1. Introduction
Precise gravity field modeling is critical for asteriod mission flight dynamics and operations [1]. The typical representation is a spherical harmonics expansion, but this approach loses its appeal as body irregularities become more important [2]. Other options like polyhedral gravity [3] can overcome some of these difficulties but introduce new requirements, such as needing a shape model and assuming homogeneous density. Mascon models [4] are also presented as an alternative, creating a complex landscape of methods for geodetic studies of irregular bodies. We propose a method to a Dynamic Lagrangian Framework for mascon model. By “Lagrangian”, we mean that mascon locations and even their existence, evolve as particles in a dynamical system, freely adapting to the field topology, rather than being confined to a fixed mesh.
2. Methodology: Topology Evolution via Split, Clone, and Prune
Our approach treats the mascon distribution as a dynamic system that self-organizes through three core Lagrangian operators, governed by a self-supervised learning strategy:
Split (Refinement): To capture high-frequency internal mass distribution details, the “Split” operator adaptively increases the model’s spatial resolution. It identifies particles in regions with high gradient residuals, which signals a poor local fit between the model and the observed gravity. These particles are then divided, creating new elements that can better represent complex gravitational features.
Clone (Intensification): For regions requiring higher mass magnitude without altering the geometric structure, we employ a “Clone” operator. By duplicating a particle at its existing location, it allows for the precise regulation of local mass density. This is physically crucial for modeling areas of high density concentration (like a core) without unnecessarily increasing the geometric complexity of the model.
Prune (Sparsification): To ensure model compactness and prevent overfitting, the “Prune” operator acts as a regularization mechanism. It dynamically removes redundant particles that are statistically insignificant to the overall gravity field or drift outside the body’s physical shape during optimization. This process guarantees that the final model is both sparse and physically constrained, improving computational efficiency and interpretability.
3. Preliminary Results
We validated this framework on the well-known asteroid (433) Eros [5]. Compared to deep learning based method, incuding the geodesyNet [6] and MasconCub [4] baselines under equal particle counts, the Split-Clone-Prune strategy achieves higer regression accuracy in the near-field. Furthermore, it also recovers low-degree Stokes coefficients with competitive fidelity. At the same time, it maintains higher computational efficiency due to its adaptive nature.
4. Conclusion
This study demonstrates that gravity field topology should be a learnable variable. By using “Split”, “Clone”, and “Prune” operators within a self-supervised Lagrangian framework, our method robustly reconstructs both the internal structure and gravity field, offering a new method for high-fidelity gravity inversion of irregular bodies.
5. Reference
[1] Calla, P., Fries, D. & Welch, C. Asteroid mining with small spacecraft and its economic feasibility. Preprint at arXiv:1808.05099 (2018).
[3] Paul, M. The gravity effect of a homogeneous polyhedron for threedimensional interpretation. Pure Appl. Geophys. 112, 553–561 (1974).
[2] Accomazzo, A. et al. The final year of the Rosetta mission. Acta Astronaut. 136, 354–359 (2017).
[4] Fanti, P., & Izzo, D. MasconCube: Fast and Accurate Gravity Modeling with an Explicit Representation. arXiv preprint arXiv:2509.08607 (2025)..
[5] Veverka, J. et al. NEAR at Eros: Imaging and spectral results. Science 289, 2088–2097 (2000).
[6] IZZO, Dario et GÓMEZ, Pablo. Geodesy of irregular small bodies via neural density fields. Communications Engineering, 48 (2022).