The Astronomical Journal, 128:1446–1454, 2004 September # 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.
FURTHER SIMPLIFICATION OF THE MANIFOLD CORRECTION METHOD FOR ORBIT INTEGRATION Toshio Fukushima National Astronomical Observatory, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan;
[email protected] Receivved 2004 April 15; accepted 2004 June 3
ABSTRACT By introducing some quantities defined on the moving orbital plane, we have developed three ways to further reduce the number of variables in our two simplified methods of manifold correction, from nine to seven or six per celestial body. Among these, the simplest option uses a set of six variables consisting of the threedimensional orbital angular momentum vector, the two independent components of the Laplace integral vector on the moving orbital plane, and a true orbital longitude measured from an origin solely determined from the orbital angular momentum vector. This scheme no longer requires any manifold correction, as does the standard method to integrate the Cartesian coordinates and velocity. However, the new method is much more precise than the standard method. For example, the longitude error of Mercury in a simultaneous integration of the Sun and nine major planets with a step size of 1.4 days exceeds 360 after a few thousand years when using the standard method but remains at the milliarcsecond level if the new method is used. In addition, the new method achieves better performance than any of the manifold correction methods we have developed, as long as round-off errors are negligible. Key words: celestial mechanics — methods: n-body simulations
1. INTRODUCTION
celestial body has been reduced from 13 to nine. One simplification uses the set of variables (x, L, P) and the other uses (n, L, P), where x is the position vector, L is the orbital angular momentum angular vector, Pmomentum is the Laplace integral vector, and n is the unit position vector. In the former case, the manifold correction is applied to maintain orthogonality between x and L and to satisfy an expression for the radius vector.2 On the other hand, the orthonormality of n is maintained in the latter case. From the viewpoint of integration cost, the reduced number of variables per celestial body, nine, is still large compared with the number of independent variables, six. The manifold correction to be applied is the key to further reduce this numberthis . If one number can find a way to do without a part of the correction, it will lead correction, to a reduction it will in the number of variables. Among the remaining manifold corrections, we focus on those that aim to maintain some orthogonality relations with respect to the orbital angular momentum, such as L = x ¼ 0. In this article, we report three ways to realize this idea. The main trick is to introduce a moving coordinate triad whose third axis is always parallel to L. Since P must be perpendicular to L even under perturbations, we can express the threedimensional P by its two independent components on the moving orbital plane. Using this technique,3 we can reduce the number of variables by one, from nine to eight. This is common to all three schemes presented here. In addition to the above, the first two methods of simplification replace the three-dimensional vector x or n with its two-dimensional components on the moving orbital plane. The number of variables per celestial body is then reduced to seven. In these cases, the remaining manifold correction is to maintain the radius vector expression for x or the normality condition on n. On the other hand, the third simplification uses a
We have recently been developing a series of powerful devices to integrate perturbed Keplerian motions numerically (Fukushimanumerically 2003b, 2003c, (Fukushima 2003d, 2004a, 2004b).1 These methods correct the integrated variables at every integration step so that the integrated orbit lies exactly on a certain manifold that is expected to contain the true solution as a subspace. Wesubspace. refer to them We as the methods of manifold correction, after Murison’s (1989) name for the idea originally proposed by Nacozy (1971). Numerical experiments show that the new methods dramatically reduce the integration errors for perturbed orbits. See Figure 1, which illustrates the performance of the manifold correction methods in a long-term integration of planetary motion where the number of force evaluations is kept the same. This gain in precision is independent of various aspects of the integration of perturbed orbits, as we studied extensively in Paper I: the eccentricity (x 3.3 in Paper I), the kind of perturbation (x 3.4 therein), the type of unperturbed orbit (whether elliptical, parabolic, or hyperbolic) (x 3.5), the type of integrator (x 3.6), the order of the integrator (x 3.7), the number of stages in the extrapolation method (x 3.8), the policy governing step-size control (x 3.9), the characteristics of each body in an n-body integration (x 3.10), and the strength of the perturbations (xx 3.11 and 3.12). This issue, independence with respect to various factors of integration, is not a straightforward or trivial matter but a serious subject to be checked in detail, as we did in Paper I. On the other hand, the cost of the manifold corrections is negligible compared with that of the force evaluation. In our last work (Paper V), we provided two simplifications of the linear transformation method developed in Paper IV, which is the best among the methods of manifold correction presented in Papers I–IV. The performance of the simplified methods is comparable to that of the original linear transformation method, while the number of integrated variables per 1
2
See x 2.2 for the details. None of the manifold correction methods we have developed has used the relation L = P ¼ 0. This is because we have no a priori knowledge which of L and P, or both, is to be corrected. However, the relation itself can be used to reduce the number of variables as we propose here. 3
Hereafter Papers I–V, respectively.
