The flattening is not a free parameter

An ellipsoid arrives in this subject as two numbers: an equatorial radius and a flattening, both fitted to measurements. That is how a datum is fitted to a region treats them, and it is how the nineteenth century treated them, and it is not the whole story.

The flattening is not free. A rotating body held together by its own gravity has a shape determined by how fast it spins and how much of it there is, and the relation between the two is a theorem — proved in 1743, a century before anybody could measure well enough to test it.

The theorem #

Clairaut’s result relates three quantities, all dimensionless:

• $f$, the geometric flattening, $(a - b)/a$ — how much shorter the body is through the poles;
• $f^*$, the gravity flattening, $(\gamma_p - \gamma_a)/\gamma_a$ — how much stronger gravity is at the pole;
• $m$, the ratio of centrifugal to gravitational acceleration at the equator, $\omega^2 a^2 b / GM$.

And the theorem is

$$f + f^* = \frac{5}{2}m$$

which is remarkable for what it does not contain. There is nothing about the density of the interior, nothing about whether the body is fluid or solid, nothing about how the mass is arranged. Two observable properties of the surface and one ratio involving the spin, and they are locked together.

On WGS84:

The two sides agree to 0.36 per cent.

The residual is the interesting number #

A theorem that holds to 0.36 per cent is either a good approximation or a bad theorem, and which one it is depends entirely on whether the 0.36 per cent can be accounted for.

It can. Clairaut’s result is first order in the flattening, meaning terms of order $f^2$ have been dropped. So the residual should be of order $f^2$, which is $1.124 \times 10^{-5}$, and the measured residual is $3.079 \times 10^{-5}$ — 2.74 times $f^2$.

That is exactly what a first-order theorem is entitled to be wrong by. A coefficient of order one, times the square of the small quantity.

This is why the assertion in the machinery is two-sided rather than one. It requires the residual to be less than four times $f^2$, which would fail if the theorem were wrong or the arithmetic broken; and it requires the residual to be more than a twentieth of $f^2$, which would fail if somebody quietly replaced the measured $f^*$ with the theorem’s own prediction and produced a perfect agreement that meant nothing.

An assertion that only checks the upper bound is satisfied by an implementation that has stopped measuring anything.

What was computed, and how #

Nothing here is a series. The quantities in the table come from the closed forms of the ellipsoid is a level surface:

$$\gamma_a = \frac{GM}{ab}\left(1 - m - \frac{m}{6}\frac{e' q_0'}{q_0}\right), \qquad \gamma_p = \frac{GM}{a^2}\left(1 + \frac{m}{3}\frac{e' q_0'}{q_0}\right)$$

with $q_0$ and $q_0'$ Legendre functions of the second kind evaluated on the ellipsoid — arctangents and rational terms, exact. So $f^*$ is computed from $a$, $f$, $GM$ and $\omega$ and is not measured gravity; the theorem is being tested against the level ellipsoid rather than against the Earth.

That is a genuine limitation and it is worth being clear about. What is verified is that the closed-form level ellipsoid satisfies Clairaut’s theorem to second order. That the Earth does is a separate claim, resting on the observation that the real gravity field’s second-degree term agrees with the level ellipsoid’s — which is the $J_2$ agreement in the previous essay, to eleven significant figures.

Running it on a different body #

The theorem contains no property of the Earth, so it should hold on anything that rotates and holds itself together — and the machinery can be pointed at a different reference system to check that it is not tuned to one case.

Beyond the Earth the theorem’s reach is a matter of how large $f$ is. Mars, at $f = 1/170$, sits comfortably inside the first-order regime. Jupiter, at $f = 1/15$, does not: $f^2$ is $4 \times 10^{-3}$, so a first-order theorem is wrong in the third significant figure, and planetary work uses higher-order theories of figure for exactly that reason.

That is the useful way to hold a first-order result. It is not approximately true; it is true to a stated order, with an error whose size is predictable, and the prediction says which bodies it can be used on.

What the flattening does to everything downstream #

It is worth seeing what a determined flattening determines, because $f$ is not a decorative constant — it is in every quantity this collection computes.

The degree length, the two radii of curvature, the auxiliary latitudes, the meridian arc, the Gauss–Krüger series and every projection’s scale factor on the ellipsoid are all functions of $f$. So a theorem that determines $f$ from the spin and the mass determines all of them, which is a strong statement about how little independent information the shape of the Earth actually contains.

Two numbers, $a$ and $f$, describe the reference figure. One is a size and the other is a consequence. What is genuinely free about the Earth’s shape, once its mass and rotation are given, is one length.

