  The observed giant outbursts of Luminous Blue Variables (LBVs) may occur when
these massive stars approach their Eddington limits. When this happens, they
must reach a point where the centrifugal force and the radiative acceleration
cancel out gravity at the equator. We call this the Omega-limit. When stars are
close to the Omega-limit, strong non-spherical mass loss should occur. This
suggests a scenario where a slow and very dense wind, strongly confined to the
equatorial plane, is followed by a fast and almost spherical wind. We compute
two-dimensional hydrodynamic models of the evolution of the nebula formed from
such interacting winds, using parameters consistent with the outburst of eta
Carinae in the last century. This outburst gave birth to the Homunculus, the
hourglass-shaped inner part of a highly structured circumstellar nebula.
Assuming the star was very close to the Omega-limit during outburst, our models
produce gas distributions that strongly resemble the Homunculus on large and
small scale. This supports the general conjecture that giant outbursts in LBVs
occur when they approach the Eddington limit. Our models constrains the average
mass loss rate since the outburst to values smaller than the present-day mass
loss rate and suggest that eta Car is approaching another outburst. Our models
imply that the occurrence of giant LBV outbursts depends on the initial stellar
rotation rate, and that the initial angular momentum is as important to the
evolution of very massive stars as their initial mass or metallicity.
