Relative Effect of Inclinations for Moonlets in the Triple Asteroidal. Systems

Transcription

1 Relative Effect of Inclination for Moonlet in the Triple Ateroidal Sytem Yu Jiang,, Hexi Baoyin, Yun Zhang. State Key Laboratory of Atronautic Dynamic, Xi an Satellite Control Center, Xi an 70043, China. School of Aeropace Engineering, Tinghua Univerity, Beijing 00084, China Y. Jiang () (correponding author) Abtract. We preent the analyi and computational reult for the inclination relative effect of moonlet of triple ateroidal ytem. Perturbation on moonlet due to the primary non-phericity gravity, the olar gravity, and moonlet relative gravity are dicued. The inclination vector for each moonlet follow a periodic elliptical motion; the motion period depend on the moonlet emi-major axi and the primary J perturbation. Perturbation on moonlet from the Solar gravity and moonlet relative gravity make the motion of the x component of the inclination vector of moonlet and the y component of the inclination vector of moonlet to be periodic.the mean motion of x component and the y component of the inclination vector of each moonlet form an ellipe. However, the intantaneou motion of x component and the y component of the inclination vector may be an elliptical dic due to the coupling effect of perturbation force. Furthermore, the x component of the inclination vector of moonlet and the y component of the inclination vector of moonlet form a quai-periodic motion. Numerical calculation of dynamical configuration of two triple ateroidal ytem (6) Kleopatra and (39) 00 SN63 validate the concluion. Key word: triple ateroidal ytem; minor celetial bodie; nonlinear dynamic;. Introduction To tudy the dynamical mechanim of triple ateroidal ytem can not only help u to undertand the origin of the Solar ytem and the formation of the ateroidal belt (Araujo et al. 0), but alo help to deign the orbit of pacecraft in the human future pace miion to triple ateroidal ytem. The firt triple ateroid (87) Sylvia wa dicovered in 00 (Marchi et al. 00), after that, there are eight uch triple

2 ateroidal ytem and one Kuiper-belt object dicovered in the olar ytem. Table how the phyical and orbital parameter of thee triple ateroidal ytem. Two of them are trinary near-earth-ateroid ytem (NEA), i.e CC (Brozović et al. 0; Fang et al. 0) and 39 00SN63 (Fang et al. 0; Araujo et al. 0). Beide, TC36 (Benecchi et al. 00) and 3608 Haumea (Pinilla-Alono et al. 009; Lockwood et al. 04) are tran-neptunian object (TNO). Other are main-belt triple ateroidal ytem. Table. Phyical and orbital parameter of triple ateroidal ytem Name of triple ateroid Primary Diameter of primary, econd ytem Ma (kg) Bulk denity Rotation period component, and third (gˑcm -3 ) (h) component (km) (4) Eugenia a-a ,, 7 (87) Sylvia b-b , 0.8, 0.6 (93) Minerva c-c , 3.6, 3. (6) Kleopatra d-d , 8.9, Balam e ,.84, TC36 f,f , 3, 3608 Haumea g-g , 30, 60 (3667) 994CC h,h , 0.3, SN63 i-i ± ±0.000.±0.3, 0.77±0., 0.43±0.4 a Beauvalet et al. 0. a Beauvalet and Marchi 04. a3 Marchi et al. 00. b Berthier et al. 04. b Fang et al. 0. b3 Frouard et al. 0. b4 Marchi et al. 00. b Winter et al c Marchi et al. 0. c Marchi et al. 03. c3 Torppa et al d Decamp et al. 0. d Jiang and Baoyin 04. d3 Jiang et al. 04. d4 Jiang 0. d Jiang et al. 0a. d6 Jiang et al. 0b. d7 Jiang et al. 0c. d8 Otro et al e Vokrouhlický009. f Benecchi et al. 00. f Mommert et al. 0. g Duma et al. 0. g Pinilla-Alono et al g3 Lockwood et al. 04. h Brozović et al. 0. h Fang et al. 0. i Fang et al. 0. i Araujo et al. 0. i3 Becker et al. 0 The calculation of dynamical parameter of triple ateroidal ytem i the bai for the tudy of dynamical mechanim for thee ytem. Marchi et al. (00) preented the two moonlet of (87) Sylvia orbiting at 70 and,360 km, and the J of the two moonlet are 0.7 and 0.8, repectively. Ragozzine and Brown (009) tudied

