Tune modulation due to synchrotron oscillations and chromaticity, and the dynamic aperture

A tracking study was done of the effects of a tune modulations, due to synchrotron oscillations and the tune dependence on momentum (chromaticity), on the dynamic aperture. The studies were done using several RHIC lattices and tracking runs of about 1 {times} 10{sup 6} turns. The dynamic aperture was found to decrease roughly linearly with the amplitude of the tune modulation. Lower order non-linear resonances, like the 1/3 and 1/4 resonance are not crossed because of the tune modulation. Three different cases were studied, corresponding to RHIC lattices with different {beta}*, and with different synchrotron oscillation amplitudes. In each case, the tune modulation amplitude was varied by changing the chromaticity. In each case, roughly the same result, was found. The result found here for the effect of a tune modulation due to chromaticity may be compared with the result found for the effect of a tune modulation due to a gradient ripple in the quadrupoles. The effect of a tune modulation due to a gradient ripple appears to be about 4 times stronger than the effect of a tune modulation due to chromaticity and synchrotron oscillations