University of Illinois at Urbana-Champaign
Differential rotation in stars generates toroidal magnetic fields whenever an initial seed poloidal field is present. The resulting magnetic stresses, along with viscosity, drive the star toward uniform rotation. This magnetic braking has important dynamical consequences in many astrophysical contexts. For example, merging binary neutron stars can form "hypermassive" remnants supported against collapse by differential rotation. The removal of this support by magnetic braking induces radial fluid motion, which can lead to delayed collapse of the remnant to a black hole. We explore the effects of magnetic braking and viscosity on the structure of a differentially rotating, compressible star. The star is idealized as a differentially rotating, infinite cylinder supported initially by a polytropic equation of state. The gas is assumed to be infinitely conducting and our calculations are performed in Newtonian gravitation. Though highly idealized, our model allows for the incorporation of magnetic fields, viscosity, compressibility, and shocks with minimal computational resources in a 1+1 dimensional Lagrangian MHD code. Our evolution calculations show that magnetic braking can lead to significant structural changes in a star, including quasistatic contraction of the core and ejection of matter in the outermost regions to form a wind or an ambient disk. These calculations serve as a prelude and a guide to more realistic MHD simulations in full 3+1 general relativity.
For a preliminary discussion on the effects of magnetic braking and viscous damping, see:
Shapiro (2000), Differential Rotation in Neutron Stars: Magnetic Braking
and Viscous Damping, ApJ, 544, 397 (astro-ph/0010493)
Cook, Shapiro & Stephens (2004), Magnetic Braking and Viscous Damping of
Differential Rotation in Cylindrical Stars, ApJ, 599, 1272 (astro-ph/0310304)
Liu & Shapiro (2004), Magnetic Braking in Differentially Rotating,
Relativistic Stars, PRD 69, 044009 (astro-ph/0312038)
We first present the results for a nearly incompressible star, where the polytropic index is n=0.001. The solution for an incompressible n=0 star is analytic, which we use as a check of our code. We demonstrate that adjusting the strength of the initial magnetic field changes only the timescale over which magnetic braking occurs, not the energy stored in the field at maximum twisting.
Here we present two simulations with a softer equation of state. The first one is nonviscous, followed by a viscous case.
We conclude the investigation with a simulation of an n=5 star. The plots are similar to the n=3 case, with the addition of a movie showing the evolution of K, the entropy parameter appearing in our equation of state (K = P/ρ^{Γ}). The movie shows a shock wave traveling out through the star, and the magnetic field exhibits a kink that traverses with the shock.
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