integral invariant relations is not limited to one or two such

The Astronomical Journal, 126:3138 3142, 2003 December # 2003. The American Astronomical Society. All rights reserved. Printed in U.S.A. EFFICIENT ORBIT INTEGRATION BY SCALING AND ROTATION FOR CONSISTENCY OF KEPLER ENERGY, LAPLACE INTEGRAL, AND ANGULAR MOMENTUM DIRECTION Toshio Fukushima National Astronomical Observatory, Osawa, Mitaka, Tokyo 181-8588, Jaan; Toshio.Fukushima@nao.ac.j Received 2003 June 10; acceted 2003 Setember 11 ABSTRACT By adding the orbital angular momentum vector as another auxiliary quantity to be integrated, we extend our scaling methods to integrate quasi-kelerian orbits numerically in order to suress the growth of integration errors in the inclination and the longitude of the ascending node. This time, the method follows the time evolution of the angular momentum vector, as well as the time develoment of the Keler energy and/or the Lalace integral, in addition to integrating the usual equation of motion. By using a rotation that is indeendent of the alication of the satial scaling, the new method adjusts the osition and velocity integrated rigorously at each integration ste in order to align both erendicular to the integrated angular momentum vector. The direction and the angle of the rotation are determined uniquely from the osition, the velocity, and the angular momentum vector integrated. As with the original scaling methods, the new method is simle to imlement, fast to comute, and alicable to a wide variety of integration methods, erturbation tyes, and comlexities of roblems. Although this addition rovides no significant decrease in the osition error, the new method is suerior to the original scaling methods in the sense that it enhances the quality of the integration by significantly reducing the errors of the orbital lane at the cost of a negligibly small amount of additional comutation. Key words: celestial mechanics methods: n-body simulations 1. INTRODUCTION Recently, we develoed a coule of owerful tools to numerically integrate erturbed Kelerian motions (Fukushima 2003a, 2003b). 1 They are based on the ideas of the integral invariant relation (Szebehely & Bettis 1970) and the manifold correction (Nacozy 1971). These methods, which we call the scaling methods for brevity, integrate not only the osition and velocity under consideration, but also some quantities that are conserved in the unerturbed case, such as the Keler energy, 2 K,and the Lalace integral, 3 P. At each ste of the integration, the integrated ositions and velocities are adjusted so as to exactly satisfy the defining relation for K, a scalar function F v µ (x µ v) derived from P, or both. The adjustments are accomlished with a single or a dual satial scale transformation, the scale factor or factors are determined by solving an associated cubic equation with Newton s method in the former case or by rigorously solving a set of two linear equations in the latter. As illustrated in Paers I and II, the scaling methods achieve a drastic decrease in the integration errors at the cost of a negligibly small increase in comutational labor. However, as stated in Paer II, K and P are not the only auxiliary quantities that can be integrated for the manifold correction. In rincile, any function of the osition, the velocity, or the time can be selected. Also, the number of 1 Hereafter Paer I and Paer II, resectively. 2 The Keler energy is the total secific energy of the two-body roblem, which is conserved under ure Kelerian motion. 3 This is the vector ointing to the ericenter. Its magnitude is roortional to the eccentricity. The vector is conserved in ure Kelerian motions. The Lalace integral is essentially the same as the Runge-Lenz vector in electromagnetic terms. 3138 integral invariant relations is not limited to one or two such as K and F. In this article, we reort on a further extension of the scaling methods by adding two scalar relations derived from the angular momentum integral. This time, we achieve the adjustments to the integrated osition and velocity not by scaling transformations but by an oeration that is indeendent of their alication a sace rotation. As will be shown below, this extension refines both the original scaling methods in the sense that errors are also reduced in the inclination and in the longitude of the ascending node, once again with a negligible increase in comutational labor. In the following, we resent the new method in x 2 and illustrate it with numerical exeriments in x 3. 2. METHOD Let us begin with a erturbed one-body roblem, the equation of motion of which is written as dv l dt ¼ r 3 x þ a ; ð1þ x is the osition, v is the velocity, l GM is the gravitational constant, r x is the radius vector, and a is the erturbing acceleration. In the scaling methods (Paers I and II), we numerically integrate not only the above equation of motion but also the time develoment of some analytic function or functions of the osition and the velocity, Q(x, v): dq @Q @Q dv dt ¼ v þ @x @v dt : ð2þ For Q, we selected the Keler energy K in Paer I, and the air of K and the Lalace integral P in Paer II. Here K and

