Gaussian approximation of the bunch lengthening in electron storage rings with a typical wake function

The validity of the Gaussian approximation of the distribution function in the study of the equilibrium bunch length in electron storage rings is considered with specific reference to the inductive wake. The Gaussian approximation can be used to describe not only localized wakes, but also uniformly distributed wakes, for which the longitudinal profiles at equilibrium are ruled by the potential-well-distortion (Haissinski) equation. Comparison of Gaussian approximation based results with those obtained from Haissinski equation shows good qualitative agreement in their common range of validity, while shedding light on the nature of the unstable regimes.