The Astronomical Journal, 126:2567–2573, 2003 November # 2003. The American Astronomical Society. All rights reserved. Printed in U.S.A.
EFFICIENT ORBIT INTEGRATION BY DUAL SCALING FOR CONSISTENCY OF KEPLER ENERGY AND LAPLACE INTEGRAL Toshio Fukushima National Astronomical Observatory, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan;
[email protected] Received 2003 May 13; accepted 2003 July 22
ABSTRACT By adding the Laplace integral as the second auxiliary quantity to be integrated, we extend our scaling method to integrate quasi-Keplerian orbits numerically in order to suppress the growth of integration errors not only in the semimajor axis but also in the other orbital elements, especially in the eccentricity and in the longitude of pericenter. This time, the method simultaneously follows the time evolution of the Kepler energy and the Laplace integral in addition to integrating the usual equation of motion. By using a dual spatial scale transformation, it adjusts the position and velocity that are integrated in order to satisfy both the Kepler energy relation and another functional relation derived from the Laplace integral rigorously at each integration step. The scale factors for the position and for the velocity are set separately and are determined by solving a set of linear equations. Just as with the original scaling method, the new method is simple to implement, fast to compute, and applicable to a wide variety of integration methods, perturbation types, and problem complexities. With the exception of the J2 perturbation, the new method is superior to the original scaling method because it achieves significantly fewer integration errors for the physical properties such as the shape and the orientation of orbits at the cost of a negligibly small amount of additional computation. Key words: celestial mechanics — methods: n-body simulation — methods: numerical We mentioned in Paper I that the Kepler energy is not the unique integral invariant relation to be used for the manifold correction. In principle, any function of the position, the velocity, and/or the time can be selected. Also, the number of integral invariant relations is not limited to one. In this short paper, we will report on an extension of the scaling method by adding a scalar relation derived from the Laplace integral. As shown below, the extension refines the original scaling method in the sense that the error is reduced not only in the semimajor axis but also in the other orbital elements specifying the shape and the orientation of the orbit,1 again at a nominal increase of computational cost. In the following, we will give a detailed explanation of the new method in x 2 and present some numerical experiments to reveal its nature in x 3.
1. INTRODUCTION
Recently we developed a new device to enhance the quality of the numerical integration of quasi-Keplerian orbits (Fukushima 2003, hereafter Paper I) by combining the ideas of the integral invariant relation (Szebehely & Bettis 1970) and the manifold correction (Nacozy 1971). The method, which we call the scaling method for brevity, integrates not only the position and velocity considered but also the Kepler energy, the total energy in the unperturbed orbit. At each step of the integration, the positions and velocities integrated are adjusted so as to satisfy the defining relation of the Kepler energy exactly. The adjustment is done by a spatial scale transformation, where the scale factor is determined by solving the associated cubic equation with Newton’s method. As illustrated in Paper I, the scaling method achieved a drastic decrease of the integration errors both in truncation and in round-off at the cost of a negligibly small increase in computational labor. Of course, this is realized by sacrificing the possibility of using the Kepler energy as a monitor to check the accuracy of the orbit integration. However, the scaling does not alter the nature of the integration error significantly. For example, Figures 1 and 2 show the step-size dependence of the integration error of the unscaled and scaled integrations, respectively. Here we show the results for the fixed-order, fixed–step-size Adams implicit method in 1 PECE (predict, evaluate, correct, evaluate) mode. Although the relations between the order and the power index are different between the unscaled and scaled integrations, the existence of a power law with respect to the step size is unchanged. Local monitoring of the integration error is then possible as usual, and therefore, the applicability of various adaptive schemes such as a variable step size, variable order, or variable extrapolation stage remains unchanged.
2. METHOD
Let us begin with a perturbed one-body problem, whose equation of motion is written as dv l ¼ xþa; ð1Þ dt r3 where x is the position, v is the velocity, l GM is the gravitational constant, r |x| is the radius vector, and a is the perturbing acceleration. There are two key points in the manifold correction using the integral invariant relations. The first is to not only integrate the above equation of motion but also to simultaneously track the time development of some analytic function of time, position, and velocity, Q(t, x, v). 1 Although we originally designed the extension in order to reduce the errors in the eccentricity and the longitude of the pericenter, the method developed turns out to reduce those in the other elements, too.
