Synodic Period Calculator

The synodic period is the time it takes for a planet to return to the same apparent position relative to the Sun as seen from Earth (e.g., from one opposition to the next).

It differs from the sidereal period (the true orbital period relative to the stars) because Earth is also moving.

Formula for an outer planet (P > 1 year):

1/P_syn = 1/P_Earth - 1/P_planet = 1 - 1/P_planet

For an inner planet (P < 1 year):

1/P_syn = 1/P_planet - 1/P_Earth = 1/P_planet - 1

Where all periods are in Earth years.

Planetary synodic periods:

Note on outer planets: The further away an outer planet is, the closer its synodic period is to 1 Earth year. This is because very distant planets hardly move relative to the stars over one year, so Earth essentially “laps” them once per year.

Oppositions (outer planets) occur when Earth is between the Sun and the planet.
That is the best time to observe: the planet is closest, fully lit, and up all night.

Conjunctions occur when the planet is behind the Sun (superior) or, for Mercury and Venus only, between Earth and the Sun (inferior).

Why Mars is the awkward one. Its synodic period of 779.9 days is the longest of any planet, so Mars oppositions come only about every 26 months, against roughly 13 months for Jupiter and barely more than 12 for Saturn and beyond. Worse, the oppositions are not equally good. Mars has an eccentric orbit, so the distance at opposition swings between about 56 and 101 million km on a roughly 15 to 17 year cycle. In August 2003 Mars came within 55.76 million km, the closest in almost 60,000 years, and reached magnitude -2.9. The March 2012 opposition was 100.8 million km out at magnitude -1.2, roughly four times fainter. Launch windows follow the same 26-month drumbeat, which is why Mars missions leave in clusters and then nothing goes for two years.

What happens at exactly one year. Feed the formula an orbital period of exactly 1.0 Earth years and it divides by zero. That is not a bug in the arithmetic, it is the physical answer: a body sharing Earth’s orbital period never changes its position relative to us, so the alignment never repeats. Earth’s quasi-satellites and the co-orbital asteroid 3753 Cruithne sit near this case, and their apparent motion drifts over decades rather than cycling in months.