Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                

Fully general relativistic simulations of rapidly rotating quark stars: Oscillation modes and universal relations

Kenneth Chen and Lap-Ming Lin
Phys. Rev. D 108, 064007 – Published 6 September 2023
PDFHTMLExport Citation
Abstract
Authors
Article Text
  • INTRODUCTION
  • FORMULATION AND NUMERICAL METHODS
  • NUMERICAL RESULTS
  • CONCLUSION
  • ACKNOWLEDGMENTS
    • References

    Abstract

    Numerical simulation of strange quark stars (QSs) is challenging due to the strong density discontinuity at the stellar surface. In this paper, we report successful simulations of rapidly rotating QSs and study their oscillation modes in full general relativity. Building on top of the numerical relativity code einstein toolkit, we implement a positivity-preserving Riemann solver and a dustlike atmosphere to handle the density discontinuity at the surface. The robustness of our numerical method is demonstrated by performing stable evolutions of rotating QSs close to the Keplerian limit and extracting their oscillation modes. We focus on the quadrupolar l=|m|=2 f-mode and study whether they can still satisfy the universal relations recently proposed for rotating neutron stars (NSs). We find that two of the three proposed relations can still be satisfied by rotating QSs. For the remaining broken relation, we propose a new relation to unify the NS and QS data by invoking the dimensionless spin parameter j. The onsets of secular instabilities for rotating QSs are also studied by analyzing the f-mode frequencies. Same as the result found previously for NSs, we find that QSs become unstable to the Chandrasekhar-Friedman-Schutz instability when the angular velocity of the star Ω≈3.4σ0 for sequences of constant central energy density, where σ0 is the mode frequency of the corresponding nonrotating configurations. For the viscosity-driven instability, we find that QSs become unstable when j≈0.881 for both sequences of constant central energy density and constant baryon mass. Such a high value of j cannot be achieved by realistic uniformly rotating NSs before reaching the Keplerian limit. The critical value for the ratio between the rotational kinetic energy and gravitational potential energy of rotating QSs for the onset of the instability, when considering sequences of constant baryon mass, is found to agree with an approximate value obtained for homogeneous incompressible bodies in general relativity to within 4%.

    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    13 More
    • Received 4 July 2023
    • Accepted 16 August 2023

    DOI:https://doi.org/10.1103/PhysRevD.108.064007

    © 2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    General relativityGravitational wave sourcesNuclear astrophysicsNuclear matter in neutron stars
    1. Physical Systems
    Neutron stars & pulsars
    1. Techniques
    Numerical relativity
    Gravitation, Cosmology & AstrophysicsNuclear Physics

    Authors & Affiliations

    Kenneth Chen* and Lap-Ming Lin†

    • Department of Physics, The Chinese University of Hong Kong, Hong Kong, China

    • *kchen@link.cuhk.edu.hk
    • †lmlin@cuhk.edu.hk

    Article Text (Subscription Required)

    Click to Expand

    References (Subscription Required)

    Click to Expand
    Issue

    Vol. 108, Iss. 6 — 15 September 2023

    Reuse & Permissions
    Access Options
    Author publication services for translation and copyediting assistance advertisement

    Authorization Required

    Log In
    Other Options
    ×

    Images

    • Figure 1
      Figure 1

      Gravitational mass is plotted against radius for nonrotating NSs modeled by the SFHo EOS and QSs modeled by four MIT bag models. Points labeled by “Seqs” correspond to the nonrotating configurations in the sequences of constant baryon mass or constant central density we used in the study of rotating stars.

      Reuse & Permissions
    • Figure 2
      Figure 2

      The spin parameter j is plotted against the angular velocity Ω normalized by the corresponding maximum rotation limit Ωmax for constant baryon mass sequences. In the figure legend, each sequence is labeled by the EOS model and the (fixed) value of the baryon mass of the sequence. For NSs, the maximal rotational frequency is its Keplerian limit, ΩK=Ωmax; but for QSs, Ωmax is larger than ΩK by about 2% [66] typically. The inset enlarges the two pairs of degenerate models of the MIT1 2M⊙ sequence in the sense that each pair has the same rotational frequency. The data set contains 115 models, including 21 SFHo models and 41 MIT1, 10 MIT2, 18 MIT3, and 25 MIT4 QS models.

