Dynamical Model of Binary Asteroid Systems Using Binary Octahedrons

Transcription

1 Dynamica Mode of Binary Asteroid Systems Using Binary Octahedrons Yu Jiang 1,, Hexi Baoyin 1, Mo Yang 1 1. Schoo of Aerospace Engineering, singhua University, Beijing , China. State Key Laboratory of Astronautic Dynamics, Xi an Sateite Contro Center, Xi an , China corresponding author: Y. Jiang () e-mai: jiangyu_xian_china@163.com H. Baoyin() e-mai: baoyin@tsinghua.edu.cn Abstract. We used binary octahedrons to investigate the dynamica behaviors of binary asteroid systems. he mutua potentia of the binary poyhedron method is derived from the fourth order to the sixth order. he irreguar shapes, reative orbits, attitude anges, as we as the anguar veocities of the binary asteroid system are incuded in the mode. We investigated the reative trajectory of the secondary reative to the primary, the tota anguar momentum and tota energy of the system, the three-axis attitude anguar veocity of the binary system, as we as the anguar momentum of the two components. he reative errors of tota anguar momentum and tota energy indicate the cacuation has a high precision. It is found that the infuence of the orbita and attitude motion of the primary from the gravitationa force of the secondary is obvious. his study is usefu for understanding the compicated dynamica behaviors of the binary asteroid systems discovered in our Soar system. Key words: Binary Asteroids; Fu wo-body Probems; Mutua Potentia 1 Introductions he discovery of the binary asteroid systems in our soar system brings a great interest to the dynamics around irreguar ceestia bodies. Severa previous iterature has studied the dynamics of the binary systems. Vasikova (005) used a triaxia eipsoid to mode the irreguar shape of asteroid, and cacuated the dynamica behavior of a massess partice orbiting around the reative equiibrium of the triaxia eipsoid. Lindner et a. (010) used the massive ine segment and a mass partice to mode the primary and the secondary of the binary asteroid system, respectivey. hey discussed the synchronous orbit, chaotic orbits, unstabe periodic orbits, and spin-orbit couping 1

2 of the system. Liu et a. (011) investigated the equiibria and orbits for a massess partice in the potentia of a rotating cube. Najid et a. (011) used a straight eongated inhomogeneous segment to mode the gravitationa potentia of the asteroid, and cacuated the trajectories and Poincarésections of a massess partice. Liu et a. (013) investigated the surface motion of a massess partice on a rotating cube. Jiang et a. (015) investigated the variety of the reative equiibria in the potentia of an asteroid, which can hep to understand the dynamica behaviors of the arge-size-ratio synchronous binary systems. Ferrari et a. (016) used three partices to mode the binary asteroids, two of them are used to mode the gravity of the primary, and the other one is used to mode the gravity of the secondary. hey refer to this mode as the patched three-body probems, and cacuated the equiibrium points and orbits around the equiibrium points for this mode. hese studies can hep us to understand the dynamica behaviors of a tiny moonet orbiting around an asteroid. However, the size and mass of the moonet in the binary asteroid systems are not zero, the assumption of the massess partice of the moonet wi cause to the oss of some important dynamica characteristic of the binary systems, incuding the spin-orbit ocked of the moonet, the topoogica cases of the reative equiibria, the motion stabiity of the moonets, etc. Besides, the shapes of the bodies in the binary asteroid system are irreguar. he assumption of the partice masses or sphere cannot mode the attitude motion of the systems. hus the spin-orbit ocked, the escape of the moonet or not, and the resonance of the binary asteroid systems cannot be investigated. Some iterature used more suitabe mode to investigate the dynamics of

