If the Earth Had Two Moons

What if the Earth had two moons instead of just one?  Would we still have high and low tides, would they be more or less extreme, at longer or shorter intervals, etc.?  This was another thought experiment, similar to others discussed here before, whose investigation led me to learn more about gravity and our moon’s influence on tidal forces here on Earth.

There is an interesting Universe Today article that describes in detail the possible effects of a hypothetical second moon, situated halfway between Earth and the Moon.  One comment in particular caught my eye:

“Humans would have to adapt to the challenges of this two-mooned Earth. The higher tides created by Luna would make shoreline living almost impossible — the difference between high and low tides would be measured in thousands of feet.”

Is this really true?  I was skeptical that the tidal forces resulting from a second moon would be that much greater.

Consider another interesting scenario: instead of a second moon between Earth and the Moon, what if we had a second moon, of the same size and distance from Earth, but on the other side of the planet?  What would be the effects on the tides in that case?  Would they be twice as large, or would they effectively “cancel out,” resulting in less extreme or no tides at all?

(This is a nice puzzle, or at least it was for me.  If you want to think it over, the solution is discussed after the break.  For what it’s worth, my initial intuition about this problem was wrong.)

To answer these questions, we first have to be able to accurately compute the tidal forces resulting from a single moon.  Donald Simanek provides a survey of the many misleading or downright incorrect explanations for tides found in some textbooks.  Wikipedia does a good job of describing the relevant mathematics: the acceleration at a point on the surface of the Earth due to the Moon’s gravity differs slightly from the acceleration at the center of the Earth, depending on whether the point is nearer to or farther from the Moon.  The vector difference between the two is the tidal acceleration, shown in the following figure.

Differential tidal acceleration at points on the surface of the Earth, due to the Moon’s gravity. The sizes of the Earth and Moon are shown to scale, but the distance between them is not.

Note that this has nothing to do with “inertia” from rotation of the Earth-Moon system; it is purely the variation in these differential gravitational accelerations across the surface of the Earth that causes the “bulging” of water on the near and far sides from the Moon.  (Indeed, these tidal forces are experienced throughout the body of the planet, not just on the surface.)

Now suppose that we add a second moon, say on the other side of the Earth, at the same distance.  What effect does this have on the tidal forces?  The key observation is that the net differential acceleration is simply additive for multiple external bodies.  So given that the field due to the single moon is roughly symmetric, the field due to a second moon located at the reflected image of the first is approximately the same; the result is that the tidal forces approximately double with two oppositely-located moons.

Differential tidal acceleration due to two “opposing” moons. The resulting vector field has approximately twice the magnitude of the field due to a single moon.

Coming back to the Universe Today article, in the case of a second moon halfway between the Earth and the Moon, we can do the same calculations.  I think most of the article’s commenters are correct: tidal forces would be “only” approximately 7 to 9 times greater, depending on the angle subtended by the two moons, which I don’t think would result in anything as extreme as “tides measured in thousands of feet.”