The models for the figure of the Earth vary in the way they are used, in their complexity, and in the accuracy with which they represent the size and shape of the Earth.
Sphere . The curvature of the Earth is evident in the
horizon across the image, and the bases of the buildings on the far shore are below that horizon and hidden by the sea. The simplest model for the shape of the entire Earth is a sphere. The Earth's
radius is the
distance from Earth's center to its surface, about . While "radius" normally is a characteristic of perfect spheres, the Earth deviates from spherical by only a third of a percent, sufficiently close to treat it as a sphere in many contexts and justifying the term "the radius of the Earth". The concept of a spherical Earth dates back to around the
6th century BC, but remained a matter of philosophical speculation until the
3rd century BC. The first scientific estimation of the radius of the Earth was given by
Eratosthenes about 240 BC, with estimates of the accuracy of Eratosthenes's measurement ranging from −1% to 15%. The Earth is only approximately spherical, so no single value serves as its natural radius. Distances from points on the surface to the center range from to . Several different ways of modeling the Earth as a sphere each yield a mean radius of . Regardless of the model, any radius falls between the polar minimum of about and the equatorial maximum of about . The difference correspond to the polar radius being approximately 0.3% shorter than the equatorial radius.
Ellipsoid of revolution , highly exaggerated relative to the actual Earth of the 2003
IERS reference ellipsoid, with north at the top. The outer edge of the dark blue line is an
ellipse with the same
eccentricity as that of Earth. For comparison, the light blue circle within has a diameter equal to the ellipse's
minor axis. The red curve represents the
Karman line above
sea level, while the yellow band denotes the
altitude range of the
ISS in
low Earth orbit.As theorized by
Isaac Newton and
Christiaan Huygens, the Earth is
flattened at the poles and
bulged at the
equator. Thus,
geodesy represents the figure of the Earth as an oblate
spheroid. The oblate spheroid, or
oblate ellipsoid, is an
ellipsoid of revolution obtained by rotating an ellipse about its shorter axis. It is the regular
geometric shape that most nearly approximates the shape of the Earth. A spheroid describing the figure of the Earth or other
celestial body is called a
reference ellipsoid. The reference ellipsoid for Earth is called an
Earth ellipsoid. An ellipsoid of revolution is uniquely defined by two quantities. Several conventions for expressing the two quantities are used in geodesy, but they are all equivalent to and convertible with each other: • Equatorial radius a (called
semimajor axis), and polar radius b (called
semiminor axis); • a and
eccentricity e; • a and flattening f. Eccentricity and flattening are different ways of expressing how squashed the ellipsoid is. When flattening appears as one of the defining quantities in geodesy, generally it is expressed by its reciprocal. For example, in the
WGS 84 spheroid used by today's GPS systems, the reciprocal of the flattening 1/f is set to be exactly . The difference between a sphere and a reference ellipsoid for Earth is small, only about one part in 300. Historically, flattening was computed from
grade measurements. Nowadays, geodetic networks and
satellite geodesy are used. In practice, many reference ellipsoids have been developed over the centuries from different surveys. The flattening value varies slightly from one reference ellipsoid to another, reflecting local conditions and whether the reference ellipsoid is intended to model the entire Earth or only some portion of it. A sphere has a single
radius of curvature, which is simply the radius of the sphere. More complex surfaces have radii of curvature that vary over the surface. The radius of curvature describes the radius of the sphere that best approximates the surface at that point. Oblate ellipsoids have a constant radius of curvature east to west along
parallels, if a
graticule is drawn on the surface, but varying curvature in any other direction. For an oblate ellipsoid, the polar radius of curvature r_p is larger than the equatorial : r_p=\frac{a^2}{b}, because the pole is flattened: the flatter the surface, the larger the sphere must be to approximate it. Conversely, the ellipsoid's north–south radius of curvature at the equator r_e is smaller than the polar : r_e=\frac{b^2}{a} where a is the distance from the center of the ellipsoid to the equator (semi-major axis), and b is the distance from the center to the pole. (semi-minor axis)
Non-spheroidal deviations Triaxiality (equatorial eccentricity) The possibility that the Earth's equator is better characterized as an ellipse rather than a circle and therefore that the ellipsoid is triaxial has been a matter of scientific inquiry for many years. Modern technological developments have furnished new and rapid methods for data collection and, since the launch of
Sputnik 1, orbital data have been used to investigate the theory of ellipticity. More recent results indicate a 70 m difference between the two equatorial major and minor axes of inertia, with the larger semidiameter pointing to 15° W longitude (and also 180-degree away).
Egg or pear shape Following work by Picard, Italian polymath
Giovanni Domenico Cassini found that the length of a degree was apparently shorter north of Paris than to the south, implying the Earth to be
egg-shaped. The theory of a slightly pear-shaped Earth arose when data was received from the U.S.'s artificial satellite
Vanguard 1 in 1958. It was found to vary in its long periodic orbit, with the Southern Hemisphere exhibiting higher gravitational attraction than the Northern Hemisphere. This indicated a flattening at the
South Pole and a bulge of the same degree at the
North Pole, with the
sea level increased about at the latter.
John A. O'Keefe and co-authors are credited with the discovery that the Earth had a significant third degree
zonal spherical harmonic in its
gravitational field using Vanguard 1 satellite data. Based on further
satellite geodesy data,
Desmond King-Hele refined the estimate to a difference between north and south polar radii, owing to a "stem" rising in the North Pole and a depression in the South Pole. The polar asymmetry is about a thousand times smaller than the Earth's flattening and even smaller than its
geoidal undulation in some regions.
Geoid gravity model and the
WGS84 reference ellipsoid) Modern geodesy tends to retain the ellipsoid of revolution as a
reference ellipsoid and treat triaxiality and pear shape as a part of the
geoid figure: they are represented by the spherical harmonic coefficients C_{22},S_{22} and C_{30}, respectively, corresponding to degree and order numbers 2.2 for the triaxiality and 3.0 for the pear shape. It was stated earlier that measurements are made on the apparent or topographic surface of the Earth and it has just been explained that computations are performed on an ellipsoid. One other surface is involved in geodetic measurement: the geoid. In geodetic surveying, the computation of the
geodetic coordinates of points is commonly performed on a
reference ellipsoid closely approximating the size and shape of the Earth in the area of the survey. The actual measurements made on the surface of the Earth with certain instruments are however referred to the geoid. The ellipsoid is a mathematically defined regular surface with specific dimensions. The geoid, on the other hand, coincides with that surface to which the oceans would conform over the entire Earth if free to adjust to the combined effect of the Earth's mass attraction (
gravitation) and the centrifugal force of the
Earth's rotation. As a result of the uneven distribution of the Earth's mass, the geoidal surface is irregular and, since the ellipsoid is a regular surface, the separations between the two, referred to as
geoid undulations, geoid heights, or geoid separations, will be irregular as well. The geoid is a surface along which the gravity
potential is equal everywhere and to which the direction of gravity is always perpendicular. The latter is particularly important because optical instruments containing gravity-reference leveling devices are commonly used to make geodetic measurements. When properly adjusted, the vertical axis of the instrument coincides with the direction of gravity and is, therefore, perpendicular to the geoid. The angle between the
plumb line which is perpendicular to the geoid (sometimes called "the vertical") and the perpendicular to the ellipsoid (sometimes called "the ellipsoidal normal") is defined as the
deflection of the vertical. It has two components: an east–west and a north–south component. == Earth rotation and Earth's interior ==