# Synodic Period Calculator

Created by Michael Darcy
Reviewed by Dominik Czernia, PhD and Jack Bowater
Last updated: Sep 27, 2022

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!

If you don't know what the sidereal and synodic periods are or would simply like to learn more about them, read on! This article will explain the difference between these two, show you how to calculate them, and present the orbital periods of the planets in the Solar System.

## What is a synodic period?

If you've ever peeked through a window while driving a vehicle or tried to look around from a merry-go-round, you must've noticed that the world outside your frame of reference had appeared to move slightly differently than it would've if you were watching it while still.

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. You can read more about them in the article accompanying the orbital period calculator.

What is the synodic period, then? Well, although determining the time elapsed may seem fairly simple - after all, we just need to monitor the planets and wait for them to return to their initial positions - it is not that easy. We must bear in mind that the Earth is also moving, affecting our observations. This is the gist of the synodic period. You could think of it as a subjective period, as it varies depending on the planet you choose as your point of reference.

On the other hand, the sidereal period is the true, objective amount of time it takes the planet to complete one full orbit of the Sun relative to the stars. This value has to be found by calculation, either using the formula applied in our sidereal period calculator or Kepler's third law calculator.

## How to calculate the synodic period and sidereal period?

Fortunately, we have much more data than our predecessors did. It means that we know either the sidereal or synodic period of many celestial bodies and can therefore calculate the other. It turns out that obtaining the sidereal period tends to be easier as it does not require any observations. They can take a really long time - for example, astronomers (supposedly a black hole!).

So, how to calculate the synodic period without spending years watching the celestial bodies, and how to calculate the sidereal period if you know the synodic period? Well, it depends on the planet's location, and it is typically represented as a relationship rather than a direct formula. In this synodic period calculator, we use three relationships:

• An inferior planet is closer to the object in the center (e.g., Sun) than the planet you observe it from (usually assumed to be Earth). The relationship is as follows:
$\frac{1}{P_{\text{sid}}} = \frac{1}{P_0} + \frac{1}{P_{\text{syn}}}.$
• A superior planet is further from the center than the reference planet. The only change in the relationship is the sign:
$\frac{1}{P_{\text{sid}}} = \frac{1}{P_0} - \frac{1}{P_{\text{syn}}}.$
• Another way to calculate the sidereal period (or the synodic one) is to use the absolute value without worrying about the planet's position:
$\frac{1}{P_{\text{syn}}} = \big |\frac{1}{P_{\text{sid}}} - \frac{1}{P_0}\big |$

where,

• $P_{\text{syn}}$ - Planet's synodic period;

• $P_{\text{sid}}$ - Planet's sidereal period; and

• $P_0$ - Reference planet's sidereal period. If the synodic period is (or is meant to be) observed from Earth, this value is equal to 1 year or 365.25 days.

Depending on what you wish to determine, it requires playing with fractions a little bit.

## Sidereal and synodic periods of the planets

It may be tricky to understand the difference - or even the fact that there is any discrepancy - between the sidereal and synodic periods without seeing the numbers for yourself. This is why we prepared the orbital periods of planets of the Solar System - using Earth as the reference planet.

Sidereal and synodic period values for chosen bodies in the Solar System.

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

29.5

As you might have noticed, 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 of planets?

Yes, although they are rarely used. The other types of orbital periods of planets are:

• The anomalistic period is the time between successive periapsis passages. The periapsis (in Solar System called the perihelion) is the point of the closest approach between a planet and the attracting body. If you would like to learn about the elliptical orbits, check out our orbital velocity calculator.
• The nodal (draconic) period applies to the satellites, both natural (such as the Moon) and artificial (e.g., navigation satellites). It's the time that elapses between successive passages of an object through successive ascending (north) nodes - when the satellite moves north from the plane of reference. For instance, the Moon's nodal period is slightly different from its sidereal and synodic periods, with a value of 27.2 days.
• The tropical period, or solar year, applies only to Earth. It's the time between two successive vernal equinoxes (around the 21st of March) when the number of hours of daytime and darkness is the same. The tropical period is .
• You might have heard about a lunar month or a lunation. These phrases usually relate to the time between two identical phases. Therefore, this is not a different type but simply the synodic period of the Moon.

## How long does the moon take to orbit the Earth? - an example

The time between two successive lunar phases (e.g., full moons) observed from Earth is approximately 29.5 days. But have you ever wondered how long does the moon take to orbit the Earth in reality?

Luckily, we already know how to calculate the sidereal period, so this shouldn't be too tricky. Since the synodic period calculator takes the Earth to be the default reference planet, all we need to do is convert this into days (365.25) and input 29.5 days as the synodic period. This way, we find that the moon needs approximately 27.3 days to orbit 360° around the Earth - which is less than we see. In other words, the sidereal period of the Moon is 27.3 days.

## FAQ

### How to find the synodic period?

You can use the following formula to find the synodic period without worrying about the relative position of the planet:

1/planet's synodic period = |1/planet's sidereal period - 1/reference planet's sidereal period|

### What is the difference between the sidereal and synodic period?

The sidereal period is the time the planet needs to complete one full orbit relative to the "fixed" stars. The synodic period is observed from a planet, so it varies depending on the point of reference.

### Can the synodic period be measured as seen from other planet than Earth?

Yes, although Earth is the usual point of reference, you can choose to consider any other planet so long as you know its sidereal period (or other related variables).

### Is the sidereal period of the Moon the same as its synodic period?

No. Although it's relatively close to Earth and we may think our observations are accurate, the sidereal period is shorter (27.3 days) than the synodic period of the Moon (29.5 days).

Michael Darcy
Planet
Custom
Reference planet's sidereal period
yrs
Planet's sidereal period
yrs
Planet's synodic period
yrs
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