The Astronomical Journal, 128:3108–3113, 2004 December # 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.
EFFICIENT INTEGRATION OF HIGHLY ECCENTRIC ORBITS BY QUADRUPLE SCALING FOR KUSTAANHEIMO-STIEFEL REGULARIZATION Toshio Fukushima National Astronomical Observatory, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan;
[email protected] Receivved 2004 Aug gust 11; accepted 2004 September 1
ABSTRACT We have extended the single scaling method for Kustaanheimo-Stiefel (K-S) regularization by applying different scaling factors to each component of the four-dimensional harmonic oscillator associated with the regularization. This is achieved by monitoring the time development of the total energy of each component of the harmonic oscillator and rescaling the magnitude of the position and velocity of the corresponding component so as to satisfy the defining relation for the harmonic energy. To perform the monitoring increases the number of variables to be integrated per celestial body from 10 to 13; however, the extra cost of computation is still negligible compared with that of the perturbing acceleration. The resulting method of quadruple scaling significantly enhances the numerical stability of orbit integrations. In the case of unperturbed orbits, the new method when applied at every integration step reduces to the machine-epsilon level the errors in all the orbital elements except the mean longitude at epoch, if sufficiently high order integrators are used with sufficiently small step sizes. This remarkable feature is lost when the scaling is applied at every apocenter. In the case of perturbed orbits, we confirm the superiority of the quadruple scaling method applied at every integration step over the other scaling methods for K-S regularized orbital motions unless round-off plays the key role in the accumulation of integration error. Key words: celestial mechanics — methods: numerical
ularization without scaling, the observed difference in growth rate leads to a large difference in the error magnitude over the long run. This feature is independent of the magnitude of the eccentricity. Figure 2 illustrates the eccentricity dependence of the longitude error of a Keplerian orbit after a certain long integration time, 32,768 orbital periods. For HALCA, this corresponds to 22 years, 3 times longer than its projected lifetime. In the figure, we omitted the results of no scaling and the E-J scaling. This is because their curves show the same flatness with respect to the eccentricity and have already been given in Figure 2 of Paper IX. The key to the success of the single scaling method, as well as the E-J scaling method, is the existence of the Kepler energy integral in the unperturbed form of the K-S regularized equation of motion (Kustaanheimo & Stiefel 1965; Stiefel & Scheifele 1971). The situation is the same as we first faced in our quest for the manifold correction methods (see Paper I). Of course, we arrived at the Kepler energy relation in terms of the K-S variables in the fictitious spacetime by translating the original Kepler energy relation expressed in terms of the real position and velocity vectors into the K-S framework (Aarseth 2003). Changing this viewpoint, however, we may interpret the Kepler energy relation in K-S variables to be nothing but the law of total energy conservation for a four-dimensional harmonic oscillator in the fictitious spacetime introduced by the K-S regularization. Since each component of the harmonic oscillator moves independently of the others in the unperturbed case, we have not one but four energy integrals corresponding to each component. This means that we can extend the single scaling method developed in Paper IX by increasing the number of energy integrals from one to four. Thus we have four scaling factors, each of which is multiplied by the pair of the position and velocity of the corresponding component of the harmonic oscillator. We therefore name the new method the
1. INTRODUCTION Triggered by practical needs to analyze the long-term evolution (on the order of a decade) of the highly eccentric (e 0:6) orbit of HALCA (Hirabayashi et al. 2000), an artificial Earth satellite of the Japanese space VLBI program, VSOP, we started an exploration of efficient techniques to numerically integrate highly eccentric orbits. Driven by the success of the methods of manifold correction for low and moderately eccentric orbits (Fukushima 2003a, 2003b, 2003c, 2004a, 2004b, 2004c, 2004d, 2004e),1 we recently applied the single scaling method, the simplest method of manifold correction, to the Kustaanheimo-Stiefel (K-S) regularization (Fukushima 2004f ).2 Through numerical experiments, we learned that the single scaling method, whether applied at every integration step or at every apocenter, is effective in integrating such highly eccentric orbits. We also confirmed that the energy–angular momentum (E-J ) scaling method designed for the K-S regularization and applied at every apocenter (Aarseth 2003) is effective as well (see Paper IX for details). Figure 1 shows the performance of these scaling methods applied to the K-S regularization in integrating a model orbit of HALCA. The growth of the longitude error, which is the largest error component, with respect to the real time changes from being quadratic for the original K-S regularization and the E-J scaling (applied at every apocenter)3 to linear for the single scaling, whether applied at every apocenter or at every integration step. Since the magnitudes of the initial errors are almost the same for these scaling methods and the original K-S reg-
