  Hot magnetic stars often exhibit incoherent circularly polarized radio
emission thought to arise from gyro-synchrotron emission by energetic electrons
trapped in the circumstellar magnetosphere. Theoretical scalings for electron
acceleration by magnetic reconnection driven by centrifugal breakout match well
the empirical scalings for observed radio luminosity with both the magnetic
field strength and the stellar rotation rate. This paper now examines how
energetic electrons introduced near the top of closed magnetic loops are
subsequently cooled by the energy loss associated with their gyro-synchrotron
radio emission. For sample assumed distributions for energetic electron
deposition about the loop apex, we derive the spatial distribution of the
radiated energy from such "gyro-cooling". For sub-relativistic electrons, we
show explicitly that this is independent of the input energy, but also find
that even extensions to the relativistic regime still yield a quite similar
spatial distribution. However, cooling by coulomb collisions with even a modest
ambient density of thermal electrons can effectively quench the emission from
sub-relativistic electrons, indicating that the observed radio emission likely
stems from relativistic electrons that are less affected by such collisional
cooling. The overall results form an initial basis for computing radio emission
spectra in future models that account for such cooling and multimode excitation
about the fundamental gyro-frequency. Though motivated in the context of
hot-stars, the basic results here could also be applied to gyro-emission in any
dipole magnetospheres, including those of ultra-cool dwarfs and even
(exo)-planets.
