Astronomy Principles and Practice Fourth Edition.pdf

158 Geocentric planetary phenomena
Proceeding in this way, it is, therefore, possible to construct an accurate model of the Solar System
in terms of the distance of the Earth from the Sun without knowing the scale absolutely. Such a model
was obtained in the 16th century. It was a far more difficult problem, only satisfactorily solved with
the advent of radar, to measure the scale.
12.8 Geocentric motion of a planet
Once the sidereal period T of a planet is found, also the mean heliocentric distance a, the velocity V
of the planet in its orbit (assumed circular) can be calculated. Thus,
V = 2πa
T .
For two planets, therefore, the ratio of their orbital velocities is given by
V 2
V 1
=
(
a2
a 1
)(
T1
T 2
)
(12.3)
where the subscripts 1 and 2 refer to the inner and outer planet respectively. Putting in values, it is
found that the velocity of the inner planet is greater than that of the outer.
As we shall see later, Kepler found that for any planet,
a 3 ∝ T 2
or
a 3 = kT 2 (12.4)
where k is a constant.
Then by equation (12.4),
( ) 1/2
a
3
T = .
k
According to Kepler, therefore, we may write for the two planets,
T 1 =
(
a
3
1
k
) 1/2
T 2 =
(
a
3
2
k
) 1/2
.
Substituting for T 1 and T 2 in equation (12.3), we obtain,
( )
V 1/2 2 a1
= . (12.5)
V 1 a 2
In figure 12.8, the orbits of the Earth and a planet are shown, the radii of the orbits (assumed
circular and coplanar) being a and b units respectively. Since the planet is a superior one, b is greater
than a.
At opposition, the positions of the planet and Earth are P 1 and E 1 and their velocity vectors are
shown as V P and V ⊕ respectively, tangential to their orbits.
Now by equation (12.5), V P < V ⊕ and so the angular velocity of the planet as observed from the
Earth is
V P − V ⊕
P 1 E 1
and is in a direction opposite to the orbital movement. It is, therefore, retrograde at opposition.