The Astronomical Journal, 128:920–922, 2004 August # 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.
SIMPLIFICATION OF A MANIFOLD CORRECTION METHOD FOR ORBIT INTEGRATION Toshio Fukushima National Astronomical Observatory, Osawa, Mitaka, Tokyo 181-8588, Japan;
[email protected] Received 2004 March 3; accepted 2004 April 20
ABSTRACT By using an expression for the velocity as a function of the unit position vector, angular momentum, and Laplace integral, we obtain two ways of simplifying our latest method of manifold correction for numerical integration of quasi-Keplerian orbits, which needs to integrate 13 variables per celestial body. Both of the simplified methods keep the integration precision comparable to that of the original method while requiring the integration of only nine variables per celestial body. Key words: celestial mechanics — methods: n-body simulations
Note that an expression for the velocity2 can be derived from the above relations as
1. INTRODUCTION Recently, we developed a series of powerful devices to numerically integrate perturbed Keplerian motions (Fukushima 2003a, 2003b, 2003c, 2004).1 These methods dramatically reduce the integration errors in various orbital elements at the cost of a negligibly small increase in computational time. They are a modern revival of Nacozy’s method of manifold correction (Nacozy 1971). Among them, the latest method, presented in Paper IV, which we refer to as the linear transformation method, has the best performance. The linear transformation method numerically integrates a set of 13 components per celestial body: the position x, the velocity v, the Kepler energy K, the orbital angular momentum vector L, and the Laplace integral P. Of course, these are not mutually independent. Actually, there exist seven analytical relations among them: K¼
v2 ; r 2
L ¼ x < v;
P ¼ v < L n;
v¼
ð3Þ
This holds even under perturbations. By using this, we can do without the integration of not only the velocity but also the Kepler energy.3 In this brief article, we report two implementations of the above idea. Both schemes integrate nine variables per body. One uses the variable set (x, L, P) and the other uses (n, L, P). The linear transformation in Paper IV is reduced to the orthogonalization and scaling of x in the former case and to the orthonormalization of n in the latter. In the following, we will present the two simplified methods in x 2 and compare their performance with the original linear transformation method in x 3.
ð1Þ 2. SIMPLIFIED METHODS Consider the perturbed two-body problem. Usually we adopt the pair of the relative position vector and the relative velocity vector, (x, v), as the basic set of variables to be integrated. Their equations of motion are
where GM is the gravitational constant and r jxj;
n x=r:
ð2Þ
At each step of the integration, the positions and velocities integrated are adjusted by a two-stage linear transformation so as to satisfy these relations exactly. The first stage is a rotation to force the position and the velocity to be perpendicular to the angular momentum. The second consists of a rotation of the velocity around the angular momentum vector and a dual scaling of the position and the velocity. As we proved in Papers I through IV, the additional integration cost of these redundant variables is rather small compared with the evaluation of the perturbation acceleration. However, the total of 13 variables to be integrated is large. Also, we use an explicit transformation to maintain the analytical relations among the redundant variables. Yet the linear transformation we adopted in Paper IV is somewhat complicated. These are small drawbacks to the linear transformation method. 1
L < ( n þ P) : L2
dx ¼ v; dt
dv ¼ 3 x þ a: dt r
ð4Þ
Here r jxj is the mutual distance between the two bodies and a is the perturbing acceleration in the relative sense. In the first scheme to simplify the linear transformation method, we adopt a set of nine variables, (x, L, P), as the basic variables to be integrated. The conversion to the new set of variables from the ordinary set is as simple as L ¼ x < v;
P¼v
