Secular dipole-dipole stability of magnetic binaries

Context. The presence of strong large-scale stable magnetic fields in a significant portion of early-type stars, white dwarfs, and neutron stars is well established. Despite this, the origins of these fields remain a subject of ongoing investigation, with theories including fossil fields, mergers, and shear-driven dynamos. One potential key for understanding the formation of these fields could lie in the connection between magnetism and binarity. Indeed, magnetism can play a significant role in the long-term orbital and precessional dynamics of binary systems. In gravitational wave astronomy, the advanced sensitivity of upcoming interferometric detectors such as LISA and the Einstein Telescope will enable the characterisation of the orbital inspirals of compact systems, including their magnetic properties. A comprehensive understanding of the dynamics of magnetism in these systems is necessary for the interpretation of the gravitational wave signals and to avoid bi the wdes in the calibration of instruments. This knowledge can additionally be used to create new magnetic population models and provide insight into the nature and origins of their internal magnetic fields. Aims. The aim of this study is to investigate the secular spin precession dynamics of binary systems under pure magnetic dipole-dipole interactions, with a focus on stars with strong, stable, and predominantly dipolar fields. Methods. We employed an orbit-averaging procedure for the spin precession equations from which we derived an effective secular description. By minimising the magnetic interaction energy of the system, we obtained the configurations of spin equilibrium and their respective stabilities. Finally, we also derived a set of conditions required for the validity of our assumptions to hold. Results. We show that among the four states of equilibrium, there is a single secular state that is globally stable, corresponding to the configuration where the spin and magnetic axes of one star are reversed with respect to the companions’, and orthogonal to the orbital plane. Our results are compared to traditional methods of finding instantaneous states of equilibrium, in which orbital motion is generally neglected. Finally, we provide analytical solutions in the neighbourhood of the stable configuration, which can be used to derive secular orbital evolution in the context of gravitational wave astronomy