Revista Mexicana de Astronomía y Astrofísica, 55, 39–54 (2019) OPTIMAL TRAJECTORIES TO KUIPER BELT OBJECTS D. M. Sanchez1, A. A. Sukhanov1,2, and A. F. B. A. Prado1 Received June 22 2018; accepted November 14 2018 ABSTRACT The present paper searches for transfers from the Earth to three of the Kuiper Belt Objects (KBO): Haumea, Makemake, and Quaoar. These trajectories are obtained considering different possibilities of intermediate planet gravity assists. The model is based on the “patched-conics” approach. The best trajectories are found by searching for the minimum total ∆V transfer for a given launch window, inside the 2023-2034 interval, and disregarding the ∆V required for the capture at the target object. The results show transfers with duration below 20 years that spend a total ∆V under 10 km/s. There is also one trajectory for each of the KBOs with ∆V under 10 km/s and duration below 10 years, using the Jupiter swingby. For the 20-year trajectories, there are also asteroids in the main belt that could be encountered with low additional ∆V , so increasing the scientific return of the mission. RESUMEN Se buscan trayectorias de transferencia entre la Tierra y tres objetos del Cin- turón de Kuiper (KBO): Haumea, Makemake y Quaoar. Las trayectorias se obienen considerando distintas posibilidades para la influencia gravitatoria de los planetas intermedios. El modelo se basa en el enfoque de “cónicas empalmadas”. Se encuen- tran las mejores trayectorias buscando la transferencia con una ∆V total mínima, para una ventana de lanzamiento en el intervalo 2023-2034, y despreciando la ∆V necesaria para la captura en la meta. Se encuentran transferencias con duración de menos de 20 años que requieren una ∆V menor que 10 km/s. También se encuentra una trayectoria para cada uno de los objetos KB con ∆V menor que 10 km/s y duración de menos de 10 años, empleando la atracción de Júpiter. Las trayecto- rias de 20 años podrían usarse también para encuentros con asteroides del cinturón central, lo cual aumentaría el valor científico de la misión. Key Words: methods: numerical — Kuiper belt objects: individual: Haumea — Kuiper belt objects: individual: Makemake — Kuiper belt objects: individual: Quaoar — space vehicles 1. INTRODUCTION However, with the advances of observational tech- © Copyright 2019: Instituto de Astronomía, Universidad Nacional Autónoma México The exploration of the Kuiper Belt Objects niques, it was discovered that several of these bod- (KBOs) is an important step to improve the theo- ies are orbited by one or more small moons, like ries of the formation of the Solar System, since these Haumea, Makemake, and Quaoar3. Another inter- bodies probably preserved material from the earlier esting point is that the New Horizons spacecraft, Solar System (Luu & Jewitt 2002). Furthermore, when passing by Pluto, discovered that Pluto has due to the large distance to these bodies from the signs of recent surface activity (less than 10 million Sun, the development of new technologies and tech- years old) (Moore et al. 2016). This fact can be niques for their exploration is required. an indication that other KBOs with sizes compa- A few years ago these objects were thought to be rable to Pluto could also present recent geological single bodies with no atmosphere, except for Pluto. activity. One example of this possibility is the po- 1 tential presence of cryovolcanism in Quaoar (Barucci National Institute for Space Research, INPE, Brazil. 2Space Research Institute of the Russian Academy of Sci- ences, IKI, Russia. 3http://www.cbat.eps.harvard.edu/minorsats.html 39 40 SANCHEZ, SUKHANOV, & PRADO et al. 2015). However, due to the large distance of during the flight.4 This was done considering var- the KBOs from the Sun, observational data were not ious possible transfer schemes for the given launch enough to significantly improve our knowledge about windows assuming that the dwarf planets will be these bodies, to carry out comparative planetology. flown by without capture. If capture is required for Spacecraft missions to one or more of these bod- these specific trajectories, a new optimization should ies beyond Pluto are necessary to provide us with be made taking into account the capture maneu- more detailed features of the KBOs. The present ver. Most of the transfer schemes considered include paper searches for optimal trajectories to (136108) various gravity assists of the planets, which lowers Haumea, (136472) Makemake, and (50000) Quaoar. the fuel consumption and/or shortens the time of These three bodies were chosen as targets because flight. Furthermore, multi-body missions are more they are good representatives of the KBOs and they interesting from the scientific point of view, since can also be