The Astronomical Journal, 127:3638–3641, 2004 June # 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.
EFFICIENT ORBIT INTEGRATION BY LINEAR TRANSFORMATION FOR CONSISTENCY OF KEPLER ENERGY, FULL LAPLACE INTEGRAL, AND ANGULAR MOMENTUM VECTOR Toshio Fukushima National Astronomical Observatory, Osawa, Mitaka, Tokyo 181-8588, Japan;
[email protected] Received 2004 January 23; accepted 2004 February 27
ABSTRACT By adopting a general linear transformation as the method of manifold correction, we modify our dual scaling method to integrate quasi-Keplerian orbits numerically. The new method adjusts the integrated position and velocity at each integration step in order to exactly satisfy the relations for the Kepler energy, angular momentum vector, and the full Laplace vector. In the case of no perturbation, the integration errors in all the orbital elements except the mean longitude at the epoch, which grows linearly with time, are reduced to the level of the machine epsilon throughout the integration. For perturbed orbits, the integration errors in position are smaller than with the previous methods of manifold correction. Since its wide applicability is unchanged and the cost of additional computation is similarly negligible, we recommend the new method as the best of our methods of manifold correction. Key words: celestial mechanics — methods: n-body simulations
1. INTRODUCTION
the machine epsilon as was the case for the semimajor axis in Paper I and the inclination and the longitude of the ascending node in Paper III. As a result, the dual scaling method is not always better than the single scaling method. See Figure 1, which compares the position error of Mercury obtained with various methods of manifold correction in long-term integrations of the Sun and nine major planets. In the earlier stages, the dual scaling method gives smaller errors than the single scaling method. However, this situation is reversed in the long run. This is mainly because the manifold correction adopted in Paper II is of the form of a dual scaling, (x, v) ! (sX x, sV v). This restriction might be overcome by extending the type of manifold correction in a more general way. In this brief article, we report our discovery of such a transformation as the combination of a single-axis rotation and a linear transformation of the type (x, v) ! (sX x, sV (v x)), where is a newly introduced parameter maintaining the angle between the integrated position and velocity properly. Fortunately, the determination of the three parameters, sX, sV, and , is simple, explicit, and well defined. On the other hand, the introduction of the third parameter makes the method more efficient, in the sense of reducing not only the errors in the eccentricity vector but the position vector as well. (See Fig. 1 again.) In the following, we present the new formulation in x 2 and illustrate the results of numerical experiments in x 3.
Recently, we developed a series of powerful devices to numerically integrate perturbed Keplerian motions (Fukushima 2003a, 2003b, 2003c).1 In short, they are an extension of Nacozy’s method of manifold correction (Nacozy 1971) using the concept of the integral invariant relation (Szebehely & Bettis 1972). These methods numerically integrate not only the position and velocity but also some quantities to be conserved in the unperturbed case, such as the Kepler energy, the orbital angular momentum vector, and/or the Laplace integral. At each step of the integration, the positions and velocities integrated are adjusted so as to exactly satisfy the defining relations of some functions derived from the above quantities. As illustrated in Papers I through III, these methods dramatically reduce the integration errors in various orbital elements at the cost of a negligibly small increase in computational time. In fact, the method presented in Paper I, which we named the single scaling method, reduces the error in the semimajor axis drastically and, therefore, the error in the mean longitude significantly, too, by maintaining the Kepler energy relation rigorously. Next, the method developed in Paper II, which we called the dual scaling method, does the same for the eccentricity and the argument of pericenter, as well as the semimajor axis. Third, the method provided in Paper III, which we denoted the rotation method, does the same for the inclination and the longitude of the ascending node by adjusting the direction of the position and velocity to be exactly perpendicular to the orbital angular momentum vector. In Paper II, however, we developed a method to satisfy not the full vectorial relation for the Laplace integral but a single scalar relation derived from it. Thus, we could not completely reduce the errors in the eccentricity or in the argument of pericenter. More specifically, in the case of unperturbed noncircular orbits the errors do not decrease down to the level of
1
2. METHOD Consider the perturbed one-body problem, the equation of motion of which is written as dx dv ¼ v; ¼ x þ a; ð1Þ dt dt r3 where x is the position, v is the velocity, GM is the gravitational constant, r |x| is the radius vector, and a is the perturbing acceleration. In the new method, we numerically integrate not only the above equation of motion but also the time development of the
Hereafter Papers I, II, and III, respectively.
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EFFICIENT ORBIT INTEGRATION. IV.
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where the quantities with an asterisk are the final ones, to be used in the next step of the integration. Here the three parameters sX, sV, and are chosen so as to satisfy the relations for K and P rigorously under the condition that both x 0 and v 0 be perpendicular to L. The solution is unique and explicitly obtained as L2 F = v0 ; ¼ ; sX ¼ F = x0 F = x0 ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2K þ 2=(sX r 0 ) sV ¼ ; (v 0 )2 2(x 0 = v 0 ) þ 2 (r 0 )2
ð7Þ
where FPþ Fig. 1.—Relative position errors |x|/|x| of Mercury in simultaneous integrations of the Sun and nine major planets using various methods of manifold correction. The errors were obtained by comparing with a reference solution that was obtained using the same integrator but with half the step size. We omitted the result for the rotation method, since its application does not reduce the position error in general. The integrator used was the PECE mode of the implicit Adams method with a fixed step size of 1.38 days, corresponding to 1/64 the nominal orbital period of Mercury. The orders of the Adams method were set as the highest among those that led to no numerical instability with the adopted step size, namely, 10 for the case of no correction, 11 for the single and dual scaling methods, and 13 for the new method. The starting tables were prepared with Gragg’s extrapolation method. In the earlier stages, the dual scaling method produces smaller errors than the single scaling method. However, this situation is reversed in the long run. What remains unchanged is the fact that the method with no manifold correction is the worst and the new method is the best.
trio of the Kepler energy K, the angular momentum vector L, and the Laplace integral P, which are defined as v2 ; L x < v; P v < L x: ð2Þ K r r 2 The equations for their time development are dK ¼ v = a; dt
dL ¼ x < a; dt
dP dL ¼a
