The ellipsoid is a level surface

The defining document for WGS84 runs to a couple of hundred pages and prints something like two dozen constants: the semi-major and semi-minor axes, the flattening and its reciprocal, the first and second eccentricities, equatorial and polar gravity, the mean gravity, the theoretical potential of the ellipsoid, the dynamical form factor, the mass of the atmosphere.

Four of them are definitions and the rest are consequences. That is stated in the document and is easy to read past, because a table of constants looks like a table of measurements.

The claim #

Fix four numbers — the equatorial radius $a$, the flattening $f$, the geocentric gravitational constant $GM$, and the angular velocity $\omega$ — and require the resulting ellipsoid to be an equipotential surface of its own gravity field. That single requirement determines the entire external field, in closed form, with no series and no fitting.

Everything else in the table follows. This essay computes five of them and compares each with the published value.

What was computed, and how #

The construction is the level ellipsoid: a body whose surface is a surface of constant potential $U = V + \Phi$, where $V$ is the gravitational part and $\Phi$ the centrifugal. The mass distribution inside is not specified and does not need to be — only the total $GM$ matters to the field outside, which is the same reason a spherical shell’s field is that of a point.

In ellipsoidal-harmonic coordinates $(u, \beta)$, where surfaces of constant $u$ are confocal ellipsoids and $u = b$ is the reference ellipsoid itself, the potential has three terms and no series:

$$U(u, \beta) = \frac{GM}{E}\arctan\frac{E}{u} + \frac{1}{2}\omega^2 a^2 \frac{q}{q_0}\left(\sin^2\beta - \frac{1}{3}\right) + \frac{1}{2}\omega^2(u^2 + E^2)\cos^2\beta$$

with $E$ the linear eccentricity and $q$ a Legendre function of the second kind, which is an arctangent and two rational terms. Differentiating along the normal at $u = b$ gives equatorial and polar gravity in closed form, and Somigliana’s formula interpolates between them exactly across latitude.

The results, against the published constants:

Every one agrees to the last digit published. The differences are the published values’ own rounding, not a disagreement.

The one that is not obvious #

Four of those five are geometry and arithmetic. The fifth is a statement about the Earth.

$J_2$ — the dynamical form factor — is the coefficient of the second-degree zonal term in the Earth’s external gravity field, and it is the largest departure of that field from a point mass’s. It is measured, and measured very well, by watching how satellite orbits precess: $J_2$ makes the plane of an orbit rotate at a rate that can be timed over years, which is how sun-synchronous orbits are designed. It is the one quantity in the table that a reader of what a coordinate refers to would not expect to be derivable from a shape at all.

The closed form for a level ellipsoid gives

$$J_2 = \frac{e^2}{3}\left(1 - \frac{2}{15}\frac{m\,e'}{q_0}\right), \qquad m = \frac{\omega^2 a^2 b}{GM}$$

and evaluating it on WGS84’s four defining constants returns $1.0826298213 \times 10^{-3}$, against a published value obtained from satellite tracking of $1.0826298213 \times 10^{-3}$.

That is the shape of the body predicting the dominant term of its own gravity field, to eleven significant figures. Shape and spin determine the field, and that agreement is the reason the level ellipsoid is a useful idealisation of the Earth rather than merely a convenient one.

The agreement is not a coincidence and not quite a discovery either: WGS84 chose $f$ so that this would hold, which is honest and is worth stating. GRS80 goes the other way — it defines $J_2$ and derives $f$ — and running the same derivation on GRS80’s constants reproduces its published flattening. Neither system is more principled; what the pair demonstrates is that the relation is exact and can be run in either direction.

Running the derivation on a system it was not tuned for #

That figure is why the generator takes the reference system as a parameter rather than drawing the Earth. A derivation that has only ever been run on the case it was written for is indistinguishable from a lookup table with extra steps. Running it on GRS80 — whose $GM$ differs in the seventh digit and whose flattening differs in the ninth — reproduces GRS80’s own published constants, which is the evidence that four numbers are going in rather than a memory of the answer.

The same figure carries a small caution. The two systems are geodetically identical to within a tenth of a millimetre, so nothing in ordinary practice distinguishes them, and treating a GRS80 coordinate as a WGS84 one is one of the very few things in this field that is genuinely harmless.

