Planetary Perturbations on the 2 : 3 Mean Motion Resonance with Neptune

PASJ: Publ. Astron. Soc. Japan 54, 493 499, 2002 June 25 c 2002. Astronomical Society of Japan. Planetary Perturbations on the 2 : 3 Mean Motion Resonance with Neptune Tetsuharu FUSE Subaru Telescope, National Astronomical Observatory of Japan, 650 North A ohoku Place, Hilo, HI 96720, U.S.A. tetsu@subaru.naoj.org (Received 2001 December 31; accepted 2002 April 2) Abstract A mean motion resonance with Neptune is one of the major dynamical characteristics of the Edgeworth Kuiper Belt Objects that spread over the outside of Neptune s orbit. We investigated the influence of Jupiter, Saturn, and Uranus on the dynamical structure of the 2 : 3 mean motion resonance with Neptune. Since the distance between the planets and objects has great influence on the dynamical structure, numerical integrations were carried out for test particles for 5 Myr under five fictitious solar-system models having different configurations. In discussion about the results, we concentrate on the size of the EKBO eccentricities, e, because they have an especially close relation to the orbital evolution; for example, larger e objects have more opportunities to encounter other objects or planets. By comparing the results of each model, it was found that the resonance structure is not largely affected by Jupiter and Saturn, and that the largest e of the resonant objects depends on the distance between Uranus and the resonance. The distance also influences the structure around the resonance because of variation in the secular resonance positions. We further found that the largest e values of the non-resonant objects at each semi-major axis do not depend on the distance from Uranus; however, they depend on the distance from Neptune. Key words: celestial mechanics Kuiper Belt solar system: formation 1. Introduction As Edgeworth (1949) and Kuiper (1951) predicted, approximately 500 objects, called Edgeworth Kuiper Belt Objects (EKBOs), have been discovered in a region outside of Neptune s orbit (called the Edgeworth Kuiper Belt; EKB) since August of 1992 (Jewitt, Luu 1993). The EKBOs are thought to be remnants of planetesimals in the early stage of the solar system formation as well as the origin of short-period comets (Everhart 1972; Fernández 1980). Some of the discovered EKBOs show two major dynamical characteristics; a mean motion resonance with Neptune is the first example. When the ratio of the orbital periods between Neptune and an EKBO can be expressed as (p + q):p, where p and q are integers, the EKBO will exhibit the (p + q) :p mean motion resonance with Neptune, and a resonant angle, called a critical argument, σ, librates around a fixed angle. The definition of σ is σ (p + q)λ N + pλ + qϖ =(p + q)[ l N +(ϖ ϖ N )] + pl, (1) where λ, ϖ,andl are the mean longitude, longitude of the perihelion, and mean anomaly, respectively, and N indicates Neptune. Among the discovered EKBOs, 39 multi-opposition EKBOs have been found in the 2 : 3 mean motion resonance with Neptune (Marsden 2002). They are called Plutinos because the motion is like Pluto and σ librates around 180.We henceforth designate a mean motion resonance with Neptune as a resonance. A secular resonance is also an important dynamical characteristic of EKBOs. When there is a commensurability between the frequencies of the precession rates of either ϖ (a longitude of perihelion) or Ω (a longitude of ascending node) among two objects, a secular resonance occurs. A ν 8 secular resonance takes place if the frequencies of the precession rates of an EKBO s and Neptune s ϖ are commensurable. When an EKBO exhibits the ν 8 secular resonance, the eccentricity, e, of the EKBO is excited. On the other hand, the ν 18 secular resonance occurs when the frequencies of the precession rates of an EKBO s and Neptune s Ω are resonant, and the EKBO s inclination, i, is excited. Knežević et al. (1991) analytically investigated the locations of secular resonances between 2 and 50 AU from the Sun, and found that a few secular resonances are present in the EKB. Gallardo and Ferraz-Mello (1997) discussed the σ libration of 2 : 3, 3 : 5 and 1 : 2 resonant objects based on a time frequency analysis. It is understood that the 2 : 3 resonance is complex because of the ν 8 and ν 18 secular resonances and the Kozai resonance (Morbidelli et al. 1995; Morbidelli 1997); the Kozai resonance couples the evolution of the EKBO s i and e, and keeps the value of a(1 e 2 )cosi constant, where a is the semi-major axis of the EKBO. Carrying out semianalytical and numerical integration methods, Gallardo and Ferraz-Mello (1998) studied stable regions in the 2 : 3 resonance. Yu and Tremaine (1999) also used three-body planar semi-analytical and numerical integration methods for the 2 : 3 resonance, and found that the stability of the resonant orbits is determined by the size of the object s e. NesvornỳandRoig (2000, 2001) investigated the 2 : 3, 1 : 2, and 3 : 4 resonances with numerical integration methods, and showed chaotic and regular motion of the resonant objects. The purpose of the present work was to extensively research into the 2 : 3 resonance and to investigate the influence of perturbing planets on the dynamical structure in and around the 2 : 3 resonance. We carried out numerical integrations of EKBO test objects under five fictitious solar system models

494 T. Fuse [Vol. 54, with developed integrator (section 2) and showed variations in the orbital elements of the objects in and around the 2 : 3 resonance (section 3). Comparing the results of the integrations, we concentrate on the eccentricity of EKBOs and discuss the influence of the perturbing planets on the dynamical structure in and around the resonance in section 4. 2. Methods of Analysis Symplectic integrators (SI) are good algorithms because solutions are area-preserving, the discretization error in the energy integral does not have a secular term, and it can treat a highly eccentric orbit without changing the step-size (Forest, Ruth 1990). It is convenient to use mixed-variable symplectic integrators (MVSI) if we can write the Hamiltonian as F = F 0 (p,q) +εr(q), where F 0 is an integrable Hamiltonian and εr is a perturbation with a small parameter ε (Kinoshita et al. 1991; Wisdom, Holman 1991). This is because we can use a longer step-size in MVSI than that in SI. We generally have to choose a step-size which is adequate for the smallest orbital period when N-body numerical integrations are carried out. Since we know that the orbital period of EKBOs is the longest in the Sun planets EKBO system, we developed MVSI to MVSI with Files (MVSI F ; see Appendix). The positions of the perturbing planets were calculated in advance with a small step-size, and we saved the data into a file. We then integrated EKBO test objects using the file as the perturbing planet positions. MVSI F can include Jupiter, Saturn, Uranus, and Neptune as the perturbing planets (the masses of the four inner planets are added to that of the Sun). We did not consider galactic tides and a passing star s perturbation, because they finally begin to have dynamical effects on objects whose a is 1000 AU (Duncan et al. 1987). The step-size was 300 days to make the planet positions file and 1500 days for the EKBO test objects. Integrations using a step-size of 300 days for the test objects were also carried out, and it was confirmed that there is no global difference between 1500 and 300 days and CPU time can be saved several dozen percent. The calculation span of the numerical integrations was 5 Myr, which is almost equal to the long-period variations in the EKBO s e and ϖ. The error of the Neptune s longitude was estimated to be 5 for 5 Myr. The initial positions, velocities, and masses of the planets were obtained from JPL DE403 planetary ephemerides at the epoch of JD 2451545.0. Since MVSI F could not treat a close approach between the test objects and planets, the calculation was stopped when the test objects entered any active spheres of the planets given in Danby (1988). 