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Author: Kristin Miller
Editor: Grace L. Deming, University Of Maryland
Our Solar System
The goals of this module: After completing this exercise, you should be able to:
Describe the properties of an ellipse.
Explain how a planet's orbital speed varies throughout its orbit.
Describe the relationship between a planet's period of revolution and its average distance from the Sun.
In this module you will explore:
The orbital motions of the planets around the Sun.
What Kepler's three laws predict for planetary motion.
Why you are doing it: As citizens of a heliocentric system, it is important for us to understand and appreciate the laws that explain the world we live in. In addition, understanding the orbital motion of the planets is key to understanding not only our own Solar System, but also the orbital properties of spacecraft, moons, binary stars, binary galaxies, etc!!
4.2Background
Nicolaus Copernicus
Tycho Brahe in his observatory
From the time of the ancient Greeks until the early 16th century, the prevailing belief among both scientists and the general public was that everything in the known universe revolved around the Earth. Then, in 1543, Nicolaus Copernicus revived a theory first proposed by the ancient Greek astronomer Aristarchus, which stated that the Earth revolves around Sun. This revolutionary idea was merely a hypothesis until the late 1500s when an observer named Tycho Brahe began to make careful and fairly accurate measurements of the orbital positions of the planets. Johannes Kepler studied Brahe's data, and used it to develop a model of planetary motions that still holds true today. This was the first scientific model of the Solar System as we now understand it to be!
Johannes Kepler
From Brahe's data, Kepler created three laws describing the orbital motions of the planets around the Sun. His first law describes the shapes of the planets' orbits. His second law explains how a planet's orbital speed varies as it travels around the Sun. His third law describes a relationship between the time it takes a planet to make one complete revolution around the Sun and the planet's average distance from the Sun.
In this activity, you will delve deeper into the significance of each of these three laws and how they apply to our Solar System.
4.3Kepler's First Law
Kepler's first law states:
Planets orbit the Sun in ellipses with the Sun at one focus.
An ellipse looks something like a squashed circle. Because it is not perfectly round, it has two axes, one longer (called the major axis) and one shorter (called the minor axis). Half of the major axis is called the semimajor axis, often represented by a lower case "a"; the semimajor axis is the average distance of the planet from the Sun. Instead of a center (such as a circle has), an ellipse has two foci. Press the play button to look at the animation below to learn more about elliptical orbits firsthand.
Question
1.
In our Solar System, what are at the two foci of the planetary orbits?
A.
B.
C.
D.
3
Look again at the animation noting what's at the two foci. Try again.
Correct. The Sun is at one focus, and there is nothing at the other.
Incorrect. The Sun is at one focus, and there is nothing at the other.
4.4Ellipses
Ellipses can be very long and thin or very nearly (or even exactly!) circular - in fact, a circle is just a special case of an ellipse. The amount of elongation an ellipse has is described by its eccentricity. Eccentricity is simply the difference in length between the major and minor axes divided by their sum:
Eccentricity = e = (major-minor)/(major + minor)
The eccentricity of a circle is zero. In our Solar System, most planetary orbits have eccentricities very close to 0. As an ellipse becomes more elongated, the eccentricity approaches a value of 1. Comets usually have very highly eccentric orbits. As you can see below, even though these ellipses have major axes of the same length, the eccentricity affects the shape of the ellipse.
Question
2.
As the eccentricity increases, the ellipse becomes more ____________.
A.
B.
C.
D.
3
Study the figure carefully and note what happens to the orbit as the eccentricity increases. Try again.
Correct. The higher the eccentricity, the more elongated the ellipse becomes.
Incorrect. The higher the eccentricity, the more elongated the ellipse becomes.
4.5Kepler's Second Law
Kepler's second law states:
A line joining the planet and the Sun sweeps out equal areas in equal times.
This next animation shows a planet orbiting the Sun. The eccentricity of this orbit is exaggerated to show Kepler's second law easily. Click on the play arrow to begin.
Question Sequence
Question
3.
At which point in its orbit does the planet in the animation above travel the fastest?
A.
B.
C.
D.
3
Replay the animation and pay close attention to the planet's speed as it moves on its orbit. Try again.
Correct. The planet moved fastest when it was closest to the Sun (perihelion).
Incorrect. The planet moved fastest when it was closest to the Sun (perihelion).
Question
4.
What Kepler's second law really tells us is that planets do not travel at a single speed as they revolve around the Sun. Instead, as you saw above, they travel much faster when they are near the Sun and much slower when they are far from the Sun.
When the planet is at perihelion, the imaginary line between it and the Sun is shorter, but it is moving fast, so it sweeps out a short wedge with a wide base. When it is at aphelion, the imaginary line connecting it to the Sun is longer, but it is moving slow, so it sweeps out a tall wedge with a narrow base. Kepler's second law tells us that the two wedges will have the same area if the time traveled is the same.
As a planet orbits the Sun, it sweeps out equal ____________ in equal times.
A.
B.
C.
D.
3
Watch the animation again. Try again.
Correct. Kepler's second law tells us that the areas will be the same in equal times.
Incorrect. Kepler's second law tells us that the areas will be the same in equal times.
4.6Kepler's Third Law
Kepler's third law states:
The square of a planet's orbital period is directly related to the cube of its semi-major axis.
