The relationship seems odd to me, that the luminosity of the entire star
somehow correlates to the broadening of an emission line
despite the fact that this luminosity reflects a combination of
surface temperature and radius, and a single relation holds
for stars of such varying radii. The line is produced in the
chromosphere and the effect is potentially a clue to the
chromosphere of "stars in general" as opposed to specifically that
of the Sun. An obvious candidate-mechanism is Doppler broadening
from chromosphere turbulence, but this passes on the mystery
to how such chromosphere turbulence could be affected by the
total luminosity in such manner.