Optimizing the length of the stores and related effects

This memo is an attempt to create an accurate analytical model for the luminosity obtained in the Fermilab Tevatron during collider operation. Other people, in particular Alan Hahn and, aledgedly, Vinod Bharadwaj and Gerry Dugan, have addressed this topic and predicted an optimum store duration. The approach taken here is slightly different from Hahn's, in particular, analytic forms for the model are written down and the optima are found directly, through differentiation. Also, specifying the parameters clearly is helpful in understanding the dependencies of this phenomenology. First, the optimum store duration is derived from first principles. Then graphs of this are made and various assumptions on the parameters are analyzed. Then some related effects are discussed. The figures are segregated at the end of this Memo