1:1 Ground-track resonance in a uniformly rotating 4th degree and order gravitational field

Abstract

Using a gravitational field truncated at the 4th degree and order, the 1:1 groundtrack resonance is studied. To address the main properties of this resonance, a 1-degree of freedom (1-DOF) system is firstly studied. Equilibrium points (EPs), stability and resonance width are obtained. Different from previous studies, the inclusion of non-spherical terms higher than degree and order 2 introduces new phenomena. For a further study about this resonance, a 2-DOF model which includes a main resonance term (the 1-DOF system) and a perturbing resonance term is studied. With the aid of Poincaré sections, the generation of chaos in the phase space is studied in detail by addressing the overlap process of these two resonances with arbitrary combinations of eccentricity (e) and inclination (i). Retrograde orbits, near circular orbits and near polar orbits are found to have better stability against the perturbation of the second resonance. The situations of complete chaos are estimated in the e − i plane. By applying the maximum Lyapunov Characteristic Exponent (LCE), chaos is characterized quantitatively and similar conclusions can be achieved. This study is applied to three asteroids 1996 HW1, Vesta and Betulia, but the conclusions are not restricted to them.

Review Article: Resonant Families of Periodic Orbits in the Restricted Three-body Problem*

“…One is the Hamiltonian approach. By focusing on the specific resonant term and eliminating other periodic terms (the so-called averaging process), the Hamiltonian is reduced to a 1°of freedom system with the resonance angle as the action variable (Henrard & Lemaitre 1983b;Feng et al 2016;Tan et al 2020). The other is the approach of periodic orbits.…”

Chaotic dynamics around cometary nuclei

“…Asteroids and cometary nuclei quite often have bilobed (dumb-bell) shapes; a spectacular example of such a shape is offered by the recent radar imaging of the near-Earth asteroid 2014 JO25 1 . The orbital dynamics around bodies with complex gravity fields (Chauvineau et al, 1993;Scheeres et al, 1996;Scheeres et al, 1998;Petit et al, 1997;Scheeres, 2002;Bartczak and Breiter, 2003;Mysen et al, 2006;Olsen, 2006;Mysen and Aksnes, 2007;Feng et al, 2017) and in particular around contact-binary solid bodies (Marchis et al, 2014;Feng et al, 2016), was explored thoroughly in the last two decades (see a brief review in Lages et al, 2017).…”

Chaotic Zones around Rotating Small Bodies

“…Small bodies of the Solar system (asteroids, trans-Neptunian objects, cometary nuclei, planetary satellites) with diameters less than one thousand kilometers usually have strongly irregular shapes (Melnikov & Shevchenko 2010;Jorda et al 2016, p. 270), in many cases resembling dumb-bells, or "contact binaries". Various models for gravity fields of the "central body" were used: that of a triaxial ellipsoid with uniform density (Chauvineau et al 1993;Mysen et al 2006;Olsen 2006;Mysen & Aksnes 2007), a rod (Bartczak & Breiter 2003), a dumb-bell or "bilobed" model (Marchis et al 2014;Feng et al 2016), a collection ("molecule") of gravitating points (Petit et al 1997), a polyhedral model (Werner 1994;, a truncated gravitational field derived from a shape model (Feng et al 2017). Orbits around actual small bodies, such as asteroids Castalia, Eros, and Hektor were extensively modeled (Scheeres et al 1996(Scheeres et al , 2000Marchis et al 2014;Yu & Baoyin 2012).…”