Numerical integration of orbits of planetary satellites

                  K.G. HADJIFOTINOU
      departement of Mathematics, faculty of science
      Aristotle University of Thessaloniki, 540 06 Thessaloniki, Greece
      e-mail:hadjifotinou@olymp.ccf.auth.gr
                        and
                     D. HARPER
      Astronomy Unit, School of Mathematical Sciences,
      Queen Mary & Westfield College, Mile End Road, London E1 4N,England
      e-mail: d.harper@qmw.ac.uk

                       ABSTRACT
   The 10th-order Gauss-Jackson backward-difference numerical
integration method [1-2] and the Runge-Kutta-Nystrom RKN12(10)17M method
[3-5] were applied to the equations of motion and variational equations
of the saturnian satellite system. The above system of variational
equations consists of the second order differential equations of the
partial derivatives of the coordinates with respect to the initial
conditions and the satellites mass-ratios over Saturn [6-7].
  
   We investigated the effect of step-size on the stability of the
Gauss-Jackson method in the two distinct cases arising from the
inclusion or exclusion of the corrector cycle in the integration of the
variational equations. In the predictor-only case, we found that an
instability occured when the step-size was greater than approximately
1/76 of the orbital period of the innermost satellite. The instability
is not related to the choice of the unit of time, nor is it affected by
the order of summations in the Gauss-Jackson scheme, or by the
machine's floating-point arithmetic, but appears to be implicit in the
Gauss-Jackson method. In the predictor-corrector case the instability
disappears, and, as a result, larger step-sizes can be used, however
with significant loss in accuracy.

  By contrast, the investigation of the RKN12(10)17M method showed that
it allows the use of much larger step-sizes and can still obtain
high-accuracy results, thus making evident the superiority of this
method for the integration of planetary satellite systems.

                        References

(1) Herrick S., 1972, Astrodynamics Vol. 2, ed. Van Nostand Reinhold
(2) Merson R.H., 1974, RAE TR 74184
(3) Dormand J.R., EL-Mikkawy M.E.A, Prince P.J.,1987, IMA J.Num,
Analysis 7, 423
(4) Brankin R.W., Dormand J.R., Gladwell l., Prince P.J.,Seward
W.L.,1987, NAG TR13/87
(5) Brankin R.W., Dormand J.R., Gladwell l., Prince P.J.,Seward W.L.,
1989, ACM Trans Math. Soft. 15, No 1, 31
(6) Sinclair A.T., Taylor D.B., 1985 Astron. Astrophys. 147, 241
(7) Hadjifotinou K.G., Harper D., 1994 Astron. Astrophys. (submitted)