Survey on studies about model uncertainties in small body explorations

Currently, the explorations of small solar system bodies (asteroids and comets) have become more and more popular. Due to the limited measurement capability and irregular shape and diverse spin status of the small body, uncertainties on the parameters of the system and s/c executions are a practical and troublesome problem for mission design and operations. The sample-based Monte Carlo simulation is primarily used to propagate and analyze the effects of these uncertainties on the surrounding orbital motion. However, it is generally time-consuming because of large samples required by the highly nonlinear dynamics. New methods need to be applied for balancing computational efficiency and accuracy. To motivate this research area and facilitate the mission design process, this review firstly discusses the dynamical models and the different methods of modeling the mostly related gravitational and non-gravitational forces. Then the main uncertainties in these force models are classified and analyzed, including approaching, orbiting and landing. Then the linear and nonlinear uncertainty propagation methods are described, together with their advantages and drawbacks. Typical mission examples and the associated uncertainty analysis, in terms of methods and outcomes, are summarized. Future research efforts are emphasized in terms of complete modelling, new mission scenarios, and application of (semi-) analytical methods in small body explorations

Dynamics around equilibrium points of uniformly rotating asteroids

Using tools such as periodic orbits and invariant manifolds, the global dynamics around equilibrium points (EPs) in a rotating second-order and degree gravitational field are studied. For EPs on the long axis, planar and vertical periodic families are computed, and their stability properties are investigated. Invariant manifolds are also computed, and their relation to the first-order resonances is briefly discussed. For EPs on the short axis, planar and vertical periodic families are studied, with special emphasis on the genealogy of the planar periodic families. Our studies show that the global dynamics around EPs are highly similar to those around libration points in the circular restricted three-body problem, such as spatial halo orbits, invariant manifolds, and the genealogy of planar periodic families

The biodiversity and stability of alpine meadow plant communities in relation to altitude gradient in three headwater resource regions

Kobresia pygmaea meadow community diversities in relation to altitude gradients (4200, 4300, 4400, 4450) on free grazing grassland was studied in the range of Chenduo county, Yushu prefecture, Qinghai province. Species richness and diversity index of vegetations in the four altitudes were comparatively analyzed. The results showed that the shape of species richness responsive curves to altitude gradient is “Bell-shape”. There were the same 11 common species in the four communities. The relative abundance of K. pygmaea decreased along increasing altitude. Moreover, the fuzzy membership functions were used to calculate the degree of stability, showing medium altitude > high altitude > low altitude, which suggested that grass land vegetation in low altitude of the sampling site had lower diversity, and the grade of species vulnerability risks may be decided with the help of the degree of stability.Key words: Alpine meadow, Yangtze, Yellow and Yalu Tsangpo river source region, altitude gradient, species diversity, membership functions

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

Using a gravitational field truncated at the 4th degree and order, the 1:1 ground-track 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

Analysis of Resonance Transition Periodic Orbits in the Circular Restricted Three-Body Problem

Resonance transition periodic orbits exist in the chaotic regions where the 1:1 resonance overlaps with nearby interior or exterior resonances in the circular restricted three-body problem (CRTBP). The resonance transition periodic orbits have important applications for tour missions between the interior and the exterior regions of the system. In this work, following the increase of the mass parameter μ in the CRTBP model, we investigate the breakup of the first-order resonant periodic families and their recombination with the resonance transition periodic families. In this process, we can describe in detail how the 1:1 resonance gradually overlaps with nearby first-order resonances with increasing strength of the secondary’s perturbation. Utilizing the continuation method, features of the resonance transition periodic families are discussed and characterized. Finally, an efficient approach to finding these orbits is proposed and some example resonance transition periodic orbits in the Sun–Jupiter system are presented

Analysis of Resonance Transition Periodic Orbits in the Circular Restricted Three-Body Problem

Resonance transition periodic orbits exist in the chaotic regions where the 1:1 resonance overlaps with nearby interior or exterior resonances in the circular restricted three-body problem (CRTBP). The resonance transition periodic orbits have important applications for tour missions between the interior and the exterior regions of the system. In this work, following the increase of the mass parameter μ in the CRTBP model, we investigate the breakup of the first-order resonant periodic families and their recombination with the resonance transition periodic families. In this process, we can describe in detail how the 1:1 resonance gradually overlaps with nearby first-order resonances with increasing strength of the secondary’s perturbation. Utilizing the continuation method, features of the resonance transition periodic families are discussed and characterized. Finally, an efficient approach to finding these orbits is proposed and some example resonance transition periodic orbits in the Sun–Jupiter system are presented

Orbit propagation in irregular and uncertain gravity field using differential algebra

Uncertainty propagation has been addressed extensively in space missions around the Earth, but much less for missions around small solar system bodies. Small bodies usually have irregular and weak gravity and our knowledge of their gravity, rotation speed and surrounding space environment is largely uncertain. These characteristics make the orbit propagation around these small bodies a challenging task. Focusing on the uncertainty of the small body's gravity, this paper applies the differential algebra (DA) technique to study the orbit propagation problem, and addresses its efficiency for a given the required accuracy. Different from traditional studies that focus on the uncertainty of the initial state, this study assumes an exact initial state and studies the influences that gravity model uncertainties have on the orbit. Taking the asteroid Steins as an example, the accuracy and the efficiency of the DA approach are firstly validated by comparison with the traditional Monte Carlo method. Then, the effects of gravity uncertainties on different types of orbits (prograde, retrograde and polar) are studied. The retrograde motion is found to be more robust to the gravity uncertainty than the prograde ones. For near polar orbits, the impact of gravity uncertainty on orbital motion depends significantly on the initial position, and it reaches the maximum if the initial position is near the polar regions. Moreover, short-term effects are found to play an important role in orbit deviation as a result of the gravity uncertainty. These discoveries can help mission designers assess the posed risk and design appropriate mission orbits.</p

Stable orbiting around small moons using J2-perturbed elliptic restricted problem

Confirmed small-body missions Martian Moons eXploration (MMX) and Hera are set to explore Martian moons and the binary asteroid Didymos’s moon Dimorphos, respectively. Orbital dynamics around these small moons differ substantially from those around previously visited targets. Simplified models, such as the circular-restricted three-body problem, cannot yield accurate predictions for orbits and their stability in real-world operations. To be specific, the orbit of the small moon and its vicinity are significantly perturbed by the oblateness of the planet and their relative positions. Realistic control constraints and the unstable 3:1 resonance of retrograde orbits further complicate orbit maintenance around a small moon. Therefore, minimizing the dynamical perturbation on baseline orbits resulting from model mismatches is crucial. This paper introduces the J2-ER3BP+GH model dedicated to describing the orbital dynamics around the small moon. It incorporates th