The tide is a difference, which is why there are two of them

There are two high tides a day, not one. That single fact defeats every intuitive account of what tides are, because the obvious explanation — the Moon pulls the water toward it — predicts one bulge, on the side facing the Moon, and says nothing at all about the other.

The second bulge is the interesting one, and getting it right means abandoning the idea that tides are caused by gravity. They are caused by differences in gravity, which is a different quantity with a different behaviour, and the moment that substitution is made both bulges appear at once.

Why the mean has to be subtracted #

The Earth is in free fall. It is not held in place while the Moon tugs at it — the whole planet is accelerating toward the Moon, continuously, and has been for four billion years.

An observer falling with the Earth does not feel the Moon’s pull, for the same reason an astronaut in a circular orbit does not feel the Earth’s. Free fall cancels a uniform gravitational field exactly, which is the observation that eventually became the equivalence principle.

What free fall cannot cancel is a field that varies from place to place. The near side of the Earth is 6,400 km closer to the Moon than the centre is, so it is pulled harder than average. The far side is 6,400 km further, so it is pulled less than average. The whole planet accelerates at the average rate, and the residual — the local pull minus the mean — is what remains to be felt.

$$\mathbf{g}_{\text{tidal}}(\mathbf{r}) = \mathbf{g}(\mathbf{r}) - \mathbf{g}(\text{centre}).$$

Subtracting the mean is the whole of it, and it produces the two bulges immediately. The near side has a residual pointing toward the Moon; the far side has one pointing away, because there the actual pull falls short of the average and the difference reverses. Around the sides, where the pull is nearly the right size but points slightly inward toward the Moon’s direction, the residual squeezes.

Two bulges, from one subtraction. The far bulge is not the water being flung outward by rotation, and it is not the water being left behind — it is the residual of a difference, and it exists whether or not anything is rotating.

Falling short of an inverse square #

Tidal force is a gradient, so it falls off faster than the force producing it. Differentiating $GM/r^2$ gives a $1/r^3$ dependence:

$$g_{\text{tidal}} \approx \frac{2GMR}{d^3},$$

for a body of radius $R$ at distance $d$. The extra power has a consequence that decides which objects matter tidally and which do not.

The Sun exerts 178 times more gravitational pull on the Earth than the Moon does. Its tidal effect is only 46% of the Moon’s, because although its mass is 27 million times greater, its distance is 390 times greater and the cube of 390 more than compensates. Distance wins, decisively, and the Moon runs the tides.

The two do combine. When Sun and Moon pull along the same line — at new and full moon — the bulges add and the tides are large, which is the spring tide. At the quarters they oppose and the range is small. That fortnightly cycle is the direct signature of two tide-raising bodies with a 1.46-to-1 ratio, and it is visible on any tide table.

The fortnightly beat is a good check on the whole account. If tides were caused by the Moon’s pull rather than its gradient, the Sun would dominate them by a factor of 178 and there would be no lunar signal at all. That the observed range tracks the Moon’s phase, at the amplitude ratio the cube law predicts, is the measurement that settles which quantity is responsible.

The line the tide will not let a body cross #

Push the tidal gradient far enough and it exceeds whatever is holding a body together.

A moon held together by its own gravity has a self-gravity that scales with its size, while the tidal stretching grows as it approaches the planet. There is a distance at which the two are equal, and inside it the moon cannot hold itself together. That distance is the Roche limit, roughly

$$d = 2.44\,R_{\text{planet}}\left(\frac{\rho_{\text{planet}}}{\rho_{\text{moon}}}\right)^{1/3}.$$

Saturn’s rings sit inside Saturn’s Roche limit and its moons sit outside it — a boundary at a computed distance, with the observed structure falling either side of it. That is not a coincidence and it is the strongest evidence that the rings are debris that was never allowed to coalesce, or a body that came too close and was pulled apart.

The mechanism has been watched. Comet Shoemaker–Levy 9, on a strongly hyperbolic approach to Jupiter, passed within its Roche limit in 1992, broke into twenty-one fragments, and struck the planet two years later in a sequence timed to the minute in advance — the fragments strung out along the orbit precisely because tidal stretching separates a body along the line to the primary.

The limit applies only to bodies held together by gravity. A rock a metre across has material strength enormously greater than its own gravity and can orbit far inside the Roche limit intact, which is why the rings are made of many small pieces and not of one large one.