3 the orbit and mae of atellite of 3608 Haumea and indicated that Haumea could have experienced a great colliion billion of year ago. Marchi et al. (00) found that the inclination of moonlet of (4) Eugenia are quite different from other known main-belt triple ateroidal ytem, the inclination of the two moonlet Petit-Prince and Princee relative to the primary equator, are 9 and 8, repectively. Fang et al. (0) found that the moonlet of (87) Sylvia orbiting at 807.±.km and 37±4.0km, and the inclination are 7.84 and 8.93, repectively. Marchi et al. (03) invetigated the triple ateroidal ytem (93) Minerva and found that the moonlet of (93) Minerva are 3km and 4km in diameter, repectively. Beauvalet and Marchi (04) analyzed the J of two triple ateroidal ytem (4) Eugenia and (87) Sylvia, and derived the internal tructure of thee two triple ytem. Jiang et al. (0a) found that the number and poition of equilibrium point around the primary of (6) Kleopatra will vary while the rotational peed of the primary change. The tudy of dynamical behaviour of triple ateroidal ytem include orbital element, pin-orbit lock, bifurcation, reonance, table and untable region, etc. Winter et al. (009) indicated that the longitude of the orbital node of the two moonlet of (87) Sylvia, Romulu and Remu, are locked to each other. Brozović et al. (0) found that the inner moonlet of (3667) 994CC i pin-orbit locked relative to the primary and the outer moonlet i not pin-orbit locked. Fang et al. (0) calculated the motion of moonlet of (39) 00SN63 and (3667) 994CC, examined the mean-motion reonance, Kozai reonance, and evection reonance for 3

4 thee two triple ateroidal ytem, the reult illutrated that the moonlet are not in thee three reonance cae. Araujo et al. (0) invetigated the table region of the three component of (39) 00SN63, they divided the region around (39) 00SN63 into four ditinct region and found that the table region are near Alpha and Beta while reonance motion with Beta and Gamma are untable. Fang et al. (0) deduced that the (87) Sylvia i not in the 8:3 mean-motion reonance, beide, they calculated the effect of a pa through 3: mean-motion eccentricity-type reonance. Frouard and Compère (0) tudied the intability zone for moonlet of the triple ateroidal ytem (87) Sylvia with conidering the non-phericity of Sylvia, and found that thi triple ytem i in a deeply table zone. Marchi et al. (03) found that the moonlet of Minerva are at % and % of the Hill radiu. Jiang et al. (0b) found four kind of bifurcation of periodic orbit familie in the potential of the primary of (6) Kleopatra. Araujo et al. (0) conidered a male particle in the vicinity of (39) 00SN63 and found that the table region of the particle retrograde orbit are much bigger that the prograde orbit. Uing the perturbation method, the motion of the moonlet relative to the primary of the large ize ratio triple ateroid ytem can be analyzed. Kozai (99) derived the perturbation of orbital element of a atellite in the gravitational potential of the Earth. Cook (96) preented the perturbation from the Sun and Moon to the orbital element of a atellite in the gravitational potential of the Earth. Allan (970) dicued the critical inclination with the J and J 4 term. For the orbit with mall inclination, the orbital element can be indicated with the inclination vector (Hiztz 4

5 008). The perturbation method can be applied to analyze the motion of moonlet relative to the primary in the binary and triple ateroid ytem. Araujo et al. (0) found that the J term of the primary ha a ignificant effect to the table retrograde orbit in the triple ateroid 00 SN63. In thi work we focu on the moonlet relative effect in the triple ateroidal ytem. In Section, the perturbation on the two moonlet due to the Solar gravity and the primary non-phericity gravity are derived, and then the relative perturbation effect between thee two moonlet have been invetigated. In Section 3, the primary J, Solar gravity, and the two moonlet relative effect are all conidered to analyze the dynamical ytem of the inclination vector of the two moonlet. We find that for each moonlet, the inclination vector form a periodic elliptical motion.. Perturbation on Moonlet Due to the Solar Gravity and the Primary Non-phericity Gravity In thi ection, we derive the formula of perturbation on moonlet due to the olar gravity and the primary non-phericity gravity. Denote J a the value of the primary J perturbation, G a the Newtonian gravitational contant, mmajor a the ma of the primary, Gmmajor, r a the primary mean radiu. Let a be the emi-major axi, n a the mean orbit angular peed, e be the eccentricity, i be 3 a the inclination, Ω be the longitude of the acending node, ω be the argument of periapi, M be the mean anomaly, m be the ma. Denote the inclination vector

