The Astronomical Journal, 129:1746 –1754, 2005 March # 2005. The American Astronomical Society. All rights reserved. Printed in U.S.A.
EFFICIENT ORBIT INTEGRATION BY KUSTAANHEIMO-STIEFEL REGULARIZATION USING TIME ELEMENT Toshio Fukushima National Astronomical Observatory, Osawa, Mitaka, Tokyo 181-8588, Japan;
[email protected] Receivved 2004 November 1; accepted 2004 November 22
ABSTRACT We have found that the introduction of the time element into the Kustaanheimo-Stiefel (K-S) regularization changes the rate of error growth from quadratic to linear with respect to the physical time, even when a scaling is not applied. It also mostly wipes out the periodic error components in the case of unperturbed orbits, independently of the application of the scaling. Since the increase in computational time due to use of the time element is negligible, we recommend that it be employed in all numerical integrations using the K-S regularization. Key words: celestial mechanics — methods: numerical
1. INTRODUCTION
step size, and (5) set the order of the Adams method as that which led to the lowest errors, namely, the ninth for the antifocal longitude method with Sundmann’s time transformation, the 13th for the quadruple scaling method, and the 10th for the others in this case. In the figure, the results from the new methods are labeled as ‘‘Single’’ for the single scaling method applied to the K-S regularization at every integration step, ‘‘Quad’’ for the quadruple scaling method applied to the K-S regularization at every integration step, ‘‘True + S’’ for the true-longitude method with Sundmann’s time transformation, and ‘‘Antifocal + S’’ for the antifocal longitude method with Sundmann’s time transformation. The performance of these methods may be compared with that of the standard K-S regularization (Kustaanheimo & Stiefel 1965), which is also shown, with a label of ‘‘KS.’’ In preparing the figure, we adopted an extraordinarily large step size, 1=30 the orbital period on average. This may seem careless for precision orbit integration in light of the large eccentricity, e ¼ 0:6, of the orbit to be integrated. Even the standard K-S regularization method is inefficient at providing initial errors at the milliarcsecond level. This initial error magnitude and the quadratic nature of the error growth with respect to the physical time make numerical integrations by the standard K-S regularization with such a large step size meaningless6 after 105 revolutions. On the other hand, the manner of growth of the longitude errors is linear for the new methods in the long run. In the case of the quadruple scaling method, which has the best performance in this specific case among the methods we developed for highly eccentric orbits in Papers IX through XI, the initial errors are so small as to be at the microarcsecond level, and the errors after 1 million revolutions remain quite small, a few tens of milliarcseconds, even for such a large step size. In order to illustrate the differences thus observed in the magnitude and growth rate of integration errors among the methods more clearly, we experimentally fitted an error formula in the form of a quadratic function with respect to the physical time in order to follow the envelope of each error curve. The determined coefficients are listed in Table 1. These show that the observed difference in growth rate leads to a large difference in the error magnitude over the long run.
Recently we have been pursuing a study of new and efficient approaches to numerically integrate the orbit of a perturbed twobody problem with large eccentricity, such as that of HALCA (Hirabayashi et al. 2000),1 an artificial Earth satellite of the Japanese space VLBI program, VSOP. These approaches are (1) the single scaling method applied to the KustaanheimoStiefel (K-S) regularization (Fukushima 2004f ),2 (2) the quadruple scaling method for the K-S regularization (Fukushima 2004g),3 and (3) the true and antifocal orbital longitude methods with Sundmann’s time transformation (Fukushima 2005).4 These works, together with our previous studies of manifold correction methods for orbits of low and moderate eccentricity (Fukushima 2003a, 2003b, 2003c, 2004a, 2004b, 2004c, 2004d, 2004e),5 have proved that the cost performance for numerical integrations of orbital motion can be dramatically improved by using these on-the-fly corrections or their equivalents to maintain certain consistencies among the integrated variables and by employing some mathematical tricks, such as Encke-like techniques to focus on the deviation of slow variables from their initial values or the modularization of secularly growing angle variables into the standard range [, ). Figure 1 shows the performance of the aforementioned methods with the K-S regularization or Sundmann’s time transformation in integrating a model orbit of HALCA. We have plotted the errors in the mean longitude at epoch, which is the largest error component among the six orbital elements, as functions of the physical time on a log-log scale. In conducting the numerical integration, we (1) adopted the implicit Adams methods in PECE ( predict, evaluate, correct, evaluate) mode as the integrator, (2) fixed the step size throughout the integration such that one orbital period in the physical time was covered by 30 steps, (3) prepared the starting tables using Gragg’s extrapolation method, (4) measured the errors at the same physical time by means of a time synchronization (Paper IX) and by comparing with reference solutions that we obtained separately using the same method, the same integrator, and the same model parameters but with half the 1 2 3 4 5
Design eccentricity e ¼ 0:6. Hereafter Paper IX. Hereafter Paper X. Hereafter Paper XI. Hereafter Papers I–VIII, respectively.
