Modified double-averaged Hamiltonian in hierarchical triple systems

In this work, we introduce a modified double-averaging approach by considering the shortterm effects and formulate a more accurate double-averaged Hamiltonian (in comparison to the classical octupole-level Hamiltonian) for hierarchial triple systems. The Hamiltonian is expressed as a power series in the ratio of the semi-major axes of the inner and outer binaries. Both the Delaunay's elements and the classical orbit elements are adopted to describe the motion. To derive the secular Hamiltonian, the short-term oscillations in the Hamiltonian are averaged out by means of a double-averaging approach. In particular, during the average over the orbital period of the outer binary, the periodic corrections to the secular motion are taken into account. Based on the double-averaged Hamiltonian, we provide two versions of equations of secular motion, given in the form of canonic relations and Lagrange planetary equations. The resulting secular evolution equations can be utilized to reproduce the long-term behaviours for those physical systems where the perturbations coming from the disturbing bodies are relatively strong. To test the approach, we use the averaged Hamiltonian to predict the longterm motions of a planet in a stellar binary system and natural satellites in Sun-planet systems. Simulation results show that the modified Hamiltonian can reproduce secular behaviours with high accuracy. Additionally, the comparison of dynamical models truncated at different orders indicates that the secular Hamiltonian with inclusion of higher order terms has better accuracy in predicting long-term evolution

Secular dynamics of stellar spin driven by planets inside Kozai-Lidov resonance

In many exoplanetary systems with `hot Jupiters', it is observed that the spin axes of host stars are highly misaligned to planetary orbital axes. In this study, a possible channel is investigated for producing such a misalignment under a hierarchical three-body system where the evolution of stellar spin is subjected to the gravitational torque induced from the planet inside Kozai--Lidov (KL) resonance. In particular, two special configurations are explored in detail. The first one corresponds to the configuration with planets at KL fixed points, and the second one corresponds to the configurations with planets moving on KL librating cycles. When the planet is located at the KL fixed point, the corresponding Hamiltonian model is of one degree of freedom and there are three branches of libration centres for stellar spin. When the planet is moving on KL cycles, the technique of Poincar\'e section is taken to reveal global structures of stellar spin in phase space. To understand the complex structures, perturbative treatments are adopted to study rotational dynamics. It shows that analytical structures in phase portraits under the resonant model can agree well with numerical structures arising in Poincar\'e sections, showing that the complicated dynamics of stellar spin are governed by the primary resonance under the unperturbed Hamiltonian model in combination with the 2:1 (high-order and/or secondary) spin-orbit resonances.Comment: 15 pages, 11 figures. Accepted for publication in MNRA

A study of the high-inclination population in the Kuiper belt -- IV. High-order mean motion resonances in the classical region

In our previous study of Neptune's 4:7 mean motion resonance (MMR), we discovered that its resonant angle can only librate within a specific eccentricity ($e$) versus inclination ($i$) region, determined by a theoretical limiting curve curve (Li et al. 2020). This ``permissible region'' is independent of time and encompasses the entire possible stable region. We now generalize this theory to investigate all high-order MMRs embedded in the main classical Kuiper belt (MCKB). We first consider the 2nd-order 3:5 MMR in the framework of planet migration and resonance capture, and have further validated our limiting curve theory for both captured and observed 3:5 resonators. It suggests that only the $(e, i)$ pairs inside the individual permissible regions should be chosen as initial conditions for studying the in-situ evolution of high-order resonators. With such a new setting, we proceed to explore the long-term stability (for 4 Gyr) of different resonant populations, and our simulations predict that: (1) the 3:5 and 4:7 resonators are comparable in number, and they could have inclinations up to $40^{\circ}$; (2) the populations of objects in the higher order 5:9, 6:11, 7:12 and 7:13 resonances is about 1/10 of the 3:5 (or 4:7) resonator population, and nearly all of them are found on the less inclined orbits with $i<10^{\circ}$; (3) for these high-order resonances, almost all resonators reside in their individual permissible regions. In summary, our results make predictions for the number and orbital distributions of potential resonant objects that will be discovered in the future throughout the MCKB.Comment: 17 pages, 8 figures, accepted for publication in MNRA

R3^3 Prompting: Review, Rephrase and Resolve for Chain-of-Thought Reasoning in Large Language Models under Noisy Context

