What if the Moon were closer?

The Moon is far away. It is very slowly moving even farther away.

But in a pretend lab, what if it came closer? What would the sea do?

The Moon is about 384,400 kilometers away, and it is actually drifting slowly farther from Earth, by about 3.8 centimeters a year. It is not going to come closer.

But imagining a closer Moon is a great way to see how tides work. Would a Moon at half the distance make tides twice as big, or much more than that? In this lab you can move the Moon in a tide model and find out.

The Moon's mean distance is 384,400 km, and it recedes by about 3.8 cm per year. Its actual distance already changes during each orbit, and over a year it ranges from about 357,000 to 407,000 km; NOAA notes that tides are larger than usual when the Moon is closest (perigee).

This lab asks how strongly tides depend on distance: what would the tidal forcing, the ideal ocean range and the length of the month be if the Moon orbited at half its present distance?

The Moon pulls on the sea. The water bulges out a little on two sides of Earth.

Earth spins through the bulges. That gives two high tides a day.

Because the Moon pulls a little harder on the near side of Earth than on the far side, it stretches the oceans into two bulges. Earth spins through them, giving most coasts two high tides a day. The Sun adds a stretch of its own, a little under half as strong.

In the model, today's Moon gives a tide strength of 1. Lined up with the Sun at new or full moon, the total is about 1.46, which on an imaginary ocean covering the whole Earth would be a range of about 0.8 meters.

In model units, today's lunar tide L = 1 and the solar tide S ≈ 0.46. They combine as √(L² + S² + 2LS cos 2θ), giving a spring tide of 1.46 and a neap tide of 0.54. The equilibrium range of the combined spring tide on a global ocean is about 0.78 m.

The synodic month, from Kepler's third law with the Sun's motion, comes out at 29.5 days.

In our pretend lab, Olly brings the Moon to half as far away.

Now the Moon pulls much harder on the sea. The tides get much, much bigger!

Choose 'half as far'. The Moon's part of the tide jumps from 1 to 8: half the distance makes its stretching 2 × 2 × 2 = 8 times stronger. With the Sun's share added, the biggest tides reach about 8.5 on the meter, nearly six times today's, and the ideal ocean range grows to about 4.5 meters.

The Moon also speeds up. A closer Moon goes round faster, so its phases would repeat about every 10 days instead of every 29 and a half.

At d = 192,200 km the lunar term becomes (384,400/192,200)³ = 8. The model's spring tide is 8.46 and neap tide 7.54, so the spring-neap contrast shrinks to about 12 percent because the solar tide is now small by comparison. The equilibrium range rises to about 4.5 m.

Kepler's third law gives a sidereal period 2^(−3/2) of today's, about 9.6 days, and a synodic month of 9.9 days. This is a thought experiment carried out with a model; the Moon cannot be moved.

Closer things pull harder. And the Moon's pull on the near side and far side of Earth gets very different.

That difference makes the sea stretch more.

Tides do not come from how hard the Moon pulls on Earth, but from the difference between its pull on the near side and on the far side. When the Moon comes closer, that difference grows very quickly: half the distance gives eight times the stretching, and one third of the distance gives 27 times.

That is also why the Moon, though small, raises bigger tides than the distant Sun.

Gravity falls as 1/d², but tidal forcing is the gradient of gravity across Earth's diameter, about 2GMR/d³, so it falls as 1/d³. That steeper dependence explains the factor of eight at half the distance and 27 at a third.

The model's limits: an ideal global ocean that responds instantly, circular orbits in the equatorial plane, and bodies drawn not to scale. A much closer Moon would also slow Earth's spin faster and change its own orbit over time; the model shows only the tide strength at one moment.

The Moon's distance changes a little every month. When it is closest, tides are a bit bigger.

Long ago, the Moon was closer than it is now.

You can see a small version of this experiment every month. The Moon's path is slightly stretched, so its distance changes by tens of thousands of kilometers. NOAA says that when the Moon is closest, the tides are higher than usual, and when it is farthest, they are smaller.

Over long times, the Moon is moving away. Tides rubbing against the Earth slowly brake its spin, and the Moon spirals outward. So, long ago, the Moon was closer than it is today.

Over a year, the lunar distance ranges from about 357,000 km at the closest perigees to 407,000 km at the farthest apogees. Put into the model, those give lunar tide factors of about 1.25 and 0.84: perigean tides are noticeably stronger, as NOAA describes.

Tidal friction lengthens Earth's day by about 0.002 s per century, and as OpenStax explains, the angular momentum lost by Earth's spin pushes the Moon outward, so the Moon was closer in the past. How close, and how big early tides were, depends on Earth's ancient oceans, an idea beyond this model.