It is useful to notice that each time an electron is “reflected” by the shock wave, its speed is increased by 2u . This is continuing until the speed is sufficient for “climbing” onto the barrier of the shock wave. Pay attention to the fact that the surplus of the speed – by how much the speed is larger than the minimally needed value – is in the range from 0 to 2u . The moment when the electron “climbs” for the first time onto the barrier is when the separatrix-crossing in the phase space happens, and hence, this is when the adiabatic invariant is not conserved: it is increased by the surface area “conquered” by the phase trajectory as a result of the separatrix-crossing. Owing to the shock wave moving slowly, the adiabatic invariant is conserved both before and after the separatrix-crossing; this fact can be used to deduce the range of the final speeds of the electrons.
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