# Tides

Having recently criticized Tristan Gooley’s explanation of the tides, I felt obliged to try to do better myself. It’s a tricky job, and there are many partial and misleading explanations out there. So here goes.

Tides happen to anything that is orbiting in a gravitational field. I’m going to hone down on the Earth in a minute; but first, an orbit:

The orbiting body (“the satellite”) would travel in a straight line if no force was being applied to it. But it is under the influence of the gravity of the central body (“the primary”). The force of gravity pulls the satellite into a curved path around the primary. For a range of speeds, this causes the satellite’s path to curve enough to make it loop right around the primary and then repeat itself. (If the satellite is moving too slowly,  its curved path will come close enough to hit the primary; too quickly, and the loop will never close, allowing the satellite to escape.)

In the diagram, the satellite has precisely the right speed to move in a circular orbit at a constant distance from the primary. The primary’s gravity pulls the satellite radially inwards with a force of constant magnitude. This generates an acceleration of constant magnitude, indicated by the blue arrow. Since the acceleration is always at right angles to the satellite’s motion, the satellite’s speed doesn’t change, only its direction of travel. This induces a constant curvature in the satellite’s path, which makes it circle endlessly.

For simplicity, I’m going to deal with only circular orbits from now one, but the logic of the tides applies equally well to all orbits.

The model of a satellite whirling in circles around a stationary central body is good enough for any satellite with a mass that’s very low compared to its primary—like the International Space Station  in orbit around the Earth, for instance. But if the satellite’s mass is comparable to the primary’s, then the primary has to follow an orbit too:

While the satellite follows a large circle, the primary moves in a small circle so that the two bodies staying exactly opposite each other on either side of their common centre of gravity, which is called the barycentre (from Greek barys, “heavy”). I’ve marked it with a little cross in the diagram. Since primary and satellite both complete one orbit in the same time, the primary has a lower speed and a smaller radial acceleration, which is provided by the weaker gravity of the less massive satellite. Like a fat man balancing a child on a see-saw, the more massive primary, huddled close to the barycentre, is in balance with the lightweight satellite moving in its more distant orbit.

The Earth-Moon system has a barycentre that is actually inside the Earth. While the Moon sweeps out its month-long orbit, the Earth describes a gentle wobble during the same time period, with its centre alway on the opposite side of the barycentre from the Moon:

The centre of the Earth is therefore always accelerating gently in the direction of the Moon as it moves around its small balancing orbit.

What may not be intuitively obvious is that at any given moment every point on the Earth’s surface and within its bulk must have exactly the same acceleration (in magnitude and direction) as the centre of the Earth does. If that didn’t happen, then the various bits of the Earth would acquire relative velocities, and the Earth would change shape. (To be strictly accurate, a little relative acceleration is allowed, as the Earth flexes under the influence of the Moon’s gravity, but the net acceleration must average out to zero over time.)

Even with that logic in place, it’s still a little difficult to see immediately why a point on the Earth’s surface on the opposite side of the barycentre from the centre of the Earth should be accelerating away from the barycentre, when the centre of the Earth is accelerating towards it.

The explanation is that every point on the Earth is tracing out its own circle in space, the same size as the Earth’s orbit around the barycentre, but displaced from it. To see how that works, let’s stop the rotation of the Earth (diagrammatically) and trace the path of a single point on its surface (marked in purple) during the course of a month.

The acceleration of the purple point is always directed towards the centre of its own (purple) circle, even though it may be directed away from the barycentre. The rotation of the Earth doesn’t make any difference to this argument—the instantaneous accelerations remain the same, they’re just handed off to different points on the surface of the Earth as it rotates.

(If you’re having trouble visualizing the circular movement of the non-rotating Earth depicted in the diagram, put a coin flat on a table, put your finger on the coin, and slide the coin around in a small circle.)

So the blue acceleration arrows show what the Earth is actually doing during the course of a lunar orbit. But does the Moon’s gravity apply forces in the right direction, and of the right magnitude, to make the Earth accelerate smoothly throughout its volume in this way?

No, it doesn’t. There are two problems:
1) The Moon’s gravity decreases with distance. While it pulls on the centre of the Earth with just the right force to induce the necessary acceleration to keep the Earth in its orbit around the barycentre, it pulls a little harder on the near side of the Earth, and a little too weakly on the far side.
2) The Moon’s gravity is a central force—it radiates out from the centre of the Moon. So it’s directed a little diagonally when it pulls on parts of the Earth that don’t lie exactly on the line connecting the centres of the Earth and Moon.

That’s all shown in this diagram, with the green arrows representing the force of the Moon’s gravity laid on top of the blue arrows representing the true acceleration:

There’s a mismatch, everywhere but at the centre of the Earth, and the difference between the applied force  and the necessary force (for uniform acceleration) must be generated by internal forces within the substance of the Earth. The nature of the mismatch between applied force and real acceleration is shown with red arrows below:

These residual forces are called tidal forces, and so at last I’ve arrived at the cause of the tides. The Earth is being stretched along an axis that runs through the Moon and the barycentre, and squeezed inwards in a plane at right angles to that axis. (Even though I’ve built this argument around the Earth and its small barycentric orbit, this is a completely general result—it applies to all bodies in orbit around other bodies. They all experience tidal forces of this sort. In fact, it should be evident that it applies equally to bodies that aren’t even in orbit, but are just falling towards each other, or even sitting next to each other—all that’s required for these internal tidal forces to show up is for an object to be maintaining its shape against the forces produced by a central gravitational field.)

Now, if the Earth was a hunk of solid metal, held together by its internal chemical bonds, it would develop a bit of tension along the “stretch axis”, and compression in the “squeeze plane”. Those internal forces would oppose the tidal forces, and ensure that all the parts of the Earth moved together with uniform acceleration.

But objects on the scale of planets aren’t held together primarily by chemical bonds—what keeps them together is their own gravity, and they settle into an equilibrium shape that evens out internal pressures. The red arrows in the diagram show that the Moon’s gravity opposes the Earth’s own gravity along the “stretch axis”, and supplements it in the “squeeze plane”. This slight alteration in the local gravitational force means that the solid body of the Earth shifts slightly in shape in order to equalize its internal pressures.

The same thing happens to the oceans—they pile up under the reduced gravity of the “stretch axis”, and squash down under the increased gravity of the “squeeze plane”:

And that’s where tides come from, and why there are two tidal bulges in the ocean, one under the Moon and one opposite it.

As the Earth rotates, it carries us past each tidal bulge in turn, so there are two high tides per day. Or, actually, not quite. By the time the Earth has completed one full rotation, the line between Earth and Moon has shifted a little, and the tidal bulge has shifted with it. The Earth therefore needs to rotate for another 50 minutes at the end of each day, in order to catch up with the position of the tidal bulges:

So instead of experiencing a high tide every 12 hours, we get one every 12 hours and 25 minutes.

The situation is actually (you guessed it) a little more complicated—the presence of landmasses distorts the even flow of water suggested in my diagram; the Sun produces its own tidal bulges; and the inclination of the Moon’s orbit to the Earth’s equator introduces its own complexities.

Those are topics for another day.

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