1
Hereafter Papers I–VIII, respectively. Hereafter Paper IX. 3 The application of the E-J scaling at every integration step frequently faces numerical instabilities. Therefore, we only deal with its application at every apocenter as Aarseth (2003) recommended. 2
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Fig. 1.—Error growth of orbital motions integrated by various methods of scaling with the K-S regularization. Illustrated are the errors in the mean longitude at epoch, L0 , of a highly eccentric (e ¼ 0:6) Keplerian orbit integrated by various scaling methods applied with the K-S regularization: (1) no scaling, (2) the energy–angular momentum (E-J ) scaling applied at every apocenter, (3) the single scaling applied at every apocenter, (4) the single scaling applied at every integration step, (5) the quadruple scaling applied at every apocenter, and (6) the quadruple scaling applied at every integration step. The notation ‘‘@A’’ means that the correction is applied at every apocenter. The adopted integrators were the implicit Adams methods in PECE mode, the step size was fixed throughout the integration and chosen such that one orbital period is covered by 90 steps, the starting tables were prepared using Gragg’s extrapolation method, and the errors were measured by comparing with reference solutions obtained by the same integrator and with the same model parameters but half the step size. The order of the Adams method was the highest among those that led to no instabilities, 14th for the case of quadruple scaling applied at every step and 11th for the others.
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Fig. 2.—Eccentricity dependence of longitude errors integrated by various methods of scaling applied with K-S regularization. Similar to Fig. 1, but the longitude errors after 32,768 periods are plotted as functions of eccentricity. The errors of the single and quadruple scaling applied at every apocenter are practically the same and larger than the cases of application at every integration step. On the other hand, if applied at every integration step, the errors of the quadruple scaling are significantly smaller than those of the single scaling. This difference mainly owes to the difference in the maximum available order of the implicit Adams method in PECE mode, 14th for the quadruple scaling applied at every step and 11th for the others.
perturbing acceleration, which is a function of uj , uj0 , and t in general. In the unperturbed case where Qj ¼ 0, the total energy of each component of the harmonic oscillator, Hj ¼ 12 ½(uj0 )2 þ 12 hK u 2j ;
quadruple scaling method. We have confirmed its superiority in the unperturbed case (Figs. 1 and 2). However, it is a different issue whether the observed superiority persists for perturbed orbits. In this paper, we report that the new scaling method further enhances the K-S regularization whether perturbations are present or not. In the following, we describe the principle of the quadruple scaling method in x 2 and present a numerical comparison with existing methods in x 3.
uj00 þ 12 hK uj ¼ Qj t 0 ¼ r;
( j ¼ 1; 2; 3; 4);
Hj0 ¼ uj0 Qj u 2j ;
hK0 ¼ 4;
j¼1
ð2Þ
4 X j¼1
4 X j¼1
u 2j ;
¼
4 1X u0Q j; r j¼1 j
ð3Þ
and primes indicate differentiation with respect to the fictitious time, s. Here r is the radius in real space, uj is the jth component of the position vector of the four-dimensional harmonic oscillator associated with the K-S regularization, t is the real time, hK is the negative Kepler energy (which takes positive values in elliptical orbits), and Qj is the jth component of the
ð5Þ
Hj ¼
; 4
ð6Þ
where G (M þ m) is the gravitational constant of the twobody problem. This relation holds whether perturbations exist or not,5 since
where r¼
j ¼ 1; 2; 3; 4:
Using these Hj , the Kepler energy relation we deployed in Paper IX can be rewritten into the simple form 4 X
ð1Þ
ð4Þ
is conserved.4 Thus, we select the Hj as the quasi integrals to be monitored during the integration in the new scaling method. The equation of time development for the Hj is simply derived from equation (1) as
2. QUADRUPLE SCALING METHOD FOR K-S REGULARIZED ORBITAL MOTION The equation of motion of a K-S regularized orbit (Stiefel & Scheifele 1971) is expressed in components as
