Lab
Low-Energy Moon Route
Why low-energy Earth-Moon transfers go slow, through the L1 gravity gate. An interactive three-body explorer with zero-velocity curves and Lagrange points.
The cheap road to the Moon runs through a gravity gate
Try a scenario
Sets the energy just below the L1 gate and launches from the Earth-side lobe onto a ballistic path that threads L1 and coasts to the Moon (about 34 days). Finding this one initial condition is the hard part the 2026 study automated; most launches miss.
Spacecraft energy
Earth-Moon rotating frame
Earth and the Moon are held still while the whole picture spins with the Moon's orbit. The bright glow is where a probe is allowed to travel; the dark region is forbidden. Click anywhere in the glow to launch a probe.
Blue = Earth, gray = Moon. The crosses are the five Lagrange (balance) points.
What's actually happening?
The energy cavern
A spacecraft coasting through the Earth-Moon system keeps one number constant: its Jacobi constant. That number fences off a region it simply cannot enter — the dark area here. The brighter the glow, the faster the probe is allowed to move there. Near the dark edge it slows to a crawl.
L1 is a gate, not a wall
Between Earth and the Moon is L1, a point where their gravity balances. When the probe has too little energy, the cavern around Earth is sealed shut at L1. Add just a touch of energy and a narrow neck opens at L1, the cheapest possible doorway to the Moon. That doorway is the whole trick.
Cheap means slow
A probe that barely squeaks through L1 moves painfully slowly near the gate, so the trip takes about a month instead of Apollo's three days. You trade time for fuel. For crewed flights that's a bad deal; for patient cargo and resupply runs it's a great one.
The 2026 result
Researchers at Coimbra and São Paulo used a method called the Theory of Functional Connections to evaluate tens of millions of possible transfers. The best route uses L1 Lyapunov-orbit manifolds and reduces the reported transfer cost by at least 58.8 m/s compared with similar routes in the literature. Small on paper, but every m/s is a lot of fuel. Read the paper (Almeida Jr. & de Oliveira, Astrodynamics), or a plain-language overview.
Challenge 1: Open the gate
Scenario: Start on "Gates sealed." Notice the dark band completely separating Earth's glow from the Moon's.
Task: Slowly raise the energy slider. Watch the L1 gate readout flip to OPEN and a neck appear between Earth and the Moon. That exact moment is the lowest-energy passageway in this simplified model.
The neck opens when the Jacobi constant drops just below C(L1) ≈ 3.188.
Challenge 2: Launch a probe through L1
Scenario: Pick "L1 gate open."
Task: Click in Earth's glow, fairly close to the L1 neck. Watch the probe wind around and try to thread the gate. It can never cross into the dark, it's trapped by its own energy, just like a real spacecraft.
Try launching from different spots. Some paths slip through; many loop back. Real mission design is exactly this search for the few that make it.
Challenge 3: Watch the clock
Scenario: Launch one probe with the gate barely open, and another with "Wide open."
Observe: The barely-open probe crawls near the gate and racks up days on the travel-time readout. The wide-open one zips across. Less energy, more time, that's the fuel-versus-speed trade at the heart of the 2026 result.
The travel-time readout converts the simulation clock into real days using the Moon's ~27.3-day orbit.
A note on the physics
This is the planar circular restricted three-body problem in the rotating frame — the standard model behind low-energy transfers. The glow, gates, and Lagrange points are computed exactly; probe paths are integrated with RK4. It's a faithful sketch of the mechanism, not a mission-grade trajectory optimizer. The real 2026 route was found by searching tens of millions of trajectories with the Theory of Functional Connections and invariant manifolds around L1.
Security model
Everything runs in your browser. No data is sent anywhere. The three-body solver, zero-velocity rendering, and trajectory integration all execute locally in JavaScript on your device.