1446
EFFICIENT ORBIT INTEGRATION. VI.
1447
numerical stability for high-order integrators caused by the change in the form of the equation of motion for the fast variables (Fig. 6 below). In what follows, we will present the above three simplified methods in x 2 and discuss their cost and performance in x 3. 2. NEW METHODS OF SIMPLIFICATION 2.1. MovvinggCoordinate Triad Orthogonality between a given vector y and the orbital angular momentum, L = y ¼ 0, can be ensured if we express y as y ¼ yA eA þ yB eB ;
ð1Þ
where eA and eB are two of three orthonormal basis vectors5 (eA, eB, eC) whose third component is always parallel to L. That is, the conditions on the triad are Fig. 1.—Position error | x|/|x| of Mercury in simultaneous integrations of the Sun and nine major planets by the standard method, labeled (x, vv), and various manifold correction methods. The errors were obtained by comparing with a reference solution computed by the same integrator but with half the step size. The attached labels denote the set of variables of integration. The integrator used was the PECE mode (predict, evaluate, correct, evaluate) of the implicit Adams method with a fixed step size of 1.38 days, corresponding to 1/64 the nominal orbital period of Mercury, and the starting tables were prepared with Gragg’s extrapolation method. The orders of the Adams method were set as high as was stable, namely, 10 for the standard method using the variables (x, vv), 11 for the single scaling method from Paper I, using the set (x, vv, K ), and for the dual scaling method from Paper II, using the set (x, vv, K, P), and 13 for the three new simplifications, using (xA, xB, L, PA, PB), (nA, nB, L, PA, PB), and ( g, L, PA, PB).
sort of true orbital longitude in place of n and requires only six variables to be integrated for each body. In this case, we need no manifold correction. Before conducting numerical experiments, we anticipated that all of the new simplifications would result in no practical change in the nature of the integration errors, as we experienced in Paper V. For the first two, this expectation is confirmed. No significant difference is produced by changing the expression for P, x, or n from three to two dimensions. However, the third simplification brings about a significant gain in precision. See Figure 1 again. In this figure, we omit the results for the linear scaling method from Paper IV, using the set (x, vv, K, L, P), and the first simplified method in Paper V, which uses (x, L, P), since both are practically the same as the result for the first new simplification, labeled (xA, xB, L, PA, PB). We also do not illustrate the second simplification from Paper V, using the set (n, L, P), since the result is nearly the same as that from the second new simplification, labeled (nA, nB, L, PA, PB). The superiority of the third new simplification, marked with ( g, L, PA, PB), over the others shown in the figure comes mainly from the difference in the expression of the fast variables—that is, the position vector x, its normalization n, and /or the velocity vector vv—versus the true orbital longitude g. One reason for the improvement is that the nonlinear time variation of the true orbital longitude is smaller than that of the position vector, the velocity vector, or the unit position vector by roughly a factor of the eccentricity.4 Another is a gain in 4 In fact, some new variables are expanded in the case of a Keplerian orbit, as xA a cos k, vA na sin k, nA cos k, and g k þ 2e sin l, where k is the mean orbital longitude, a is the semimajor axis, n is the mean motion, e is the eccentricity, and l is the mean anomaly.
