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Passive Sunlight-Synchronized Relative Motion for Geostationary Collocation

This paper proposes a passive sunlight-synchronized relative motion framework for geostationary collocation that optimizes lighting geometry by aligning orbital parameters and eccentricity/inclination vectors, enabling multiple spacecraft to maintain favorable illumination conditions without active control.

Original authors: Hong-Xin Shen, Zhi-Sheng Duan

Published 2026-09-03

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CC BY 4.0

Original authors: Hong-Xin Shen, Zhi-Sheng Duan

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, in a ring of orbit roughly 36,000 kilometers up, sits a crowded neighborhood of satellites known as the geostationary belt. These machines hover over the same spot on the planet, acting as the backbone for global communications, weather monitoring, and television broadcasting. Because this orbital real estate is finite and incredibly valuable, satellites often have to park very close to one another. To keep them from colliding, engineers give them slightly different paths, causing them to drift in slow, predictable loops around a central target satellite. This relative motion is essential for safety, but it creates a new problem for anyone trying to look closely at these machines. Whether inspecting a satellite for damage or preparing to service it, a camera needs good light. If the angle between the camera, the target, and the Sun is too wide, the target falls into shadow or appears as a dark, featureless silhouette. For years, keeping a satellite in a position where it is always well-lit for inspection required constant, fuel-burning thruster adjustments, a costly and temporary solution.

Researchers at Peking University have proposed a different approach that relies on the natural laws of physics rather than active fuel consumption. They developed a method called passive sunlight-synchronized relative motion, a way to design the path of an inspector satellite so that it naturally stays aligned with the Sun and its target for days at a time. The core idea is to match the speed and average position of the inspector satellite exactly with the target satellite. By doing this, the inspector does not drift away but instead traces a stable, repeating loop around the target. The researchers then carefully tuned the shape of this loop and its tilt relative to the Earth's equator. They found that by setting the loop's orientation to face the Sun directly, the inspector satellite would naturally maintain a sharp, clear view of the target's surface as it moved along its path. This alignment ensures that the Sun, the target, and the inspector remain in a nearly straight line, bathing the target in direct light without the need for any engine burns.

The study reveals that the most effective way to achieve this perfect lighting is to keep the inspector satellite in the exact same flat plane as the target. While it might seem intuitive to tilt the path to catch the light from different angles, the researchers demonstrated through their calculations that any tilt actually worsens the worst-case lighting conditions. The optimal design is completely flat, relying solely on the shape of the orbit to manage the light. When the Sun is directly over the Earth's equator, this flat path allows the lighting angle to stay between zero and about 19.5 degrees, a range that provides excellent visibility. Even when the Sun moves to its highest or lowest points in the sky during the year, the angle remains manageable, never exceeding roughly 30 degrees. This passive stability means that once the satellites are placed in their correct positions, they can perform long-duration optical inspections without using a single drop of fuel.

To make this system even more robust, the team showed how to place multiple inspector satellites on the same orbital loop, spaced out like runners on a track. By shifting their starting positions slightly, these satellites can take turns being in the perfect spot to view the target. When one satellite moves into a position where the light is slightly less ideal, another satellite in the formation is already moving into the optimal spot. This cooperative arrangement ensures that there is always at least one satellite with a clear, well-lit view of the target, effectively creating a continuous, fuel-free observation window. The researchers tested this concept using high-fidelity computer simulations that accounted for the complex gravitational pulls of the Earth, Moon, and Sun, as well as the pressure of sunlight itself. In these simulations, the satellites maintained safe distances from one another while keeping the target illuminated for three consecutive days, proving that the theory works even in the messy reality of space.

This work offers a practical solution for the growing need to monitor and service the crowded geostationary belt. By using the natural geometry of orbits to solve a lighting problem, engineers can extend the life of satellites and improve the quality of space situational awareness without the burden of carrying extra fuel. The findings suggest that future missions can rely on these passive, synchronized paths to keep their cameras focused and their targets bright, turning a complex orbital challenge into a simple, elegant alignment of celestial mechanics.

Technical Summary: Passive Sunlight-Synchronized Relative Motion for Geostationary Collocation

Problem Statement
The increasing density of space debris in the geostationary orbit (GEO) regime necessitates robust space situational awareness and on-orbit servicing capabilities. Optical observation is a primary enabling technology for these missions, yet its efficacy depends heavily on the "lighting angle"—the angle between the relative position vector of an inspector spacecraft and the Sun-target line. A small lighting angle ensures the target is fully illuminated, whereas large angles result in shadows or featureless outlines. Existing relative motion designs for proximity operations, such as hovering or fly-around maneuvers, typically rely on active thrusting to maintain specific geometries or focus on fuel minimization without explicitly addressing illumination constraints. Furthermore, strategies that do incorporate Sun-angle constraints often require continuous active control or numerical optimization, leading to increased propellant consumption and operational complexity. The core problem addressed is how to design a relative motion trajectory for a secondary (inspector) spacecraft that naturally maintains a small lighting angle relative to a primary (target) spacecraft over extended periods without requiring active control or propellant.

