Question

The outer planets (Uranus and Neptune) have synodic periods that are only slightly longer than a year. Explain why this is so.

Short answer

Uranus and Neptune have long sidereal periods, making their synodic periods only slightly over a year.

Step-by-step solution

Define Synodic Period

The synodic period of a planet is the time it takes for a planet to return to the same position in the sky relative to the Earth and the Sun. For outer planets (those beyond Earth's orbit), like Uranus and Neptune, it is the time between consecutive oppositions.

Understand the Synodic Period Formula

For outer planets, the formula for the synodic period \( S \) is given by: \[ \frac{1}{S} = \frac{1}{P} - \frac{1}{E} \] where \( P \) is the sidereal period of the planet and \( E \) is the Earth's sidereal period (1 year).

Identify Uranus and Neptune's Sidereal Periods

Uranus has a sidereal period of about 84 years, while Neptune's sidereal period is about 165 years. Compared to Earth's 1-year sidereal period, both are significantly longer.

Calculate Uranus's Synodic Period

Using the formula: \[ \frac{1}{S} = \frac{1}{84} - 1 \] Solving for \( S \), the result is a synodic period slightly over 1 year.

Calculate Neptune's Synodic Period

Using the formula: \[ \frac{1}{S} = \frac{1}{165} - 1 \] Solving for \( S \), the result is also a synodic period slightly over 1 year.

Explain Why the Synodic Period is Slightly Longer than a Year

Since Uranus and Neptune have very long sidereal periods compared to Earth's, the subtraction of \( 1 \) in the formula results in a very small value of \( \frac{1}{S} \). Thus, \( S \) becomes slightly longer than 1 year, reflecting the small difference between each planet's sidereal period and Earth's.

Key concepts

Outer Planets

The outer planets in our solar system, like Uranus and Neptune, are those that orbit the Sun beyond Earth. These planets have distinct characteristics due to their distance from the Sun. They are much larger than Earth, predominantly composed of gases like hydrogen and helium, and have longer orbital periods. When we explore these planets' motion, one concept that arises is the synodic period, which relates to how we observe them from Earth. Understanding the movement of these outer planets involves both their sidereal and synodic periods.
Unlike planets closer to the Sun, outer planets take much longer to complete an orbit, leading to significant time spans between alignments with the Earth and Sun. This is why their synodic periods, which are the time intervals between common apparent positions viewed from Earth, tend to be longer, yet they remain relatively stable just over a year for each alignment.

Sidereal Period

The sidereal period of a planet is crucial for understanding its motion in space. It refers to the time a planet takes to complete one full orbit around the Sun, relative to the distant stars. For Uranus, this period is about 84 years, and for Neptune, it's around 165 years. These are remarkably longer than Earth's 1-year sidereal period.
The long sidereal periods of outer planets mean that they move slowly across the sky from our viewpoint. This slow movement relative to the background stars is essential in calculating the synodic period. When we consider these extensive sidereal periods in the context of Earth's orbit, we understand why the synodic periods of Uranus and Neptune are just slightly over a year.

Astronomical Formulas

Astronomical formulas serve as invaluable tools for comprehending the movement of planetary bodies. To determine an outer planet’s synodic period, astronomers use a specific formula: \[\frac{1}{S} = \frac{1}{P} - \frac{1}{E} \]Here, \(S\) denotes the synodic period, \(P\) is the planet’s sidereal period, and \(E\) represents Earth's sidereal period, which is 1 year. This formula helps bridge an understanding between a planet's individual orbit and its periodic return to similar positions relative to Earth and the Sun.
The formula shows that to find the synodic period, we assess the difference between the frequency of Earth's orbit and that of the other planet. This subtraction of Earth's orbital frequency from that of Uranus or Neptune results in a very small fraction that leads to a synodic period just over a year for these distant outer giants.