Kepler’s Laws of Planetary Motion

Purpose: The purpose of this lab is to further investigate Kepler’s Laws through laboratory experimentation. You will be learning about how planets revolve around the sun, and how each planet’s revolution is different, and the distance from the sun influences the rate and shape of the revolution.

Introduction: For thousands of years people believed that the Earth was the center of the universe, and everything else in the solar system revolved around this. However in the 1500s, Nicholaus Copernicus challenged this geocentric (earth-centered) model of the solar system and instead proposed a heliocentric model, in which Earth and the other planets orbit the sun. Other scientists, including Galileo Galilei confirmed these observations using his telescope in the early 1600’s. Another astronomer, Johannes Kepler, also began to examine the positions of the stars and planets, and proposed the motion of these celestial bodies using mathematical terms. Together, these are known as Kepler’s Laws.

Kepler’s First Law

Kepler’s First Law, the “Law of Ellipses” states that all objects that orbit the Sun, including planets, asteroids and comets, follow elliptical paths. An ellipse is an oval-shaped geometric figure whose shape is determined by two points within the figure. Each point is called a focus (plural: foci). In the solar system, the Sun is at one focus of the orbit of each planet; the second focus is empty. In this portion of the experiment you will be studying the eccentricity of the various planets. The eccentricity measures how far away from a perfect circle a planetary body is. The range for eccentricity is from 0-1; if the orbit is a perfect circle the eccentricity will be 0, if the eccentricity is 1.0 there is no orbit. To calculate the eccentricity (e), it is the distance between the foci (F) divided by the length of the major axis (A). e = F/A.

Materials:

Ruler Cotton String (22cm) Push Pins (6) Pencil CalculatorTape Piece of Paper Cardboard Scissors

Procedure:

1. Pick up a piece of string and a ruler. Measure 22 cm of the string and cut using scissors. Return the excess string back up to the teacher’s desk.

2. Form a loop with the 22cm of string, and hold in place with tape.3. Place a piece of plain white paper on top of the cardboard paper and fasten using 4 of the 6 push pins. One for

each corner.4. On the piece of plain white paper, use the ruler to determine the exact center of your paper (remember: it is 8 ½

x 11 in paper). Mark this point using your pencil.5. Place the other 2 push pins exactly 2 cm apart from one another at the center of the paper (each push pin

should be 1 cm away from the center of the paper). These are your foci. Label these points F1 and F2

6. Place the looped string over the two foci.7. With one hand the push pins and the other on your pencil. Stretch the string as far as it will go using the pencil,

and begin to move your pencil around the two push pins. BE SURE TO KEEP THE STRING TIGHT. You have drawn an ellipse (it will not be a perfect circle). Label the ellipse “Ellipse 1.”

8. Now that your ellipse for 2 cm is drawn, repeat steps 5-7 for when the foci are: 5 cm (label F3 and F4), 7 cm (label F5 and F6) and 9 cm (label F7 and F8).

Calculations:

1) For each of the four ellipses, you’re now going to measure the eccentricity. What is the general formula for eccentricity? What does each of the variables represent?

2) To measure the length of the foci, use the ruler to measure the distance between the push pins (you should have labeled these on your piece of plain white paper). To measure the axis, measure the diameter of the ellipse (from one end of the ellipse to the other):

Ellipse Distance Between Foci (cm)

Length of Major Axis (cm)

1234

3) Using the formula you have written in Calculations #1 and the data you have collected in Calculations #2, calculate the eccentricity of each of the 4 ellipses below. SHOW ALL WORK. Round answers to the nearest thousandth.

Ellipse 1 Ellipse 2 Ellipse 3 Ellipse 4

4) Now that you have calculated the eccentricities of the ellipses above, answer the following questions:a) Describe the relationship between the foci and the eccentricity of the ellipse.

b) What is the maximum value that an eccentricity can be? What shape would this be?

c) What is the minimum value an eccentricity can be? What shape would this be?

d) If there was only one push pin used in this experiment, what shape would the shape of the ellipse be? What eccentricity is that?

e) Using the data below, calculate the eccentricity of Earth’s orbit to the nearest thousandth. Show all of your work below:

Length of Major Axis 299,000,000 kmDistance Between Foci 5,083,000 km

f) How does the eccentricity of Earth’s orbit compare with the eccentricity of the ellipses you drew?

KEPLER’S SECOND LAWKepler’s Second Law, the “Law of Equal Areas” states that a line drawn from the Sun to a planet sweeps equal areas in equal time, as illustrated on the diagram on the next page. A planet’s orbital velocity (the speed at which it travels around the Sun) changes as its position in its orbit changes. Its velocity is fastest when it is closest to the Sun and slowest when it is farthest from the sun.

5) If Area X = Area Y on the diagram above, what can be inferred about the orbital velocities as the planet travels along its orbit through Area X compared to Area Y? (Which is faster?)__________________________________________________________________________________________________________________________________________

6) A planet’s orbital velocity is fastest at the position it its orbit called __________________ (perihelion/aphelion). Look back to your notes for the date when Earth is at this position.

7) During what season (in the Northern Hemisphere) is Earth at this position? ___________________________Therefore, Earth moves ________________________ (faster/slower) in summer than in winter, so summer in the Northern Hemisphere must be ___________________________ (longer/shorter) than winter.

8) Isaac Newton later determined that the force of GRAVITY holds the planets in orbit around the Sun. When a planet is closer to the Sun, the force of the Sun’s gravitational attraction on the planet is _________________________ (stronger/weaker) than when the planet is farther from the Sun.

KEPLER’S THIRD LAWKepler’s Third Law, the “Law of Periods” relates a planet’s period of revolution (the time it takes to complete one orbit of the Sun) to its average distance from the Sun. Kepler determined the mathematical relationship between period and distance and concluded that the square of a planet’s period is proportional to the cube of its mean distance from the Sun. The formula used to determine this relationship for any planet is: T2 = R3, where T is the planet’s period in Earth years and R is the planet’s mean distance from the Sun in astronomical units (AU, where 1 AU equals the mean distance from the Earth to the Sun = 150 million km).

Sample Problem: Planet X has an average distance from the Sun of 1.76 AU. What is the planet’s period of revolution, in Earth years?T2 = R3 T2 = (1.76)3 = 5.45 T =√ 5.45 = 2.33 Earth years

9) Calculate the period of revolution of each of the following planets:Planet Mean Distance to Sun (AU) Period of Revolution (Earth

Years)Mercury 0.387Venus 0.712Earth 1.000

Mars 1.524Jupiter 5.216Saturn 9.539Uranus 19.232

10) Haley’s comet has an average distance of 17.91 AU from the Sun. Calculate the period of Haley’s comet. SHOW YOUR WORK BELOW!

11) Draw a graph that shows the relationship between a planet’s period of revolution in Earth years (some planets will need to be converted from days to years) and its average distance from the Sun (in AU). Look up the data on the Planet Data Table in your notes. Plot period on the x-axis and distance on the y-axis. Label each planet on the graph. Be sure to label the axes and include a title.

12) Describe the graph. What is the relationship between period and distance from the Sun?