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.

Analytical expression for distant retrograde orbits around a small natural satellite - Advances in Engineering
Geometry of DRO in the Mars–Deimos system is plotted in this figure, on which the numerical, analytical, and semi-analytical solutions are compared. In the analytical method, the orbit is calculated using the approximation that the variation of angular velocity of DRO is small, whereas in the semi-analytical method, some variables are derived by Newton’s method.

About the author

Masaya Kimura received his B.S. and M.S. degrees in aerospace engineering from Nagoya University in 2009 and 2011, respectively, and earned his Ph.D. in engineering from Osaka University in 2019. He has been a research engineer at the Advanced Technology R&D Center at Mitsubishi Electric Corporation since 2011 where he is engaged in the research and development of industrial automation products such as servo systems and programmable controllers. He is working to improve the accuracy, productivity, and operation of industrial machinery.

His research interests include orbital dynamics and control of spacecraft, factory automation, industrial application of machine learning technologies, enhancement of user interface usability of products, and high-performance edge-level computing.

About the author

Masanori Kawamura obtained his Master of Science in Mechanical Engineering from Osaka University, Japan in 2017. He is a researcher at Advanced Technology R&D Center of Mitsubishi Electric Corporation, where he is engaged in the development and application of Satellite Attitude and Orbit Control System (AOCS) and automotive navigation system.

In AOCS, he is working to improve the control accuracy and fuel efficiency for spacecraft. As for automotive navigation system, he is developing a robust state estimation algorithm to enhance the accuracy of the on-board sensors. His areas of interest are orbital dynamics and control of spacecraft, and state estimation such as Kalman Filter.

About the author

Katsuhiko Yamada received the B.S., M.S., and Dr.Eng. degree from the University of Tokyo, Tokyo, Japan, in 1978, 1980, and 1989, respectively. In 1980, he joined Central Research Laboratory, Mitsubishi Electric Corporation, Japan, where he worked for the dynamics analysis and controller design of spacecraft. In 2005, he was the professor of aerospace engineering of Nagoya University, Nagoya, Japan. From 2013 to present, he is the professor of mechanical engineering of Osaka University, Osaka, Japan.

His research interests include modeling, stability analysis, optimization, controller design of spacecraft system.


Masaya Kimura, Masanori Kawamura, Katsuhiko Yamada. Analytical expression for distant retrograde orbits around a small natural satellite. Advances in Space Research, volume 63 (2019) page 1336–1346.

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