Where the model stops #

Clairaut’s theorem assumes hydrostatic equilibrium. The body has to be in balance under its own gravity and rotation, with surfaces of constant density coinciding with surfaces of constant potential. The Earth is close to that and not exactly: mantle convection maintains density variations that are not hydrostatic, and the observed flattening exceeds the hydrostatic prediction by about one part in three hundred. That discrepancy is a real geophysical measurement — it constrains the mantle’s viscosity — and it is a hundred times larger than the second-order term this essay accounts for.

The theorem is not the whole relation. Clairaut’s is the first-order result; Radau’s approximation and the higher-order theories of figure extend it, and the modern treatment is a numerical integration of the Clairaut differential equation through a density model. The clean two-line statement is the leading term of something with more in it.

$f^*$ here is the level ellipsoid’s, not the Earth’s. Stated above and worth repeating, because it is the difference between a mathematical check and a physical one.

The rotation rate is not constant. The length of the day varies by milliseconds on seasonal timescales and is lengthening by about 2 milliseconds per century from tidal friction. Both change $m$, and both are far too small to matter here — but the Earth’s shape does respond to changes in rotation, over geological time, which is one of the reasons the hydrostatic figure is not quite the observed one.

Why the theorem has no density in it #

The absence of any interior property from the statement is the part worth dwelling on, because it looks too good to be true and there is a clean reason for it.

The external gravity field of any body depends on its mass distribution only through the moments of that distribution. To first order in the flattening, the field outside is fixed by two numbers: the total mass, and the difference between the polar and equatorial moments of inertia. The shape of an equilibrium surface depends on the same two, because an equilibrium surface is a level surface of that field plus the centrifugal term.

So both sides of Clairaut’s relation are functions of the same two moments, and the moments cancel between them. A body with a dense core and one with uniform density will have different values of $f$, different values of $f^*$, and the same $f + f^*$ for a given $m$.

That is a general and useful shape of argument. When two observables depend on a hidden quantity in the same way, their combination is independent of it — and finding that combination turns two measurements that each need a model into one that needs none.

The higher-order theories lose the property, which is why they are harder. At second order the field depends on the fourth moment as well, the two sides stop depending on the interior identically, and the relation acquires a term that carries a density-distribution parameter. That parameter is what the residual measured here is mostly made of, and measuring it is how the mantle’s density profile is constrained.

The generalisation #

Two things worth carrying out of this, and the second is the more useful.

A parameter that looks free may not be. An ellipsoid is presented as two fitted numbers, and one of them is a consequence of the body’s mass and spin to within a part in three hundred. That changes what a measurement of it means: measuring the flattening well is not merely characterising a shape, it is testing whether the body is in hydrostatic equilibrium, and the residual is the interesting output.

The general habit is to ask, of any fitted parameter, whether something else in the system already determines it. If it does, the fit is over-parameterised and the discrepancy between the fitted and the predicted value is a measurement in its own right — often a more informative one than the parameter.

A first-order result should be checked against its own order. This is the discipline that runs through this whole collection and it has a precise form:

• the neglected terms are $O(\varepsilon^2)$, so the residual should be $O(\varepsilon^2)$;
• a residual much larger than that means an implementation error;
• a residual much smaller means either a cancellation worth understanding or, more often, that the check has stopped being independent.

The third case is the one that catches people, and it caught this collection once already: the Chebyshev criterion reported a scale spread three parts in a thousand below a proved bound, and the explanation was a sampling artefact rather than a better-than-theoretical result. A result better than a theorem is a bug every time.

Which is why the assertion here has a floor as well as a ceiling. An agreement that is too good is evidence of a broken test, and testing for it costs one line.

Reading a table of ellipsoids after this #

The practical consequence is a way of looking at the list of historical ellipsoids that a datum is fitted to a region sets out.

Airy 1830 gives $1/f = 299.32$. Bessel 1841 gives 299.15. Clarke 1866 gives 294.98. The International Ellipsoid of 1924 gives exactly 297. WGS84 gives 298.257.

Two things are visible once the theorem is in hand. First, the spread — from 295 to 299.3 — is about 1.5 per cent, which is far larger than the second-order residual of Clairaut’s relation and is therefore measurement error rather than any real variation. There is one flattening and these are attempts at it.

Second, the 1924 figure being exactly 297 is a tell. A physical constant does not come out at a round number; that value was chosen, from Hayford’s work, for the convenience of a table. It is the clearest signal in the list that these are conventions adopted by committees as much as measurements.

