idereal and Synodic Periods. In this problem, you will derive the relationship b

Question

idereal and Synodic Periods. In this problem, you will derive the relationship between a planet’s sidereal and its synodic period. For the purposes of this problem, assume that the planets orbit the Sun in perfect circles.

a) Suppose that at time t = 0, the Earth and a superior planet are both at angular position = 0. Find an equation for (t) and (t) in terms of the two planets’ sidereal periods.

b) Write down the relationship between (t) and (t) after one synodic period has elapsed.

c) Combine the results of (a) and (b) to arrive at the desired equation.

Explanation / Answer

Earth makes S/P_ orbits about the Sun during the time required for another planet to make S/P orbits.

If that other planet is a superior planet then Earth must make one extra trip around the Sun to overtake it, hence S/P_ = S/P + 1.

Similarly, for an inferior planet, that planet must make the extra trip, or S/P = S/P_ + 1.

The synodic period of a planet is given by this formula:

S = 1 / abs(1/E - 1/P)
where
S = synodic period of planet
E = sidereal year of earth
P = sidereal year of planet
abs() = absolute-value function

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