Inferring the distribution of exoplanetary system architectures from survey data increasingly relies on forward modeling with N-body simulations, but the computational cost of these models makes rigorous statistical inference impractical. We develop a statistical emulator for the distribution of outcomes of planet formation simulations, enabling fast conditional sampling in place of expensive integrations. The emulator targets the joint distribution of detected multiplicity and three observables derived from neighboring planet pairs: orbital period ratios, period-normalized transit duration ratios, and orbital spacing in mutual Hill radii. Our approach combines a calibrated model for multiplicity, flexible regression models for conditional distributions, dependence modeling in a transformed probability space, and methods to capture rare events in the tails. We introduce diagnostics to assess calibration, dependence preservation, and reproduction of rare events. The resulting emulator accurately captures heteroskedasticity, dependence structure, and tail behavior of the simulator outputs while reducing computational cost by several orders of magnitude relative to direct N-body simulation. This framework enables integration of detailed dynamical models into simulation-based inference for exoplanet populations.
The spin–orbit coupled motion of a rigid body on a Keplerian orbit is a classical problem in celestial mechanics, historically motivated by the rotational dynamics of the Moon. A Hamiltonian formulation is developed in this work to describe the motion of a slow-rotating rigid body subject to a gravity-gradient potential. Modified Serret–Andoyer variables and Delaunay elements, combined with canonical transformations and averaging over the fast orbital mean anomaly, reduce the original four-degree-of-freedom system to a three-degree-of-freedom system governing the slow resonant dynamics. In this formulation, the fourth canonical pair consists of the orbital mean anomaly and its conjugate action, while the nodal and apsidal rates are prescribed as external orbital precessions. Fourier–Bessel expansions of the orbital motion identify the dominant spin–orbit commensurabilities and drive the reduction. Relative equilibria corresponding to generalized Cassini states are computed, and closed-form algebraic conditions for the ecliptic obliquity are derived across the five dominant commensurabilities. Applied to the Earth–Moon system, linearization about the 1 : 1 resonant equilibrium recovers longitudinal, latitudinal, and wobble libration periods of 2.895, 24.076, and 74.686 years, with relative errors of 0.13% , 0.33% , and 0.08% , respectively, relative to the values reported by Rambaux and Williams (2011).
We prove that the elliptic photo-gravitational Hill problem is not integrable except in one case, when the gravitational force of the lighter primary acting on the infinitesimal mass is balanced by the radiation-pressure force of this primary. In effect, the infinitesimal mass moves under the influence of the more massive primary localised at infinity. We show that in this case the equations of motion are integrable and can be solved explicitly in terms of elementary functions. Moreover, we distinguish a three-dimensional subspace of initial conditions for which the solutions are periodic and the corresponding orbits are algebraic curves defined by a polynomial of degree four. All of them are of genus zero.
The relative equilibria of the planar Newtonian 1+4 -body problem, when the four masses tend to zero, become coorbital around the central mass. In this paper, we investigate convex coorbital central configurations in the planar 1+4 -body problem. For the homogeneous potential ( s>1 ), we show that convex 1+4 coorbital central configurations possess an axis of symmetry under the symmetric mass conditions m_1 = m_4>0 and m_2 = m_3>0 . Furthermore, we prove that the convex 1+4 coorbital central configuration with m_1 = m_4>0 and m_2 = m_3>0 is linearly stable under some constraints.
This work presents the development of a hybrid symbolic-numerical framework dedicated to the automation of perturbation theories in celestial mechanics, with a particular emphasis on the theory and dynamics of artificial satellites and space mission design. The system is founded upon the interoperability between two complementary tools: symcelmech, developed in Maxima, and CelestialMechanics.jl, implemented in the Julia language. symcelmech implements the Hori-Deprit recursive method based on Lie transformations, employing Poisson series, closed form and expansions in powers of eccentricity for the systematic elimination of short periodic and, where applicable, long periodic terms in high-order Hamiltonians. The resulting analytical expressions are exported via text files to CelestialMechanics.jl, a high-performance toolkit designed for high-fidelity numerical propagation and the validation of analytical results. The framework’s modular architecture focuses on managing symbolic complexity (expression swell) and creating an automated pipeline that converts complex algebraic expressions into optimized numerical functions. The system’s feasibility is demonstrated through the processing of Hamiltonians perturbed by zonal and tesseral harmonics, highlighting that the proposed algorithmic structure enables the systematic processing of high-order dynamical models with analytical rigor and computational efficiency. The framework is open-source, providing a robust platform for the development of modern perturbation theories and for the design and analysis of space missions.
