Clairaut's theorem

Not to be confused with Clairaut's relation.

For the theorem concerning symmetry of second derivatives of a mapping $f \colon \mathbb{R}^n \to \mathbb{R}$, see Symmetry of second derivatives.

Figure 1: An ellipsoid

File:Gnuplot ellipsoid.svg Clairaut's theorem is a general mathematical law applying to spheroids of revolution. Published in 1743 by Alexis Claude Clairaut in his Théorie de la figure de la terre, tirée des principes de l'hydrostatique, which synthesized physical and geodetic evidence that the Earth is an oblate rotational ellipsoid, it was initially used to relate the gravity at any point on the Earth's surface to the position of that point, allowing the ellipticity of the Earth to be calculated from measurements of gravity at different latitudes.

Formula

Clairaut's formula for the acceleration due to gravity g on the surface of a spheroid at latitude φ, was:

$g(\varphi) = \mathbb{G}_e \left[ 1 + \left(\frac{5}{2} m - f\right) \sin^2 \varphi \right] \ ,$

where $\mathbb{G}_e$ is the value of the acceleration of gravity at the equator, m the ratio of the centrifugal force to gravity at the equator, and f the flattening of a meridian section of the earth, defined as:

$f = \frac {a-b}{a} \ ,$

(where a = semimajor axis, b = semiminor axis).

Clairaut derived the formula under the assumption that the body was composed of concentric coaxial spheroidal layers of constant density. This work was subsequently pursued by Laplace, who relaxed the initial assumption that surfaces of equal density were spheroids. Stokes showed in 1849 that the theorem applied to any law of density so long as the external surface is a spheroid of equilibrium. A history of the subject, and more detailed equations for g can be found in Khan.

Somigliana equation

The above expression for g has been supplanted by the Somigliana equation (after Carlo Somigliana):

$g(\varphi) = G_e \left[ \frac{1+k\sin^2 \varphi}{\sqrt{1-e^2 \sin^2 \varphi }} \right],\,\!$

where,

• $e^2=\frac{a^2-b^2}{a^2}$ is the spheroid's eccentricity, squared;
• $\mathbb{G}_e,\mathbb{G}_p$ is the defined gravity at the equator and poles, respectively;
• $k=\frac{b\mathbb{G}_p-a\mathbb{G}_e}{a\mathbb{G}_e}$ (formula constant);

For Earth, $\mathbb{G}_e$ = 9.7803253359 ms⁻²; $\mathbb{G}_p$ = 9.8321849378 ms⁻²; k = 0.00193185265241 ; e² = 0.00669437999013:

$g(\varphi)=9.7803253359\left[\frac{1+0.00193185265241\sin^2\varphi}{\sqrt{1-0.00669437999013\sin^2\varphi}}\right] \,\mathrm{ms}^{-2}.$

Geodesy

See also: Theoretical gravity

The spheroidal shape of the Earth is the result of the interplay between gravity and centrifugal force caused by the Earth's rotation about its axis. In his Principia, Newton proposed the equilibrium shape of a homogeneous rotating Earth was a rotational ellipsoid with a flattening f given by 1/230. As a result gravity increases from the equator to the poles. By applying Clairaut's theorem, Laplace was able to deduce from 15 gravity values that f = 1/330. A modern estimate is 1/298.25642. See Figure of the Earth for more detail.

For a detailed account of the construction of the reference Earth model of geodesy, see Chatfield.