Looking At Retrograde Motion

 Recall from Kepler’s 3^rd 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 V_p, tangential to their orbits. 

From the expression for (V2/V1),  if   V_p < V,   then the angular velocity of the planet as observed from Earth is:

 (V_p - 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 V_p has a component V_p sin (φ) perpendicular to E'P'. The other component, V_p 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:

V_p 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  V_p 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: 

 - (V_p - V)/ PE and parallel  to the orbital motion?