Geophysics Gravity field of the Earth Theorem

Normal gravity and Somigliana's formula

Builds on Reference ellipsoid Geopotential Geodetic coordinates Eccentricity

Statement

Let $\mathcal{E}$ be an ellipsoid of revolution with semi-axes $a > b > 0$ , rotating with constant angular velocity $\omega$ about its minor axis. Let $U$ be a normal potential for $\mathcal{E}$ . This is the sum of a gravitational potential of masses inside $\mathcal{E}$ and the centrifugal potential $\tfrac12\omega^2(x^2 + y^2)$ , and it is constant on $\mathcal{E}$ . The normal gravity is $\gamma = \norm{\nabla U}$ on $\mathcal{E}$ . Let $\gamma_a$ and $\gamma_b$ be its values at the equator and at the poles. Then at each point of $\mathcal{E}$ with geodetic latitude $\varphi$ , $\gamma(\varphi) = \frac{a\gamma_a \cos^2\varphi + b\gamma_b \sin^2\varphi}{\sqrt{a^2\cos^2\varphi + b^2\sin^2\varphi}}.$

Remarks

The formula is due to Somigliana (1929). Outside $\mathcal{E}$ the gravitational part of $U$ is harmonic and is fixed by its boundary values on $\mathcal{E}$ . So it has a closed form in ellipsoidal coordinates. Differentiating along the ellipsoid normal and simplifying gives the formula. An equivalent form is $\gamma(\varphi) = \gamma_a (1 + k\sin^2\varphi)/\sqrt{1 - e^2\sin^2\varphi}$ , with $k = b\gamma_b/(a\gamma_a) - 1$ and $e^2 = 1 - b^2/a^2$ , where $e$ is the eccentricity. For the GRS 80 ellipsoid, $\gamma_a = 9.7803267715$ m/s $^2$ and $\gamma_b = 9.8321863685$ m/s $^2$ . At $\varphi = 0$ the formula gives $a\gamma_a/a = \gamma_a$ , and at $\varphi = \pm\pi/2$ it gives $\gamma_b$ , as it must.

Going deeper

For a student

The level ellipsoid of a geodetic reference system, such as GRS 80, is determined by four constants: $a$ , $f$ , $GM$ and $\omega$ . Its exterior potential is unique, and Somigliana's formula is the closed form of its gravity on the surface. Expanding it in powers of the flattening recovers, to first order, Clairaut's theorem. The difference between observed gravity and normal gravity is the gravity anomaly, the main data of physical geodesy.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Eccentricity concept
Geodetic coordinates concept
Geopotential concept
Reference ellipsoid concept
Normal gravity and Somigliana's formula statement

Tag	`Y35VMCD5K0`
URI	`m2://Y35VMCD5K0`
Kind	statement, theorem
Writing	W0 stub
MSC	86A30, 86A20
Status	This copy of the site has no build data. Run `m2 status Y35VMCD5K0`.
About	Reference ellipsoid, Geopotential
Uses	Reference ellipsoid, Geopotential, Geodetic coordinates, Eccentricity
Changed	3 November 2026