Recall from Kepler’s 3rd or harmonic law:
(P1/ P2)2 = k(a1/ a2)3
where P1, P2 are the periods, related to a1, a2 - the semi-major axes, as shown.
Now, it should be clear that once the sidereal period P of a planet is known, and also the semi-major axis a(or mean heliocentric distance) then the velocity of the planet in its orbit (assumed circular) can be computed, or:
V= 2π a/ P
Hence, for two planets, the ratio of their orbital velocities is:
V2/V1 = (a2/a1) (P1/P2)
where we intentionally allow the numbers 1 and 2 to refer to the inner and outer planets, respectively. As may deduced form Kepler's 3rd law:
P1 = [(a1)3/k] ½ and T2 = [(a2)3/k]½
Substituting for T1 and T2 in the earlier form:
V2/ V1 = (a1/a2)½
In Fig. 1, the orbits for the Earth and a superior planet are shown, and the semi-major axes are denoted by a and b, respectively. For any superior planet, b > a.
At opposition (the alignment SEP) the
positions of Earth and planet are given as E and P, with velocity vectors V and
Vp, tangential to their orbits.
From the expression for
(V2/V1), if Vp < V, then the angular velocity of the planet as observed
from Earth is:
(Vp - V)/ PE
and is in a direction opposite to the
orbital motion , and hence is retrograde at opposition.
At the following quadrature, shown by configuration SE' P', the Earth's orbital velocity V is now along the line P'E' but the planet's velocity Vp has a component Vp sin (φ) perpendicular to E'P'. The other component, Vp cos (φ) lies along the line P'E' and - like the Earth's velocity -doesn't contribute to the observed angular velocity of the planet.
The geocentric angular velocity at quadrature is then:
Vp sin (φ)/ E'P'
Suggested Problem:
1 a) Compare the orbital velocities of Venus and Earth, if the sidereal period for Venus, T1, is 224.69 d, and for Earth (T2) is 365.25 d.
b) Verify this by using a Table of orbital velocities for the planets - given in km/s
2) (a) Why doesn't the component Vp cos (φ) contribute to the observed angular velocity of the planet, (i.e. in Fig. 1) ?
(b) What if the angular velocity of the planet as observed from Earth is:
- (Vp - V)/ PE and parallel to the orbital motion?
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