Differential algebraic determination of high-order off-energy closed orbits, chromaticities, and momentum compactions

For some modern particle accelerators, including the planned muon collider, the accurate analysis of non-linear time-of-flight effects in the form of momentum compactions is critical for the preservation of bunch structure. A Differential Algebra-based (DA) method is presented that allows the determination of off-energy closed orbits and chromaticities to any order. By performing a coordinate transformation to the off-energy closed orbit, it is possible to compute momentum compactions analytically. This method has been implemented in the code COSY INFINITY and is tested for two cases where analytical solutions can be obtained by hand; agreement to machine precision is found. By contrast, comparisons are made with several codes that use conventional numerical methods for the determination of momentum compaction, and it is found that these approaches sometimes yield rather inaccurate results, especially for higher orders