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Orbital Dynamics, Chaos in

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

By the term orbital dynamics we mean the study of the motion of one or more bodies. Motion is one of the first things that a human being noticed, since the very early stages of human life. Apart from the motion of himself, walking around, he also realized that everything around him is not still, but changes position, being it a wild animal, a dry leaf drifting in the wind, the motion of clouds in the sky, or the change of the position of the celestial bodies, most notably of the Sun and the Moon.

Evidently, motion is one of the most important aspects in everyday life. By the term motion we mean the change of the position of one or more bodies in space, with respect to the other bodies in that region. If only one body existed in the universe, motion could not be defined. This makes necessary the introduction of an important notion in physics, the frame of reference. The surface of the Earth, for example, defines a frame of...

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Abbreviations

Reference frame :

A reference frame can be determined by a set of solid bodies, through which we can define a three dimensional geometric figure, for example a triedron (three non planar axes starting from a point). The surface of the Earth can be used to define a reference system. A moving car can be also used to define a reference frame, different from the first one.

Inertial frame :

An inertial frame is a special class of reference frames, in which the basic laws of motion (Newton's laws) are valid. According to Galileos' Principle of Relativity , any frame of reference moving uniformly (with constant velocity without rotation) with respect to an inertial frame is also an inertial frame. A frame of reference which is rotating with respect to an inertial frame is not inertial. The criterion for a frame to be inertial is Newton's first law to be valid. This means that in an inertial frame a free body is either at rest or moves in a straight line with constant velocity. The best approximation in nature of an inertial frame is that frame which is defined by a triedron whose origin is at the center of mass of our Solar System and its three axes are in three fixed directions in space, defined by three distant stars.

Degrees of freedom:

The number of independent variables that are needed to determine the position of a dynamical system is called the number of degrees of freedom . For example, a particle moving freely in space has three degrees of freedom, since its position is determined by its three Cartesian coordinates (\( { x_1,x_2,x_3 } \)), which are independent.

Phase space:

Consider a space whose coordinates determine exactly the state of the system. This space is called the state space or the phase space of the system. Each point of the phase space determines uniquely the initial conditions of the motion. The evolution of the system in the phase space is represented by a smooth curve, which is called the phase curve. The phase curves do not intersect, otherwise the point of intersection would correspond to two different solutions. The set of all phase curves is called the phase diagram and gives important information of the stable and unstable regions of the phase space. For gravitational systems, the phase space is the space of coordinates and velocities of all the bodies. Usually, instead of the velocities, the moments are used in the definition of the phase space. In a gravitational system with n degrees of freedom, the phase space has \( { 2n } \) dimensions. For example, a body moving in the plane under the action of a force, has two degrees of freedom (coordinates x, y) and its phase space is the four-dimensional space x, y, \( { p_x=m\dot x } \), \( { p_y=m\dot y } \).

Orbit :

An orbit of a body, or a set of bodies, considered as point masses, is the path that the bodies describe in a reference frame. The orbit of the same body or set of bodies is different in different frames of reference.

Periodic orbit :

A periodic orbit is the orbit of one or more bodies that repeats itself after a certain time, which is called the period of the periodic orbit. The periodicity property is closely related to the frame of reference to which the motion is referred to. For example, an orbit may be periodic in a rotating frame, but not in the inertial frame. In this latter case, for two or more bodies, it is the relative configuration that is repeated in the inertial frame.

Poincaré map :

The Poincaré map is a method by which we transform the continuous flow of a dynamical system in its n-dimensional phase space, into a discrete map in a reduced phase space. The map is obtained by taking the intersections of the continuous flow in the original phase space with a  surface of section , defined properly. This surface of section is (\( { n-1 } \))-dimensional, in general, or (\( { n-2 } \))-dimensional if an integral of motion exists, which is the energy integral in gravitational systems. These will be explained in detail in Sects. “The Poincaré Map,” “Poincaré Map in Hamiltonian Systems.” A periodic orbit appears as a fixed point on the Poincaré surface of section. The Poincaré map is very useful in the study of ordered and chaotic motion in a dynamical system, especially in systems with few degrees of freedom.

