# Dynamics and Control of Typical Orbits around Saturn
## 1 Introduction
[22] studied a semi-analytical acquisition algorithm for repeating ground track orbits' maintenance. Studies analyzing the dynamics around other terrestrial planets in the solar system had developed rapidly as well. [PERSON] et al. [23] analyzed five typical orbits around Mars by using the zonal harmonic coefficients \(J_{2}\), \(J_{4}\) and a tesseral harmonic coefficient \(J_{22}\). In addition, [PERSON] et al. [24] further studied the recursive orbits around stationary points of Martian gravitational field. [PERSON] et al. [25] designed recursive sun-synchronous orbits of \(J_{2}\) predominant planets, including Mars. [PERSON] et al. [26] and [PERSON] et al. [27] studied distant quasi-periodic orbits and artificial frozen orbits around Mercury. Actually, these dynamic models are not suitable for Saturn, as Saturn is not a \(J_{2}\) predominant planet. At the same time, the tesseral harmonic coefficient of Saturn \(J_{22}\approx 0\). Though this research on terrestrial planets is inspiring, the more relevant studies are the dynamic analysis around gaseous planets. [PERSON] et al. [28] analyzed the dynamics of several typical orbits around Jupiter. [PERSON] et al. [29] studied the initial inclination prebiased method and semimajor axis compensation strategy for sun-synchronous repeating ground track orbits around Jupiter. [PERSON] et al. [29]'s control strategy only considered \(J_{2}\) term, while Jupiter is not a \(J_{2}\) predominant planet. Though his control strategy is instructive, we still insist that the combination of \(J_{2}\) and \(J_{4}\) is necessary for gaseous planets, especially for non predominant \(J_{2}\) celestial bodies. In this paper, we analyze the dynamics of some typical orbits around Saturn using the aspheric gravitational model considering \(J_{2}\) and \(J_{4}\) terms. Then we design the sun-synchronous orbits, repeating ground track orbits, frozen orbits, and stationary orbits around Saturn based on the mean element theory [15; 16]. While analyzing the perturbations, we divide them into three types of terms: secular terms, long-period terms, and short-period terms [15; 16]. After that, we mainly consider secular perturbations caused by atmospheric drag, solar gravitation, and solar radiation pressure. As for sun-synchronous orbits, we first analyze the local time drift at the descending node caused by atmospheric drag and solar gravitation. Then, we take two methods, including initial inclination biased method and periodic inclination biased method, to damp the local time drift. The main perturbation we have analyzed for repeating ground track orbits is atmospheric drag. Afterwe have chosen a reasonable repetition parameter, we could calculate the decay rate of orbital semimajor axis with an initial condition of semimajor axis and eccentricity. To maintain the recursive feature, we give a solution of semimajor axis compensation and control period. When we use \(J_{2}\) and \(J_{4}\) terms to design frozen orbits around Saturn, we naturally see a novel result that the argument of perigee of frozen orbits around Saturn must be 270\({}^{\circ}\) to make eccentricity meaningful. At last, we give some strategies to offset the influence on inclination caused by the solar gravitation and the influence on eccentricity caused by solar radiation pressure. In this paper, we use some traditional symbols to denote the basic parameters of Saturn and the orbital elements of spacecraft motion. \(a\) is the semi-major axis of orbit. \(e\) is the orbital eccentricity. \(i\) is the orbital inclination. \(J_{2}\) is the second-order zonal harmonic. \(J_{3}\) is the third-order zonal harmonic. \(J_{4}\) is the fourth-order zonal harmonic. \(J_{22}\) is the second-degree and -order tesseral harmonic. \(M\) is the mean anomaly. \(f\) is the true anomaly. \(n\) is the mean angular velocity. \(n_{s}\) is the mean angular velocity of Saturn around the Sun. \(p\) is the semiparameter. \(R_{e}\) is the reference equatorial radius of Saturn. \(r\) is the position of the spacecraft.