eA = eB ¼ eB = eC ¼ eC = eA ¼ 0; e2A ¼ e2B ¼ e2C ¼ 1;
eC k L:
ð2Þ
It is trivial that the third component is expressed as eC L=L;
ð3Þ
where L jLj. Once the direction of one of the two other components, say, eA, is fixed, the remaining one is determined by eB ¼ eC < eA :
ð4Þ
Then the problem is reduced to the question how to specify eA while keeping eA = eC ¼ 0. This question is none other than the issue of selecting a longitude origin on a moving plane, which has a long history of study (see Fukushima 2001 and references therein). Various options have been proposed. In celestial mechanics, one often adopts the longitude origin in the direction of pericenter as eA ¼ P=P, where P jPj. In this case, the true orbital angle becomes the true anomaly, f. However, this choice faces trouble in nearly circular orbits, where P / e is so small that the pericenter is poorly determined. This is the well-known difficulty with small eccentricities. An alternative is to select the direction of the ascending node, namely, the direction of the intersection of the orbital plane and the x-y plane of the adopted coordinate system. (See the Appendix of Fukushima 2001 for the detailed scheme in this case.) This is equivalent to the addition of the argument of pericenter, !, to the orbital angle so we have f þ ! instead of f. Unfortunately, another problem occurs if the angular momentum direction is close to the z-axis of the adopted coordinate system. This is known as the problem of small inclination. A common technique to avoid this problem is to add in the longitude of the ascending node, , as f þ ! þ . For beginners, the geometric meaning of this quantity is difficult to grasp, in the sense that one is measuring an angle along two arcs of different great circles. This also makes the vectorial expression quite complicated. A correct understanding, as well as a simple vector expression, requires the concept of the departure point. This is exactly the same as the nonrotating origin (NRO) and is useful in astrometry, for example, 5
Hereafter, we refer to these as a coordinate triad.
1448
FUKUSHIMA
as a proper device to define UT1 for the rotation of Earth. From the point of view of economy, however, this option has a serious drawback. It is not geometrically but kinematically defined, such that its specification requires another independent integration. This means that we must increase the number of variables to be integrated by one. Thus, we choose a new type of longitude origin solely and geometrically determined from the moving L. It is defined as eA ¼ A=A:
ð5Þ
Here A jAj and the defining vector, A, is expressed as A ¼ B0 < L;
ð6Þ
B 0 L 0 < x0
ð7Þ
where
and the subscript zero denotes initial values. It is easily to show that this definition satisfies the orthonormality conditions of equation (2). In fact, eA is initially in the direction of x0. This choice faces no singularity unless L departs so far from its initial value L0 that it becomes very close to B0 , which is 90 apart from L0. Such a critical situation is rare in perturbed Keplerian orbits. An unlucky example happens in a nearly polar orbit under the J2 perturbation. In that case, the angular momentum vector slowly circulates nearly on the equatorial plane. Then if the initial direction x0 happens to be in the direction of the poles, the resulting B0 is located nearly on the equatorial plane and L may become very close to B0 in the long run. Of course, this is avoided by setting B0 ¼ x0 instead. In any case, one can apply a ‘‘rectification’’ when such a critical situation occurs, as in Encke’s classic method. Namely, one may redefine B0 from L and x at that time so as to reconstruct the triad. The coordinate triad thus defined is a function of time in general. Once the time derivative of L is given, those of the triad are evaluated as deA 1 d dA eA = ¼ eA ; ð8Þ A dt dt dt deB deC deA ¼ < eA þ eC < ; ð9Þ dt dt dt deC 1 dL dL eC = ¼ eC ; ð10Þ L dt dt dt where dA dL ¼ B0 < : dt dt
ð11Þ
2.2. First Simplification Consider the perturbed two-body problem. Usually one adopts the pair of the relative position vector and the relative velocity vector referred to the heavier body, (x, vv), as the basic set of variables to be integrated. Their equations of motion are dx dv ¼ v; ¼ 3 x þ a: ð12Þ dt dt r
Vol. 128
Here GM is the gravitational constant of the two-body problem, r jxj is the mutual distance between the two bodies, and a is the perturbing acceleration in the relative sense, which is expressed as a function of x, vv, and the time t in general as a ¼ a(x; v; t):
ð13Þ