3. Results Not only the existence of perturbing planets, but also the distance between a planet and an EKBO is important to study the influence on the orbital stability of the objects. Since Uranus is especially the closest planet to the resonant EKBOs with Neptune, the distance between them should have an effect on the dynamical structure of the EKBOs. Therefore, we integrated 50960 EKBO test objects placed in and around the 2:3 resonance for 5 Myr assuming five fictitious solar system models: [1] the perturbing planets are Jupiter, Saturn, Uranus, and Neptune; the semi-major axis of Uranus a U is obtained from DE403 ( a U0 ) : Model J N; [2] the perturbing planets are Uranus and Neptune, and a U = a U0 : Model U + N; [3] the perturbing planets are Uranus and Neptune, and a U = a U0 3AU : Model U +N; [4] the perturbing planets are Uranus and Neptune, and a U = a U0 + 3AU : Model U + +N;and [5] the perturbing planet is Neptune only : Model N. The initial conditions of the test objects were as follows: 38.35 AU a 40.79 AU ( a =0.01 AU), 0.01 e 0.4625 ( e =0.0025), i =1, which refer to invariable planes consisting of the Sun four planets for [1], of the Sun Uranus Neptune for [2], [3], and [4], and of the Sun Neptune for [5], ϖ ϖ N =0, Ω = Ω N +90,andσ = 180. The two initial secular angles were chosen from the viewpoint of close pericenters and far nodes. The initial a and e sets are shown on an a e plane in figure 1. We also draw the 2 : 3 resonance borders using the semi-analytical methods developed based on Yoshikawa (1990, 1991). The two arcs crossing the resonance borders indicate Neptune-crossing and Uranus-crossing curves. [1] Model J N We show the initial a and e values of the test objects that survive for 5 Myr in figure 2. The colors of σ stand for the amplitude of σ,definedas σ σ 180, (2) because σ of the 2 : 3 resonant objects librates around 180, like Pluto. It is clear that the resonant σ distribution forms a layer structure at the center of a =39.3AU, and that the inner region has a smaller amplitude than the outer. This is consistent with Gallardo and Ferraz-Mello (1998) as well as Nesvornỳ and Roig (2000). The smallest value of the resonant σ is approximately 6, which means that the amplitude of σ is 174 σ 186. The total number of the surviving objects is 8102 compared with the initial 50960 objects. We can see difference between the Neptune crossing-curve and the largest non-resonant initial e, which is found to be approximately 0.014. Gladman et al. (2001) also discuss the boundary between the stable and chaotic orbits on the initial a e plane. They empirically imply that it is q =30+0.09(a 30), where q is the object s perihelion distance. Because of the relation q = a(1 e), the difference between the Neptunecrossing curve and the stable maximum e was calculated at 0.019 around the 2 : 3 resonance. The value in figure 2 supports the empirical rule by Gladman et al. (2001). In order to investigate the dynamical structure of the resonance on the a e plane, we give the following time-average values of a and e: N / N / a = a i N, e = e i N, (3) i=1 i=1 where N stands for the number of output steps for the integration time. The colors indicate σ defined by equation (2).

No. 3] The 2 : 3 Mean Motion Resonance with Neptune Fig. 1. Initial a and e conditions on the a e plane in and around the 2 : 3 resonance for numerical integrations. a is 0.01 AU and e is 0.0025. The arcs show the Neptune-crossing and Uranus-crossing curves. Fig. 2. Initial a and e of test objects that survived for 5 Myr for [1] Model J N. The colors of σ denote σ = σ 180. Fig. 3. Time-averaged a and e of test objects that survived for 5 Myr for [1] Model J N. The colors of σ denote σ = σ 180. The test objects in the 2 : 3 resonance form a line. Fig. 4. Initial a and e of test objects which survived for 5 Myr for [2] Model U + N. The colors of σ denote σ = σ 180. Fig. 5. Time-averaged a and e of test objects which survived for 5 Myr for [2] Model U + N. The colors of σ denote σ = σ 180. We can see that the a and e values in figure 3 form a line at a 39.45 AU and this is the 2 : 3 resonance center. Since figure 2 shows the osculating orbital elements for the initial conditions, there is a difference in the positions of the layer and line