Stable magnetic equilibria and their evolution in the upper main sequence, white dwarfs, and neutron stars

Abstract

Context. Long-lived, large-scale magnetic field configurations exist in upper main sequence, white dwarf, and neutron stars. Externally, these fields have a strong dipolar component, while their internal structure and evolution are uncertain but highly relevant to several problems in stellar and high-energy astrophysics. Aims. We discuss the main properties expected for the stable magnetic configurations in these stars from physical arguments and the ways these properties may determine the modes of decay of these configurations. Methods. We explain and emphasize the likely importance of the non-barotropic, stable stratification of matter in all these stars (due to entropy gradients in main-sequence envelopes and white dwarfs, due to composition gradients in neutron stars). We first illustrate it in a toy model involving a single, azimuthal magnetic flux tube. We then discuss the effect of stable stratification or its absence on more general configurations, such as axisymmetric equilibria involving poloidal and toroidal field components. We argue that the main mode of decay for these configurations are processes that lift the constraints set by stable stratification, such as heat diffusion in main-sequence envelopes and white dwarfs, and beta decays or particle diffusion in neutron stars. We estimate the time scales for these processes, as well as their interplay with the cooling processes in the case of neutron stars. Results. Stable magneto-hydrostatic equilibria appear to exist in stars whenever the matter in their interior is stably stratified (not barotropic). These equilibria are not force-free and not required to satisfy the Grad-Shafranov equation, but they do involve both toroidal and poloidal field components. In main sequence stars with radiative envelopes and in white dwarfs, heat diffusion is not fast enough to make these equilibria evolve over the stellar lifetime. In neutron stars, a strong enough field might decay by overcoming the compositional stratification through beta decays (at the highest field strengths) or through ambipolar diffusion (for somewhat weaker fields). These processes convert magnetic energy to thermal energy, and they occur at significant rates only once the latter is less than the former; therefore, they substantially delay the cooling of the neutron star, while slowly decreasing its magnetic energy.

Relaxed equilibrium configurations to model fossil fields

“…7 in Braithwaite (2008), a selective decay of the total helicity (H) and of the magnetic energy (U Mag ) occurs during the initial relaxation with a stronger decrease in U Mag than that of H. This hierarchy, which is well known in plasma physics (see for example Biskamp 1997;Shaikh et al 2008) justifies the variational method used to derive our configuration (Montgomery & Phillips 1988) while the introduction of I I;1 is justified by the non force-free character of the field in stellar interiors (Reisenegger 2009) and by the stratification, which inhibits the transport of flux and mass in the radial direction (see Sect. 3.2 and Braithwaite 2008).…”

“…We then focus on its minimum energy eigenmodes for a given mass and helicity, which are derived and applied to modeling relaxed stellar fossil magnetic fields, which are found to be non forcefree. Arguments in favor of the stability of the obtained configurations are finally discussed (Wright 1973;Tayler 1980;Braithwaite 2009;Reisenegger 2009), and we compare their properties with those of relaxed fields obtained in numerical simulations (Braithwaite 2008). The case of general baroclinic equilibrium states will be studied in Paper II (Wright 1969;Moss 1975).…”

“…This point has to be discussed here. First, Reisenegger (2009) demonstrates that the magnetic field cannot be forcefree everywhere in stellar interiors (see the demonstration in appendix A of his paper). In this context, the "force-free" configurations obtained by Broderick & Narayan (2008) verify this theorem because they have current sheets with a non-zero Lorentz force on the stellar surface.…”

MHD relaxation of fossil magnetic fields in stellar interiors

“…First, Reisenegger (2009) reminds us that no configuration can be force-free everywhere. Although there do exist "force-free" configurations, they must be confined by some region or boundary layer with non-zero or singular Lorentz force.…”

Dynamics of fossil magnetic fields in massive star interiors

“…Several reasons inclined us to focus on such equilibria instead of force-free ones, which are often studied in plasma laboratory experiments. First, Reisenegger (2009) shows us that no configuration can be force-free everywhere. Although there do exist "force-free" configurations, these induce discontinuities such as current sheets, which are unlikely to appear in nature except in a transient manner.…”