How much a world gives

A tide is a difference of gravity, and everything this collection has said about tides so far treats the body being pulled on as a passive object: the field is computed, the two bulges follow, the couple is worked out. The body’s own response has been assumed rather than derived.

It should not be. A body pulled on from one side stretches, and how much it stretches is a property of the body — of the arrangement of its density, and, if it is small enough, of the strength of the material it is made of. That response is one dimensionless number, and it is remarkably informative for something so compact: it runs over four orders of magnitude across ordinary astronomical objects, and where it can be measured it is one of the very few direct statements anybody has about an interior.

What the number is a ratio of #

The tide-raising body imposes an external potential on the body being deformed, and that potential is, to leading order, a quadrupole: it stretches along the line joining the two and squeezes across it. The deformed body then acquires a quadrupole of its own, because its mass has moved, and that new quadrupole has its own potential.

$k_2$ is the ratio of the second to the first, evaluated at the surface. It is not a length, a mass or a time; it is a number, and the whole of the body’s structure has been compressed into it.

Two conventions collide here and they differ by a factor of two, which has caused a great deal of confusion. The planetary literature’s $k_2$ is the ratio just defined and gives $3/2$ for a uniform fluid. The stellar literature’s $k_2$ is the apsidal-motion constant, half of that, and gives $0.75$. Both are correct; neither is universal; every paper has to say which. This essay uses the first, and the figures label it.

Solving it, by an equation that is nearly a hundred and fifty years old #

The response of a self-gravitating fluid to a quadrupole forcing reduces, remarkably, to a single first-order differential equation. Radau’s equation follows the logarithmic derivative of the deformation outward,

$$r\frac{d\eta}{dr} + 6\frac{\rho(r)}{\bar\rho(r)}\left(\eta+1\right) + \eta(\eta-1) - 6 = 0,$$

with $\bar\rho(r)$ the mean density inside $r$, and $\eta$ zero at the centre. The Love number then follows from its surface value alone.

Two special cases are exact and are what the integration in the hero figure was checked against. If the density is uniform, $\rho/\bar\rho = 1$ everywhere, $\eta \equiv 0$ solves the equation, and $k_2 = 3/2$. If all the mass is at the centre, $\rho/\bar\rho \to 0$, the equation relaxes to $\eta^2 - \eta - 6 = 0$, so $\eta \to 3$ and $k_2 \to 0$.

Everything else is between those, and the ordering has a physical reading. A body deforms because the tidal potential moves mass around; a body whose mass is concentrated at the centre has very little mass out where the tidal forcing is strong, so there is very little to move. Central condensation stiffens a fluid body without giving it any strength at all. There is a shortcut worth knowing, because it converts a shape into a Love number without any integration at all. To first order in the flattening, the same Radau equation connects $\eta$ at the surface to the body’s observable oblateness — so a photograph of a rotating planet and a rotation period give $\eta$, and $\eta$ gives both $k_2$ and the moment of inertia. That is the Radau–Darwin relation, and it is why the giant planets had estimated interior structures long before any spacecraft measured a gravity field. Its weakness is exactly its strength: it assumes hydrostatic equilibrium, and a body whose shape is partly frozen strength rather than fluid response returns a confident wrong answer.

For anything smaller than a planet, density is not the variable #

The Radau treatment assumes a fluid. That is right for a star, tolerable for a giant planet, and badly wrong for a moon, because a moon has strength.

For a homogeneous elastic sphere the answer is classical and short:

$$k_2 = \frac{3/2}{1 + \dfrac{19\mu}{2\rho g R}},$$

where $\mu$ is the rigidity. The group in the denominator compares the material’s strength with the pressure its own weight generates, and it decides everything. Rock and ice have rigidities of tens of gigapascals; a body has to be very large before $\rho g R$ approaches that. For the Earth the denominator is about 4.9 and $k_2$ comes out near 0.3. For a moon two thousand kilometres across it is in the hundreds, and $k_2$ collapses to a per cent or two.

That control is what makes the next figure an argument rather than an assertion.

