The sidereal period of a planet in its motion around the Sun, or in its rotation on its axis, is the length of time it takes for one revolution or rotation, as viewed by a fixed (non-moving) external observer.
Suppose that we have two cars going around a circular racetrack. This is illustrated in the figure. One of the cars is A, the other is B. Car A takes a period P hours to go around the racetrack once, and car B takes a shorter time, Q hours, to go once around. Suppose we start the two cars out at time T=0 at the starting gate. We want to know how long it takes car B to "lap" car A.
First, we notice that we could be talking about planets instead of cars. In that case, the sidereal periods of the two planets would be P and Q, respectively. The length of time that it takes one of the planets to "lap" the other is called the synodic period of the one planet as observed from the other. In the figure, car A goes from A to A' in the same time that car B goes from B to B'. So we see that the speed that each car has in its race around the track, as seen from someone sitting at the starting gate, is 1/P and 1/Q laps per hour, respectively. Thus, if car A were to take 1/2 hour to go around the track once, its speed around the track is 2 laps per hour. But to decide how long it takes car B to "lap" car A, we need to know the relative speeds of the two cars. That is, sitting in car A, we see that car B is moving ahead of us, but not as rapidly as it is relative to the observer at the starting gate, since we are trying to catch it. In terms of laps per hour, the speed of car B relative to car A is the smaller number
1/R = relative speed (laps/hour) = 1/Q - 1/P
laps per hour, that is, the difference between the speeds. This is illustrated by the heavy line in the figure. (It is just like when you are travelling down the highway at, say, 50 miles/hour, and someone passes you going, say 55 miles per hour. The car that passes you is going 5 miles per hour faster than you, and that car's speed, relative to you, is 5 miles per hour.) Now, it should be easy to see that the length of time it is going to take for car B to "lap" car A is just the reciprocal of the relative speed:
Time to lap = R = QP/(P - Q)
This is because if the two cars are separating at a speed of 1/R laps per hour, it will take R hours to make 1 lap. Example: The minute and hour hands on a clock. The hour hand has a period P of 12 hours and the minute hand has a period Q of 1 hour. The relative speed is therefore
1/R = 1/1 - 1/12 = 11/12 laps/hour
and the time it takes for the minute hand to catch up to the hour hand is
R = 12/11 hour, or 1 hour 5 minutes 27+ seconds
The tropical periods (measured relative to the Vernal Equinox) of Venus, Earth, and Mars are, respectively, 224.69544 days, 365.24219 days, and 686.92971 days. As viewed from the Earth, what are the synodic periods of Venus and Mars? Let's do Mars. You can do Venus for practice. We would have
1/R = 1/365.24219 - 1/686.92971 = 0.00128216
so that the synodic perod of Mars is
779.936 days, a little over 2 years.
Here's another example. The tropical period of the Sun "around" the Earth is given above. The tropical period of the Moon around the Earth is 27.321582 days. What is the synodic period of the Moon relative to the Sun? (This period is the length of time it takes to go from new Moon to new Moon, i.e., our ordinary lunar month. It is the synodic month that the ancients used in their calendar systems.) The answer to this calculation is 29.53059 days, by the way, so you can check if you got it right. Finally, the rotation of the Earth itself gives an example of this. The Earth actually rotates once on its axis in one sidereal day of 0.99726956633 solar days. However, there is a point of confusion here, since the sidereal day is actually measured with respect to the Vernal Equinox. Thus, for the year we must still use the tropical year since it is also measured with respect to the vernal equinox. What is the length of the day, as measured from solar noon to noon (that is, length of time it takes the Sun to appear to go around the Earth once, as seen from the rotating Earth)? You should get a number very close to 1.000000 solar days, since that is indeed the time from noon to noon. In this example, the Earth rotates about 366.25 times on its axis for every 365.25 times that we have an ordinary noon-to-noon day.
This JavaScript calculates the synodic period of a planet given the periods of the planet and the planet you are observing from. To use it, you have to be using a JavaScript-aware browser such as Netscape v. 2.0 or later, or Microsoft Explorer. You may use it for practice.
Period 1
Period 2
Synodic Period
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