In the first simplification scheme, we adopt a set of seven quantities, (xA, xB, L, PA, PB), as the basic variables to be integrated. Here xA and xB are the two independent components of x on the orbital plane, while PA and PB are those of the Laplace integral vector P. The procedure for converting to the new set of variables from the ordinary set is pffiffiffiffiffiffiffiffiffiffi L ¼ x < v; A ¼ B0 < L; A ¼ A = A; eA ¼ A=A; pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi L ¼ L = L; eC ¼ L=L; eB ¼ eC < eA ; r ¼ x = x; P ¼ v < L x=r; xA ¼ x = eA ; xB ¼ x = eB ; PA ¼ P = eA ; PB ¼ P = eB ;
ð14Þ
where we assume that B0 has been determined from equation (7) beforehand. The procedure for the reverse transformation becomes qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi xA xB r ¼ x2A þ x2B ; nA ¼ ; nB ¼ ; L ¼ L = L; r r (nB þ PB ) nA þ PA ; vB ¼ vA ¼ ; L L x ¼ xA eA þ xB eB ; v ¼ vA eA þ vB eB : ð15Þ The equations for the time development of the new variables are dxA dxB dL ¼ N; ¼ vA F xB ; ¼ v B þ F xA ; dt dt dt dPA dPB ¼ QA F PB ; ¼ QB þ F PA ; ð16Þ dt dt where we assume that the velocity components vA and vB, the coordinate triad (eA, eB, eC), the position vector x, and the velocity vector vv have already been evaluated according to equation (15), and the other quantities needed are computed as N ¼ x < a; C ¼ B0 < N; pffiffiffiffiffiffiffiffiffiffi A ¼ B0 < L; A ¼ A = A; F ¼ C = eB =A; Q ¼ a < L þ v < N; QA ¼ Q = eA ; QB ¼ Q = eB : a ¼ a(x; v; t);
ð17Þ
The quantity F is the angular velocity of the frame rotation associated with the inertial motion of the longitude origin. The vectors C and Q are the time derivatives of the longitudeorigin defining vector and the Laplace vector, namely, C dA=dt and Q dP=dt. (See the Appendix for a detailed derivation of the equations of motion for the newly introduced variables xA, xB, PA, and PB.) At each integration step before the reverse transformation (eq. [15]), begins, we modify the integrated position components xA and xB so as to satisfy the radius vector expression r¼
L2 : þ P = x=r
ð18Þ
No. 3, 2004
EFFICIENT ORBIT INTEGRATION. VI.
This apparently difficult problem is simply solved by applying a scaling xA ¼ sX xA ;
xB ¼ sX xB ;
ð19Þ
where L2 sX ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2A þ x 2B þ PA xA þ PB xB
ð20Þ
is the scale factor and the quantities with asterisks are the final position components, to be used in the next step of the integration. The effect of the scaling depends on when it is applied. We recommend that it be done at the very beginning of the reverse transformation (eq. [15]). The second scheme is an application of the first simplification to the second set of variables we introduced in Paper V, (n, L, P), where n x=jxj. Thus, the new set of variables consists of seven components, (nA, nB, L, PA, PB). The forward transformation is the same as equation (14) except the part for nA and nB: pffiffiffiffiffiffiffiffiffiffi L ¼ x < v; A ¼ B0 < L; A ¼ A = A; eA ¼ A=A; pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi L ¼ L = L; eC ¼ L=L; eB ¼ eC < eA ; r ¼ x = x; n ¼ x=r; P ¼ v < L n; nA ¼ n = eA ; PA ¼ P = eA ;
PB ¼ P = eB ;
ð21Þ
where we again assume that B0 is determined from equation (7) beforehand. On the other hand, the reverse transformation becomes simpler than equation (15), as pffiffiffiffiffi L2 ¼ L = L; L ¼ L2 ; (nB þ PB ) nA þ PA ; vB ¼ vA ¼ ; L L L2 r¼ ; þ PA nA þ PB nB x ¼ r(nA eA þ nB eB ); v ¼ vA eA þ vB eB : ð22Þ Similarly, the equations for the time development of the new variables are the same as equation (16) except those for nA and nB: dnA dnB dL ¼ N; ¼ C nB ; ¼ C nA ; dt dt dt dPA dPB ¼ QA F PB ; ¼ QB þ F PA : dt dt
ð23Þ
(See the Appendix for the derivation.) The parts for nA and nB are of a simpler form, that of a harmonic oscillator with a time-varying frequency C, the speed of the orbital angular velocity including the contribution of frame rotation. The quantities on the right-hand sides of the above equations of motion are computed similarly to those in equation (17), as N ¼ x < a; C ¼ B0 < N; pffiffiffiffiffiffiffiffiffiffi A ¼ A = A; F ¼ C = eB =A;
a ¼ a(x; v; t); A ¼ B0 < L;
C ¼ L=r 2 þ F ; QA ¼ Q = eA ;
Q ¼ a < L þ v < N; QB ¼ Q = eB ;
where we assume that the magnitude of the orbital angular momentum L, the radius r, the coordinate triad (eA, eB, eC), the position vector x, and the velocity vector vvhave already been computed according to equation (22). This time, the radius is explicitly evaluated as in equation (22). Thus, the condition on the radius vector cannot be used for the manifold correction as in the first scheme. Rather, we maintain the normality of the unit position vector: n2A þ n2B ¼ 1:
ð25Þ
This is simply done by a renormalization nA ¼ sN nA ;
