Translunar injection

Reproduce the translunar injection manoeuvre of Artemis II in simulation and find the speed of the Earth-Moon-Earth free return trajectory.

Activity overview:

The student configures in the Orbits and Gravitation simulation Earth at the centre, a probe of negligible mass at 50,000 km below Earth with an initial horizontal speed, and the Moon to the right at 385,000 km launched at 1.022 km/s upward. The student varies the probe's speed and observes four possible trajectory regimes: fall back to Earth, high orbit, escape, or free return passing near the Moon. Activity suited to high school, Year 11 level.

Level:

Author:

High school

FizziQ

Duration (minutes) :

45-60

What students will do :

'- Identify the relevant parameters (position, speed, mass) of a three-body gravitational problem
- Measure the effect of initial speed on a probe's trajectory in the Earth-Moon system
- Verify that a free return trajectory exists in a narrow speed window
- Compare the experimental speed to the theoretical Hohmann transfer value
- Understand the principle of translunar injection used by Apollo and Artemis

Scientific concepts:

'- Newton's law of universal gravitation (F = G·m₁·m₂/r²)
- Three-body problem
- Free return trajectory
- Translunar injection (TLI)
- Hohmann transfer
- Escape velocity
- Gravitational sphere of influence
- Sensitivity to initial conditions and launch window

Sensors:

'- FizziQ Web Orbits and Gravitation simulation

Material needed:

'- Computer, tablet or smartphone with FizziQ Web
- FizziQ experiment notebook

Experimental procedure:

Open the Orbits and gravitation simulation in FizziQ Web (Experiment → Simulations → Orbits and gravitation).
Set the distance scale to about 1,000 km/pixel to visualize the entire Earth-Moon system, and the time scale to 5 minutes per frame.
Configure body 1 — Earth: mass = 1 M⊕ (Earth), speed = 0 km/s, angle = 0°. Leave it at the centre of the screen.
Add body 2 — Probe (click the + tab): mass = space probe, initial speed = 4.3 km/s, angle = 0° (rightward). Drag the probe 40,000 km to the right of Earth.
Add body 3 — Moon: mass = 0.012 M⊕ (Moon), speed = 1 km/s, angle = -90° (upward). Drag the Moon 385,000 km to the right of Earth with the three bodies aligned.
Verify the distances in the upper-right panel: probe-Earth ≈ 40,000 km and Moon-Earth ≈ 385,000 km.
Centre the view on Earth (button 1 in the Centering area) to track the probe in the Earth reference frame.
Click Start and observe the probe's trajectory. What does the probe do?
By varying the angle and then the speed, create a trajectory that allows the probe to pass near the Moon and return to Earth without firing the engine.
For each attempt, click IMG to archive a trajectory snapshot in the experiment notebook, noting the speed tested.
Once a free return trajectory is identified (the probe swings around the Moon and returns to Earth), click REC to record the probe's x and y positions over time.
Let the simulation run until the probe returns near Earth, then click REC again to export the data to the experiment notebook.
In the experiment notebook, create a computed quantity: probe-Earth distance = √(x_probe² + y_probe²). Plot this distance versus time.
Compare to the Hohmann transfer speed and conclude.

Expected results:

The student finds that the probe's initial speed must lie within a narrow window, of order 3.7 to 3.8 km/s, for it to reach the Moon's vicinity. Below this value, the probe traces an elliptical orbit that falls back towards Earth. Above it, it moves away from the system permanently. For a precise speed (around 3.75 km/s, to be adjusted according to the Moon's starting position), the probe swings around the Moon and returns near Earth, drawing a characteristic figure-of-eight or wide loop. The required precision on speed is typically a few tens of metres per second, less than 1% error. Total trip duration is about 6 to 8 simulated days. The result is sensitive to the Moon's initial position at launch, which illustrates the need for a precise launch window in real missions. The simulator's symplectic Euler numerical integration introduces a slight energy drift that can be visible over long durations.