6 ix iin. The ubcript M, M, and repreent orbital parameter of Moonlet, iy ico mm Moonlet and Sun, repectively. Denote M m m M major mm and M m m M major.. Perturbation on Moonlet Due to the Primary Non-phericity Gravity Conider the primary J perturbation acting on the two moonlet, the rate of average change (Kozai 99) of inclination and right acenion of the acending node are di 0 dt d 3nJ r dt a e. () coi For the orbit with mall inclination, ue the Lagrange planetary equation (Cook 96), we have dix di d in i co dt dt dt diy di d co i in dt dt dt. () Subtituting Eq. () into Eq. () and uing mall angle approximation, then the inclination vector ecular variation for moonlet can be expreed a dixm 3Jr i dt nmam diym 3Jr i dt nmam ym xm, (3) and the inclination vector ecular variation for moonlet can be expreed a 6

7 dixm 3Jr i dt nmam diym 3Jr i dt nmam ym xm. (4) where n M and n M are mean orbit angular peed for moonlet and moonlet, repectively. ix M and iy M are component of inclination vector of moonlet, while ix M and iy M are component of inclination vector of moonlet. a M and a M are emi-major axe for moonlet and moonlet, repectively. and where Thee two equation can be rewritten by d ixm ixm K dt i ym i, () ym d ixm ixm K dt i ym i, (6) ym K 0 3Jr nmam 3J r n MaM 0 and K 0 3Jr nmam 3J r n MaM 0. (7) Eigenvalue of K are 3J r j while eigenvalue of n M a M K are 3J r j, n M a M where j. Thu we can conclude that the primary J perturbation make each moonlet inclination vector to be periodic motion. The motion trajectory of the extremal point of the inclination vector i an ellipe. The motion period are 4 n 3J a M M r and 4 n 3J a M M r, repectively. 7

8 . Perturbation on Moonlet Due to the Solar Gravity and Moonlet Relative Gravity Here we only conider the olar gravity and moonlet relative gravity. The rate of average change of inclination and right acenion of the acending node due to the third body gravity (Cook 96) are where di 3 dt n d 3 dt nin i, (8) GM (9) r d 3 d and d ud id ud d co co co in in in cou coi in u co coi in i ini inu in d coud coid in ud co d ini coi inid inud d d d d d d d. (0) Here the ubcript d repreent orbital parameter of the third body. u f, f i the true anomaly. The olar gravity and moonlet relative gravity acting on the inclination vector of moonlet i (the derivation i preented in appendix A) di 3 n 3 n n in in i n in i co di 3 n 3 n n in in i n in i in xm M M M M M M dt 4 nm 8 nm ym M M M M M M dt 4 nm 8 nm, () where n repreent the mean orbit angular peed for the Sun in the primary 8

9 centroid inertial coordinate ytem, which equal to the triple ateroidal ytem mean orbit angular peed relative to the Sun; and i repreent the true anomaly and the inclination of the Sun in the primary centroid inertial coordinate ytem, repectively. im and M repreent the inclination and the longitude of the acending node of moonlet in the primary centroid inertial coordinate ytem, repectively. Conider the ecular item, one can eaily obtain di 3 n 3 n n in i n in i co diym 3 n M M nm in im in M dt 8 nm xm M M M M M M dt 8 nm 8 nm, () Where in and in 0 i applied to the above equation. In like manner, the Solar gravity and moonlet relative gravity acting on the inclination vector of moonlet i di 3 n 3 n n in i n in i co diym 3 n M MnM in im in M dt 8 nm xm M M M M M M dt 8 nm 8 nm, (3) where im and Mrepreent the inclination and the longitude of the acending node of moonlet in the primary centroid inertial coordinate ytem, repectively. Conider the moonlet are in the orbit which i near the equator of the primary. Thi aumption i atified for mot of the triple ateroidal ytem (Beauvalet and Marchi 04; Fang et al. 0; Decamp et al. 0; Vokrouhlický009). With thi aumption, in Eq. (), one have 9

