EXTENDING NACOZY S APPROACH TO CORRECT ALL ORBITAL ELEMENTS FOR EACH OF MULTIPLE BODIES

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; xwu@ncu.edu.cn 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 headinggs: celestial mechanics methods: n-body simulations methods: numerical 1. INTRODUCTION 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 (P x, P y, and P z )ofthe Laplace vector P, and the three components (L x, L y, and L z )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 L x, L y, and L z. Finally, the argument of perihelion can be obtained from the z-component P z 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 P 2 2KL 2 ¼ 2, where is the gravitational constant of the two-body problem ( Fukushima 2003c). So, there are only five independent integrals, K, L x, L y, L z, and P z. 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 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

EXTENDING NACOZY S APPROACH 1295 (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 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)! ½s X x; s V (v x)š so that the three parameters s X, s V,and satisfy exactly the relations associated with the three quantities K, P, andl 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, L x, L y, L z, and P z 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 c i (i ¼ 1; :::;s). That is to say, i (x) ¼ i (x) c i ¼ 0 for a true state vector x in the phase space. But, i (h) 6¼ 0due to a numerical solution h yielding errors in the computation. Hence, one gets a nonzero functional vector of the form e ¼ ½ 1 (h); :::; s (h) Š T, where the T superscript indicates transpose. ð1þ

1296 MA, WU, & ZHONG Vol. 687 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 with x ¼ h þ #h ð2þ #h ¼ W 1 E T EW 1 E T 1e ð3þ becomes closer to the true solution x than h because i (x) 2 i (h): 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 K ¼ 1 2 v 2 r : ð4þ ð5þ Besides this, there are integrals of the Laplace vector and angular momentum vector in the forms P ¼ v < L r r; L ¼ r < v: Here, we specify that r ¼ (x; y; z), v ¼ (ẋ; ẏ; ż), G(M þ m), and r ¼jrj denote the position, velocity, gravitational constant, and radius, respectively. It should be emphasized that these integrals determine directly five orbital elements, namely, the semimajor axis a, the eccentricity e, the inclination I, the longitude of ascending node, and the argument of perihelion!. Relations between the integrals and the elements are described as ð6þ ð7þ a ¼ 2K ; e ¼ P ; ð8þ L z L x I ¼ arccos ; ¼ arctan ; ð9þ L L y P z! ¼ arcsin : ð10þ e sin I For an illustration, the mean anomaly M, which includes information on the location in the orbit, is associated with the mean motion specified by the Keplerian energy. The orientation of on the orbital plane should be given by the signs of L x and L y. Similarly, the orientation of! is based on the sign of sin! and that of P x cos þ P y sin. Therefore, the conservation of K, L x, L y, L z, and P z means the improvement of the five elements (note that if K and L are controlled, then P is also controlled automatically). Naturally, the mean anomaly can be improved. Matching with the quantities in x 2.1, the state vector x ¼ (x; y; z; ẋ; ẏ; ż) and five constants, c 1 ;:::;c 5, are respectively K 0, L 0x, L 0y, L 0z, and P 0z,as initial values of K, L x, L y, L z, and P z. Now, the application of Nacozy s approach to the case becomes easy by taking 0 @K @L x @L y E ¼ @L z B @ @P z 0 1 K K 0 L x L 0x e ¼ L y L 0y ; ð11þ B C @ L z L 0z A P z P 0z 1 @K @K @K @K @K @L x @L x @L x @L x @L x @L y @L y @L y @L y @L y : ð12þ @L z @L z @L z @L z @L z C @P z @P z @P z @P z @P z A The concrete expressions of all the components of E are of the following forms, 0 x y z 1 r 3 r 3 r 3 ẋ ẏ ż 0 ż ẏ 0 z y E ¼ ż 0 ẋ z 0 x ; ð13þ B C @ ẏ ẋ 0 y x 0 A Q 1 Q 2 Q 3 Q 4 Q 5 Q 6 with Q 1 ¼ ẋżþxz/r 3, Q 2 ¼ ẏż þ yz/r 3, Q 3 ¼ ẋ 2 þ ẏ 2 (x 2 þy 2 )/r 3, Q 4 ¼ 2zẋ xż, Q 5 ¼ yżþ2zẏ,andq 6 ¼ xẋ yẏ. Obviously, the matrix E is of rank 5. It should be mentioned that the components of the E matrix are of different dimensions, so that the resulting minimization process like the steepest descent method becomes disastrous. It is important to adopt the choice of nondimensionalization in the actual application of Nacozy s approach. For nondimensionalization, we carry out scale transformations, such as r! r/r 0, v! v/v 0, t! t/t, etc., where constants r 0, v 0, and T are the initial radius, the magnitude of the initial velocity, and the average orbital period, respectively. The weight matrix W is a unit matrix through the operation of nondimensionalization. Note that the original quantities are still used before and after the implementation of the manifold correction. Of course, the correction method can further be extended to treat the perturbed case. 2.3. The Algorithm Extended to a Perturbed Keplerian Problem When a perturbing acceleration a exists, the perturbed one body evolves according to the equation of motion dv dt ¼ r 3 r þ a: ð14þ

