Longitudinal stability of Flat Bunches with Space-Charge or Inductive Impedance

We study the loss of Landau damping for the longitudinal plane via the "Sacherer formalism". Stability limits are calculated for several longitudinal beam distributions, in particular for two types of flat bunches, which could be of interest to the LHC upgrade. The resulting stability diagrams are computed and displayed for different azimuthal modes. A general recipe is given for calculating the threshold intensity in the case of a capacitive impedance below transition or, equivalently, for a purely inductive impedance above transition. The formalism was applied to the case of the PS Booster, as an example of space-charge impedance below transition, and to the SPS, as an example of inductive impedance above transition.