# Optimal Trajectories of Diffractive Sail to Highly Inclined Heliocentric Orbits
## 1 Introduction  
Observing the Sun's polar regions is a goal of great scientific interest and a necessary step towards understanding some fundamental physical phenomena and answering not yet clear questions concerning, for example, the creation of the solar wind, the release and evolution of coronal mass ejections, and the creation and control of the heliosphere. Another objective of great importance is to obtain detailed magnetic maps of the polar regions to improve understanding of the Sun's magnetic dynamo and to obtain new information on the fast solar wind originating from polar coronal holes [1, 2]. Unfortunately, the Sun's polar regions are not visible from Earth or from spacecraft moving on orbits in the plane of the Ecliptic or at a small inclination to it. For this reason, a typical requirement of missions designed to obtain images of the Sun's polar regions is to achieve a circular orbit of a sufficiently small radius, typically less than or equal to 0.48 au, and an inclination of at least 60 deg, so as to ensure a sufficiently long residence time at high solar latitudes during which observations of the poles can be made. Reaching such high inclinations relative to the Ecliptic, however, requires extremely large \(\Delta V\)s, which are impossible to achieve with today's launch vehicles. A mission based on the use of chemical thrusters therefore needs the aid of a planetary gravity assist in order to achieve the high orbital inclinations required [3]. This, for example, was the choice made in the case of theESA-NASA Ulysses spacecraft, which, launched in 1990, achieved an inclination of 80.2 deg with respect to the Ecliptic via a Jupiter gravity-assist maneuver. The final heliocentric orbit, however, was not able to get very close to the Sun, having a perihelion located at a distance of 1.3 au [4]. February 2020 saw the launch of the Solar Orbiter, a mission involving an international collaboration between ESA and NASA to explore the Sun's polar regions and the heliosphere from close range. The probe uses chemical propulsion and exploits seven Venus gravity-assist maneuvers to decrease the perihelion distance to 0.28 au and increase the orbital inclination [5]. The final orbit has a maximum heliolatitude of 18 deg in the nominal mission and 33 deg in the extended mission, which, taking into account the axial tilt of the Sun polar axis, corresponds to an inclination of approximately 40 deg on the Ecliptic. Due to the complications and long mission times caused by trajectories with multiple gravity-assist maneuvers, possible alternatives to the use of chemical propulsion have been proposed, as, for example, electric thrusters [6] or more advanced propellantless propulsion systems such as solar sails [7; 8]. In fact, solar sails were suggested as a means of propulsion in the technical feasibility study for a Solar Polar mission conducted by JPL in 1996 [9] and have subsequently been considered in various mission analyses and optimal trajectory analyses [10; 11; 12]. In a recent conceptual study of the Solar Polar Imager mission, concluded by NASA's Advanced Concepts Office, a square solar reflective sail with a side of 83.6 m was designed to achieve a circular orbit around the Sun of radius 0.48 au and an inclination of 75 deg [13]. The estimated total mission time was 13 years, including 3 years of observations and in situ measurements. Optical diffraction is another physical phenomenon that allows momentum transfer between photons from the Sun and the surface of a solar sail. This phenomenon is the basis of the concept of diffractive sail, recently introduced by [PERSON] [14; 15; 16; 17]. A distinctive feature of a diffractive sail, which uses a metamaterial-based membrane to diffract the incoming solar rays, is its ability to generate an steerable thrust vector even in a Sun-facing condition, that is, when the nominal plane of the sail is normal to the Sun-spacecraft line.