SYMPLECTIC INTEGRATORS - ROTATIONS AND ROUNDOFF ERRORS

We investigate the numerical implementation of a symplectic integrator combined with a rotation las in the case of an elongated rotating pri mary). We show that a straightforward implementation of the rotation a s a matrix multiplication destroys the conservative property of the gl obal integrator, due to roundoff errors. According to Blank et al. (19 97), there exists a KAM-like theorem for twist maps, where the angle o f rotation is a function of the radius. This theorem proves the existe nce of invariant tori which confine the orbit and prevent shifts in ra dius. We replace the rotation by a twist map or a combination of shear s that display the same kind of behaviour and show that we are able no t only to recover the conservative propel-ties of the rotation, but al so make it more efficient in term of computing time. Next we test the shear combination together with symplectic integrator of order 2, 4, a nd 6 on a Keplerian orbit. The resulting integrator is conservative do wn to the roundoff errors. No linear drift of the energy remains, only a divergence as the square root of the number of iterations is to be seen, as in a random walk. We finally test the three symplectic integr ators on a real case problem of the orbit of a satellite around an elo ngated irregular fast rotating primary. We compare these integrators t o the well-known general purpose, self-adaptative Bulirsch-Stoer integ rator. The sixth order symplectic integrator is more accurate and fast er than the Bulirsch-Stoer integrator. The second- and fourth- older i ntegrators are faster, but of interest only when extreme speed is mand atory.