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Chapter 3: Problem 32

The distance that Neptune has to travel in its orbit around the Sun is approximately 30 times greater than the distance that Earth must travel. Yet it takes nearly 165 years for Neptune to complete one trip around the Sun. Explain why.

Short Answer

Expert verified
Neptune takes 165 years to orbit the Sun because its orbit is 30 times larger, and according to Kepler's Third Law, greater distances result in much longer orbital periods.

Step by step solution

01

- Understand the problem

Neptune travels a much larger distance around the Sun compared to Earth. Despite this, it takes significantly longer to complete one orbit. The goal is to explain why.
02

- Kepler's Third Law

Recall Kepler's Third Law which states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. Mathematically, this is expressed as: \[\frac{T_1^2}{R_1^3} = \frac{T_2^2}{R_2^3} \] where \( T \) is the orbital period and \( R \) is the distance from the Sun (semi-major axis).
03

- Apply Kepler's Third Law

Given that Neptune's distance from the Sun is 30 times that of Earth's, we let \( R_2 = 30 R_1 \). Substituting in Kepler's law we get: \[\frac{T_1^2}{R_1^3} = \frac{T_2^2}{(30 R_1)^3} \] Which simplifies to: \[\frac{T_1^2}{R_1^3} = \frac{T_2^2}{27,000 R_1^3} \]
04

- Solve for Neptune's Period

Since Earth’s orbital period \( T_1 \) is 1 year, let's solve for Neptune’s orbital period \( T_2 \): \[\frac{1^2}{1^3} = \frac{T_2^2}{27,000} \] Thus: \[\frac{1}{1} = \frac{T_2^2}{27,000} \] And: \ T_2 = \sqrt{27,000} = 164.32 \text{ years} So it takes Neptune about 165 years to complete one orbit around the Sun.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Orbital Period
The orbital period is the time a planet takes to complete one full orbit around the Sun. It is an important concept in astronomy, helping us understand the dynamics of planetary movement.
Kepler's Third Law plays a crucial role in determining the orbital period. According to this law, the square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit.
This means that if we know the distance of a planet from the Sun, we can calculate how long it takes to orbit the Sun.
For example, Earth takes about one year to orbit the Sun because its distance, or semi-major axis, is defined as 1 Astronomical Unit (AU).
Semi-Major Axis
The semi-major axis is the longest radius of an ellipse, which is the shape of a planet's orbit around the Sun. It is essentially the average distance between the planet and the Sun.
In mathematical terms, the semi-major axis is symbolized by \( R \).
According to Kepler's Third Law, the cube of the semi-major axis is directly related to the square of the orbital period.
For Neptune, the semi-major axis is 30 times greater than that of Earth. This gives us a clear understanding of why Neptune’s orbit takes so much longer.
Planetary Orbits
Planetary orbits are the paths that planets follow around the Sun. These paths are not perfect circles; they are ellipses.
Kepler's laws offer a deep understanding of these orbits. His Third Law, in particular, is vital to grasp why different planets have varying orbital periods.
The further a planet is from the Sun, the larger its orbit, and consequently, the longer its orbital period. This relationship is mathematically captured in the expression:
\( \frac{T_1^2}{R_1^3} = \frac{T_2^2}{R_2^3} \)
Neptune's Orbit
Neptune's orbit around the Sun is a fascinating example of how Kepler's Third Law works. Despite being 30 times farther away from the Sun than Earth, Neptune takes nearly 165 years to complete one orbit.
To understand this, we apply Kepler's Third Law. Given that Earth's distance from the Sun is defined as 1 AU, Neptune's distance is 30 AU.
By substituting into Kepler's formula, we see that if Earth's orbital period is 1 year, Neptune's orbital period, \( T_2 \), becomes:
\( \frac{1^2}{1^3} = \frac{T_2^2}{30^3} \)
Simplifying, we get \( T_2 = \sqrt{27,000} \approx 164.32 \text{ years} \). This tells us why Neptune takes almost 165 years to orbit the Sun.

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Most popular questions from this chapter

You are riding along on your bicycle at \(20 \mathrm{km} / \mathrm{h}\) and eating an apple. You pass a bystander. a. How fast is the apple moving in your frame of reference? b. How fast is the apple moving in the bystander's frame of reference? c. Whose perspective is more valid?

What is the eccentricity of a circular orbit?

Flybynite Airlines takes 3 hours to fly from Baltimore to Denver at a speed of \(800 \mathrm{km} / \mathrm{h}\). To save fuel, management orders the airline's pilots to reduce their speed to \(600 \mathrm{km} / \mathrm{h}\) How long will it now take passengers on this route to reach their destination?

Most ancient Greek astronomers were skeptical about the idea that Earth revolved around the Sun because a. of their religious beliefs. b. it feels as though Earth is stationary. c. Earth is so important. d. the heavens were thought to be perfect.

For Earth, \(P^{2} / A^{3}=1.0\) (in appropriate units). Suppose a new dwarf planet is discovered that is 14 times as far from the Sun as Earth is. For this planet, a. \(P^{2} / A^{3}=1.0\) b. \(P^{2} / A^{3}>1.0\) c. \(P^{2} / A^{3}<1.0\) d. you can't know the value of \(P^{2} / A^{3}\) without more information.
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