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Regularity and reentry basins of low Earth orbits in the J2--solar radiation pressure problem

This paper numerically investigates the long-term dynamics of low Earth orbits under solar radiation pressure and Earth's oblateness, revealing that while reentry basins can exhibit fractal structures for less-stringent mitigation guidelines, most reentry trajectories for large area-to-mass ratios follow regular motion on short timescales, offering insights for developing dynamically informed deorbiting strategies.

Original authors: Cassandra Barbis, Jérôme Daquin, Elisa Maria Alessi, Charalampos Skokos

Published 2026-10-13

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CC BY 4.0

Original authors: Cassandra Barbis, Jérôme Daquin, Elisa Maria Alessi, Charalampos Skokos

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

High above the Earth, thousands of satellites and pieces of space junk circle the planet in a vast, invisible traffic system. While we often think of these objects as moving in perfect, predictable loops, the reality is far more complex. Their paths are constantly nudged by the Earth's slightly lumpy shape and by the pressure of sunlight hitting their surfaces. For some objects, especially those that are light and have large surfaces like solar panels or sails, these gentle pushes can add up over time, causing their orbits to stretch and eventually plunge them back into the atmosphere. This process is not just a matter of space debris cleaning up itself; it is a critical tool for engineers who want to design satellites that can safely return to Earth at the end of their lives without needing expensive fuel. The challenge lies in predicting exactly when and where this will happen, because the forces involved can create a chaotic environment where tiny differences in starting conditions lead to vastly different outcomes.

A team of researchers has recently taken a deep dive into this chaotic environment to understand how these natural forces interact over decades. They focused on a specific type of satellite orbit, one that is low enough to be affected by the Earth's shape but high enough to avoid the thick atmosphere that usually slows objects down quickly. Using powerful computer simulations, they modeled the long-term behavior of these orbits, treating the satellite as a simple object influenced by two main factors: the Earth's oblateness, which is the fact that our planet is slightly squashed at the poles, and the constant push of solar radiation pressure. They were particularly interested in how the ratio of an object's surface area to its mass affects its journey. A satellite with a large solar sail, for instance, has a high area-to-mass ratio, making it much more sensitive to the sun's push than a heavy, compact satellite.

The researchers used a sophisticated method to map out which orbits would remain stable and which would eventually crash into the Earth. They ran millions of simulations, tracking the paths of virtual satellites over a period of one hundred years. Their results revealed a clear pattern: as the area-to-mass ratio increased, the amount of chaotic behavior in the system grew significantly. For objects with a low ratio, the orbits were mostly predictable and orderly. However, for those with larger ratios, the paths became tangled and unpredictable, with chaotic regions spreading across the map of possible orbits. Despite this underlying chaos, the team discovered a surprising fact about the satellites that actually managed to reenter the atmosphere. When they looked at the specific trajectories that led to a crash within a standard twenty-five-year safety window, they found that the vast majority of these paths were actually regular and predictable. Even though the broader system was chaotic, the specific routes that led to a timely return were smooth and stable, at least for the duration of the mission.

The study also examined the boundaries between orbits that would reenter and those that would stay in space forever. These boundaries act like the edges of a map, separating safe zones from danger zones. The researchers found that the shape of these edges depends heavily on how strict the safety rules are. If the goal is to have a satellite reenter within five years, the boundary between success and failure is relatively smooth and easy to define. However, if the goal is relaxed to a twenty-five-year window, the boundary becomes incredibly complex and jagged. In mathematical terms, this boundary becomes fractal, meaning it has intricate patterns that repeat at every scale, much like the edge of a coastline. This complexity has a practical consequence: predicting whether a satellite will reenter within twenty-five years requires far more precise knowledge of its starting position than predicting a five-year reentry. A tiny error in knowing where the satellite starts could lead to a completely wrong prediction about its fate if the timeline is long.

Ultimately, this research offers a new perspective on how to manage the end of a satellite's life. It suggests that while the long-term dance of space objects can be chaotic, the specific steps needed to bring them home safely are often straightforward. The findings indicate that designing satellites to reenter within a shorter timeframe, such as five years, is not only a stricter environmental goal but also a more reliable one from a physics standpoint. By targeting these shorter windows, mission planners can avoid the unpredictable, fractal-like boundaries that make long-term predictions so difficult. This work provides a clearer roadmap for the future of space sustainability, showing that by understanding the natural rhythms of orbital mechanics, we can design better strategies for cleaning up the space around our planet.

Technical Summary: Regularity and Reentry Basins of Low Earth Orbits in the J2–Solar Radiation Pressure Problem

Problem Statement
This study investigates the long-term dynamical structure of Low Earth Orbits (LEOs) under the coupled secular effects of Earth's oblateness ( $J_2$ ) and Solar Radiation Pressure (SRP). The research focuses on objects with high area-to-mass ratios ( $A/m$ ), such as satellites equipped with large solar sails or debris with significant surface area, where SRP becomes a dominant perturbation. The primary objective is to characterize the transition between regular and chaotic motion, map the regions leading to atmospheric reentry, and analyze the geometric complexity of reentry basins within the context of space debris mitigation guidelines (specifically the 25-year rule and stricter 5-year targets). The study aims to determine whether natural eccentricity growth mechanisms can facilitate passive deorbiting and to understand the predictability of such trajectories given uncertainties in initial conditions.

