The geometry of magnetic fields in plasma around magnetars, described by the Grad-Shafranov equation.Magnetar plasma geometry concerns the shape and configuration of magnetic fields in the plasma surrounding magnetars—neutron stars with extremely strong magnetic fields. The Grad-Shafranov equation describes the equilibrium of such plasma-magnetic field systems, balancing pressure gradients against magnetic forces. Understanding this geometry is crucial for explaining observed phenomena like pulsar wind nebulae and magnetar flares.
The Grad-Shafranov equation was formulated independently by Harold Grad and Vitaly Shafranov in the late 1950s for controlled fusion research, later adapted to astrophysical plasmas.
The Grad-Shafranov equation is a partial differential equation that solves for the poloidal flux function in axisymmetric plasma configurations. By specifying the plasma pressure and magnetic field profiles, the equation yields the magnetic field lines and current distributions. In the magnetar context, this model helps predict how the magnetic field twists and deforms the star's crust and how energy is released in bursts.
Applied in modeling magnetic confinement in fusion reactors (tokamaks) and in astrophysical simulations of neutron star magnetospheres.
The same equation used to design fusion reactors also explains the zebra-like patterns in radio emissions from the Crab pulsar.
| β | variable | Plasma beta — plasma beta |
| p | variable | Plasma pressure — pressure pascal |
| B | variable | Magnetic field strength — magnetic field tesla |
| π | constant | Mathematical constant — pi = 3.14159 |
| \rho_{\rm GJ} | variable | Goldreich-Julian charge density — charge density coulomb per cubic metre |
| \mathbf{\Omega} | variable | Angular velocity vector of the neutron star — angular velocity radian per second |
| \mathbf{B} | variable | Magnetic field vector — magnetic field tesla |
| c | constant | Universal constant, speed of light in vacuum — speed of light = 299792458 metre per second |
| 2\pi | constant | Mathematical constant, 2π — two pi = 6.283 |
| \cdot | operator | Scalar product of two vectors — dot product |