No. 2, 2008 EXTENDING NACOZY S APPROACH 1297 Fig. 1. Variations of errors in all elements of a pure Keplerian orbit with time. In this case, K, P, and L in equations (5)Y(7) remain no longer invariant, but vary slowly with time in the following forms (Fukushima 2003c), dp dt dk dt ¼ v = a; ¼ 2(a = v)r (r = a)v (r = v)a; ð16þ dl dt ¼ r < a: ð15þ ð17þ Equations (15) Y(17) are so-called integral invariant relations about these quasi-integrals of the Keplerian energy, the Laplace vector, and the angular momentum vector. Note that the righthand sides of equations (15)Y(17) are small quantities, so the left-hand sides would use K ¼ K K 0, P ¼ P P 0,and L ¼ L L 0 rather than K, P, and L in order to greatly reduce the round-off errors. The values of K, P, and L from direct integration of equations (14)Y(17) should be more accurate than the corresponding values of them by equations (5)Y(7) with a numerical solution of equation (14). We mark them as K, P, and L in sequence. They are regarded as standard or reference values for correcting the quasi-integrals. Let the constants K 0, L 0x, L 0y, L 0z, and P 0z in equation (11) give way to K, L x, L y, L z, and Pz, respectively. Then the manifold correction algorithm of Nacozy does still work for the perturbed Keplerian problem. Hereafter, we call this extension method 1 (M1). It is worth emphasizing that the generalization, M1, is also suitable to the case of multiple bodies. As stated above, M1 gives corrections to the integrated velocities and positions, which approximately and synchronously satisfy the relations of the Keplerian energy, the three components of the angular momentum vector, and the z-component of the Laplace vector like equation (4). Before this method, a