nB ¼ sN nB ;
ð26Þ
where
2.3. Second Simplification
nB ¼ n = e B ;
1449
ð24Þ
sN ¼ 1=
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2A þ n2B :
ð27Þ
Once again, we note that the application of the scaling is not always commutative with the above procedures for the reverse transformation. Thus, one must take care in when the scaling is applied. Our recommendation is before the start of the procedure of reverse transformation (eq. [22]). 2.4. Third Simplification The third scheme is a modification of the second. The normalization condition in equation (25) is automatically satisfied by expressing nA and nB in terms of the trigonometric functions of a certain angle, as nA ¼ cos g;
nB ¼ sin g:
ð28Þ
The angle g thus defined has the meaning of a true orbital longitude measured from the direction of A. Then we obtain a set of six variables, (g, L, PA, PB). This set is similar to the set ( , L, LX, LY , eJ , eK) that we once invented for the orbital motion in work on a generalization of Encke’s method (Fukushima 1996; see x 4.2 and Appendix B therein). The major difference is in the choice of three independent components of the orbital angular momentum, and a minor one is in the definition of the adopted longitude origin. The forward transformation is the same as equation (14) with the addition of an expression for g at its end: g ¼ tan1 (nB =nA ):
ð29Þ
Similarly, the reverse transformation is the same as equation (22) with the addition of equation (28) at its beginning. The equations of motion for the new variables become the same as equation (23) after replacing the parts for nA and nB with that for g: dg ¼ C ; dt dPA ¼ QA F PB ; dt
dL ¼ N; dt dPB ¼ QB þ F PA : dt
ð30Þ
The quantities needed in the right-hand sides of the equations of motion are given by equation (24). We stress that no manifold correction is necessary in this case. However, as will be shown below, the simplified method achieves better performance than the existing methods with
1450
FUKUSHIMA
manifold corrections. This is surprising; we think that the introduction of a true orbital longitude that satisfies the normalization condition, equation (25), indirectly enforces the radius expression, equation (18), and therefore ensures that the integrated orbit lies in the underlying invariant manifold of Keplerian orbits. 2.5. General Notes All three new variable sets introduced in the previous subsections are universal, in the sense that they are well defined whether the osculating orbit is elliptical, parabolic, or hyperbolic. Also, they have no singularities caused by the smallness of the eccentricity or the orientation of the coordinate system adopted. Of course, this is true for the third simplification introduced in this paper only so long as the perturbations do not grow so much that the magnitude of the vector defining the longitude origin, A, approaches zero. In the actual implementation, we do not integrate L, PA, or PB themselves but their deviations from the initial values L L L0 ; PA PA (PA )0 ;
PB PB (PB )0 ;
ð31Þ
as dL ¼ x < a; dt d PA d PB ¼ QA F PA ; ¼ QB F PB : dt dt
ð32Þ
L ¼ L0 þ L; PB ¼ (PB )0 þ PB :
ð33Þ
As we saw in Fukushima (1996), this trick greatly reduces the round-off error. For the last scheme, we further add a couple of notes. First, it is well known that the evaluation of trigonometric functions is time-consuming. In order to suppress this computational load as much as possible, we recommend use of the tangent function instead of calling both the sine and cosine functions. By using the half-angle formula, we can quickly evaluate cos g and sin g as cos g ¼
12 2 ; sin g ¼ 1þ2 1þ2
when tan
g 1; 2 ð34Þ
and as cos g ¼
˜ 2 1 2˜ ; sin g ¼ 2 1 þ ˜ 1 þ ˜ 2
when ˜
1 < 1:
is sufficiently small, say, less than 103. In other words, the method will lose its efficiency in integrations with a small step size. Finally, we omit the explanation of how to extend the schemes presented above to the case of multiple bodies, since it is trivial as described in Papers I through V. Note that the coordinate triad differs body by body. This flexibility in the choice of moving coordinate systems helps us avoid the geometric singularities that would be faced if we adopted a common coordinate system. 3. DISCUSSION For the three new methods, we have performed a number of numerical experiments to integrate various types of perturbed orbits. First of all, let us explain our method of measuring the integration errors. As detailed in x 3.1 of Paper I, there are many techniques to evaluate the integration errors. Among them, we have adopted a method that compares a reference solution obtained with the same-order integrator and the same parameters of integration but with half the step size. It is well known that the precision of a given integrator follows some power law with respect to the step size, as X (h) X (0) ¼ Ch p þ ;
