Synodic Period Calculator: Why Mars Oppositions Come Round When They Do
Synodic Period Calculator
Mars goes round the Sun in 687 days, but a Mars opposition comes round every 780. The difference is the synodic period: how long one body takes to lap another as seen from the Sun, and it is the quantity that sets launch windows, eclipse seasons and when a planet is worth looking at. One subtraction of angular rates does the whole job.
Time for one body to lap another: 1/S = |1/T₁ − 1/T₂|
How long between one Mars opposition and the next? The Earth’s sidereal orbit period is 365.256 days and Mars’ is 686.980 days, both from the NASA fact sheets.
Subtract the rates, then take the reciprocal
- S
- synodic period: the time from one line-up to the next. Also the time from one opposition to the next, or one conjunction to the next — the formula does not care which configuration you start from.
- T₁, T₂
- the two sidereal orbital periods, measured against the stars, in the same unit. Days here.
- ω = 360/T
- angular rate in degrees per day. Subtracting these is the whole derivation, and it is the same trick as the minute hand lapping the hour hand.
- | |
- the absolute value, which is why the order of the two bodies makes no difference. A synodic period belongs to a pair.
Worked example
How long between one Mars opposition and the next? The Earth’s sidereal orbit period is 365.256 days and Mars’ is 686.980 days, both from the NASA fact sheets.
Turn each period into an angular rate. The Earth sweeps 360 ÷ 365.256 = 0.985 610 degrees a day. Mars sweeps 360 ÷ 686.980 = 0.524 033 degrees a day. Same units, same direction, both measured against the stars.
Subtract them. The Earth gains on Mars at 0.985 610 − 0.524 033 = 0.461 577 degrees a day. This is the only quantity that matters; the individual rates do not appear again.
How long to gain a whole lap? 360 ÷ 0.461 577 = 779.93 days, or 2.135 years. That is the synodic period.
The same thing in reciprocals. 1/S = 1/365.256 − 1/686.980 = 0.002 737 81 − 0.001 455 60 = 0.001 282 21 per day, and S = 1/0.001 282 21 = 779.93 days. Identical arithmetic, fewer keystrokes.
Check it against the published figure. NASA publishes Mars’ synodic period as 779.94 days. The calculation gives 779.934, which agrees to five figures — and it was done from two numbers taken off a completely different row of the same table.
And the sanity check that catches the common slip. The answer must be longer than either orbital period, always, because 1/S is a difference of two positive numbers and so is smaller than either. If your answer comes out shorter than 687 days you have added the rates instead of subtracting them.
Every published synodic period, rebuilt from two sidereal periods
| Pair with the Earth | Its sidereal period (d) | Synodic from this page (d) | NASA published (d) | Difference |
|---|---|---|---|---|
| Mercury | 87.969 | 115.8771 | 115.88 | 0.003% |
| Venus | 224.701 | 583.9236 | 583.92 | 0.001% |
| Mars | 686.980 | 779.9343 | 779.94 | 0.001% |
| Jupiter | 4332.589 | 398.8836 | 398.88 | 0.001% |
What a synodic period is called when you are standing on the Earth
| Configuration | Where the planet is | How often |
|---|---|---|
| Opposition (outer planets only) | opposite the Sun in the sky, so highest at midnight | once per synodic period — Mars every 780 days |
| Conjunction | behind or in front of the Sun, lost in the glare | once per synodic period, halfway between oppositions |
| Greatest elongation (inner planets only) | as far from the Sun as it ever gets, so best seen at dawn or dusk | twice per synodic period — Venus every 584 days, east then west |
| Launch window | at the phase angle a transfer orbit needs | once per synodic period — which is why Mars missions go roughly every 26 months |
Rates subtract, and that is the whole idea
This is the clock-hands problem. The minute hand goes round in 60 minutes and the hour hand in 720, and they line up not every 60 minutes but every 65⅔ — because by the time the minute hand gets back to twelve, the hour hand has moved on and has to be caught. Planets do exactly this. The synodic period is how long the faster body takes to gain one whole lap on the slower one, and it is always longer than either orbital period.
The derivation is two lines. Body one turns at ω₁ = 360/T₁ degrees a day and body two at ω₂ = 360/T₂. The angle between them changes at ω₁ − ω₂, so a whole 360° takes S = 360/(ω₁ − ω₂) days, and dividing through by 360 gives the memorable form 1/S = 1/T₁ − 1/T₂. Take the absolute value and the order of the two bodies stops mattering.
Why Mars is the famous case. Mars takes 687 days to go round, but you cannot see a Mars opposition every 687 days, because the Earth has also moved. The Earth has to catch up the extra lap, and that takes until day 780. Those 780 days are also the gap between Mars launch windows, which is the real reason Mars missions arrive in clusters and then nothing happens for two years. The same number controls when Mars is bright in the sky and when it is invisible behind the Sun.
The behaviour at the two extremes is worth holding on to. When the two periods are close, the synodic period explodes. Two bodies at 365 and 366 days lap each other once every 366 years. When the two periods are very different, the synodic period collapses onto the shorter one: Jupiter takes 4333 days to go round but its synodic period with the Earth is only 399 days, barely more than a year, because Jupiter has hardly moved by the time the Earth comes back. Every outer planet has a synodic period a little over a year for that reason, and the further out the planet, the closer to exactly a year it gets.
