The Astrophysical Journal, 687:1294Y1302, 2008 November 10 # 2008. The American Astronomical Society. All rights reserved. Printed in U.S.A.
EXTENDING NACOZY’S APPROACH TO CORRECT ALL ORBITAL ELEMENTS FOR EACH OF MULTIPLE BODIES Da-Zhu Ma, Xin Wu, and Shuang-Ying Zhong Department of Physics, Nanchang University, Nanchang 330031, China;
[email protected] Received 2008 May 26; accepted 2008 July 9
ABSTRACT For each object of an n-body problem in planetary dynamics, orbital elements except the mean anomaly are directly determined by five independently slow-varying quantities or quasi-integrals, which include the Keplerian energy, the three components of the angular momentum vector, and the z-component of the Laplace vector. The mean anomaly depends on the mean motion specified by the Keplerian energy. Decreasing integration errors of these quasiintegrals at every integration step means improving the accuracy of all the elements to a great extent. Because of this, we take reference values of these quantities in terms of the integral invariant relations as control sources of the errors and then give an extension of Nacozy’s idea of manifold correction. The technique is almost the same as the linear transformation method of Fukushima in its explicit validity of correcting all elements, if the adopted basic integrators can give a necessary precision to the stabilizing sources considered. Especially it plays a more important role in significantly suppressing the growth of numerical errors in high eccentricities. Subject headingg s: celestial mechanics — methods: n-body simulations — methods: numerical
1. INTRODUCTION
As emphasized, the corrected orbit does not lie exactly on the hypersurface. In fact, the constraint through the correction is only accurate to the second order of the uncorrected counterpart. Similar to this idea, a scheme on how to keep all invariants (like those in the two-body problem above) during a process of numerical integration was also given by Nacozy. In principle, the manifold correction is able to treat an n-body gravitational system with 10 constants of motion, made of the total energy, the total angular momentum, and integrals of the center of mass, in an inertial coordinate system. Nacozy (1971) found that an application of the correction method for maintaining the 10 integrals of a 25-body problem to numerical integrations of the system is successful at obtaining a significant gain in precision. However, Hairer et al. (1999) pointed out that the approach with the constancy of both the total energy and the total angular momentum fails to work well in a five-body integration of the Sun and four outer planets. Numerical explorations and analytical interpretations of the results like this have appeared in a series of references (e.g., Wu et al. 2006, 2007; Wu & He 2006). The reason stated in these articles is mainly that corrections of integrals, such as the total energy or the total angular momentum, in a full system do not bring great corrections to independent integrals for each subsystem or to individual varying quantities resembling the individual energy or the individual angular momentum for each object. This tells one that it is more important to correct individual nonconstant quantities of each body of an n-body problem, corresponding to the integrals of the above pure two-body problem, than to stabilize the 10 integrals of the n-body system. Unfortunately, Nacozy’s original method of manifold correction has difficulty in treating the case of dissipative systems, because it is merely limited to the use of conservative systems with many constraints. If the individual varying quantities that need to be corrected have more accurate reference values at any time, the quantities integrated can be adjusted to the reference values by corrections of the positions and velocities. Namely, the manifold correction method is still valid in this case. For an illustration, the related reference values are obtained from integral invariant relations of the varying quantities
A pure Keplerian two-body problem is the simplest case in celestial mechanics and dynamical astronomy. In the relative coordinates, there are seven conserved quantities, involving the Keplerian energy K, the three components (Px , Py , and Pz ) of the Laplace vector P, and the three components (Lx , Ly , and Lz ) of the angular momentum vector L. They are closely associated with orbital elements. The Keplerian energy determines directly the semimajor axis as well as the mean motion, and the eccentricity is given by the magnitude of the Laplace vector, namely, the Laplace integral P (=jPj). In addition, the inclination and the longitude of the ascending node depend on the magnitude L (=jLj) of the angular momentum vector and the three components Lx , Ly , and Lz . Finally, the argument of perihelion can be obtained from the z-component Pz of the Laplace vector and the inclination. It should be worth noting that