With the help of Chain-of-Thought (CoT) prompting, Large Language Models (LLMs) have achieved remarkable performance on various reasoning tasks. However, most of them have been evaluated under noise-free context and the dilemma for LLMs to produce inaccurate results under the noisy context has not been fully investigated. Existing studies utilize trigger sentences to encourage LLMs to concentrate on the relevant information but the trigger has limited effect on final answer prediction. Inspired by interactive CoT method, where intermediate reasoning steps are promoted by multiple rounds of interaction between users and LLMs, we propose a novel prompting method, namely R$^3$ prompting, for CoT reasoning under noisy context. Specifically, R$^3$ prompting interacts with LLMs to perform key sentence extraction, variable declaration and answer prediction, which corresponds to a thought process of reviewing, rephrasing and resolving. The responses generated at the last interaction will perform as hints to guide toward the responses of the next interaction. Our experiments show that R$^3$ prompting significantly outperforms existing CoT prompting methods on five reasoning tasks under noisy context. With GPT-3.5-turbo, we observe 3.7% accuracy improvement on average on the reasoning tasks under noisy context compared to the most competitive prompting baseline. More analyses and ablation studies show the robustness and generalization of R$^3$ prompting method in solving reasoning tasks in LLMs under noisy context

Prompting Large Language Models with Chain-of-Thought for Few-Shot Knowledge Base Question Generation

The task of Question Generation over Knowledge Bases (KBQG) aims to convert a logical form into a natural language question. For the sake of expensive cost of large-scale question annotation, the methods of KBQG under low-resource scenarios urgently need to be developed. However, current methods heavily rely on annotated data for fine-tuning, which is not well-suited for few-shot question generation. The emergence of Large Language Models (LLMs) has shown their impressive generalization ability in few-shot tasks. Inspired by Chain-of-Thought (CoT) prompting, which is an in-context learning strategy for reasoning, we formulate KBQG task as a reasoning problem, where the generation of a complete question is splitted into a series of sub-question generation. Our proposed prompting method KQG-CoT first retrieves supportive logical forms from the unlabeled data pool taking account of the characteristics of the logical form. Then, we write a prompt to explicit the reasoning chain of generating complicated questions based on the selected demonstrations. To further ensure prompt quality, we extend KQG-CoT into KQG-CoT+ via sorting the logical forms by their complexity. We conduct extensive experiments over three public KBQG datasets. The results demonstrate that our prompting method consistently outperforms other prompting baselines on the evaluated datasets. Remarkably, our KQG-CoT+ method could surpass existing few-shot SoTA results of the PathQuestions dataset by 18.25, 10.72, and 10.18 absolute points on BLEU-4, METEOR, and ROUGE-L, respectively.Comment: Accepted by EMNLP 2023 main conferenc

A sample return mission to Ceres

The goal of Project GAUSS (Genesis of Asteroids and evolUtion of the Solar System) is to return samples from the dwarf planet Ceres. Ceres is the most accessible candidate of ocean worlds and the largest reservoir of water in the inner Solar System. It shows active volcanism and hydrothermal activities in recent history. Recent evidence for the existence of a subsurface ocean on Ceres and the complex geochemistry suggest past habitability and even the potential for ongoing habitability. GAUSS will return samples from Ceres with the aim of answering the following top-level scientific questions: What is the origin of Ceres and what does this imply for the origin of water and other volatiles in the inner Solar System? What are the physical properties and internal structure of Ceres? What do they tell us about the evolutionary and aqueous alteration history of dwarf planets? What are the astrobiological implications of Ceres? Is it still habitable today? What are the mineralogical connections between Ceres and our current collections of carbonaceous meteorites

Dynamical Structures Associated with High-Order and Secondary Resonances in the Spin–Orbit Problem

In our solar system, spin–orbit resonances are common under Sun–planet, planet–satellite, and binary asteroid configurations. In this work, high-order and secondary spin–orbit resonances are investigated by taking numerical and analytical approaches. Poincaré sections as well as two types of dynamical maps are produced, showing that there are complicated structures in the phase space. To understand numerical structures, we adopt the theory of perturbative treatments to formulate resonant Hamiltonian for describing spin–orbit resonances. The results show that there is an excellent agreement between analytical and numerical structures. It is concluded that the main V-shape structure arising in the parameter space $(\dot{\theta },\alpha )$ is sculpted by the synchronous primary resonance, those minute structures inside the V-shape region are dominated by secondary resonances and those structures outside the V-shape region are governed by high-order resonances. Finally, the analytical approach is applied to binary asteroid systems (65803) Didymos and (4383) Suruga to reveal their phase-space structures