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Exoplanetary Tour Design with Solar Sails: TheAntipodes Results in the GTOC13 Problem

This paper presents the third-place solution by the team "TheAntipodes" to the GTOC13 exoplanetary tour problem, detailing a hybrid approach that combines large-scale beam search for gravity assist structures with sequential convex programming and a lossless control-convex formulation to optimize continuous solar sail trajectories for maximizing scientific return.

Original authors: Jack Yarndley, Adam Evans, Xingyu Zhou, Minduli Wijayatunga, Cristina Parigini, Roberto Armellin

Published 2026-07-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Jack Yarndley, Adam Evans, Xingyu Zhou, Minduli Wijayatunga, Cristina Parigini, Roberto Armellin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are the captain of a spaceship, but you've forgotten to pack any fuel. No rockets, no chemical boosters. The only thing you have is a giant, shimmering sail that catches the wind of light streaming from a distant star. This is the challenge faced by the team "TheAntipodes" in the 13th Global Trajectory Optimization Competition (GTOC13). They had to design a 200-year tour of a fictional alien solar system called Altaira, visiting planets, comets, and asteroids to collect scientific points, all while relying solely on that light-sail and the gravity of the planets to steer.

Their final result? They came in third place with a score of 337.878, collecting data from 133 different flybys. Here is how they pulled it off, explained without the heavy math.

The Big Idea: A Two-Step Dance

The team realized they couldn't just guess the whole path at once. It was too messy. Instead, they broke the problem into two distinct phases, like planning a road trip by first looking at a map of major highways, and then figuring out the exact turns and detours later.

Step 1: The "Ballistic" Map (The Gravity Train)
First, they pretended the solar sail didn't exist. They asked: "If we just coast and bounce off planets like a pinball, what paths look promising?" They used a computer search method called "beam search" (think of it as a tree that grows thousands of branches, but the computer only keeps the top 100,000 most promising ones at each step) to find sequences of gravity assists.

They found some cool structures, like a "triangle" path involving three outer planets and a fast loop around the inner planets. However, these paths had a fatal flaw: they were too rigid. To visit the tiny, weightless asteroids and comets (which are worth points but have no gravity to help you turn), you need to be able to steer. A purely "ballistic" path (coasting only) couldn't hit all the targets efficiently.

Step 2: The Solar Sail Refinement (The Fine-Tuning)
This is where the magic happened. Once they had a rough "skeleton" of a tour from Step 1, they brought back the solar sail. They used a powerful math tool called Sequential Convex Programming (SCP).

Think of SCP as a super-smart GPS that can take a rough, bumpy route and smooth it out in real-time. It adjusted the timing of every flyby and the angle of the solar sail to fix the mismatches. The paper highlights a specific trick they used: a "lossless convex formulation." In plain English, this means they found a way to turn the tricky, curved rules of solar sailing into a straight-line math problem that computers can solve instantly, without losing any accuracy.

The Secret Weapon: The "Vulcan" Resonance

One of their biggest discoveries was how to use the innermost planet, Vulcan. Vulcan is tiny but super heavy and orbits its star incredibly fast.

The team realized they could use Vulcan like a trampoline. They would bounce off Vulcan, fly out to catch a comet, and then use the timing of their orbit to bounce off Vulcan again. Because Vulcan orbits so fast, they could set up a "resonant" loop: fly out, wait for Vulcan to catch up, bounce again, fly out to another comet, and repeat.

They found a sequence where they could visit 74 comets just by hopping between Vulcan and the comets. This "resonant tour" became the middle section of their journey, lasting about 75 years and earning them a huge chunk of their score.

