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

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.