j ¼ 1; 2; 3; 4;
Hj0 ¼
4 X j¼1
uj0 Qj
4 X
u 2j ¼ 0:
ð7Þ
j¼1
Using this identity, we can reduce the number of harmonic energies to be integrated from four to three. As for selection of the component whose harmonic energy is not to be integrated, we adopt the policy of assigning the component with the largest value of Hj initially. This is in order to minimize the 4
Hereafter we call the Hj harmonic energies for short. Rather, the equation of motion of the Kepler energy is derived from the requirement that this relation holds even in the perturbed case. 5
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error in integrating the harmonic energies, since the errors of the quasi integrals are generally proportional to their magnitude. Hereafter, we number such component as the fourth without loss of generality. In the course of numerical integration of equations (1), (2), and (5), the harmonic energy relation (eq. [4]) may not always hold. In this case, we assume that the errors in the fast variables, uj and uj0 , are the cause of the observed inequality and correct both of them to satisfy the relation. As the method of correction, we select the single scaling (uj ; uj0 ) ! (j uj ; j uj0 );
j ¼ 1; 2; 3; 4;
ð8Þ
where the scaling factor j is different from component to component. As in Paper IX, the scaling factor is uniquely determined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Hj j ¼ ; ð9Þ 2 0 2(uj ) þ hK u 2j where the quantities uj , uj0 , and hK on the right-hand side are those integrated. As discussed in Paper IX, this scaling is almost always feasible. The Kepler energy relation (eq. [6]) is automatically satisfied since we arranged for the four harmonic energies to satisfy the relation. Thus the semimajor axis, a, is expected to be error-free in the new method for the unperturbed case. As in Paper IX, we again skip the extension to n-body integrations because it is trivial. One only has to determine the scaling factor for each component of the harmonic oscillator associated with each body and apply the scaling, component by component and body by body. Finally, let us reprise some practical techniques to enhance the performance of the new method. First, because the quasi integrals, Hj and hK , remain almost constant throughout the integration, in order to reduce the accumulation of round-off error it is better not to integrate them but their deviation from the initial values, hK hK (hK )0 ; Hj Hj (Hj )0 ( j ¼ 1; 2; 3):
ð10Þ
In practice, we replace the second component of equation (2) and all of equation (5) with ( hK ) 0 ¼ 4; ( Hj ) 0 ¼ uj0 Qj u 2j ( j ¼ 1; 2; 3);
ð11Þ
and evaluate hK and Hj by ð12Þ
whenever needed. We derive the fourth6 component of the harmonic energies from the Kepler energy relation as 3 X
Hj ;
ð13Þ
j¼1
where we assume that the initial values of Hj satisfy the Kepler energy relation (eq. [6]). Next, in order to reduce the accumulation of round-off errors in the integration of the secularly growing component, t, 6
we separately treat its integer (k) and fractional (t ) parts measured in some unit, T, by applying the following procedure at each integration step: if (t* > T) { k += 2; t* -= 2T; } else if (t* < T) { k -= 2; t* += 2T; }; (see Paper IX for details). As for when to apply the quadruple scaling, we consider two options—at every integration step and at every apocenter. Here the phrase ‘‘at apocenter’’ is meant to be at the integration step when the radius r reaches a local maximum (see Appendix C of Paper IX). Before concluding this section, we stress that the quadruple scaling described above leaves the universality of the original K-S formulation unchanged. Thus it is applicable to all types of orbits: elliptical, parabolic, hyperbolic, and linear. This fact implies that the results of the new scaling method will not significantly change with respect to the orbital eccentricity. 3. NUMERICAL EXPERIMENTS
hK ¼ (hK )0 þ hK ; Hj ¼ (Hj )0 þ Hj ( j ¼ 1; 2; 3)
H4 ¼ (H4 )0
Fig. 3.—Element errors of a Keplerian orbit regularized by the K-S transformation with the quadruple scaling applied at every step. The errors in position and in the modified orbital elements of a highly eccentric Keplerian orbit are plotted as functions of the real time on a log-log scale. The adopted integrator was the 12th-order implicit Adams method in PECE mode and the step size was chosen such that one orbital period is covered by 90 steps. Most of the curves are offset by some factor to avoid overlap. All the element errors except that of the mean longitude at epoch, L0 , remain at the level of the machine epsilon throughout the integration period. Compare with Figs. 3–6 of Paper IX, which illustrate the cases of no scaling, the E-J scaling applied at every apocenter, the single scaling applied at every apocenter, and the single scaling applied at every step, respectively.
More specifically, the component whose harmonic energy is not integrated.
