New Open-Source Framework Automates Satellite Orbit Theories Using Maxima and Julia

A team of Brazilian researchers has unveiled an ambitious open-source software ecosystem that promises to transform how scientists build the analytical theories behind satellite orbits. The framework, described in the journal Celestial Mechanics and Dynamical Astronomy, links two complementary programs: symcelmech, a symbolic engine written in the Maxima computer algebra system, and CelestialMechanics.jl, a high-performance numerical toolkit built in the Julia language. Together they automate a notoriously laborious process, deriving perturbation theories for artificial satellites that would otherwise require months of hand calculation and be vulnerable to algebraic mistakes that no human could reliably detect.

The mathematical heart of the new system is the Hori-Deprit method, a canonical perturbation technique based on Lie transformations that was developed in the 1960s by Gen-ichiro Hori and André Deprit. The method works on a Hamiltonian, the energy function that governs orbital motion, expanded in powers of a small parameter representing the strength of a perturbing force. Through a sequence of recursive canonical transformations, the algorithm systematically strips away the short periodic oscillations in the satellite’s motion, leaving behind a simplified, averaged Hamiltonian that captures only the long-term, secular evolution of the orbit. The approach rose to fame when Deprit and Rom used it in 1970 to produce an automated third-order solution to the main problem of artificial satellite theory, eliminating the small-eccentricity singularities that had plagued earlier analytical solutions.

What the new framework adds is a fully automated, open pipeline for carrying out such calculations at modern scale. The symcelmech package represents Hamiltonians as Poisson series, multivariate Fourier series whose coefficients are themselves Laurent series in the orbital variables, and it manipulates them within a closed algebraic domain so that differentiation, averaging and integration never produce unmanageable new mathematical objects. A dedicated Poisson bracket engine, equipped with memoization that caches previously computed brackets and exploits antisymmetry to avoid redundant work, drives the recursive Lie transformation. Because high-order analytical theories are often defeated not by physics but by expression swell, the explosive growth of symbolic terms, the authors devoted considerable effort to controlling that growth, including targeted rational simplification, selective trigonometric expansion on a single chosen variable, and eccentricity truncation that caps the number of terms entering each stage of the recursion.

The automation extends to the most computationally delicate step: integration. Maxima’s general-purpose symbolic integrator relies on heuristic searches that can buckle when asked to integrate expressions containing thousands of trigonometric terms. The researchers therefore wrote specialized routines that exploit the structure of Poisson series. Averaging is performed through a divide-and-conquer routine that classifies each term as secular or periodic in linear time, while quadrature for the generating functions is carried out by mapping trigonometric terms into the complex exponential domain, where integration becomes simple division by the angular frequency. For perturbations naturally expressed in the true anomaly, such as third-body gravity and solar radiation pressure, the framework offers closed-form averaging routes based on Hansen coefficients or on a change of variable to the eccentric anomaly, preserving exact eccentricity dependence rather than relying on truncated series.

Once the algebra is complete, the analytical expressions are exported as optimized text files and parsed by CelestialMechanics.jl, which converts them into fast Julia functions ready for numerical evaluation. The Julia toolkit assembles full-fidelity force models, including zonal harmonics up to degree 18, tesseral and sectorial gravity terms, third-body perturbations and solar radiation pressure with shadow models, drawing ephemerides from NASA’s SPICE library and gravity fields from standard planetary data files. It supports three distinct propagation modes, Cartesian Cowell integration, integration of Hamilton’s equations in Delaunay variables, and integration of Lagrange’s planetary equations in classical orbital elements, all feeding a unified post-processing pipeline that converts states into comparable orbital elements and generates scientific visualizations. For long-duration runs, trajectory data are streamed to disk to avoid exhausting memory.