The level surfaces are not parallel #

One consequence of having the potential in closed form is worth taking immediately, because it is the reason the vertical has its own machinery at all.

A surface of constant potential converges towards the pole, and it converges by 5.28 metres in a thousand — which is $5.28 imes 10^{-3}$, and is the gravity flattening $f^* = (\gamma_p - \gamma_a)/\gamma_a$ to within second-order terms. So a quantity defined as a fractional difference in a force turns out to be the geometric flattening of the surfaces a spirit level follows. That is a satisfying identity and it is asserted rather than admired: the measured convergence and $f^*$ are required to agree to two per cent.

The second curve is the reason the potential was written out exactly rather than expanded. The first-order height $\Delta U/\gamma$ is what a textbook gives, and it is 15.8 centimetres low at a kilometre — four times the tolerance a first-order levelling network is run to. It gets the convergence right to three millimetres, so it is a good approximation to the shape and a poor one to the offset, which is exactly the kind of distinction that gets lost when only one number is quoted.

Somigliana against Gauss #

Everything above shares one derivation, so agreeing with itself proves nothing. The site’s habit is two independent routes, and gravity has an excellent second one.

Gauss’s divergence theorem relates the flux of a field out of a closed surface to what is inside it. For gravity on a rotating body:

$$\oint_S \gamma \,\mathrm{d}S = 4\pi G M - 2\omega^2 V$$

The left-hand side is Somigliana’s formula integrated over the ellipsoid’s surface. The right-hand side is a statement about the mass and the volume inside, and it shares no line of algebra with the formula for the field at a point.

They agree. Integrated at 16,000 latitude bands, the two sides differ by $1.6 \times 10^{-9}$ relative — and the assertion does not stop there, because a single resolution cannot tell a small disagreement from a coarse integral. The check is run at two resolutions and requires the residual to fall by four when the sample count doubles, which is the signature of midpoint quadrature and is not available to a genuine gap between the two sides. It comes out at 4.000008.

That is the difference between “the residual is below a number somebody chose” and “what is left is the integration, demonstrably”.

What the field does with height #

The free-air gradient is the derivative of the field with respect to height, and it is worth two paragraphs because getting it out of the machinery took a correction that is instructive.

The first attempt differenced the exact potential across a metre. That is mathematically correct and numerically hopeless: the potential is $6.3 imes 10^7$ and changes by 9.8 across a metre, so differencing throws away nine of the sixteen digits double precision offers. It returned $-0.13$ milligal per metre at the equator and $\mathbf{+0.39}$ at the pole — the wrong size and, at one end, the wrong sign, which would have had gravity increasing with altitude.

The fix is to use the closed-form expansion in height, and the check is the interesting part: the expansion is compared against the exact potential over a two-kilometre baseline, where the differencing is conditioned well enough to be worth believing, and against the mean of the expansion over that interval rather than its value at either end — because the mean is what a potential difference measures, and it is sensitive to the second-order term that separates a proper expansion from a first-order one. The two agree to a part in $10^7$.

This is the same failure mode the transverse Mercator series records from the other direction: an expansion that looked right and was wrong by 11 millimetres, caught only because an independent quadrature was run beside it.

Where the model stops #

The level ellipsoid is not the Earth. It is the best equipotential ellipsoid, and the Earth’s actual equipotential surface — the geoid — departs from it by up to about a hundred metres. Everything in this essay is exact about a body that is not quite the planet, which is exactly why it is useful: it supplies a reference field known to eleven digits, against which the real field’s departure can be measured rather than merely described.

Nothing here has been fitted to gravity data. The four defining constants are published inputs, and $GM$ and $\omega$ are measured quantities — $GM$ from satellite orbits, $\omega$ from timing the rotation. What is computed is the field they imply. A reader who wants the Earth’s own field needs a spherical-harmonic model, which is a data product with a truncation degree in it, and height above what? sets out why this site does not use one.

The atmosphere is inside $GM$. WGS84’s $GM$ includes the mass of the atmosphere, so the field computed here is the field outside all of it. There is a separate published constant for the mass of the atmosphere alone, used when the two need separating for satellite work, and neglecting the distinction is a part in $10^6$ of $GM$.

A level surface is not the geoid. The surfaces computed here are level surfaces of the normal field, which is smooth by construction. The Earth’s own level surfaces are bumpy at the same scale as the geoid, so the convergence computed here is the systematic part of the effect and not the whole of it — a distinction that matters for a levelled height and does not change its order of magnitude.