6
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In the sense that the longitude error becomes greater than 1 radian.
log10 ∆L0
EFFICIENT ORBIT INTEGRATION. XII. 2 0 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46 -48 -50 -52 -54 -56 -58 -60 -62 -64 -66 -68 -70 -72 -74 -76 -78 -80 -82 -84 -86
30 Steps/Period Kepler, e=0.6
KS 10−6 KS+PT
10−20 True+S
10−30 Antifocal+S
10−38 Single
10
−44
Quad
10−52 KS+T 10−60 Single+T
10−66 Quad+T
-2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 log10 (Physical Time/Period) Fig. 1.—Error growth of a highly eccentric (e ¼ 0:6) Keplerian orbit integrated by various regularization methods. Illustrated are the errors in the mean longitude at epoch, L0 , as functions of the physical time in log-log plots. The notation ‘‘KS’’ means that the Kustaanheimo-Stiefel regularization was used, ‘‘Single’’ denotes the single scaling method applied at every integration step, ‘‘Quad’’ denotes the quadruple scaling method applied at every integration step, ‘‘True + S’’ stands for the true-longitude method with Sundmann’s time transformation, and ‘‘Antifocal + S’’ stands for the antifocal longitude method with Sundmann’s time transformation; ‘‘+T’’ indicates use of the time element, and ‘‘+PT’’ indicates use of the time element in the primitive form.
Unfortunately, the new methods still inherit one of the drawbacks of the original K-S formulation: the existence of periodic errors of nonnegligible magnitude, which are dominant in the initial time period, the duration of which ranges from several orbital periods for the original K-S formulation to some thousands of orbital periods for the quadruple scaling method in this specific case. The existence of this periodic error component is also the major weak point of the two orbital longitude methods. (See Fig. 1 and Table 1 again.) In order to find the cause of these periodic errors, we examined the details of the error growth in all the variables with respect to not the physical but the fictitious time. We found that the cause lies not in the integration of the harmonic oscillator itself but in the time integration part. Because the integration of
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TABLE 1 Coefficients of Experimental Error Formula for Regularized Keplerian Motion: Integration with 30 Steps per Orbital Period for an Orbit with e ¼ 0:6 Method
log |C0|
log |C1|
log |C2|
Reference
KS ..................... KS + PT ............ True + S ............ Antifocal + S..... Single ................ KS + T............... Single + T ......... Quad.................. Quad + T ...........
9.2 ... 3.4 3.8 9.2 ... ... 11.1 ...
... ... 6.7 6.7 11.2 11.3 11.2 14.6 14.7
10.5 10.3 ... ... ... ... ... ... ...
Stiefel & Scheifele (1971) This work Paper XI Paper XI Paper IX Stiefel & Scheifele (1971) This work Paper X This work
Note.—Listed are the three coefficients of the error formulae obtained experimentally from the integration errors in Fig. 1, namely, those for a Keplerian orbit with e ¼ 0:6 integrated with a fixed step size of 1=30 the orbital period. The formulae are of the form jL0 j ¼ jC0 þ C1 T þ C2 T 2 j, where T is the physical time measured in units of the orbital period.