Methodology
The authors employ a closed-form, averaged Hamiltonian model that incorporates $J_2$ and SRP perturbations. The model assumes a "cannonball" approximation for the satellite (radiation normal to the surface) and averages the Hamiltonian over the satellite's mean anomaly, reducing the system to three degrees of freedom (DoF) with the semi-major axis ( $a$ ) as a conserved parameter. The dynamics are governed by six resonant harmonics dependent on the orbital inclination, argument of perigee, and solar longitude.

To analyze the system, the authors utilize:

Smaller Alignment Index (SALI): A fast numerical chaos indicator used to distinguish between regular and chaotic motion. SALI is computed by integrating the equations of motion alongside variational equations for two deviation vectors. Regular orbits exhibit bounded fluctuations in SALI, while chaotic orbits show exponential decay to machine precision.
Dynamical Cartography: The authors generate stability maps over fine Cartesian meshes of initial conditions in the inclination ( $i$ ) versus semi-major axis ( $a$ ) space, as well as in $i-e$ and $a-e$ spaces, for $A/m$ values of 1, 5, and 10 $m^2kg^{-1}$ .
Reentry Basin Analysis: Orbits are classified as "reentry orbits" if they reach a critical eccentricity ( $e^*$ ) corresponding to a perigee altitude of 120 km within a specified timeframe (5, 10, 15, 20, or 25 years).
Uncertainty Exponent Method: To quantify the fractality of reentry basin boundaries, the authors compute the uncertainty fraction $f(\epsilon)$ for perturbed initial conditions. The scaling $f(\epsilon) \propto \epsilon^\alpha$ yields the uncertainty exponent $\alpha$ , where $\alpha=1$ indicates smooth boundaries and $0 < \alpha < 1$ indicates fractal boundaries.

Key Contributions and Results

Chaos and $A/m$ Dependence: The study demonstrates that the extent of chaotic motion is directly governed by the area-to-mass ratio. For low $A/m$ (1 $m^2kg^{-1}$ ), the phase space is predominantly regular. As $A/m$ increases to 5 and 10 $m^2kg^{-1}$ , connected chaotic seas emerge, particularly around inclinations of $\sim 100^\circ$ and semi-major axes of $\sim 11,000$ km. These chaotic regions grow and develop filamentary structures along resonant lines as $A/m$ increases.
Orbital Lifetime Distribution: For $A/m \ge 5 m^2kg^{-1}$ , the number of reentry orbits ( $N_*$ ) within a given time window follows an inverse-square power law with respect to the reentry time ( $t_*$ ), i.e., $N_* \propto t_*^{-2}$ . This relationship holds across various initial eccentricities.
Effective Stability vs. Long-Term Chaos: A critical finding is the distinction between long-term stability and "effective" stability. While the system exhibits widespread chaos over a 100-year timescale, the vast majority of orbits that satisfy the 25-year reentry guideline (99.7% for $A/m=5$ and 99.8% for $A/m=10$ ) exhibit regular dynamics up to the moment of reentry. Chaotic behavior in these specific trajectories often manifests only after the critical eccentricity is reached or on timescales exceeding the mitigation window.
Fractality of Reentry Basins: The geometry of reentry basins is highly dependent on the chosen mitigation guideline.
- 5-Year Guideline: The basin boundaries are relatively smooth, with uncertainty exponents ( $\alpha$ ) close to 1 (e.g., $\alpha \approx 0.99$ ).
- 25-Year Guideline: The boundaries develop complex, fractal-like structures. The uncertainty exponent drops significantly (e.g., $\alpha \approx 0.393$ in the primary domain), indicating that tiny variations in initial conditions can drastically alter whether an orbit reenters within the 25-year window.
- Implication: Achieving a 25-year reentry prediction requires significantly higher precision in initial state knowledge compared to a 5-year target, as the fractal nature of the basin boundaries amplifies prediction errors.

Significance and Scope
The paper claims that this numerical dynamical study offers insights for developing "dynamically informed deorbiting strategies." By isolating the effects of $J_2$ and SRP without the complexity of atmospheric drag or higher-order geopotential terms, the model serves as a testbed for understanding resonant eccentricity growth mechanisms. The authors note that while the model omits drag, the resonant structures identified are robust and persist in higher-fidelity models, as evidenced by the model's ability to support the observed evolution of the historical Echo I satellite.

The study concludes that for high $A/m$ objects, natural reentry is feasible within short timescales (a few years) where motion remains regular, avoiding the unpredictability associated with fractal basin boundaries. However, as the allowed disposal timeframe extends (e.g., to 25 years), the system's sensitivity to initial conditions increases due to the emergence of fractal structures, making the robust design of reentry trajectories more challenging. The work suggests that from a dynamical perspective, stricter mitigation guidelines (shorter timeframes) may be advantageous for mission design and disposal strategies due to the smoother, more predictable nature of the corresponding reentry basins.

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