On the Fixed Homogeneous Circle Problem

Azevedo, Célia Lopes; Cabral, Hildeberto E.; Ontaneda, Pedro

doi:10.1515/ans-2007-0103

Cited by 13 publications

(22 citation statements)

References 4 publications

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“…Remark 3 In Azevêdo et al (2005), it is proved that the potential associated to the fixed homogeneous circle problem is exactly the potential of the circular Sitnikov problem (see Belbruno et al 1994). But, in our case the potential is different of the circular Sitnikov problem, however the flows are equivalent, although these problems are completely different.…”

Section: Corollary 2 the Flow Associated To The Homogeneous Annulus Dmentioning

confidence: 65%

“…We will procedure in the same way as in the deduction of the potential of a homogeneous circle (see Alberti 2003, Azevêdo et al 2005, Kellog 1929, MacMillan 1958. Let P = (r, θ , z) be a point in cylindrical coordinates not in the massive annulus A.…”

Section: The Potential Of a Homogeneous Circular Ringmentioning

confidence: 99%

“…2π 0 dθ √ (r+a) 2 cos 2 θ +(r−a) 2 sin 2 θ +z 2 which corresponds to the potential associated to the fixed homogeneous circle problem, which was considered in Alberti (2003), Azevêdo et al (2005), Broucke and Elipe (2005).…”

Section: Properties Of the Potentialmentioning

confidence: 99%

“…Another motivation is related with the idea of extending the cases considered until now in the literature, for example, when the massive body is a straight segment (see Elipe and Riaguas 2003, Elipe et al 1999, Elipe and Lara 2003 and in the case of a solid circular ring (see Alberti 2003, Azevêdo et al 2005, Broucke and Elipe 2005, Lass and Blitzer 1983. We emphasize that most of the research has used discrete particles, for example in the circular case, while we are using a continuous massive ring.…”

mentioning

confidence: 99%

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Dynamics of a particle in a gravitational field of a homogeneous annulus disk

Alberti

Vidal

2007

Celestial Mech Dyn Astr

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We formulate the problem of an infinitesimal particle moving in the space under the influence of the gravitational force induced by a homogeneous annulus disk fixed on a plane. We compute using different coordinates and in terms of an elliptic integral the potential associated to this problem. Also we study the symmetries of the associated potential. After that, we look at the dynamics in some particular cases, namely: in the line perpendicular to the plane that contains the annulus disk and passes through the center of the ring and on the plane that contains the massive annulus.

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Section: Corollary 2 the Flow Associated To The Homogeneous Annulus Dmentioning

confidence: 65%

Section: The Potential Of a Homogeneous Circular Ringmentioning

confidence: 99%

Section: Properties Of the Potentialmentioning

confidence: 99%

mentioning

confidence: 99%

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Dynamics of a particle in a gravitational field of a homogeneous annulus disk

Alberti

Vidal

2007

Celestial Mech Dyn Astr

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“…The gravitational field of a cube was apparently first calculated in (MacMillan, 1958) and later extended to cuboid masses (Nagy, 1966;Mufti, 2006a), including varying density (Garcia-Abdeslem, 2005;Hansen, 1999) and also for general polyhedra (Paul, 1974;Coggon, 1976), as well as for a range of simpler planar objects such as straight line segments as well as disks and annular shapes (Riaguas et al, 1999;2001;Azevedo and Ontaneda, 2007;Azevedo et al, 2005;Fukushima, 2010;Alberti and Vidal, 2007;Blesa, 2005;Gutierrez-Romero et al, 2004;Palacian et al, 2006;Iorio, 2007;2012). These results were found to have application in the calculation of the gravitational anomalies on the Earth (Mufti, 2006b) and the slowdown of the Earth's rotation rate due to tidal drag (Celnikier, 1990).…”

Section: Introductionmentioning

confidence: 99%

The Gravity Field of a Cube

Chappell

Iqbal

et al. 2012

Physics International

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We calculate the Newtonian gravitational potential and field of a cubic, homogeneous asteroid and we apply it to the orbit of possible satellites. Large astronomical objects such as stars or planets, naturally tend to form spherical shapes due to the dominance of the gravitational forces, but as a thought experiment, we consider the properties of a planet in the form of a perfect cube. We investigate the formation of stable orbits around such cubic objects, for the case of a static as well as a rotating cube employing the method of Poincare sections. The calculation of the gravitational field around non-spherical objects has a significant role in space missions to investigate asteroid belt objects that require calculating orbits around a large non-spherical mass. The calculation of such non-spherical fields also has relevance in identifying deposits or beds of ores inside the Earth, by measuring gravitational anomalies.

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Equilibria, periodic orbits around equilibria, and heteroclinic connections in the gravity field of a rotating homogeneous cube

Liu

Baoyin

2011

Astrophys Space Sci

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This paper investigates the dynamics of a particle orbiting around a rotating homogeneous cube, and shows fruitful results that have implications for examining the dynamics of orbits around non-spherical celestial bodies. This study can be considered as an extension of previous research work on the dynamics of orbits around simple shaped bodies, including a straight segment, a circular ring, an annulus disk, and simple planar plates with backgrounds in celestial mechanics. In the synodic reference frame, the model of a rotating cube is established, the equilibria are calculated, and their linear stabilities are determined. Periodic orbits around the equilibria are computed using the traditional differential correction method, and their stabilities are determined by the eigenvalues of the monodromy matrix. The existence of homoclinic and heteroclinic orbits connecting periodic orbits around the equilibria is examined and proved numerically in order to understand the global orbit structure of the system. This study contributes to the investigation of irregular shaped celestial bodies that can be divided into a set of cubes.

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On the Fixed Homogeneous Circle Problem

Cited by 13 publications

References 4 publications

Dynamics of a particle in a gravitational field of a homogeneous annulus disk

Dynamics of a particle in a gravitational field of a homogeneous annulus disk

The Gravity Field of a Cube

Equilibria, periodic orbits around equilibria, and heteroclinic connections in the gravity field of a rotating homogeneous cube

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