Equilibriums of extremely magnetized compact stars with force-free magnetotunnels

TL;DR

This work extends GRMHD equilibrium modeling of ultramagnetized compact stars by incorporating force-free magnetospheres and a new differential-rotation law, enabling self-consistent solutions with interior ideal-MHD regions and exterior force-free/ magnetospheric regions. The formulation uses a master potential Υ to relate electromagnetic potentials and current, ensuring smooth coupling across regions, and introduces a differential rotation Ω=Ω_c Ξ'(A_φ) to support rotating magnetospheres. Numerical solutions reveal that extreme mixed poloidal-toroidal fields concentrate near the equatorial surface and can expel matter to form a toroidal magnetotunnel, with four model families (EV-MT-UR, EV-MT-DR, MS-MT-DR, MS-DR) exploring vacuum and magnetosphere exteriors under uniform and differential rotation; toroidal fields may extend into the magnetosphere in some cases (MS-DR), while stability and astrophysical relevance remain open questions. The results provide initial data frameworks for GRMHD simulations and offer insights into the extreme magnetic-field limits of compact-star equilibria, with implications for magnetar physics and merger remnants, albeit with caveats about stability and realism of such extreme configurations.

Abstract

We present numerical solutions for stationary and axisymmetric equilibriums of compact stars associated with extremely strong magnetic fields. The interior of the compact stars is assumed to satisfy ideal magnetohydrodynamic (MHD) conditions, while in the region of negligible mass density the force-free conditions or electromagnetic vacuum are assumed. Solving all components of Einstein's equations, Maxwell's equations, ideal MHD equations, and force-free conditions, equilibriums of rotating compact stars associated with mixed poloidal and toroidal magnetic fields are obtained. It is found that in the extreme cases the strong mixed magnetic fields concentrating in a toroidal region near the equatorial surface expel the matter and form a force-free toroidal magnetotunnel. We also introduce a new differential rotation law for computing solutions associated with force-free magnetosphere, and present other extreme models without the magnetotunnel.

Figures (4)

• Figure 1: Solutions for uniformly rotating extremely magnetized compact stars associated with an electromagnetic vacuum outside and magnetotunnel (EV-MT-UR type). The panels in the first row correspond to the rapidly rotating model EV-MT-UR-1. The left panel: contours of $p/\rho$ (black closed curves), the poloidal magnetic field (orange arrows), color density map for the toroidal magnetic fields (red and blue), and the boundary of the magnetotunnel (green circles) are shown. The contours of $p/\rho$ are drawn at $p/\rho=0.001,0.002,0.005,0.01,0.02,0.05,0.1$. The middle panel: the rest mass density $\rho/\rho_c$ (red curve) and the angular velocity $\Omega/\Omega_c$ are plotted along the equatorial radius ($x$-axis). An inset is a close-up of $\rho/\rho_c$ near the surface. The 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 along the equatorial radius ($x$-axis). The panels in the second row are the same as the first row but for the slowly rotating model EV-MT-UR-2. In the third row, the first panel from the left, the metric potentials are shown, 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). In the second panel from the left, the components of electromagnetic 1-form are shown, which are the contours of $A_\phi$ (green curves), the contours of $A_t$ (dashed red (positive), purple (zero), blue (negative)), and the surface of the star (black closed curve) for the model EV-MT-UR-1. The third and fourth panels of the third row are the same as the first and the second panels, respectively, but for the model EV-MT-UR-2.
• Figure 2: Same as Fig.\ref{['fig:EV-MT-UR']} but for differentially rotating and extremely magnetized compact stars associated with a magnetosphere and a magnetotunnel, MS-MT-DR-1 (rapidly rotating model) and MS-MT-DR-2 (slowly rotating model).
• Figure 3: Same as Fig.\ref{['fig:EV-MT-UR']} but for differentially rotating and extremely magnetized compact stars associated with an electromagnetic vacuum outside and a magnetotunnel, EV-MT-DR-1 (rapidly rotating model) and EV-MT-DR-2 (slowly rotating model).
• Figure 4: Same as Fig.\ref{['fig:EV-MT-UR']} but for differentially rotating and extremely magnetized compact stars associated with a magnetosphere, MS-DR-1 (supramassive model) and MS-DR-2 (normal mass model), whose toroidal magnetic fields are distributed across the stellar support and magnetosphere.