Reference ellipsoid

Ingeodesy, areference ellipsoidis a mathematically-defined surface that approximates thegeoid, the truerfigure of the Earth, or other planetary body. Because of their relative simplicity, reference ellipsoids are used as a preferred surface on whichgeodetic networkcomputations are performed and point coordinates such aslatitude,longitude, andelevationare defined.

Ellipsoid parametersIn 1687Isaac Newtonpublished thePrincipiain which he included a proof[1][not in citation given]that a rotating self-gravitating fluid body in equilibrium takes the form of an oblateellipsoidof revolution which he termed anoblatespheroid. Current practice (2012)[2]

HYPERLINK "http://en.wikipedia.org/wiki/Reference_ellipsoid" \l "cite_note-flattening-3" [3]uses the word 'ellipsoid' alone in preference to the full term 'oblate ellipsoid of revolution' or the older term 'oblate spheroid'. In the rare instances (someasteroidsandplanets) where a more general ellipsoid shape is required as a model the term used istriaxial(or scalene) ellipsoid. A great many ellipsoids have been used with various sizes and centres but modern (postGPS) ellipsoids are centred at the actualcenter of massof the Earth or body being modeled.

The shape of an (oblate) ellipsoid (of revolution) is determined by the shape parameters of thatellipsewhich generates the ellipsoid when it is rotated about its minor axis. Thesemi-major axisof the ellipse,a, is identified as the equatorial radius of the ellipsoid: thesemi-minor axisof the ellipse,b, is identified with thepolardistances (from the centre). These two lengths completely specify the shape of the ellipsoid but in practice geodesy publications classify reference ellipsoids by giving the semi-major axis and theinverseflattening,1/f, The flattening,f, is simply a measure of how much the symmetry axis is compressed relative to the equatorial radius:

For theEarth,is around 1/300 corresponding to a difference of the major and minor semi-axes of approximately 21km. Some precise values are given in the table below and also inFigure of the Earth. For comparison, Earth'sMoonis even less elliptical, with a flattening of less than 1/825, whileJupiteris visibly oblate at about 1/15 and one ofSaturn'striaxial moons,Telesto, is nearly 1/3 to 1/2.

A great many other parameters are used ingeodesybut they can all be related to one or two of the seta,bandf. They are listed inellipse.

CoordinatesA primary use of reference ellipsoids is to serve as a basis for a coordinate system oflatitude(north/south),longitude(east/west), andelevation(height). For this purpose it is necessary to identify azeromeridian, which for Earth is usually thePrime Meridian. For other bodies a fixed surface feature is usually referenced, which for Mars is the meridian passing through the craterAiry-0. It is possible for many different coordinate systems to be defined upon the same reference ellipsoid.

The longitude measures the rotationalanglebetween the zero meridian and the measured point. By convention for the Earth, Moon, and Sun it is expressed as degrees ranging from 180 to +180 For other bodies a range of 0 to 360 is used.

The latitude measures how close to the poles or equator a point is along a meridian, and is represented as angle from 90 to +90, where 0 is the equator. The common orgeodetic latitudeis the angle between the equatorial plane and a line that isnormalto the reference ellipsoid. Depending on the flattening, it may be slightly different from thegeocentric (geographic) latitude, which is the angle between the equatorial plane and a line from the center of the ellipsoid. For non-Earth bodies the termsplanetographicandplanetocentricare used instead.

The coordinates of a geodetic point are customarily stated as geodetic latitude and longitude, i.e., the direction in space of the geodetic normal containing the point, and the heighthof the point over the reference ellipsoid. SeeGeodetic systemfor more detail.

Historical Earth ellipsoidsCurrently the most common reference ellipsoid used, and that used in the context of the Global Positioning System, is the one defined byWGS 84.

Traditional reference ellipsoids orgeodetic datumsare defined regionally and therefore non-geocentric, e.g.,ED50. Modern geodetic datums are established with the aid ofGPSand will therefore be geocentric, e.g., WGS 84.

The following table lists some of the most common ellipsoids:

NameEquatorial axis (m)Polar axis (m)Inverse flattening,

Airy18306 377 563.46 356 256.9299.324 975 3

Clarke 18666 378 206.46 356 583.8294.978 698 2

Bessel 18416 377 397.1556 356 078.965299.152 843 4

International 19246 378 3886 356 911.9297

Krasovsky 19406 378 2456 356 863298.299 738 1

GRS 19806 378 1376 356 752.3141298.257 222 101

WGS 19846 378 1376 356 752.3142298.257 223 563

Sphere (6371km)6 371 0006 371 000

Ellipsoids for other planetary bodiesReference ellipsoids are also useful for geodetic mapping of other planetary bodies including planets, their satellites, asteroids and comet nuclei. Some well observed bodies such as theMoonandMarsnow have quite precise reference ellipsoids.

For rigid-surface nearly-spherical bodies, which includes all the rocky planets and many moons, ellipsoids are defined in terms of the axis of rotation and the mean surface height excluding any atmosphere. Mars is actuallyegg shaped, where its north and south polar radii differ by approximately 6km, however this difference is small enough that the average polar radius is used to define its ellipsoid. The Earth's Moon is effectively spherical, having no bulge at its equator. Where possible a fixed observable surface feature is used when defining a reference meridian.

For gaseous planets likeJupiter, an effective surface for an ellipsoid is chosen as the equal-pressure boundary of onebar. Since they have no permanent observable features the choices of prime meridians are made according to mathematical rules.

Small moons, asteroids, and comet nuclei frequently have irregular shapes. For some of these, such as Jupiter'sIo, a scalene (triaxial) ellipsoid is a better fit than the oblate spheroid. For highly irregular bodies the concept of a reference ellipsoid may have no useful value, so sometimes a spherical reference is used instead and points identified by planetocentric latitude and longitude. Even that can be problematic fornon-convexbodies, such asEros, in that latitude and longitude don't always uniquely identify a single surface location.