The eccentricity matters more than it looks, and the cube law is the reason. An 11% swing in distance becomes a 37% swing in tidal force, so the monthly variation in tidal range is comparable to the fortnightly spring–neap cycle. When perigee happens to coincide with a new or full moon the two reinforce, and the resulting tides are the ones that flood car parks.

The torque that stops a moon turning #

The bulges do more than rise and fall. They apply a torque, and over long times that torque rearranges the whole system.

The Earth rotates once a day while the Moon takes a month to go round, so the planet’s rotation drags the tidal bulges slightly ahead of the Moon. The leading bulge is closer to the Moon than the trailing one, so it pulls the Moon forward along its orbit, adding angular momentum. The Moon’s reaction pulls back on the bulge, braking the Earth’s rotation.

Both effects are measured. The Moon is receding at 3.8 cm per year, obtained by bouncing lasers off the retroreflectors left by Apollo. The day is lengthening by about 1.8 milliseconds per century, obtained from ancient eclipse records — a Babylonian observation of a total eclipse constrains the Earth’s rotation two and a half thousand years ago, because a slower rotation puts the shadow track in the wrong place.

Angular momentum is conserved across the exchange. What the Earth loses in spin, the Moon gains in orbit, and the third law converts a larger orbit into a longer month. The transfer is the second law’s conservation statement applied to a pair rather than to one body, with friction in the oceans as the mechanism that moves it from one account to the other. That last point catches almost everyone. The tidal torque adds energy and angular momentum to the Moon, and the Moon responds by receding and slowing down. Orbits behave this way generally: adding energy to a bound orbit raises it and reduces its speed, which is the same paradox that makes orbital rendezvous so unintuitive to fly.

The endpoint of the process is tidal locking. Once a body’s rotation matches its orbital period, the bulge stops sweeping and the torque vanishes. The Moon reached that state long ago, which is why it shows one face. Pluto and Charon have both locked to each other, so each hangs motionless in the other’s sky. The Earth is on the same road and will not get there before the Sun makes the question moot.

What is actually measured #

The theory above describes an ocean in equilibrium with the tidal field: two bulges, following the Moon smoothly round the planet. No coastline in the world sees that.

What a tide gauge sees is a wave. The equilibrium bulge cannot keep up with the Moon — a shallow-water wave in an ocean four kilometres deep travels at about 200 m/s, and the tidal bulge would need to travel at 460 m/s at the equator to stay under the Moon. It falls behind, and what results is a forced oscillation running round the ocean basins, reflecting off continents and rotating about fixed points where the range is nearly zero.

So real tides are basin resonances driven by the tidal field, and the local range depends on the shape of the coast far more than on the Moon. The Bay of Fundy reaches 16 metres because its natural period is close to twelve hours; the Mediterranean manages a few centimetres.

Prediction is therefore empirical. A harmonic analysis fits a few dozen sinusoids — each at a frequency the astronomy supplies, with an amplitude and phase the local record supplies — and the resulting model is accurate to centimetres. The frequencies come from celestial mechanics; the coefficients come from watching. Every tide table in the world is that combination.

The heat, which was predicted nine days early #

The largest tidal effect in the solar system is not a tide at all. It is a furnace, and its discovery is one of the cleanest cases on record of a computation getting there first.

Flexing a body dissipates energy. A tide raised on a satellite in a circular orbit, tidally locked, is static — the bulge points permanently at the primary and nothing moves, so nothing dissipates. But give the orbit some eccentricity and the situation changes completely: the distance varies, so the bulge’s height varies, and the satellite is kneaded once per orbit. It also librates, rocking back and forth about its locked orientation because the orbital angular rate varies while the spin does not, which sweeps the bulge across the surface. Both effects turn orbital energy into heat inside the body.

Io is the extreme case. It orbits Jupiter every 1.77 days at a distance of 422,000 km, with an eccentricity of only 0.0041 — which sounds negligible and is not, because the tidal amplitude goes as the inverse cube of a very small distance and the flexing is enormous in absolute terms. Io’s surface rises and falls by about 100 metres. The dissipated power is roughly $10^{14}$ watts, some forty times the entire heat flow of the Earth, delivered into a body a quarter of the Earth’s diameter.

