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Appendix B
Orbital dynamics (Kepler’s laws)
For many purposes it suffices to assume that
planetary and satellite orbits are circular.
Johannes Kepler (1571–1630) first recognized
that they are elliptical from his analyses of obser-
vations by Tycho Brahe. The departure from a
circle is expressed by the eccentricity
e ¼ 1 b
2
=a
2
1=2
; (B:1)
where a, b are the semi-major and semi-minor
axes. Some asteroids have very elliptical orbits,
but orbital eccentricities of the planets are
mostly slight, being greatest for Pluto (0.250)
and Mercury (0.2056). For the Earth’s orbit,
e ¼0.016 73.
Kepler summarized his conclusions in three
empirical laws:
1. the orbit of each planet is an ellipse with the
Sun at one focus;
2. the line between a planet and the Sun sweeps
out equal areas in equal times;
3. the square of the orbital period of a planet is
proportional to the cube of its mean distance
from the Sun (which is equal to the semi-
major axis of the orbit).
The second of these laws is a statement of the
principle of conservation of angular momentum.
The proof that the first and third laws are con-
sequences of the inverse square law of gravita-
tional attraction is the most widely known of
Isaac Newton’s discoveries.
A planetary orbit is confined to the plane
defined by the instantaneous orbital velocity
and the planet–Sun line. Since there is no velocity
component or force on the planet perpendicular
to this plane, the planet cannot leave it. Thus
plane polar coordinates (r, y) are most convenient
to use in analysing the orbital problem, with the
origin at the centre of mass of the Sun-plus-
planet. The displacements of the planet, mass m,
and Sun, mass M,fromthisoriginremainina
fixed ratio, being inversely proportional to their
masses, so that, whatever path a planet follows,
the Sun follows its converse, appropriately dimin-
ished in size. If, at any instant, the planet is at a
radial distance r (from the origin), the Sun is at a
distance (m/M)r in the opposite direction and the
planet–Sun separation is r(1 þm/M). The mutual
attractive force is
F ¼
GMm
r
2
ð1 þ m=MÞ
2
: (B:2)
Thus the planet moves as though attracted to the
coordinate origin by a mass fixed there, of
magnitude
M
0
¼ M=ð1 þ m=MÞ
2
: (B:3)
In the case of the Earth and Sun, or, more
correctly, (Earth plus Moon) and Sun,
m/M ¼3 10
6
and the orbital motion of the
Sun is slight. The outer planets have bigger
effects and, both directly and through their influ-
ence on the Sun, perturb the motion of the Earth
from its simple two-body interaction with the
Sun. In what follows, the motion of a planet, m,
is analysed as though attracted to a fixed central
mass M, acknowledging that this should, strictly,
be M
0
by Eq. (B.3).