Simulations of Merging Helion Bunches on the AGS Injection Porch

During the setup of helions for the FY2014 RHIC run it was discovered that the standard scheme for merging bunches on the AGS injection porch required an injection kicker pulse shorter than what was available. To overcome this difficulty, K. Zeno proposed and developed an interesting and unusual alternative which uses RF harmonic numbers 12, 4, 2 (rather than the standard 8, 4, 2) to merge 8 helion bunches into 2. In this note we carry out simulations that illustrate how the alternative scheme works and how it compares with the standard scheme. This is done in Sections 13 and 14. A scheme in which 6 bunches are merged into 1 is simulated in Section 15. This may be useful if more helions per merged bunch are needed in future runs. General formulae for the simulations are given in Sections 9 through 12. For completeness, Sections 1 through 8 give a derivation of the turn-by-turn equations of longitudinal motion at constant magnetic field. The derivation is based on the work of MacLachlan. The reader may wish to skip over these Sections and start with Section 9.