Extensions of the longitudinal envelope equation

Recently, longitudinal space charge effects have become of increased importance in a variety of dynamical situations. The CEBAF FEL injector beam dynamics shows large space-charge effects, even at 10 MeV ({gamma} {approx} 20). Space-charge dominated longitudinal motion has also been studied in the IUCF ion storage ring. Previously a longitudinal envelope equation with a self-consistent phase-space distribution has been developed, and has been of considerable use in analyzing the motion of these cases. Longitudinal motion in detailed agreement with this envelope equation has been observed at the U. of Maryland Laboratory for Plasma Research, and at the GSI electron cooling storage ring ESR, as well as at the IUCF. However, the initial presentation in ref. 4 used non-relativistic linear- accelerator bunching motion as a simplifying approximation in order to avoid inadvertent errors and minimize misprints, and must be adapted to include relativistic and/or synchrotron effects. In the present note we extend the envelope equation formulae to include relativistic, synchrotron, and acceleration effects, and define the various factors in the equations in explicit detail. The object is to obtain a set of debugged formulae for these extended cases, with all of the various factors defined explicitly, so that the formulae can be used as a reference without repetitive rederivations. The usual ambiguities over emittance definitions and units and {beta}, {gamma}, g factors should be resolved. The reader (or readers) is invited to discover any remaining errors, ambiguities or misprints for removal in the next edition. 12 refs