Concept

Dynamical form factor — where it appears

The coefficient describing how a body's mass is distributed between its equator and its poles, measurable from the orbits of satellites around it. Clairaut's theorem ties it to the flattening and the spin, so a shape can be predicted from a gravity field rather than assumed.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Normal gravity, derived from four constants. Gravity on the surface of the level ellipsoid, by Somigliana's closed form, for WGS84. Nothing here is measured: a, f, GM and ω go in and the whole curve comes out, rising 5186 milligal — 0.53 per cent — from equator to pole. The two open marks are the published values of equatorial and polar gravity for WGS84, which the derivation reproduces to ten significant figures rather than borrowing.

The ellipsoid is a level surface

WGS84 publishes two dozen constants and defines four of them. The other twenty are consequences — polar gravity, the potential of the ellipsoid, the coefficient that dominates the Earth's gravity field — and every one comes back here from a, f, GM and ω to the last digit published.

datums · Height

Clairaut's theorem, and the term it drops. Clairaut's theorem says the flattening of a rotating body plus the flattening of the gravity on it equals five halves of the ratio of centrifugal to gravitational acceleration at the equator. Measured on WGS84: f = 3.3528e-3, f* = 5.3024e-3, and their sum is 8.65525e-3 against the theorem's 8.62447e-3. The residual is 3.08e-5, which is 2.74 times f² — second order, which is what a first-order theorem is entitled to be wrong by.

The flattening is not a free parameter

An ellipsoid is usually presented as two numbers somebody fitted. One of them is not free — Clairaut's theorem relates the shape of a rotating body to the gravity on it, and the relation holds on WGS84 with a residual of 3.1×10⁻⁵ — which is 2.74 times f², exactly what a first-order theorem is entitled to.

datums · Ellipsoid

Shape predicted from field, against shape as published. Clairaut's theorem gives a body's flattening from two numbers of its gravity field: J₂, which is how its mass is arranged, and m = ω²a³/GM, which is how fast it spins. For a body in hydrostatic equilibrium the prediction is the shape, and the diagonal is where such a body sits. Earth is on it to 0.05 per cent — 12 metres at the pole, out of twenty-one kilometres of flattening. Mars is 12.5 per cent off it, which is 2.23 kilometres, and the excess is Tharsis: a body carrying a continent-sized volcanic load is not a fluid figure, so its ellipsoid is not one of its own level surfaces, and its zero of height has to be chosen rather than found.

A body with no sea level

On Earth the zero of height is found rather than chosen — water settles onto the equipotential surface by itself. Nowhere else has one, and the difference is measurable: Clairaut's theorem predicts the Earth's flattening from its own gravity field to twelve metres at the pole and misses Mars's by 2.2 kilometres.

datums · Bodies

Named alongside it

The objects these essays reach for when they reach for this one.

Equipotential Flattening Assertion Clairaut theorem Ellipsoid Geoid Normal gravity Verification WGS84 Closed form Conservation Convention

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