The Moon is the same formula in disguise. The Moon goes round the Earth against the stars in 27.3217 days, which is the sidereal month. But a new moon to the next new moon is 29.53 days, because the Earth has moved along its own orbit and the Moon has to catch up on the Sun’s new direction. Put 27.3217 and 365.256 into this page and you get 29.5306 days, which is the synodic month to six figures. That is the calendar month you actually live by.
Where this goes wrong is feeding in the wrong kind of period. The formula needs sidereal periods, measured against the fixed stars. Fact sheets print both the sidereal and the synodic period in the same table, a few rows apart, and taking the synodic one as an input gives an answer that is wrong and not obviously so. The other trap is mixing units — one period in days and one in years — which the formula will not catch because it is dimensionally happy either way.
Frequently asked questions
Why is the synodic period longer than both orbital periods?
Because 1/S is the difference of two positive numbers, so 1/S is smaller than either 1/T, so S is bigger than either T. Physically: the faster body has to gain a whole extra lap on a target that is itself moving away. The only exception is the degenerate case where the two periods are equal, when S is infinite. If your answer comes out shorter than either period, you have added instead of subtracting.
Does it matter which body I put first?
No. The formula takes the absolute value of the difference, so swapping them gives the same number. That is worth saying out loud, because it means a synodic period is a property of a pair of orbits and not of either body — “the synodic period of Mars” is shorthand for “the synodic period of Mars and the Earth”, and Mars has a different one with Jupiter.
What is the difference between a sidereal and a synodic period?
Sidereal is measured against the fixed stars; synodic is measured against the Sun, or against another body. The Earth’s sidereal rotation is 23 h 56 min 4 s and its synodic rotation — the solar day — is 24 hours. The Moon’s sidereal month is 27.32 days and its synodic month is 29.53. In both cases the synodic one is longer, for the same reason: the thing you are measuring against has moved.
How does this set a launch window to Mars?
A Hohmann transfer only works if Mars will be at the far end of the transfer ellipse when the spacecraft arrives, which pins the Earth-Mars angle at departure to a single value — about 44° ahead. That geometry comes round once per synodic period, so once every 780 days, or about every 26 months. Miss it and you wait more than two years. The Hohmann transfer page computes the required angle.
Can I use this for two moons of Jupiter, or for a satellite and the Moon?
Yes for two moons of Jupiter, because they both orbit the same body. The formula knows nothing about the Sun; it only needs two angular rates about a common centre. It is not valid for two bodies orbiting different centres, and the relative orbit size this page prints assumes a common centre too, since it comes from Kepler’s third law.
Why do the published and calculated synodic periods differ in the last digit?
Because the sidereal periods going in are rounded. NASA publishes the Earth at 365.256 days and Mars at 686.980, and those feed a result good to about one part in a hundred thousand — 779.9343 against a published 779.94. Nothing is wrong with either figure; the difference is simply what rounding the inputs costs you. Quote a school answer as 780 days.
Related calculators
References
- NASA Goddard, National Space Science Data Coordinated Archive, Planetary Fact Sheet (Metric) and the individual planet fact sheets, curated by D. R. Williams, page updated 18 March 2025. Source of the semi-major axes, sidereal orbit periods, eccentricities and published synodic periods tabulated on this page.
nssdc.gsfc.nasa.gov/planetary/factsheet/ - Paris Observatory, Numerical standards and constants (IERS/EOP product centre). Sidereal year 365.256 363 004 mean solar days, after Simon et al., Astron. Astrophys. 282 (1994) 663; sidereal day 86 164.090 530 832 88 s exactly, after Aoki et al., Astron. Astrophys. 105 (1982) 359; geocentric gravitational constant 3.986 004 418(8) × 1014 m3 s−2.
- NASA Science, Basics of Space Flight, Chapter 4: “Trajectories”. Describes the Hohmann transfer from Earth’s orbit at perihelion to Mars’ orbit at aphelion, the departure burn “in the direction of Earth’s revolution around the Sun” and the orbit-insertion burn at arrival, and gives the launch-window spacing as roughly 25 months for Mars and 19 for Venus. It prints no Δv figures, so the ones on this page are computed, not quoted.
- NCERT, Physics: Textbook for Class XI, Part I, the chapter “Gravitation”, which states all three of Kepler’s laws and derives the orbital and escape speeds from them. The chapter number differs between the pre-2023 edition and the rationalised one, and
ncert.nic.indisallows automated reading, so it is cited here by name only rather than by a number that could be wrong. - E. Mamajek et al., IAU 2015 Resolution B3 on Recommended Nominal Conversion Constants for Selected Solar and Planetary Properties, arXiv:1510.07674. Source of every GM and radius used here: nominal solar mass parameter 1.3271244 × 1020 m3 s−2, terrestrial 3.986004 × 1014, jovian 1.2668653 × 1017; nominal solar radius 6.957 × 108 m, terrestrial equatorial radius 6.3781 × 106 m, jovian equatorial 7.1492 × 107 m. The resolution is explicit that these are conversion factors fixed by convention, not measurements with uncertainties.