MasconCubes: Efficient Gravitational Field Modeling

• MasconCubes is a computational framework that models gravity fields using a fixed grid of optimized point masses within a defined shape model.
• It leverages explicit mass assignments and normalization on a Cartesian lattice to rapidly invert gravitational observations with high fidelity.
• The approach supports heterogeneous interior structures while significantly reducing computation time compared to traditional methods.

MasconCubes represent a computational framework for modeling gravitational fields of irregularly shaped small celestial bodies using an explicit three-dimensional grid of point masses (“mascons”). Unlike traditional approaches relying on spherical harmonics, polyhedral discretizations, or implicit neural representations, MasconCubes optimize mass assignments at fixed grid points within a known shape model, directly encoding physical mass distributions and enabling rapid, physically interpretable inversion of gravitational observations. The method addresses critical limitations in prior art by supporting heterogeneous interior structures, maintaining computational efficiency, and delivering high fidelity in gravity field recovery, particularly relevant for deep space missions to asteroids and comets.

1. Formulation and Conceptual Basis

MasconCubes define the gravity inversion as a direct optimization over a regular Cartesian grid of mascons enclosed within the asteroid’s volume. Each mascon has a fixed spatial location and a trainable mass parameter, initialized randomly and subsequently updated to minimize discrepancy between modeled and observed gravitational accelerations. This explicit representation stands in contrast to continuous/implicit neural models (e.g., GeodesyNets, PINN-GM), which require learning arbitrary functional mappings and extensive data sampling.

A known shape model (typically a mesh or volumetric representation) is leveraged to constrain mascon allocation: only those grid centers inside the surface are included, typically numbering fewer than one million for a $100\times100\times100$ grid. Mascon positions remain fixed; only masses are optimized, with normalization ensuring $\sum_j m_j = 1$, focusing inversion on relative distribution without conflating total mass scaling.

2. Context and Technical Challenges in Gravity Field Modeling

Standard paradigms in gravitational modeling—spherical harmonic expansions and polyhedral models—exhibit fundamental deficiencies for irregular bodies:

• Spherical harmonics: Analytical solutions to Laplace’s equation diverge within the Brillouin sphere, i.e., at radii interior to the smallest circumscribing sphere, where spacecraft typically operate. These expansions also lack sufficient spatial resolution for heteromorphic geometries.
• Polyhedral models: While stable up to the surface, they impose homogeneous density for the entire interior volume, an assumption rarely satisfied for asteroids or comets with compositional layering, voids, or density anomalies.

Emergent machine learning-based methods, notably GeodesyNets and PINN-GM, demonstrate improved sample efficiency and field compactness via implicit representations and physics-informed constraints. However, these approaches require orders of magnitude more training samples (often thousands or millions) and significant computation resources—leading to extended training times and practical limitations for rapid mission operations.

3. Methodological Framework

Grid Initialization and Constraints

• Grid definition: A cubic lattice (e.g., $100\times100\times100$) is instantiated to fully enclose the target body.
• Shape masking: Only mascons with centers inside the surface mesh are retained.
• Initialization: Mascon masses $m_j$ initialized with uniform random values.

Gravity Field Prediction

For a gravity observation at $\mathbf{r}_i$: $$\hat{\mathbf{g}}_i = -G \sum_{j=1}^{N} \frac{m_j(\mathbf{r}_i - \mathbf{r}_j)}{||\mathbf{r}_i - \mathbf{r}_j||^3},$$ where $G$ is the gravitational constant and $N$ is the number of active mascons. Mass normalization ($\sum_j m_j = 1$) decouples inversion from global scale.

Loss Function and Scaling

To match observed ($\mathbf{g}_i$) and predicted ($\hat{\mathbf{g}}_i$) accelerations, a normalized L$_1$ loss is employed: $$\mathcal{L} = \frac{1}{M} \sum_{i=1}^{M} ||\mathbf{g}_i - c\hat{\mathbf{g}}_i||_1,$$ with the optimal scaling factor

$$c = \frac{\sum_i \mathbf{g}_i \cdot \hat{\mathbf{g}}_i}{\sum_i ||\hat{\mathbf{g}}_i||^2}.$$

Optimization Procedure

• Mini-batches: Randomly sample $1000$ points per batch within a fixed-radius sphere.
• Iterations: Train for $1000$ steps using the Adam optimizer with initial learning rate $1\times 10^{-5}$ and scheduled decay.

4. Quantitative Evaluation and Empirical Results

Comprehensive evaluation of MasconCubes was conducted across a range of test bodies, including asteroid Bennu, Eros (with uniform, two-region, and three-region density fields), Itokawa (both smooth and two-region scenarios), and synthetic planetesimals, utilizing ground truth from tetrahedral meshes or “cubified” grid discretizations.

Key Metrics

• Mean Absolute Error (MAE) on normalized Stokes coefficients (up to degree $7$): MasconCubes outperformed GeodesyNets by large margins.
• Gravitational acceleration accuracy: Cosine distance, Euclidean norm distance, and relative Euclidean norm distance all favored MasconCubes, with consistently lower errors across scenarios.
• Trajectory simulations: Spacecraft position predictions using MasconCube-modeled gravity fields were highly accurate, demonstrating robustness for mission applications.

Computational and Interpretability Advantages

• Efficiency: Training converges in under $7$ minutes on GPUs, approximately $40\times$ faster than GeodesyNets (which require hundreds of minutes).
• Physical interpretability: The explicit mascon grid enables immediate visualization and quantitative analysis of internal mass distributions—critical for geophysical investigations.

5. Applications in Space Missions and Scientific Research

Mission Operations

High-fidelity gravity models are central to proximity operations (e.g., orbit determination, fly-bys) around small bodies. The adaptable, rapid training paradigm of MasconCubes facilitates near real-time updates, making them suitable for evolving mission profiles and dynamic operational constraints.

Scientific Insights

Direct access to explicit internal mass distributions permits inference of density heterogeneities, detection of voids, and identification of stratified compositions. This level of physical insight informs models of asteroid formation, evolutionary history, and resource localization for future exploration or in-situ utilization.

6. Future Prospects and Extensions

MasconCubes’ explicit framework enables further methodological refinements. A plausible implication is the integration of regularization strategies to relax reliance on shape constraints, supporting “shape-free” training for poorly characterized bodies or incomplete shape data. Additionally, the explicit mass representation supports extension to uncertainty quantification and multi-modal inversion problems.

7. Summary and Significance

MasconCubes establish a rigorous, efficient, and interpretable paradigm for gravity field modeling in irregular celestial bodies, overcoming analytic and statistical limitations of spherical harmonics, polyhedral discretizations, and implicit neural architectures. With demonstrated empirical superiority—lower MAE on Stokes coefficients, rapid convergence, and direct physical interpretability—the framework is suited for both operational mission needs and deep scientific inquiry into small body interiors, contributing distinctly to gravitational modeling and planetary science (Fanti et al., 10 Sep 2025).