## A2 b2 c2 2bc cos A 22 The Earth

A position on the Earth is usually given by two spherical coordinates (although in some calculations rectangular or other coordinates may be more convenient). If neces-

sary, also a third coordinate, e. g. the distance from the centre, can be used.

The reference plane is the equatorial plane, perpendicular to the rotation axis and intersecting the surface of the Earth along the equator. Small circles parallel to the equator are called parallels of latitude. Semicircles from pole to pole are meridians. The geographical longitude is the angle between the meridian and the zero meridian passing through Greenwich Observatory. We shall use positive values for longitudes east of Greenwich and negative values west of Greenwich. Sign convention, however, varies, and negative longitudes are not used in maps; so it is usually better to say explicitly whether the longitude is east or west of Greenwich.

The latitude is usually supposed to mean the geographical latitude, which is the angle between the plumb line and the equatorial plane. The latitude is positive in the northern hemisphere and negative in the southern one. The geographical latitude can be determined by astronomical observations (Fig. 2.7): the altitude of the celestial pole measured from the hori-

Celestial pole k

Celestial pole k Fig. 2.7. The latitude 0 is obtained by measuring the altitude of the celestial pole. The celestial pole can be imagined as a point at an infinite distance in the direction of the Earth's rotation axis

zon equals the geographical latitude. (The celestial pole is the intersection of the rotation axis of the Earth and the infinitely distant celestial sphere; we shall return to these concepts a little later.)

Because the Earth is rotating, it is slightly flattened. The exact shape is rather complicated, but for most purposes it can by approximated by an oblate spheroid, the short axis of which coincides with the rotation axis (Sect. 7.5). In 1979 the International Union of Geodesy and Geophysics (IUGG) adopted the Geodetic Reference System 1980 (GRS-80), which is used when global reference frames fixed to the Earth are defined. The GRS-80 reference ellipsoid has the following dimensions:

equatorial radius polar radius flattening a = 6,378,137 m, b = 6,356,752 m, f = (a — b)/a = 1/298.25722210.

The shape defined by the surface of the oceans, called the geoid, differs from this spheroid at most by about 100 m.

The angle between the equator and the normal to the ellipsoid approximating the true Earth is called the geodetic latitude. Because the surface of a liquid (like an ocean) is perpendicular to the plumb line, the geodetic and geographical latitudes are practically the same.

Because of the flattening, the plumb line does not point to the centre of the Earth except at the poles and on the equator. An angle corresponding to the ordinary spherical coordinate (the angle between the equator and the line from the centre to a point on the surface), the geocentric latitude < is therefore a little smaller than the geographic latitude < (Fig. 2.8).

We now derive an equation between the geographic latitude < and geocentric latitude p', assuming the Earth is an oblate spheroid and the geographic and geodesic latitudes are equal. The equation of the meridional ellipse is

22 x2 y2

The direction of the normal to the ellipse at a point (x, y) is given by dx a2 y tan p = -— = —-.

dy b2 x Fig. 2.8. Due to the flattening of the Earth, the geographic latitude p and geocentric latitude p' are different

The geocentric latitude is obtained from tan p' = y/x .

Hence b

is the eccentricity of the ellipse. The difference Ap = < — p' has a maximum 11.5' at the latitude 45°.

Since the coordinates of celestial bodies in astronomical almanacs are given with respect to the centre of the Earth, the coordinates of nearby objects must be corrected for the difference in the position of the observer, if high accuracy is required. This means that one has to calculate the topocentric coordinates, centered at the observer. The easiest way to do this is to use rectangular coordinates of the object and the observer (Example 2.5).

One arc minute along a meridian is called a nautical mile. Since the radius of curvature varies with latitude, the length of the nautical mile so defined would depend on the latitude. Therefore one nautical mile has been

defined to be equal to one minute of arc at \$ = 45°, whence 1 nautical mile = 1852 m. 