Numerical model of Phobos’ motion incorporating the effects of free rotation

¹ Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, PR China
e-mail: yang.yongzhang@ynao.ac.cn
² State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, Wuhan 430070, PR China
e-mail: jgyan@whu.edu.cn
³ Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, PR China
⁴ National Astronomical Observatory of Japan, 2-12 Hoshigaoka, Mizusawa, Oshu, Iwate 023-0861, Japan
⁵ Observatoire géodésique de Tahiti, BP 6570, 98702 Faa’a, Tahiti, French Polynesia, France
⁶ The Graduate University for Advanced Studies, SOKENDAI, Shonan Village, Hayama, Kanagawa 240-0193, Japan
⁷ Key Laboratory of Lunar and Deep Space Exploration, Chinese Academy of Sciences, Beijing 100012, PR China

Received: 12 September 2020
Accepted: 6 February 2024

Abstract

Context. High-precision ephemerides are not only useful in supporting space missions, but also in investigating the physical nature of celestial bodies. This paper reports an update to the orbit and rotation model of the Martian moon Phobos. In contrast to earlier numerical models, this paper details a dynamical model that fully considers the rotation of Phobos. Here, Phobos’ rotation is first described by Euler’s rotational equations and integrated simultaneously with the orbital motion equations. We discuss this dynamical model, along with the differences with respect to the model now in use.

Aims. This work is aimed at updating the physical model embedded in the ephemerides of Martian moons, considering improvements offered by exploiting high-precision observations expected from future missions (e.g., Japanese Martian Moons exploration, MMX), which fully supports future studies of the Martian moons.

Methods. The rotational motion of Phobos can be expressed by Euler’s rotational equations and integrated in parallel with the equations of the orbital motion of Phobos around Mars. In order to investigate the differences between the two models, we first reproduced and simulated the dynamical model that is now used in the ephemerides, but based on our own parameters. We then fit the model to the newest Phobos ephemeris published by Institut de Mécanique Céleste et de Calcul des Éphémérides (IMCCE). Based on our derived variational equations, the influence of the gravity field, the Love number, k₂, and the rotation behavior were studied by fitting the full model to the simulated simple model. Our revised dynamic model for Phobos was constructed as a general method that can be extended with appropriate corrections (mainly rotation) to systems other than Phobos, such as the Saturn and Jupiter systems.

Results. We present the variational equation for Phobos’ rotation employing the symbolic Maple computation software. The adjustment test simulations confirm the latitude libration of Phobos, suggesting gravity field coefficients obtained using a shape model and homogeneous density hypothesis should be re-examined in the future in the context of dynamics. Furthermore, the simulations with different k₂ values indicate that it is difficult to determine k₂ efficiently using the current data.