the harmonic oscillator controls the numerical stability of the system, we had at first ignored the contribution of the time integration to the final error budget. In other words, if we can decrease the error produced in the integration of the physical time, then it might lead to a reduction in the overall error. Tracing back the history, we learned that Stiefel & Scheifele (1971) had already noted this issue and provided a remedy, the introduction of the ‘‘time element.’’ This approach is a kind of time transformation defined not kinematically, as is Sundmann’s, but analytically as an explicit function of the physical time and other K-S variables (see eq. [1] below). As we expected, introduction of the time element into the existing two scaling methods with the K-S regularization leads to an almost complete reduction of the periodic error components mentioned above. To our surprise, it also changes the error growth rate of the original K-S formulation from quadratic to linear without the use of any scaling at all. See Figure 1 and Table 1 again, where the methods into which the time element has been introduced are indicated with an additional ‘‘+T.’’ Further investigation revealed that this change in growth rate is brought about by Stiefel & Scheifele’s (1971) ingenious trick of rewriting the main part of the equation of motion for the new time variable, which is originally a complicated function of the other K-S variables, into a simpler form by using the Kepler energy relation. This can be understood as an implicit use of the manifold correction, something that we realized in creating the orbital longitude methods in Papers VI through VIII, where we implicitly used the radius vector relation by introducing new variables, the orbital longitudes. This conjecture was confirmed by a test integration based on the K-S regularization with the time element, but with the equation of motion kept in its original form without the above recasting. The curve in Figure 1 and the row in Table 1 both tagged additionally with ‘‘+PT’’ show the result of the test integration, which is practically the same as that of the original K-S formulation in the long run. In any event, we showed that introduction of the time element is beneficial, as already seen from Figure 1 and Table 1. However, this is for the case of Keplerian motion. Whether it works even under perturbations is an essentially different issue to be investigated carefully. In this paper, we report that the introduction of the time element significantly reduces the errors of all the methods based on the K-S regularization, whether the single or quadruple
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FUKUSHIMA
scaling is applied or not. In the following, we summarize how to incorporate the time element into the existing methods using the K-S regularization in x 2 and examine its effect through numerical comparison with the existing methods in x 3. 2. INTRODUCTION OF TIME ELEMENT INTO K-S REGULARIZED ORBITAL MOTION The time element, , is a replacement for the physical time, t, to be integrated simultaneously with the four-dimensional harmonic oscillator u and the (negative) Kepler energy hK associated with the K-S regularization of a perturbed two-body problem (Stiefel & Scheifele 1971). It is explicitly defined as an analytic function of u, t, and hK as tþ
u = u0 ; hK
ð1Þ
where primes represent differentiation with respect to the fictitious time, s. Let us recall that the equations of motion7 of the K-S variables (u, t, hK) are expressed as 0 hK u =Q 00 0 u þ ; ð2Þ u ¼ Q; hK ¼ 4 u2 2 t 0 ¼ u2 ;
ð3Þ
where Q is the perturbing acceleration in terms of the K-S variables, which is a function of u, u 0 , and t in general. Using these, we formally rewrite the equation of motion of the physical time, equation (3), into that for as 0 ¼ þ p = Q;
ð4Þ
where 2(u 0 )2 þ hK u2 ; 2hK u u = u0 p þ4 u0: hK (hK )2 u2
ð6Þ
ð7Þ
where G(M þ m) is the bodycentric gravitational constant of the two-body problem. Note that this is valid even under perturbations (see Paper IX). Then they rewrote the term into a simpler form,8 as 1 2
¼ =hK :
ð8Þ
We call the use of this rewritten form for the -term in equation (4) the standard form of the equation of motion for the time element. 7
In the unperturbed case, where Q ¼ 0, the right-hand side of equation (4) reduces to a constant with respect to the fictitious time. Then any proper integrator can integrate the equation without truncation errors, as Stiefel & Scheifele (1971) stressed. This fact implies that, even under perturbations, numerical integration of the time element would be more efficient than integration of the physical time. Once the time element is determined together with the other K-S variables, the physical time is easily obtained as t¼
u = u0 : hK
ð9Þ
Note that the replacement of t by requires only a negligible amount of additional computational time, since the bulk of it is occupied by the evaluation of the perturbing acceleration, Q. Finally, we remark upon the computational treatment of the integrated time element. In Papers IX through XI, we employed a special treatment for the integrated physical time t, expressing it as the pair of an integer and a double-precision variable, (k, t*), such that t is expressed as t = kT + t* while satisfying the condition |t*| T, where T is a certain unit of time, say, half the nominal orbital period. During the course of test integrations of the methods using the time element, we found that this treatment is not adequate to ensure zero integration error in when the standard expression for , equation (8), is used in the case of unperturbed orbits. We thus decided to express the integrated time element as a so-called double-length double-precision variable: a pair of double-precision variables, ( 1, 2), satisfying simultaneously the summation condition ¼ 1 þ 2 and the magnitude condition j2 j < j1 j, where denotes the doubleprecision machine epsilon. See Appendix B of Fukushima (2001) for details. 3. NUMERICAL EXPERIMENTS
ð5Þ
We refer to the use of equation (5) for the -term in equation (4) as the primitive form of the equation of motion for the time element. Now, Stiefel & Scheifele (1971) noticed the existence of the integral 2(u 0 )2 þ hK u2 ¼ ;
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See eq. (1) of Paper IX. This term is exactly half the osculating semimajor axis of the orbit, ¼ a=2, since the negative Kepler energy is expressed as hK ¼ 12 =a.