      Reuse & Permissions
    • Figure 3
      Figure 3

      The spin parameter j is plotted against the angular velocity Ω normalized by the f-mode frequency σ0 of the corresponding nonrotating stars for constant central energy density sequences. In the figure legend, each sequence is labeled by the EOS model and the baryon mass of the nonrotating star in the sequence. The data set contains 52 models, including 6 SFHo models and 37 MIT1 and 9 MIT4 QS models.

      Reuse & Permissions
    • Figure 4
      Figure 4

      Evolutions of the percentage changes of the total baryon mass of a nonrotating QS for five different combinations of the Riemann solvers and reconstruction methods using a medium grid resolution Δx=0.16 (≈236  m) at the finest refinement level. The default scheme PP+PPM employed in this study can preserve the mass conservation to high accuracy, about 0.034% at t≈9.85  ms.

      Reuse & Permissions
    • Figure 5
      Figure 5

      L2-norm of the Hamiltonian constraint violation ∥H∥ in the evolutions of a nonrotating QS for three different grid resolutions. The inset plots the stable plateau values ∥H∥s of the constraint violation and demonstrates linear-order convergence for ∥H∥s.

      Reuse & Permissions
    • Figure 6
      Figure 6

      Fourier spectra of the rest-mass density for the evolutions of a nonrotating QS using three different grid resolutions. The vertical dashed lines indicate the frequencies of the radial oscillation modes, from the fundamental F0 mode to the tenth overtone F10, determined by the perturbative normal-mode analysis. The inset enlarges the region between F3 and F10.

      Reuse & Permissions
    • Figure 7
      Figure 7

      Spectra around the first peak in Fig. 6. The smooth black curve (without data points) represents the slope of a quadratic curve fit to the high-resolution result (Δx=0.12). The fundamental mode frequency 2745 Hz obtained by the position at which the slope passes through zero agrees to the known normal-mode value (indicated by the vertical dashed line at F0=2778  Hz) to about 1.2%.

      Reuse & Permissions
    • Figure 8
      Figure 8

      L2-norms of the Hamiltonian constraint violations ∥H∥ for a sequence of 2M⊙ baryon mass MIT1 QSs are plotted against time. The rotational frequencies of the chosen models span from 300 to 1200 Hz, where the maximal rotation limit of the sequence is near 1228 Hz. The results are obtained using the same resolution Δx=0.16.

      Reuse & Permissions
    • Figure 9
      Figure 9

      Snapshots of the rest-mass density in the first quadrant of the x–z plane for two MIT1 QS models with rotational frequencies 300 Hz (top left) and 1200 Hz (top right). The density profiles of the two models along the x axis are plotted in linear (middle panels) and logarithmic (bottom panels) scales.

      Reuse & Permissions
    • Figure 10
      Figure 10

      Plot of the velocity profile vy along the direction θ=π/4 for the 1200 Hz rotating model studied in Fig. 9 at t=0, 4.93 ms, and 9.85 ms on the x–z plane.

      Reuse & Permissions
    • Figure 11
      Figure 11

      Fourier spectra of fluid variables ρ, vr, vθ, and vϕ of the sequence of MIT1 QSs with constant baryon mass 2M⊙ rotating at (300, 450, 600) Hz. The gray dashed lines label the quadrupole fundamental mode f0=1897  Hz, the first pressure mode p0=7868  Hz, and the fundamental quasiradial mode F0=2778  Hz of the corresponding nonrotating model. In each panel, two red (blue) dashed lines track the m=±2 f-modes (p-modes).

