Fundamental Astronomy

12
2. Spherical Astronomy
angles correspond to each other in a unique way, it is
customary to give the central angles instead of the sides.
In this way, the radius of the sphere does not enter into
the equations of spherical trigonometry. An angle of
a spherical triangle can be defined as the angle between
the tangents of the two sides meeting at a vertex, or as
the dihedral angle between the planes intersecting the
sphere along these two sides. We denote the angles of
a spherical triangle by capital letters (A, B, C) and the
opposing sides, or, more correctly, the corresponding
central angles, by lowercase letters (a, b, c).
The sum of the angles of a spherical triangle is always
greater than 180 degrees; the excess
E = A + B + C − 180 ◦
(2.1)
is called the spherical excess. It is not a constant, but
depends on the triangle. Unlike in plane geometry, it is
not enough to know two of the angles to determine the
third one. The area of a spherical triangle is related to
the spherical excess in a very simple way:
Area = Er 2 , [E]=rad . (2.2)
This shows that the spherical excess equals the solid
angle in steradians (see Appendix A.1), subtended by
the triangle as seen from the centre.
Fig. 2.3. If the sides of a spherical triangle are extended all
the way around the sphere, they form another triangle ∆ ′ ,
antipodal and equal to the original triangle ∆. The shaded
area is the slice S(A)
To prove (2.2), we extend all sides of the triangle ∆
to great circles (Fig. 2.3). These great circles will form
another triangle ∆ ′ , congruent with ∆ but antipodal to
it. If the angle A is expressed in radians, the area of the
slice S(A) bounded by the two sides of A (the shaded
area in Fig. 2.3) is obviously 2A/2π = A/π times the
area of the sphere, 4πr2 . Similarly, the slices S(B) and
S(C) cover fractions B/π and C/π of the whole sphere.
Together, the three slices cover the whole surface
of the sphere, the equal triangles ∆ and ∆ ′ belonging
to every slice, and each point outside the triangles, to
exactly one slice. Thus the area of the slices S(A), S(B)
and S(C) equals the area of the sphere plus four times
the area of ∆, A(∆):
A + B + C
4πr
π
2 = 4πr 2 + 4A(∆),
whence
A(∆) = (A + B + C − π)r 2 = Er 2 .
As in the case of plane triangles, we can derive relationships
between the sides and angles of spherical
triangles. The easiest way to do this is by inspecting
certain coordinate transformations.
Fig. 2.4. The location of a point P on the surface of a unit
sphere can be expressed by rectangular xyz coordinates or by
two angles, ψ and θ.Thex ′ y ′ z ′ frame is obtained by rotating
the xyz frame around its x axis by an angle χ