Hydrodynamics of rotating liquid mirrors. I. Synchronous disturbances

The effects of axis alignment errors, planetary rotation, and tidal forces on rotating liquid mirrors are analyzed. These produce a surface distortion that decreases exponentially with distance inward from the rim with a characteristic length l= square root of 3hf/2, where h is the thickness of the fluid and f is the focal length. Even a small tilt of the rotation axis can produce a significant deformation of the optical surface. The maximum surface height error is 3epsilonl, where epsilon is the tilt angle and is typically of the order of 1.5 microm for a 1 arc sec tilt. The main optical effect of the wave is to produce a ring, with angular diameter 6epsilon, offset by half of the diameter in the direction opposite the tilt. This diamond ring aberration can be avoided by accurate alignment of the rotation axis or by masking the outer few centimeters of the mirror. Planetary rotation produces a small deformation of the order of 100 nm for a 10 m telescope at low latitude on Earth. This deformation can be canceled by a small tilt of the rotation axis. Tidal forces produced by the Moon, or by the Earth in the case of a lunar telescope, produce an inconsequential, subnanometer, surface deformation.