We next examine the effects of the introduction of the time element into the original K-S formulation and its two variants using the single or the quadruple scaling. Before going further, let us explain how we measure the integration errors. As in Papers I through XI, we measure the errors by taking the difference from a reference solution that was obtained using the same integrator, under the same initial conditions and with the same model parameters, but with half the step size. In the case of regularized orbital motions with the K-S regularization or Sundmann’s time transformation, we must ensure that the differences are taken for the same physical time in order for them to be meaningful. To do this, we make a correction due to the errors in integrating the physical time, detailed in Appendix B of Paper IX. 3.1. Unperturbed One-dimensional Harmonic Oscillator In order to study the nature of error growth of K-S regularized orbit, we begin with a one-dimensional harmonic oscillator with no perturbations as the simplest model. The equations of motion for the system we considered are u 00 þ 12 h K u ¼ 0
ð10Þ
for the harmonic oscillator and
8
t 0 ¼ u2
ð11Þ
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EFFICIENT ORBIT INTEGRATION. XII.
for the physical time integration, where we have assumed that hK is a positive constant, which corresponds to an elliptical orbit. In this case, the (single) scaling is done as 0
0
(u; u ) ! (u; u );
ð12Þ
¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2(u 0 )2 þ h K u2
ð13Þ
and where u and u 0 on the right-hand side are the integrated values. See Paper IX for their extension to the full four-dimensional case. In the current case, the time element is defined as tþ
uu 0 : hK
ð14Þ
Its equation of motion in the primitive form becomes 0 ¼
(u 0 )2 u 2 þ ; hK 2
ð15Þ
and that in the standard form is 0 ¼ 12 =h K :
ð16Þ
We conducted test integrations of this system in six different ways: (1a) direct integration of equations (10) and (11) without application of the scaling; (1b) direct integration of equations (10) and (15) without scaling; (1c) direct integration of equations (10) and (16) without scaling; (2a) the same as case 1a but with the scaling (eq. [12]) applied at every integration step; (2b) the same as case 1b but with the scaling; and (2c) the same as case 1c but with the scaling. Figure 2 illustrates the errors in the physical time obtained for three of these cases, 1a, 1b, and 2a. We omit the results for the remaining three cases since they led to zero errors, thanks to the use of double-length double-precision variables for the time element. In Figure 2 we also show the dimensionless combined position and velocity errors of the harmonic oscillator for cases 1a and 2a. The dimensionless combined error is defined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hK (u)2 þ 2(u 0 )2 hK (u)2 þ 2(u 0 )2 ¼ : ð17Þ hK u2 þ 2(u 0 )2 We do not show the results for the other four cases, since we have confirmed that the nature of the errors is independent of the choice of the form of time integration. Compare cases 1a (labeled ‘‘Unscaled’’) and 1b (labeled ‘‘Unscaled + PT’’). The introduction of the time element itself suppresses the initial periodic errors in the physical time with respect to the fictitious time. The errors in the harmonic oscillator show no periodic features, whichever scaling is used. Hence, the periodic features with respect to the physical time (as shown in Fig. 1) should come from periodic features in the fictitious-time integration. Thus the presence of the initial periodic features is proved to be caused by incorrectness in the direct integration of the physical time. On the other hand, the integration of the physical time in the primitive form without scaling, that is, case 1b, does not change
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where the scaling factor is determined as
2 0 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42
Harmonic Oscillator 60 Steps/Period
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∆t Unscaled 10−6∆t Unscaled+PT
10−12∆t Scaled 10−20∆ρ Unscaled 10−24∆ρ Scaled
-2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 log10 (Fictitious Time/Period) Fig. 2.—Similar to Fig. 1, but plotted are the dimensionless combined position and velocity errors of a one-dimensional harmonic oscillator, f½hK (u)2 þ 2(u 0 )2 =g1=2 , and of the physical time associated with the oscillator, t. We compare the results obtained with and without scaling. For t, we also include the results of the unscaled integration with the introduction of the time element but integrated in the primitive form of its equation of motion, which we designate by adding the label ‘‘+PT.’’ The step size in the fictitious time was fixed at 1=60 the oscillation period, such that the orbital period in the physical time of an associated orbit is covered by 30 steps. The orders of the implicit Adams methods were chosen as the highest among those that led to no numerical instabilities, namely, the 10th for the unscaled case and the 13th for the scaled one.
