The most efficient method for traveling to the Moon, it appears, is to pass directly by it. Not to enter orbit or to make a landing approach—simply passing by, coming within 73 kilometers of the lunar surface before circling back through a gravitational sweet spot located midway between the two celestial bodies. This unexpected conclusion was reached by a team of researchers from Portugal, France, and Brazil after analyzing approximately 30 million simulated pathways, a figure that far exceeds any prior efforts in planning lunar trajectories. The results, published in the journal *Astrodynamics*, reduces fuel costs by at least 58.80 meters per second compared to the most cost-effective route previously documented, a statistic that seems small until you factor in the weight of rocket fuel.
The effort to return humans to the Moon and establish a permanent presence is quickening on multiple fronts. About 250 missions are anticipated to target the lunar surface post-2030, ranging from scientific missions to cargo deliveries and crewed habitats, and each kilogram of propellant conserved during the journey translates to more capacity for the instruments, supplies, or personnel that justify the investment in the mission.
### A Gravitational Waypoint Between Worlds
The path designed by the researchers does not travel directly from Earth orbit to lunar orbit. Instead, it takes a pause. Approximately 85% of the distance between these two bodies lies the L1 Lagrangian point, one of five regions within the Earth-Moon system where the gravitational forces of the two bodies and the centrifugal forces from the rotating system reach a precarious balance. L1 is an unusual area: a spacecraft stationed there is technically balanced, yet because this equilibrium is unstable, even a slight push can send it off course. This instability is what makes L1 significant for mission design. Orbits around it (flat, looping trajectories known as Lyapunov orbits) are connected by invisible pathways termed invariant manifolds, surfaces in space where a spacecraft can glide freely, moved by gravity instead of engines, toward or away from the Lagrangian point itself.
The challenge is to find the least expensive entry point to one of those pathways. Here is where the team’s methodology markedly diverges from prior research.
Previous studies examining Earth-to-Moon trajectories via L1 explored databases of around 280,000 potential routes. Allan Kardec de Almeida Júnior from the University of Coimbra and his team evaluated more than 30 million. This remarkable scale was achieved through a mathematical approach known as the theory of functional connections (TFC), which reformulates trajectory optimization issues into a format that is significantly less expensive to compute. Rather than incrementally approaching a solution numerically, repeatedly adjusting a trajectory until it somewhat fits the mission parameters, TFC integrates those conditions directly and analytically into the equations of motion. Any potential solution generated is already assured to comply with the boundary requirements; the optimizer merely selects the most economical option from a much broader selection.
### The Wrong Branch Was the Right Answer
The conventional belief in trajectory planning is that a spacecraft aiming for L1’s stable manifold from Earth should approach from the section of the manifold on the Earth’s side of the system. This seems reasonable: geographically nearer and conceptually neat. The simulations revealed a different story. The route on the Moon’s side of L1 proved to yield lower overall fuel expenses, a conclusion that becomes evident when considering the geometry: that route passes closely to the Moon, sufficiently for a gravity assist. “Instead of presuming it’s simpler to select the segment of the variate closest to Earth,” remarks Vitor Martins de Oliveira from the University of São Paulo, a co-author of the study, “we can apply systematic analysis with expedited methods to seek out nontrivial solutions.”
The resulting trajectory occurs in three segments. A spacecraft in a low Earth orbit at an altitude of 167 kilometers fires its engines to launch, a maneuver requiring about 3,142 meters per second in velocity adjustment. It then travels for roughly 3.69 days, drawing within 73 kilometers of the Moon’s surface. A small supplementary burn near the Moon places it into the stable manifold, after which natural dynamics guide it toward the Lyapunov orbit around L1. It can remain there as long as necessary, with the orbit’s 13.75-day period permitting the spacecraft to linger in multiples of that duration until optimal conditions arise for the next phase. A concluding series of burns, totaling just below 649 meters per second, positions it into a circular 100-kilometer orbit around the Moon.
The Moon leg burn occurred at merely 0.767 meters per second above the theoretical minimum. Little room remains to optimize that segment of the journey.
### Always in Touch with Earth
In addition to fuel savings, there is a functional advantage to the L1 waypoint. A spacecraft orbiting the Lagrangian point never disappears from the view of Earth;