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case, and therefore their integration errors also become zero in that case. Consequently, the above assumption is shown to be valid when the perturbation is negligible. In principle, it is possible that any number of functions could be integrated simultaneously, that any functional form could be allowed to define such functions, and that any method of correction is permissible as long as it is deterministic. In Paper I, we (1) fixed the number of functions at one, (2) adopted the Kepler energy K as the function, and (3) used the single4 spatial scale transformation as the method of correction. Here K is defined as K T þU ;
Fig. 1.—Step-size dependence of integration error: no scaling. Illustrated are the integration errors of a pure Kepler orbit as functions of the number of steps per period. Integrated are a pure Keplerian orbit of e = 0.10 and I = 23 for 128 orbital periods. The adopted integrators are the PECE mode of the implicit Adams method of various orders, and no scaling is applied. The starting tables were prepared by Gragg’s extrapolation method. The integration error of a certain order follows some power law with respect to the step size.
The other point is, at each integration step, to modify the position and velocity integrated so as to completely2 satisfy the relations QA = QI. Here QA Q(t, xI, vI) on the lefthand side is evaluated from the integrated position and velocity, xI and vI, and the time t by using the defining relation. The quantity QI on the right-hand side is obtained by integrating its time development equation. If the integrator used is error-free, these two must coincide with each other. In general, however, there remain inequalities. In that case, we venture to say that these are produced by the errors in the integrated position and velocity, although this is a one-sided vision. However, it would work when Q is a quasi-conserved quantity.3 In fact, the time variation of such quantities is zero in the unperturbed 2 Computationally speaking, with only round-off errors on the order of the machine epsilon or less. 3 The quasi-conservative quantities are those conserved in the unperturbed orbit.
T v2 =2 ;
U l=r :
ð2Þ
The time development equation of K is written simply as dK/dt = v x a. Note that this choice is only one of a huge number of possibilities. We consider that our success shown in Paper I can be attributed to the selection of a quasi-conserved quantity as the functional relation to be maintained during the integration. A natural extension would be to add other functions that are conserved in a purely Keplerian orbit. One candidate for such an additional quantity is the magnitude of the orbital angular momentum, L |L|, where L x µ v. However, as will be shown in the Appendix, we faced difficulties in finding suitable methodss of correction to satisfy the relations of K and L simultaneously. We explored a different possibility by using another quasiconserved quantity, the Laplace integral:5 P v µ L þ Ux ;
ð3Þ
whose equation of time development is dP ¼ 2ðv x aÞx ðx x aÞv ðx x vÞa : dt
ð4Þ
Note that the magnitude of the Laplace vector is in proportion to the orbital eccentricity e, as P |P| = le. Unfortunately, we could not find a proper method of scaling to keep the consistency of a scalar relation of K and a vector relation of P at the same time, mainly because the number of conditions to be satisfied was as large as four. Instead, we decided to use a scalar quantity derived from the Laplace vector, F | F |, where F is the first term of P: F v µ L ¼ ðv2 Þx ðx x vÞv :
ð5Þ
Note that this quantity is multiplicative with respect to scalings. Now we have two quantities, K and F, which depend differently on the position and velocity. Thus we introduce two scale factors to be determined, sX and sV, such that the scaling transformation is dual: ðx; vÞ ! ðsX x; sV vÞ :
ð6Þ
Then the equations to be satisfied become s2V T þ U=sX ¼ K ;
Fig. 2.—Step-size dependence of integration error: single scaling. Same as Fig. 1, but for the single scaling method. The adopted integrators are the PECE mode of the Adams implicit integrator of various orders. The powerlaw feature is the same. However, the relation between the order and the power index is different.
sX s2V F ¼ D ;
ð7Þ
4 This means that the scaling factors of the position and velocity are the same. 5 Hereafter, we call it the Laplace vector to emphasize the fact that it is a three-dimensional vector.