Then L, PA, and PB are computed as PA ¼ (PA )0 þ PA ;
Vol. 128
ð35Þ
This roughly halves the time required to compute cos g and sin g (see x 9.1 of Fukushima 2003a). Another drawback of the introduction of trigonometric functions is an increase of round-off errors. Our experience shows that a single call of any trigonometric function provided in the standard library of mathematical functions implemented in the usual CPUs such as the Intel Pentium 4 causes a roundoff of few tens to hundreds of machine epsilons. Therefore, we predict that the third simplification will be weak against the accumulation of round-off errors unless the eccentricity
ð36Þ
where X(h) is the result of a test integration with step size h, X(0) is the true solution, which is unknown in general, C is a certain constant, and p is the order of the integrator. Usually the order is sufficiently high, 8 to 15, that the difference from the results obtained by halving h gives a good approximation of the error itself, as p ð37Þ X (h) X (h=2) ¼ Ch p ½1 12 Ch p : Throughout the experiments, we initially confirmed that the first scheme, using the set (xA, xB, L, PA, PB), yields almost the same results as produced by the first option in Paper V, using the set (x, L, P), which we have shown to provide nearly the same results as given by the linear transformation method using the set (x, vv, K, L, P). This situation is basically the same for the second scheme, using the set (nA, nB, L, PA, PB), and the second option in Paper V using the set (n, L, P). This is natural because the form of the equations of motion remains essentially unchanged. Also, we confirmed that all of the new schemes face no difficulties caused by the choice of coordinate system. On the other hand, the third option provides a significant improvement in precision when round-off is negligible. Figure 1 has already shown this for Mercury in the integration of the Sun and major planets. Figure 2 shows a typical case of strong perturbation, the position errors of the Moon in the main problem of lunar theory. In every case, where truncation errors are dominant the third new scheme gives the best result and the first new simplification is the second best. In order to examine the differences between the three new simplifications, we prepared Figures 3 through 5, showing the errors in the modified Keplerian elements of the same problem under perturbations of moderate strength. The test problem is an artificial satellite under J2 perturbation. The initial elements are taken such that e ¼ 0:1, I ¼ 23 , and the perigee distance is equal to the radius of Earth. The difference between the first two cases is small in general. However, the errors in the normalized perigee longitude, e $, obtained by the first new
No. 3, 2004
EFFICIENT ORBIT INTEGRATION. VI.
Fig. 2.—Same as Fig. 1, but for the Moon in the restricted three-body problem of the Sun, Earth, and the Moon, the so-called main problem of lunar theory. This time the maximum stable order for the third new simplification is 15, and the orders for the other methods are 10 for the variable set (x, vv), 11 for (x, vv, K ), and 13 for (xA, xB, L, PA, PB) and (nA, nB, L, PA, PB). We omit the result for the dual scaling method using the variable set (x, vv, K, P), since it is almost the same as that for the single scaling method, using (x, vv, K ).
simplification are smaller than those from the second new simplification by a factor of around 2. As a result, the position errors in the first new simplification are also smaller than those of the second new one. In any sense, the superiority of the third new simplification is obvious. The difference in performance mainly comes from the form of the equations of motion for the fast variables: (xA, xB) in the first option, (nA, nB) in the second scheme, and g in the last. In fact, we find that the stability of the new simplifications is not the same: see Figure 6, which compares the maximum stable order of the PECE mode of the implicit Adams method for various methods of manifold correction. (See also Fig. 4 of Paper IV, showing the case of a circular orbit.) In this figure, we only show the curves that are significantly different from each other. In particular, we omit (1) the graph for the dual scaling method using the set (x, vv, K, P), since it is mostly the same as that of the single scaling method using
1451
Fig. 4.—Same as Fig. 3, but for the second new simplification.