What They Tried (And What Didn't Work)

The paper is very honest about what they tried and discarded:

The Asteroid Belt Tour: They thought about building a tour just inside the asteroid belt to grab points quickly. They simulated this and found it could work, but it was too slow. It would take about 30 years to get enough points to be worth the time, and the score rate was too low compared to the comet hopping. So, they ruled it out for their final solution.
Direct Entry: They tested entering the system by diving straight toward the star to slow down. While they calculated they could enter at speeds up to 151.419 km/s using a close pass, it turned out this saved too little time to be worth the risk. It was better to take a slightly longer path that visited a planet first to set up the rest of the trip.
The "Grand Tour" Bonus: To get a massive score boost, you have to visit every major planet, the dwarf planet Yandi, and at least 13 asteroids or comets. Their final solution did exactly this, earning a 20% score multiplier.

The Final Result

The winning trajectory is a masterpiece of timing.

The Entry: They started with a path visiting the outermost planet, PlanetX, and then Planet 7 before diving toward the star.
The Middle: They spent 75 years bouncing between Vulcan and 74 comets, using the solar sail to tweak their path just enough to hit the next target.
The End: They finished with a high-speed loop around the inner planets (specifically Planets 5, 6, and 7), squeezing in as many flybys as possible before the 200-year clock ran out.

The final trajectory visits 57 planets, 2 dwarf planets, and 74 comets. The total mission time was 191.177 years.

How Sure Are They?

The authors are very confident in their numbers because they didn't just guess; they simulated the entire thing.

They ran beam searches with frontiers of up to 2,500,000 nodes to find the best structures.
They used Sequential Convex Programming to refine the paths, solving problems with up to 46,770 variables in less than a minute on a standard computer.
The score of 337.878 is a calculated result based on their specific trajectory, which they submitted to the competition.

However, the paper admits that the "resonant" part of their solution (the Vulcan-comet hopping) was the hardest to perfect. They had to manually check and block bad paths during their search because the computer's initial guesses weren't always perfect. They suggest that if they had a better way to handle that specific part, they might have pushed the score even higher, perhaps toward 360–370.

In short, TheAntipodes didn't just find a path; they built a flexible, self-correcting system that turned a giant, impossible puzzle into a third-place victory, proving that with the right math, a solar sail can indeed take you on a grand tour of the stars.

Technical Summary: Exoplanetary Tour Design with Solar Sails (GTOC13)

Problem Statement
This paper details the solution developed by the team "TheAntipodes" for the 13th Global Trajectory Optimization Competition (GTOC13). The challenge involved designing a trajectory for a spacecraft equipped solely with a solar sail to explore the fictional Altaira exoplanetary system. The system contains 10 major planets, a dwarf planet (Yandi), 257 asteroids, and 42 comets. The objective was to maximize a scientific score derived from flybys of these bodies within a 200-year time window.

Key constraints included:

Propulsion: The spacecraft relies entirely on solar radiation pressure (SRP) for maneuvering, with no chemical or electric propulsion.
Gravity Assists (GAs): Only the 10 major planets possess gravitational parameters and can be used for GAs. Encounters with asteroids, comets, and the dwarf planet are modeled as massless; they contribute to the score but cannot alter the spacecraft's velocity vector without SRP.
Scoring: The score is a weighted sum of flybys, penalized by flyby velocity (too fast or too slow) and seasonal diversity (repeated observations from similar viewing angles). Global bonuses are awarded for visiting all major planets and a minimum number of small bodies (Grand Tour) and for early submission (Time Bonus).
Dynamics: The solar sail acceleration is highly underactuated and non-convex, as the achievable thrust vector is constrained by the sail's orientation relative to the star.

Methodology
The team employed a multi-stage approach, decomposing the intractable global optimization problem into sequential search, refinement, and assembly stages.