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where D jP Uxj :
ð8Þ
Note that T, U, and F are analytic functions of the integrated position and the velocity, while K and D are obtained by the associated integration of K and P. Fortunately, this set of equations has a unique, meaningful,6 and analytical solution: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TD þ FU KD ; sV ¼ : ð9Þ sX ¼ FK TD þ FU As a result, the procedure to determine the scale factors is different from the original (single) scaling method. Note that the above form of solution is robust, in the sense that it has no singularity with respect to the value of eccentricity, unless the orbit considered is close to being rectilinear, namely, when F is nearly zero. Also note that the new method allows one to use different kind of integrators for the position and the velocity, such as the Sto¨rmer-Cowell method for the former and the Adams method for the latter, since the scaling factors are determined separately. At this stage, the reader may wonder why we do not integrate a scalar D directly in place of a vector P. Of course, such an approach would decrease7 the amount of additional cost of computation. However, if we do that, we violate the assumption that the error of D is much smaller than those of the integrated position and velocity. In actuality, the quantity D varies as much as the position or the velocity; therefore, its integration error is expected to be as large as those of the position and velocity. Let us add an important note. Just as with K, the magnitude of variation in P is small when compared with its own value. Thus, it is wise to integrate not K and P directly but their deviations from the initial values, DK K K0 and DP P P0, instead. Note that the time development equation remains virtually unchanged by this trick, as d DK ¼ vxa ; dt
d DP ¼ 2ðv x aÞx ðx x aÞv ðx x vÞa : ð10Þ dt
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This generalization does not apply to the case of multiple bodies, which is straightforward, just as in the original scaling method; simply integrating DK and DP for each body and adjusting their x and v separately by the scale factors is different body by body. Note that the above procedure requires only a negligible increase in the actual computation. This is mainly because the computation of the right-hand sides of equation (10) is very easy—its major part, the calculation of the perturbative accelerations, has already been done in evaluating the right-hand side of the equation of motion. 3. NUMERICAL EXPERIMENTS
Here we show some results of numerical experiments using the new scaling method. First of all, we confirmed that the new method has almost the same good properties as the original single scaling method does. Namely, the method is applicable to (1) any type of problem, from a full N-body system to restricted problems and up to the pure Keplerian orbits; (2) any kind of perturbations, including dissipative forces and velocitydependent accelerations; (3) any method of integration covering the Runge-Kutta families, the extrapolation methods, and the symmetric and nonsymmetric linear multistep methods; (4) any parameter of the integration methods, such as the step size, the order, the number of extrapolation stages, and the policy of iterations in realizing the implicit integrators as the PECE usage of the linear multistep methods; and (5) any number of celestial bodies to be integrated simultaneously. We omit the graphs that show these confirmations, since they are just the same as we gave in Paper I. Below, we concentrate on the difference between the single and dual scaling methods. Let us begin with the pure Keplerian orbits. Figures 3–5 show the normalized errors in the orbital elements, (Da)/a,
As we experienced in the generalization of Encke’s method (Fukushima 1996), this technique greatly reduces the rounding off. Summarizing the above, we explicitly state the procedure at each integration step: 1. Integrate the equation of motion (eq. [1]) and the time development equations for DK and DP (eq. [10]) simultaneously. 2. Compute K and P from the integrated deviations, DK and DP, and the initial values, K0 and P0, as K ¼ K0 þ DK ;
P ¼ P0 þ DP :
ð11Þ
3. Evaluate T, U, F, and D by using equations (2), (5), and (8) from the integrated position and velocity and the computed Laplace vector. 4. Determine sX and sV by using equation (9) from T, U, F, D, and K. 5. Correct x and v by the dual scaling transformation, equation (6). 6 In the sense that both the scale factors are in general positive and close to unity. 7 From a practical viewpoint, this brings little gain, since the additional cost is in any case negligibly small.
Fig. 3.—Element error of Keplerian orbit: no scaling. Shown are the normalized errors in the orbital elements, (Da)/a, De, DI, (sin I )D , e D$, and DL0, for the unscaled integration. Integrated are a pure Keplerian orbit of e = 0.10 and I = 23 for around 106 orbital periods. The integrator used was the PECE mode of the implicit 10th-order Adams method with a step size of 1/128 the orbital period. With respect to time, the error of the mean longitude at the epoch, L0 M0 + $, grows quadratically with respect to time, those of (Da)/a, De, and e D$ do so linearly, and those of DI and (sin I )D increase in proportion to the 12 power.
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Fig. 4.—Element error of Keplerian orbit: single scaling. Same as Fig. 3, but for the single scaling method. Growing linearly with respect to time are all the element errors except that of the semimajor axis, a, which remains randomly on the order of the machine epsilon.
Fig. 6.—Eccentricity dependence of integration error: eccentricity. Compared are the eccentricity dependence of the integration error without scaling and with single and dual scalings. Shown are the errors in the eccentricity of a pure Keplerian orbit after 106 revolutions. The dual scaling method is the best.