(x, vv, K ), and (2) the graphs for the linear transformation method using the set (x, vv, K, L, P) and the first simplified method in Paper V, using (x, L, P), because they give practically the same result as the first new simplification using the set (xA, xB, L, PA, PB) does. We also do not present the results for the second option in Paper V using the set (n, L, P) and the second new simplification using (nA, nB, L, PA, PB), since they are almost the same as the minimum of the two curves in the figure labeled with (xA, xB, L, PA, PB) and (g, L, PA, PB). For practical ranges of step sizes, say, larger than 1/500 the nominal orbital period, the last new simplification provides the greatest stability. For smaller step sizes, however, the first new simplification is best. This difference in the stability produces a difference in precision, as shown in Figure 2. Then what causes the small but nonnegligible difference in precision of the first two simplifications? The answer is the existence of radial motion in the process of numerically integrating the fast variables. A circular motion is not efficiently integrated by ordinary numerical integrators, including the Adams method, because they are designed to reproduce quasi-linear motion exactly.6 In noncircular orbits, the existence of a radial component makes the motion to be integrated 6 By quasi-linear motion, we mean that the motion is described by a polynomial of time.
Fig. 3.—Errors in the modified Keplerian elements of an artificial satellite around Earth under J2 perturbation obtained with the first new simplification. The integrator is the implicit Adams method in PECE mode, and the step size is 1/64 the orbital period. The order of the integrator was chosen as the highest among the stable ones, the 13th. The initial eccentricity and inclination of the orbit are e ¼ 0:1 and I ¼ 23 .
Fig. 5.—Same as Fig. 3, but for the third new simplification. This time the order of the integrator is 15.
1452
FUKUSHIMA
Fig. 6.—Highest stable order of the PECE mode of the implicit Adams method as a function of step size. We regarded ‘‘stability’’ as showing no exponential error growth for the first 32 periods of a pure Keplerian orbit with eccentricity e ¼ 0:1. Attached labels denote the set of variables used.
more quasi-linear. Imagine the extreme case of a comet-like orbit, where the orbit is well approximated by a linear motion except near pericenter passage. It is true that the third simplification is better than any of the methods of manifold correction; however, one should be aware of its weakness against round-off errors, as we predicted. Figure 7 shows the step-size dependence of errors in the last scheme. In all of the cases shown in Figure 7, the errors are almost constant with time for the first few periods. Then they grow roughly in proportion to the 3/2 power of time. This power index and the independence of the magnitude of the growing component with respect to step size suggest that this increase is caused by the accumulation of round-off errors. Unfortunately, the magnitude of this round-off component is much larger than in the other simplifications—see Figure 8, which shows similar graphs in the case of the first scheme, using the set (xA, xB, L, PA, PB). Therefore it is essential to understand the relative behavior of the three new simplifications and obtain optimal results
Vol. 128
Fig. 8.—Same as Fig. 7, but for the first new simplification.
depending on the problem at hand. Compare Figures 7 and 8. If one needs ultimate precision and is ready to pay the high cost of taking 180 steps per 88 days, then the first new simplification using (xA, xB, L, PA, PB) is suitable. On the other hand, if milliarcsecond-level precision is sufficient for a few thousand years, as is usually the case when creating a planetary ephemeris, then the third new simplification using (g, L, PA, PB) is recommended, and the integration cost is as low as only 64 steps per 88 days. Finally, we mention the cost of the three new simplifications compared with the two simplified methods in Paper V. Clearly, the cost of the first two new simplifications is significantly less than that of their counterparts in Paper V. This is due to the reduction in the number of variables to be integrated. When compared with these two new schemes, the third simplification is more time-consuming in the point of requiring one call of the tangent function per body per step, while it is less time-consuming in the sense that the number of variables per celestial body is reduced by one and no manifold correction is needed. In sum, the third simplification runs faster than the other two. In any case, such a difference in computational overhead is minute, since the overhead itself is much smaller than the evaluation of the perturbing force. This is especially true when dealing with a large-scale n-body problem or integrating the orbit of a low-altitude artificial satellite. 4. CONCLUSION By introducing some new quantities defined on the moving orbital plane, we have developed three ways to reduce the number of variables in the simplified methods of manifold correction in Paper V, from nine to seven or six per celestial body. The first method adopts a set of seven variables, (xA, xB, L, PA, PB), the second uses another set of seven variables, (nA, nB, L, PA, PB), and the last employs a set of six variables, (g, L, PA, PB). In the last case, we need no manifold correction. In this sense, the third scheme cannot be classified as a variation of the manifold correction method.7 Rather, it is a method of transformation of variables using an orbital longitude.
