One over S is the difference of the two mean motions
If two bodies go around the same primary in coplanar circular orbits with sidereal periods T1 and T2, the time between successive similar alignments is 1/S = |1/T1 − 1/T2|. Earth (365.256 days) and Mars (686.980 days) therefore have S ≈ 779.9 days, a little over 26 months — the familiar spacing of Mars launch windows in this idealized model. Two custom periods of 10 and 15 days meet every 30 days, the same identity in a tighter orbit pair.
Eccentricity, inclination, and phasing all move a real window off this mean interval. For the Earth–Mars Hohmann time of flight and heliocentric Δv, see the planetary transfer calculator; for a Hohmann transfer around a single body, see Hohmann transfer.
Sources
- The synodic-period identity 1/S = |1/T1 − 1/T2| (Danby, Fundamentals of Celestial Mechanics; Vallado, Fundamentals of Astrodynamics and Applications).
- Mean sidereal periods from standard astronomical constants (approximate).
Frequently asked questions
Is 780 days the only Earth–Mars launch opportunity?
It is the mean spacing of geometrically similar Hohmann windows in a circular coplanar model. Real trajectories use a broader window around that date, and a fast or slow transfer has a different geometry.
What if the two periods are equal?
There is no synodic period — the relative geometry never repeats because it never changes. The page returns nothing.
Can I use this for two satellites around Earth?
Yes. Enter their periods as custom values. The same identity is the time between successive close approaches if the orbits are coplanar and circular.