Ballistic Beam Search for Structural Identification:
- Before applying solar sail optimization, the team conducted large-scale beam searches over ballistic (unpowered) gravity assist tours. This identified favorable planetary encounter sequences and resonant structures.
- Planetary Tours: A beam search explored sequences of major planets, using a patched-conic model to find feasible transfers where the required departure velocity matched the incoming velocity after a GA turn.
- Resonant Vulcan Tours: A specialized search focused on the innermost planet, Vulcan, which has a short orbital period and high gravity. The team generated "resonant" transfer options where the spacecraft returns to Vulcan after a specific number of orbits ( $R = p/q$ ), potentially intersecting comets or other planets in between. This created a catalog of feasible "Vulcan-return" arcs.
Sequential Convex Programming (SCP) for Refinement:
- The core of the trajectory optimization was a lossless convexification of the solar sail dynamics. The non-convex control constraints (cone and clock angles) were relaxed into a convex set using a second-order cone (SOC) formulation.
- Lossless Convexification: By introducing a small objective penalty on the radial acceleration component, the relaxation was forced to bind to the true non-convex boundary at optimality. This allowed the use of efficient convex solvers (Clarabel via JuMP) to optimize large-scale trajectories.
- Applications: SCP was used to:
  - Refine ballistic planetary tours by optimizing flyby timing and sail control.
  - Analyze entry strategies (e.g., direct capture vs. planetary flyby chains).
  - Refine the "terminal sequence" (the final high-scoring phase involving repeated flybys of outer planets).
  - Transform the discrete, mismatched resonant Vulcan beam search results into dynamically feasible trajectories by correcting velocity mismatches via sail control and timing adjustments.
Solution Assembly and Selection:
- The final trajectory was assembled by joining distinct phases: an entry sequence (10-7-sun), a long resonant Vulcan phase (approx. 75 years), and a high-scoring terminal sequence.
- Phase boundaries were optimized using SCP to ensure dynamic continuity.
- A genetic algorithm was used in the final step to select which flybys to designate as "scientific" (maximizing the score while minimizing seasonal penalties), as the trajectory contained more potential flybys than the scoring limit allowed.

Key Contributions

Lossless Convex Formulation: The paper demonstrates the application of a lossless control-convex formulation for solar sail dynamics to a large-scale, multi-leg trajectory problem. This allowed the simultaneous optimization of hundreds of gravity assist geometries and flyby timings.
Hybrid Search Strategy: The work highlights a successful decomposition strategy where combinatorial beam searches (to identify topological structures like resonant chains) are coupled with continuous convex optimization (to refine dynamics).
Resonant Tour Construction: The development of a specific beam search for Vulcan-resonant tours, which leverages the inner planet's dynamics to access comets and small bodies efficiently, proved central to the solution's success.

Results
The Antipodes' solution placed third in the competition with a final score of 337.878.

Trajectory Structure: The trajectory consisted of 252 total flybys, with 133 selected as scientific. It included 57 planetary flybys, 2 dwarf planet flybys, and 74 comet flybys. No asteroid flybys were included in the final submission.
Phases:
- Entry: A 10-7-sun sequence entering the system.
- Resonant Phase: A 75-year phase dominated by repeated Vulcan flybys and comet encounters, utilizing resonant transfers to maintain high score rates.
- Terminal Sequence: A final phase featuring an 8-9-8 triangular structure and a high-scoring chain of flybys (5-7-6-7-6-7) involving outer planets.
Performance: The solution achieved the Grand Tour bonus (visiting all 10 planets, Yandi, and >13 small bodies). The terminal sequence alone achieved a score rate of approximately 1.5 unbonused points per year.

Significance and Claims
The paper claims that the results demonstrate the scalability of Sequential Convex Programming for very large trajectory design problems involving complex combinatorial structures and continuous control. The authors note that while the ballistic beam search provided the necessary topological "skeleton" of the solution, the SCP refinement was essential for making these structures dynamically feasible and maximizing the score.

The authors modestly suggest that the resonant Vulcan phase was the area with the most potential for improvement. They observe that the first-place team (THU-LAD) achieved a higher score (approx. 360–370), likely due to a more sophisticated handling of the resonant phase and the interaction between sail control and GA timing. The paper concludes that their approach successfully balanced broad combinatorial exploration with precise continuous optimization, validating the decomposition strategy for future large-scale exoplanetary mission designs.