3 the binary systems. he most common mode is used by the two finite straight segments to simuate the gravity of the binary systems. Guirao et a. (011) used the two finite orthogona straight segments to approximate the irreguar binary systems, and studied the noninear stabiity of the equiibrium points of the system. Jain and Sinha (014a) studied the positions and stabiity of the equiibrium points in the restricted three-body probem with the assumption of both finite straight segments. Under the same assumption, Jain and Sinha (014b) investigated the non-inear stabiity of L4 for a massess partice. Baikie et a. (014) aso used two massive ine segments to mode the binary systems; they derived the potentia energy, kinetic, force, and torque of the system, and presented the unstabe periodic orbits and chaotic orbits of the systems. Shang et a. (015) used the doube eipsoids to mode the gravitationa environment of the synchronous binary systems, and cacuated the periodic orbits for a massess partice in the systems. Nadoushan and Assadian (016) studied the spin-orbit resonance of the binary asteroids with assuming the shape mode of the two asteroids as a sphere and an eipsoid. he Poincare section for a fictitious binary asteroid is cacuated. Eshaboury et a. (016) used two triaxia rigid bodies to mode the two primaries in the restricted three-body probem, and cacuated the positions and stabiity of the equiibrium points in a specia case. hey aso found three unstabe coinear equiibria. In this paper, we use two octahedrons to mode the irreguar shapes of binary asteroid systems. With this mode, irreguar shapes, reative orbits, attitude anges, and the anguar veocities of the binary asteroid system can be cacuated. he mutua 3

4 potentia, force terms and torque terms of the system are cacuated by the method of two homogeneous poyhedra. he dynamica equation is integrated by the 7/8-order Runge-Kutta method. his paper is organized as foows: Section deas with motion of the Fu wo-body Probems. Section 3 presented the simuation of the binary asteroid system using the binary octahedron. Section 4 gives a brief review of this study. Motion Equations of the Fu wo-body Probems We focus on the binary asteroida system which comprised of two irreguar-shaped bodies. Let 1, be the -th body, G be the gravitationa constant, r be the position vector from the origin of the inertia reference frame to the center of mass of the -th body, dm D be the mass eement at, D be the position vector from the center of mass of to the mass eement, D be the density at the mass eement which satisfies dmd D dv D, dv D be the voume eement, p mr be the inear momentum vector of, q AD r be the position vector from the origin of the inertia reference frame to the mass eement dm D, m means the mass of, A be the attitude matrix from the inertia reference frame to the principa reference frame of, K r p AI be the anguar momentum vector of, d A D, and G I. Using arbitrary vector v vx, vy, v z, the antisymmetric matrix of v is denoted as 0 vz v y v vz 0 vx. (1) vy vx 0 4

5 Denote R Ar, p mr, mm m 1 m m 1, P A p,, A A1A Γ, 1 AG 1 and Γ G. hen, G A g I 1,. And the tota mutua gravitationa k k k k k k potentia energy can be expressed with the parameters in the body-fixed frame of as: U 1 D D D D G dv dv 1 1 A D1D R. () he motion equation of the binary asteroida system (Maciejewski 1995; Naidu and Margot 014; Jiang et a. 016, 017; Wang and Xu 018) can be written in the body-fixed frame of by U P P R P R R m Γ1 Γ1 μ1 Γ Γ μ A A 1A A A, (3) where k 1,, I Γ and 1, I AΓ he kinetic energy of the binary asteroida system can be written as 1 mk rk k, I k k, (4) k1 hen, the tota energy of this binary asteroida system can be given by H U 1 D D D D G dv dv 1 1 mk rk k, Ik k 1 k 1 A D1D R. (5) he grativationa force acting on k reads 5

6 f k G 1 D D dv D dv D A D1D R. (6) he resutant gravitationa torque acting on k expressed in the inertia space can be written as k G A D1R D n D dv dv 3 1 D D1 D 1 A D1D R. (7) he inear momentum integra and the anguar momentum integra of the system are p p p, K K K. (8) 1 1 Using the infinite series, one can cacuate the mutua potentia of two homogeneous poyhedra. Werner and Scheeres (005) presented the series of the mutua potentia to the third order as foows: ij i j i Q Qw Qijr 3Qijw w i U G a bab a 1 b R R R R. (9) ij k i j k 3Qijkr w 5Q ijkw w w 5 7 R R he forth order to the sixth order of the mutua potentia can be cacuated by U U U 3Q r 15Q r w w 35Q w w w w 8R 4R 8R ijk ij k i j k ijk ijk ijk Q r w 70Q r w w w 63Q w w w w w ijk m ij k m i j k m ijkm ijkm ijkm R 8R 8R 5Q r 105Q r w w 35Q r w w w w 31Q w w w w w w ijkmn ijk m n ij k m n i j k m n ijkmn ijkmn ijkmn ijkmn R 16R 16R 16R (10) Here a and b are densities of the simpexes a and b on the poyhedra, respectivey; a and b are Jacobian determinant of the transformation for the 6

