Analytical expression for distant retrograde orbits around a small natural satellite

A distant retrograde orbit (DRO) is a periodic orbit in the circular restricted three-body problem that, in the rotating frame, looks like a large quasi-elliptical retrograde orbit around the secondary body. Generally, a DRO is a highly stable lunar orbit. The recent surge in space exploration activities has sparked immense attention from both researchers and engineers, particularly due to the inherent design properties that may aid in spacecraft mission design. Majority of DRO related studies are mainly numerical. The stability of planar DROs has also been shown numerically, however, they are yet to be analytically expressed. This is a drawback since both the numerical integration and iterative calculations are vital when obtaining the geometry of planar DROs.

A fundamental step has recently been reported where the planar DROs have been expressed in polar coordinates and approximated by a truncated Fourier series. This approach has proven its convenience in recent space mission designs. Nonetheless, orbital data of the position and velocity are needed to estimate the Fourier series coefficients. Practically, a simple method for deriving planar DROs for each Jacobi constant is required. Unfortunately, no published study has focused on addressing the issue.

Recently, Mitsubishi Electric Corporation scientists Dr. Masaya Kimura, Mr. Masanori Kawamura and led by Professor Katsuhiko Yamada at Osaka University developed a new method to analytically obtain the geometries of planar DROs corresponding to each Jacobi constant. The team searched for an analytical approximation for planar DROs in the Hill’s three-body problem, after which the planar DRO around Deimos was analyzed as an example. Their work is currently published in the research journal, Advances in Space Research.

The research team first derived the equations of motions under the assumption of Hill’s problem. Owing to the fact that a planar distant retrograde orbit is a closed orbit and can be expressed as an approximately elliptical orbit, the scientists analytically calculated the respective geometries and periods of DROs. They then determined the switching point, where various properties of planar DROs change abruptly with an increase in the orbital radius. Finally, the Mars–Deimos system was considered as a case study.

The authors confirmed that the geometries and periods of DROs could be expressed analytically by the proposed method and the results of the same were in good agreement with the results of the numerical calculations. In addition, they confirmed that the switching point was near the Lagrange point of the Hill’s three-body system.

In summary, the study presented an analytical expression of DROs. Remarkably, using the presented method geometries and periods of DROs were easily calculated, whereas the numerical integration and iterative computation were heretofore required. Altogether, the proposed approach can be applied to other cases where the mass of a second celestial body is smaller than the mass of the main celestial body.