centers in figures 2 and 3. The maximum resonant e at the center is approximately 0.40, and the value is approximately 1.67-times larger than the Neptune-crossing curve. This is because the ν8 secular resonance is not around the resonance center (Morbidelli 1997), and the 2 : 3 resonant objects can be stable at a large-e region for a long time. Gallardo and Ferraz-Mello (1998) reported that the maximum e is 0.37 in their integrations. The slight difference might be due to the different initial conditions. According to a list of EKBO orbital elements at the website of the Minor Planet Center ( http://cfa-www.harvard.edu/iau/lists/tnos.html on 2002/2/24), the largest e of the 2 : 3 resonant EKBO is 0.336 of 1996 TP66. It is consistent with the results in figure 3. Furthermore, several lines are, for example, seen at a = 38.20, 38.48, 39.98, 40.42, and 40.70 AU. According to the Kepler s third law, they correspond to the 7 : 10 (38.20 AU), 9 : 13 (38.48 AU), 9 : 14 (40.42 AU), and 7 : 11 (40.70 AU) resonances with Neptune, and the 1 : 3 (39.98 AU) 495

496 T. Fuse [Vol. 54, resonance with Uranus. The minimum e values around 40 AU are larger than those in other regions, because e is excited by the ν 8 secular resonance pointed out by Knežević et al. (1991) and Morbidelli (1997). Most of the largest e values of nonresonant objects (blue marks) are approximately 0.05 smaller than the Neptune-crossing curve for any a-region. This means surviving objects that have a larger e than the curve must be in the 2 : 3 resonance. On the other hand, the value is inconsistent with figure 2 and Gladman et al. (2001), because of the difference between the initial conditions and time average values. When we discuss a more realistic configuration of the outer solar system, we should use the value of 0.05 led by figure 3. The i values of the test objects were also calculated, and the maximum and minimum i were found to be 7. 5and 0. 5, respectively. In the left region of the resonance borders, most i values are smaller than approximately 3,andthe i values are between 1 and 5 in the right region of the resonance borders (around 40 AU). The phenomena are consistent with the presence of the ν 18 secular resonance (Knežević et al. 1991; Morbidelli 1997). Because of the existence of the ν 8 and ν 18 secular resonances outside the 2 : 3 resonance borders, the small-e and -i EKBOs should not be there in a real EKB. [2] Model U +N Figure 4 describes the initial a and e values of objects that survive during integrations under the Sun Uranus Neptune system. We can see that the shape of the resonant σ structure in figure 4 is fairly similar to that in figure 2. The total surviving objects are 8357, which is also close to the results of the model [1]. The a and e values under the model [2] are shown in figure 5. The maximum resonant e is approximately 0.39 at a 39.44 AU, where the a and e values form a line. The difference between this value and that of the model [1] might be quite small, and is approximately 0.02. We can say from these similarities that the influence of Jupiter and Saturn on the dynamical structure of the 2 : 3 resonance on the a e plane is small. It is seen that the e values outside the resonance borders are fairly larger than those of the model [1]. The difference should be due to the change in the positions of the ν 8 secular resonance because of the absence of Jupiter and Saturn, or the direct effects of the two planets on the objects. We need a further investigation to conclude that the difference is due to the direct or indirect influence of Jupiter and Saturn. On the other hand, most maximum e values of the non-resonant objects are approximately 0.05 smaller than the Neptune-crossing curve at any a values. Since this model does not include Jupiter and Saturn, it seems that the influence of the two planets on these values is also small. [3] Model U +N The next model contains two perturbing planets, the same as the model [2], but a U is by 3 AU smaller than a U0. It is expected that the effect of Uranus on the objects becomes weak compared with the model [2], and the resonance