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

Abstract

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.

Periodic orbits in the gravity field of a fixed homogeneous cube

“…For fixed massive bodies, periodic orbits and other dynamics characteristics around a straight segment (Riaguas et al 1999;Arribas and Elipe 2001), a solid ring (Broucke and Elipe 2005;Fukushima 2010), a homogeneous annulus disk (Alberti and Vidal 2007;Fukushima 2010), and simple planar plates including square and triangular plates (Blesa 2006) were investigated. The dynamics around a rotating segment were extensively studied (Riaguas et al 2001;Elipe and Riaguas 2003;Gutiérrez-Romero et al 2004;Palacián et al 2006).…”

The Gravity Field of a Cube

“…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).…”

Computation of families of periodic orbits and bifurcations around a massive annulus

“…One of the pioneers, Maxwell (1859) proposed a model for the motion of the particles surrounding Saturn considering a polygonal configuration for the planar (n + 1) body problem, in such a way that n bodies of equal mass are located at the vertices of a regular n-gon centered at the remaining body. This model attracted the interest of researchers (Scheeres 1992;Breiter et al 1996;Kalvouridis 1999;Arribas and Elipe 2005;Arribas et al 2007; Alberti and Vidal 2007;Elipe et al 2007) in the last years because of the possibility of considering this type of configuration for different dynamical systems. The dynamical models proposed consider a gravitation potential created by a ring, where the ring is described as a finite number of particles placed in a ring configuration, or a solid circular wire.…”

“…We focus our research on the models proposed by Scheeres (1992), Kalvouridis (1999), Arribas and Elipe (2005), Alberti and Vidal (2007), with the aim of extend the dynamical systems proposed in their works to a planetary ring. Work is currently underway to analyze the dynamics when a central elliptical body is introduced into the model.…”

“…They provide expressions that are mathematically correct, but not appropriated for numeric evaluation and do not cover the whole space. Works by Alberti and Vidal (2007) or Fukushima (2010) deal with the dynamics of the problem but makes use of integral form for the potential and no numerical computation of orbits are made. This paper investigates the dynamics of a particle orbiting the annulus through the search of periodic orbits.…”