7 simpexes a and b, respectivey; 1 Q is the rank-0 tensor of rationa numbers, 36 symmetric in a index pairs, and Q is the rank-k tensor of rationa numbers, i1 ik symmetric in a index pairs; R R ; i w is the exporting variabe from R and the negative weight of simpexes; ij r and ijk r are the rank- and rank-4 tensors generated by the negative weight of simpex with itsef, respectivey. 3 Simuation of the Binary Asteroid Systems Using Binary Octahedrons We use two octahedrons to mode the irreguar shapes of binary asteroida systems. Generay, the two components of the binary asteroida system are not the same; we refer to the arger one as the primary (denoted as A in the figures), and the smaer one as the secondary (denoted as B in the figures). he voumes of the octahedrons are set to be m 3 and m 3. Densities of them are.5 g cm -3. he three axis coordinates of the primary are [-1.0, 1.0]km, [-1.5,1.5]km, and [-0.9, 0.9]km in x, y, and z axis, respectivey. he three axis coordinates of the secondary are [-1.0, 1.0]km, [ , ]km, and [ , ]km in x, y, and z axis, respectivey. o see the shape and gravitation of the octahedrons, we potted the shape of the primary in Fig. 1, and showed the structure of the gravitationa potentia for the primary in Fig.. he gravitationa potentia is cacuated by the binary poyhedra method (Werner and Scheeres 1997; souis 01; D Urso 014) to the fourth order. he octahedron in Fig. has the same size and shape with the primary, and its mass is set to be a unit mass because we are ony interested in the structure of the gravitationa potentia. From Fig., one can see that the gravitationa potentias in the 7

8 different coordinate panes are different, and are reated to the shape of the octahedron. he maxima vaue of the gravitationa potentia is m s -. Figure 1. he 3D shape of the primary. (a) 8

9 (b) (c) (d) 9

10 Figure. he structure of the gravitationa potentia for the primary in the coordinate system of the principa axis of inertia, the unit of the gravitationa potentia is m s -. (a) he gravitationa potentia in the xy pane; (b) he gravitationa potentia in the yz pane; (c) he gravitationa potentia in the zx pane; (d) the 3D structure of the gravitationa potentia. o simuate the dynamica behaviors of the binary asteroida systems with the binary octahedrons, the initia positions and moments need to set. he initia reative position and moment of the secondary are [-1.107, 4.186,.5]km and [-0.085, -0.08, 0.0]km s -1, respectivey. he initia anguar momentums of the secondary and the primary are [ , , ] kg km s -1 and [ , , 0.141]kg km s -1, respectivey; the attitude ange of the secondary and the primary are [0.0, 0.0, 0.0]deg and [10.0, 0.0, 0.0]deg, respectivey. Here a the initia reative position, initia reative veocity, initia moment, and initia anguar momentum are expressed in the body-fixed frame of the primary. hus, the initia position and the initia moment of the primary are zero. he 7/8-order Runge-Kutta method has been used to integrate the dynamica equation. Athough the motion of the secondary reative to the primary is not the Keperian motion, we sti use the orbita eements to anayze the motion, and discuss the strong perturbation of the fu two-body probems. he gravitationa force and torque can be cacuated by the expansion in series of the mutua potentia of the binary poyhedra. he gravitationa force (Fahnestock and Scheeres 006; Yu et a. 017a, b) acting on body A and B can be expressed reative to the inertia reference frame as and F U U U (11) A 0 1 G a ba b aa bb A A A 10

