Flattening (oblateness) of a spheroid
Flattening (or oblateness) measures how much a rotating spheroid is squashed at the poles relative to its equator, important in geodesy, planetary science and mapping.
The term flattening (also called oblateness or ellipticity) describes how much an otherwise spherical body is compressed along its axis of rotation, so that the distance from center to the poles is smaller than the distance to the equator. It applies to rotating fluid bodies and to geometric models such as an oblate spheroid.
Definition and formulas
Flattening is commonly denoted by f and defined by the simple ratio f = (a - b)/a, where a is the equatorial radius and b the polar radius. Related measures include the geometric eccentricity e, with e² = f(2 - f). For small f the two are approximately proportional, but they quantify different geometric properties.
Physical origin
Rotation produces a centrifugal force that is strongest at the equator and acts to redistribute mass outward, producing an equatorial bulge. Classical studies by Newton and later developments by Maclaurin and Jacobi described how a rotating, self-gravitating fluid reaches an equilibrium shape that is oblate rather than perfectly spherical.
Examples and scale
Planetary bodies exhibit measurable flattening. Earth is slightly oblate: its equatorial radius exceeds its polar radius by only a few parts per thousand. Giant gas planets, which rotate faster and are less rigid, show much larger flattening; for instance, Jupiter and Saturn display pronounced equatorial bulges. Stars and rapidly rotating exoplanets can also be oblate.
Uses and importance
- Geodesy and mapping: flattening is a basic parameter of terrestrial reference ellipsoids used for coordinate systems and maps.
- Orbital dynamics: the equatorial bulge perturbs satellite orbits and affects precession.
- Planetary science: measuring a body's oblateness provides constraints on its interior structure and rotation.
Related concepts and distinctions
Flattening should not be confused with terms like "polar flattening" or simply "eccentricity"; while related, eccentricity is a parameter of conic sections and ellipses, whereas flattening is a normalized difference of radii. Observationally, the pole–to–equator difference matters for climate modeling, satellite measurements and precise surveying.
Causes
Gravity alone forms spherical bodies, the oblateness results from the centrifugal force 
, which results from rotation:
with
- the mass
of a mass element - its angular velocity ω
- its distance
from the axis of rotation.
The deformation of two orbiting celestial bodies into elongated ellipsoids, which is caused by tidal friction and is directed in the direction of the mutual gravitational forces, runs counter to the flattening. Mostly, however, the flattening of the bodies caused by the centrifugal force predominates, so that the shape of the elongated ellipsoid can hardly be observed.
Interrelationships
If one considers rigid bodies of - simplifying - constant dense material, then both the gravitational force and the centrifugal force (each on a sample mass) increase linearly with the radius of the body. Therefore, the flattening of such bodies is determined only by the frequency of rotation, regardless of their radius: If a body rotates faster (= smaller period of revolution or higher angular velocity), it will flatten more than another of the same structure.
Differences of density cause, if flows are possible, concentric layering - lightest (at earth: air and water) at top, most dense at core at centre of sphere. Such bodies with different density-layers within behave likely at surface, thus flatten same amount, as long as average densities of each complete body correspond.
However, a lower average density causes a lower gravity at the surface and thus a larger oblateness at the same rotation frequency (example gas planets).
A core of higher density will therefore flatten less than the overall body with lighter higher layers.
More complex are observations of gaseous stars with zones of different rotation frequency, as they occur on the Sun (differential rotation).
When a body contracts, expands, or changes its density locally, its angular velocity generally changes with the moment of inertia, and so does the oblateness, while the angular momentum remains constant. This is particularly relevant when stars or galaxies change greatly in diameter. See also Coriolis force.
Similar to regional density irregularities causing gravity anomalies, the flattening of an entire body or even its core causes gravity flattening, i.e. the gravitational force on the surface of the body is not the same everywhere, but depends on the location.
Related articles
Author
AlegsaOnline.com Flattening (oblateness) of a spheroid Leandro Alegsa
URL: https://en.alegsaonline.com/art/35044