Stability :

The notion of stability refers to the behavior of the orbits in the vicinity of a periodic orbit. If a slight change in the initial conditions results to a new orbit, called the perturbed orbit, which deviates much from the periodic orbit, then the periodic orbit is called unstable. In the gravitational systems that we will study, this deviation is exponential. If on the other hand, the perturbed orbit stays close to the periodic orbit, the periodic orbit is called stable. But there are different aspects of stability. For example, if the perturbed orbit, considered as a geometrical figure, is close to the periodic orbit, then the periodic orbit is called orbitally stable. However, in this latter case it may happen that two bodies, one on the periodic orbit and one on the perturbed orbit, which start very close to each other, may deviate much as each one moves on its own orbit, although the geometric figures of the two orbits are close to each other. In this aspect, the orbit is considered as unstable. A Keplerian elliptic orbit, in the two-body problem, belongs to this latter category. A different type of stability is the asymptotic stability . In this case any perturbed orbit, not only stays in the vicinity of the periodic orbit, but tends asymptotically to the periodic orbit. In gravitational systems asymptotic stability does not appear, unless there exists a dissipation to the system.

Ordered and chaotic motion:

The notion of chaoticity is related to the deviation of a perturbed orbit from a given orbit. It may happen that the perturbed orbit does not deviate much as time goes on. In this case we say that we are in an ordered region . The prediction of the evolution of the system in this case is possible. In some cases however, the perturbed orbit deviates exponentially from the original orbit. Prediction is not possible for a long time. In this latter case we are in a  chaotic region . In general, both ordered and chaotic regions exist in the same dynamical system.

Bibliography

Primary Literature

  1. Arnold VA, Avez A (1968) Ergodic Problems of Classical Mechanics. WA Benjamin, New York

    Google Scholar 

  2. Beaugé C, Michtchenko T (2003) Modelling the high-eccentricity planetary three-body problem. Application to the GJ876 planetary system. MNRAS 341:760

    Google Scholar 

  3. Beaugé C, Ferraz-Mello S, Michtchenko T (2003) Extrasolar Planets in Mean-Motion Resonance, Apses Alignment and Asymmetric Stationary Solutions. Ap J 593:1124

    Google Scholar 

  4. Beaugé C, Callegari N, Ferraz-Mello S, Michtchenko T (2005) Resonances and stability of extra-solar planetary systems. In: Knezevic Z, Milani A (eds) Dynamics of Populations of Planetary Systems. Cambridge Univ Press, p 3

    Google Scholar 

  5. Beaugé C, Ferraz-Mello S, Michtchenko TA (2006) Planetary Migration and Extrasolar Planets in the \( { 2/1 } \) Mean-motion Resonance. MNRAS 365:1160–1170

    Google Scholar 

  6. Celletti A, Kotoulas T, Voyatzis G, Hadjidemetriou JD (2007) The dynamical stability of a Kuiper belt-like region. MNRAS 378:1153–1164

    ADS  Google Scholar 

  7. Froeschlé C (1991) Modelling: An aim and a tool for the study of the chaotic behaviour of asteroidal and cometary orbits. In: Roy AE (ed) Predictability, Stability and Chaos in N-Body Dynamical Systems. Plenum Press, pp 125–155

    Google Scholar 

  8. Ferraz-Mello S, Beaugé C, Michtchenko T (2003) Evolution of migrating planet pairs in resonance. Cel Mech Dyn Astr 87:99–112

    Google Scholar 

  9. Gomes R, Levison HF, Tsiganis K, Morbidelli A (2005) Origin of the Cataclysmic Late Heavy Bombardment period of the terrestial planets. Nature 435:466

    ADS  Google Scholar 

  10. Gozdziewski K, Bois E, Maciejewski A (2002) Global dynamics of the Gliese 876 planetary system. MNRAS 332:839

    ADS  Google Scholar 

  11. Hadjidemetriou JD (1975) The continuation of periodic orbits from the restricted to the general three-body problem. Cel Mech 12:155–174

    MathSciNet  ADS  MATH  Google Scholar 

  12. Hadjidemetriou JD (1976) Families of Periodic Planetary Type Orbits in the Three-Body Problem and their Stability. Astrophys Sp Sci 40:201–224

    MathSciNet  ADS  MATH  Google Scholar 

  13. Hadjidemetriou JD (1982) On the Relation between Resonance and Instability in Planetary Systems. Cel Mech Dyn Astr 27:305–322

    Google Scholar 

  14. Hadjidemetriou JD (1992) The Elliptic Restricted Problem at the 3:1 Resonance. Celest Mech 53:151–183

    MathSciNet  ADS  MATH  Google Scholar 

  15. Hadjidemetriou JD (1993) Asteroid Motion near the 3:1 Resonance. Cel Mec Dyn Astr 56:563–599

    MathSciNet  ADS  MATH  Google Scholar 

  16. Hadjidemetriou JD (1998) Symplectic Maps and their use in Celestial Mechanics. In: Benest D, Froeschlé C (eds) Analysis and Modeling of Discrete Dynamical Systems, ch 9. Gordon and Breach Publ, pp 249–282

