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The amplitude dependent tune shifts from either space charge or beam-beaminteractions are calculated analytically with the inclusion of synchrotronoscillations and multiple interactions around the ring. Simpler formulae arederived under limits of bunches longer than the transverse sizes, equal andunequal transverse sizes etc. This is used to derive semi-analytical forms forthe density distribution of the tune shifts. The tune spread and the densitydistribution are needed to understand beam decoherence or Landau damping witheither interaction. The tune footprints due to space charge in IOTA aresimulated using pyorbit and found to be in good agreement with the theoreticalpredictions.

Density distributions of tune shifts from space charge or beam-beam interactions in Gaussian bunches