the quadratic growth. This can be understood as follows: (1) the errors in the harmonic oscillator, u or u 0 , grow linearly with respect to the fictitious time as seen in Figure 2; (2) they cause linear growth of the errors in the -term if the primitive form of the equation of motion for the time element was selected; and (3) the resulting errors in the time element grow quadratically with the fictitious time after the integration. On the contrary, the standard form, equation (16), enables us to integrate the physical time in an error-free fashion whether the magnitude of errors in the harmonic oscillator is reduced by scaling or not, as seen in the figure. These results explain well the differences in the manner of error growth in the K-S regularized orbital motion already shown in Figure 1 and Table 1. In conclusion, introducing the time element with the standard form for its equation of motion is essential to suppressing the integration error in the physical time associated with the harmonic oscillator. Before ending this subsection, we highlight one important aspect of the scaling: enhancement of the stability of numerical integrators. Figure 3 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. There is a large difference in the available highest order between the unscaled and scaled cases. This difference for the one-dimensional harmonic oscillator has a cause similar to that for the difference between the quadruple scaling method and others in the K-S regularized orbital motion, depicted in Figure 6 of Paper X.
FUKUSHIMA
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Fig. 3.—Stability limit curves for a harmonic oscillator. Illustrated are the highest stable order of the PECE mode of the implicit Adams method in integrating a harmonic oscillator as a function of the fixed step size. We regard ‘‘stability’’ as showing no exponential error growth for the first 1024 periods.
3.2. Unperturbed Orbital Motion
-2 -4 AM10, 30 Steps/Period o ∆x /r -6 Kepler, e=0.6, I=31 -8 KS+T 10−4∆L0 -10 -12 10−8e∆ϖ -14 -16 10−12∆e -18 -20 -22 -24 -26 10−16(sinI)∆Ω -28 -30 10−20∆I -32 -34 10−30(∆a)/a -36 -38 -40 -42 -44 -46 -48 -2 -1 0 1 2 3 4 5 6 7 8 9 10 log10 (Physical Time/Period)
Fig. 4.—Element errors of a Keplerian orbit regularized by the K-S transformation using the time element and with no scaling.
-2 -4 AM10, 30 Steps/Period o ∆x /r -6 Kepler, e=0.6, I=31 Single+T -8 10−4∆L0 -10 -12 -14 -16 10−8e∆ϖ -18 -20 10−12∆e -22 -24 10−16(sinI)∆Ω -26 -28 -30 10−20∆I -32 -34 -36 -38 -40 10−30(∆a)/a -42 -44 -46 -48 -2 -1 0 1 2 3 4 5 6 7 8 9 10 log10 (Physical Time/Period)
Fig. 5.—Same as Fig. 4, but with the single scaling applied at every integration step.
integrated in the standard form; (3a) K-S regularization with the quadruple scaling but without introduction of the time element; and (3c) K-S regularization with the quadruple scaling and with the time element integrated in the standard form. The integration errors of the first two cases, 1a and 1b, grow quadratically with respect to the physical time. The main reason for this quadratic growth is the incorrectness of the integration of the physical time itself or of the time element, as we proved by examining the similar results for a one-dimensional
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Now we deal with the orbital motions. From here on, we restrict ourselves to the perturbed two-body problem for elliptical orbits. First, we discuss the unperturbed case. Figure 1 has already compared the longitude error growth for seven cases based on the K-S regularization: (1a) the original K-S regularization without introduction of the time element; (1b) K-S regularization with the time element, but using its equation of motion in the primitive form; (1c) K-S regularization with the time element and its equation of motion in the standard form; (2a) K-S regularization with the single scaling but without introduction of the time element; (2c) K-S regularization with the single scaling and with the time element
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-2 -4 AM13, 30 Steps/Period o -6 Kepler, e=0.6, I=31 -8 Quad+T ∆x /r -10 -12 10−4∆L0 -14 -16 -18 10−8e∆ϖ -20 -22 10−12∆e -24 -26 10−16(sinI)∆Ω -28 -30 10−20∆I -32 -34 -36 -38 -40 -42 -44 10−30(∆a)/a -46 -48 -2 -1 0 1 2 3 4 5 6 7 8 9 10 log10 (Physical Time/Period)
Fig. 6.—Same as Fig. 4, but with the quadruple scaling applied at every integration step.
2 0 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46 -48 -50 -52 -54 -56 -58 -60 -62 -64 -66 -68 -70 -72 -74 -76 -78 -80 -82 -84 -86
EFFICIENT ORBIT INTEGRATION. XII.