    Google Scholar 

  17. Hadjidemetriou JD (1999) A symplectic mapping model as a tool to understand the dynamics of the \( { 2/1 } \) resonant asteroid motion. Cel Mec Dyn Astr 73:65–76

    MathSciNet  ADS  MATH  Google Scholar 

  18. Hadjidemetriou JD (2002) Resonant periodic motion and the stability of extrasolar planetary systems. Cel Mech Dyn Astr 83:141–154

    MathSciNet  ADS  MATH  Google Scholar 

  19. Hadjidemetriou JD (2006) Symmetric and Asymmetric Librations in Extrasolar Planetary Systems: A global view. Cel Mech Dyn Astron 95:225–244

    MathSciNet  ADS  MATH  Google Scholar 

  20. Hadjidemetriou JD (2006) Periodic orbits in gravitational systems. In: Steves BA et al (eds) Chaotic Worlds: From Order to Disorder in Gravitational N-Body Dynamical Systems. Springer, pp 43–79

    Google Scholar 

  21. Hadjidemetriou JD, Voyatzis G (2000) The \( { 2/1 } \) and \( { 3/2 } \) resonant asteroid motion: A symplectic mapping approach. Cel Mech Dyn Astron 78:137–150

    ADS  MATH  Google Scholar 

  22. Henrard J, Watanabe N, Moons M (1995) A bridge between Secondary and Secular Resonances inside the Hecuba Gap. Icarus 115:336–346

    ADS  Google Scholar 

  23. Israelian G, Santos N, Mayor M, Rebolo R (2001) Evidence for planet engulfment by the star HD82943. Nature 411:163

    ADS  Google Scholar 

  24. Jordan DW, Smith P (1988) Nonlinear Ordinary Differential Equations. Clarendon Press, Oxford

    Google Scholar 

  25. Laskar J (1988) Secular evolution of the Solar System over 10 million years. Astron Astrophys 198:341–362

    ADS  Google Scholar 

  26. Laskar J (1989) A numerical experiment on the chaotic behaviour of the Solar System. Nature 338:237–238

    ADS  Google Scholar 

  27. Laskar J (1994) Large scale chaos in the Solar System. Aston Astrophys 287:I9–12

    ADS  Google Scholar 

  28. Lee MH (2004) Diversity and Origin of 2:1 Orbital Resonance in Extrasolar Planetary Systems. Ap J 611:517

    ADS  Google Scholar 

  29. Lee MH, Peale S (2002) Dynamic and origin of the 2:1 orbital resonances of the GJ 876 planets. Ap J 567:596–609

    ADS  Google Scholar 

  30. Malhotra R (2002) A dynamical mechanism for establishing apsidal resonance. Ap J 575:L33–36

    ADS  Google Scholar 

  31. Marcy G, Butler P, Fischer D, Vogt S, Lissauer J, Rivera E (2001) Pair A of Resonant Planets Orbiting GJ 876. Ap J 556:296

    ADS  Google Scholar 

  32. Marcy GW, Butler RP, Fischer DA, Laughlin G, Vogt SS, Henry GW, Pourbaix D (2002) A planet at 5AU around 55Cnc. Ap J 581:1375–1388

    ADS  Google Scholar 

  33. Mayor M, Udry S, Naef D, Pepe F, Queloz D, Santos NC, Burnet M (2004) The CORALIE survey for southern extra-solar planets. XII Orbital solutions for 16 extra-solar planets discovered with CORALIE. A& A 415:291

    Google Scholar 

  34. Meyer KR, Hall GR (1992) Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. Springer

    Google Scholar 

  35. Michtchenko TA, Ferraz-Mello S (1996) Comparative study of the asteroidal motion in the 3:2 and 2:1 resonances with Jupiter. I planar model. Astron Astrophys 310:1021–1035

    Google Scholar 

  36. Michtchenko TA, Beaugé C, Ferraz-Mello S (2006) Stationary Orbits in Resonant Extrasolar Planetary Systems. Cel Mech Dyn Astron 94:381–397

    Google Scholar 

  37. Morbidelli A (1996) On the Kirkwood Gap at the \( { 2/1 } \) Commensurability with Jupiter: numerical results. Astron J 111:2453–2461

    ADS  Google Scholar 

  38. Morbidelli A, Giorgilli A (1990) On the Dynamics in the Asteroids belt. Part I: General Theory. Celest Mech 47:145–172

    MathSciNet  ADS  MATH  Google Scholar 

  39. Morbidelli A, Giorgilli A (1990) On the Dynamics in the Asteroids belt. Part II: Detailed Study of the Main Resnances. Celest Mech 47:173–1204

    MathSciNet  ADS  MATH  Google Scholar 

  40. Morbidelli A, Levison HF, Tsiganis K, Gomes R (2005) Chaotic capture of Jupiter's Trojan Asteroids in the early Solar System. Nature 435:462

