ABSTRACT. This describes our integrator RADAU, which has been used by several groups in the U.S.A., in Italy, and in the U.S.S.R. over the past 10 years in the numerical integration of orbits and other problems involving numerical solution of systems of ordinary differential equations. First-and second-order equations are solved directly, including the general second-order case. A self-starting integrator, RADAU proceeds by sequences within which the substeps are taken at Gauss-Radau spacings. This allows rather high orders of accuracy with relatively few function evaluations. After the first sequence the information from previous sequences is used to improve the accuracy. The integrator itself chooses the next sequence size. When a 64-bit double word is available in double precision, a 15th-order version is often appropriate, and the FORTRAN code for this case is included here. RADAU is at least comparable with the best of other integrators in speed and accuracy, and it is often superior, particularly at high accuracies.

New osculating orbits are presented for 110 nearly parabolic comets. Combining these with selected orbit determinations from other sources, we consider a total of 200 orbits where the available observations yield a result of very good (first class) or good (second class) quality. For each of these, the original and future orbits (referred to the barycenter of the solar system) are calculated. The Oort effect (a tendency for original 1/a values to range from 0 to-f-lOOXlO-6 AU-1) is clearly seen among the first-class orbits but not among the second-class orbits. Modifications in original 1/a values due to the effects of nongravitational forces are considered.

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Abstract. The solutions of .~ = F(x, t), and also .~ --F (x, t), are developed in truncated series in time t whose coefficients are found empirically. The series ending in the t n term yields a position at a final prechosen time that is accurate through 9th order in the sequence size. This is achieved by using Gauss-Radau and Gauss-Lobatto spacings for the several substeps within each sequence. This timeseries method is the same in principle as implicit Runge-Kutta methods, and the present algorithm generates coefficients for families of implicit Runge-Kutta forms, including some not described previously. In some orders these methods are unconditionally stable (A-stable). In the time-series formulation the implicit system converges rapidly. For integrating a test orbit the method is found to be about twice as fast as high-order explicit Runge-Kutta-Nystr6m-Fehlberg methods at the same accuracies. Both the Cowell and the Encke equations are solved for the test orbit, the latter being 3 5 faster. It is shown that the Encke equations are particularly well-adapted to treating close encounters when used with a single-sequence integrator (such as this one) provided that the reference orbit is re-initialized at the start of each sequence. This use of Encke equations is compared with the use of regularized Cowell equations.

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