The continuation of periodic orbits from the restricted to the general three-body problem

Hadjidemetriou, John D.

doi:10.1007/bf01230209

Cited by 103 publications

(60 citation statements)

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“…It can be proved (Hadjidemetriou 1975) that families of periodic orbits exist in the rotating frame x Oy. In particular, we are interested in periodic orbits of the planetary type, that we shall study in the next section.…”

Section: The Rotating Framementioning

confidence: 99%

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Symmetric and asymmetric librations in extrasolar planetary systems: a global view

Hadjidemetriou

2006

Celestial Mech Dyn Astr

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We present a global view of the resonant structure of the phase space of a planetary system with two planets, moving in the same plane, as obtained from the set of the families of periodic orbits. An important tool to understand the topology of the phase space is to determine the position and the stability character of the families of periodic orbits. The region of the phase space close to a stable periodic orbit corresponds to stable, quasi periodic librations. In these regions it is possible for an extrasolar planetary system to exist, or to be trapped following a migration process due to dissipative forces. The mean motion resonances are associated with periodic orbits in a rotating frame, which means that the relative configuration is repeated in space. We start the study with the family of symmetric periodic orbits with nearly circular orbits of the two planets. Along this family the ratio of the periods of the two planets varies, and passes through rational values, which correspond to resonances. At these resonant points we have bifurcations of families of resonant elliptic periodic orbits. There are three topologically different resonances: (1) the resonances (n + 1):n, (2:1, 3:2, …), (2) the resonances (2n + 1):(2n − 1), (3:1, 5:3, …) and (3) all other resonances. The topology at each one of the above three types of resonances is studied, for different values of the sum and of the ratio of the planetary masses. Both symmetric and asymmetric resonant elliptic periodic orbits exist. In general, the symmetric elliptic families bifurcate from the circular family, and the asymmetric elliptic families bifurcate from the symmetric elliptic families. The results are compared with the position of some observed extrasolar planetary systems. In some cases (e.g., Gliese 876) the observed system lies, with a very good accuracy, on the stable part of a family of resonant periodic orbits.

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Section: The Rotating Framementioning

confidence: 99%

“…It can be proved (Hadjidemetriou 1975) that families of periodic orbits in the planar general three body problem exist, in a rotating frame which will be defined in Sect. 2.…”

Section: Introductionmentioning

confidence: 99%

Symmetric and asymmetric librations in extrasolar planetary systems: a global view

Hadjidemetriou

2006

Celestial Mech Dyn Astr

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“…The planar version of this theorem is due to Hadjidemetriou [11]. There are similar theorems about non-degenerate symmetric periodic solutions -see [18].…”

Section: Theorem Let η = φ(T) ξ = ψ(T) Be a Periodic Solution With mentioning

confidence: 87%

From the restricted to the full three–body problem

Meyer

Schmidt

2000

Trans. Amer. Math. Soc.

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Abstract. The three-body problem with all the classical integrals fixed and all the symmetries removed is called the reduced three-body problem. We use the methods of symplectic scaling and reduction to show that the reduced planar or spatial three-body problem with one small mass is to the first approximation the product of the restricted three-body problem and a harmonic oscillator. This allows us to prove that many of the known results for the restricted problem have generalizations for the reduced three-body problem.For example, all the non-degenerate periodic solutions, generic bifurcations, Hamiltonian-Hopf bifurcations, bridges and natural centers known to exist in the restricted problem can be continued into the reduced three-body problem. The classic normalization calculations of Deprit and Deprit-Bartholomé show that there are two-dimensional KAM invariant tori near the Lagrange point in the restricted problem. With the above result this proves that there are three-dimensional KAM invariant tori near the Lagrange point in the reduced three-body problem.

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“…Hadjidemetriou 1975). Degrees of freedom are defined here as the set Q(t) of positions and velocities of both the planet and asteroid.…”

Section: Periodic Orbitsmentioning

confidence: 99%

Linking long-term planetaryN-body simulations with periodic orbits: application to white dwarf pollution

Antoniadou¹,

Veras²

2016

Mon. Not. R. Astron. Soc.

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Mounting discoveries of debris discs orbiting newly-formed stars and white dwarfs (WDs) showcase the importance of modeling the long-term evolution of small bodies in exosystems. WD debris discs are in particular thought to form from very long-term (0.1-5.0 Gyr) instability between planets and asteroids. However, the time-consuming nature of N -body integrators which accurately simulate motion over Gyrs necessitates a judicious choice of initial conditions. The analytical tools known as periodic orbits can circumvent the guesswork. Here, we begin a comprehensive analysis directly linking periodic orbits with N -body integration outcomes with an extensive exploration of the planar circular restricted three-body problem (CRTBP) with an outer planet and inner asteroid near or inside of the 2:1 mean motion resonance. We run nearly 1000 focused simulations for the entire age of the Universe (14 Gyr) with initial conditions mapped to the phase space locations surrounding the unstable and stable periodic orbits for that commensurability. In none of our simulations did the planar CRTBP architecture yield a long-timescale ( 0.25% of the age of the Universe) asteroid-star collision. The pericentre distance of asteroids which survived beyond this timescale (≈ 35 Myr) varied by at most about 60%. These results help affirm that collisions occur too quickly to explain WD pollution in the planar CRTBP 2:1 regime, and highlight the need for further periodic orbit studies with the eccentric and inclined TBP architectures and other significant orbital period commensurabilities.

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The continuation of periodic orbits from the restricted to the general three-body problem

Cited by 103 publications

References 5 publications

Symmetric and asymmetric librations in extrasolar planetary systems: a global view

Symmetric and asymmetric librations in extrasolar planetary systems: a global view

From the restricted to the full three–body problem

Linking long-term planetaryN-body simulations with periodic orbits: application to white dwarf pollution

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