions are (i) demonstrating near-linear scaling of $f_F$ and $f_{^2f}$ with compactness and radius ratio across $\tilde{A}$, and (ii) providing explicit linear fits $f^{\mathrm{pred}}_{F}(T/|W|) \approx 1.45-3.42\frac{T}{|W|}$ and $f^{\mathrm{pred}}_{^2f}(T/|W|) \approx 1.56-0.65\frac{T}{|W|}$ with small residuals, improving over Cowling-based estimates. These results advance gravitational-wave asteroseismology of proto-neutron stars and merger remnants by incorporating differential rotation and dynamical spacetime, and establish a foundation for future inclusion of realistic EOS, 3D effects, and magnetic fields.

Abstract

Violent astrophysical events, including core-collapse supernovae and binary neutron star mergers, can result in rotating neutron stars with diverse degrees of differential rotation. Oscillation modes of these neutron stars could be excited and emit strong gravitational waves. Detecting these modes may provide information about neutron stars, including their structures and dynamics. Hence, dynamical simulations were employed to construct relations for quantifying the oscillation mode frequency in previous studies. Specifically, linear relations for the frequencies of fundamental $l=0$ quasi-radial mode $f_{F}$ and fundamental $l=2$ quadrupolar mode $f_{^2f}$ were constructed by simulations with the Cowling approximation. Nevertheless, these relations can overestimate $f_{F}$ and underestimate $f_{^2f}$ up to $\sim 30\%$. Furthermore, it has yet to be fully studied how the degree of differential rotation affects $f_{F}$ and $f_{^2f}$. Here, for the first time, we consider both various degrees of differential rotation $\Tilde{A}$ and dynamical spacetime to construct linear relations for quantifying $f_{F}$ and $f_{^2f}$. Through 2D axisymmetric simulations, we first show that both $f_{F}$ and $f_{^2f}$ scale almost linearly with the stellar compactness $M/R$ for different values of $\Tilde{A}$. We also observe the quasi-linear relations for both $f_{F}$ and $f_{^2f}$ with the kinetic-to-binding energy ratio $T/|W|$ for different $\Tilde{A}$ values. Finally, we constructed linear fits that can quantify $f_{F}$ and $f_{^2f}$ by $T/|W|$. Consequently, this work updated the relations for the fundamental modes of rotating neutron stars with differential rotations in dynamical spacetime.

Figures (3)

• Figure 1: Plot of fundamental mode frequency $f$ against stellar compactness $M/R$ (left panel) and radius ratio $r_\mathrm{p}/r_\mathrm{e}$ (right panel), where $M$ is the stellar mass, $R$ is the circumferential radius, $r_\mathrm{p}$ is the polar radius and $r_\mathrm{e}$ is the equatorial radius. The data points are arranged into 5 sequences with $\Tilde{A} \in \{0.0,1.0,2.0,3.0,4.0\}$, where $\Tilde{A}$ is the degree of differential rotation. $\Tilde{A}=0.0$ refers to uniformly rotating models. Data points connected by solid lines denote the data for the frequency of fundamental $l=0$ quasi-radial mode $f_{F}$ while the points connected by dashed lines denote the data for the frequency of fundamental $l=2$ quadrupolar mode $f_{^2f}$. Both $f_{F}$ and $f_{^2f}$ increase approximately linearly with $M/R$ and $r_\mathrm{p}/r_\mathrm{e}$, with the slopes varying only slightly as $\Tilde{A}$ changes. Hence, this demonstrates a quasi-linear relation between fundamental mode frequency, stellar compactness and radius ratio for rotating neutron stars with different degrees of differential rotation.
• Figure 2: Plots of fundamental mode frequencies $f$ (top panels) and the frequency deviations between the simulation data $f^\mathrm{data}$ and the predictions by linear fits $f^\mathrm{pred}$ (bottom panels) against kinetic-to-binding energy ratio $T/|W|$. In particular, we plot fundamental $l=0$ quasi-radial mode frequency $f_{F}$ (top left panel), deviation of $f_{F}$ (bottom left panel), fundamental $l=2$ quadrupolar mode frequency $f_{^2f}$ (top right panel), and deviation of $f_{^2f}$ (bottom right panel) against $T/|W|$. The data points are arranged into 5 sequences with $\Tilde{A} \in \{0.0,1.0,2.0,3.0,4.0\}$, where $\Tilde{A}$ is the degree of differential rotation. $\Tilde{A}=0.0$ refers to uniformly rotating models. Both $f_{F}$ and $f_{^2f}$ for different $\Tilde{A}$ decrease almost linearly with $T/|W|$. Hence, we perform linear regressions using our simulation data to obtain 2 linear relations of $f^\mathrm{pred}_{F}(T/|W|)$ in Eq. (\ref{['eqn6']}) and $f^\mathrm{pred}_{^2f}(T/|W|)$ in Eq. (\ref{['eqn7']}) respectively. We find that only a slight deviation between our simulation data and our linear fits with $f^\mathrm{data}_{F}/f^\mathrm{pred}_{F} - 1 \lesssim 1 \%$ and $f^\mathrm{data}_{^2f}/f^\mathrm{pred}_{^2f} - 1 \lesssim 2 \%$. We also compare our linear fits with the data points of the previous study of 2006MNRAS.368.1609D (purple diamonds), in which the fundamental modes of uniformly rotating and differentially rotating ($\Tilde{A}=1.0$) neutron stars in dynamical spacetime were considered. We find that only minor deviations between the simulation data in 2006MNRAS.368.1609D and our linear fits with $f^\mathrm{data}/f^\mathrm{pred}- 1 \lesssim 2 \%$ for both fundamental modes. Therefore, we have constructed linear relations that can quantify $f_{F}$ and $f_{^2f}$ for rotating neutron stars with differential rotations in dynamical spacetime.
• Figure A1: The power spectral density (PSD) of the $r$-component $v_r$ with $l=0$ perturbation imposed (left panels) and the $\theta$-component $v_\theta$ with $l=2$ perturbation imposed (right panels) of the three-velocity field in arbitrary units for the non-rotating model A0 (top panels) and model E25 (bottom panels). We first identify the peaks in the PSDs that correspond to the fundamental $l=0$ quasi-radial mode frequency $f_F$ and the fundamental $l=2$ quadrupolar mode frequency $f_{^2f}$ for our model A0 (top panels) by comparing them with the well-tested mode frequencies reported by Dimmelmeier et al. (dotted lines). After identifying the peaks of $f_F$ and $f_{^2f}$ in the PSDs, we track how $f_F$ and $f_{^2f}$ change in each sequence as the kinetic-to-binding energy ratio $T/|W|$ increases, as illustrated in the example of model E25 (bottom panels).