We next examine the effects of the quadruple scaling method for K-S regularized orbits and compare them with the other scaling methods. In the following, we restrict ourselves to perturbed two-body problems of elliptical but highly eccentric (0:5 < e < 1) orbits. As in Papers I–IX we measure the integration errors by taking the difference from a reference solution obtained using the same integrator under the same initial conditions with the same model parameters but with half the step size. In K-S regularized motions, we must ensure that the differences are taken for the same real time to obtain meaningful results. To do this, we make a correction due to the errors in integrating the real time (see Appendix B of Paper IX). We begin with the unperturbed case. Figure 3 illustrates an example of the error growth of a highly (e ¼ 0:6) eccentric
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Fig. 4.—Step-size dependence of the error growth of the eccentricity of a Keplerian orbit regularized by the K-S transformation with the quadruple scaling applied at every integration step. This is the same as the e curve in Fig. 3 but for various step sizes. The attached numbers are the number of steps per orbital period. Most of curves are offset by some factor to avoid overlap.
Keplerian orbit integrated in K-S regularized form with application of the quadruple scaling at every integration step. Plotted are the position error and the errors in six modified orbital elements. All the element errors except that in the mean longitude at epoch, L0 , are completely suppressed. (More rigorously speaking, they randomly fluctuate at the level of the machine epsilon for a sufficiently long time, say, half a million orbital periods.) The exception, L0 , grows linearly with respect to time. As a result, the position errors also increase linearly in the long run. This is a surprising feature if one considers the fact that the number of scaling factors used is four whereas the number of elements whose errors are reduced down to the level of machine epsilon is five. However, this good property is not always achieved. Consider Figure 4, which shows the step-size dependence of e with the integrator fixed as the 12th-order implicit Adams method in PECE ( predict, evaluate, correct, evaluate) mode, and Figure 5, which illustrates the dependence of I on the
Fig. 5.—Order dependence of the error growth of the inclination of a Keplerian orbit regularized by K-S transformation with quadruple scaling. Same as the I curve of inclination in Fig. 3, but for various orders of the implicit Adams method as labeled.
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Fig. 6.—Stability enhancement of the implicit Adams method due to applying the quadruple scaling to K-S regularized orbital motions at every step. Illustrated is the highest stable order of the implicit Adams method as a function of the size. Note that the maximum stable order in the quadruple scaling method applied at every step grows more rapidly than the other scaling methods for the K-S regularization: no scaling, the E-J scaling applied at every apocenter, the single scaling applied at every apocenter and at every step, and the quadruple scaling applied at every apocenter. We also plot a separation curve for the quadruple scaling method applied at every integration step, which separates the region where the element errors other than the mean longitude at epoch grow linearly and the region where the errors are periodic at the machine-epsilon level, so that they remain finite for a long time. All the curves illustrated are independent of the magnitude of the eccentricity.
order of the implicit Adams method in PECE mode with the step size fixed so as to cover one orbital period every 90 steps. These figures indicate that the element errors are completely suppressed if integrated by a sufficiently high order integrator with a sufficiently small step size. This is the same as we experienced with the orbital longitude methods in Papers VII and VIII. The main part of the K-S regularized orbit integration is nothing but the integration of a harmonic oscillator. Therefore, the above phenomenon implies that the numerical integration of a harmonic oscillator by the (single) scaling method is almost error-free if one uses a sufficiently high order integrator with a sufficiently small step size. Another remarkable feature of the quadruple scaling is the enhancement of the stability of numerical integrators. Figure 6 shows the maximum stable order of the implicit Adams method in PECE mode as a function of the number of steps per orbital period for various scaling methods with the K-S regularization. The result of the quadruple scaling method when applied at every integration step is significantly superior to the other cases, all of which are practically the same: (1) no scaling, (2) the E-J scaling applied at every apocenter, (3) the single scaling applied at every apocenter, (4) the single scaling applied at every integration step, and (5) the quadruple scaling applied at every apocenter. The difference comes from both the nature of the scaling to be applied and the frequency of its application. In this diagram, we also plot a separation curve for the quadruple scaling method applied at every integration step, which separates the region where the element errors (aside from L0) grow linearly and the region where they remain finite at the machine-epsilon level. Note that all the features shown in the figure are independent of the magnitude of the orbital eccentricity. Unfortunately, the desirable characteristics observed above no longer hold when the times of scaling are limited to every apocenter. Figure 7 shows the growth of the element errors for
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Fig. 7.—Same as Fig. 3, but with the application of the quadruple scaling limited to every apocenter.