structure on the a e plane may be different from those of the two models, [1] and [2]. In figure 6, we indicate the initial a and e of the test objects that survive for 5 Myr. The Uranus-crossing curve disappears from the figure because of the change in a U. According to figure 6, the resonant σ distribution is fairly different from those in figures 2 and 4; i.e. more resonant objects that initially have large-e remain stably in the resonance. It was found that 10548 objects survive, and the number is approximately 1.3-times larger than those of models [1] and [2]. The results reflect the weakness of the influence of Uranus. We can see the a and e values in figure 7. The maximum resonant e is approximately 0.42 at a 39.44 AU, which is obviously larger than those of models [1] and [2]. Because of the distance from Uranus, the maximum resonant e should be large. On the other hand, most objects outside the resonance borders stay near the initial positions, expect for some resonant objects. These show that the distance between Uranus and the resonance must be important for determining the largest-e value of the resonant objects and the dynamical structure around the 2 : 3 resonance. The largest e values of the non-resonant objects are approximately 0.05 smaller than the Neptune-crossing curve, which is the same as those in models [1] and [2]. From the results, we can say that the values are not determined by the distance from Uranus, but from Neptune. In figure 6, objects forming islands are observed around the initial a =38.20 AU and e =0.32 0.4; lines can also be seen near the same positions in figure 7. Such objects have far larger e values than the Neptune-crossing curve; however, they are stable because of being in the 7 : 10 resonance with Neptune. [4] Model U + +N Although Uranus and Neptune are included in this model in the same way as in models [2] and [3], a U is 3 AU larger than a U0. It is possible that we will see a large effect on the structure on the a e plane in and around the 2 : 3 resonance Figure 8 contains the initial a and e values of the objects that survive during the integrations. Since a U is larger, the Uranuscrossing curve comes down in figure 8. The smaller e region of the resonant σ structure around a 39.4AU is similar to those of models [1], [2], and [3] in figures 2, 4, and 6, respectively. It is, however, seen that the maximum initial e value of the surviving resonant objects is significantly smaller than those of other models. The total number of the surviving objects is 4856, which is the minimum value, and the smallest distance between Uranus and the resonance must make the differences. We show a and e on the a e plane in figure 9. The maximum resonant e has the smallest value among the previous four models, and is approximately 0.29 at a 39.45AU. The structure outside the resonance borders is entirely different from the others, and there are especially few surviving objects at the small-e region at 38AU a 39AU. Comparing these figures, we can say that the existence of Uranus determines the structure in and around the 2 : 3 resonance on the a e plane. Most of the values of the maximum non-resonant e are, in contrast, the same as those in models [1] [3] and it is approximately 0.05. It becomes clear that it does not depend on the distance from Uranus.

No. 3] The 2 : 3 Mean Motion Resonance with Neptune 497 Fig. 7. Time-averaged a and e of test objects that survived for 5 Myr for [3] Model U + N. The colors of σ denote σ = σ 180. Fig. 8. Initial a and e of test objects that survived for 5 Myr for [4] Model U + + N. The colors of σ denote σ = σ 180. Fig. 9. Time-averaged a and e of test objects that survived for 5 Myr for [4] Model U + + N. The colors of σ denote σ = σ 180. Fig. 10. Initial a and e of test objects that survived for 5 Myr for [5] Model N. The colors of σ denote σ = σ 180. Fig. 11. Time-averaged a and e of the test objects that survived for 5 Myr for [5] Model N. The colors of σ denote σ = σ 180. Fig. 6. Initial a and e of test objects that survived for 5 Myr for [3] Model U + N. The colors of σ denote σ = σ 180.

