Kepler’s Laws
Introduction
In this experiment, you will gain an understanding of Kepler’s three laws of planetary motion by plotting and analyzing the orbit of a lunar satellite, as suggested by D. B. Hoff, et al. Activities in Astronomy, 2nd edition, Kendall/Hunt Publishing Company, 1984.
Equipment and Materials
1. Centimeter ruler; push-pins (2)
2. Loop of string, 10 inch circumference
3. Pencil, red pencil, yellow felt-tip marker
4. Double thickness of corrugated cardboard
Important Concepts
Johannes Kepler (1571-1630), a German mathematician and teacher, was deeply interested in the design of the universe. In 1600, religious persecution forced him to leave his home, and he journeyed to Prague to work with Tycho Brahe, who had spent years accumulating vast quantities of accurate astronomical data by naked-eye observations with giant sextants and the like.
Kepler’s first task was to use Tycho’s data to find the orbit of Mars. Most people thought Earth was the center of the solar system, which was then thought to be the entire universe. Fortunately, Kepler had studied under one of the few teachers who accepted Copernicus’ hypothesis, published in 1543, that the Sun was the center and that Earth and the other planets revolved around the Sun.
At first, Kepler was guided by the ancient idea that planets, being heavenly bodies, must move in that most perfect of all geometric figures, the circle. Even Copernicus had endorsed circular orbits. But try as he might, Kepler could not get Tycho’s observations of Mars’ positions to quite fit a circular orbit. Predictions of Mars’ position using the circular orbit hypothesis disagreed with the observational data by only a little bit, but by too much to be due to observational error. Kepler, bothered by the discrepancy between theory and observation, worked for years with tedious mathematical calculations to resolve the situation. The observational data forced him to abandon circular orbits, and in 1609 he published his solution, the first two laws of planetary motion.
Kepler’s first law: Each planet travels in an elliptical orbit around the Sun, with the Sun at one focus of the ellipse. (See Figure 1.) The position where a planet (or comet or asteroid) is closest to the Sun is called its perihelion, while the position where it is farthest away is known as its aphelion (pronounced a-FEE-lee-on).
Figure 1: Earth is shown traveling around the Sun in an (exaggerated) elliptical orbit. The Sun is at one focus; nothing is at the other focus.
Figure 2 illustrates the nomenclature associated with an ellipse. An ellipse has two foci located in such positions that you get the same value for the sum of the lengths of two lines drawn to the foci from any point on the circumference of the ellipse. (See Figure 3.) The major axis of an ellipse is the line that passes through both foci and runs from one edge of the ellipse to the other. The semimajor axis is just half of the major axis. The minor axis is the line within the ellipse that bisects the major axis and is perpendicular to it. The semiminor axis is half the minor axis. Figure 3 tells how to draw an ellipse.
Figure 2: The parts of an ellipse. The major axis is the entire horizontal line. The eccentricity is 0.77.
Figure 3: An ellipse is drawn by placing a loop of string over two pins (the foci). Move a pencil within the taut loop of string.
If one focus lies on top of the other, the major and minor axes are equal, and the ellipse is a circle. How far an ellipse departs from being a circle is described by the term eccentricity, defined in Equation 1 as the distance between the foci divided by the length of the major axis. (A circle’s eccentricity is 0; the maximum eccentricity an ellipse could have would approach 1.) Actually, the orbits of the planets in our solar system, except those of Mercury and Pluto, are not far from being circles; but the orbits of comets and some asteroids are often highly elliptical.
Equation 1:
Kepler’s second law: A line drawn between the Sun and a planet sweeps out equal areas in equal intervals of time. (See figure 4.) The second law could not hold unless the planet moves at faster speeds when near perihelion and at slower speeds at aphelion. It is obvious from the spacing of the dots in Figure 4 that the planet does change speeds in that fashion, since the time interval between dots is the same.
Figure 4: A line connecting a planet and the Sun sweeps out equal areas (the alternating clear and shaded zones) in equal times. The orbital speed of the planet is fastest when the planet is closest to the Sun, so that’s when it travels the greatest distance in one time interval.
Kepler’s third law deals with the period of an orbiting body; that is, the time it takes to complete one revolution around the body it orbits. Thus, the period of Earth is one year. Kepler discovered this third law of planetary motion in 1619, ten years after the first two laws. It states that the square of the period (P) of revolution of a planet is directly proportional to the cube of the semimajor axis (a) of its orbit. (Often, the average distance from the planet to the Sun can be used instead of the length of the semimajor axis. An astronomical unit, AU, is the average distance from the Earth to the Sun, or 93 x 106 miles. The AU is a commonly used distance unit when dealing with the solar system.)