Tides are not in this at all. The solid Earth deforms under the Sun and Moon by decimetres twice a day, and the potential changes with it. There are three separate conventions for whether a published height or gravity value includes that effect, and they differ by centimetres in height — enough to matter and small enough to be dropped silently.

The generalisation #

Two portable statements, and the second is the one this collection keeps returning to.

Distinguish what a table defines from what it derives. The same distinction runs through the seven parameters, where three published units hide the fact that all seven quantities are the same size of thing. A published set of constants usually mixes the two, and the mixture is invisible because both are printed to the same number of digits in the same units. The consequence of not distinguishing them is a false sense of how much is being assumed: a system with four free parameters looks like a system with twenty-four, and a reader has no way to tell which numbers could have come out differently.

There is a practical test. Re-derive the derived ones. If the derivation reproduces them, the structure is confirmed and the defining set is identified; if it does not, either the derivation is wrong or the table is not what it claims to be. That is a cheap check on any published parameter set and it is almost never run, because a table of constants does not look like a claim.

A verification is only worth its independence. Recomputing Somigliana’s formula a second way would prove nothing. Integrating it over a closed surface and comparing with the mass inside shares no algebra with it, which is why the agreement is evidence. The same discipline runs through measuring curvature from inside, where Gaussian curvature is computed both from the embedding and from the metric alone and the two are required to agree — which is Gauss’s Theorema Egregium exercised on every build rather than cited.

And the corollary, which is the part that took an attempt to get right: a numerical check needs to distinguish its own error from the effect it is testing. Requiring a residual to fall as the resolution improves does that; requiring it to be below a threshold does not.

That figure is the bridge to the other half of this argument. Everything above takes $f$ as given and derives the field. The flattening is not a free parameter runs the relation the other way: the spin and the mass predict the shape, to within a term whose size can be measured, and the prediction was made a century before anybody could check it.

What the field says about the shape it came from #

Setting the geometry beside the field is the whole argument in two pictures. The shape determines the field exactly, through the requirement that the surface be an equipotential — and the test of that is $J_2$, which is measured from orbits and reproduced here from the shape.

Who found it, and when #

Isaac Newton argued in the Principia that a rotating fluid Earth must bulge at the equator, and estimated the flattening at 1/230 by balancing two columns of fluid. Christiaan Huygens got 1/578 from a different assumption about where the mass sits. Both were reasoning from equilibrium rather than from measurement, and both were closer than most of the surveying that followed.

Alexis Clairaut’s Théorie de la figure de la Terre of 1743 turned this into a theorem relating the shape of a rotating body to the gravity measured on its surface, which is the subject of the flattening is not a free parameter. Pierre-Simon Laplace and Adrien-Marie Legendre supplied the functions that make the exterior field expressible; the $q_0$ in the formula above is a Legendre function of the second kind.

Carlo Somigliana published his closed formula for gravity on a level ellipsoid in 1929, replacing the series expansions that had been used until then. It is exact rather than truncated, and it is the reason nothing in this essay is a first-order result in the flattening.

The Geodetic Reference System of 1967 was the first to define a system by four constants and derive the rest, and GRS80 and WGS84 follow that pattern. It is worth appreciating what the pattern asserts: that the external gravity field of the Earth is determined, to the precision anybody needs, by its size, its shape, its mass and its spin.

That assertion is worth reading precisely, because it is exactly true and about the wrong Earth. Four constants determine the normal field completely — the field of the level ellipsoid, which is a mathematical body chosen to be close to the planet. They determine the real Earth’s field not at all. Everything the actual gravity field does beyond those four numbers is, by definition, the disturbing potential, and the geoid is its level surface. Read the other way, the definition is what makes the departure measurable at all: a difference needs something to be different from.

So the elegance of the four-constant definition is not a claim that the planet is simple; it is the construction that isolates how far the planet departs from a body that is.

None of which is a criticism of the definition; it is what the definition is for.

Where this goes next #

The field is now available exactly, so its consequences can be computed. The first is that the surfaces of constant potential are not parallel to each other, which means a levelled height is not a distance — and the correction is larger than the error of the survey that produced it. That is a levelled height is not a distance.