We have an astronomy lesson for you this week. It’s about the terms *sidereal period* and *synodic period*. The terms are mostly used to describe the motion of planets, but in general apply to anything orbiting the Sun. We’ve used these terms in earlier posts, but only linked to online explanations. Here we animate them, just to make those concepts crystal clear.

We’ll start with the easier term: *sidereal period*. This is the length of time it takes for the planet to make one orbit around the Sun with respect to the distant stars — sometimes called the *fixed* (in place) *stars*. For Earth, the sidereal period is 365.25 days or 1 Earth-year. For Venus, it’s 225 days (Earth-days, not Venus-days which last 243 Earth-days). In the animation above (and below) both planets start at the 12 o’clock position, so however long it takes them to return to 12 o’clock is that planet’s sidereal period.

There are, however, several interesting astronomical events that are based on sidereal periods. For example, a *conjunction of Venus*, as is shown here, or the *greatest eastern elongation of Venus*, when it’s dubbed the Evening Star, or a *transit of Venus*, all of which are being seen by observers on a moving Earth. You end up with Earth “chasing” the other planet, like runners on a track, until the the geometry lines up again and the conjunction/elongation/transit repeats.

There’s a simple formula for calculating synodic periods ( T_{SYN}) from the two planets’ sidereal periods (T_{SID1} and T_{SID2}) :

T_{SYN} = 1/|(1/T_{SID1} − 1/T_{SID2})| ⇐ note the absolute value brackets

where T_{SID1} is that of the planet being observed, and T_{SID2} is that of the planet from which the observations are made. If the observations are being made from Earth (which they usually are), then T_{SID2} = 1 year and the formula becomes even simpler.

If you use that formula for Venus, you’ll get T_{SYN} = 1.6 years. Watch the animation again and note that Earth makes 1.6 orbits before inferior conjunction recurs.

Interestingly, synodic periods are symmetric with regard to the two moving planets. For example, the synodic period of Mars as seen from Earth is 780 days, so the synodic period of Earth as seen from Mars is also 780 days. For future Martian colonists, that would be the time between greatest elongations of Earth, aka the Pale Blue Dot.

The formula works for all planets, but in the case of Mars, or other planets farther from the Sun, they are doing the chasing and not vice versa. That’s why the absolute value function needs to be part of the formula. And since inferior conjunctions are impossible for these planets, the other key event is called opposition — the point every 780 days when the planet is closest to Earth and provides the best telescopic views. Watch Mars chase Earth:

Mars has a sidereal period of 1.88 years. If you use that formula again for Mars, you’ll find T_{SYN} = 2.14 years (780 days) as cited earlier. In this second animation you can see Earth orbit 2.14 times between two Mars oppositions.

These celestial motions were recorded and predicted for ages by indigenous cultures around the world, but it wasn’t until the 16th century that Nicholas Copernicus derived the formula presented here. It followed logically from his *heliocentric theory*, which put the planets in orbit around than Sun, and not around the Earth. And by rearranging the formula to solve for T_{SID1} Copernicus was able to determine the true sidereal periods of the planets from their known synodic periods.

Next Week in Sky Lights ⇒ Sunset Slide Show