10 in i co in i co i co i co i i O i i in i in in i co i in i co i i O i i M M M M M ym M ym M ym M M M M M xm M xm M xm (4) Subtituting Eq. (4) into Eq. () yield the following equation di 3 n 3 n n in i n i diym 3 n M M nm ixm dt 4 nm xm M M M M ym dt 8 nm 4 nm In the ame way, we have. () di 3 n 3 n n in i n i diym 3 n M MnM ixm dt 4 nm xm M M M M ym dt 8 nm 4 nm From Eq. () and Eq. (6), we obtain two planar dynamical ytem. (6) and di 3 n 3 n n in i n i diym 3 n M MnM ixm dt 4 nm xm M M M M ym dt 8 nm 4 nm, (7) di 3 n 3 n n in i n i diym 3 n M M nm ixm dt 4 nm xm M M M M ym dt 8 nm 4 nm. (8) Eq. (7) indicate that the x component of the inclination vector of moonlet and the y component of the inclination vector of moonlet form a planar dynamical ytem, while Eq. (8) indicate that the x component of the inclination vector of 0

11 moonlet and the y component of the inclination vector of moonlet form a planar dynamical ytem. Thee two planar dynamical ytem can be expreed a Eq. (9) and Eq. () where d ixm ixm A B dt i i, (9) ym ym A 3 0 n M MnM 4 nm 3 4 n M M 0 n n M M and 3 n B 8 0 M n n M in i. (0) d ixm ixm C D dt i i, () ym ym where C 3 0 n M MnM 4 nm 3 4 n M M 0 n n M M and 3 n D 8 0 M n n M in i. () Uing the theory from Strogatz (994, ee page 0-), for a two-dimenional nonlinear ytem, the linear tability of the ytem can be determined by the linearized ytem. The linearized ytem of Eq. (9) i d ixm ixm A dt i i, The Jacobian ym ym matrix i A, which i a contant matrix. Eigenvalue of A are 3 4 n n j, which mean that the planar dynamical ytem Eq. (9) i M M M M linearly table. Eigenvalue of C are alo 3 4 n n j, which mean M M M M that the planar dynamical ytem Eq. () i alo linearly table.

12 3 K n n 4 Let M M M M, then Eq. (9) and Eq. () can alo be expreed a dixm Ki xm dt di ym 9 n MnM nm in i K iym dt 3 nm, (3) and dixm Ki xm dt di ym 9 n M nm nm in i K iym dt 3 nm. (4) The form of Eq. (3) and Eq. (4) look like the equation of harmonic ocillator which ha no frictional damping. For intance, Kiy M i like the linear retoring 3 force in the harmonic ocillator. The frequency i K MM nm nm, and the 4 period i T. The motion of the x component of the inclination vector of K moonlet and the y component of the inclination vector of moonlet i periodic. The motion x component of the inclination vector of moonlet and the y component of the inclination vector of moonlet i periodic. 3. Relative Effect on Inclination Vector Between the two Moonlet In thi ection, the primary J, Solar gravity, and the two moonlet relative effect are all calculated. Conider the primary J, Solar gravity, and moonlet gravity acting on the inclination vector of moonlet a well a moonlet gravity acting on the inclination vector of moonlet, then combine Eq. (3), (4), (7), and (8), one can obtain the following equation

13 di di J r n x M 3Jr 3 n 3 n M i ym nm in i M nm iym dt nm am 8 nm 4 nm ym 3 3 M i xm M nm ixm dt nm am 4 nm di diym 3Jr 3 n M i n i dt nm am 4 nm x M 3Jr 3 n 3 nm i ym nm in i M nm dt nm am 8 nm 4 nm Thi equation can be implified into x M M M xm i ym. () ixm ixm d i y M i ym E F, (6) dt i i xm xm iym iym where 3Jr 3 n M 0 0 MnM nm am 4 nm 3Jr 3 n M 0 MnM 0 nm am 4 n M E 3 n M 3Jr 0 MnM 0 4 nm nm am 3 n M 3Jr MnM nm nm am (7) 3 n 8 0 F 3 n 8 0 M M n n M n n M in i in i. (8) For the triple ateroidal ytem, the influence on the inclination vector from the 3

