The effect of collisional cooling of energetic electrons on radio emission from the centrifugal magnetospheres of magnetic hot stars

Abstract This paper extends our previous study of the gyro-emission by energetic electrons in the magnetospheres of rapidly rotating, magnetic massive stars, through a quantitative analysis of the role of cooling by Coulomb collisions with thermal electrons from stellar wind material trapped within the centrifugal magnetosphere (CM). For the standard, simple CM model of a dipole field with aligned magnetic and rotational axes, we show that both gyro-cooling along magnetic loops and Coulomb cooling in the CM layer have nearly the same dependence on the magnitude and radial variation of magnetic field, implying then that their ratio is a global parameter that is largely independent of the field. Analytic analysis shows that, for electrons introduced near the CM layer around a magnetic loop apex, collisional cooling is more important for electrons with high pitch angle, while more field-aligned electrons cool by gyro-emission near their mirror point close to the loop base. Numerical models that assume a gyrotropic initial deposition with a gaussian distribution in both radius and loop co-latitude show the residual gyro-emission is generally strongest near the loop base, with highly relativistic electrons suffering much lower collisional losses than lower-energy electrons that are only mildly relativistic. Even for cases in which the energy deposition is narrowly concentrated near the loop apex, the computed residual emission shows a surprisingly broad distribution with magnetic field strength, suggesting that associated observed radio spectra should generally have a similarly broad frequency distribution. Finally, we briefly discuss the potential applicability of this formalism to magnetic ultracool dwarfs (UCDs), for which Very Long Baseline Interferometry (VLBI) observations indicate incoherent radio emission to be concentrated around the magnetic equator, in contrast to our predictions here for magnetic hot stars. We suggest that this difference could be attributed to UCDs having either a lower ambient density of thermal electrons, or more highly relativistic non-thermal electrons, both of which would reduce the relative importance of the collisional cooling explored here.