Sidereal and Synodic Periods

What they measure

A sidereal period is the time an object takes to complete a rotation or revolution relative to a distant celestial reference. It describes the object's own motion against the background of the sky. A synodic period is the time between successive returns to the same relative configuration as seen from a chosen reference body or direction. It depends on the motions of both objects involved.

These periods answer different questions. A sidereal period asks when an object returns to the same orientation relative to the distant sky. A synodic period asks when the geometry between moving objects repeats. For example, the interval between two successive oppositions of a planet is its synodic period relative to Earth. For Earth's rotation, the corresponding solar recurrence is the mean solar day.

Key results

For two planets with mean sidereal orbital periods $P_1$ and $P_2$, let $P_1<P_2$, so body 1 is the inner planet. Their mean angular speeds are $2\pi/P_1$ and $2\pi/P_2$. The inner planet gains one full revolution on the outer planet during one synodic period $S$. Use the same time unit for all periods, usually days. The relative angular displacement during that interval is one complete turn:

$$\left(\frac{2\pi}{P_1}-\frac{2\pi}{P_2}\right)S=2\pi.$$

Dividing by $2\pi S$ gives

$$\boxed{\frac{1}{S}=\frac{1}{P_1}-\frac{1}{P_2}}.$$

For an arbitrary ordering, use the magnitude of the relative angular speed:

$$S=\frac{1}{\left|1/P_1-1/P_2\right|}.$$

The same idea applies to a planet's rotation relative to the Sun. For a prograde-rotating planet, let $\tau_*$ be its sidereal rotation period, $P$ its orbital period and $\tau$ its mean solar day, all measured in days. These satisfy

$$\frac{1}{\tau}=\frac{1}{\tau_*}-\frac{1}{P}.$$

For retrograde rotation, the relative angular speeds add instead:

$$\frac{1}{\tau}=\frac{1}{\tau_*}+\frac{1}{P}.$$

Earth illustrates the distinction. Its sidereal day is about 23 hours, 56 minutes and 4 seconds, while its mean solar day is 24 hours. Earth advances along its orbit while rotating, so it must turn slightly more than 360 degrees relative to the stars before the Sun returns to the same meridian. The actual apparent solar day varies slightly; 24 hours refers to the mean solar day.

Select the planetary-orbit mode and compare Earth and Jupiter. Change either sidereal period, then watch the relative angle return to its starting value; switch to Earth's-day mode to see the same calculation applied to rotation and the Sun.

Worked example

Calculate the interval between successive oppositions of Jupiter, using Earth's sidereal orbital period $P_E=365.2564\,\mathrm{d}$ and Jupiter's sidereal orbital period $P_J=4330.6\,\mathrm{d}$. Since Earth is the inner planet,

$$\begin{aligned} \frac{1}{S} &=\frac{1}{365.2564}-\frac{1}{4330.6},\\ S&\approx 398.901\,\mathrm{d}. \end{aligned}$$

The synodic period is therefore about $398.9$ days. This is shorter than Jupiter's sidereal orbital period because Earth moves around the Sun in the same direction and gradually catches up with Jupiter's orbital longitude.

Common traps

• Using a planet's sidereal orbital period when a question asks for the interval between repeated oppositions or conjunctions.
• Forgetting that a synodic period depends on relative angular speed, not on either object's angular speed alone.
• Subtracting reciprocal periods without checking which body moves faster, or omitting the absolute value when the ordering is unspecified.
• Treating every apparent solar day as exactly 24 hours. The 24-hour clock day is a mean solar day.
• Confusing Earth's sidereal rotation period with its sidereal orbital period around the Sun. One is a day-scale rotation; the other is a year-scale revolution.

Further reading

Karttunen et al., Fundamental Astronomy, 6th edition, Section 2.13, pp. 34–36; Section 7.2, pp. 143–145; Example 7.1, pp. 175–176; Tables C.4 and C.11, pp. 479 and 482.