there exist two identical relations among the seven quantities. One is the orthogonality of two vectors, P = L ¼ 0. The other is the relation P2 2KL2 ¼ 2 , where is the gravitational constant of the two-body problem (Fukushima 2003c). So, there are only five independent integrals, K, Lx , Ly , Lz , and Pz . These facts imply that the consistency of the five independent integrals over the whole course of the numerical integration leads to the high accuracy of the orbital elements. The manifold correction of Nacozy (1971), as a pioneering work, is considered to be a preferable and convenient tool for arriving at this aim. First, Nacozy considered the case of a known integral. The motion remains on the hypersurface of the integral in phase space, but jumps to a different hypersurface for the presence of a numerical error. In order to compensate for the deviation, Nacozy used a Lagrange multiplier to find the minimum value of the square of the magnitude of a correction vector subjected to the constraint. Adding the least-squares correction to the numerical solution is a basic concept of Nacozy’s approach. From the geometrical point of view, the method pulls the integrated orbit back to the original integral hypersurface along the shortest path or the perpendicular to the hypersurface. It is usually called the steepest descent method. 1294
EXTENDING NACOZY’S APPROACH (Szebehely & Bettis 1971; Huang & Innanen 1983; Mikkola & Innanen 2002), in which the equations of motion and the time derivatives of the quantities are integrated together. Based on this point, several kinds of extensions to Nacozy’s approach have been developed recently. Next, let us introduce and remark on some of them according to different correction aims. Case 1: Corrections of individual Keplerian energy.— Clearly, stabilization or correction of the Keplerian energy of each planet or asteroid in an n-body problem is rather significant, since it is the best help in the struggle against Lyapunov’s instability of Keplerian motion so as to monitor the accumulation of the in-track error of the orbit ( Baumgarte 1972; Avdyushev 2003). Meanwhile, the accuracy of the semimajor axis and the mean anomaly can be raised drastically. Although the Keplerian energy is no longer invariant, it varies slowly. In this sense, it is still viewed as a quasi-integral. As mentioned above, its reference value at every step is given by the integral invariant relation. This just provides a chance to correct the Keplerian energy. From the theoretical point of view, there are various paths of manifold correction of the Keplerian energy. Two types of corrections are worth noting. One is the rigorous methods for exactly satisfying the relation of the Keplerian energy. The scaling method of Liu & Liao (1988, 1994), the single scaling method of Fukushima (2003a), and the velocity scaling method and the position scaling method of Ma et al. (2008b) are some typical examples. In light of the constraints of a two-body problem, Liu & Liao used two distinct scale factors, and , to adjust the position and velocity of the body, respectively. The two factors obey the relation 2 ¼ 1. Unlike this correction, Fukushima’s single scaling method adopts the same scale factor for the integrated position and velocity. The factor is given by solving a certain cubic equation related to the Keplerian energy. As a refined version of the single scaling method, the velocity scaling method (or the position scaling method) uses only a scale factor for the integrated velocity vector (or the integrated position vector). Here, the determination of the scale factor becomes simpler and more explicit. The other is the approximate methods with the least-squares correction of the Keplerian energy. As an example, a direct extension to Nacozy’s approach, which Wu et al. (2007) gave, is to only correct three components of the velocity vector. As Ma et al. (2008b) concluded, in spite of the five corrections from different directions, they all are almost effective in the sense of drastically improving the semimajor axis as well as the mean anomaly when the uncorrected integrator is full of sufficient precision. In addition, they are nearly the same at a negligible increase of the computational cost. In particular, both the velocity scaling method and the method of Wu et al. (2007) are the most convenient to apply. Case 2: Corrections of individual Keplerian energy and Laplace integral.—A small object is closer to the centered body at the pericenter when orbital eccentricity becomes larger. This gives rise to the fast accumulation of numerical errors. To diminish it, Fukushima (2003b) gave a dual spatial scale transformation to the integrated positions and velocities of each object, where both the Keplerian energy relation and another functional relation associated with the Laplace integral are exactly satisfied. The dual scaling method is also a rigorous method. By using the Laplace integral instead of the functional relation, an approximate method with velocity corrections to the two integrals or quasiintegrals is presented (Ma et al. 2008a). Still, the approximate scheme is nearly as valid as the rigorous method in raising the accuracy of the semimajor axis and the eccentricity when an
