Mass-radius relationship and gravitational wave emission from magnetized spheroidal quark stars

TL;DR

This work addresses how strong magnetic fields and color superconductivity influence the structure and gravitational-wave signals of magnetized quark stars. By extending the MIT Bag model to include pressure anisotropy and using the $\gamma$-metric to capture oblate deformations, the authors compute mass-radius relations, ellipticity, redshift, quadrupole moments, tidal deformability, and continuous GW amplitudes $h_0$ for MSQM and MCFL phases with density-dependent magnetic fields. They find that magnetic-field-induced anisotropy and pairing stiffen the EoS, yielding more massive, larger, and more deformed stars, with MCFL configurations often dominating in mass and GW output; observable signatures arise in $Z_{rs}$, $\Lambda$, $Q$, and $h_0$ potentially detectable by ET/CE and testable with NICER-like constraints. The results offer multimessenger pathways to probe dense-quark matter EoS, magnetic field profiles, and color superconductivity in compact stars, while outlining validity limits of the $\gamma$-metric for extreme deformations. Future work could extend to rotation, finite temperature, and color-magnetic interactions to refine predictions for gravitational waves and X-ray observables.

Abstract

In this work, we investigate the structure and gravitational wave (GW) signatures of strongly magnetized, oblate spheroidal quark stars by employing an anisotropic equation of state (EoS) derived from the MIT Bag model, extended to include the effects of density-dependent strong magnetic fields and the resulting pressure anisotropy arising from the breaking of spatial symmetry. Both magnetized strange quark matter (MSQM) and magnetized color-flavor locked (MCFL) phases are examined within the framework of the $γ$-metric formalism, which captures the deviation from spherical symmetry. We compute the mass-radius relation, ellipticity, gravitational redshift, mass quadrupole moment and tidal deformability for representative bag constants of $\rm{65\,MeV/fm^3}$ and $\rm{75\,MeV/fm^3}$. Using the obtained quadrupole moments, we further estimate the continuous gravitational wave strain amplitude ($h_{0}$) for isolated deformed rotating quark stars. Our results indicate that density-dependent strong magnetic fields and color superconductivity can significantly alter stellar compactness and yield gravitational wave signals, potentially detectable by next generation observatories like the Einstein Telescope and Cosmic Explorer.

Figures (13)

• Figure 1: Energy per baryon $E/A$ as a function of the baryon density ratio $n_{B}/n_{0}$ and total perpendicular pressure $P_{\perp T}$ for a central magnetic field of $\rm{5 \times 10^{17}}\,G$. The upper row panels correspond to $B_{bag}=\rm{65\,MeV/fm^{3}}$ and $B_{bag}=\rm{75\,MeV/fm^{3}}$, respectively. The upper row panels show the variation of $E/A$ with the baryon density ratio $n_{B}/n_{0}$, while the lower row panels depict the corresponding variation with the total perpendicular pressure $P_{\perp T}$ for the same bag constants. Results are shown for different phases of quark matter, namely strange quark matter (SQM), color-flavor locked (CFL) matter with pairing gaps $\Delta=\rm{30\,MeV}$ and $\Delta=\rm{50\,MeV}$, magnetized strange quark matter (MSQM), and magnetized color-flavor locked (MCFL) matter with the same pairing gaps. The horizontal dashed line denotes the energy per baryon of the ${^{56}{Fe}}$ nucleus, and the black dots mark the points where the total perpendicular pressure vanishes.
• Figure 2: Variation of pressure $P$ with baryon number density $n_{B}$ under different physical conditions. The upper row panels correspond to the magnetized strange quark matter (MSQM) phase and show the behavior of the total parallel pressure $P_{\parallel T}$ and total perpendicular pressure $P_{\perp T}$ for central magnetic field strengths $B_{cen}=\rm{5 \times 10^{17}\,G}$ and $\rm{1\times 10^{18}\,G}$ at fixed bag constants $B_{bag}=\rm{65\,MeV/fm^3}$ and $\rm{75\,MeV/fm^3}$, respectively. The black solid line represents the non-magnetized case ($B=0$). The lower row panels correspond to the magnetized color-flavor locked (MCFL) phase and illustrate the effect of different pairing gaps, $\Delta=\rm{30\,MeV}$ and $\Delta=\rm{50\,MeV}$, on both $P_{\parallel T}$ and $P_{\perp T}$ for the same bag constants at a central magnetic field strength of $B_{cen}=\rm{5\times 10^{17}}\,G$.
• Figure 3: Variation of central pressure with central magnetic field strength for a fixed central baryon number density $n_{B}=2.5\, n_{0}$ and bag constant $B_{bag}=\rm{65 \; MeV/fm^{3}}$.
• Figure 4: Variation of the stellar mass $M$ and corresponding equatorial radius $R$ with central magnetic field strength $B_{cen}$ for a fixed central baryon density $n_{B0}=\rm{0.5 \,fm^{-3}}$. The left panel shows the dependence of mass on $B_{cen}$, while the right panel presents the corresponding variation of equatorial radius. Results are shown for magnetized strange quark matter (MSQM) and magntized color-flavor locked (MCFL) phases with bag constants $B_{bag}=\rm{65\,MeV/fm^3}$ and $B_{bag}=\rm{75\,MeV/fm^3}$, and pairing gaps $\Delta=\rm{30\,MeV}$ and $\Delta=\rm{50\,MeV}$.
• Figure 5: Variation of the stellar mass $M$ with equatorial radius $R$ for different central magnetic field strengths $B_{cen}=\rm{[5 \times 10^{17}, 7\times 10^{17}, 1\times 10^{18}, 1.5 \times 10^{18}]\,G}$. The top row shows the mass-radius relations for the magnetized strange quark matter (MSQM) phase with $B_{bag}=\rm{65\,MeV/fm^3}$ (left) and $B_{bag}=\rm{75\,MeV/fm^3}$ (right), together with the non-magnetized case ($B=0$). The middle row presents the corresponding results for the magnetized color-flavor locked (MCFL) phase for the same bag constants and a pairing gap of $\Delta=\rm{30\,MeV}$. The lower row compares the MSQM and MCFL phases. The dots indicate the maximum mass configurations in each case.