Stability diagram for Landau damping with a beam collimated at an arbitrary number of sigmas

The general formula for the dispersion equation used to draw the stability diagram for Landau damping with two-dimensional betatron tune spread from octupoles is given for the nth order distribution function. It is solved in the particular case of the 15th order distribution function, which is consistent with the nominal collimator settings in the LHC at top energy, i.e. extending up to 6 sigmas in transverse space. The new stability diagram is compared to the ones already obtained with both the 2nd order distribution function, which extends up to 3.2 sigmas in transverse space, and the Gaussian distribution, which extends to infinity. The case of a distribution extending up to 6 sigmas in transverse space but with more populated tails than the Gaussian is also discussed. This case may apply in reality in proton machines, where several diffusive mechanisms can take place.