EFFICIENT ORBIT INTEGRATION. III. 3139 P are defined as K T þ U ; P v µ L þ Ux ; ð3þ T v 2 =2 ; U l=r ; L x µ v : ð4þ The equations for the time develoment of K and P are dk dt ¼ v x a ; dp dt ¼ 2ðv x aþx ðx x aþv ðx x vþa : ð5þ If the numerical integrations conducted are erfect, then identity relations on certain functions of x, v, and Q must hold: f ðqðx I ; v I Þ; x I ; v I Þ¼f ðq I ; x I ; v I Þ ; the subscrit I denotes the quantity integrated. As for f, we chose K itself in Paer I, and the air of K and F jp Uxj ¼jv µ Lj ¼jðv 2 Þx ðx x vþvj in Paer II. In general, however, there remain inequalities in the identity relations (eq. [6]) due to the errors in numerical integrations. We dare to judge that these inequalities are roduced by the errors in x I and v I. Thus, we adjust x I and v I in order to satisfy the identity relations. This adjustment was done with a single satial scale transformation, ðx; vþ!ðsx; svþ ; in Paer I, and with a dual satial scale transformation, ðx; vþ!ðs X x; s V vþ ; in Paer II. We determined the scale factors s in Paer I and the air s X and s V in Paer II rigorously from the identity relations. This adjustment on the fly will work when the Q are constants of the motion, since the integration of Q is error-free in that case. We observed that this treatment in fact significantly reduces the integration errors in Paers I and II. In the case of ure Kelerian motion, the single scaling method kees the integration error in the semimajor axis, a, at the level of the machine esilon. This leads to suression of the growth of the integration error in the mean longitude at the eoch, L 0 M 0 + $, which becomes only linear with resect to time. (Here M 0 is the mean anomaly at the eoch, $ +! is the longitude of ericenter, is the longitude of the ascending node, and! is the argument of ericenter.) On the other hand, the dual scaling method achieves the same imrovements, but it also kees the integration errors in the eccentricity e and in $ bounded. The magnitude of the bounded errors can be reduced by choosing smaller ste sizes or higher order integrators. From the viewoint of integration errors in the orbital elements, those in the remaining two elements, and the inclination I, remain the challenge to be addressed. Thus, we take aim at a remaining integral of the Kelerian motion, L, as an additional quantity to be integrated. The equation of time evolution is simly written as dl dt ¼ x µ a : ð10þ However, it is not easy to find a suitable functional relation ð6þ ð7þ ð8þ ð9þ or relations derived from L for roer adjustment of the integrated osition and velocity. In fact, as we described in the aendix to Paer II, we once failed to simultaneously maintain a relation for its magnitude, L L, as well as for K with a dual scaling. The remaining two orbital elements are angle variables and, therefore, are insensitive to scaling in general. Thus, we abandon the direction of extending the method by using scaling transformations only. Instead, we note that rotational oerations are rather suitable to align the directions of the osition and the velocity. Then we choose two orthogonal relations as the functional relations derived from L: x x L ¼ v x L ¼ 0 : These two conditions are satisfied by a rotation ðx; vþ!ðrx; RvÞ ; ð11þ ð12þ the rotation matrix R is given by s x b Rb ¼ cb þ s µ b þ s : ð13þ 1 þ c Here b denotes x or v, and c and s are defined as c ffiffiffiffiffiffiffiffiffiffiffiffi 1 s 2 ; s L L µ L I ; ð14þ L I L x I µ v I and L is its magnitude, while L I is the integrated angular momentum vector and L I is its magnitude. See the Aendix for the derivation of R. The above rotational oeration is well defined for any combination of L and L I, including the case in which L or L I vanishes, the conditions are satisfied without alying any rotation. This rotational oeration is also indeendent of the alication of a single or dual scaling. In fact, there is no difference in the adjusted osition and velocity whether the scaling is alied after or before the rotation. Therefore, the rotation can be done by itself or in combination with either of the scalings. At this stage, the reader may question the use of the magnitude of the angular momentum, L. In the single scaling method, we could not find a roer use for it, as described in the aendix of Paer II. In the dual scaling method, this is a surlus quantity to be maintained, since there is a well-known relation 4 among L, K, and P P, P 2 2KL 2 ¼ l 2 : ð15þ This means that if one controls K and P, then L is also controlled automatically. Let us add a note that is imortant in ractice. As with K and P, the magnitude of the variations in L is small comared with the value of L itself. Thus, it is wise to integrate not L directly but its deviation from the initial value, DL L L 0, instead. The time develoment equation is substantially unchanged by this trick, as d DL dt ¼ x µ a : ð16þ 4 This is easily derived from their exressions in terms of orbital elements as K = 1 2 l/a, L =[la(1 e2 )] 1/2, and P = le.

