Modeling the resonant dynamics of infinitesimal particles orbiting nonspherical rotating bodies

Abstract This work proposes a model to investigate the resonant dynamics of infinitesimal particles orbiting a nonspherical rotating body (a planet, a dwarf planet or an asteroid), under the additional perturbations induced by the attraction of the Sun and the solar radiation pressure. Assuming that the nonspherical body rotates uniformly around a fixed axis and revolves around the Sun on a Keplerian orbit, the spherical harmonics approach is considered to model the gravitational potential of the nonspherical body and various expansions in terms of the orbital elements are used to model the third-body perturbations and the disturbances induced by the solar radiation pressure. Classifying the terms of expansions, two types of resonances that influence the dynamics around a nonspherical body are discussed. One type of resonance occurs whenever there is a commensurability between the orbital period of the infinitesimal particle and the rotation period of the nonspherical body. This is called a sectoral–tesseral resonance and its effects can be noticed on moderate time scales. We show that each sectoral–tesseral resonance splits into a multiplet of resonances that could overlap and lead to complex evolution of the orbital elements. The location of each component of the resonance multiplet is estimated analytically and the occurrence of phenomenon of resonance overlap is predicted. The other type of resonance, called secular, involves the frequency of slow angles and is due to the complex interaction between the oblateness of the nonspherical body, the attraction of the Sun and the solar radiation pressure. Although secular resonances are extensively studied in the Earth’s environment, and recently in the Moon’s environment, little attention is paid to understand their effects in case of dynamics around other solar system bodies. We classify the secular resonances and discuss their location in the parameter space and in the physical space. An averaged Hamiltonian, capable of characterizing the sectoral–tesseral resonances $$j:\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>:</mml:mo> <mml:mi>ℓ</mml:mi> </mml:mrow> </mml:math> , with $$j\in \{1,2,3,4\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> and $$\ell \in \{1,2,3\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> , and the secular resonances is derived. Finally, as an application of the derived Hamiltonian, the 1 : 1 sectoral–tesseral resonance around Vesta is discussed, showing how the components of the resonance multiplet overlap to generate a complex dynamical evolution of the orbital elements. In particular, a mechanism that leads to very fast and large-amplitude variations of eccentricity is analyzed, although the usual effect of sectoral–tesseral resonances is the variation of the semimajor axis.