How to calculate sidereal period

Question

How do I calculate the sidereal period of planets when I'm not given the orbital period?

I know the Earth has an orbital period of 365.25 days, but then we are only given the time between successive inferior conjunctions of Earth and another hypothetical planet (200 days), and the time between successive oppositions of Earth and a hypothetical planet (500 days). How do I calculate the sidereal periods of the two planets?

Answer 1

Let T be the longer period, and t be the shorter period. The time between conjunctions is the synodic period of the faster planet.

When the faster planet has completed one revolution in time t, the other planet will have completed t/T fraction of one period. In order to catch up, the second planet will need to travel that same fraction of a period, but that will take it t/T * t, in which time, the longer period planet will have traveled an additional t *t/T *t/T, and the speedier planet will have to travel that additional period. The total is the infinite sum of these periods.

The sum is a geometric series with the total time being

t + t * t/T + t * (t/T) * (t/T) ..... t * (t/T)^n

That is a geometric series with starting value of t, and ratio = t/T. Notice that t/T is always less than 1, so the series converges. The formula for the infinite sum of a geometric series is a/(1 - r) or t / (1 - t/T). Let's let S be the synodic period.

S = t/(1 - t/T). Cross multiplying gives t = S * (1 - t/T). t = S - St/T. Collecting the t terms on the left gives

t + St/T = S; t ( 1 + S/T) = S; t = S/(1 + S/T);

inverting both sides gives 1/t = 1/S + 1/T. This is the formula we need to work both problems.

For the first problem, T = 365.25, and synodic period is 200. We want to find the period t of the shorter period (inner planet)

1/t = 1/200 + 1/365.25 => t - 129.23 days

For the second problem, the earth is the shorter period t, and we want to solve for T.

1/365.25 = 1/500 + 1/T 1/T = 1/365.25 - 1/500. T = 1355.3 days;