### Abstract

More and more Trojan asteroids are discovered in our Solar system, suggesting that they are probably a common spinoff of planet formation and evolution. Till date we know thousands of Trojan followers of Jupiter, and Trojan asteroids and moons of other major planets have also been discovered. The large number of such celestial bodies can mean that Trojan companions to extrasolar planets may also be common. Trojan celestial bodies are examples for the Lagrangian triangular solutions of the three-body problem. In this paper, the stability of the Lagrangian point L_{4} is investigated in the elliptic restricted three-body problem by using Floquet's theory. Stable and unstable domains are determined in the parameter plane of the mass parameter and the eccentricity by computing the characteristic exponents. Frequencies of motion around L_{4} have been determined both in the stable and unstable domains, and fitting functions for the frequencies are derived depending on the mass parameter and the eccentricity. Resonances between the frequencies are studied in the whole parameter plane. It is shown that the 1:1 resonances are not restricted only to single curves but extend to the whole unstable domain. In the unstable domains, longer escape times of the test particle from the neighbourhood of L_{4} are related to certain resonances, but changing the parameters the same resonances may lead to faster escape.

Original language | English |
---|---|

Pages (from-to) | 1988-1998 |

Number of pages | 11 |

Journal | Monthly Notices of the Royal Astronomical Society |

Volume | 443 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2014 |

### Fingerprint

### Keywords

- Celestial mechanics
- Methods: analytical
- Methods: numerical
- Planets and satellites: dynamical evolution and stability

### ASJC Scopus subject areas

- Space and Planetary Science
- Astronomy and Astrophysics

### Cite this

_{4}in the elliptic restricted three-body problem.

*Monthly Notices of the Royal Astronomical Society*,

*443*(3), 1988-1998. https://doi.org/10.1093/mnras/stu1212

**Frequencies and resonances around L _{4} in the elliptic restricted three-body problem.** / Rajnai, R.; Nagy, I.; Érdi, B.

Research output: Contribution to journal › Article

_{4}in the elliptic restricted three-body problem',

*Monthly Notices of the Royal Astronomical Society*, vol. 443, no. 3, pp. 1988-1998. https://doi.org/10.1093/mnras/stu1212

_{4}in the elliptic restricted three-body problem. Monthly Notices of the Royal Astronomical Society. 2014;443(3):1988-1998. https://doi.org/10.1093/mnras/stu1212

}

TY - JOUR

T1 - Frequencies and resonances around L4 in the elliptic restricted three-body problem

AU - Rajnai, R.

AU - Nagy, I.

AU - Érdi, B.

PY - 2014

Y1 - 2014

N2 - More and more Trojan asteroids are discovered in our Solar system, suggesting that they are probably a common spinoff of planet formation and evolution. Till date we know thousands of Trojan followers of Jupiter, and Trojan asteroids and moons of other major planets have also been discovered. The large number of such celestial bodies can mean that Trojan companions to extrasolar planets may also be common. Trojan celestial bodies are examples for the Lagrangian triangular solutions of the three-body problem. In this paper, the stability of the Lagrangian point L4 is investigated in the elliptic restricted three-body problem by using Floquet's theory. Stable and unstable domains are determined in the parameter plane of the mass parameter and the eccentricity by computing the characteristic exponents. Frequencies of motion around L4 have been determined both in the stable and unstable domains, and fitting functions for the frequencies are derived depending on the mass parameter and the eccentricity. Resonances between the frequencies are studied in the whole parameter plane. It is shown that the 1:1 resonances are not restricted only to single curves but extend to the whole unstable domain. In the unstable domains, longer escape times of the test particle from the neighbourhood of L4 are related to certain resonances, but changing the parameters the same resonances may lead to faster escape.

AB - More and more Trojan asteroids are discovered in our Solar system, suggesting that they are probably a common spinoff of planet formation and evolution. Till date we know thousands of Trojan followers of Jupiter, and Trojan asteroids and moons of other major planets have also been discovered. The large number of such celestial bodies can mean that Trojan companions to extrasolar planets may also be common. Trojan celestial bodies are examples for the Lagrangian triangular solutions of the three-body problem. In this paper, the stability of the Lagrangian point L4 is investigated in the elliptic restricted three-body problem by using Floquet's theory. Stable and unstable domains are determined in the parameter plane of the mass parameter and the eccentricity by computing the characteristic exponents. Frequencies of motion around L4 have been determined both in the stable and unstable domains, and fitting functions for the frequencies are derived depending on the mass parameter and the eccentricity. Resonances between the frequencies are studied in the whole parameter plane. It is shown that the 1:1 resonances are not restricted only to single curves but extend to the whole unstable domain. In the unstable domains, longer escape times of the test particle from the neighbourhood of L4 are related to certain resonances, but changing the parameters the same resonances may lead to faster escape.

KW - Celestial mechanics

KW - Methods: analytical

KW - Methods: numerical

KW - Planets and satellites: dynamical evolution and stability

UR - http://www.scopus.com/inward/record.url?scp=84906061774&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84906061774&partnerID=8YFLogxK

U2 - 10.1093/mnras/stu1212

DO - 10.1093/mnras/stu1212

M3 - Article

AN - SCOPUS:84906061774

VL - 443

SP - 1988

EP - 1998

JO - Monthly Notices of the Royal Astronomical Society

JF - Monthly Notices of the Royal Astronomical Society

SN - 0035-8711

IS - 3

ER -