This law is often written as P2∝a3 , where P is the period, a is the semimajor axis of the orbit and ∝ means "is proportional to." The equation becomes an equality when we use units of years for P and AU (AU stands for an "astronomical unit", which is the average distance between the Earth and the Sun) for a; then we can write P2=a3
Planet
Sidereal period P (years)
Semimajor axis a (AU)
p2
a3
0.24
0.39
0.06
0.06
0.61
0.72
0.37
0.37
1.00
1.00
1.00
1.00
1.88
1.52
3.53
3.51
11.86
5.20
140.7
140.6
29.46
9.55
867.9
871.0
84.10
19.19
7,072
7,067
164.86
30.07
27,180
27,190
Table : Table Caption
Question
5.
Of those listed below, which planet has the longest year?
A.
B.
C.
D.
3
Try clicking on each of these planets in the simulation above and see which has the largest period. Try again.
Correct. Saturn has the largest semimajor axis of the listed planets and thus the largest period of those listed.
Did you notice that as we look at planets that are increasingly farther from the Sun, the period becomes very long very quickly? This happens because the relationship between period and semimajor axis is not linear (in other words, when "a" doubles, the value of "P" more than doubles).
Incorrect. Saturn has the largest semimajor axis of the listed planets and thus the largest period of those listed.
Did you notice that as we look at planets that are increasingly farther from the Sun, the period becomes very long very quickly? This happens because the relationship between period and semimajor axis is not linear (in other words, when "a" doubles, the value of "P" more than doubles).
4.7Quick Check Quiz
Indepth Activity: Kepler's Laws
Question
6.
As a planet orbits the Sun:
A.
B.
C.
D.
E.
Correct. Kepler's second law tells us that a planet travels much faster at perihelion than at aphelion.
Incorrect. Kepler's second law tells us that a planet travels much faster at perihelion than at aphelion.
Question
7.
An ellipse becomes more ____________ as the eccentricity ____________.
A.
B.
C.
D.
E.
Correct. The eccentricity of a circle is zero. The amount of elongation an ellipse has is described by its eccentricity. Eccentricity is simply the difference in length between the major and minor axes divided by their sum:
Eccentricity = e = (major – minor)/(major + minor)
Incorrect. The eccentricity of a circle is zero. The amount of elongation an ellipse has is described by its eccentricity. Eccentricity is simply the difference in length between the major and minor axes divided by their sum:
Eccentricity = e = (major – minor)/(major + minor)
Question
8.
What does Kepler's first law state?
A.
B.
C.
D.
E.
Correct. Kepler's first law: planets orbit the Sun in ellipses with the Sun at one focus.
Incorrect. Kepler's first law: planets orbit the Sun in ellipses with the Sun at one focus.
Question
9.
Kepler's second law states a line joining the planet and the Sun sweeps out equal areas in equal times. What does this tell us about the speed of the planet as it orbits the Sun?
A.
B.
C.
D.
E.
Correct. When the planet is at perihelion, the imaginary line between it and the Sun is shorter, but it is moving fast, so it sweeps out a short wedge with a wide base. When it is at aphelion, the imaginary line connecting it to the Sun is longer, but it is moving slowly, so it sweeps out a tall wedge with a narrow base. Kepler's second law tells us that the two wedges will have the same area if the time traveled is the same.
Incorrect. When the planet is at perihelion, the imaginary line between it and the Sun is shorter, but it is moving fast, so it sweeps out a short wedge with a wide base. When it is at aphelion, the imaginary line connecting it to the Sun is longer, but it is moving slowly, so it sweeps out a tall wedge with a narrow base. Kepler's second law tells us that the two wedges will have the same area if the time traveled is the same.
Question
10.
At which point in its orbit does a planet travel the slowest?
A.
B.
C.
D.
E.
Correct. Kepler's second law implies that planets do not travel at a single speed as they revolve around the Sun. Instead, they travel faster when they are near the Sun and slower when they are far from the Sun.
Incorrect. Kepler's second law implies that planets do not travel at a single speed as they revolve around the Sun. Instead, they travel faster when they are near the Sun and slower when they are far from the Sun.
Question
11.
Based on Kepler’s third law, which of the following statements is true?
A.
B.
C.
D.
E.
Correct. Kepler's third law states the square of a planet's orbital period is directly related to the cube of its semi-major axis. Even though the original formulation was limited to planets, Newton’s generalization showed that this law applies to all objects in orbit, including moons around planets.
Incorrect. Kepler's third law states the square of a planet's orbital period is directly related to the cube of its semi-major axis. Even though the original formulation was limited to planets, Newton’s generalization showed that this law applies to all objects in orbit, including moons around planets.
Question
12.
In our Solar System, most planetary orbits have eccentricities that are approximately equal to:
A.
B.
C.
D.
Correct. In the Solar System, planetary orbits are very close to 0; Mercury is the highest with an eccentricity of 0.206.
Incorrect. In the Solar System, planetary orbits are very close to 0; Mercury is the highest with an eccentricity of 0.206.
Question
13.
The average distance of a planet from the Sun is given by which orbital property?
A.
B.
C.
D.
Correct. The semimajor axis, half of the long axis of the ellipse, is also the average distance of the planet from the Sun.
Incorrect. The semimajor axis, half of the long axis of the ellipse, is also the average distance of the planet from the Sun.
Question
14.
The Earth is 1 AU from the Sun and has a period of 1 year. What would be the period for an imaginary planet located at 2 AU from the Sun?
A.
B.
C.
D.
Correct. 2 cubed (2 x 2 x 2) is 8, and the square root of 8 is about 2.8.
Incorrect. 2 cubed (2 x 2 x 2) is 8, and the square root of 8 is about 2.8.
Question
15.
Which is the correct form of Kepler's third law?
A.
B.
C.
D.
Correct. Kepler found that the square of the period was proportional to the cube of the semimajor axis.
Incorrect. Kepler found that the square of the period was proportional to the cube of the semimajor axis.