498 T. Fuse [Vol. 54, Table 1. Surviving object ratios and maximum resonant e values. Model Survived (%) Max e [1] Model J N 15.9 0.40 [2] Model U+N 16.4 0.39 [3] Model U + N 20.7 0.42 [4] Model U + + N 9.53 0.29 [5] Model N 25.6 0.49 Note. Given are the ratios of the surviving object numbers to the initial object numbers, and the maximum resonant e values for the numerical integrations under the five fictitious solar system models. [5] Model N This model consists only of Neptune as the perturbing planet. Because the test object is massless, a system comprising the Sun and Neptune is a two-body problem and the orbital elements of Neptune do not change, except for the mean anomaly, l N. Figure 10 indicates the initial a and e of the objects that survive for 5 Myr; we can see that the resonant σ layer structure is the same as in the other models. The smallest resonant σ is approximately 2. 1. Since Uranus is not included in the system, the motion of the objects is not affected by Uranus, and the maximum initial e of the surviving objects is the largest initial value of 0.4625; however, a further investigation is required to confirm the real stable maximum e in this model. The total number of surviving objects is 13045, which is the maximum value among the five fictitious solar system models. The largest resonant e is approximately 0.47 at a 39.45 AU in figure 11, which is also the maximum value among the five models. It can be seen that the resonance in this model is quite stable, and the resonant objects will be able to have large e values. On the other hand, the values between the maximum non-resonant e and the Neptune-crossing curve is approximately 0.05; we should notice that this value is the same as that for the other models. Therefore, we conclude that Jupiter, Saturn, and Uranus do not have any influence on the largest e values of non-resonant objects; however, the distance from Neptune determines them. As is the case with figures 6 and 7, we can see some islands and lines above the Neptune-crossing curve. They correspond to the 7:10 (38.20 AU), 9:13 (38.48 AU), and 7:11 (40.70 AU) resonances with Neptune; also, the resonant objects are stable even if they have large e values. 4. Discussion We summarize the results of our numerical integrations under the five solar system models for the 2 : 3 resonance in table 1. Comparing the maximum resonant e in table 1, we can say that the values are determined by the distance between Uranus and the resonance; for example, a further distance from Uranus makes the resonant maximum e larger. We also tabulate the ratio of the surviving object number to the initial number (= 50960) in table 1. It can be seen that the distance Fig. 12. Distribution of a and e of the discovered EKBOs between 37.8 AU and 41 AU. The orbital elements are based on the website of the Minor Planet Center on 2002/2/24. between Uranus and the resonance has a close relation to the ratio; if it is larger, the ratio becomes smaller. There is no large difference in both the resonance structure and the surviving number ratio between [1] Model J N and [2] Model U + N. This indicates that the influence of Jupiter and Saturn on the resonance structure is small, and we need a further investigation to confirm that it has direct or indirect effects of the planets. We cannot see the dependence of the distance from Uranus on the largest e values of the non-resonant objects; however, it is found that the whole structure outside the resonance would also be affected by Uranus according to the results of [2] Model U + N, [3] Model U + N, and [4] Model U + +N. This is because the motion of Neptune should be different if Uranus s orbit differs, and the positions of the ν 8 secular resonance will change. Therefore, it is concluded that the structure around the resonance is determined by the distance between Uranus and the resonance, while the maximum e values of the nonresonant objects are restricted by the distance from Neptune. It is thought that planets may have migrated outward during the solar system formation (Malhotra 1995). From the above discussion, if Uranus migrated during the solar system formation, the structure of the resonance was affected by the variation in the distance between Uranus and the resonance. This means that the evolution of Uranus will be determined if the resonance structure is revealed. According to the orbital distribution of the observed EKBOs between 37.8 41 AU in figure 12, we can presume that the 2 : 3 resonant EKBOs having larger e values should exist, and that non-resonant small-e EKBOs ought to be around both 38.7 AU and 39.8 AU if Uranus did not migrate. Further survey observations for those undiscovered EKBOs are expected to clarify the real dynamical structure in and around the resonance and to understand the possibility of the migration of Uranus. 