Tune Modulation Due to Gradient Ripple and the Dynamic Aperture

A tracking study was done of the effects of a tune modulation, due to a gradient ripple in the quadrupoles, on the dynamic aperture in RHIC lattices. Using tracking runs of about 1 x 106 turns, the dynamic aperture was found to decrease roughly linearly with the amplitude of the tune modulation, and may be represented by A = A0 (1 - 42 Δv) where A0 is the dynamic aperture for Δv = 0, and v is the tune modulation amplitude. Two RHIC lattices were studied. One with 6 insertions having β* = 6m, and one with 6 insertions having β* = 2m. Roughly the same result, Eq. (1), was found for both lattices. The dependence of the dynamic aperture on the frequency of the gradient ripple was also studied. No appreciable dependence on the ripple frequency was found for the range covered in the ripple frequency and in the tune modulation amplitude.