1298 MA, WU, & ZHONG rigorous method with correction of the Keplerian energy, the magnitude of the angular momentum vector, and the Laplace integral was the linear transformation method ( M2) of Fukushima (2004). See this article for more information. In sum, a typical differencebetweenm1andm2liesinthatm1isapproximate to maintain the consistency of the relations of the five quantities, but M2 is rigorous to satisfy the three quantities. Thus, it is strongly desired to evaluate the validity of M1 and to especially know whether the difference has an effect on enhancing the quality of orbit integrations. In-depth numerical explorations will be necessary. For an overall comparison of various manifold corrections in an n-body problem, we consider the approximately individual Keplerian energycorrectionmethod(m3) of Wu et al. (2007). 3. EVALUATION OF THE METHOD In our numerical tests, we mainly take pure Keplerian twobody problems with various eccentricities as test models of numerical simulations. In addition, we pay attention to the case of a three-body problem made of the Sun, Jupiter, and Saturn. 3.1. Pure Keplerian Orbits Let us begin with the simplest case, numerical integration of a pure Keplerian problem (eq. [5]) with a massless particle. We select an orbit with initial conditions of x ¼ 1, y ¼ 0:005, z ¼ 0:005, ẋ ¼ 0:005, ẏ ¼ 1, and ż ¼ 0:005, and a fifth-order Runge-Kutta (RK5) algorithm with an invariant time step being 1/100 of the orbital period T, asabasicintegrator. Errors in all the Keplerian elements of RK5 and its corrections, M1, M2, and M3, are plotted in Figure 1. It is clear in Figure 1a that RK5 gains the linear growth of the error in the semimajor axis a with respect to time, but any of the three corrections achieves the error on the order of the machine epsilon. This displays that the three corrections have almost the same performance in controlling the Keplerian energy. Similarly, they should not produce a great difference in the error of the mean anomaly M (Fig. 1f ). However, the three corrections give distinct answers to the errors in the eccentricity e and the argument of pericenter!,as shown in Figures 1b and 1d. Both M1 and M2 can greatly improve the accuracy of the eccentricity. In addition, M2 is slightly better than M1, but M3 (like RK5) is the worst. This shows that the adjustments of K and L lead to the automatic adjustment of P for M1, and it is successful at correcting! by controlling the z-component of the Laplace vector. As to M3, it is only used to correct the Keplerian energy. On the other hand, it can be seen from Figures 1c and 1e that both M1 and M2 play the same role in the adjustment of the orbital inclination I and the longitude of the ascending node. This shows that M1 and M2 are equally valid in keeping with the angular momentum vector. Besides the accuracies of the orbital elements, we observe the relative position error of the orbit for each of the above methods. At the end of integration, both M1 and M3 raise the orbital accuracy by 4 orders of magnitude, as shown in Figure 2. As a slight difference between M1 and M3, M1 becomes slightly better. In particular, M2 is the best. Now, let us concentrate on the eccentricity dependence of element error for a Keplerian orbit when one of the corrections is applied. We fix initial orbital elements a ¼ 2 AU, I ¼ 20, ¼ 50,! ¼ 30, and M ¼ 40, but vary e from 0 to 0.7 at intervals of 0.01. The integrator used is still the same as that in Figure 1, and each orbit stops computing after 10 4 periods. Plotted are the errors in the orbital elements as functions of eccentricity Fig. 2. Relative position errors for the same orbit from Fig. 1. for the numerical schemes above. Like Figure 1, Figure 3 illustrates that M1, M2, and M3 have the same effectiveness in improving a and M. Compared with M3, both M1 and M2 drastically decrease the errors in I and, down to the limit of the machine epsilon. There are not even any differences in the errors of e and! between M1 and M2. On the other hand, the magnitude of the error in a does not depend on the eccentricity for any of the three corrections; neither do the magnitudes of the errors in I and for M1 and M2. Especially for M1 and M2, the accuracy of e and! seems to become better as e gets larger. But the error in M increases with e for each correction. In order to evaluate the case of the high eccentricities near 1, we use a fifth- to sixth-order Runge- Kutta-Fehlberg algorithm of variable time step as a basic test tool and obtain results (not plotted) similar to those of Figure 3. But there is only one difference, namely, that the error of M does not depend on the eccentricity at all for any correction. For the computations of various orbits in Figure 3, let us provide a comparison of computational cost, say, CPU times. At the end of numerical integration for all 70 orbits, the CPU times of RK5, M1, M2, and M3 are 94, 193, 103, and 96 s, respectively. The costs of the correction methods M2 and M3 increase slightly. As expected, the computational efficiency of M3 is the best, and that of M1 is the worst because M1 needs some additional time to solve the inverse matrix (EE T ) 1. A series of numerical experiments about manifold corrections are carried out in the case that the uncorrected integrators, e.g., RK5, give the accuracy of a certain integral on the order of not larger than 10 8 in a double-precision environment. These tests turn out to show that M1 and M2 are almost the same in improving the orbital elements. Without loss of generality, we shall compare M1 with M2 using real physical models. 3.2. Three-Body Problem We apply RK5 and its corrections M1, M2, and M3 to a general three-body problem of the Sun, Jupiter, and Saturn as an experiment model. The related physical parameters and initial conditions are quoted from those at J2000.0 in JPL s planetary ephemeris, DE405. The fixed time step adopted is 36.525 days, about 1/120 of Jupiter s orbital period. A 12th-order Cowell method is used to get a higher precision reference orbit for the construction of the errors in the orbital elements and the positions of each planet. In the heliocentric coordinate system, individual quantities K, P, andl givenbyequations(5)y(7)

Fig. 3. Eccentricity dependence of element error for a Keplerian orbit with the uncorrected integrator, RK5, of a fixed time step.

Fig. 4. Errors of all elements for Jupiter in the three-body problem of the Sun, Jupiter, and Saturn.

