New analytical model revolutionizes spacecraft orbit prediction under third-body gravitational effects

As the complexity of space missions—such as Earth–Moon transport and deep-space exploration—continues to increase, the classical Keplerian two-body model can no longer meet the precision requirements of modern trajectory design. In addition to the dominant gravitational attraction of the central body, spacecraft motion is also affected by the gravitational forces of other celestial bodies, such as the Moon or the Sun.

These third-body perturbations cause long-period oscillations in orbital eccentricity and inclination, especially near the critical inclination of 39.23°, leading to nonlinear behaviors known as Lidov–Kozai resonance. This makes orbit prediction and control far more complex than traditional perturbation models can handle.

Developing efficient analytical propagation models for such scenarios is crucial for the next generation of autonomous spacecraft. These models can provide fast, accurate orbit predictions for mission planning and real-time onboard control, significantly reducing computational costs while maintaining high fidelity.

This paper, published in the Chinese Journal of Aeronautics, proposes a new analytical propagation method for spacecraft orbits perturbed by a third body. The method decomposes orbital evolution into long-term and periodic components:

1. Long-term variations are solved analytically using elliptic integrals;
2. Periodic corrections are introduced via von Zeipel transformation, linking instantaneous orbital elements with averaged dynamics.

This hybrid framework enables step-size–independent analytical computation of orbital states at any epoch, overcoming a core limitation of numerical integration.

To validate the method, the researchers simulated lunar orbits under the influence of a 50×50 lunar gravity field, Earth–Moon–Sun perturbations, and solar radiation pressure. Compared with a fourth-order Runge–Kutta numerical benchmark, the analytical model achieved remarkable precision: For a circulating lunar orbit (inclination = 20°), the maximum position error was 26.28 km, and velocity error 0.003 3 km/s. For a librating orbit (inclination = 80°), errors were 39.39 km and 0.005 1 km/s, respectively. This demonstrates that the analytical model retains high accuracy while greatly reducing computation time—making it practical for onboard or real-time applications.

The new approach enables accurate orbit prediction at any epoch without dependence on integration step size. It supports both circulating and librating orbital configurations and accounts for the inclination and eccentricity of the perturbing body.

Such analytical efficiency is particularly valuable for deep-space exploration, cislunar transport, and satellite constellation maintenance, where onboard computational resources are limited. By providing a universal, physics-consistent analytical framework, this research lays the foundation for future autonomous orbit control and trajectory optimization in complex gravitational environments.

Original Source

Tao NIE, Jinfeng LI, Shijie ZHANG, Jiadong REN, Rui XU. Analytical propagation for third-body perturbed orbits [J]. Chinese Journal of Aeronautics, 2025, https://doi.org/10.1016/j.cja.2025.103865.

About Chinese Journal of Aeronautics 

Chinese Journal of Aeronautics (CJA) is an open access, peer-reviewed international journal covering all aspects of aerospace engineering, monthly published by Elsevier. The Journal reports the scientific and technological achievements and frontiers in aeronautic engineering and astronautic engineering, in both theory and practice. CJA is indexed in SCI (IF = 5.7, Q1), EI, IAA, AJ, CSA, Scopus.