What this calculator does
Mars takes 687 days to orbit the Sun, but Earth does not wait for it. By the time Mars returns to the same place, Earth has moved on, so the interval between successive alignments is longer than either orbit: about 780 days.
That interval is the synodic period, and it governs when planets can be observed at opposition and when launch windows to them open. It is the same arithmetic that makes the lunar month 29.5 days rather than the 27.3 days the Moon actually takes to orbit.
The formula
Take the difference between the two orbital rates, expressed as one over each period, and invert the result. The absolute value handles the case where the reference body is the faster one.
| Term | Meaning |
|---|---|
| Sidereal period | The true orbital period measured against the fixed stars. |
| Synodic period | The interval between successive alignments as seen from the reference body. |
| Opposition | The alignment at which a planet is opposite the Sun in the sky, recurring each synodic period. |
The inputs explained
| Field | What to enter |
|---|---|
| Sidereal period of the body (Psid) (days) | The orbital period of the body of interest, in days. Mars is 687; Venus is 224.7. |
| Reference body sidereal period (P0) (days) | The orbital period of the reference body. Earth is 365.25 days. |
When to use it
Finding the next Mars launch window
Transfer opportunities recur once per synodic period, which is why Mars missions cluster roughly every 26 months.
Predicting an opposition
A planet returns to opposition once per synodic period, which is when it is closest and brightest.
Understanding the lunar month
The Moon's 29.5 day cycle of phases is its synodic period, not its 27.3 day orbit.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How often does each body align with Earth?
Alignment intervals for bodies with different orbital periods.
| Sidereal period | Synodic period (Psyn) | Synodic period |
|---|---|---|
| 224.7 days | 583.93 days | 1.599 years |
| 687 days | 779.88 days | 2.135 years |
| 4333 days | 398.87 days | 1.092 years |
| 27.32 days | 29.53 days | 0.081 years |
Questions
Why is the synodic period longer than the orbital period?
For an outer planet, because Earth has moved on by the time it returns to the same spot, so Earth must catch up. The gap between alignments is therefore longer than either orbit on its own.
Why is Jupiter's synodic period so close to a year?
Because Jupiter moves so slowly that it barely shifts while Earth completes an orbit. Earth only needs a little extra to catch it, giving 398.87 days rather than 365.
Why is the lunar month 29.5 days and not 27.3?
The Moon orbits Earth in 27.32 days, but Earth has moved around the Sun in that time, so the Moon needs about two more days to return to the same Sun-Earth-Moon alignment that defines the phase.
What happens if the two periods are equal?
The bodies never lap each other, so there is no recurring alignment and the synodic period is infinite. The calculator reports that rather than dividing by zero.
For the orbital period itself, see the Kepler's third law calculator. For orbital speed at a given radius, see the orbital velocity calculator.