Moon’s Perturbation KS 30 Steps/Period HALCA 10−6 KS+PT
10−20 True+S
10−30 Antifocal+S
10−38 Single −44
10
Quad
10−52 KS+T 10−60 Single+T 10−66 Quad+T
-2 -1 0 1 2 3 4 5 6 7 8 9 10 11 log10 Days Fig. 7.—Same as Fig. 1, but for a model HALCA under the Moon’s perturbation. The orbit of the Moon was given as a fixed Keplerian orbit.
harmonic oscillator in the previous subsection. Let us thus move to the difference in the other five cases, for which the errors grow linearly with respect to time in the long run. First, there is around four digits’ difference in the error magnitude over the long term between two groups, that is, cases 1c, 2a, and 2c and cases 3a and 3c. This difference is explained by the difference in the available highest order of the integrator— the 10th for the first group and the 13th for the quadruple-scaled case. The averaged time step of 1=30 the orbital period of the two-body problem corresponds to 1=60 the oscillation period of the associated harmonic oscillator, so the step size in the fictitious time is around 0.1 radians. Thus the difference of 3 in the order of the integrator will lead to a difference of some three digits in the magnitude. Next, the initial periodic errors seen in the methods integrating the physical time directly, that is, cases 1a, 2a, and 3a, almost disappear by virture of the introduction of the time element, as seen in cases 1b, 1c, 2c, and 3c. This is because the periodic errors are mainly caused by the incorrectness of the
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No. 3, 2005
2 0 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46 -48 -50 -52 -54 -56 -58 -60 -62 -64 -66 -68 -70 -72 -74 -76 -78 -80 -82 -84 -86
Air Drag 30 Steps/Period HALCA
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KS 10−6 KS+PT
10−20 True+S
10−30 Antifocal+S
10−38 Single 10−44 Quad 10−52 KS+T
10−60 Single+T 10−66 Quad+T
-2 -1 0 1 2 3 4 5 6 7 8 9 10 11 log10 Days Fig. 8.—Same as Fig. 7, but under perturbations due to a model air drag.
physical-time integration, as we learned in the case of the onedimensional harmonic oscillator. On the other hand, there is no difference in the manner of error growth in the long run. This seems to give the impression that all three methods using the time element are the same, aside from the magnitude of the errors. However, this is not true. See Figures 4–6, which illustrate the errors in the position and the six modified orbital elements. Compare these with the results without introduction of the time element, Figure 3 of Paper IX for case 1a, Figure 6 of Paper IX for case 2a, and Figure 3 of Paper X for case 3a (although the step size adopted there is much smaller, 1=90 the orbital period on average). The manner of error growth in the orbital elements other than the mean longitude at epoch differs from method to method. These difference are caused not by the introduction of the time element, but by the introduction of scaling and differences in the type of scaling. In the unscaled case, that is, in Figure 4, the errors in the semimajor axis, a, the eccentricity, e, and the longitude of pericenter, e $, are initially of the same order of magnitude
FUKUSHIMA
2 0 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46 -48 -50 -52 -54 -56 -58 -60 -62 -64 -66 -68 -70 -72 -74 -76 -78 -80 -82 -84 -86
J2 Perturbation 60 Steps/Period HALCA
KS 10−6 KS+PT
10−20 True+S
10−30 Antifocal+S 10−38 Single 10−44 Quad 10−52 KS+T 10−60 Single+T 10−66 Quad+T
-2 -1 0 1 2 3 4 5 6 7 8 9 10 11 log10 Days Fig. 9.—Same as Fig. 7, but under the J2 perturbation of Earth. The inclination of HALCA is as moderate as I ¼ 31 . This time we chose a small step size in order to keep the initial errors due to the perturbations at the machine-epsilon level. As a result, the order of the implicit Adams method that led to the lowest errors changed to the 11th for the true-longitude method with Sundmann’s time transformation, the 13th for the two quadruple scaling methods and the antifocal longitude method with Sundmann’s time transformation, and the 12th for the others.