Scientific questions:

'- Why is the speed window allowing free return so narrow?
- What happens if the probe is launched at the wrong moment (Moon at top, bottom, left)?
- How does the Moon's mass influence the return trajectory? What would an "Earth without Moon" yield?
- Why was the free return trajectory essential for saving Apollo 13?
- How does the experimental speed deviate from the theoretical Hohmann transfer value, and why?
- In practice, what would propulsion of a spacecraft be useful for if the free return trajectory is free?
- When launching the probe at angle 180° (retrograde), why does the Moon no longer manage to deflect the trajectory? Compare the relative speed at flyby in both cases (prograde and retrograde) to explain.
- If the retrograde probe eventually returns near Earth, can it be called a free return in the Apollo and Artemis sense? What is the Moon's expected role in a real mission?

Scientific explanations:

Translunar injection (TLI, Trans-Lunar Injection) is the manoeuvre that takes a spacecraft from an Earth orbit to a trajectory intercepting the Moon. It was used by all the Apollo missions (1968-1972) and by Artemis II in April 2026. The motion obeys Newton's universal law of gravitation: F = G·m₁·m₂/r², with G = 6.674 × 10⁻¹¹ N·m²/kg².

In the Earth-Moon-probe system, each body experiences the vector sum of the forces exerted by the other two: this is a three-body problem, which has no general analytical solution. The probe's motion is dominated by Earth's gravity near Earth, then by the Moon's gravity once it enters the Moon's sphere of influence (about 66,000 km around it).

For a circular orbit around Earth at distance r, the theoretical speed is v_circ = √(G·M_E/r). At r = 40,000 km, this speed equals 3.16 km/s. The escape velocity at this same distance is v_esc = √(2·G·M_E/r) ≈ 4.46 km/s. The translunar injection speed must necessarily lie between these two values: large enough to reach the Moon's orbit, but not so large as to escape Earth's potential well permanently.

Ignoring the Moon's mass, a Hohmann transfer brings a probe from r₁ = 40,000 km to r₂ = 385,000 km along an ellipse of semi-major axis a = (r₁ + r₂)/2 = 217,500 km. The speed at perigee is then v_p = √(G·M_E·(2/r₁ − 1/a)) ≈ 4.25 km/s. This is the order of magnitude sought experimentally.

The Moon's presence modifies this result. As the probe approaches, lunar attraction curves its trajectory: the probe swings around the Moon and is sent back towards Earth. This free return trajectory requires no speed correction, which makes it an essential safety trajectory for crewed missions.

The result depends very strongly on the initial speed and on the Moon's position at launch. A 50 m/s variation can turn a free return into a Moon collision or an ejection into interplanetary space. This is what defines the launch window of a lunar mission: the spacecraft must be launched at a precise instant so the Moon is in the right place during flyby.

It is precisely this property that saved the Apollo 13 crew in April 1970: after an oxygen tank explosion, the service module could no longer perform lunar orbit insertion. Since the trajectory had been calculated as a free return, the Moon's gravity naturally sent the spacecraft back towards Earth. Today, Artemis II follows the same principle for safety.

Extension activities:

Frequently asked questions:

Q: Why does the probe have such a small mass (space probe ≈ 1000 kg)?
A: The probe's mass is negligible compared with Earth's (5.97 × 10²⁴ kg) and the Moon's (7.35 × 10²² kg). It barely influences the motion of the two large bodies, which simplifies the analysis: the probe is attracted but does not significantly attract.

Q: What exact speed value will I find?
A: The exact value depends on the precise initial position of the Moon and the probe, and on the simulator's numerical method. It lies between 3.7 and 3.8 km/s. The objective is not to find a single value but to understand the sensitivity of the result to initial conditions.

Q: Why animate the Moon when the blog article keeps it fixed?
A: With a fixed Moon, the trajectory depends only on the probe's speed. With an animated Moon, you must also account for when the probe will arrive in the Moon's neighbourhood: this is the real problem of the space launch window, as for Apollo or Artemis.

Q: Why doesn't the trajectory close exactly back on Earth?
A: The Earth-Moon-probe system is a three-body problem. The free return trajectory is never exactly periodic: the probe passes close to Earth but not at an identical point. In a real mission, a small correction manoeuvre is added on arrival.

Q: Does the simulator account for atmospheric drag?
A: No, the model is purely gravitational and pointlike. There is no atmosphere, no body rotation, no relativistic effects. This is consistent with the physics of interplanetary travel in space's vacuum.