14 primary J perturbation i bigger than from the Solar gravity and the two moonlet relative effect. The Solar gravity and the two moonlet relative effect make the x component of the inclination vector of moonlet and the y component of the inclination vector of moonlet form a planar dynamical ytem, meanwhile, they make the x component of the inclination vector of moonlet and the y component of the inclination vector of moonlet form a planar dynamical ytem. However, the primary J perturbation make the x component and the y component of the inclination vector of moonlet form a planar dynamical ytem, meanwhile, it make the x component and the y component of the inclination vector of moonlet form a planar dynamical ytem. Generally peaking, for the inclination vector, the influence from the J perturbation of the primary i much bigger than from the Solar gravity and the two moonlet relative effect. Thi implie that the mean motion of x component and the y component of the inclination vector of each moonlet form an ellipe; however, the intantaneou motion of x component and the y component of the inclination vector of each moonlet may form an elliptical dic. In addition, the x component of the inclination vector of moonlet and the y component of the inclination vector of moonlet form a quai-periodic motion, and the x component of the inclination vector of moonlet and the y component of the inclination vector of moonlet form a quai-periodic motion. Two triple ateroidal ytem, (6) Kleopatra and (39) 00 SN63 are taken a example to verify the above theory. The gravitational field and irregular 4

15 hape of the primary i computed with hape model data uing the polyhedral model (Neee 004). The primary gravitational potential (Werner 994; Werner and Scheere 997) can be computed by U G re Ee re Le G rf Ff r f f, (9) eedge f face the primary gravitational force i calculated by U G E r L G F r, (30) e e e f f f eedge f face while the Heian matrix of the primary gravitational potential can be calculated by ( U) G Ee Le G F f f, (3) eedge f face where G= m 3 kg - - repreent the Newtonian gravitational contant, σ repreent the primary bulk denity; r e and r f are body-fixed vector, r e i from the field point to the point on the edge e while r f i from the field point to the point on the face f; E e and F f are geometric parameter, E e i related to edge while F f i related to face; L e i the integration factor while ω f i the olid angle. We apply the above reult to two triple ateroidal ytem (6) Kleopatra and (39) 00 SN63. Moonlet of (6) Kleopatra are Alexhelio and Cleoelene, while moonlet of (39) 00 SN63 are Beta and Gamma. Table how the initial orbital parameter for the moonlet of two triple ateroidal ytem ued in the calculation. To compare with the theoretical reult of the previou content, we ue the gravitational model and integrate the dynamical equation to calculate the inclination vector. The dynamical equation are

16 T where ψk AkIk Ak Kk rk p k, k pk fk pk rk mk Kk nk A ψ A k k k, k,,3, (3) r repreent the poition vector of the k-th body, pk mkr k repreent the linear momentum vector, f k repreent the gravitational force acting on the k-th body, K k repreent the angular momentum vector, Ak i the attitude matrix. n k i the reultant gravitational torque acting on the k-th body. All the vector are expreed in the inertial pace. ψ k i calculated with the following method. For a vector v vx, vy, v z T, define the matrix 0 vz v y v vz 0 vx. (33) vy vx 0 Table. Initial orbital parameter for the moonlet of two triple ateroidal ytem a) (6) Kleopatra (Decamp et al. 0) Orbital parameter Alexhelio Cleoelene Semi-major axi: a (km) Eccentricity: e 0 0 Inclination: i (deg) Long. of acend. node: Ω (deg) Arg. periapi: ω (deg) 0 0 Mean anomaly: M (deg) 0 0 Ma: (kg) b) (39) 00 SN63 (Fang et al. 0) Orbital parameter Beta Gamma Semi-major axi: a (km) Eccentricity: e Inclination: i (deg) Long. of acend. node: Ω (deg) Arg. periapi: ω (deg) Mean anomaly: M (deg)

17 Denity (g cm -3 ).0.3 Fig.. The dynamical configuration of the two moonlet relative to the primary for the triple ateroidal ytem (6) Kleopatra, the imulation duration i 8d. (a) (b) 7

18 (c) (d) Fig.. The numerical calculation of the component of inclination vector of two moonlet of the triple ateroidal ytem (6) Kleopatra, (a) the trajectory of two component of the inclination vector of moonlet, (b) the trajectory of two component of the inclination vector of moonlet, (c) the trajectory of the x component of the inclination vector of moonlet and y component of the inclination vector of moonlet, (d) the trajectory of the x component of the inclination vector of moonlet and y component of the inclination vector of moonlet. Fig. 3. The dynamical configuration of the two moonlet relative to the primary for the triple 8

