Astronomy Principles and Practice Fourth Edition.pdf

Position on the Earth’s surface 47
If the radius of the sphere is taken as unity,
s = θ
showing that the length of a great circle arc on a sphere of unit radius is equal to the angle (in circular
measure) subtended by this arc at the centre of the sphere.
The length of a small circle arc, such as FG, is related simply to the length of an arc of the great
circle whose plane is parallel to that of the small circle.
In figure 7.1, let r be the radius of the small circle EFGHE.Then
FG = r × FKG.
Also
BC = R × BOC.
Both OB and KF lie on plane PFBQ; KF also lies on plane EFGH while OB lies on plane
ABCD. Therefore, KF must be parallel to OB, since plane EFGH is parallel to plane ABCD.
Similarly, KG is parallel to OC.Then
FKG = BOC.
Hence,
FG = BC × r R .
In the plane triangle KOF, right-angled at K , KF = r; OF = R. Hence,
FG = BC × sin KOF.
But POB = 90 ◦ so that we may write alternatively
FG = BC cos FB.
If the radius of the sphere is unity,
PF = POF = KOF
and
so that we have
and
FB = FOB,
FG = BC sin PF
FG = BC cos FB. (7.2)
7.3 Position on the Earth’s surface
To illustrate these concepts we consider the Earth. A point on the surface of the Earth is defined by
two coordinates, longitude and latitude, based on the equator and a particular meridian (half of a
great circle) passing through the North and South Poles and Greenwich, England. The equator is the
great circle whose poles are the North and South Poles. The longitude, λ, of the point is measured
east or west along the equator. Its value is the angular distance between the meridian passing through
the point and the Greenwich meridian. The longitude may be expressed in angular measure or in time
units related to each other by table 7.1.