Zero velocity surfaces for the general planar three-body problem

Bozis, George

doi:10.1007/bf00640013

Cited by 52 publications

(13 citation statements)

References 4 publications

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“…The topology of these surfaces and hence the regions of possible motion are controlled in this case by the Jacobi constant. For full three-body systems, where all the masses have finite values, the theory of Hill stability was developed using different approaches by Gobulev (1967Gobulev ( , 1968, Marchal and Saari (1975), Zare (1976Zare ( , 1977, and Bozis (1976). For this full three-body case the important parameter is c 2 E, where E is the energy and c the angular momentum of the system of three masses.…”

Section: Full Three Body Hill Stabilitymentioning

confidence: 99%

Limits on the Orbits of Possible Eccentric and Inclined Moons of Extrasolar Planets Orbiting Single Stars

Donnison

2014

Earth Moon Planets

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Limits are placed on the range of orbits and masses of possible moons orbiting extrasolar planets which orbit single central stars. The Roche limiting radius determines how close the moon can approach the planet before tidal disruption occurs; while the Hill stability of the star-planet-moon system determines stable orbits of the moon around the planet. Here the full three-body Hill stability is derived for a system with the binary composed of the planet and moon moving on an inclined, elliptical orbit relative the central star. The approximation derived here in Eq. (17) assumes the binary mass is very small compared with the mass of the star and has not previously been applied to this problem and gives the criterion against disruption and component exchange in a closed form. This criterion was applied to transiting extrasolar planetary systems discovered since the last estimation of the critical separations (Donnison in Mon Not R Astron Soc 406:1918, 2010a) for a variety of planet/ moon ratios including binary planets, with the moon moving on a circular orbit. The effects of eccentricity and inclination of the binary on the stability of the orbit of a moon is discussed and applied to the transiting extrasolar planets, assuming the same planet/moon ratios but with the moon moving with a variety of eccentricities and inclinations. For the non-zero values of the eccentricity of the moon, the critical separation distance decreased as the eccentricity increased in value. Similarly the critical separation decreased as the inclination increased. In both cases the changes though very small were significant.

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Section: Full Three Body Hill Stabilitymentioning

confidence: 99%

Limits on the Orbits of Possible Eccentric and Inclined Moons of Extrasolar Planets Orbiting Single Stars

Donnison

2014

Earth Moon Planets

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“…The theory of Hill stability has been developed for full threebody systems by Gobulev (1967Gobulev ( , 1968, and Marchal and Saari (1975) using Sundman's inequality; Zare (1976Zare ( , 1977 using Hamiltonian dynamics and reduction; Bozis (1976) using algebraic manipulation of integrals. These methods extended the concept of zero velocity surfaces, introduced by Hill (1878) and used extensively in the restricted three-body problem, to the problem of general three-body motions.…”

Section: Hill Stabilitymentioning

confidence: 99%

The Hill stability of inclined bound triple star and planetary systems

Donnison

2009

Planetary and Space Science

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“…For some values of the energy and of the angular momentum integrals, the seventies, with the integrals in the 3-body the level manifolds of the 10 classical integrals are topologically disconnected as subsets of the phase space; moreover, the projections of these disconnected components on the configuration space are also disconnected, hence forbidden configurations do form a boundary that separates regions of'trapped' motions (Golubev, 1968;Smale, 1970a;Marchal, 1971;Smale, 1970b;Easton, Marchal and Saari, 1975;Zare, 1976Zare, , 1977Bozis, Although the relevance of this result for the systems was perceived (see e.g. 1977;Roy, 1979;Walker et al, 1980;Walker and Roy, 1981) there have been some problems in fully exploiting this discovery in assessing the 'stability' of such systems.…”

Section: Introductionmentioning

confidence: 96%

On topological stability in the general three-body problem

Milani

Nobili

1983

Celestial Mechanics

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Abstract. The confining curves in the general three-body problem are studied; the role of the integral c2h (angular momentum squared times energy) as bifurcation parameter is established in a very simple way by using symmetries and changes of scale. It is well known (Birkhoff, 1927) that the bifurcations of the level manifolds of the classical integrals occur at the Euler-Lagrange relative equilibrium configurations. For small values of the mass ratio e = m3/m 2 both the positions of the collinear equilibrium points and the c2h integral are expanded in power series of e. In this way the relationship is found between the confining curves resulting from the c2h integral in the general problem and the zero velocity curves given by the Jacobi integral in the corresponding restricted problem. For small values of e the singular confining curves in the general and in the restricted problem are very similar, but they do not correspond to each other: the offset of the two bifurcation values is, in the usual system of units of the restricted problem, about one half of the eccentricity squared of the orbits of the two larger bodies. This allows the definition of an approximate stability criterion, that applies to the systems with small e, and quantifies the qualitatively well known destabilizing effect of the eccentricity of the binary on the third body. Because of this destabilizing effect the third body cannot be bounded by any topological criterion based on the classical integrals unless its mass is larger than a minimum value. As an example, the three-body systems formed by the Sun, Jupiter and one of the small planets Mercury, Mars, Pluto or anyone of the asteroids are found to be 'unstable', i.e. there is no way of proving, with the classical integrals, that they cannot cross the orbit of Jupiter. This can be reliably checked with the approximate stability criterion, that gives for the most important threebody subsystems of the Solar System essentially the same information on 'stability' as the full computation of the c2h integral and of the bifurcation values.

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Zero velocity surfaces for the general planar three-body problem

Cited by 52 publications

References 4 publications

Limits on the Orbits of Possible Eccentric and Inclined Moons of Extrasolar Planets Orbiting Single Stars

Limits on the Orbits of Possible Eccentric and Inclined Moons of Extrasolar Planets Orbiting Single Stars

The Hill stability of inclined bound triple star and planetary systems

On topological stability in the general three-body problem

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