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Low-thrust rendezvous with four asteroid groups (GTOC2)
Design a low-thrust tour that rendezvouses with one asteroid from each of four groups, maximizing final mass per year of flight.
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| 1 | +# Low-thrust rendezvous with four asteroid groups (GTOC2) | |
| 2 | + | |
| 3 | +Plan a tour in which an electric-propulsion spacecraft leaves Earth and rendezvouses with one asteroid from each of four very different families, keeping as much mass as possible per year of flight. Choosing which four of 910 asteroids to visit, and in what order, is as much a part of the problem as shaping the thrust profile. | |
| 4 | + | |
| 5 | +## Task | |
| 6 | + | |
| 7 | +The asteroids are listed in the statement's data file with Keplerian elements (heliocentric J2000 ecliptic frame), in four groups: | |
| 8 | + | |
| 9 | +| Group | Asteroids | Members | | |
| 10 | +|---|---|---| | |
| 11 | +| 1 | Jupiter former comets (Tisserand parameter 2 to 3, low albedo) | 96 | | |
| 12 | +| 2 | C- or M-class asteroids | 176 | | |
| 13 | +| 3 | S-class main-belt asteroids | 300 | | |
| 14 | +| 4 | Aten asteroids | 338 | | |
| 15 | + | |
| 16 | +- **Launch:** from Earth (Keplerian orbit given in the statement) in the years 2015 to 2035, with hyperbolic excess speed of up to 3.5 km/s in any direction; initial mass 1500 kg, of which 1000 kg is usable propellant. | |
| 17 | +- **Propulsion:** specific impulse 4000 s, thrust up to 0.1 N in any direction, on or off at will. | |
| 18 | +- **Dynamics:** two-body motion about the Sun ($\mu=1.32712440018\times10^{11}\ \mathrm{km^3/s^2}$) plus thrust; no gravity assists. | |
| 19 | +- **Tour:** rendezvous (matching position and velocity) with exactly one asteroid from each group, in any order, staying at least 90 days at each of the first three. The flight time $t_f$, from launch to arrival at the fourth asteroid, is at most 20 years. | |
| 20 | + | |
| 21 | +Maximize | |
| 22 | + | |
| 23 | +$$J=\frac{m_f}{t_f}\quad[\mathrm{kg/year}],$$ | |
| 24 | + | |
| 25 | +where $m_f$ is the spacecraft mass on arrival at the fourth asteroid. | |
| 26 | + | |
| 27 | +## Background | |
| 28 | + | |
| 29 | +GTOC2 was organized in 2006 by Anastassios Petropoulos of JPL's Outer Planets Mission Analysis Group, winner of the first edition, and was inspired in part by the Dawn mission to Vesta and Ceres. The groups were chosen so that no tour is obviously best and every good tour needs large changes in most orbital elements. All bodies follow fixed conics, so the problem is completely specified and a solution can be checked without any external ephemeris. | |
| 30 | + | |
| 31 | +## Data and evaluation | |
| 32 | + | |
| 33 | +- **Instance:** one fixed problem: the [problem statement](https://web.archive.org/web/20131101000511/http://sophia.estec.esa.int/gtoc_portal/wp-content/uploads/2012/11/gtoc2_problem.pdf) and its [asteroid file](https://web.archive.org/web/20131101000514/http://sophia.estec.esa.int/gtoc_portal/wp-content/uploads/2012/11/ast_ephem.txt) (910 asteroids, plus a row for Earth). | |
| 34 | +- **Evaluation:** there is no public verifier. A check must re-integrate the submitted trajectory, given at one-day steps for each leg as the statement requires, confirm the rendezvous conditions, stay times and limits, and recompute $J$. | |
| 35 | +- **Hardware:** none assumed; the score does not depend on run time. | |
| 36 | + | |
| 37 | +## What is known | |
| 38 | + | |
| 39 | +- Final ranking (Petropoulos, 2007): Politecnico di Torino won with $J=98.64$ kg/year, ahead of the Moscow Aviation Institute with Khrunichev (87.93) and ESA's Advanced Concepts Team (87.05). Eleven of the 15 returned solutions were ranked. | |
| 40 | +- The winning team chose its asteroid sequence by hand rather than by algorithm, starting from the aphelion of the final asteroid, and optimized each leg with an indirect method (GTOC portal). | |
| 41 | +- All teams visited group 4 first and group 1 last, in order of increasing orbital energy (Petropoulos, 2007). | |
| 42 | + | |
| 43 | +## What would count as progress | |
| 44 | + | |
| 45 | +- A tour with $J>98.64$ kg/year meeting every constraint, with its asteroid sequence and full trajectory file for re-integration. | |
| 46 | +- An open-source verifier for GTOC2 trajectories, validated on a published solution. | |
| 47 | +- A systematic search showing whether any asteroid sequence other than the winner's can beat it, with the best value found for each. | |
| 48 | + | |
| 49 | +## References | |
| 50 | + | |
| 51 | +1. A. E. Petropoulos. Problem description for the 2nd Global Trajectory Optimisation Competition. Jet Propulsion Laboratory, 2006. [Archived PDF](https://web.archive.org/web/20131101000511/http://sophia.estec.esa.int/gtoc_portal/wp-content/uploads/2012/11/gtoc2_problem.pdf) | |
| 52 | +2. A. E. Petropoulos. Final rankings and brief descriptions of the returned solutions and methods used for the 2nd Global Trajectory Optimisation Competition. Jet Propulsion Laboratory, 2007. [Archived PDF](https://web.archive.org/web/20131101000438/http://sophia.estec.esa.int/gtoc_portal/wp-content/uploads/2012/11/ACT-RPT-MAD-GTOC2-ranks-1.pdf) | |
| 53 | + | |
| 54 | +## Source | |
| 55 | + | |
| 56 | +Imported from [GTOC Portal, GTOC 2 – Multiple Asteroid Rendezvous](https://sophia.estec.esa.int/gtoc_portal/?page_id=15), curated by the ESA Advanced Concepts Team. The competition closed in December 2006; the problem stays public and is used as a benchmark. The portal could not be reached on 2026-10-20, so this write-up relies on an [archived copy of the page](https://web.archive.org/web/20131031234713/http://sophia.estec.esa.int/gtoc_portal/?page_id=15). The statement above is written in our own words; see the source for the original wording, history and discussion. | |