3140 FUKUSHIMA Vol. 126 As we exerienced in the generalization of Encke s method (Fukushima 1996), this technique greatly reduces the rounding off. We omit the generalization to the case of multile bodies, which is straightforward, as in the original scaling methods; simly integrate DL for each body and adjust its x and v searately, with the rotations being different body by body. The above rocedure requires only a negligible increase in actual comutation. This is mainly because comutation of the right-hand side of equation (16) is very easy, since its major art, the calculation of the erturbation accelerations, has already been done in evaluating the right-hand side of the equation of motion. 3. NUMERICAL EXPERIMENTS In the following, we show some results from numerical exeriments on the new method. Before we go further, let us exlain how the integration errors are estimated. We measure the error of a numerical integration by comaring with a reference solution that was obtained with the same integrator, the same initial conditions, and the same model arameters but with half the ste size. As was illustrated in Figures 1 and 2 of Paer II, the results from the scaling methods obtained using the Adams method follow ower laws with resect to the ste size. Since the indices of the ower laws are sufficiently large, at 10 or 12, we conclude that any difference is mostly due to the incorrectness of the integration with the larger ste size (see x 3.1 of Paer I for details). First, we omit the results confirming the same roerties as we exerienced with the scaling methods, such as the wide alicability with resect to the tye of roblem, the kind of erturbation, the method of integration, the arameters of the integrator, and the number of celestial bodies. Hereafter, we will concentrate on the difference obtained by alying the rotational oeration. Let us begin with the ure Kelerian orbits. Figures 1 and 2 illustrate the nature of the error growth for the cases of the single scaling with the rotation and the dual scaling with the rotation, resectively. Plotted are the normalized errors in the orbital elements, (Da)/a, De, DI, e D$, and DL 0. Some quantities have been scaled to illustrate the manner of the variation more clearly. The results for D are omitted, since they were always less than the machine esilon. The orbit integrated is of moderate eccentricity and inclination, with e = 0.1 and I =23. The integrator used was the PECE mode (redict, evaluate, correct, evaluate) of the imlicit 10th-order Adams method, the starting tables were reared using Gragg s extraolation method and the ste size was fixed at 1/128 of the orbital eriod. When comared with the case of using only the scalings (Figs. 4 and 5 of Paer II), we find that the errors in and I have decreased drastically, down to the level of the machine esilon. This situation is indeendent of the tye of scaling alied together with the rotation. Also, the henomenon has no relation to the magnitude of the eccentricity, as shown in Figure 3, the magnitude of the errors in e, $, andl 0 increase with resect to e while the others do not change. From the viewoint of reducing the total error, that is, the error in osition, however, the rotation does not contribute significantly. This is because the main error source is other than the errors in and I namely, (1) that in L 0, Fig. 1. Element error for a Kelerian orbit. Shown are the normalized errors in the orbital elements, (Da)/a, De, DI, e D$, and DL 0, when the single scaling and the rotation are alied. The errors were obtained by comaring with the reference solution, which was obtained using the same integrator but with half the ste size. We omitted the result for D, since it was always less than the machine esilon. Integrated is a ure Kelerian orbit with e = 0.10 and I =23 for around 10 6 orbital eriods. The integrator used was the PECE mode of the imlicit 10th-order Adams method with a ste size of 1/128 the orbital eriod. The starting tables were reared using Gragg s extraolation method. With resect to time, De, e D$, and DL 0 all grow