Methodology
The authors propose a framework termed Passive Sunlight-Synchronized Relative Motion (SSRM). The methodology is grounded in the kinematics of relative motion between two spacecraft in near-circular, near-equatorial GEOs, parameterized by nonsingular equinoctial elements.

Fundamental Constraints: The framework imposes two primary constraints: the semi-major axis difference ( $\Delta a$ ) and the mean longitude difference ( $\Delta l$ ) between the secondary and primary are set to zero. This eliminates secular drift and centers the relative motion on the primary spacecraft, resulting in a purely periodic trajectory.
Degrees of Freedom: With $\Delta a$ and $\Delta l$ fixed, the remaining degrees of freedom—the eccentricity difference vector ( $\Delta \mathbf{e}$ ) and the inclination difference vector ( $\Delta \mathbf{i}$ )—are utilized to control the lighting geometry.
Analytical Derivations:
- In-Plane Minimax Design: For scenarios where the Sun lies in the orbital plane (or for average performance), the authors derive an optimal phase condition for the eccentricity difference vector. By setting the phase $\varphi_e = \alpha_{\odot} + \pi$ (where $\alpha_{\odot}$ is the Sun's right ascension), the maximum lighting angle over one orbit is minimized.
- Out-of-Plane Analysis: The paper analytically demonstrates that introducing a non-zero relative inclination ( $\Delta i \neq 0$ ) generally worsens the worst-case lighting angle performance compared to a coplanar configuration. Thus, the optimal average design is strictly coplanar ( $\Delta i = 0$ ).
- Instantaneous Zero-Angle Design: For specific epochs requiring perfect illumination (e.g., the start of an inspection), the authors derive closed-form conditions using a deliberate out-of-plane bias ( $\Delta i \neq 0$ ) and specific phase adjustments to achieve an instantaneous lighting angle of zero.
- Phase-Offset Collocation: A method is developed allowing multiple secondary spacecraft to share the same relative elliptical trajectory by assigning different initial phases ( $\varphi_e$ ). This enables cooperative coverage where multiple satellites maintain favorable lighting conditions simultaneously without active coordination.

Key Contributions
The paper presents two primary contributions:

Concept of Passive SSRM: The introduction of a natural, periodic relative motion framework that requires no active control or propellant to maintain favorable illumination. This contrasts with existing methods that depend on frequent maneuvers or active thrusting to satisfy Sun-angle constraints.
Closed-Form Analytical Conditions: The derivation of explicit formulas that fully characterize the SSRM framework, eliminating the need for iterative numerical optimization. These include:
- The optimal in-plane design condition ( $\varphi_e = \alpha_{\odot} + \pi$ ) which minimizes the maximum lighting angle to a theoretical bound of approximately $19.47^\circ$ .
- The proof that the optimal average design is coplanar ( $\Delta i = 0$ ), regardless of solar declination.
- Exact conditions for achieving an instantaneous zero lighting angle at a chosen epoch for non-zero solar declination.
- A phase-offset collocation theorem establishing that multiple spacecraft can share the same relative trajectory shape by varying only their initial phase.

Results
Numerical simulations were conducted using a high-fidelity orbital dynamics model (including Earth's non-spherical gravity, third-body perturbations, and solar radiation pressure) over three-day periods.

Coplanar Configuration (Equinox): In a scenario with near-zero solar declination, a single secondary spacecraft designed with the optimal in-plane condition maintained a lighting angle oscillating between $0^\circ$ and $\approx 19.5^\circ$ . When a phase-offset configuration of three spacecraft was employed (with $\pm 20^\circ$ phase offsets), the formation maintained a lighting angle below $10^\circ$ for almost the entire duration, demonstrating that cooperative phasing significantly improves coverage.
Out-of-Plane Configuration (Solstice): For a winter solstice scenario with a large solar declination ( $\approx -23.4^\circ$ ), a purely coplanar design resulted in a maximum lighting angle of $\approx 30.2^\circ$ . By combining one coplanar spacecraft with two additional spacecraft designed for instantaneous zero-lighting angles at different epochs (offset by 12 hours), the formation successfully maintained a maximum lighting angle below $20^\circ$ for the majority of the mission. The simulations confirmed that inter-satellite distances remained above safety thresholds (5 km) throughout the simulation.

Significance
The paper claims that the proposed SSRM framework offers a passive, fuel-free approach for optical inspection and space situational awareness in the geostationary ring. By leveraging natural dynamical structures rather than active control, the method reduces operational complexity and propellant consumption. The analytical results provide mission designers with explicit tools to configure multi-spacecraft formations that can sustain favorable illumination conditions over extended periods, even under varying solar declinations, without the need for continuous thrusting or complex numerical optimization. This capability is particularly significant for the economically valuable and crowded GEO regime, where robust, long-duration observation is critical.

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