Beam-beam stability in the SSC. A preliminary report

The stability of proton orbits in the SSC is threatened by two principle elements in the beam system: magnet errors and the beam-beam interaction. Of these, magnet errors are considered to be of greater importance because they are more tenacious. Beam-beam effects can be reduced if necessary by widening the beam crossing angle. Magnet errors, however, cannot be removed and their effects are difficult to compensate for. The stability of the beam system is characterized by a curve in the two-dimensional amplitude plane which separates dynamically stable trajectories, at small amplitudes, from dynamically unstable trajectories, at large amplitudes. The area of stable motion is refered to here as the ``stability aperture``. Protons outside the stability aperture exhibit chaotic motion and are lost rapidly from the beam. Those inside the aperture, in the absence of single particle scattering, remain confined indefinitely. Although the edge of the stability aperture is not a ``fine line``, as it would be for a one-degree-of-freedom system with time dependence, the amplitudes representing the transition from stable to unstable motion are fairly well defined in the SSC. The primary purpose of this investigation is to roughly determine the size and shape of the stability aperture induced by the beam-beam interaction alone, and its dependence on the principle machine parameters. This information, together with a determination of the magnet error aperture and the combined total aperture, is needed to select the nominal beam crossing angle.