linearly, while the remaining two remain randomly on the order of the machine esilon or less. Comare this with Figs. 1 3 of Paer II, which illustrate the cases of no scaling, the single scaling only, and the dual scaling only, resectively. which grows quadratically with resect to time in the case of no scaling, (2) those in e, $, and L 0, all of which grow linearly with resect to time under the single scaling method, or (3) that in L 0, which grows linearly with resect to time in the dual scaling method. In any event, it is obvious that the alication of the rotation is useful in the sense of suressing the integration errors in the direction of the orbital lane. Let us move to the erturbed case. First, we confirm that we face a similar situation in the case that the erturbing acceleration conserves the direction of the angular momentum vector, such as standard air drag. Also, when the Fig. 2. Same as Fig. 1, but for the dual scaling and the rotation. This time, only DL 0 grows linearly with time and De and e D$ remain bounded.

No. 6, 2003 EFFICIENT ORBIT INTEGRATION. III. 3141 Fig. 3. Eccentricity deendence of element error for a Kelerian orbit. Comared are the eccentricity deendence of the integration errors when the dual scaling and the rotation are alied. Shown are the relative errors in the orbital elements of a ure Kelerian orbit after 3.3 10 4 revolutions. erturbation is sufficiently weak, the results are ractically the same as in the ure Kelerian case. Then, as a tyical case of a moderate-strength erturbation, we integrated an orbit affected by the J 2 erturbation, as deicted in Figure 4. For this lot we used the single scaling with the rotation, since the total error in osition is smaller in the single scaling method than in the dual scaling method, as we found in Paer II. What we learned is that Figure 4 is ractically the same as the results obtained using the single scaling and no rotation, in every element. In other words, the effect of the rotation is insignificant. As an examle of a third-body attraction, we integrated a general three-body roblem of the Sun, Juiter, and Saturn using the dual scaling and the rotation. The conditions of the integration were the same as in the ure Kelerian case. The initial conditions were quoted from those at J2000.0 in the latest lanetary and lunar ehemeris, DE405. Also, the lanetary masses were set to be the same as used in creating Fig. 5. Inclination error of Juiter in the general three-body roblem of the Sun, Juiter, and Saturn. Comared are the results of no scaling, the single scaling only, the dual scaling only, and the dual scaling and the rotation. The last is the best, although it becomes the same as the second to last after a long time of integration. A similar figure was obtained for the longitude of the ascending node. DE405. Figure 5 shows a comarison of the errors in Juiter s inclination with other cases: with no scaling or rotation, with the single scaling only, and with the dual scaling only. Although the effect of the rotation diminishes after some long time, the combination of the dual scaling and the rotation kees the error at the level of the machine esilon for the first 10 4 yr or so. This is also the case for. For the other elements, we see no significant changes from the case without rotation. To summarize, we conclude that although the overall gain in recision for the osition is not significant, use of the rotation to maintain the consistency relations connected to the orbital angular momentum enhances the quality of orbit integrations, esecially in weak-erturbation cases and for some secific erturbation tyes. Thus, the addition of the rotation to the manifold correction imroves the quality of orbit integration somewhat. Since the additional cost to erform the rotation is small, we generally recommend its use in combination with the single or dual scaling methods when they are alied. Fig. 4. Single scaling and rotation for a erturbed Kelerian orbit. Same as Fig. 1, but erturbed by the effect of J 2 with a relative magnitude of 10 3. There is ractically no