Oblate spheroid: geometry, causes, and examples

Geometric definition and parameters

Formally an oblate spheroid is a special case of an ellipsoid: two principal axes are equal (the equatorial radii) and the third (the polar radius) is shorter. That relationship places oblate spheroids within the family of ellipsoids but distinguishes them from triaxial ellipsoids and from prolate spheroids (which are elongated, not flattened). Typical notation uses a for the equatorial radius and c for the polar radius with a > c. Important derived quantities include the flattening f = (a - c)/a and the eccentricity e = sqrt(1 - c^2/a^2). The volume of an oblate spheroid has a simple closed form: V = (4/3)π a^2 c. The surface area is more complicated and involves elliptic functions.

Physical cause: rotation and equilibrium shapes

Oblateness arises when a self-gravitating body rotates. Centrifugal force reduces the effective gravity at the equator, so an initially spherical fluid body adjusts to a figure of equilibrium that is flattened. Slowly rotating bodies approximate an oblate spheroid; faster rotation can lead to larger flattening or to other equilibrium figures. Mathematical models of rotating, self-gravitating fluid masses include the Maclaurin spheroids (axisymmetric oblate shapes) and, beyond certain spin thresholds, non-axisymmetric figures such as Jacobi ellipsoids. The axis of symmetry is the rotation axis and the symmetry property itself is known as rotational symmetry.

Examples and scale

• Planets: Many planets are well approximated by oblate spheroids. The planets of the Solar System display varying degrees of flattening; gas giants typically show greater bulging than rocky worlds.
• Earth: The actual shape of the Earth is close to an oblate spheroid — its departure from a perfect sphere is small, on the order of one part in a few hundred. This modest flattening has measurable effects on gravity and sea level.
• Stars and compact objects: Rotating stars, including the neutron stars and other rapidly spinning stellar objects, can become noticeably oblate. The extent of flattening increases with angular velocity.

Importance and applications

Recognizing that planets and many celestial bodies are oblate spheroids is crucial in geodesy, cartography and satellite navigation. Models that use an oblate reference ellipsoid allow accurate mapping, altitude measurement and orbit prediction. The non-spherical mass distribution also affects gravitational potential, causing latitude-dependent gravity, small shifts in satellite orbits and precession of orbital planes.

Distinctions and notable facts

Oblate spheroids differ from prolate spheroids (stretched along the rotation axis) and from general ellipsoids (no equal axes). Although the term describes a simple mathematical form, real planets deviate in detail because of topography, crustal density variations and tides. For further introductions and technical treatments see basic references and survey material on planetary figures and geodesy; for example a general discussion of oblate forms and their properties is available via a basic planetary overview oblate spheroids and broader resources on planetary shapes and measurements equator, circumference, and mapping considerations ellipsoids. For historical and observational context consult standard texts on rotating fluid equilibria and planetary astronomy rotational symmetry, axis, and comparative examples in the Solar System planets. For extreme astrophysical cases see materials on fast stellar rotation and compact object shapes neutron stars.