19 ateroidal ytem (39) 00 SN63, the imulation duration i 600d. (a) (b) (c) (d) Fig. 4. The numerical calculation of the component of inclination vector of two moonlet of the triple ateroidal ytem (39) 00 SN63, (a) the trajectory of two component of the inclination vector of moonlet, (b) the trajectory of two component of the inclination vector of moonlet, (c) the trajectory of the x component of the inclination vector of moonlet and y component of the inclination vector of moonlet, (d) the trajectory of the x component of the inclination vector of moonlet and y component of the inclination vector of moonlet. Decamp et al. (0) preented the orbit parameter of two moonlet of (6) Kleopatra in mean J000 equator, ee Table. The frame ued here i defined a follow, the origin i the ma center of the primary, the xy plane i the equator of the primary, and z axi i the pin axi of the primary. In our frame, the inclination of Alexhelio and Cleoelene are.6 deg and 3.8 deg, repectively. So the inclination 9

20 of thee two moonlet are mall and the reult here can be ued to analyze the orbit of thee two moonlet. Fig. how the dynamical configuration of the two moonlet relative to the primary for the triple ateroidal ytem (6) Kleopatra while Fig. preent the component of inclination vector of two moonlet. From Fig., one can conclude that the mean motion of i x and i y of each moonlet form an ellipe, and the amplitude of the intantaneou motion of the elliptical trajectory for Cleoelene i bigger than for Alexhelio. Beide, Alexhelio i x and Cleoelene i y form a quai-periodic motion, and Alexhelio i y and Cleoelene i x form a quai-periodic motion. For the motion near the urface of ateroid like Kleopatra, the perturbation method with low Legendre coefficient can t model the orbital motion accurately. The reaon i that the higher order term of the Legendre coefficient need many iteration to converge (Elipe and Riagua 003). Beide, there exit ome orbit where the minimal ditance between the ma center of Kleopatra and the orbit i maller than Kleopatra mean radiu (Jiang et al. 0c). Thi mean that the perturbation method with low Legendre coefficient can t be ued to model the motion near the urface of Kleopatra. However, if the orbit i far from the urface of Kleopatra, the perturbation method with low Legendre coefficient can alo be ued. The ratio of the emi-major axi of the moonlet and the mean radiu of Kleopatra are 6.7 and 0 (Decamp et al. 0). The numerical method ue the polyhedral model to model the gravity of Kleopatra. The numerical reult fit the theoretical reult well becaue the orbit are far from Kleopatra, and the ma ratio of the moonlet and Kleopatra are only 0

21 and Fang et al. (0) preented the orbit parameter of two moonlet of (39) 00 SN63 in mean J000 equator, ee Table. In our frame, the inclination of Beta and Gamma are 0.33 deg and 3.3 deg, repectively. The inclination of thee two moonlet are alo mall and the reult here can be ued to analyze the orbit of thee two moonlet. Fig. 3 how the dynamical configuration of the two moonlet relative to the primary for the triple ateroidal ytem (39) 00 SN63 while Fig. 4 preent the component of inclination vector of two moonlet. Fig. 4 implie that the mean motion of i x and i y of each moonlet form an ellipe, and the amplitude of the intantaneou motion of the elliptical trajectory for Gamma i much bigger than for Beta. The intantaneou motion of the elliptical trajectory for Beta form an ellipe while the intantaneou motion of the elliptical trajectory for Gamma form an elliptical dic. Additionally, Beta i x and Gamma i y form a quai-periodic motion, and Beta i y a well a Gamma i x form a quai-periodic motion. The numerical calculation validate the above theoretical derivation. The theoretical reult ay that the mean motion of x component and the y component of the inclination vector of each moonlet form an ellipe. From Fig. (a), (b), 4(a), and 4(b), one can ee that the inclination vector of each moonlet form an ellipe. The relative amplitude of the trajectory in Fig. 4(b) i maller than that in Fig. 4(a), becaue the inclination of Beta and Gamma are 0.33 deg and 3.3 deg in the equator of the primary, repectively. Fig. 4(a) how the inclination vector of Beta while Fig. 4(b) how the inclination vector of Gamma. In addition, the theoretical

