# Continuous-Thrust Circular Orbit Phasing Optimization of Deep Space CubeSats
## 1 Introduction  
Since the pioneering and successful mission of NASA's Mars Cube One (MarCO) [1; 2], which in 2018 provided an in situ communication relay to NASA's InSight lander [3; 4; 5] during its descent phase through the atmosphere of the Red Planet, CubeSats have demonstrated their capabilities in supporting future scientific missions beyond near-Earth space [6]. In this context, the scientific outcomes obtained by the panoramic pictures of the impact between NASA's DART spacecraft and asteroid Dimorphos [7; 8; 9], which have been provided by the Light Italian CubeSat for Imaging of Asteroids (LICIACube) of the Italian Space Agency [10], are probably the clearest proof of the importance and the potentialities of CubeSats in the design of complex and ambitious interplanetary missions [11]. The near future hopefully will see another two CubeSats investigating the binary asteroid 65803 Didymos after the impact of the DART spacecraft, thanks to the ESA's Hera mission whose launch date is planned in October 2024 [12; 13]. In particular, Hera spacecraft will carry the two CubeSats named Milani and Juventas, the first one dedicated to the memory of Prof. [PERSON] from the University of Pisa, which are designed to obtain the geophysical characterization and the global mapping of the asteroid by operating in close proximity to this celestial body [14; 15; 16]. To date, CubeSats used in interplanetary missions have been designed to be transported inside a sort of main, mother, spacecraft and then released near the target celestial body. This is the case, for example, of the two pioneering (twin) CubeSats MarCO-A and MarCO-B, the more recent LICIACube, and the upcoming Milani and Juventas. The next step in the practical employment of CubeSats in deep space scientific missions is, therefore, the use of an own propulsion system to perform the entire (or a consistent part of the) interplanetary transfer required to reach the target solar system's body or the target heliocentric orbit. In this context, the proposed ESA's The Miniaturised Asteroid Remote Geophysical Observer (M-ARGO) mission can be considered a milestone in the use of CubeSats for the solar system's scientific exploration [17; 18]. In fact, the M-ARGO demonstration mission, whose launch is currently scheduled for 2026, will use the electric propulsion system installed on board a 12U CubeSat (parked at the Sun-Earth collinear Lagrangian point \(L_{2}\)) to trace a heliocentric trajectory that will achieve a rendezvous with a small and rapidly spinning (near-Earth) asteroid. The latter has yet to be selected in a set of recently identified targets as discussed in the interesting work by [PERSON] et al. [19]. The (continuous-thrust) miniaturized electric thruster (i.e., a single radiofrequency grided ion thruster) is one of the CubeSat advanced subsystems [20] as, for example, a multispectral camera and a laser altimeter, which will be used to obtain the in situ observation of the target celestial body, or the flat-panel antenna (also called \"reflectarray\" [21; 22; 23]) for deep space communications. Figure 1 shows an artistic concept of the CubeSat M-ARGO approaching a potential target asteroid. Starting from the procedure described by [PERSON] et al. [18], which is detailed at the beginning of Section 2, this paper uses the surrogate model for the propulsive performance of the M-ARGO electric thruster in order to develop an approximated (and simplified) analytical thrust model, which can be used to optimize the heliocentric transfer trajectory of a M-ARGO-type CubeSat (MTC), when the classical calculus of variations is employed to solve the optimization problem. In particular, as will be discussed in Section 3.1, the proposed (analytical) thrust model is used to obtain a closed form of the Euler-Lagrange equations which are employed, together with the optimal control law obtained with the aid of the well-known Pontryagin's maximum principle, to simulate the optimal performance of a MTC in a pair of typical two-dimensional heliocentric mission scenarios.