Good Rotations

Numerical integrations in celestial mechanics often involve the repeated computation of a rotation with a constant angle. A direct evaluation of these rotations yields a linear drift of the distance to the origin. This is due to roundoff in the representation of the sine s and cosine c of the angle θ. In a computer, one generally gets c 2 + s 2 = 1, resulting in a mapping that is slightly contracting or expanding. In the present paper we present a method to find pairs of representable real numbers s and c such that c 2 +s 2 is as close to 1 as possible. We show that this results in a drastic decrease of the systematic error, making it negligible compared to the random error of other operations. We also verify that this approach gives good results in a realistic celestial mechanics integration.