11 F U U U. (1) B 0 1 G a ba b aa bb B B B Here A and B represent the bodies and θ represent a tensor index; a and b are densities of the simpices a and b, respectivey; a and b are Jacobian determinants of the simpices a and b, respectivey; U0 i Q Qw R, i U1 3 R, U Q r 3Q w w. A and R R ij i j ij ij 3 5 B represent the position vectors from the origin of the inertia space to the mass centre of the bodies A and B, respectivey. he torque acting on the each body can be expressed in its body-fixed frame by M E E E, (13) A P P P and M E E E. (14) B S S S Here the matrix E E E E, and E, E, E are coumn vectors of the matrix, and the eement of matrix E is cacuated by U1 U U 3 E G a ba b, (15) aa bb where P S, is the eement of matrix, P is the transformation matrix from the body-fixed frame of A to the inertia space, and S is the transformation matrix from the body-fixed frame of B to the inertia space. he coumn vectors of the matrices P respectivey. and S are defined by P P P P and S, S S S Let v P a, a, a, S b, b, b r r r r r r. (16) 11

12 Here each of the variabe in the form of a, b i r is a coumn vector, and is the vertex of the i-th face of the simpex a or b. hen in Eq. (11), using the Einstein convention of summation, we have U 0 3 A R i i U1 3QiR w i 5 3 R Q v A R U A U A 3 QR 3Q r R 15Q R w w 3Q w v R R R ij i j i j ij ij ij Q r R w 3Q r v 35Q w w w 15Q w w v R R R R ij k ij k i j k i j j ijk ijk ijk ijk (17) And in Eq. (15), we have j i U Q 1 ir D j 3 R i j i p i U Q 3 ijv pdp Qijw R Dp 3 5 R R i j p k U 3Q 15 3 ijk Q i j k ij p k ijkw w R Dp 5 v pdpw r R Dp 7 R R U 3Q 15Q r w R w w v D 35Q w w w R D 4 R R R ij k p i j k i j k p ijk ij k ijk p p ijk p r v 5 pdp (18) he attitude ange means the attitude of the body-fixed frame of the octahedrons reative to the inertia frame, and is the 3--1 rotationa order of the Euer ange. Fig. 3 shows the trajectory of the secondary reative to the primary in the inertia space. Fig. 4 gives the distance between B and A. he reative trajectory of the secondary reative to the primary is not cosed, and has a significanty variety. he minimum and maximum distances between these two bodies are.35 km and 6.51 km, respectivey. When the binary system is moving, the orbits of the two components and the attitudes of the two components reative to the inertia space a vary. If one chose the

13 body-fixed frame of the primary as the reference frame, the motion of the secondary reative to the primary can be expressed in the transating and rotating frame. In the frame of the primary, the orbit and attitude of the secondary aso have obvious changes. o anayze the cacuation precision here, we first consider the moment and energy of the binary system and their errors. Fig. 5 iustrates the moment of system reative to center of mass of A in the inertia whie Fig. 6 iustrates the variety of the potentia energy, kinetic energy, and tota energy for the binary system. Because the system is a Hamitonian system, the tota moment and tota energy are conservative. In Fig. 5, the moment of the system reative to the center of mass of A in the inertia is expressed in three different coordinate axes, and each component of the moment is a const. In Fig. 6, one can see that athough the tota energy is conservative, the potentia energy and kinetic energy are changing during the movement of the binary system. From Fig.7 and Fig.8, one can see that the reative error of the tota anguar momentum and tota energy remain sma. he dynamica system is integrated for s. he reative error of the tota anguar momentum in inertia has the order of 10-10, and the reative error of the tota energy has a smaer order of he dynamica system consists of binary octahedrons are cacuated by the two-homogeneous-poyhedra method and the 7/8-order Runge-Kutta method, thus one can concude that the methods can be used to simuate the dynamics of the binary asteroid systems. he origin of error of the system incudes two parts, one is from the 13

14 truncating of the series in two-homogeneous-poyhedra method, and the other one is from the non-sympectic structure of the 7/8-order Runge-Kutta method. Fig. 9 reveas the three-axis attitude anguar veocity of the binary asteroid system in the inertia space, which incudes the attitude anguar veocity of the primary A and the secondary B. he components of the attitude anguar veocity vary quasi-periodicay. From Fig. 9(a), one can see that the infuence of the attitude motion of the primary from the gravitationa force of the secondary is obvious. he three-axis attitude anguar veocity of the primary and the secondary vary non-periodic, which indicates that the attitude anges of the two components of the system vary non-periodic. Fig. 10 iustrates the reative momentum of B. he reative momentum of B aso varies non-periodic, and its maximum ampitude is not a constant. Fig. 11 presents the anguar momentum of A reative to masscenter of A in the inertia system and Fig. 1 presents the anguar momentum of B reative to center of mass of B in the inertia system. Athough the tota anguar momentum of the system is conservative, the anguar momentum of A and B, as we as the components of the two bodies are changing. 14

