Trans-lunar injection

• 1. Introduction

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  A trans-lunar injection, the rocket burn that sends a spacecraft toward the Moon, once lit up the pre-dawn sky south of Waikiki. That burn belonged to Apollo 8, and people across the Hawaiian Islands watched it happen. It was photographed, and the newspapers reported it the next day. The same maneuver, usually shortened to TLI, has launched probes, landers and crews for more than sixty years. Yet it raises puzzling questions. How can one burn near Earth aim at a target that is itself circling Earth? How can a crew loop around the Moon and come home without firing another engine? And why do some spacecraft reach the Moon quickly while others drift toward it for months?

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• 2. Aiming At A Moving Target

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  Every trans-lunar injection begins in a low circular parking orbit around Earth. From that circle, a large burn, usually from a chemical rocket engine, raises the spacecraft's velocity. The orbit stretches from a circle into a highly eccentric ellipse. The burn is sized and timed to hit the Moon where it will be, not where it is. Engineers schedule it so the spacecraft nears the highest point of its new orbit, called apogee, just as the Moon approaches. That apogee sits close to the radius of the Moon's own orbit. After the burn, the spacecraft flies passively on its own momentum, tugged by both Earth's gravity and the Moon's. This stretch of the mission is called translunar coast. Eventually the craft crosses into the Moon's sphere of influence, the region where lunar gravity dominates, and makes a hyperbolic swingby. Typical lunar transfers approximate a Hohmann transfer, the classic fuel-efficient path between two orbits. For short missions without significant disturbances from outside the Earth-Moon system, a fast Hohmann transfer is usually the more practical choice. Some of these swingbys can be designed so that they carry the spacecraft straight back home.

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• 3. Free Return

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  Apollos 8, 10 and 11 each began their journeys on a free return trajectory. In this design, the TLI burn is aimed so that the spacecraft loops around behind the Moon and comes back toward Earth with no further propulsive maneuvers. The idea adds a margin of safety to human spaceflight, because a crew that loses its engine after the burn still swings home. It also saves money, since the spacecraft returns to Earth "for free" after that single burn. Later Apollo missions used a hybrid trajectory that works in a functionally similar way. It requires a midway course correction to reach the Moon, so it is not quite free. Artemis missions 1 and 2 are both free return missions as well.

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• 4. Apollo By The Numbers

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  Roughly 350 seconds is how long the Apollo TLI burn lasted. The restartable J-2 engine in the third stage of the Saturn V rocket did the work. That burn added between 3.05 and 3.25 kilometers per second to the spacecraft's velocity, or 10,000 to 10,600 feet per second. When it ended, the spacecraft was traveling at about 10.4 kilometers per second relative to Earth, or 34,150 feet per second. Those numbers were visible from the ground. In 1969, observers at Cloncurry, Australia, saw the pre-dawn TLI of Apollo 10. They compared it to car headlights coming over a hill in fog. The spacecraft itself looked like a bright comet with a greenish tinge.

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• 5. Modeling The Path

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  Patched conics is the simplest method for sketching a lunar transfer. It treats the spacecraft as moving under classical two-body dynamics, dominated by Earth until it reaches the Moon's sphere of influence. Motion in that model is deterministic and simple to calculate. That makes it suited to rough mission design and "back of the envelope" studies. Real flight is messier. Earth and Moon dominate the spacecraft's acceleration, and the spacecraft's own mass is negligible next to theirs. So its path is better approximated as a restricted three-body problem. That model is closer to reality but has no analytic solution, so it needs numerical integration. The most detailed simulations add the Moon's true orbital motion, gravity from other astronomical bodies, and the uneven gravity fields of Earth and Moon. They also include solar radiation pressure. Propagating a spacecraft through such a model is numerically intensive, but true mission accuracy demands it. Two named numerical methods, Runge-Kutta-Fehlberg and Dormand-Prince, are associated with this kind of calculation.

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• 6. Early Probes

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  On the 2nd of January 1959, the Soviet Union's Luna 1 attempted the first TLI, intending to hit the Moon. The burn did not go exactly as planned. The probe missed the Moon by more than three times the Moon's radius and ended up in orbit around the Sun. Luna 2 repeated the maneuver more accurately on the 12th of September 1959 and crashed into the Moon two days later. The Soviets repeated that success with 22 more Luna missions and 5 Zond missions that reached the Moon's vicinity between 1959 and 1976. The United States launched its first lunar impactor attempt, Ranger 3, on the 26th of January 1962, and it failed to reach the Moon. Ranger 4 followed on the 23rd of April 1962 and became the first American success. From 1962 to 1973, another 27 American missions headed to lunar space. They included five Surveyor soft landers, five Lunar Orbiter surveillance probes and nine Apollo missions. Those Apollo flights landed the first humans on the Moon.

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• 7. Patient Paths

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  Japan's first lunar mission, launched in 1990, used the Hiten satellite to fly by the Moon and place the Hagoromo microsatellite in lunar orbit. Hiten then explored a novel low delta-v route, meaning one that demands a smaller change in velocity. The cost was time, with a transfer of six months. In 1994 the American Clementine spacecraft, built to showcase lightweight technologies, used a TLI lasting three weeks, with two intermediate Earth flybys before it entered lunar orbit. Three years later, Asiasat-3 became the first commercial satellite to reach the Moon's sphere of influence. After a launch failure, it swung by the Moon twice as a low delta-v way to reach its desired geostationary orbit. It passed within 6200 kilometers of the lunar surface. In 2003 the European Space Agency's SMART-1 technology demonstrator launched into a geostationary transfer orbit and relied on solar-powered ion engines. Its extremely low delta-v TLI took over 13 months to reach lunar orbit and 17 months to reach its desired orbit. SMART-1 became the first European satellite to orbit the Moon.

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• 8. Recent Arrivals

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  China launched its first Moon mission in 2007, placing the Chang'e 1 spacecraft in lunar orbit. It used multiple burns to raise its apogee slowly until it reached the Moon's vicinity. India followed in 2008 with Chandrayaan-1, which was launched into a geostationary transfer orbit. Like the Chinese spacecraft, it increased its apogee over a number of burns. In 2011 the NASA GRAIL satellites took a different low delta-v route that passed the Sun-Earth L1 point. That journey took over three months. The Israel Aerospace Industries lander Beresheet used this kind of maneuver in 2019 but crashed on the Moon. Its failure shows that a well-aimed injection delivers a spacecraft to the Moon but does not guarantee a safe landing.

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