5. Conclusion We carried out numerical integrations for EKBO test objects to study the dynamical structure in and around the 2 : 3 resonances under five fictitious solar system models. We reached the following conclusions based on our investigations:

No. 3] The 2 : 3 Mean Motion Resonance with Neptune 499 Comparing the results of [1] Model J N and [2] Model U + N, we can say that the effect of Jupiter and Saturn on the dynamical structure of the 2 : 3 resonance is small. The maximum e fairly depends on the distance between Uranus and the 2 : 3 resonance. The distance between Uranus and the 2 : 3 resonance also has an effect on the dynamical structure around the 2 : 3 resonance, because of the variation in the positions of the ν 8 secular resonance. The values between the largest non-resonant objects e and the Neptune-crossing curve are approximately 0.05 under the five solar system models. They are determined by the distance from Neptune. If Uranus did not migrate, undiscovered larger-e 2:3 resonant and small-e non-resonant EKBOs should exist. The orbital evolution of Uranus will be confirmed when we reveal the dynamical structure in and around the 2 : 3 resonance by further observations. I thank Hiroshi Kinoshita and Hiroshi Nakai for fruitful discussions. I am also grateful Makoto Yoshikawa for useful comments. Workstations at the Astronomical Data Analysis Center of the National Astronomical Observatory of Japan were used for the data analysis. Symplectic Integrator for Hamiltonian includ- Appendix. ing Time We assume a Hamiltonian system, H, that includes time, t, explicitly: H (q,p,t)=k(p) +U(q,t), (A1) where q and p are a canonical set. After introducing T,which is a momentum conjugated to t, and the new Hamiltonian F Danby, J. M. A. 1988, Fundamentals of Celestial Mechanics, (Virginia: Willmann-Bell), 352 Duncan, M., Quinn, T., & Tremaine, S. 1987, AJ, 94, 1330 Edgeworth, K. E. 1949, MNRAS, 109, 600 Everhart, E. 1972, Astrophys. Lett., 10, 131 Fernández, J. A. 1980, MNRAS, 192, 481 Forest, E., & Ruth, R. D. 1990, Physica, 43D, 105 Gallardo, T., & Ferraz-Mello, S. 1997, AJ, 113, 863 Gallardo, T., & Ferraz-Mello, S. 1998, Planet. Space Sci., 46, 945 Gladman, B., Holman, M., Grav, T., Kavelaars, J., Nicholson, P., Aksnes, K., & Petit, J.-M. 2001, astro-ph/0103435 Jewitt, D., & Luu, J. 1993, Nature, 362, 730 Kinoshita, H., Yoshida, H., & Nakai, H. 1991, Celest. Mech. Dyn. Astron., 50, 59 can be written as F = T + H = K(p) +T + U(q,t) = K (p,t )+U(q,t). The second order SI is q 1 = q 0 + c 1 τ K p (p 0,T 0 )=q 0 + c 1 τ K p (p 0), References t 1 = t 0 + c 1 τ K T (p 0,T 0 )=t 0 + c 1 τ, p 1 = p 0 d 1 τ U q (q 1,t 1 ), T 1 = T 0 d 1 τ U t (q 1,t 1 ), (A2) (A3) q 2 = q 1 + c 2 τ K p (p 1)=t 0 + c 1 τ, t 2 = t 1 + c 2 τ = t 0 +(c 1 + c 2 )τ, p 2 = p 1 d 2 τ U (A4) q (q 2,t 2 ), T 2 = T 1 d 2 τ U t (q 2,t 2 ), where τ is the step size, and (q 0,p 0 )and(q 2,p 2 ) are the initial values and numerical solutions after τ, respectively. We note that we do not have to evaluate T 1 and T 2 because they do not actually appear in the system. If we do not calculate the positions of perturbing planets simultaneously, the Hamiltonian includes the time explicitly, as above. The positions of the perturbing planets are, in practice, calculated with a small time step; they are saved into a data file beforehand for integrating for test objects. Knežević, Z., Milani, A., Farinella, P., Froeschlé, Ch., & Froeschlé, Cl. 1991, Icarus, 93, 316 Kuiper, G. P. 1951, Astrophysics, ed. J. A. Hynek (New York: McGraw-Hill), 357 Malhotra, R. 1995, AJ, 110, 420 Marsden, B. G. 2002, Minor Planet Electron. Circ., 2002-C08 Morbidelli, A. 1997, Icarus, 127, 1 Morbidelli, A., Thomas, F., & Moons, M. 1995, Icarus, 118, 322 Nesvornỳ, D., & Roig, F. 2000, Icarus, 148, 282 Nesvornỳ, D., & Roig, F. 2001, Icarus, 150, 104 Wisdom, J., & Holman, M. 1991, AJ, 102, 1528 Yoshikawa, M. 1990, Icarus, 87, 78 Yoshikawa, M. 1991, Icarus, 92, 94 Yu, Q., & Tremaine, S. 1999, AJ, 118, 1873