I have the privilege of working at two major observatories in the Boston area, and I get a variety of questions from the general public as they are looking - or, more commonly, waiting to look - through a telescope. One of the most common questions is whether there are more planets visible in the summer or winter, or whether they planets we are looking at are visible at the same time every year. The answer is: well, not quite.
In order to really explain the cycles of the planets, we need to introduce the idea of "synodic periods." A synodic period is the time it takes a body to return to the same location relative to the Earth and Sun, i.e., the same point in the night sky.
Let's consider the case of Mars. Obviously, Earth and Mars both orbit the Sun, in different periods of time. When the Sun, Earth, and Mars are positioned in a line, Mars is considered to be in opposition. When does the next opposition occur? Earth, being the inner of the two planets, moves faster and completes an orbit more quickly - every 12 months. But by the time Earth has returned to the position of the previous opposition, Mars will have moved on some additional distance along its orbit. Earth will have to move for twelve months plus the additional distance Mars has travelled to duplicate the conditions of the opposition.
There is a formula to calculate such "synodic periods" among the planets.
Earth's year is, of course, 1 year long. That of Mars is 1.88 Earth years. The formula for calculating the synodic period is as follows:
1/Psyn = 1/P1 - 1/P2
where P1 is Earth's period and P2 is that of Mars.
Plugging in the values above, we find the synodic period of Mars to be about 2.1 years. (This is actually an approximation, since the orbits of Earth and Mars are non-circular and not co-planar.)
An interesting effect is that, the farther the planet is from the Sun, the closer its synodic period comes to being 1 year - since the planet moves more slowly as its distance from the Sun increases.
Here is a table of synodic periods in the Solar System, relative to Earth:
||Sidereal Period ("Year")
From the table we can see that Saturn returns to its same approximate position in our sky after about 1.035 years, or about 12.42 months.
The positions of the planets in our sky, as seen from our observatories, return to the same place after approximately one synodic period. So there is the answer to the visitors' question - albeit a somewhat complicated one!
-- John Sheff, CfA docent