the quadruple scaling applied at every apocenter. This is almost the same as the single scaling methods applied at every integration step or at every apocenter, as shown in Figures 5 and 6 of Paper IX. In conclusion, for unperturbed orbits the quadruple scaling method applied at every integration step gives the best performance, as has already been manifested in Figures 1 and 2. Let us turn to the perturbed orbits. Having confirmed that the situation observed in case of unperturbed orbits remains practically the same under sufficiently weak perturbations, we concentrate on the moderate and strong cases. Precise and fast integration of the effects of outer perturbers is the most important issue in conducting n-body integrations. Figure 8 illustrates a typical such case, Icarus under Jupiter’s third-body perturbations. In preparing the figure, we set the initial conditions of Icarus and Jupiter as those at J2000.0. Also, the orbit of Jupiter was fixed as its osculating one at J2000.0. The conditions of the integration are the same as in Figure 1, namely, the 14th-order implicit Adams method in PECE mode with the step size set to cover one nominal orbital period of Icarus every 90 steps. Thus the averaged step size is as large as
Fig. 8.—Effect of the scaling on the K-S regularized orbital motion under a third body’s perturbations. Compared are the longitude errors of a model Icarus under Jupiter’s perturbation. The orbit of Jupiter was given as a fixed Keplerian orbit. The step size in the fictitious time was fixed so as to cover one nominal orbital period in 90 steps. This corresponds to an averaged step size of 3.6 days in real time.
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Fig. 9.—Same as Fig. 8, but integrated is the orbit of HALCA, an artificial Earth satellite with e ¼ 0:600, under perturbations due to a model air drag. The perigee altitude was as low as 560 km. The step size was chosen such that 90 steps cover the nominal orbital period, 6.3 hr.
3.6 days. We confirm the superiority of the quadruple scaling applied at every integration step. The difference is mainly due to the difference in the maximum available order of the integrators shown in Figure 6. The quadruple scaling at every integration step is also effective for dissipative perturbations, as shown in Figure 9. This is a comparison of integration errors for HALCA under the perturbations of air drag. In the case of oblateness perturbations, the superiority of the quadruple scaling diminishes. Figure 10 shows the similarity of the integration errors for HALCA under the J2 perturbation of Earth for various scaling methods with K-S regularization. We must point out a weak point of the quadruple scaling. Figure 11 plots the integration errors of Icarus under the Sun’s general relativistic perturbations with a step size chosen small enough that round-off plays the key role. We omit the cases of no scaling and the E-J scaling applied at every apocenter, since their errors grow in proportion to the 3=2 power of time and thus are much larger than the results of the single and quadruple scalings shown here. Roughly speaking, the four curves shown in the figure have the same behavior, namely, the errors
Fig. 10.—Same as Fig. 9, but under the J2 perturbations of Earth. This time the step size was halved. The inclination of HALCA is as moderate as I ¼ 31 . In order to show the similarity of the curves clearly, we offset the results that employ scaling.
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find that the results of the quadruple scaling are somewhat worse than those of the single scaling whether they are applied at every integration step or at every apocenter. Thus the best combination over the long run is the single scaling method applied at every apocenter, which is the same conclusion we reached in Paper IX. This is because the total number of arithmetic operations and transcendental function calls per orbital period is minimized in that combination. 4. CONCLUSION
Fig. 11.—Same as Fig. 8, but under the Sun’s general relativistic perturbation. This time the step size was set so small as to cover one orbital period by 256 steps. The order of the implicit Adams methods was set to the highest that led to no numerical instabilities for the given step size, the 17th for the quadruple scaling applied at every step and the 14th for the others.
first increase in proportion to the square root of time for some length of time, say, a few thousand to a few tens of thousands of years, and then grow more rapidly. If we examine the square-root growing part carefully, the result of scaling applied at every integration step gives lower errors than those applied at every apocenter, whether the scaling is single or quadruple. On the other hand, if we compare the errors in the long run, we
By monitoring the time development of the four total energies of the four-dimensional harmonic oscillator associated with the K-S regularization, we have extended the single scaling method provided in Paper IX. The manifold correction is accomplished by multiplying four scaling factors with the pair of the position and velocity of each component of the associated harmonic oscillator. The implementation of monitoring increases the number of variables to be integrated per celestial body from 10 to 13. However, the extra cost of computation is negligible compared with that of the perturbing acceleration. The new method, which we call the quadruple scaling method, significantly enhances the numerical stability of orbit integrations when it is applied at every integration step. As a result, the quadruple scaling so applied is significantly better than any existing manifold correction method with the K-S regularization, whether perturbations are present or not, as long as the truncation errors are dominant. If round-off is the main source of integration error, however, we prefer the single scaling method applied at every apocenter, as we have seen in Paper IX.
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