Astronomy & Celestial Mechanics
the sky, computed
Gravitation

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

The Moon pulls the ocean toward it. That explains one bulge. The second one, on the far side, is the whole of the physics — and it comes from subtracting.
12 min read 6 figures Small perturbations, large consequencesTwo bodies, and no more

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.

The tidal field is a differenceThe pull of a distant body at each point of a sphere, minus its pull at the sphere's centre. What remains stretches along the line to the source and squeezes across it — two bulges, not one.toward the sourcethe near side is pulled harder than the centrethe far side is pulled less — and so falls behindnot to scale: the source is far outside this frame
Fig. 1 The pull of a distant body at each point of a sphere, minus its pull at the sphere’s centre. Every arrow is that subtraction evaluated, and what remains stretches along the line to the source and squeezes across it.

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.

gtidal(r)=g(r)−g(centre).\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/r2GM/r^2 gives a 1/r31/r^3 dependence:

gtidal≈2GMRd3,g_{\text{tidal}} \approx \frac{2GMR}{d^3},

for a body of radius RR at distance dd. 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.

Phases are a viewing angle, not a shadowA satellite at eight points of its orbit. Exactly half of it is lit at every one of them; what changes is how much of the lit half faces the centre. Nothing is in shadow except at an eclipse.sunlightnewwaxing crescentfirst quarterwaxing gibbousfullwaning gibbouslast quarterwaning crescentas seen from the centrehalf lit, always
Fig. 2 The lunar cycle, which is a viewing angle rather than a shadow. Spring tides fall at the two ends of this diagram — new and full — where Sun, Earth and Moon are in line and the two tidal fields reinforce.

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 tidal field is a differenceThe pull of a distant body at each point of a sphere, minus its pull at the sphere's centre. What remains stretches along the line to the source and squeezes across it — two bulges, not one.toward the sourcethe near side is pulled harder than the centrethe far side is pulled less — and so falls behindnot to scale: the source is far outside this frame
Fig. 3 The same body with the source three times further away. The residual arrows have shrunk by a factor of nearly thirty — the cube law — which is why the Sun, overwhelmingly more massive, is the junior partner in the Earth’s tides.

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 Rplanet(ρplanetρmoon)1/3.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.

An orbit at eccentricity 0.055An orbit of eccentricity 0.055. The primary sits at a focus, offset from the centre by 0.055 of the semi-major axis, and the closest and furthest points differ by a factor of 1.12.empty focusrperiapsisapoapsis
Fig. 4 The Moon’s orbit at its true eccentricity of 0.055. The distance varies by 11% over a month, and since the tidal field goes as the inverse cube, the tide-raising force varies by about a third — which is why perigean tides are noticeably larger.

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.

Speed against distance, for orbits of the same periodOrbital speed against distance from the primary. Every orbit with the same semi-major axis follows the same curve; the eccentricity decides only which stretch of it the body uses.00.511.50123distance from the primary (a = 1)e = 0e = 0.4e = 0.8circular speed at r = a
Fig. 5 Speed against distance for orbits of the same size. Adding angular momentum to the Moon moves it outward — and, counterintuitively, makes it move more slowly, because a larger orbit is a slower one.

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.

Two bodies at a mass ratio of 81 to 1Both bodies orbit their common centre of mass, on similar ellipses whose sizes are in inverse proportion to the masses — here 81 to 1, so the heavier body's path is 81 times smaller.barycentrethe line joining them always passes through it
Fig. 6 The Earth–Moon pair at its true mass ratio of 81 to 1. The barycentre lies inside the Earth but not at its centre, so the planet genuinely circles a point 4,700 km from its own middle, once a month.

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.

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 10710^7, 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.

The ladder from here

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. Tidal heating, which keeps Io volcanic and Europa’s ocean liquid — the most consequential tidal effect in the solar system. 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.

Where this goes next

barycentrethe line joining them always passes through it Neither body is still, and the wobble is how planets are found A planet does not orbit its star. Both orbit a point between them, and the star's share of that motion is small, measurable, and the reason thousands of planets are known. sunlightnewwaxing crescentfirst quarterwaxing gibbousfullwaning gibbouslast quarterwaning crescentas seen from the centrehalf lit, always Phases are not shadows, and eclipses are Half the Moon is lit at every instant of every month. The phase is which part of the lit half faces the Earth — and confusing that with a shadow is the commonest error in astronomy. L₁L₂L₃L₄L₅L₄ and L₅ are exactlyequilateral with both bodiesthe collinear three areroots of a quintic Five places that keep station, in a problem with no solution Three bodies under gravity cannot be solved. Restrict the problem slightly and five exact answers fall out anyway — three of them roots of a quintic, two of them perfect equilateral triangles.