The Magnetic Early B-type Stars IV: Breakout or Leakage? H$alpha$ emission as a diagnostic of plasma transport in centrifugal magnetospheres

Rapidly rotating early-type stars with strong magnetic fields frequently show H$alpha$ emission originating in Centrifugal Magnetospheres (CMs), circumstellar structures in which centrifugal support due to magnetically enforced corotation of the magnetically confined plasma enables it to accumulate to high densities. It is not currently known whether the CM plasma escapes via Centrifugal Breakout (CB), or by an unidentified leakage mechanism. We have conducted the first comprehensive examination of the H$alpha$ emission properties of all stars currently known to display CM-pattern emission. We find that the onset of emission is dependent primarily on the area of the CM, which can be predicted simply by the value $B_{rm K}$ of the magnetic field at the Kepler corotation radius $R_{rm K}$. Emission strength is strongly sensitive to both CM area and $B_{rm K}$. Emission onset and strength are {em not} dependent on effective temperature, luminosity, or mass-loss rate. These results all favour a CB scenario, however the lack of intrinsic variability in any CM diagnostics indicates that CB must be an essentially continuous process, i.e. it effectively acts as a leakage mechanism. We also show that the emission profile shapes are approximately scale-invariant, i.e. they are broadly similar across a wide range of emission strengths and stellar parameters. While the radius of maximum emission correlates closely as expected to $R_{rm K}$, it is always larger, contradicting models that predict that emission should peak at $R_{rm K}$.