Mars at opposition in 2001, as seen by the Hubble Space Telescope
An apparition of an object is a period of weeks or months during which it is visible in the night sky.
All of the planets go through periods when they are well-placed for observation, punctuated by other times when they pass too close to the Sun to be readily observed.
The apparitions of the outermost planets in the solar system – Saturn, Uranus and Neptune – last roughly 6-9 months, and repeat roughly once a year.
By contrast, the planets that orbit closer to the Sun than the Earth – the inferior planets, Mercury and Venus – move across the sky in a very different way. They never venture very far from the Sun. Venus's maximum separation from the Sun is 48°, while Mercury's is 28°. They appear alternately in the evening or morning skies, depending whether they lie to the east of the Sun – setting shortly after sunset – or to the west of the Sun – rising shortly before sunrise.
The geometry of the solar system
This comes about because of the different geometry of the orbits of Mercury and Venus as compared to the other planets.
The outer planets are optimally placed for observation when they lie on the same side of the solar system to the Earth. When the Sun and Earth lie in an approximately straight line with the object being observed, with the Earth in the middle, the object appears in almost exactly the opposite part of the sky to the Sun. This configuration is known as opposition, and means the object is visible almost all night.
By contrast, when Mercury or Venus lie on the same side of the solar system to the Earth, they do not lie opposite to the Sun in the sky, but instead pass between the Earth and Sun. Rather than appearing high in the night sky, they disappear into the Sun's glare. This configuration is termed inferior solar conjunction. In rare cases of very exact alignment, the planet can pass directly in front of the Sun, creating a transit.
Between the times when Mercury and Venus pass in front of or behind the Sun, they appear alternately in the evening or morning skies, depending whether they lie to the east or to the west of the Sun.
Calculating a planet's synodic period
The synodic period of an astronomical body is the period of time over which its observability cycles, with durations varying from a few months to just over two years.
Technically, it can be defined as the interval between the times when the object passes around the far side of the solar system, such that the Sun lies between us and it.
The outer planets have one apparition within each synodic cycle. The inner planets have two, appearing once in the evening sky, and once in the morning sky.
Each planet has its own unique synodic period. This is partly because they orbit the Sun at different speeds, but is also complicated because our own planet is also moving. The observability of a planet depends on its position relative to the Earth, not its position as a hypothetical observer outside the solar system would see it.
In consequence, the time interval between successive solar conjunctions of a planet depends specifically on how often it overtakes the Earth in its orbit, rather than how fast it goes around the Sun.
The plot above shows the theoretically-calculated synodic periods of planets in circular orbits around the Sun, as a function of their distance from the Sun.
The outermost planets in the solar system, for example, move very slowly around their orbits. Each time the Earth completes an orbit, these planets have not moved very far from where they were a year earlier. As a result, the Earth overtakes them roughly once a year. The movement of these planets between the morning and evening skies happens primarily because of the Earth's movement around its orbit, rather than their own.
At the opposite extreme, Mercury circles the Sun once every 88 days, after which time the Earth has completed less than a quarter of a revolution. Mercury overtakes the Earth every 116 days, the additional 28 days being needed for it to catch up with the Earth's own orbital progress in the intervening time. This period – the time interval between any planet's closest approaches to the Earth – is another way to define its synodic period.
Those planets which are closest to the Earth – Venus and Mars – have very long synodic periods. This is because they orbit the Sun at speeds very similar to that of the Earth. If there were to be another planet that orbited the Sun at exactly the same distance as the Earth, it would orbit the Sun at exactly the same rate as the Earth and never overtake it – its synodic period would be infinitely long.