    ADS  Google Scholar 

  41. Murray CD, Dermott SF (1999) Solar System Dynamics. Cambridge University Press

    Google Scholar 

  42. Psychoyos D, Hadjidemetriou JD (2005) Dynamics of \( { 2/1 } \) resonant extrasolar systems. Application to HD82943 and Gliese 876. Cel Mech Dyn Astr 92:135–156

    MathSciNet  ADS  MATH  Google Scholar 

  43. Rivera EJ, Lissauer JJ (2001) Dynamical models of the resonant pair of planets orbiting the star GJ 876. Ap J 558:392–402

    ADS  Google Scholar 

  44. Roy AE (1982) Orbital Motion, 2nd edn. Adam Hilder

    Google Scholar 

  45. Sandor ZS, Suli A, Erdi B, Pilat-Lohinger E, Dvorak R (2007) A stability catalogue of the habitable zones in extrasolar planetary systems. MNRAS 375:1495–1502

    Google Scholar 

  46. Szebehely V (1967) Theory of Orbits. Academic Press

    Google Scholar 

  47. Tsiganis K, Varvoglis H, Hadjidemetriou JD (2000) Stable chaos in the 12:7 mean motion resonance and its relation to the stickiness effect. Icarus 146:240–252

    ADS  Google Scholar 

  48. Tsiganis K, Varvoglis H, Hadjidemetriou JD (2002) Stable chaos in Higher-order Jovian Resonances. Icarus 155:454–474

    ADS  Google Scholar 

  49. Tsiganis K, Varvoglis H, Hadjidemetriou JD (2002) Stable chaos versus Kirkwood Gaps in the Asteroid Belt: A comparative Study of mean Motion Resonances. Icarus 159:284–299

    ADS  Google Scholar 

  50. Tsiganis K, Gomes R, Morbidelli K, Levison HF (2005) Origin of the orbital architecture of the Giant Planets of the Solar System. Nature 435:459

    ADS  Google Scholar 

  51. Voyatzis G, Hadjidemetriou HD (2005) Symmetric and asymmetric librations in planetary and satellite systems at the \( { 2/1 } \) resonance. Cel Mech Dyn Astr 93:263–294

    MathSciNet  ADS  MATH  Google Scholar 

  52. Voyatzis G, Hadjidemetriou HD (2006) Symmetric and asymmetric 3:1 resonant periodic orbits. An application to the 55Cnc extra-solar system. Cel Mech Dyn Astr 95:259–271

    MathSciNet  ADS  MATH  Google Scholar 

  53. Wisdom J (1982) The Origin of the Kirkwood gaps. Astron J 87:577–593

    MathSciNet  ADS  Google Scholar 

  54. Wisdom J (1983) Chaotic behaviour and the Origin of the Kirkwood gaps. Icarus 56:51–74

    ADS  Google Scholar 

  55. Wisdom J (1985) A Perturbative Treatment of Motion Near the \( { 3/1 } \) Commensurability. Icarus 63:272–289

    ADS  Google Scholar 

  56. Wisdom J (1987) Urey Prize lecture. Chaotic Dynamics in the Solar System. Icarus 72:241–275

    ADS  Google Scholar 

  57. Wisdom J, Peale SJ, Mignard F (1984) The chaotic rotation of Hyperion. Icarus 58:137–152

    ADS  Google Scholar 

  58. Yakubovich VA, Starzhinskii VM (1975) Linear Differential Equations with Periodic Coefficients, vol 1,2. Halsted Press

    Google Scholar 

Books and Reviews

  1. Contopoulos G (2002) Order and Chaos in Dynamical Systems. Springer

    Google Scholar 

  2. Dvorak R, Freistetter F, Kurths J (eds) (2005) Chaos and Stability in Planetary Systems. Lecture Notes in Physics. Springer

    Google Scholar 

  3. Ferraz-Mello S (2007) Canonical Perturbation Theories. Degenerate Systems and resonance. Springer

    Google Scholar 

  4. Hagihara Y (1970–1972) Celestial mechanics vol 1, 2(I), 2(II). MIT Press, Cambridge

    Google Scholar 

  5. Hagihara Y (1974–1976) Celestial mechanics, vol 3(I), 3(II), 4(I), 4(II), 5(I), 5(II). Japan Society For the Promotion of Science, Tokyo

    Google Scholar 

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Authors and Affiliations

  1. Department of Physics, University of Thessaloniki, Thessaloniki, Greece

    John D. Hadjidemetriou

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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Hadjidemetriou, J.D. (2009). Orbital Dynamics, Chaos in. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_379

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