De, DI, (sin I )D , e D$, and DL0, for the cases without scaling, and after the single and dual scalings, respectively. Here $ + ! is the longitude of pericenter and L0 M0 + $ is the mean longitude at the epoch. The integrated orbit is of moderate eccentricity and inclination, with e = 0.1 and I = 23 . The integrator used is the PECE mode of the implicit 10th-order Adams method, where the starting tables were prepared by Gragg’s extrapolation method. The step size was fixed as 1/128 of the orbital period. Compared with the case of no scaling, the errors of a and L0 drastically decrease in both the scaling methods. This is the result of the maintenance of Kepler energy consistency, which we experienced in Paper I. On the other hand, the dual scaling method significantly reduces the errors in two more elements, the eccentricity and the longitude of pericenter. More specifically, the new method keeps them bounded at the level of 1012 as shown in Figure 5. Both of these elements have a deep connection to the Laplace vector and therefore are tightly related to the
newly introduced quantity, F. However, it is puzzling why two elements are kept accurate by adjusting only one quantity, F. In the single scaling method, the error in e is the largest, as illustrated in Figure 4. Namely, it is larger than that of L0 by around one digit. Also, e D$ is comparable to the error in L0. Since the dual scaling method succeeds in suppressing the two largest error components remaining in the single scaling method, the total error in the relative position is reduced by around one digit in the dual scaling method. In fact, Figures 6 and 7 show the eccentricity dependence of the integration errors in the eccentricity and in the position after 106 revolutions, respectively. It is clear that the dual scaling method is the best method. Let us move on to the perturbed case. First, we confirmed that when the perturbation is sufficiently weak, the results are practically the same as in the pure Keplerian case. A good example is Figure 8, which shows the relative position error of an orbit of the nominal eccentricity of e = 0.1 and
Fig. 5.—Element error of Keplerian orbit: dual scaling. Same as Fig. 3, but for the dual scaling method. This time, the errors in the eccentricity, e, and in the longitude of pericenter, $ + !, remain bounded with a nonzero mean.
Fig. 7.—Eccentricity dependence of integration error: relative position. Same as Fig. 6, but for the error in the position. We omit the graph for the no-scaling case, since it is too large at more than 1. Again, the dual scaling has the least error.
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Fig. 8.—Relative position error of perturbed Keplerian orbit: general relativity. Compared are the time growth of integration errors without scaling and with single and dual scalings. Shown are the position of an orbit of nominal eccentricity of e = 0.1 and the perturbation by the post-Newtonian acceleration of relative magnitude 108.
the perturbation by the post-Newtonian acceleration whose relative magnitude is of the order of 108. Then, as a typical case of moderate strength perturbation, we integrated an orbit perturbed by the air drag in a standard form. Figure 9 manifests a great increase in precision by the dual scaling method. Unfortunately, the dual scaling method is not always superior to the single method, as in the case of J2 perturbation shown in Figure 10. Thus, one must take care in its application. As an example of the N-body problems, we integrated a general three-body problem of the Sun, Jupiter, and Saturn and examined the errors in Jupiter’s osculating elements. Each of Figures 11–16 compares the normalized integration error of each element obtained without scaling to the single and dual scalings, respectively. The condition of the integration was the same as those in the pure Keplerian case. The initial conditions were quoted from those at J2000.0 in the latest planetary/lunar ephemeris, DE405. The planetary masses were set the same as those used in creating DE405 also.
Fig. 9.—Relative position error of perturbed Keplerian orbit: air drag. Same as Fig. 8, but for the perturbation of air drag in a standard form.
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Fig. 10.—Relative position error of perturbed Keplerian orbit: J2 perturbation. Same as Fig. 8, but for the perturbation of J2 of relative magnitude 103.
This time the growth of errors in e, , I, and $ was significantly reduced in the case of the dual scaling method, while that of a is fundamentally unchanged. Unfortunately, we could not clarify why the integration errors of the orientation of the angular momentum, and I, were also suppressed despite the fact that we did not control the angular momentum directly. Of course, the increase in L0 of the dual scaling method, when compared with that of the single scaling one, is large enough that the error in relative position also increases. From the physical point of view, however, the shape and the orientation of the orbit is roughly a few digits better maintained in the dual scaling method than in the single one. This will enhance the reliability of orbit integrations, especially when the interactions between inner and outer orbits such as those of Jupiter and Saturn are concerned. On the other hand, when the perturbation is strong enough, as in the Moon’s main problem, we could not find any significant difference between the two scaling methods.
Fig. 11.—Element error of three-body problem: semimajor axis. Compared are the normalized integration errors of the semimajor axis of Jupiter in the Sun-Jupiter-Saturn general three-body problem integrated without scaling and with single and dual scalings, respectively.
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Fig. 12.—Element error of three-body problem: eccentricity. Same as Fig. 11, but for the eccentricity of Jupiter.
Fig. 14.—Element error of three-body problem: longitude of node. Same as Fig. 11, but for the longitude of the ascending node of Jupiter.
In summarizing the above, we conclude that the dual scaling method is overall superior to the single scaling method mainly because the physical properties of the orbit are better maintained in the former than in the latter.
