Magnetically supramassive and hypermassive compact stars

TL;DR

This work demonstrates that strong internal magnetic fields in relativistic compact stars with mixed poloidal-toroidal configurations can significantly raise the maximum sustainable mass beyond non-magnetized TOV limits. Using the cocal code to compute stationary equilibria with EMV (electromagnetic vacuum) and FF (force-free magnetosphere) exterior conditions, the authors show magnetically supramassive configurations (exceeding the TOV maximum by up to ~$12\%$) and magnetically hypermassive configurations (up to ~$31\%$ excess) in carefully chosen parameter sequences. The results reveal distinct structural and mass–energy characteristics, including toroidal-field concentrations near the equator and exterior-field influence on field topology, while highlighting that hypermassive states are accessible in certain magnetic-field geometries and deformations. These magnetically supported equilibria offer valuable initial data for GRMHD simulations and provide theoretical benchmarks for interpreting potential observational signatures in gravitational waves and X-ray timing, though stability and broader parameter coverage remain important follow-up challenges.

Abstract

It is known that the mass of magnetized relativistic compact star is larger than that of non-magnetized one for the same equation of state and central density, albeit the excess of mass is sizable only if the magnetic fields are strong enough B~10^17-10^18G. Using our recently developed numerical code COCAL, we systematically compute such compact star solutions in equilibrium associated with mixed poloidal and toroidal magnetic fields, and show the magnetically supramassive solutions whose masses exceed by more than 10% of the maximum mass of the static and spherically symmetric solutions. For some extremely strong magnetic field configurations, we also obtain solutions more massive than the maximum mass of the uniformly rotating solutions at the Kepler (mass-shedding) limit, namely magnetically hypermassive solutions.

Figures (4)

• Figure 1: A (non-rotating) extremely magnetized supramassive compact star associated with mixed poloidal and toroidal magnetic field (EMV$^-$IV model) and a uniformly rotating (non-magnetized) compact star are compared for the models with the same axis ratio $R_z/R_0=0.6875$ and the central $(p/\rho)_{\rm c}=0.3$. First row, left panel: contours of $p/\rho$ (black closed curves), vector plots of poloidal magnetic field (orange arrows), color density map for the toroidal magnetic fields (red and blue), and the boundary of the magnetotunnel (green circles) for the magnetized star are shown. Middle panel: contours of $p/\rho$ for the rotating star are shown. The contours are drawn at $p/\rho=0.001,0.002,0.005,0.01,0.02,0.05,0.1,0.2$. Right panel: the profiles of $p/\rho$ for both stars are plotted along the normalized equatorial coordinate $x/R_0$, and a close-up of the region $x/R_0 \geq 0.8$ is shown in an inset. Second row, left panel: the contours of the components of electromagnetic 1-form $A_\phi$ (green curves), the contours of $A_t$ (dashed red (positive) purple (zero), blue (negative)) are shown for the magnetized star. The black closed curve is the surface of the star. Right panel: components of the magnetic fields, $B_{\rm pol} = F_{xy}$ (dashed purple curve) and $B_{\rm tor} = -F_{xz}$ (dark green curve) are plotted with respect to the normalized equatorial radius $x/R_0$. Third row, left two panels: the metric potentials for the magnetized star in $xz$ and $xy$ planes are shown in left and right panels, respectively, which are the contours of $\psi$ (green closed curves), the color density map for $\tilde{\beta}_y$ (red and blue), the contours of $h_{xz}$ (red and blue curves), and the surface of the star (black closed curve). Right two panels: the same as the left two panels but for the uniformly rotating (non-magnetized) star. The coordinate lengths in all panels are normalized by the equatorial radius $R_0$ of the magnetized star, which is about $4.5\%$ smaller than that of the rotating star.
• Figure 2: Solutions for magnetically hypermassive compact stars associated with an electromagnetic vacuum outside (EMV$^+$ model) and with a force-free magnetosphere outside (FF model). First row: the EMV$^+$ model with $\Lambda_1=0.432$, $(p/\rho)_{\rm c}=0.2558203$ ($M_0=1.5$), and $R_z/R_0=0.7$. Left panel: same as the left panel of the first row of Fig.\ref{['fig:MNS-RNS']}. Middle panel: the normalized rest mass density $\rho/\rho_c$ (red curve) and the normalized angular velocity $\Omega/\Omega_{\rm c}$ are plotted along the equatorial radius ($x$-axis). Right panel: same as the right panel of the second row of Fig. \ref{['fig:MNS-RNS']}. Second row: the panels are the same as the first row but for the FF model with $(p/\rho)_{\rm c}=0.2558203$ ($M_0=1.5$), and $R_z/R_0=0.55$. Third row: first and second panels from the left are those of the EMV$^+$ model of the first row, and the third and fourth the FF model of the second row. First panel from left: same as the first (left) panel of the third row of Fig. \ref{['fig:MNS-RNS']}. Second panel: same as the left panel of the second row of Fig. \ref{['fig:MNS-RNS']}. Third and fourth panels: same as the first and the second panels, respectively.
• Figure 3: Sequence of solutions for EMV$^-$I--IV models. Top panel: plots of the ADM mass $M$ with respect to the ratio $(p/\rho)_{\rm c}$ of magnetized compact stars (plus-crosses, EMV$^-$). A solid (red, TOV) and a dashed (blue, UR) curves correspond, respectively, to the solution sequences of TOV equation and to the maximally rotating (Kepler rotation) models of uniformly rotating relativistic stars with the same EOS. The maximum mass of each curve is shown with a filled circle, and the horizontal dashed lines are drawn at the maximum mass of the TOV solutions and at the 10% and 20% higher values from bottom to top, respectively. Bottom panel: plots of the ratio of magnetic to gravitational energy ${\cal M}/|{\cal W}|$ with respect to $(p/\rho)_{\rm c}$ for the same EMV$^-$I--IV models. In both panels, data points (plus-crosses) from bottom to top at the same $(p/\rho)_{\rm c}$ correspond to EMV$^-$I--IV models, respectively.
• Figure 4: The same as Fig. \ref{['fig:M_prhoc']} but for the EMV$^+$ (diamonds purple) and FF (plus-crosses) models that include hypermassive solutions. In top panel, the horizontal dashed lines are drawn at values of the maximum mass of the TOV solutions (red) and at the maximum mass of the uniformly rotating solutions (blue).