as that of the mean longitude at epoch, L0, say, some milliarcseconds in this specific case. Then they grow linearly with respect to the physical time. However, the errors in the other two elements, the inclination, I, and the longitude of the ascending node, (sin I ), first grow in proportion to the square root of the physical time, for the first few thousand periods in this specific case, and then increase linearly. In the case of single scaling, that is, in Figure 5, the errors in the semimajor axis remain at the machine-epsilon level for a long period of time. This is the effect of the single scaling to maintain the Kepler energy relation in terms of the K-S variables. However, after some time, tens of thousands of periods in this case, the errors grow quadratically with respect to the physical time. We do not fully understand the reason behind this error growth; however, the magnitude of this quadratic error growth is
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2 0 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46 -48 -50 -52 -54 -56 -58 -60 -62 -64 -66 -68 -70 -72 -74 -76 -78 -80 -82 -84 -86
Jupiter’s Perturbation KS 30 Steps/Period Icarus 10−6 KS+PT
10−20 True+S
10−30 Antifocal+S
10−38 Single 10−44 Quad 10−52 KS+T 10−60 Single+T 10−66 Quad+T
-2 -1 0 1 2 3 4 5 6 7 8 9 10 11 log10 Years Fig. 10.—Similar to Fig. 7, but for a model Icarus under Jupiter’s third-body perturbation. This time we halved the step size because the perturbation is strong.
too small to induce significant changes in the errors of the other orbital elements, including that of the mean longitude at epoch. As for the remaining four orbital elements, e, e $, I, and (sin I ), all are initially at the machine-epsilon level and increase linearly with respect to the physical time. Thus, compared with the unscaled case, the errors of the single-scaled case are (1) significantly smaller for the Kepler energy and the eccentricity vector, (2) almost the same for the longitude and, therefore, for the position, and (3) significantly larger for the orientation of the orbital plane. Finally, in the case of the quadruple scaling, that is, in Figure 6, the errors in the semimajor axis are always at the machine-epsilon level. As a result, the longitude errors, and therefore the position errors too, grow linearly with respect to the physical time. On the other hand, the magnitudes of the other four elements remain at the level of the machine epsilon for some amount of time, say, a few thousand periods in this specific case, and then grow linearly. In any event, for every error component this method is the best among those we have tested.
2 0 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46 -48 -50 -52 -54 -56 -58 -60 -62 -64 -66 -68 -70 -72 -74 -76 -78 -80 -82 -84 -86
EFFICIENT ORBIT INTEGRATION. XII.
General Relativity 30 Steps/Period Icarus
KS 10−6 KS+PT
10−20 True+S
10−30 Antifocal+S 10−38 Single 10−44 Quad 10−52 KS+T 10−60 Single+T 10−66 Quad+T
-2 -1 0 1 2 3 4 5 6 7 8 9 10 11 log10 Years Fig. 11.—Same as Fig. 10, but under the Sun’s general relativistic perturbation.
We have confirmed that nothing in the above situation changes significantly with respect to the magnitude of the eccentricity, from zero to unity. Thus, these facts indicate that the quadruple scaling method using the time element provides the best performance for unperturbed orbital motions. 3.3. Perturbed Orbital Motion We now turn to the perturbed orbits. First of all, we confirmed that the behavior observed in the case of unperturbed orbits remains practically the same when the perturbation is sufficiently weak. See Figure 7, showing the case of a model HALCA under the Moon’s third-body perturbation only. This is an example of a weak perturbation. When compared with the unperturbed case, Figure 1, the only significant difference is the appearance of some periodic error components in the case of the quadruple scaling using the time element, which turns out to be caused by the Moon’s periodic perturbations. Next let us examine the case of perturbations of moderate strength. Figure 8 plots the error growth of a model HALCA
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2 0 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 -32 -34 -36 -38 -40 -42 -44 -46 -48 -50 -52 -54 -56 -58 -60 -62 -64 -66 -68 -70 -72 -74 -76 -78 -80 -82 -84 -86
Moon’s Perturbation 60 Steps/Period HALCA
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KS 10−6 KS+PT
10−20 True+S
10−30 Antifocal+S
10−38 Single 10−44 Quad
10−52 KS+T 10−60 Single+T 10−66 Quad+T
-2 -1 0 1 2 3 4 5 6 7 8 9 10 11 log10 Days Fig. 12.—Same as Fig. 7, but with the step size halved so that round-off errors are dominant initially. The orders of the implicit Adams method in PECE mode were the 14th for the two quadruple scaling methods and the antifocal longitude method with Sundmann’s time transformation, and the 12th for the others.
