Multi-Target Low-Thrust Interplanetary Trajectory Design of DESTINY
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a feasible trajectory. This is particularly more challenging for the PGC members, as their highly eccentric and inclined orbits would impose very narrow time windows of flyby opportuni- ties with very high relative velocities. In DESTINY+’s case, this is combined with low relative velocities at Earth escape and limited thrust capability of ion engines. Additionally, the spacecraft design imposes constraints on the trajectory, making the interplanetary trajectory design of DESTINY+ challenging altogether. The interplanetary trajectory design of DESTINY+ was pre- viously investigated by Sarli et al.10, 11) Sarli et al. (2015) presented ballistic transfer opportunities by solving Lambert’s problem and applying gravity-assist maneuvers by Earth and Mars for multiple asteroid visits to the PGC.10) In a more recent work, Sarli et al. (2017) computed low-thrust optimal trajecto- ries for Earth-Phaethon transfers only and presented potential targets for mission extension regardless of their relation to the PGC.11) (a) xy-plane Building on these works, the present study explores the opti- mal transfers between Earth and the PGC members by consid- ering low escape energy, limited thrust and mission constraints of DESTINY+. A large database of ballistic transfers for Earth-Phaethon (E1-P2), Phaethon-Earth (P2-E3) and Earth- Secondary Asteroid (2005UD, 1999YC) are generated. Can- didate low-thrust solutions for full mission sequence are iden- tified with a simple “minimum ∆V” criterion and C3-matching algorithm. Transfer opportunities are expanded as compared to Sarli et al. (2015) by the addition of low-thrust propulsion. The chosen initial conditions are then inputted to GALLOP in order to generate optimal solutions. Robustness of a selected trajec- tory is assessed in a sensitivity study. The paper is structured as follows: In Sec. 2., the Phaethon- Geminid Complex is briefly introduced. In Sec. 3., initial guess generation is explained in more detail. In Sec. 4., optimal so- lutions are presented. Finally, conclusions are presented in Sec. 5.. (b) xz-plane Fig. 1.: Orbits of the target bodies 2. Phaethon-Geminid Complex The main scientific target DESTINY+, 3200 Phaethon (for- impose large ∆Vs for a rendezvous mission, therefore only fly- merly known as 1983TB), is an Apollo asteroid with cometary bys are possible. Due to ion engine of DESTINY+, the velocity features.12) It is also the source of Geminids meteroid shower, change that can be given to the spacecraft is limited. This paper which occurs mid-December every year. It was also identified therefore investigates the low-∆V transfers to Phaethon and the to be a potentially hazardous asteroid due to its minimum in- other target bodies. This begins with selection of suitable initial tersection distance with Earth. Extended observations since its conditions that are presented in the next section. discovery allowed researchers to constrain its orbit and proper- ties.13) According to this, Phaethon is 5.1 km ±0.2 km aster- Table 1.: Orbital properties of the Phaethon-Geminid Complex 14) oid with a spin period of ∼3.6 h.13) Phaethon orbits the Sun in Ecliptic J2000 frame (as of April 27, 2019) on a ∼22o inclined, highly eccentric orbit (∼0.89) with respect Property 3200 Phaethon 2005UD 1999YC to ecliptic plane, with 0.14 AU at the closest point to the Sun Eccentricity 0.89 0.87 0.83 Semi-major with an orbit period of approximately 1.4 years.14) Asteroids 1.27 1.27 1.43 axis [AU] (155140) 2005UD and (225416) 1999YC are suggested to orig- Inclination [o] 22.26 28.67 38.23 inate from Phaethon due to their spectroscopic and orbital simi- Arg. of 3, 4) 322.18 207.58 156.40 larity. Both 2005UD and 1999YC are ∼1.5 km-sized objects perihelion [o] on roughly similar orbits Phaethon. 2005UD has an orbit pe- Right ascension of 265.21 19.73 64.78 riod very similar to Phaethon, i.e. 1.4 years, whereas 1999YC the ascending node [o] has longer orbit period (∼1.7 years). Figure 1 shows their or- Period [yr] 1.43 1.44 1.70 bits, and Table 1 tabulates their orbital properties. As can be inferred from the Fig. 1, the orbit characteristics of the PGC 3. Initial guess solutions they show the best patching opportunities, they do not specify whether there exists a suitable low v1 transfer at Earth depar- Finding a globally optimal solution is a challenging task for ture for a given low patching ∆V. It only specifies that patching mission designers. One deals with multiple local minima for between incoming and outgoing trajectory arcs is possible with given bounds in parameters and for that one needs good ini- low ∆V. We therefore derive a final relation