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uncorrected integrator has sufficient precision. For emphasis, the former is superior to the latter in the correction of eccentricity in general. This is because the former turns out to give the Laplace integral (equivalently, the eccentricity) a direct correction, while the latter only gives an indirect correction to the Laplace integral. Case 3: Corrections of individual Keplerian energy, Laplace integral, and angular momentum vector.—In order to suppress the growth of integration errors in the inclination and the longitude of the ascending node, Fukushima (2003c) designed a rotation for consistency of the individual orbital angular momentum vector. Thus, the combination of the dual scaling and the rotation is completely in keeping with the individual Keplerian energy, Laplace integral, and angular momentum vector during the numerical integration, so that the errors in all the orbital elements of each body can be decreased. As the author claimed, this rotational operation is independent of the application of the dual scaling. It means that one can use a dual spatial scale transformation to the integrated positions and velocities with correction of both the Keplerian energy and the Laplace integral, and then carry out a space rotational transformation to the corrected ones for satisfying the angular momentum. From the theoretical point of view, the method with such independent operations of the two transformations, called a two-step correction method, does not keep the simultaneous consistency of these quasi-integrals. For the completeness of the correction theory, Fukushima (2004) constructed such a transformation as the combination of a singleaxis rotation and a linear transformation of the type (x; v) ! ½sX x; sV (v x) so that the three parameters sX , sV , and satisfy exactly the relations associated with the three quantities K, P, and L in a one-step correction. This is a linear transformation method. The main purpose of the present paper is to give an approximate method with the least-squares correction of the five independent quasi-integrals of K, Lx , Ly , Lz , and Pz like cases 1 and 2. Our paper is organized as follows. Section 2 gives a new extension to Nacozy’s approach with integral invariant relations. Then by some numerical simulations we evaluate and compare it with the linear transformation method of Fukushima (2004) and the method of Wu et al. (2007) in x 3. Finally, x 4 concludes our results. 2. A NEW EXTENSION TO NACOZY’S CORRECTION SCHEME Following the manifold correction algorithm of Nacozy (1971) with the integral invariant relations (Szebehely & Bettis1971), we generalize this idea to maintain the consistency of the Keplerian energy, the angular momentum vector, and the z-component of the Laplace vector for the case of a perturbed Keplerian problem. Several details are described in the following. 2.1. The Manifold Correction Algorithm of Nacozy Suppose an m-dimensional dynamical system has s integrals, i (x) ¼ ci (i ¼ 1; : : : ; s). That is to say, i (x) ¼ i (x) ci ¼ 0
ð1Þ
for a true state vector x in the phase space. But, i (h) 6¼ 0 due to a numerical solution h yielding errors in the computation. Hence, one gets a nonzero functional vector of the form e ¼ ½1 (h); : : : ; s (h)T , where the T superscript indicates transpose.
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Let E be an s ; m matrix, with the ith row and jth column element, @i (h)/@ j ( j ¼ 1; : : : ; m). Nacozy (1971) applied Lagrangian multipliers to find a correction vector #h to the solution h, such that the following corrected solution x ¼ h þ #h
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ties in x 2.1, the state vector x ¼ (x; y; z; x˙ ; y˙ ; z˙ ) and five constants, c1 ; : : : ; c5 , are respectively K0, L0x , L0y , L0z , and P0z , as initial values of K, Lx , Ly , Lz , and Pz . Now, the application of Nacozy’s approach to the case becomes easy by taking 0
ð2Þ
B B Lx L0x B e¼B B Ly L0y B @ Lz L0z Pz P0z
with
T 1
#h ¼ W E EW E 1
T
1
e
ð3Þ 0
becomes closer to the true solution x than h because i (x) i2 (h):
ð4Þ
Here, W is a weighting matrix, and the matrix E is required to have rank s. In a word, x, adjusting each integral (eq. [1]) with equation (4), is just what one wants. Next, we shall consider the application of the correction scheme. 2.2. The Algorithm Applied to a Pure Keplerian Problem A pure Keplerian problem in the relative coordinates can be simplified to a one-body problem with the Keplerian energy K¼
1 2 v : 2 r
ð5Þ
Besides this, there are integrals of the Laplace vector and angular momentum vector in the forms P¼v