As an extension of the single scaling method to keep the Kepler energy consistent, we have invented another new approach to integrate the quasi-Keplerian orbits precisely. The new method integrates simultaneously not only the usual equation of motion and the time evolution of the Kepler energy but also the development of the Laplace vector. The method then corrects at each integration step the position and velocity by a dual spatial scale transformation, which rescales the position and vector integrated by different scale factors, respectively. The correction is done to satisfy the defining relations of the Kepler energy, K, and a new quantity F |v µ (x µ v)|. The latter has a deep relation to the Laplace integral or the so-called eccentricity vector. The two scales, one for the position and the other for the velocity, are carefully deter-
mined from the numerical values of K and F. In this instance, we needed no iterative procedure to reach the solution of the consistency equations. In fact, the scale factors are explicitly described as functions of the integrated position and velocity, K and F evaluated, and other quantities related. We note that the new method inherits a number of good features from the original single scaling method, such as easy implementation, a negligible additional cost of computation, and wide applicability to integrators, perturbations, and problems. Numerical experiments showed that, with the exception of a few cases, such as the J2 perturbation, the method of dual scaling in general produces a physically better orbit solution than the single scaling method. In case of pure Keplerian orbits, the errors in a, e, and $ remain finite in the dual scaling method, while only that in a does in the single scaling method. In the perturbed cases, the errors in the other elements as and I are also reduced, depending on the strength of the perturbation. In conclusion, we generally recommend the use of the dual scaling method in place of the single scaling method, because it promises the possibility of a significant gain in
Fig. 13.—Element error of three-body problem: inclination. Same as Fig. 11, but for the inclination of Jupiter.
Fig. 15.—Element error of three-body problem: longitude of pericenter. Same as Fig. 11, but for Jupiter’s longitude of perihelion.
4. CONCLUSION
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different from each other, we try to introduce a dual scale transformation as we did in the main text. This time, the equations to be satisfied become s2V T þ U=sX ¼ K ;
sX sV L ¼ J ;
ðA2Þ
where J is the same as L in its definition but is obtained by integrating the equation of the time development on L as dJ 1 ¼ ½ðx µ vÞ x ðx µ aÞ : dt J
ðA3Þ
Note that there is no difficulty in integrating this quantity simultaneously with that of K and the equation of motion. Now, solving the second part of the consistency equations (eq. [A2]), we have sX ¼ Fig. 16.—Element error of three-body problem: mean longitude at epoch. Same as Fig. 11, but for Jupiter’s mean longitude at the epoch.
precision and in the physical quality of orbit integrations when the perturbation is weak, while requiring a negligibly small amount of additional computational labor. APPENDIX
In the main text, we showed the feasibility of satisfying the simultaneous consistency of the Kepler energy K and a scalar quantity F, the latter of which was derived from the Laplace vector, P. The reader may wonder why we adopted a quantity based not on the orbital angular momentum, L, but on P. However, this is more difficult than it appears. In this appendix, we show how we failed to find a suitable procedure of dual scaling in this case. Let us replace F with L, the magnitude of L: ðA1Þ
This quantity is multiplicative with respect to scalings as F, but in a different way. Since we have two quantities, K and L, whose dependence on the position and velocity are
Fukushima, T. 1996, AJ, 112, 1263 ———. 2003, AJ, 125, 1097 Nacozy, P. E. 1971, Ap&SS, 14, 40
ðA4Þ
Using this, we eliminate sX in the first part of equation (A2) and obtain a quadratic equation in sV as TJs2V þ LUsV KJ ¼ 0 :
ðA5Þ
Unfortunately, there is no assurance that this equation will always have a unique real solution. In fact, the discriminant of the quadratic equation, W ðLUÞ2 þ 4TKJ 2 ;
DIFFICULTY IN DUAL SCALING FOR CONSISTENCY OF KEPLER ENERGY AND ANGULAR MOMENTUM
L jLj ¼ jx µ vj :
J : sV L
ðA6Þ
is not nonnegative definite. Of course, T is always greater than zero. However, K < 0 for elliptical orbits. Therefore, W can be either positive or negative in this case depending on the ratio of TKJ2 and (LU)2. For example, if the integrations are all error-free, then W must be zero. This is because sX = sV = 1 should be the solution in that case. This fact means that the signature of W can be either positive or negative depending on the manner of the integration error accumulation, which is unpredictable in principle. In fact, numerical experiments have shown a 50-50 chance of negative W. Further, there exist two real solutions even if W happens to be positive. Both of them are close to each other, since W is generally small. In this case, we could not find any condition to select one from the two. All in all, our experience was that the replacement of F by L was not easy.
REFERENCES Szebehely, V., & Bettis, D. G. 1970, in IAU Colloq. 10, Gravitational N-Body Problem, ed. M. Lecar (Dordrecht: Reidel), 136