to assess the amount tial guess for a converging optimal solution. One of the most of impulsive ∆V that needs to be provided during E1-P2-E3 se- common methods to generate initial conditions is the solution quence, described as below: of the Lambert problem. The Lambert problem deals with the conic trajectory between a two given position in space for a ∆V = Vdep + V − V (2) given time-of-flight (ToF) between them.15) This is generally total 1 miss M2M dep a good approximation for high-thrust near-impulsive transfers. where V1 is the magnitude of the velocity at the Earth escape However, solution to the the low-thrust transfer problem is not and VM2M is the magnitude of velocity gain at the end of mul- as straightforward, because transfer trajectories include large tiple lunar flyby phase. In its current form, Equation 2 assumes dep thrusting arcs in extended periods of time due to limited thrust- that departure V1 and Vmiss have equal weighting because of ing capability. Nevertheless, the solution to the Lambert trans- the impulsive ∆V assumption. But Eq. 2 is likely to include dep fers can provide hints about the near-ballistic or low-∆V trans- weighting coefficients to scale the contributions from V1 and fers that can be used in the initial guess generation for the op- Vmiss as a function of, for example, transfer time to Phaethon. timization phase. Therefore, the Lambert transfer problem is This was not considered in the current paper and left as a future solved for each leg of the DESTINY+ mission. The Lambert work. Equation 2 can then be used to compute total ∆V values solver in this paper can account for multiple revolution trajec- for given Phaethon flyby dates and a range of Earth-departure tories around the central body (in this case, the Sun).16) Given and Earth-arrival dates, similar to mismatch ∆V maps. In prin- the limited amount of ∆V that can be provided by DESTINY+, ciple, minimum values in this map should provide converging only prograde solutions are taken into account. Figure 2 shows initial guesses to the optimization process, as shown in Fig. 4 the lowest-∆V transfers between Earth and Phaethon (E1-P2) for the flyby date Jan 4, 2028. for Earth departure dates between Sep, 1 2025 and July 27, Total ∆V up to 3 km/s is shown in the figure. The vertical dep 2029, and Earth arrival dates between July 01, 2026 and June lines in the figure shows departure V1 values therefore one 05, 2031, in accordance with DESTINY+ schedule. can quickly calculate Vmiss with VM2M = 1.5 km/s assumption. Note that Fig 2 only show transfer ∆Vs lower than 3 km/s. The results show that there exist periodic low-∆V transfer with 3.1. Secondary target solutions about 1.4 years, i.e. Phaethon’s orbit period, in descending of Transfer opportunities to secondary targets are first investi- Phaethon (not shown in the figure, see Ref. [10]). The inset gated individually. Lambert problem is first solved for transfer in Fig 2 provides a more detailed overview of the first flyby opportunities between 2028-2032 Earth departure and 2028- opportunity in Dec 2027 – Jan 2028 time frame with less than 2038 time frame for asteroid flyby. Note that Earth depar- ∆V = 1 km/s, i.e. lower than the upper boundary provided by ture dates are different than what is selected for Earth-Phaethon the multiple lunar flyby phase of the mission. transfers. This is due to the selected mission sequence and po- As mentioned earlier, the Lambert problem is solved each leg tential low-thrust opportunities identified previously.10) Figure of the mission. As the first leg of the interplanetary phase shown 5 and 6 show the Lambert problem solutions for Earth-2005UD in Fig 2, Earth-return leg (P2-E3) of the transfers also have mul- and Earth-1999YC transfers, respectively. tiple solutions. The best combination between E1-P2 and P2-E3 Note that color bars in both figures are scaled such that low- legs can be found when the magnitude of the difference between est value found in the analysis is minimum, and the maximum incoming and outgoing velocity vectors at Phaethon flyby point, value is 5 km/s. Earth-2005UD transfers show plenty of low ∆V defined as Vmiss in Eq. 1: transfer opportunities. The lowest value found is ∼2 km/s. This transfer window was also identified in Sarli et al. (2015) as potential ballistic transfer between Earth and 2005UD.10) The V = jVout − Vin j (1) miss 1 1 orbital geometry in this time frame between Earth, Phaethon in out where V1 and V1 incoming and outgoing V1 velocities at the and 2005UD is such that it is possible to complete a Phaethon flyby point. Velocity vectors are used in Eq. 1 to ensure the flyby, Earth gravity-assist and 2005 UD flyby in less than a directions of the velocity vectors are also nearly-aligned.