The authors validated the pipeline in two increasingly demanding applications. In the first, they used symcelmech to normalize the Hamiltonian of the main J2 problem, the dominant perturbation caused by Earth’s equatorial bulge, through two successive Lie transformations that removed both the fast orbital angle and the argument of perigee. The entire symbolic computation, including a second-order generating function with thousands of terms, ran in a fraction of a second on an ordinary laptop. Predictions of the secular precession rates of perigee and ascending node derived from the doubly averaged Hamiltonian were then checked against two independent numerical propagations of the full, non-averaged J2 dynamics, agreeing to within roughly a part in a thousand, exactly consistent with the expected size of the neglected third-order contributions.

The second application targeted frozen orbits around the Moon, special orbits whose eccentricity and argument of perilune remain constant over time, making them ideal for stable low-altitude observation missions. The team built a high-fidelity lunar gravity model including zonal harmonics from J2 through J9 together with the leading tesseral and sectorial coefficients, and normalized the Hamiltonian in the Moon’s rotating body-fixed frame. Solving the frozen orbit conditions across a grid of semi-major axes and inclinations revealed distinct families of equilibria: bands of low-eccentricity frozen orbits at intermediate inclinations, a polar family with eccentricities below 0.01, and narrow strips of higher-eccentricity solutions near the equator driven by the odd zonal harmonics. Phase portraits confirmed the expected libration structure, with stable equilibria surrounded by closed energy contours and separatrices dividing librating from circulating orbits.

A representative near-polar frozen orbit was then propagated for 800 days, roughly nine thousand lunar revolutions, using an eighth-order symplectic integrator with a fixed one-minute step. The argument of perilune librated around 270 degrees with an amplitude of about 70 degrees, confirming that the orbit remained trapped in the libration island predicted by the analytical theory, while the periapsis altitude stayed confined between 165 and 185 kilometers. Crucially, the Jacobi constant, an exact integral of motion in the rotating frame, was conserved to about five parts in one hundred million with no secular drift, demonstrating that the long-term evolution observed in the simulation reflected genuine physics rather than numerical artifacts. The authors also documented a known limitation honestly: near-circular frozen orbits expose a coordinate singularity in Delaunay variables that will require reformulating the transformation in Poincare variables, a development already partially implemented.

Beyond the immediate results, the significance of the work lies in its openness and its architecture. The ecosystem of analytical celestial mechanics has long been dominated by proprietary mission analysis suites whose internal algorithms cannot be audited or extended by the community, and by aging research codes tied to obsolete languages. By releasing both symcelmech and CelestialMechanics.jl as open-source projects, the researchers hope to democratize access to sophisticated perturbation methods and invite collaborative refinement. Planned extensions include the elimination of the parallax in polar-nodal variables to tame memory demands at third order, computation of Birkhoff normal forms for stability analysis, non-singular Poincare formulations for near-circular orbits, and integration with the SciML ecosystem to explore physics-informed neural networks and machine-learned surrogate models for long-term orbit prediction. For an era of rapidly multiplying lunar and planetary missions, the framework offers a rigorous, transparent and computationally efficient foundation for designing the orbits of the future.

Subject of Research: An open-source hybrid symbolic-numerical framework automating Hori-Deprit perturbation theories for artificial satellite dynamics and space mission design.

Article Title: A hybrid symbolic-numerical framework for artificial satellite theory and dynamics using Maxima and Julia

Article References: de Oliveira Paes, G., Berton, L., & de Moraes, R. V. (2026). A hybrid symbolic-numerical framework for artificial satellite theory and dynamics using Maxima and Julia. Celestial Mechanics and Dynamical Astronomy, 138(5), Article 56. https://doi.org/10.1007/s10569-026-10331-0

Image Credits: AI Generated

DOI: 10.1007/s10569-026-10331-0

Keywords: celestial mechanics, artificial satellite theory, Hori-Deprit method, Lie transformations, computer algebra, Maxima, Julia, Poisson series, frozen orbits, lunar orbits, Hamiltonian dynamics, orbital perturbation theory