Stability and bifurcations of Sitnikov motions

Perdios, E. A.; Markellos, V. V.

doi:10.1007/bf01232956

Cited by 43 publications

(24 citation statements)

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“…The stability of the rectilinear motion will be determined by the roots of the characteristic equation of (25) (Perdios and Markellos 1988;Belbruno et al 1994). Let us consider the matrix B = X −1 (t)X(t + T ), where X(t) is a fundamental solution of system (24) and T is the period of a specific solution of (5).…”

Section: Bifurcations Into 3d Periodic Orbitsmentioning

confidence: 99%

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On Sitnikov-like motions generating new kinds of 3D periodic orbits in the R3BP with prolate primaries

Douskos

Kalantonis

Markellos

et al. 2011

Astrophys Space Sci

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The existence of new equilibrium points is established in the restricted three-body problem with equal prolate primaries. These are located on the Z-axis above and below the inner Eulerian equilibrium point L 1 and give rise to a new type of straight-line periodic oscillations, different from the well known Sitnikov motions. Using the stability properties of these oscillations, bifurcation points are found at which new types of families of 3D periodic orbits branch out of the Z-axis consisting of orbits located entirely above or below the orbital plane of the primaries. Several of the bifurcating families are continued numerically and typical member orbits are illustrated.

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Section: Bifurcations Into 3d Periodic Orbitsmentioning

confidence: 99%

“…motions (Sitnikov 1960;Perdios and Markellos 1988;Dvorak 1993;Belbruno et al 1994;Lara and Buendía 2001). They originate at the inner collinear equilibrium point L 1 , and are of special interest as generators of families of threedimensional periodic orbits (Perdios 2007;Perdios et al 2008).…”

mentioning

confidence: 98%

On Sitnikov-like motions generating new kinds of 3D periodic orbits in the R3BP with prolate primaries

Douskos

Kalantonis

Markellos

et al. 2011

Astrophys Space Sci

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“…For such a system there exist several studies of periodic orbits in the circular resticted three body problem. The first investigations for the three dimensional case date back to Goudas (1961Goudas ( , 1963, followed by the work of Markellos (1977Markellos ( , 1978, Michalodimitrakis (1979), Perdios & Markellos (1988) and Zagouras (1977) and more recently Broucke (2001). In the present study we determine numerically (a) by means of orbital computations and (b) with the aid of the Fast Lyapunov Indicator (FLI) (Froesché et al 1997) the variation of the stable zone by increasing the eccentricity of the binary from 0 to 0.5.…”

Section: Introductionmentioning

confidence: 99%

Stability limits in double stars

Pilat-Lohinger

Funk

Dvořák

2003

A&A

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Abstract.We treat the problem of the stability of planetary orbits in double stars with the aid of numerical studies in the model of the elliptic restricted three-body problem. Many investigations exist for the limits of the stability of the S-type planetary motion (i.e. the motion around one component of the binary). In this paper we present a numerical investigation of the so-called P-type orbits which are defined as orbits of the planet around both primary masses. The growing interest of such stability studies is due to the discovery of the numerous extra-solar planets, where five of them move in double star systems. Two different methods are used; in a first study the stability was investigated systematically for different initial conditions of the planet, which is regarded to be massless. We vary the initial starting distance from the barycenter (always with velocities corresponding to circular orbits at the beginning), the angle to the connecting line of the binary and the starting position of the primaries (peri-astron and apo-astron); the primaries' masses are considered to be equal. The numerical stability criterion is set to the condition that during the whole integration time (50 000 periods of the primaries) the planet should not suffer from a close encounter to one of the primaries. The binary's eccentricity (e) is increased form 0 to 0.5 with ∆e = 0.05, and the initial distance (A) to the barycenter is taken of the range from 1.8 to 3.5 AU (for e ≤ 0.15) or to 4.5 AU (for e ≥ 0.2) with ∆A = 0.05 AU. For the first time, we study inclined P-type orbits (0 • ≤ i ≤ 50 • with a step of ∆i = 2.5 • ) for which we analyse the border line of the stable zone. Additionally we also record the escape times for all orbits. In the second part of our study we apply the Fast Lyapunov Indicator (FLI) to distinguish between the differnt types of motion, taking the same initial conditions, only the grid for the initial distance to the barycenter was finer (∆A = 0.01 AU). Since the FLI is known to be a fast chaos indicator we reduced the integration time to 10 4 periods of the primaries. A comparison of the results of the two studies show that they are in good agreement. It turned out that the inclination does not influence the stability significantly. The stability limit itself varies between A = 2.1 AU (for e = 0 and i = 42, 5 • ) and A = 3.85 AU (for e = 0.5 and i = 50 • ).

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“…For such a system several studies of periodic orbits exist in the circular restricted three-body problem. The first investigations for the three-dimensional case date back to Goudas [43] (1961, 1963), followed by the works of Markellos [44, 45] (1977, 1978), Michalodimitrakis [46] (1979), Perdios and Markellos [47] (1988) and Zagouras and Markellos [48] (1977) and more recently Broucke [49] (2001).…”

Section: P-type Motionmentioning

confidence: 99%

Planets in Double Stars

Pilat-Lohinger

Dvořák

2007

Extrasolar Planets

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Stability studies of planetary motion in binary systems are very important, since we expect that more and more planets in such stellar systems will be discovered in the future -due to the fact that most of the stars in the solar neighborhood form double or multiple star systems. Currently (January 2007) we know only few planetary systems, which have a "close" stellar companion, out of a -meanwhile long -list of extrasolar planetary systems. The total number of binaries with planets is about 30. In most of these double-star systems the distance of the two stellar components is between 100 and more than 12000 AU. Therefore, it is obvious that the detected planets were found to move around one stellar component.This chapter gives an overview about planetary orbits in binary systems, where the different types of motion are discussed. First, we present some general stability studies that define the stable region for planetary motion in such systems, where different mass ratios of the stellar components were considered. In this context we discuss the influence of both eccentricities -that of the binary and that of the planet. We show that circular planetary motion cannot always give the necessary information, whether a planet is in a stable zone or not, especially when high eccentric planetary orbits are close to the border of stable motion. Some applications to real binary systems will underline the importance of such general investigations. Moreover, we show some results about the influence of the secondary companion on the region between the host star and the giant planet. IntroductionThe famous discovery of the first planet orbiting the sun-like star 51 Peg more than 10 years ago [1] was certainly a milestone in astronomy. Since that time, we have known that the planets of our solar system are not the only ones.

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Stability and bifurcations of Sitnikov motions

Cited by 43 publications

References 5 publications

On Sitnikov-like motions generating new kinds of 3D periodic orbits in the R3BP with prolate primaries

On Sitnikov-like motions generating new kinds of 3D periodic orbits in the R3BP with prolate primaries

Stability limits in double stars

Planets in Double Stars

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