  An important class of formation theories for hot Jupiters involves the
excitation of extreme orbital eccentricity (e=0.99 or even larger) followed by
tidal dissipation at periastron passage that eventually circularizes the
planetary orbit at a period less than 10 days. In a steady state, this
mechanism requires the existence of a significant population of super-eccentric
(e>0.9) migrating Jupiters with long orbital periods and periastron distances
of only a few stellar radii. For these super-eccentric planets, the periastron
is fixed due to conservation of orbital angular momentum and the energy
dissipated per orbit is constant, implying that the rate of change in
semi-major axis a is \dot a \propto a^0.5 and consequently the number
distribution satisfies dN/dlog a\propto a^0.5. If this formation process
produces most hot Jupiters, Kepler should detect several super-eccentric
migrating progenitors of hot Jupiters, allowing for a test of high-eccentricity
migration scenarios.