difference from the result with the single scaling only. 4. CONCLUSION By introducing a rotation as a second kind of transformation to adjust the integrated osition and velocity, we have extended our single and dual scaling methods for numerically integrating quasi-kelerian orbits. This time, we added the orthogonality of the osition and velocity to the orbital angular momentum as the consistencies to be maintained during the numerical integration. The new method simultaneously integrates not only the usual equation of motion and the time evolution of the Keler energy, with or without the Lalace vector, but also the time develoment of the orbital angular momentum vector. Then, at each integration ste the method corrects the integrated osition and velocity by a single or dual satial scale transformation lus the rotation, which aligns the integrated osition and velocity so as to be erendicular to the integrated angular momentum vector. The new method behaves as well as the original scaling methods in the

3142 FUKUSHIMA sense that it has wide alicability among integrators, erturbations, and roblems. In the case of ure Kelerian orbits, the errors in I and are reduced to the level of the machine esilon. For erturbed orbits, these errors grow with resect to time. However, their rate of growth is slower than or equal to the case without the rotation. The gain in recision is significant when the erturbation is small. In conclusion, we generally recommend the addition of the rotation to the single or the dual scaling method, since it enhances the quality of the scaling methods somewhat, at the cost of a negligibly small increase in the comutational time and labor. APPENDIX ROTATION FOR CONSISTENCY OF ANGULAR MOMENTUM DIRECTION Let us derive the exression for the rotation matrix, equation (13) in the main text. Assume that L I, the integrated angular momentum vector, is not the same as L x I µ v I, the angular momentum vector analytically comuted from x I, the osition integrated, and v I, the velocity integrated. Note that x I and v I are erendicular to L by definition. Then we only have to find a rotation matrix R that transforms L so as to be arallel to L I,as ðrlþ µ L I ¼ 0 : ða1þ This is because such a matrix automatically transforms x I and v I so as to satisfy the defining relation of the angular momentum, ðrx I Þ µ ðrv I Þ¼L I ; and therefore to satisfy the orthogonality relations, ðrx I Þ x L I ¼ðRv I Þ x L I ¼ 0 ; ða2þ ða3þ as a natural result. The condition in equation (A1) can be rewritten in terms of unit vectors as RN ¼ N I ; ða4þ N L=jLj ; N I L I =jl I j : ða5þ This rotational oeration is accomlished by a rotation of angle h around the axis n. Here h is the angle between N and N I, while n is a vector erendicular to both N and N I. Namely, ¼ sin 1 jsj ; n ¼ s=jsj ; ða6þ s N µ N I ¼ðsin Þn : ða7þ It is well known 5 that the rotation of a vector b around a unit vector n by an angle h is exressed in terms of vector roducts as Rb ¼ b þðsin Þn µ b þð1 cos Þn µ ðn µ bþ : ða8þ Let us denote 6 s sin ¼ jj; s c cos ¼ ffiffiffiffiffiffiffiffiffiffiffiffi 1 s 2 : ða9þ Then, by using the identities n µ ðn µ bþ ¼ bþðnxbþn ; 1 cos ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffi 1 s 2 s 2 ¼ 1 þ ffiffiffiffiffiffiffiffiffiffiffiffi ; ða10þ 1 s 2 we can rewrite the above exression as s x b Rb ¼ cb þ s µ b þ s ; ða11þ 1 þ c which is what we aimed to rove. Note that the final exression contains no trigonometric functions but one square root. This means that it is fairly fast to evaluate. Also, the final form is robust against the situation in which h or s is small, which is usually the case in recise integrations. 5 See, e.g., eq. (74.1) of MacMillan (1960). 6 Do not confuse this s with the single scaling factor in Paer I. Fukushima, T. 1996, AJ, 112, 1263. 2003a, AJ, 126, 1097 (Paer I). 2003b, AJ, 126, 2567 (Paer II) MacMillan, W. D. 1960, Dynamics of Rigid Bodies (New York: Dover) REFERENCES Nacozy, P. E. 1971, A&SS, 14, 40 Szebehely, V., & Bettis, D. G. 1970, in Gravitational N-Body Problems, ed. M. Lecar (Dordrecht: Reidel), 136