The fractal dynamics of satellite capture in the circular restricted three-body problem

Murison, M. A.

doi:10.1086/115303

Cited by 41 publications

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“…As in Murison (1989), some fractal structures can be seen in Figure 1. Figure 3, whose gray scale was adjusted in order to exhibit self-similarity structures, is a sequence of magniÐcations of box 1 in the Ðrst Ðgure.…”

Section: The Conservative Problemmentioning

confidence: 84%

“…As usual, the units are normalized, and following Murison (1989), we take the mass of the primaries to be given by and which gives ). selection of the initial conditions, we followed Murison (1989) and set and (direct motion…”

Section: The Problemmentioning

confidence: 99%

“…In such cases we usually observe chaotic motions, fractal structures, and also power laws or scaling laws that can appear together in many dynamical systems. For example, considering conservative problems, Murison (1989), in a paper about escape/capture in the circular restricted three-body problem, and Contopoulos, Kandrup, & Kaufmann (1993), in a work about escapes in a twodimensional galactic potential, show the association between some chaotic orbits with fractal objects, via selfsimilarity arguments. For nonconservative systems, there is the conjecture of Kaplan & Yorke (see, e.g., Lichtenberg & Lieberman 1992), which establishes a relation between the fractal dimension and the Lyapunov exponents.…”

Section: Introductionmentioning

confidence: 99%

“…Following Murison (1989), all our investigations were performed in the space given by (Jacobi constant) ] (initial position) ] (capture time). For the conservative case, the regions associated with capture and with noncapture are shown in this space.…”

Section: Introductionmentioning

confidence: 99%

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Complexity of Capture Phenomena in the Conservative and the Dissipative Restricted Three-Body Problems

Cordeiro¹,

Martins

Leonel³

1999

The Astronomical Journal

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Some dynamical properties of satellite capture in the conservative and the dissipative circular and planar restricted three-body problems are studied. The investigations were performed in the space given by (Jacobi constant) ] (initial position) ] (capture time). For the conservative case, we identify the regions with capture and with noncapture and we associate them with the orbital resonances. The fractal dimensions of their contours also are computed. Moreover, it is shown that the rate of capture falls o † as a power law with time. For the dissipative forces, we study the e †ect of gas drag on the structures that are observed in the space of initial conditions. We inspect how the fractal dimensions of the transition regions (the contour of the capture regions) and the power law associated with the time of capture are related to a change in the drag coefficient.

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Section: The Conservative Problemmentioning

confidence: 84%

Section: The Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Complexity of Capture Phenomena in the Conservative and the Dissipative Restricted Three-Body Problems

Cordeiro¹,

Martins

Leonel³

1999

The Astronomical Journal

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“…2 Phobos and Deimos give every appearance of being captured asteroids of the carbonaceous chondritic type (Veverka 1977, Pang et al 1978, Pollack et al 1979, Tolson et al 1978. Given the considerations summarised in detail by Murison (1988), it seems least unlikely that the Martian satellites were formerly asteroids that were captured by the mechanism of gas drag. If so, this must have occurred early in the history of the solar system while the gas disk was substantial enough to effect a capture.…”

Section: Goldreich (1965)mentioning

confidence: 99%

Long-Term Evolution of Orbits About A Precessing Oblate Planet: 1. The Case of Uniform Precession

Efroimsky

2005

Celestial Mech Dyn Astr

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It was believed until very recently that a near-equatorial satellite would always keep up with the planet's equator (with oscillations in inclination, but without a secular drift). As explained in Efroimsky and Goldreich (2004), this misconception originated from a wrong interpretation of a (mathematically correct) result obtained in terms of non-osculating orbital elements. A similar analysis carried out in the language of osculating elements will endow the planetary equations with some extra terms caused by the planet's obliquity change. Some of these terms will be nontrivial, in that they will not be amendments to the disturbing function. Due to the extra terms, the variations of a planet's obliquity may cause a secular drift of its satellite orbit inclination. In this article we set out the analytical formalism for our study of this drift. We demonstrate that, in the case of uniform precession, the drift will be extremely slow, because the first-order terms responsible for the drift will be short-period and, thus, will have vanishing orbital averages (as anticipated 40 years ago by Peter Goldreich), while the secular terms will be of the second order only. However, it turns out that variations of the planetary precession make the first-order terms secular. For example, the planetary nutations will resonate with the satellite's orbital frequency and, thereby, may instigate a secular drift. A detailed study of this process will be offered in the subsequent publication, while here we work out the required mathematical formalism and point out the key aspects of the dynamics. 11 Physical motivation and the statement of purpose Ward (1973Ward ( , 1974 noted that the obliquity of Mars may have suffered large-angle motions at long time scales. Later, Laskar and Robutel (1993) and Touma and Wisdom (1994) demonstrated that these motions may have been chaotic. This would cause severe climate variations and have major consequences for development of life.It is a customary assumption that a near-equatorial satellite of an oblate planet would always keep up with the planet's equator (with only small oscillations of the orbit inclination) provided the obliquity changes are sufficiently slow (Goldreich 1965, Kinoshita 1993. As demonstrated in Efroimsky and Goldreich (2004), this belief stems from a calculation performed in the language of non-osculating orbital elements. A similar analysis carried out in terms of osculating elements will contain hitherto overlooked extra terms entailed by the planet's obliquity variations. These terms (emerging already in the first order over the precession-caused perturbation) will cause a secular angular drift of the satellite orbit away from the planetary equator.The existence of Phobos and Deimos, and the ability of Mars to keep them close to its equatorial plane during obliquity variations sets constraints on the obliquity variation amplitude and rate. Our eventual goal is to establish such constraints. If the satellites' secular inclination drifts are slow enough that the sate...

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Capture Statistics of Short-Period Comets: Implications for Comet D/Shoemaker–Levy 9

Kary

Dones

1996

Icarus

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The fractal dynamics of satellite capture in the circular restricted three-body problem

Cited by 41 publications

References 0 publications

Complexity of Capture Phenomena in the Conservative and the Dissipative Restricted Three-Body Problems

Complexity of Capture Phenomena in the Conservative and the Dissipative Restricted Three-Body Problems

Long-Term Evolution of Orbits About A Precessing Oblate Planet: 1. The Case of Uniform Precession

Capture Statistics of Short-Period Comets: Implications for Comet D/Shoemaker–Levy 9

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