The theorem also supplies the correct order of magnitude independently. With $m = 3.45 \times 10^{-3}$ and $f^*$ around $5.3 \times 10^{-3}$, the theorem gives $f \approx 3.3 \times 10^{-3}$, or $1/f \approx 300$ — from the length of the day, the mass of the Earth and a gravity measurement at two latitudes, with no surveying at all.

The measurement the theorem replaces #

Clairaut’s route to the flattening — from gravity — and the surveying route — from arc lengths — are genuinely independent, and comparing what each costs is instructive.

The gravimetric route needs a pendulum at two widely separated latitudes and a value for $m$, which needs the length of the day and the mass of the Earth. The geodetic route needs two triangulation chains, each running a degree of latitude across difficult country, each with its own baseline.

The gravimetric route is enormously cheaper and it is also better conditioned: $f^*$ is $5.3 imes 10^{-3}$ of a quantity that can be measured to parts in $10^5$ with a pendulum, so the flattening comes out to a per cent or so from a single campaign. The geodetic route amplifies its input error by 118. The figure of the Earth was measured follows that amplification to the point where the answer changes sign.

Both were pursued through the eighteenth and nineteenth centuries, and the geodetic one got the attention because it was heroic — expeditions to Lapland and Peru — while the pendulum work looked like instrument-tending. The conditioning says the instrument-tending was the better experiment.

The relation on the other reference system #

What a determined flattening determines is then the whole geometry: the curvature of the reference figure at every latitude is a function of $a$ and $f$ alone, and Clairaut’s theorem has just fixed the second of them from the spin and the mass.

That closes the loop this essay opens. The spin and the mass fix the flattening; the flattening fixes the geometry; and the geometry, put back through the requirement that the surface be an equipotential, reproduces the gravity field that was the starting point.

Who found it, and when #

Alexis Claude Clairaut was nineteen when he was elected to the Académie des Sciences and twenty-three when he went to Lapland with Maupertuis to measure the arc that settled the Earth’s shape — the expedition in the figure of the Earth was measured. Théorie de la figure de la Terre followed in 1743.

What makes it a landmark is that it derives a property of the Earth from mechanics rather than from surveying. Newton had argued that a rotating fluid body must bulge and estimated $f = 1/230$ from a hydrostatic argument about two columns of fluid; Huygens got $1/578$ from a different assumption about the mass distribution. Both were reasoning about a specific model. Clairaut’s theorem is model-independent: whatever is inside, the surface flattening and the surface gravity are related in the same way.

The test had to wait for gravity to be measurable at widely separated latitudes, which meant pendulum campaigns through the nineteenth century — a pendulum’s period gives $g$, and a pendulum can be carried on a ship. Those campaigns confirmed the relation and then, as precision improved, revealed the departure from it, which is where geophysics enters.

The vocabulary is worth a note. “Clairaut’s theorem” refers to at least three different results in mathematics, and the other two — on the equality of mixed partial derivatives, and on the constancy of $\cos\varphi \sin\alpha$ along a geodesic, which appears in the great circle vertex — have nothing to do with this one. Same person, three theorems, one name.

What the residual is evidence about #

The theorem’s model-independence is what makes its residual worth having, and the point is worth stating on its own because it inverts what the relation appears to be for.

A relation that holds whatever is inside cannot say what is inside. That is the strength of the theorem — it needed no assumption about density, which is why Clairaut’s result outlived Newton’s and Huygens’s competing estimates, both of which did.

So a departure from it is evidence, and evidence of one specific thing. The theorem describes a body in hydrostatic equilibrium: a fluid whose surface has settled into the shape its own rotation and gravity require. Every part of the derivation is about that settling. A real planet that does not satisfy the relation is therefore not in hydrostatic equilibrium, and the size of the departure measures how far from it the body sits.

The Earth’s departure is about one per cent of its flattening, which is small enough to have been invisible until gravity could be measured precisely and large enough to be one of the standing facts of geophysics. What holds the extra bulge up is the strength and the circulation of the mantle — a solid that flows over geological time and does not flow over the time a shape would take to relax.

Which turns the residual into an instrument rather than an error. A body’s flattening compared against its own hydrostatic value says whether it is behaving as a fluid, and that is a question about the interior asked entirely from the outside, with no assumption about the interior anywhere in it.

Where this goes next #

The theorem says the flattening is determined. It does not say anybody knew what it was: measuring it took two expeditions to opposite ends of the Earth, and the measurement is so badly conditioned that a hundred metres of error in an arc moves the answer by ten per cent. The figure of the Earth was measured inverts two arcs for the shape and finds the point at which the answer changes sign.