EXTENDING NACOZY S APPROACH 1301 Fig. 5. Same as Fig. 4, but for Saturn. vary slowly with time and satisfy equations (15)Y(17). As stated above, M1, M2, and M3 are still able to work in this sense. Unlike Figure 1, Figures 4 and 5 show that the three corrections arrive at the same effect of improving all the orbital elements for each planet by a great measure. The main agreement of M1, M2, and M3 lies in the errors in a, M, I, and. But M1 as well as M2 is better than M3 in correcting the errors of the eccentricity and the argument of pericenter (see Figs. 4b and 4d or Figs. 5b and 5d for more details). In a word, Figures 4 and 5 show that M1 is basically the same as M2 in the improvement of each orbital element. Here, three main points are worth noting. 1. The correction of a (besides M ) is the most valid of all the orbital elements. The reason can be found from equations (15)Y (17). Generally speaking, a is a small quantity, and jvj is rather smaller than jrj. Thus, the right hand-side of equation (16) is larger than that of equation (15), but smaller than that of equation (17). This seems to show that the accuracy of the reference value K, the accuracy of the reference value P, and the accuracy of the reference value L change from high to low. 2. M1 with M2 does not have an advantage over M3 in controlling the errors of I and, because L is nearly equal to the numerical value L given by equation (7). 3. The validity of orbital corrections for Jupiter is more explicit than that for Saturn. This is because the orbital accuracies for Jupiter are worse than those for Saturn in the uncorrected case with the same step size. The relative position error of each planet in Figure 6 is unlike that of the one-body in Figure 2. As an emphasis, here the three

1302 MA, WU, & ZHONG Fig. 6. Relative position errors of two planets in the three-body problem. corrections, M1, M2, and M3, are nearly the same after a short time of integration. 4. CONCLUSIONS Based on the integral invariant relations and the manifold correction of Nacozy, some correction schemes, with the adjustments of the varying Keplerian energy, Laplace integral, and/or angular momentum vector for a perturbed Keplerian one-body problem or each of multiple bodies, were recently developed along two different paths. One is the rigorous methods. One of the typical works is the linear transformation method (M2) of Fukushima. The other is the approximate methods with the leastsquares corrections. Along this line, the present paper has given a one-step correction method ( M1), where five independently slowvarying quantities or quasi-integrals, the Keplerian energy, the three components of the angular momentum vector, and the z-component of the Laplace vector, are approximately and simultaneously satisfied. The consistency of these quasi-integrals means the improvement of all the orbital elements of each object. By comparing between M1 and M2, we found that they have almost the same performance in improving all the orbital elements when the uncorrected integrators give the chosen integral a necessary precision on the order of not larger than 10 8 in a double-precision environment. The authors thank the referee for valuable suggestions. The research is supported by the Natural Science Foundation of China under contract 1056300. It is also supported by the Science Foundation of Jiangxi Province (0612034) and the Program for Innovative Research Team of Nanchang University. REFERENCES Avdyushev, V. A. 2003, Celest. Mech. Dyn. Astron., 87, 383 Liu, L., & Liao, X.-H.. 1994, Celest. Mech. Dyn. Astron., 59, 221 Baumgarte, J. 1972, Celest. Mech., 5, 490 Ma, D.-Z., Wu, X., & Liu, F. Y. 2008a, Int. J. Mod. Phys. C, in press Fukushima, T. 2003a, AJ, 126, 1097 Ma, D.-Z., Wu, X., & Zhu. J.-F. 2008b, NewA, 13, 216. 2003b, AJ, 126, 2567 Mikkola, S., & Innanen, K. 2002, AJ, 124, 3445. 2003c, AJ, 126, 3138 Nacozy, P. E. 1971, Ap&SS, 14, 40. 2004, AJ, 127, 3638 Szebehely, V., & Bettis, D. G. 1971, Ap&SS, 13, 365 Hairer, E., Lubich, C., & Wanner, G. 1999, Geometric Numerical Integration Wu, X., & He, J.-Z. 2006, Int. J. Mod. Phys. C, 17, 1613 ( Berlin: Springer) Wu, X., Huang, T.-Y., Wan, X.-S., & Zhang, H. 2007, AJ, 133, 2643 Huang, T.-Y., & Innanen, K. 1983, AJ, 88, 870 Wu, X., Zhu, J. F., He, J. Z., & Zhang, H. 2006, Comput. Phys. Commun., 175, Liu, L., & Liao, X.-H. 1988, Chinese Astron. Astrophys., 12, 26 15