  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 $R_{\rm K}$.

Solar and Stellar Astrophysics