      Reuse & Permissions
    • Figure 12
      Figure 12

      Fourier spectra of fluid variables ρ, vr, vθ, and vϕ of the sequence of MIT1 QSs with constant baryon mass 2M⊙ rotating at (900, 1100, 1200, 1225) Hz. Similar to Fig. 11, the red and blue dashed lines track the m=±2 f- and p-modes. The green dotted line in each panel tracks the position of twice the rotation frequency of the star. The maximal rotation rate of this sequence is about 1228 Hz. See text for the identification of the onsets of CFS and viscosity-driven instabilities from the spectra.

      Reuse & Permissions
    • Figure 13
      Figure 13

      Plot of Σ^i [see Eq. (24)] against the scaled angular velocity Ω^ for a total of 167 star models, including 27 SFHo NSs, and 78 MIT1, 10 MIT2, 18 MIT3, and 34 MIT4 QSs. The predictions from Eq. (23) [see also Eq. (6) in 54] for the counterrotating and corotating f-modes are given by the lower and upper gray lines, respectively.

      Reuse & Permissions
    • Figure 14
      Figure 14

      Plot of the scaled mode frequencies σi/σ0 observed in the inertial frame for sequences of constant central energy density. Data points contain the same models as in Fig. 3. The predictions from Eq. (25) [see also Eq. (4) in 54] for the counterrotating and corotating f-modes are given by the lower and upper gray lines, respectively. The purple horizontal dashed line represents the zero-frequency line, on which the counterrotating mode becomes unstable to the CFS instability.

      Reuse & Permissions
    • Figure 15
      Figure 15

      Plot of the scaled mode frequencies σc/σ0 observed in the rotating frame for sequences of constant baryon mass. Data points contain the same models as in Fig. 2. The predictions from Eq. (26) [see also Eq. (5) in 54] for the counterrotating and corotating f-modes are given by the upper and lower gray lines, respectively. The purple horizontal dashed line represents the zero-frequency line, on which the corotating mode becomes unstable to the viscosity-driven instability.

      Reuse & Permissions
    • Figure 16
      Figure 16

      Normalized frequency Σ˜c [Eq. (27)] is plotted against the spin parameter j in the rotating frame. It contains both constant central density and constant baryon mass sequences, including 167 models as in Fig. 13. The quadratic fitting curve [Eq. (28)] crosses the zero-frequency point at j≈0.881.

      Reuse & Permissions
    • Figure 17
      Figure 17

      Plot of the scaled corotating mode frequency σc/σ0 in the rotating frame against the normalized energy ratio λ=(T/|W|)/(T/|W|)crit for constant baryon mass sequences. The data set contains 94 QS models used in Fig. 15. The fitting curve [Eq. (30)] crosses the zero-frequency line at λ≈1.04.

      Reuse & Permissions
    • Figure 18
      Figure 18

      Plot of the scaled corotating mode frequency σc/σ0 in the rotating frame against the eccentricity ζ for constant baryon mass sequences. The data set contains 94 QS models used in Fig. 15. The fitting curve [Eq. (31)] crosses the zero-frequency line at ζ≈0.842.

      Reuse & Permissions
    • Figure 19
      Figure 19

      Plot of scaled frequency Mp0 of nonrotating models against compactness χ=M/R for the class of MIT bag EOSs with the square of the speed of sound css ranging from 1/10 to 1. The gray fitting curves are based on Eq. (32).

      Reuse & Permissions
    • Figure 20
      Figure 20

      Plot of p^i±−GΩ¯2 against χ/cssΩ¯ for constant baryon mass sequences of rotating QSs. The fitting relations [Eq. (33)] for the m=±2 p-mode frequencies pi± observed in the inertial frame are given by the solid lines.

      Reuse & Permissions
    ×

    Sign up to receive regular email alerts from Physical Review D

    Log In

    Cancel
    ×

    Search

    Article Lookup

    Paste a citation or DOI
    Enter a citation
    ×