22 reult ay that the x component of the inclination vector of moonlet and the y component of the inclination vector of moonlet form a quai-periodic motion, and the x component of the inclination vector of moonlet and the y component of the inclination vector of moonlet form a quai-periodic motion. From Fig. (c), (d), 4(c), and 4(d), one can ee that the component of the inclination vector between different moonlet are coupled and form a quai-periodic motion. The reult hown above agree with previou work baed on obervational data that concluded periodical variety of the orbital parameter of different triple ateroidal ytem. Marchi et al (00) calculated orbital parameter of the triple ateroidal ytem (4) Eugenia, and found that the inclination of thee two moonlet of (4) Eugenia are about 9 deg and 8 deg relative to the equator of the primary, and have a periodical variety. Fang et al. (0) calculated the change rate of the argument of pericenter and the longitude of the acending node for the two moonlet of (39) 00 SN63. Our reult alo indicate that the longitude of the acending node have a variety. Fang et al. (0) alo invetigated the emi-major axi and eccentricity of Remu and Romulu relative to Sylvia of the triple ateroidal ytem (87) Syivia, and found both of them have a periodical variety, and the variety period are different. The previou tudie only conider the inclination of the moonlet in the triple ateroidal ytem. However, the moonlet of (6) Kleopatra and (39) 00 SN63 are in the orbit which i near the equator of the primary, the inclination vector i much better to analyze the motion of thee moonlet than inclination of thee moonlet (Hintz 008). Fig. and Fig. 4 preent the coupling motion of the inclination vector of two

23 moonlet in the triple ateroidal ytem. 4. Concluion The nonlinear dynamical behaviour in the triple ateroidal ytem are complicated. The primary ha irregular hape and the moonlet have relative effect. The primary non-phericity gravity, the olar gravity, and moonlet relative gravity are all conidered in thi paper. It i found that the inclination vector for each moonlet form a periodic elliptical motion. The Solar gravity and moonlet relative gravity lead to the periodic motion for i x of moonlet and iy of moonlet, and the periodic motion for iy of moonlet and i x of moonlet. The mean motion of i x and i y of the inclination vector of each moonlet form an ellipe. The intantaneou motion of i x and i y may be elliptical due to the coupling effect of thee force. The coupling effect of thee force alo make i x of moonlet and i y of moonlet form a quai-periodic motion, and quai-periodic motion. i x of moonlet and i y of moonlet form a The numerical computation of orbital motion of two triple ateroidal ytem (6) Kleopatra and (39) 00 SN63 further illutrate the reult. The numerical reult are compared with the reearch in exiting literature. The moonlet of (6) Kleopatra and (39) 00 SN63 motion near the equator of the primary, then the inclination of thee moonlet are mall. To analyze the motion of thee moonlet, uing the inclination vector i better than the inclination. We alo compare the numerical reult with the theoretical reult. It i found that the amplitude of the 3

24 intantaneou motion of the elliptical trajectory for the moonlet Cleoelene i bigger than for the moonlet Alexhelio in the triple ateroidal ytem (6) Kleopatra. The intantaneou motion of the elliptical trajectory for Gamma look like an elliptical dic in the triple ateroidal ytem (39) 00 SN63. Acknowledgement Thi reearch wa upported by the National Natural Science Foundation of China (No. 370& No. 766), the State Key Laboratory of Atronautic Dynamic Foundation (No. 06ADL-00) and the National Science Foundation for Ditinguihed Young Scholar (08). Appendix A In thi ection, we preent the derivation of Eq. (). For the firt moonlet in the nearly circular orbit near the equatorial plane of the primary, the perturbation force on moonlet due to the Solar gravity and the econd moonlet relative gravity (Tremaine et al. 008) i r' r F rn c 3co r' r, (A) where all the vector are expreed in the equatorial inertial frame of the firt moonlet, r repreent the firt moonlet poition vector, r i the norm of r, r ' repreent the Sun poition vector or the econd moonlet poition vector, r ' i the norm of r ', repreent the angle between r and r '. For the Solar gravity, nc n; for the econd moonlet gravity, n n. c M M The component of F in the normal direction of the firt moonlet orbital plane i 4

25 F n ' 0 3rnc co r Fn n 0, (A) r ' Where in iin n 0 in i co, coi for the Solar gravity, co u co in u in co i r co u in in u co co i r in umin im M M M M M M M M M M, (A3) co r ' in coi r ', (A4) in in i while for the econd moonlet gravity, co co in in co i r ' co in in co co i r ' in Min im M M M M M M M M M M. (A) Thu the perturbation of the inclination vector due to the Solar gravity i di F n in M xm dt rnm 3 n n in coi co in i in in coi in i co in in i diym Fn co M dt rnm 3 n nm co co in i in in coi in i co in in i nm M nm. (A6) Conidering that in i, thu di 3 n xm nm in in i dt 4 nm diym 3 n n in in i dt 4 M nm. (A7) The perturbation of the inclination vector due to the econd moonlet gravity i

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