under the perturbations of air drag in a standard form. This time, the initial error magnitude has become much larger, at the milliarcsecond level even for the three cases of K-S regularized orbits with the time element integrated in the standard form of the equation of motion. To our surprise, the errors of the unscaled K-S regularization using the time element first increase linearly but grow more rapidly, in a cubic manner, after some time, say, a few thousand orbital periods. The case of the single scaling with the time element indicates a similar rise after nearly 1 million orbital periods. In any case, the supremacy of the quadruple scaling with use of the time element is unchanged. Let us discuss the case of strong perturbations. Figure 9 illustrates the error growth of a model HALCA under the J2 perturbation of Earth, which is the strongest affecting the orbital motions of Earth’s artificial satellites. This time we halved the step size so as to accommodate the resulting integration errors in a range similar to that for the previous cases of weak and
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moderate perturbations. Under such strong perturbations, there seem to be no practical differences among all the methods we examined here. In fact, all the error curves grow linearly with respect to the physical time for a considerably long time, say, a few thousand to a few tens of thousands of periods. After that they grow more rapidly, typically being quadratic with respect to the physical time. Now, we examine the effect of nominal eccentricity. Figure 10 illustrates a highly eccentric case under typical perturbations, namely, a model Icarus of initial eccentricity e ¼ 0:827, perturbed by Jupiter. Such an eccentric orbit under outer third-body perturbations is the most typical case we face in conducting n-body integrations. Very frequently we have to track the orbital evolution of such a compact binary subsystem under the influence of other bodies for a considerable time, say, a few hundred to some tens of thousands of orbital periods of the binary system. In preparing this 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 averaged step size in this case is as large as 20 days. In this case, the last five methods provide almost the same performance, as shown in the figure. Another typical perturbation to be seriously considered in highly eccentric orbits is that due to general relativistic effects. Figure 11 shows similar error curves for a model Icarus, but under the Sun’s post-Newtonian perturbations. This case also falls into the category of moderate-strength perturbations if the large eccentricity is taken into account. Again we confirm that no significant difference has been brought about by the introduction of the time element. Finally, let us examine a case in which round-off errors play the key role. Figure 12 is the same as Figure 7, but we halved the step size so that almost all the truncation errors are reduced to well below the machine-epsilon level. In this case, the orders of
Fukushima, T. 2001, AJ, 121, 1768 ———. 2003a, AJ, 126, 1097 ( Paper I) ———. 2003b, AJ, 126, 2567 ( Paper II) ———. 2003c, AJ, 126, 3138 ( Paper III ) ———. 2004a, AJ, 127, 3638 ( Paper IV ) ———. 2004b, AJ, 128, 920 ( Paper V ) ———. 2004c, AJ, 128, 1446 ( Paper VI) ———. 2004d, Celest. Mech. Dyn. Astron., submitted ( Paper VII)
the integrators were the 14th for the two quadruple scaling methods and the antifocal longitude method with Sundmann’s time transformation, and the 12th for the others. The errors increase in proportion to the 3=2 power of the physical time for the original K-S regularization and its variant with the time element but integrated using the primitive form of equation of motion. On the contrary, the errors grow linearly in the case of the two orbital longitude methods. For the remaining cases, the manner of error growth is much slower, roughly in proportion to the square root of the physical time for a considerable period of time. The best performance is achieved by the single scaling method without the introduction of the time element. This is probably because the number of arithmetic operations is the least in this method. 4. CONCLUSION By changing the time variable to be integrated from the physical time to a function of it and the other K-S variables, named the time element, we have enhanced the original K-S regularization and its variations using the single or quadruple scaling. The additional cost for this replacement is negligible. The introduction greatly reduces the errors of the K-S regularization in two ways: it suppresses the initial periodic error components, and it changes the error growth rate from quadratic to linear with respect to the physical time for the unscaled K-S regularization. In the case of perturbed orbits, the introduction provides better results if the perturbations are weak and yields almost the same performance when the perturbations are moderate or strong. In other words, by introducing the time element one will lose almost nothing in terms of cost and gain significantly in the sense of performance. Therefore, we recommend use of the time element in any numerical integration based on the K-S regularization, whether a scaling is applied or not.
REFERENCES Fukushima, T.———. 2004e, AJ, 128, 1455 ( Paper VIII ) ———. 2004f, AJ, 128, 3114 ( Paper IX ) ———. 2004g, AJ, 128, 3108 ( Paper X ) ———. 2005, AJ, 129, 1171 ( Paper XI) Hirabayashi, H., et al. 2000, PASJ, 52, 955 Kustaanheimo, P., & Stiefel, E. L. 1965, J. Reine Angew. Math., 218, 204 Stiefel, E. L., & Scheifele, G. 1971, Linear and Regular Celestial Mechanics ( New York: Springer)
