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[abbreviated] Long-lived, large-scale magnetic field configurations exist inupper main sequence, white dwarf, and neutron stars. Externally, these fieldshave a strong dipolar component, while their internal structure and evolutionare uncertain, but highly relevant for several problems in stellar andhigh-energy astrophysics. We discuss the main properties expected for thestable magnetic configurations in these stars from physical arguments, and howthese properties may determine the modes of decay of these configurations.Stable magneto-hydrostatic equilibria are likely to exist in stars whenever thematter in their interior is stably stratified (not barotropic). Theseequilibria are not force-free and not required to satisfy the Grad-Shafranovequation, but they do involve both toroidal and poloidal field components. Weargue that the main mode of decay for these configurations are processes thatlift the constraints set by stable stratification, such as heat diffusion inmain-sequence envelopes and white dwarfs, and beta decays or particle diffusionin neutron stars. In the former, heat diffusion is not fast enough to makethese equilibria evolve over the stellar lifetime. In neutron stars, a strongenough field might decay by overcoming the compositional stratification throughbeta decays (at the highest field strengths) or through ambipolar diffusion(for somewhat weaker fields). These processes convert magnetic energy tothermal energy, and they occur at significant rates only once the latter issmaller than the former, and therefore they substantially delay the cooling ofthe neutron star, while slowly decreasing its magnetic energy.

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