Astronomy Principles and Practice Fourth Edition.pdf

Planetary orbits 167
Figure 13.1. Method of drawing an ellipse.
Figure 13.2. An elliptical planetary orbit.
a non-spherical planet, do they fail to describe in their usual highly accurate manner the behaviour of
such bodies. Even then, however, they may be used as a first approximation.
We now examine them in more detail.
13.2.2 Kepler’s first law
The law states that the orbit of a planet is an ellipse with the Sun at one focus.
A well-known way of drawing an ellipse is to insert two pins some distance apart at F 1 and F 2 in
a sheet of paper (see figure 13.1), place a loop of thread over them, hold it taut by means of a pencil
and then run the pencil along the path allowed by the tight loop. The figure obtained is an ellipse. The
two positions occupied by the pins are called the foci.
If the two pins are placed nearer to each other and the operation is repeated with the same loop, it
is found that the ellipse is more circular than the previous one. The ellipse, in fact, becomes a circle in
the limiting case where the two pins occupy the same position, i.e. only one pin is used.
Figure 13.2 shows an elliptical planetary orbit, with the Sun, S, at one of the foci, as Kepler’s first
law states. The other focus, F, is often called the empty focus.
Then the line AA ′ is the major axis of the ellipse, C is the centre and, therefore, CAand CA ′ are
the semi-major axes. Likewise BB ′ is the minor axis, with CB and CB ′ the semi-minor axes. Ifa
and b denote the lengths of the semi-major and semi-minor axes respectively, then
b 2 = a 2 (1 − e 2 )