The prediction came before the observation, and only just. Stanton Peale, Patrick Cassen and Ray Reynolds published a paper on 2 March 1979 computing the tidal dissipation in Io and concluding that it should be “the most intensely heated terrestrial-type body in the solar system” and that widespread melting was likely. Voyager 1 reached Jupiter on 5 March and returned images on which Linda Morabito, checking navigation frames, found a plume 300 kilometres high on 8 March. Nine days between the prediction and the discovery, in an era when a paper took months to appear.

The eccentricity itself is the part that closes the argument. Tidal dissipation should circularise an orbit — that is what dissipation does — and Io’s eccentricity ought to have decayed away long ago. It survives because Io, Europa and Ganymede are locked in a 1:2:4 resonance that pumps it back, so the heating is maintained by the other two moons rather than by anything about Io. Europa’s subsurface ocean and Enceladus’s south-polar jets have the same origin, and the possibility of liquid water hundreds of millions of kilometres beyond the region a star can warm rests entirely on a difference of two gravitational pulls.

The bulge is not where the equations put it #

One correction deserves stating before the model’s limits, because the picture of two bulges following the Moon is the most familiar image in the subject and the ocean does not do that.

The equilibrium theory computes the shape a frictionless ocean covering the whole Earth would settle into, and no such ocean exists. The real oceans are interrupted by continents, are shallow compared with their width, and are being driven at a period close to their own natural periods of oscillation — so what actually happens is a forced resonant response in a set of irregular basins.

The result is that the tide propagates as a wave around a set of amphidromic points, where the range is nearly zero and the high water rotates around them once per cycle. The tidal range varies from a few centimetres in parts of the Mediterranean to sixteen metres in the Bay of Fundy, where a basin’s natural period is close to twelve hours and the response is resonant.

None of that changes the forcing, which is exactly the difference this essay is about. What it changes is the response, and the distinction is the same one that runs through every driven system: the tide-generating potential is a clean piece of celestial mechanics, and the water’s answer to it is fluid dynamics in an awkwardly shaped container.

Where the model stops #

Equilibrium. Covered above, and it is the largest failure. The two-bulge picture gets the forcing right and the response wrong.

A fluid, uniform Earth. The solid Earth also flexes, by about 30 cm twice a day. That solid tide has to be corrected for in GPS and in gravimetry, and it is one of the few places where the ground underfoot is measurably not still.

Two bodies. The Sun’s contribution is not a small correction — it is 46% — and a full treatment carries dozens of terms with periods from twelve hours to 18.6 years. The three-body character of the Earth–Moon–Sun system shows up here as it does everywhere else it appears: as a sum of small terms with no closed form behind them.

A rigid body’s response. Real bodies dissipate energy when they flex, and the dissipation is what drives the whole locking story — which is why the Moon shows one face and always has. How much a body dissipates is summarised in a number that is poorly known for almost everything.

The figures share a limitation worth naming: the bulge drawn is exaggerated by orders of magnitude. The real equilibrium tide is about half a metre on a planet 12,700 km across — a distortion of one part in $10^7$, far thinner than the line used to draw the sphere. Every tidal figure ever published has this problem, and the honest response is to say the number rather than to try to draw it.

It is worth stating once more what the subtraction buys, because it is the whole content of the subject. A uniform field produces no tide however strong it is; a weak field with a steep gradient produces a large one. That is why the Sun, pulling 178 times harder than the Moon, raises less than half the tide — and why a body falling toward a black hole is destroyed not by the pull but by the difference between the pull on its near side and its far side, a quantity that grows without bound while the field itself is still perfectly ordinary.

The ladder from here #

At a hundredth of an astronomical unit the same difference is some $10^5$ times stronger than the Sun raises on the Earth, and it is what circularises the orbit of a planet that arrived violently.

Later rungs: the tidal potential and its expansion in Legendre polynomials. Spring and neap tides worked through properly. The Roche limit derived, for fluid and rigid bodies. The Laplace resonance that maintains Io’s eccentricity, and what happens to a resonance that is broken. Tidal locking timescales. The lunar recession and its history, which has the Moon impossibly close under a naive extrapolation. Amphidromic systems and harmonic prediction. Tidal disruption of stars by black holes. And the gradient’s appearance in relativity, where tidal force is what a gravitational field is, and free fall removes everything else.

Newton got the two bulges right in the Principia and the timing badly wrong, because he assumed equilibrium. Laplace fixed it a century later by treating the ocean as a driven oscillator, which is a good illustration of how far a correct force law gets on its own.