Flattening

Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is f and its definition in terms of the semi-axes of the resulting ellipse or ellipsoid is

The compression factor is b/a in each case. For the ellipse, this factor is also the aspect ratio of the ellipse.

There are two other variants of flattening (see below) and when it is necessary to avoid confusion the above flattening is called the first flattening. The following definitions may be found in standard texts and online web texts

Definitions of flattening

In the following, a is the larger dimension (e.g. semimajor axis), whereas b is the smaller (semiminor axis). All flattenings are zero for a circle (a=b).

Identities involving flattening

The flattenings are related to other parameters of the ellipse. For example:

where is the eccentricity.

Numerical values for planets