Fig. 7.—Same as Fig. 1, but for the third new simplification with various step sizes. Labels denote the number of steps per Mercury’s nominal orbital period, 88 days.
7 The irony is not lost on us that we finally arrived at a method requiring no manifold correction after a long advance toward the best method of manifold correction.
No. 3, 2004
EFFICIENT ORBIT INTEGRATION. VI.
When the main source of error is truncation, the last option gives the best performance and the first is second best. As we anticipated, however, the last method is weaker against the accumulation of round-off errors than the other two new simplified methods of manifold correction. Since the cost of computation is almost the same for these three options, our recommendation is (1) the third scheme when truncation error is dominant and (2) the first one otherwise.
We thank Professor Jose´ M. Ferra´ndiz for his valuable suggestions to improve the quality and readability of this article.
1453
Substituting the expression for deA/dt from equation (8), we finally obtain 1 dA = eB :
F ¼ ðA10Þ A dt This is what we aimed to prove for xA. We omit the cases for xB, PA, and PB, since the line of reasoning is almost the same as for xA. Next let us deal with nA. By differentiating the expression for the unit position vector, n ¼ nA eA þ nB eB ;
ðA11Þ
APPENDIX TIME DEVELOPMENT OF THE NEW VARIABLES Let us derive the equations of time development for the newly introduced variables: xA, xB, PA, PB, nA, nB, and g. We begin with xA. By differentiating both sides of the expression for the position vector, x ¼ xA eA þ xB eB ;
ðA1Þ
with respect to t, we obtain dn dnA deA dnB deB ¼ eA þ nA þ eB þ nB : dt dt dt dt dt
Substituting this and equation (A11) into the time development equation for n from Paper V, dn L ¼ 2 < n; dt r
with respect to t, we obtain v¼
dxA deA dxB deB eA þ xA þ eB þ xB : dt dt dt dt
ðA2Þ
Taking the scalar product of the both sides with eA, we obtain the expression dxA deB þ xB eA = vA ¼ : ðA3Þ dt dt
eA = eB ¼ 0
e2A ¼ 1;
ðA4Þ
and the time derivative of the first orthonormality condition, eA =
deA ¼ 0: dt
ðA5Þ
By rewriting equation (A3), we obtain dxA ¼ vA F xB ; dt
ðA6Þ
where
F ¼ eA =
deB : dt
ðA7Þ
From the time derivative of the second orthonormality condition above, eA =
deB deA þ = eB ¼ 0; dt dt
ðA8Þ
deA = eB : dt
dnA deA dnB deB L eA þ nA þ eB þ nB ¼ 2 (nA eB nB eA ): r dt dt dt dt ðA14Þ Here we have used the expression ðA15Þ
and the vector product relations eC < eA ¼ eB ;
eC < eB ¼ eA :
ðA16Þ
Taking the scalar product of both sides of equation (A14) with eA, we obtain dnA deB L þ nB eA = ðA17Þ ¼ 2 nB : r dt dt By a similar procedure as was used for xA, we rewrite this as dnA L ¼ 2 þ F nB ; ðA18Þ r dt which is what we aimed to prove. Again, we omit the proof for nB since it is mostly the same as that for nA. Finally, let us focus on g. By substituting the expression for the unit vector components in terms of g, nA ¼ cos g;
nB ¼ sin g;
ðA19Þ
into equation (A18), we easily show that
we rewrite this expression for F as
F ¼
ðA13Þ
we can rewrite
L ¼ LeC
Here we have used the orthonormality conditions
ðA12Þ
ðA9Þ
dg L ¼ 2 þ F : dt r
ðA20Þ
1454 Fukushima, T. 1996, AJ, 112, 1263 ———. 2001, AJ, 122, 482 ———. 2003a, AJ, 126, 494 ———. 2003b, AJ, 126, 1097 (Paper I) ———. 2003c, AJ, 126, 2567 (Paper II)
FUKUSHIMA REFERENCES Fukushima, T.———. 2003d, AJ, 126, 3138 (Paper III) ———. 2004a, AJ, 127, 3638 (Paper IV) ———. 2004b, AJ, 128, 920 (Paper V) Murison, M. A. 1989, AJ, 97, 1496 Nacozy, P. E. 1971, Ap&SS, 14, 40