15 (a) (b) (c) 15

16 (d) Figure 3. he trajectory of the secondary reative to the primary in the inertia space viewed from different direction. (a) the 3D view; (b) the projection viewed in the xy pane; (c) the projection viewed in the yz pane; (d) the projection viewed in the zx pane. Figure 4. he distance between B and A, the unit of the distance is km. Figure 5. he moment of the system reative to the center of mass of A in the inertia, the unit of the moment is 10 1 kg m s

17 Figure 6. he variety of the potentia energy, kinetic energy, and tota energy for the binary system, the unit of the energy is kg m s -. Figure 7. he reative error of the tota anguar momentum in the inertia space. 17

18 Figure 8. he reative error of the tota energy for the binary system. (a) 18

19 (b) Figure 9. he three-axis attitude anguar veocity of the binary system in the inertia space, the unit of the anguar veocity is rad s -1. (a) he three-axes attitude anguar veocity of the primary A; (b) he three-axes attitude anguar veocity of the secondary B. Figure 10. he reative momentum of B, the unit of the moment is 10 1 kg m s -1. Figure 11. he anguar momentum of A reative to the center of mass of A in the inertia system, the unit of the anguar momentum is kg m rad s

20 Figure 1. he anguar momentum of B reative to the center of mass of B in the inertia system, the unit of the anguar momentum is kg m rad s Concusions he dynamica behaviors of binary asteroid systems have been modeed by the binary octahedrons. he orbita parameters and attitude parameters caused by the mutua potentia of the irreguar shapes are cacuated. We presented the binary poyhedron formuas of the mutua potentia from the fourth order to the sixth order. he mutua potentia of the system is cacuated by the two-homogeneous-poyhedra method to the fourth order with the 7/8-order Runge-Kutta method. he resut has a high precision. With the method used here, the reative error of tota anguar momentum in inertia has the order of when the orbita time is s. he reative error of tota energy for the binary system has the order of when the orbita time is s. he reative trajectory of the secondary reative to the primary is not cosed. he moment of the system reative to center of mass of the primary in the inertia is 0

21 conservative. he tota energy of the system is aso conservative. A of the orbits and attitude of the two bodies reative to the inertia space vary, incuding the distance, the three-axis attitude anguar veocity, anguar momentum, reative momentum, potentia energy, and kinetic energy. he resuts show that the infuence of the attitude motion of the primary affected by the gravitationa force of the secondary is obvious. For both of the primary and the secondary, the three-axis attitude anguar veocities vary non-periodic. he reative momentum of the secondary varies non-periodic. he resut is hepfu for understanding the compicated dynamica behaviors of the binary asteroid systems caused by the irreguar shapes. Acknowedgements his project is funded by China Postdoctora Science Foundation- Genera Program (No. 017M610875), China Postdoctora Science Foundation-Specia Funding (No ), and the Nationa Natura Science Foundation of China (No ). References Baikie, A., et a. 014, Phys Rev E, 89(4), D Urso, M. G. 014, J. Geodesy 88(1), Eshaboury, S. M., et a. 016, Astrophys. Space Sci. 361(9), 315. Fahnestock, E. G., Scheeres, D. J. 006, Ceest. Mech. Dynam. Astron. 96(3-4), Guirao, J. L. G., Rubio, R. G., Vera, J. A. 011, J. Comput. App. Math. 35, Jain, R., Sinha, D. 014a, Astrophys. Space Sci. 351(1), Jain, R., Sinha, D. 014b, Astrophys. Space Sci. 353(1), Lindner, J. F., et a. 010, Phys. Rev. E, 81(), Liu, X., Baoyin, H., Ma, X. 011, Astrophys. Space Sci. 333(), Liu, X., Baoyin, H., Ma, X. 013, Sci. China Phys. Mech. Astron. 56(4),

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