Enceladus makes the point more sharply still, and by a different observable. It is a tenth of Titan’s radius, so $\rho g R$ is a hundred times smaller and a solid Enceladus would have $k_2$ of about two thousandths — a body essentially rigid against tides. What is observed instead is a physical libration of a tenth of a degree, four times what a body frozen through could show, and a plume of water vapour leaving its south pole. The three observations agree on a conclusion none of them states: there is a global liquid layer, and the shell above it is not attached to the core.

Notice what has and has not been shown. The measurement does not see the ocean; it sees a deformation too large for a solid. What rules out every alternative is that the alternatives were enumerated first — a formula with one free parameter, swept across four decades of it, never reaching the observed value. An argument of that shape is only as good as the enumeration, and the enumeration here rests on a homogeneous model, which no icy moon is. Layering changes the answer by factors of a few and never by the factor of thirty the conclusion turns on.

There is a second reason the small-body case is worth separating out. For a fluid body the response is instantaneous in the sense that matters — the material has no memory — and $k_2$ is a purely structural quantity. For a solid one it is not: rock and ice are viscoelastic, so what a body does under a tide depends on how fast the tide is applied. The same moon has one Love number at the orbital period and a different one at the period of its own free oscillations, and quoting a $k_2$ for a solid body without saying at what frequency is quoting half a number.

Reading it off a binary #

The stellar application is older than the planetary one and works by a completely different observable. Two stars close enough to raise tides on each other are not point masses, so their mutual orbit does not close: the quadrupole each acquires torques the other, and the line of apsides turns.

The rate depends on the fifth power of the ratio of radius to separation and linearly on the Love numbers, and on essentially nothing else about the stars. So a century of eclipse timings on a close binary, giving an apsidal period, is a measurement of stellar central condensation — one of very few there are.

Two things make this harder than it looks. The first is the subtraction just described: the same forty-three arcseconds a century that broke Newtonian gravity at Mercury is, in a close binary, a large fraction of the whole signal, and getting it wrong contaminates the interior directly. There is a famous case in which the measured apsidal motion came out several times slower than theory allowed, and the discrepancy was attributed in turn to relativity, to the stars’ internal structure, and finally — correctly — to their spin axes being tilted out of the orbital plane, which adds a third term nobody had allowed for.

The second is that the rate carries the fifth power of $R/a$, so a ten per cent error in a stellar radius is a sixty per cent error in the predicted rate. Radii from eclipsing light curves are the best there are, and they are still the limiting uncertainty.

The imaginary part, which is heat #

Everything above treats the response as instantaneous. It is not: a real body takes time to deform, so its bulge lags the forcing, and the lag is where all of tidal dissipation lives. Formally the Love number becomes complex, and the imaginary part is what is usually written $k_2/Q$.

That figure belongs to a separate argument about Io’s mantle and is quoted here only to make the point that $k_2$ has a phase as well as a magnitude, and that the two say different things. The magnitude is about how the mass is arranged; the phase is about what the material does under stress. A body can have a large $k_2$ and dissipate nothing, and the Moon’s tidal recession depends on the second while the Roche limit depends on the first.

What has actually been measured, and how #

Four kinds of observation give a Love number, and it is worth separating them because their failure modes are unrelated.

Satellite ranging. The solid Earth’s tidal bulge changes the gravity field a satellite orbits in, and tracking the satellite for years recovers the periodic term. This is the most precise $k_2$ anywhere, at a few parts in a thousand.

Spacecraft gravity science. A flyby’s Doppler residual carries the target’s quadrupole, and repeating the flyby at different points of the target’s eccentric orbit separates the part that varies — which is the tidal response. This is how Titan’s was obtained, and the measurement chain is a subject of its own.

Apsidal motion. Described above; the only method that works on stars.

Transit timing and shape. A close-in giant planet raises a tide on itself, becoming slightly ellipsoidal, which changes its transit shape at the level of parts in $10^{5}$ — and its apsidal motion, which is measurable as a drift in transit timing over a decade. A handful of hot Jupiters now have $k_2$ measured this way, and the values imply cores of a few tens of Earth masses.

