prolateness

Prolateness is the property of a three-dimensional body's shape that describes elongation along one axis relative to the others. In geometry and astronomy, it is used to characterize ellipsoids and astrophysical objects whose mass distributions depart from sphericity toward a prolate spheroid, where one axis (the major axis) is longer than the other two, which are equal or nearly equal.

For a triaxial ellipsoid with semi-axes a ≥ b ≥ c, prolateness can be quantified by the dimensionless index P = (a^2 − b^2) / (a^2 − c^2). This index equals 1 for a perfect prolate spheroid (b = c, a > b), equals 0 for an oblate spheroid (a = b > c), and lies between 0 and 1 for intermediate shapes with a > b > c.

Measurement: Prolateness is typically inferred from the inertia tensor of the object’s mass distribution or from imaging data by fitting an ellipsoid to the observed shape. Projection effects can complicate interpretation; additional information from kinematics or gravitational lensing can help recover the three-dimensional shape.

Applications: In astronomy, prolateness describes the shapes of galaxies and dark matter halos, which affects dynamics and lensing signals. In geoscience and planetary science, similar concepts describe the elongation of bodies such as asteroids or planets relative to a reference sphere.

See also: Ellipsoid, Oblateness, Triaxiality, Inertia tensor.