What if Earth had two moons?

Earth has one Moon. It goes round and round us.

What if Earth had two moons? Would they go round nicely too?

Every month our Moon makes one trip around Earth, and each trip follows almost the same path as the one before. Earth's pull keeps it on that steady loop.

But what if Earth had a second moon, just as big as ours, a little farther out? Would both moons keep to tidy loops? In this lab you can add a second moon to an orbit model and watch.

The Moon completes one orbit relative to the stars every 27.3 days at a mean distance of 384,400 km, and its path repeats closely from month to month.

This lab asks what happens when a second body of the same mass shares the space around Earth. Gravity acts between every pair of bodies, so the question becomes a three-body problem. Such systems can be chaotic, which is why this lab explores it with a step-by-step simulation.

Our Moon is always moving sideways. Earth's pull bends its path into a loop.

One loop takes about four weeks. Then it starts again.

The Moon is always moving sideways at about one kilometer every second. Earth's gravity pulls it inward, bending its path into a loop. In the model, one loop takes about 27 days, and the Moon stays at the same distance, about 384,400 kilometers, all the way round.

With only Earth and one moon, nothing else disturbs the loop, so it repeats again and again.

For two bodies, Newton's gravity gives closed orbits: the model starts the Moon at the circular speed √(G(M+m)/r), about 1.02 km/s, and its distance stays at 384,400 km for orbit after orbit. Its period comes out at 27.29 days (slightly shorter than the observed 27.32 days because the model uses the mean distance).

The real Moon's orbit is a slightly stretched ellipse, ranging between about 357,000 and 407,000 km, and is disturbed by the Sun; the model leaves both out.

In our pretend lab, Olly adds a second moon. It is just like ours, but farther out.

Now the moons pull on each other. Their paths start to wobble!

Switch the system to Earth and two moons. The new moon is a copy of ours, starting on the opposite side of Earth, 1.6 times as far away. Now watch the distance readout for our Moon: it no longer stays fixed. In the model, over the first year it drifts between about 330,000 and 420,000 kilometers.

The loops no longer repeat neatly, because each moon tugs on the other as they pass.

With the second moon at 1.6 × 384,400 km, the leapfrog integration shows the first moon's distance swinging between about 329,000 and 423,000 km within a year, instead of holding at 384,400 km. The orbits keep changing size and shape instead of repeating.

Move the second moon closer (try 1.3) and the disturbances grow much larger. Move it out to 2 and they shrink. The model's 'one orbit takes' figure is a two-body prediction, so it does not capture these changes.

Everything with mass pulls on everything else. So the two moons pull on each other.

Each pull nudges the other moon a little. The nudges add up.

Gravity is not only a pull between Earth and each moon. Every object pulls on every other object, so the two moons pull on each other too. Each time they pass, the outer moon gives the inner one a small tug, and the inner moon tugs back.

One tug is tiny, but the tugs keep happening at different points around the orbits. They add up, and the neat loops turn into wobbly ones.

The model sums Newtonian accelerations for every pair, aᵢ = Σ G mⱼ (rⱼ − rᵢ)/|rⱼ − rᵢ|³, and integrates with a symplectic leapfrog so energy errors stay bounded. With one moon the orbit closes and repeats; with a third body the pulls keep changing, and some three-body systems are chaotic.

The model's limits: motion in one plane, point masses, no Sun, and an imaginary second moon. Because three-body motion can be sensitive to tiny changes, long runs are illustrations, not forecasts. Very close passes are stopped as a crash rather than modeled.

Mars has two moons! They are small and not round like a ball.

Some big planets have lots and lots of moons.

Some planets really do have more than one moon. Mars has two small ones, Phobos and Deimos. Phobos is so close to Mars that it goes all the way round in less than eight hours. Jupiter and Saturn have dozens of moons each.

A second moon as big as ours would also pull on Earth's oceans, so scientists expect it would add tides of its own, an idea our orbit model does not calculate.

Mars's moons are tiny compared with ours: Phobos orbits at about 9,400 km from Mars's center every 0.32 days, and Deimos at about 23,500 km every 1.26 days. NASA describes Phobos as orbiting Mars three times a day.

A second Moon-sized body would also raise its own tides, which would vary as the two moons lined up or spread apart. That is an idea, not a model result: the orbit model ignores tides entirely.