Although he trained to be a Lutheran priest, from an early age Johannes Kepler was fascinated by astronomy. Along with his contemporary Galileo, he was one of the first to embrace the theory that the Earth and other planets orbited the Sun.
In 1600, Kepler moved to Prague to work for the great astronomer Tycho Brahe. When Brahe died a year later, Kepler began to analyze his vast collection of planetary observations, starting with the orbit of Mars. After nine long years of crunching numbers, plotting positions, and discarding flawed theories, Kepler finally discovered the underlying geometry at work. His description of planetary orbits can be summarized by three elegant laws.

Kepler's First Law

In this activity, you will observe the shape of planetary orbits.

The SIMULATION pane of the Gizmo™ shows a sun, represented by a yellow sphere, and a planet, represented by a purple sphere. Check that the Show vectors and Show trails checkboxes are turned on. The purple arrow represents the direction and magnitude of the planet's velocity (v), and the green arrow represents the pull of the sun's gravity (g) on the planet. Click Play (

play button

).

As the planet revolves around the sun, its orbit is traced by a series of dots. What is the shape of the orbit?

What do you notice about the green gravitation vector? Where does this force always point?

Click Reset (

reset button

) and drag the planet to a new position, closer to the sun. Click Play. What is the shape of the orbit made by the planet now?

Experiment by placing the planet in four different positions and dragging the purple vector to create different starting velocities. Use the + and - zoom controls on the right side of the simulation as needed. Sketch each orbit, or take a snapshot using the camera icon and paste each image into a single document.

The flattened circle you see in most cases is called an ellipse. Ellipses have many properties analogous to circles. For example, every point on a circle is equally distant from the center of the circle. On an ellipse, the sum of distances from two focus points (foci) to any point on the edge is always the same. On the diagram below, a1 + a2 = b1 + b2. Turn on the Show foci and center checkbox. Play through several simulations, using a variety of starting velocities and positions.

Ellipse

In each case, is the sun located at the center of the ellipse, on a focus of the ellipse, or somewhere else? Kepler's First Law states that planetary orbits are ellipses with the sun at one focus.

Adjust the initial settings so that the foci are very close together, and click Play. What is the shape of the orbit when the foci are close together? Most planets in our solar system have orbits with this shape.

Adjust the initial settings so that the foci are relatively far apart. What is the shape of the orbit in this case? This type of orbit is said to be eccentric. The orbits of comets and some other objects are usually very eccentric.

How does the mass of the planet or sun affect the orbit? To answer this question, click Reset, change the Planet mass to large and click Play.

Did changing the mass of the planet affect its orbit?

Click the + zoom control several times to zoom in on the sun. How does a very massive planet affect the sun? Describe what you see.

How will changing the mass of the sun affect the planet's orbit? Make a prediction, and test your hypothesis using the Gizmo.

Kepler's Second Law

In this activity, you will focus on the speed of the planet at various points in its orbit.

Click Play, and adjust the zoom controls until the entire orbit is in view. Be sure the orbit is clearly elliptical. (If not, adjust the initial velocity until this is the case.)

At what point in its orbit does the planet travel fastest? Where does it travel slowest?

In an elliptical orbit, the planet moves fastest when it is nearest the source of gravity (the sun), and slowest when it is farthest away. How is this analogous to a ball rolling up and down a hill on Earth?

Imagine a line connecting the planet to the sun. Over time, this line will sweep out a wedge-shaped area in space. Kepler discovered that the area swept out in a given time obeys an interesting rule.

While the simulation is playing, click the Sweep area button at bottom right. How much area was swept out in a 100–day time interval?

Without stopping the simulation, click Sweep area three more times to compare various parts of the orbit. Record the area swept out each time in your notes.

What do you notice about each area swept out by the planet?

Use the slider to adjust the time interval and experiment with different orbits to see if this pattern persists. Write a rule to summarize your observations. This rule is known as Kepler's Second Law.

Kepler's Third Law

Kepler also discovered a relationship between a planet's average distance from the sun, a, and the period of time required to revolve around the sun, T.

Click Reset. Set the Sun mass to small and turn on the Show grid checkbox. Drag the planet to a position (r) of approximately 2i + 2j and set the velocity vector to approximately 9i − 9j. (The current position and velocity are reported below the Show vectors checkbox.)

Click the POINTER tray near the bottom-right of the Gizmo. Drag a pointer to mark the initial position of the planet. Click Play, and then Pause (

pause button

) when the planet returns to its initial position. How long did it take the planet to go around the sun? This is the approximate period (T) of the planet. The time unit is Earth days.

