“It is not unlikely that the first remark of many who see my title The tides and kindred phenomena in the solar system will be that so small a subject as the Tides cannot demand a whole volume; but, in fact, the subject branches out in so many directions that the difficulty has been to attain to the requisite compression of my matter… The problems involved in the origin and history of the solar and of other celestial systems have little bearing upon our life on the Earth, yet these questions can hardly fail to be of interest to all those whose minds are in any degree permeated by the scientific spirit.”
—George Darwin (1898)

Our obligation to the Sun as the source of energy for life on this planet needs no elaboration. Should our appreciation also extend to the Moon? A journalist once said, “We live with the Moon, but not by it.”[1] That may be true of our attitudes toward our Moon and our awareness of it, but if the Moon did not exist, how different would life on this planet be?[2]

Astronomical bodies are connected by gravitation, which holds satellites in their orbits about stars and planets. To understand the orbits themselves we initially model these bodies as point masses. But planets and moons have finite sizes. Gravitational forces vary with distance, and gradients in the field produced by one body generate internal stresses in other bodies. For instance, the Moon’s gravity is slightly stronger on the Moon-facing side of Earth than on the opposite side, producing strains throughout Earth, the “tidal forces.” The distortions produced by these stresses show up most dramatically in the oceans, forming two bulges of water on opposite sides of the planet. The bulges lie approximately on the line connecting the centers of the Earth and the Moon.

The bulge on the side of the Earth facing the Moon makes intuitive sense, but why is there a bulge on the opposite side? The authors of one introductory astronomy textbook write that the second bulge exists “because the Moon pulls more strongly on the Earth’s center than on the far side. Thus the Moon pulls the Earth away from the oceans, which flow into a bulge away from the Moon....”[3] Other authors write that “the high tide on the far side of the Earth from the Moon occurs because the water there is ‘left behind’ as the Moon pulls the solid center of the Earth toward it.”[4] Such explanations were written for nonmathematical audiences, so the authors had to resort to plausibility arguments. “Elegant Connections” readers may prefer discussions linked directly to elementary but quantitative physics. Let’s have a go at it.

To get at the essentials of the tidal mechanism and the appearance of two tidal bulges, we will begin with an oversimplified model of the Earth. Then we will add realistic complications, including friction created by the Earth’s crust rotating beneath the oceans. The friction acts like a brake applied to Earth’s spin and slightly displaces the bulges from the Earth–Moon axis. This offset allows Earth and the Moon to exert torques on one another. The consequences include Earth’s day gradually getting longer as the Moon slowly recedes from Earth.

Geological and paleontological evidence has long supported the hypothesis that the days were shorter in the distant past. To clinch the the torque-coupling hypothesis, confirmation was needed that the Moon’s orbit is receding. This became possible in 1969, when the Apollo 12 astronauts left reflectors on the lunar surface, enabling the distance between Earth and the Moon to be measured to millimeter precision with pulsed lasers...

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