classified as Trans-Neptunian Objects scientific data could be acquired during the flybys. (TNOs). Haumea and Makemake were recognized However, since the limited time of flight is consid- by the International Astronomical Union (IAU) as ered, the number of bodies in a single trajectory dwarf planets, but Quaoar is just a candidate to this may be limited. Thus, the number of the planets classification. Haumea is probably the most intrigu- was maximized for trajectories with 20 years of to- ing of the TNOs, because it is a triaxial ellipsoid tal time of flight, leaving a smaller number of plan- with fast rotation (3.9154 h), possesses two moons, ets for trajectories with total time of flight less than Namaka and Hi’iaka, and a recently discovered ring 20 years. The main difficulty in the design of these (Ortiz et al. 2017). The thin layer of carbon depleted trajectories is to find feasible combinations of plan- ice that surrounds the rocky core of Haumea is an- ets, within the proposed time interval, in a realistic other important characteristic of this dwarf planet time of flight. Some combinations are not always (Pinilla-Alonso et al. 2009). possible. For example, the combination used by the mission of the Voyager 2 (Kohlhase & Penzo 1977), Makemake is the third largest TNO, after Pluto Earth-Jupiter-Saturn-Uranus-Neptune-Outer space, and Eris. It also has a recently discovered moon, cannot be reproduced in this century anymore. This which has no official name yet, and its general desig- technique was widely used in missions to the outer nation is “S/2015 (136472) 1” (Parker et al. 2016). planets and outer space, such as in the Voyager 1 Quaoar also has a small moon, named Weywot. (Kohlhase & Penzo 1977), with the scheme E-J-S- This moon seems to be in an eccentric orbit. Like Outer space, Galileo (D’Amario et al. 1982), with Haumea, Quaoar has a rocky core covered by a thin the scheme E-V-E-E-A-A-J, Ulysses (Wenzel et al. layer of ice, but differently from Haumea, Quaoar 1992), with the scheme E-A-DV-E-A, Cassini (Per- has a high density and probably its core is en- alta & Flanagan 1995), with the scheme E-V-V-E-A- tirely formed by silicate material (Fraser et al. 2013). J-S, and the New Horizons (Guo & Farquhar 2005), Then, Quaoar probably is the densest TNO, which with the scheme E-A-J-P-Kuiper Belt, where E, V, makes it a good target for spacecraft exploration. J, S, U, N, P, A stand for Earth, Venus, Jupiter, Table 1 presents some orbital and physical charac- Saturn, Uranus, Neptune, and anăasteroid, respec- teristics of Haumea (Ragozzine & Brown 2009; Ortiz tively.ăDV means a deep space propulsive maneu- et al. 2017), Makemake (Brown 2013; Parker et al. ver. All of these schemes, except E-V-V-E, which 2016), and Quaoar (Fraser et al. 2013). The val- areăpossible for launch in 2023-2034 areăanalyzed in ues of the semi-major axes (a), eccentricities (e) and this paper. Also transfer schemes E-V-E-E-S andăE- © Copyright 2019: Instituto de Astronomía, Universidad Nacional Autónoma México inclinations (I) are approximate. The inclinations V-E-DV-E-S are considered. are given with respect to the ecliptic plane. Table 1 The method of patched-conics is a well-known also presents the masses (m), densities (ρ), and the technique and was used for the planning of several moons of these bodies. interplanetary missions (Kohlhase & Penzo 1977; D’Amario et al. 1982; Sukhanov 1999; Strange & We analyzed the dates of launch in the 2023- Longusky 2002; Solórzano et al. 2008). The descrip- 2034 interval. The only exception was direct flight to tion of the method can be found in several publi- Haumea with launch in 2058. The optimal transfer trajectories in terms of the minimum fuel consump- 4 tion were found. Instead of the fuel consumption In fact this is not quite correct because the launch and an equivalent parameter was considered, namely the deep space maneuvers will be made by different engines and ∆ ∆ cannot be simply summed. Although since the characteristics total V , which is the sum of the launch V in of the engines are not known in advance the sum of the ∆V s the low Earth parking orbit and all necessary ∆V s is the only way to estimate optimality of the transfers. OPTIMAL TRAJECTORIES TO KUIPER BELT OBJECTS 41 TABLE 1 SOME PHYSICAL AND ORBITAL CHARACTERISTICS OF HAUMEA, MAKEMAKE, AND QUAOAR Body m (kg) ρ (g/cm3) a (AU) e I (deg) Moon(s) Haumea 4.006 × 1021 1.885 43 0.19 28.2 Namaka, Hi’iaka Makemake < 4.4 × 1021 1.4 – 3.2 46 0.15 29.0 S/2015 (136472) 1 Quaoar 1.3 – 1.5×1021 4.2 44 0.04 8.0 Weywot cations, like Escobal (1968). The transfer trajecto- planets are considered, which increases the space- ries shown here can be useful for future missions to craft velocity and decreases the launch ∆V .