A new expansion of planetary disturbing function and applications to interior, co-orbital and exterior resonances with planets

In this study, a new expansion of planetary disturbing function is developed for describing the resonant dynamics of minor bodies with arbitrary inclinations and semimajor axis ratios. In practice, the disturbing function is expanded around circular orbits in the first step and then, in the second step, the resulting mutual interaction between circular orbits is expanded around a reference point. As usual, the resulting expansion is presented in the Fourier series form, where the force amplitudes are dependent on the semimajor axis, eccentricity and inclination, and the harmonic arguments are linear combinations of the mean longitude, longitude of pericenter and longitude of ascending node of each mass. The resulting new expansion is valid for arbitrary inclinations and semimajor axis ratios. In the case of mean motion resonant configuration, the disturbing function can be easily averaged to produce the analytical expansion of resonant disturbing function. Based on the analytical expansion, the Hamiltonian model of mean motion resonances is formulated, and the resulting analytical developments are applied to Jupiter’s inner and co-orbital resonances and Neptune’s exterior resonances. Analytical expansion is validated by comparing the analytical results with the associated numerical outcomes.

Orbital Design and Control for Jupiter-Observation Spacecraft

This paper investigates the evolution of orbits around Jupiter and designs a sun-synchronous repeating ground track orbit. In the dynamical models, the leading terms of the Jupiter’s oblateness are J2 and J4 terms. A reasonable range of ground track repetition parameter Q is given and the best observation orbit elements are selected. Meanwhile, the disturbing function acting on the navigation spacecraft is the atmospheric drag and the third body. The law of altitude decay of the spacecraft’s semimajor orbit axis caused by the atmospheric drag is studied, and the inclination perturbation caused by the sun’s gravity is analyzed. This paper designs a semimajor axis compensation strategy to maintain the orbit’s repeatability and proposes an initial inclination prebiased strategy to limit the local time at the descending node in a permitted range. In particular, these two methods are combined in the context of sun-synchronous repeating ground track orbit for better observation of the surface of Jupiter.

Modelling the Triaxial Compression Behavior of Loess Using the Disturbed State Concept

This paper investigates the triaxial compression behaviour of Q3 loess soil and the construction of a constitutive model accounting for the structural effect of loess on the basis of the disturbed state concept. By analyzing the triaxial compression testing results, we have established a new disturbing function with respect to the volumetric and shear moduli parameters. A research into the evolution laws of the disturbing function was also conducted, followed by the construction of a constitutive model for loess soil as well as the verification of the constitutive model with model test results. The results indicate that the double-parameter disturbing function evolves in an exponential form, capturing well the effect of moisture content and confining pressure on the loess structural behavior. The parameters of the constructed constitutive model based on the disturbed state are easy to be obtained and have clarified physical meanings. Considering the effectiveness in capturing the structural behavior of the loess, the constructed constitutive model has a great potential to be applied in the engineering practice in the loess area. The constructed constitutive model based on the disturbed state concept provides new ideas for the study of the structural constitutive model of loess, which is theoretically significant.

Resonance libration and width at arbitrary inclination

ABSTRACT We apply the analytical disturbing function for arbitrary inclination derived in our previous work to characterize resonant width and libration of mean motion resonances at arbitrary inclination obtained from direct numerical simulations of the three-body problem. We examine the 2:1 and 3:1 inner Jupiter and 1:2 and 1:3 outer Neptune resonances and their possible asymmetric librations using a new analytical pendulum model of resonance that includes the simultaneous libration of multiple arguments and their second harmonics. The numerically derived resonance separatrices are obtained using the mean exponential growth factor of nearby orbits (megno chaos indicator). We find that the analytical and numerical estimates are in agreement and that resonance width is determined by the first few fundamental resonance modes that librate simultaneously on the resonant time-scale. Our results demonstrate that the new pendulum model may be used to ascertain resonance width analytically, and more generally, that the disturbing function for arbitrary inclination is a powerful analytical tool that describes resonance dynamics of low as well as high inclination asteroids in the Solar system.

Potential softening and eccentricity dynamics in razor-thin, nearly Keplerian discs

ABSTRACTIn many astrophysical problems involving discs (gaseous or particulate) orbiting a dominant central mass, gravitational potential of the disc plays an important dynamical role. Its impact on the motion of external objects, as well as on the dynamics of the disc itself, can usually be studied using secular approximation. This is often done using softened gravity to avoid singularities arising in calculation of the orbit-averaged potential – disturbing function – of a razor-thin disc using classical Laplace–Lagrange theory. We explore the performance of several softening formalisms proposed in the literature in reproducing the correct eccentricity dynamics in the disc potential. We identify softening models that, in the limit of zero softening, give results converging to the expected behaviour exactly, approximately or not converging at all. We also develop a general framework for computing secular disturbing function given an arbitrary softening prescription for a rather general form of the interaction potential. Our results demonstrate that numerical treatments of the secular disc dynamics, representing the disc as a collection of N gravitationally interacting annuli, are rather demanding: for a given value of the (dimensionless) softening parameter, ς ≪ 1, accurate representation of eccentricity dynamics requires N ∼ Cς−χ ≫ 1, with C ∼ O(10), 1.5 ≲ χ ≲ 2. In discs with sharp edges a very small value of the softening parameter ς (≲ 10−3) is required to correctly reproduce eccentricity dynamics near the disc boundaries; this finding is relevant for modelling planetary rings.

High-Order Analytic Expansion of Disturbing Function for Doubly Averaged Circular Restricted Three-Body Problem

Terms in the analytic expansion of the doubly averaged disturbing function for the circular restricted three-body problem using the Legendre polynomial are explicitly calculated up to the fourteenth order of semimajor axis ratio (α) between perturbed and perturbing bodies in the inner case (α<1), and up to the fifteenth order in the outer case (α>1). The expansion outcome is compared with results from numerical quadrature on an equipotential surface. Comparison with direct numerical integration of equations of motion is also presented. Overall, the high-order analytic expansion of the doubly averaged disturbing function yields a result that agrees well with the numerical quadrature and with the numerical integration. Local extremums of the doubly averaged disturbing function are quantitatively reproduced by the high-order analytic expansion even when α is large. Although the analytic expansion is not applicable in some circumstances such as when orbits of perturbed and perturbing bodies cross or when strong mean motion resonance is at work, our expansion result will be useful for analytically understanding the long-term dynamical behavior of perturbed bodies in circular restricted three-body systems.

Semianalytic Integration of High-Altitude Orbits under Lunisolar Effects

The long-term effect of lunisolar perturbations on high-altitude orbits is studied after a double averaging procedure that removes both the mean anomaly of the satellite and that of the moon. Lunisolar effects acting on high-altitude orbits are comparable in magnitude to the Earth’s oblateness perturbation. Hence, their accurate modeling does not allow for the usual truncation of the expansion of the third-body disturbing function up to the second degree. Using canonical perturbation theory, the averaging is carried out up to the order where second-order terms in the Earth oblateness coefficient are apparent. This truncation order forces to take into account up to the fifth degree in the expansion of the lunar disturbing function. The small values of the moon’s orbital eccentricity and inclination with respect to the ecliptic allow for some simplification. Nevertheless, as far as the averaging is carried out in closed form of the satellite’s orbit eccentricity, it is not restricted to low-eccentricity orbits.