# Circular Orbit Flip Trajectories Generated by E-Sail
## 1 Introduction  
A plane change maneuver, i.e., a maneuver that changes the orbital inclination of a Keplerian orbit while keeping the semimajor axis and eccentricity (i.e., both its shape and size) unchanged [1], is usually considered to be one of the most demanding maneuvers in astrodynamics from the standpoint of velocity change and, therefore, propellant expenditure [2]. A possible solution to perform such an orbital maneuver is given by the use of a propellantless propulsion system, such as a (photonic) solar sail [3; 4; 5] or an Electric Solar Wind Sail (E-sail) [6], which exploit either the solar radiation pressure (solar sail case) or the solar wind momentum flux (E-sail case) to generate a propulsive acceleration. The solar sail capabilities in a plane change maneuver of a circular heliocentric orbit have been analyzed by the authors [7] as a special case of a more general orbit-to-orbit interplanetary transfer between two mutually inclined circular trajectories. A plane change maneuver with an E-sail-based spacecraft has not yet been investigated and is not even the goal of this work because, in this regard, a detailed study is left for future research. Instead, the aim of this paper is to analyze a more specific mission scenario in which the orbital inclination of a given heliocentric Keplerian orbit is changed by exactly 180 degrees. In other terms, this particular plane change maneuver allows the spacecraft to invert its direction of rotation around the Sun, thus obtaining a sort of artificial (thrust-induced) 'orbit flip mechanism' [8; 9; 10; 11], which preserves the shape and the dimension of the original trajectory. More precisely, the focus of this study is on the analysis of the orbit flip mechanism of a circular heliocentric orbit of assigned radius, by means of a two-dimensional propelled trajectory coplanar to the initial circular orbit. Such a COFT is obtained by using the propulsive acceleration given by an E-sail propulsion system. An E-sail is a propellantless thruster concept proposed by [PERSON] [12; 13; 14; 15] at the beginning of this century. Such a propulsion system uses a number of conducting tethers [16; 17; 18] to deflect the solar wind and obtain a thrust in the interplanetaryspace [18; 19; 20]. The E-sail thrust vector can be steered [21; 22; 23] by suitably changing the sail attitude with respect to the solar wind stream direction [24; 25]. The main characteristics of the E-sail concept are thoroughly illustrated in a recent review paper [6], while an interesting collection of papers on E-sail mission applications is reported in a special issue of the companion Aerospace journal (The Aerospace special issue named \"Advances in CubeSat Sails and Tethers\" can be found at the URL [[https://www.mdpi.com/journal/aerospace/special_issues/2319](https://www.mdpi.com/journal/aerospace/special_issues/2319) OV36 DR]([https://www.mdpi.com/journal/aerospace/special_issues/2319](https://www.mdpi.com/journal/aerospace/special_issues/2319) OV36 DR), accessed on 23 August 2023). The new features of this work are essentially two. The first one concerns the study of the COFT from an optimal point of view, using a general mathematical model that can be employed in a wide range of mission scenarios. In fact, the proposed model provides a set of graphs that allow the reader to quickly estimate the E-sail performance in a COFT without the need of numerical simulations. The optimization process also points out that a generic COFT can be thought of as a special case of Vulpetti H-reversal trajectory, the main features of which are illustrated in the review by [PERSON] et al. [26]. The second and most important contribution of this work lies in a particular shape of the COFTs, which has been identified during the numerical study of the trajectories. In fact, the optimization process has shown that there exists a family of COFTs that presents a close passage near the Sun and a zero magnitude of the inertial velocity in correspondence of the aphelion point.