How this instrument works
The synodic period answers a different question than the orbital period does. T1 and T2 describe how long each body takes to complete one full circuit against the fixed stars — its own year. Tsyn describes how long it takes the pair to return to the same relative arrangement as seen from one to the other, which is the number that actually matters to anyone standing on one body and watching the other.
The formula falls out of angular speed. A body sweeps 2π radians every orbital period, so its angular speed is 2π ⁄ T. The faster body gains angle on the slower one at the rate given by the difference of their angular speeds, 2π(1 ⁄ T1 − 1 ⁄ T2), and it has lapped the slower body once that gained angle reaches a full turn — which takes 1 ⁄ |1 ⁄ T1 − 1 ⁄ T2| of whatever time unit T1 and T2 are given in. The reciprocals subtract because it's the difference in rotation rate, not the difference in period, that sets the beat.
Two limits are worth knowing. If T1 equals T2 the denominator hits zero and Tsyn is undefined — the two bodies share an angular speed and never lap each other, the situation Jupiter's Trojan asteroids sit in permanently, clustered near a fixed 60° lead or lag on Jupiter's own orbit. And Tsyn is not always longer than both inputs: it always exceeds the shorter period, but only exceeds the longer one when the two periods are reasonably close. Earth and Saturn, wildly mismatched at 365.25 days against a 10,759-day orbit, realign every 378.1 days — barely a fortnight past Earth's own year, because a nearly stationary outer planet hardly needs lapping at all.
- Enter the orbital period of the faster or inner body — Earth's 365.25 days, say — into 'Orbital period of body 1'.
- Enter the orbital period of the slower or outer body, such as Mars's 687 days, into 'Orbital period of body 2'.
- Keep both fields in the same time unit, days or years, so the reciprocal subtraction compares like quantities.
- Read 'Synodic period' for the interval between successive alignments — Earth and Mars work out to about 779.9 days.
- Swap the two inputs if you like; the absolute value in the formula means the order of T1 and T2 never changes the result.
Worked example — Earth, Mars, and the 26-month launch window
Put Earth's orbital period in seconds, 31,557,600 (365.25 days), into T1, and Mars's, 59,356,800 seconds (687 days), into T2. The instrument returns Tsyn = 67,381,728.6713 seconds. Divide by 86,400 seconds per day and that is 779.88 days — about two years and two months, not either planet's own year.
That 779.88-day rhythm is the real clock behind Mars exploration. NASA's InSight lander left Earth in 2018 and Perseverance in 2020, each waiting for an opposition-aligned window because a direct, fuel-efficient transfer orbit only lines up once every synodic period — not every Earth year, and not every 687-day Mars year either. Miss a window and the wait is a full Tsyn, not a T2.
Questions
What happens if I enter the same period for T1 and T2?
The formula divides by zero, because 1/T1 − 1/T2 vanishes when the periods match — two bodies with identical periods share the same angular speed and never lap each other at all. Jupiter's Trojan asteroids live in exactly this state, holding a roughly fixed 60° lead or lag on Jupiter forever. The calculator flags equal inputs as invalid rather than return a meaningless infinite result.
Is the synodic period always longer than both orbital periods?
No. It is always longer than the shorter of the two periods, but only longer than the longer one when the two periods are fairly close together, as with Earth and Mars. When they're mismatched, Tsyn drifts toward the shorter period instead: Earth and Saturn (365.25 days versus 10,759 days) realign every 378.1 days, barely past Earth's own year, since a near-stationary outer planet is easy to lap.
Does it matter which body I put in T1 versus T2?
No. The formula takes an absolute value, |1/T1 − 1/T2|, so swapping the two fields returns an identical Tsyn. This instrument's examples put the faster body in T1 by convention, purely for readability — the arithmetic treats the two fields symmetrically.
How does this work for an inferior planet like Venus?
Identically — the same formula, no special case. Earth's 365.25-day year and Venus's 224.7-day year give a synodic period of 583.9 days, about 19 months, which is the cycle length behind Venus's alternating runs as morning star and evening star as it laps past Earth from the inside track.
How is synodic period different from orbital period?
Orbital, or sidereal, period is how long one body takes to complete a circuit relative to the distant stars — Mars's own year, 687 days. Synodic period is how long two bodies take to return to the same relative alignment as seen from one to the other — 779.9 days for Earth watching Mars. A sidereal period belongs to a single orbit; a synodic period belongs to a pair.
Why does Mars seem to move backward before opposition?
Because Earth, moving faster on the inner track, overtakes Mars once every synodic period. As Earth passes on the inside near each 779.9-day alignment, Mars's position against the background stars appears to reverse for several weeks before resuming its normal eastward drift — the retrograde loop amateur astronomers track every opposition.