What is a synodic period?
The synodic period calculator is a tool designed to help you calculate the time between two successive identical configurations as seen from any planet. In other words, if you missed the last time Mars came to opposition, you can find out when the next opportunity to see the Red Planet will be. It can also be used in reverse as a sidereal period calculator!
As we know, the planets in our system move in elliptical orbits about the Sun. The orbital period is the time needed to complete one revolution. The synodic period, however, is subjective: it depends on which planet you observe from, because the Earth (or your reference planet) is also moving. You could think of it as a relative period — the time it takes for two bodies to return to the same configuration as seen by an observer.
On the other hand, the sidereal period is the true, objective amount of time it takes a planet to complete one full orbit of the Sun relative to the stars. This value has to be found by calculation, either with the relationships used in this calculator or with Kepler's third law.
How to calculate the synodic period and sidereal period?
If you know either the sidereal or synodic period of a celestial body, you can calculate the other. The relationship depends on the planet's location, and it is typically written as a relationship between reciprocals rather than a direct formula. This calculator uses three relationships:
Inferior planet
An inferior planet is closer to the central body (e.g., the Sun) than the planet you observe it from (usually Earth):
1 / P_sid = 1 / P₀ + 1 / P_syn
Superior planet
A superior planet is farther from the center than the reference planet. Only the sign changes:
1 / P_sid = 1 / P₀ − 1 / P_syn
Absolute relationship
You can also find the synodic period without worrying about the planet's position by using the absolute value:
1 / P_syn = | 1 / P_sid − 1 / P₀ |
where:
- P_syn — the planet's synodic period;
- P_sid — the planet's sidereal period; and
- P₀ — the reference planet's sidereal period. If the synodic period is observed from Earth, this value equals 1 year, or 365.25 days.
Sidereal and synodic periods of the planets
The difference between the sidereal and synodic periods is easier to grasp with real numbers. The table below lists the orbital periods of the Solar System planets, using Earth as the reference planet.
| Celestial body | Sidereal period [yrs] | Synodic period [days] |
|---|---|---|
| Mercury | 0.241 | 116 |
| Venus | 0.616 | 584 |
| Earth | 1.0 | N/A |
| Mars | 1.9 | 780 |
| Jupiter | 11.9 | 399 |
| Saturn | 29.5 | 378 |
| Uranus | 84.1 | 370 |
| Neptune | 164.9 | 368 |
| Moon | 27.3 days | 29.5 |
The differences can be huge — the synodic period of Neptune is approximately a year, whereas in reality it takes nearly 165 years to complete one revolution around the Sun. Similarly, the sidereal period of Saturn is 29.5 years, but its synodic period is almost the same as Neptune's.
Are there other types of orbital periods?
Yes. Besides the sidereal and synodic periods, astronomers use the anomalistic period (from perihelion to perihelion), the draconic (nodal) period (between successive passes of an orbital node), and the tropical period (relative to the vernal equinox). For most everyday purposes, the sidereal and synodic periods are the two you will meet most often.
How long does the Moon take to orbit the Earth? — an example
The Moon's sidereal period — one orbit relative to the stars — is about 27.3 days. But because the Earth is also moving around the Sun, the Moon needs a little longer to return to the same phase (for example, from one full moon to the next). That synodic period is about 29.5 days. To reproduce this with the calculator, set the reference period P₀ to 27.32 days (Earth's motion relative to the Moon over a month) and the Moon's sidereal period to 27.3 days.
A note on units
Orbital periods are measured in time, which uses the same units worldwide — there is no separate "metric" or "US/imperial" unit of time. To keep things flexible the calculator lets you enter each period in either days or years, and it always reports the result in both, plus a friendly years-and-days breakdown.
FAQs
What is the difference between a synodic and a sidereal period?
The sidereal period is the true time for one orbit measured against the fixed stars. The synodic period is the apparent time between two identical configurations (such as two oppositions) as seen from a moving reference planet like Earth.
Why is the synodic period of Mars 780 days?
Mars has a sidereal period of about 1.9 years. Using 1 / P_syn = | 1 / P_sid − 1 / P₀ | with P₀ = 1 year, the synodic period works out to roughly 780 days — that is the average time between successive Mars oppositions.
Can the synodic period be infinite?
Yes. If a planet's sidereal period equals the reference planet's period, the two never change their relative configuration, so the synodic period is infinite. The calculator will warn you in that case.
Which formula should I use, inferior or superior?
If you are calculating the synodic period from a known sidereal period, the calculator detects automatically whether the planet is inferior (closer to the Sun) or superior (farther) than the reference planet. When working in reverse — from synodic to sidereal — you choose the position yourself.