Flattening - AbsoluteAstronomy.com

The flattening, ellipticity, or oblateness of an oblate spheroid is a measure of the "squashing" of the spheroid's pole

Geographical pole

A geographical pole is either of the two points—the north pole and the south pole—on the surface of a rotating planet where the axis of rotation meets the surface of the body...

, towards its equator

Equator

An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

. If a is the distance from the spheroid center to the equator and b the distance from the center to the pole then

First order

The first, primary flattening, ƒ, is the versine

Versine

The versine or versed sine, versin, is a trigonometric function equal to and 2sin2. It appeared in some of the earliest trigonometric tables and was once widespread, but it is now little-used...

of the spheroid's angular eccentricity
Angular eccentricity
In the study of ellipses and related geometry, various parameters in the distortion of a circle into an ellipse are identified and employed: Aspect ratio, flattening and eccentricity....

α, equalling the relative difference between its equatorial radius, a, and its polar radius, b:

Second and third orders

There is also a second flattening, f' ,

and a third flattening, f' ' (sometimes denoted as "n" – a notation first used in 1837 by Friedrich Bessel

Friedrich Bessel

-References:* John Frederick William Herschel, A brief notice of the life, researches, and discoveries of Friedrich Wilhelm Bessel, London: Barclay, 1847 -External links:...

on calculation of meridian arc

Meridian arc

In geodesy, a meridian arc measurement is a highly accurate determination of the distance between two points with the same longitude. Two or more such determinations at different locations then specify the shape of the reference ellipsoid which best approximates the shape of the geoid. This...

length – that is the squared half-angle tangent of α:

First order flattening of planets

The flattening of the smoothed Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

's surface in the World Geodetic System
World Geodetic System
The World Geodetic System is a standard for use in cartography, geodesy, and navigation. It comprises a standard coordinate frame for the Earth, a standard spheroidal reference surface for raw altitude data, and a gravitational equipotential surface that defines the nominal sea level.The latest...

(WGS-84) is 1:298.257223563 (which corresponds to a radius difference of 21.385 km (13 mi) of the Earth radius
Earth radius
Because the Earth is not perfectly spherical, no single value serves as its natural radius. Distances from points on the surface to the center range from 6,353 km to 6,384 km...

6378.137 – 6356.752 km) and would not be realized visually from space, since the difference represents only 0.335 %.
The flattening of Jupiter
Jupiter
Jupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...

(1:16) and Saturn
Saturn
Saturn is the sixth planet from the Sun and the second largest planet in the Solar System, after Jupiter. Saturn is named after the Roman god Saturn, equated to the Greek Cronus , the Babylonian Ninurta and the Hindu Shani. Saturn's astronomical symbol represents the Roman god's sickle.Saturn,...

(nearly 1:10), in contrast, can be seen even in a small telescope
Telescope
A telescope is an instrument that aids in the observation of remote objects by collecting electromagnetic radiation . The first known practical telescopes were invented in the Netherlands at the beginning of the 1600s , using glass lenses...

;
Conversely, that of the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

is less than 1:1000 and that of the Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

barely 1:900.

The amount of flattening depends on

the relation between gravity and centrifugal force;

and in detail on

size and density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

of the celestial body (see Figure of the Earth
Figure of the Earth
The expression figure of the Earth has various meanings in geodesy according to the way it is used and the precision with which the Earth's size and shape is to be defined. The actual topographic surface is most apparent with its variety of land forms and water areas. This is, in fact, the surface...

);
the rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

of the planet or star;
and the elasticity
Elasticity (physics)
In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....

of the body.

First order

Second and third orders

First order flattening of planets

See also