Dynamic aperture extrapolation in the presence of tune modulation

In hadron colliders, such as the Large Hadron Collider (LHC) to be built at CERN, the long-term stability of the single-particle motion is mostly determined by the field-shape quality of the superconducting magnets. The mechanism of particle loss may be largely enhanced by modulation of betatron tunes, induced either by synchrobetatron coupling (via the residual uncorrected chromaticity), or by unavoidable power supply ripple. This harmful effect is first investigated in a simple dynamical system model, the H\'enon map with modulated linear frequencies. Then a realistic accelerator model describing the injection optics of the LHC lattice is analyzed. Orbital data obtained with long-term tracking simulations (10⁵--10⁷ turns) are post-processed to obtain the dynamic aperture. It turns out that the dynamic aperture can be interpolated using a simple empirical formula, as it decays proportionally to a power of the inverse logarithm of the number of turns. Furthermore, the extrapolation of tracking data at 10⁵ turns gives reliable estimates of the dynamic aperture for 10⁷ turns, which represent the expected duration of the LHC injection plateau.