About how far from the sun was the planet, on average? The mean distance from the planet to the sun is called the semi–major axis of the orbit (a). The unit for distance is the astronomical unit (AU), equal to the Earth-Sun distance.

Select the TABLE tab and click Record data. What were the exact values of a and T? (Note: T is listed in Earth–days and Earth–years.)

Use the Record data button to find a and T for a variety of different starting positions and velocities. (Do NOT change the mass of the sun during these experiments.) There should be some examples in which the planet is close to the sun and some in which it is farther away. If you press Play and the planet flies off without returning, click Reset and try a smaller initial velocity.

Select the GRAPH tab. There are 9 graphs you may examine by choosing different options from the dropdown menus. On which graph do your data points fall on a perfectly straight line?

The graph of T2 versus a3 is linear. This means that the relationship a3 = kT2 holds for some value of k. This is Kepler's Third Law.

Select the TABLE tab. Choose one row of the table, and find the values of a (AU) and T (years) in that experiment. To determine the value of k, substitute these values for a and T into the equation above and solve for k.

Drag the planet to a position of approximately 2j and drag the velocity vector so that it has the value of about 15i. This will result in a nearly circular orbit with a close to 2 AU. From Kepler's Third Law and the value of k you just determined, predict the approximate period of this planet.

Use the Gizmo to check your answer. Were you close?

Use the Gizmo to determine if either the planet's mass or the sun's mass affects the period (T). Challenge: What is the value of k when the Sun mass is medium (2.0 • 1030 kg), close to the mass of our own Sun?

## ORBITAL MOTION E-LEARNING

Orbital Motion E-Learning

Although he trained to be a Lutheran priest, from an early age Johannes Kepler was fascinated by astronomy. Along with his contemporary Galileo, he was one of the first to embrace the theory that the Earth and other planets orbited the Sun.DIRECTIONS FOR USE OF THE GIZMO:In 1600, Kepler moved to Prague to work for the great astronomer Tycho Brahe. When Brahe died a year later, Kepler began to analyze his vast collection of planetary observations, starting with the orbit of Mars. After nine long years of crunching numbers, plotting positions, and discarding flawed theories, Kepler finally discovered the underlying geometry at work. His description of planetary orbits can be summarized by three elegant laws.

## Kepler's First Law

In this activity, you will observe the shape of planetary orbits.SIMULATIONpane of the Gizmo™ shows a sun, represented by a yellow sphere, and a planet, represented by a purple sphere. Check that the Show vectors and Show trails checkboxes are turned on. The purple arrow represents the direction and magnitude of the planet's velocity (v), and the green arrow represents the pull of the sun's gravity (g) on the planet. Click Play (a1 +a2 =b1 +b2. Turn on the Show foci and center checkbox. Play through several simulations, using a variety of starting velocities and positions.## Kepler's Second Law

In this activity, you will focus on the speed of the planet at various points in its orbit.## Kepler's Third Law

Kepler also discovered a relationship between a planet's average distance from the sun,a, and the period of time required to revolve around the sun,T.r) of approximately 2i+ 2jand set the velocity vector to approximately 9i− 9j. (The current position and velocity are reported below the Show vectors checkbox.)POINTERtray near the bottom-right of the Gizmo. Drag a pointer to mark the initial position of the planet. Click Play, and then Pause (T) of the planet. The time unit is Earth days.a). The unit for distance is the astronomical unit (AU), equal to the Earth-Sun distance.TABLEtab and click Record data. What were the exact values ofaandT? (Note:Tis listed in Earth–days and Earth–years.)aandTfor a variety of different starting positions and velocities. (Do NOT change the mass of the sun during these experiments.) There should be some examples in which the planet is close to the sun and some in which it is farther away. If you press Play and the planet flies off without returning, click Reset and try a smaller initial velocity.GRAPHtab. There are 9 graphs you may examine by choosing different options from the dropdown menus. On which graph do your data points fall on a perfectly straight line?T2 versusa3 is linear. This means that the relationshipa3 =kT2 holds for some value ofk. This is Kepler's Third Law.TABLEtab. Choose one row of the table, and find the values of a (AU) and T (years) in that experiment. To determine the value ofk, substitute these values foraandTinto the equation above and solve fork.jand drag the velocity vector so that it has the value of about 15i. This will result in a nearly circular orbit withaclose to 2 AU. From Kepler's Third Law and the value ofkyou just determined, predict the approximate period of this planet.T). Challenge: What is the value ofkwhen the Sun mass is medium (2.0 • 1030 kg), close to the mass of our own Sun?