Optimisation of average luminosity in pp and pp colliders

Algorithms are presented for the optimisation of the luminosity in pp and pp colliders. Their average luminosity is lower than the peak luminosity because the transverse dimensions of the beams increase due to intra-beam scattering, and the beam currents decay due to the collisions with the rest gas and the opposite beam. Assuming that the p are produced during the collision time links the p production rate to the maximum p current. Either the average luminosity or the average number of collisions with exactly one beam-beam event are optimised. The free parameters are the collision time, the initial luminosity, and the density of the beams. Numerical solutions are based on the parameters of the pp option of the Large Hadron Collider (LHC) and on the p production rate of the p source (ACOL) at CERN. Approximate analytical solutions are also presented.