Exact optics - IV. Small `trumpet' correctors for large spheres

Spherical-mirror telescopes could look in diverse directions simultaneously as the sphere has no intrinsic axis. However, this remarkable ability is lost if the secondary is too big, as then the different secondaries required for the different directions get in each other's way. Small correctors for a fast sphere may allow a very large Arecibo-like fixed sphere to look in up to 1000 different directions simultaneously with each corrector using a changing 15 per cent of the mirror at a time. Among two-mirror spherical-aberration-and-coma-free telescopes the smallest corrector mirror is the `trumpet' here investigated. Although its radius is only 1.1 per cent of that of the patch of the primary that it uses, it suffers from a small field and a significant central light loss albeit to another focus. As an example we consider the possible application of these correctors to the Hobby-Eberly telescope or the South African Large Telescope (SALT). We also give the general theory of coma-ameliorating correctors.