The four disagree about what is easy. Ranging needs a satellite and a laser and gives the best number anywhere for the one body already best measured. Gravity science needs a spacecraft at another planet and several encounters, and returns a number whose formal precision is far better than its real one, because the tidal term is correlated with everything else in the fit. Apsidal motion needs a century, and is the only one of the four that reaches a star. Transit shape needs photometry at parts in a hundred thousand and reaches objects nobody will visit. What they share is the structure of the inference: in every case the quantity measured is a small periodic term riding on a much larger signal, and the whole difficulty is separating it from the things it is degenerate with.

Both of those extensions leave the definition untouched, which is the reason a quantity introduced for the Earth’s tides transfers to them at all.

What the number cannot say #

A single $k_2$ is one number about a whole interior, and it is worth being clear about the size of that constraint.

It cannot give a core radius. The same $k_2$ is produced by a small dense core with a stiff mantle and by a larger less dense one with a softer mantle, in the same way that one bulk density is produced by many compositions. What it gives is a single integral of the interior against a known weighting, and inverting a single integral for a function is not a well-posed problem.

It cannot separate structure from strength on a small body, because the two enter the same denominator. A moon with an unexpectedly large $k_2$ might have a liquid layer or might be made of something far weaker than assumed, and only the size of the discrepancy decides — which is why the Titan argument above turns on a factor of thirty rather than a factor of two.

And it is defined by a linear response, so it stops meaning anything when the deformation stops being small. Close to the distance at which a body comes apart, the induced quadrupole is no longer proportional to the forcing, higher harmonics matter, and the whole framework has to be replaced by a full calculation of the equilibrium shape. Two extensions of the same quantity are worth separating before the ladder, and only one of them is about small bodies.

The same number at nuclear density #

The strong-field version mentioned above deserves its own section, because it is the one place the quantity is being used to answer a question no other measurement can reach.

Two neutron stars spiralling together deform each other, exactly as any two bodies do, and the deformation grows steeply as the separation shrinks. A deformed star carries a quadrupole, and a quadrupole changes the orbital dynamics — so the rate at which the orbit sweeps through the last few hundred cycles differs from what two point masses would give.

The difference appears in the phase of the emitted signal. It is small, it accumulates, and it enters at a high order in the expansion, which means it becomes visible only in the last part of the inspiral where the separation is a few stellar radii. What is measured is a combination usually written as a dimensionless tidal deformability, and it is the same Love number scaled by the fifth power of the ratio of the star’s radius to its mass in geometrical units.

The interest is entirely in what that constrains. A neutron star’s radius is set by how stiff matter is above nuclear density, which is a question about the strong interaction that no terrestrial experiment reaches — laboratories can produce that density fleetingly and not in equilibrium. A stiff equation of state gives a larger star at fixed mass, and a larger star deforms more, so the tidal deformability is a direct readout of the stiffness.

The first measurement came from the neutron-star merger detected in 2017, and it was an upper limit rather than a determination: the signal was consistent with no tidal effect and inconsistent with the largest deformabilities, which excluded the stiffest equations of state and implied a radius below about thirteen and a half kilometres for a star of 1.4 solar masses.

That is a modest constraint by the standards of a laboratory measurement and a remarkable one by any other standard. The same quantity, defined by the same ratio of an induced quadrupole to an imposed one, is measured at one gram per cubic centimetre on a moon of Saturn and at $10^{15}$ on an object nobody will ever visit — and the two measurements are made by watching a radio signal in one case and a strain in a laser interferometer in the other.

One more body extends the comparison to the most dissipative object in the solar system.

Where the ladder goes #

The obvious next rungs are the two halves of the complex number. The magnitude leads to layered interiors: what an ocean’s depth and a shell’s thickness do to $k_2$, and why an ice shell floating free responds differently from one frozen to the seafloor. The phase leads to rheology, dissipation and the histories of orbits, which is where tidal heating and orbital evolution meet.

There is also a strong-field version. A neutron star in a binary is deformed by its companion in the last seconds before merger, and the deformation changes the gravitational waveform’s phase in a way that is measurable — so the same quantity, defined the same way, is now measured for objects at nuclear density by detectors that watch two stars spiral together. The number that told Cassini there was water under Titan tells a gravitational-wave observatory what happens to matter above nuclear density, which is not a connection anybody designed.