Equation 2:
where k is a constant. If P is put in Earth years and a in astronomical units, then k = 1 and Equation 2 becomes
Equation 3:
The third law indicates that the farther the planet from the Sun, the longer it takes to make one circuit around it. For example, Mercury is located at an average distance of 0.387 AU from the Sun, so its period would be found like this: ; years. Neptune, at an average distance of 30.0 AU, is much farther out; its period would be: P2 = (30.0)3 = 2.70 x 104; P = = 164 years. Of course, the equation can also be used in reverse; knowing that Mars has a period of 1.88 Earth years would allow its average distance from the Sun to be calculated as a3 = P2 = (1.88)2 = 3.53; a = =1.52 AU.
Enter Isaac Newton
Kepler’s laws of planetary motion describe the orbits of objects that are moving in accordance with Newton’s laws of motion and the law of universal gravitation. Knowing that gravity is the force applied to any two bodies in orbital motion around each other, namely,
Equation 4:
Where m1and m2 are the two masses, P is the period of revolution, G is the universal gravitational constant, and a is the semimajor axis of the orbit. When Body 1 is a planet and Body 2 is the Sun, m1 + m2is basically just equal to m2 (the mass of the Sun is 1.99 x 1030kg). That is why Kepler did not realize that both masses should be included in his equation. Also, if m1 is dropped and m2 is put in units of solar masses, then m1 + m2 = 1. Also, if P is put in Earth years and a in AU, 4p2/G becomes equal to 1, and Newton’s general equation reduces to P2=a3, the commonly used form of Kepler’s third law.
When dealing with a satellite orbiting a much more massive body other than the Sun, observations of the satellite’s motion enables Equation 4 to be used to calculate the mass of the body being orbited. In this experiment, when finding the mass of the Moon from the Explorer35 data, the mass of Explorer 35 (105 kg) may be ignored relative to the mass of the Moon, m2. You will put 4.57 x 1013 kg h2/km3. Thus Equation 4, rearranged for m2, becomes
Equation 5:
Instructions
Part A. Drawing and Measuring an Ellipse
- Use the two push-pins, the 10-inch circumference loop of string, the corrugated cardboard backing, and a pencil to draw an ellipse on the data sheet. The two black dots on the data sheet are the foci where the pins should be inserted. Refer to Figure 3 for further instructions.
- Draw the major axis in red pencil and the minor axis in plain pencil.
- Use the centimeter ruler to find the length of the lines called for in the Part A.1 data table (measure to the nearest 0.5 mm). Calculate the eccentricity of your ellipse.
- On the circumference of the ellipse, choose three points spread out through one quadrant, and label them A, B, and C. Record in the Part A.2 data table the measured length from each focus to Point A. Repeat for Points B and C. Answer the question.
Part B. The Orbit of Explorer 35 Around the Moon
- Use the positional data given in Table 1 for the Explorer 35 satellite to plot its orbit on the graph. The axes have already been scaled and labeled. Use a small, penciled dot for each point. The intersection of the X and Y axes is the center of the Moon, and the scale is 2 cm = 1 lunar radius, or 1.74 x 103 km. Note that the position is given for each 0.25 hour (15 minute) interval. Do not draw a line connecting the points.
- Draw a line in pencil for the major axis, which passes through the origin and points on the orbit at (X = -2.25, Y = +4.80) and (X = +0.66, Y = -1.40). Measure the major axis to the nearest 0.05 cm, and calculate the length of the semimajor axis.
- Using that calculated length, place an X in pencil at the spot on the major axis where the minor axis will intersect. Measure to make sure each semimajor axis is the same length, and record in the data table. Draw the minor axis in pencil, measure it, and record in the data table. Measure and record the length of the semiminor axis.
- Measure the distance between the Moon and the intersection of the major and minor axes. Use that length to locate the second focus. Show it by a circled pencil dot, ʘ. Measure and record the distance between the foci. Answer Questions B.1, 2, and 3.
- To test Kepler’s second law, you will find the area in mm2 of two triangular segments swept out in equal time intervals of 0.25 h.
Draw a red line from the origin to the dot at (X = -0.85, Y = -1.58). Draw a similar line from the origin to the dot at (X = -0.03, Y = -1.59). Complete the triangle by drawing a red line from the first dot to the second. Fill in the triangle with yellow.
- Draw similar lines to the points (X – 0.00, Y = +4.74) and (X = -0.27, Y = +4.86). To make this a right triangle and simplify calculations, draw a short red line from X = +4.80 to X = -0.27 (4.80 is halfway between 4.74 and 4.86, so the number of squares gained equals the number of squares lost). Fill in this triangle with yellow.
- The area of a right triangle is found by taking one-half of the product of the base times the height. A = ½ bh. Use the X and Y readings to calculate the area of each triangle in mm2. Answer Questions B.4 and 5. From data in the Part B table, answer Question B.6.
- Use Equation 5 and your experimental values for P (in h) and a (in km) to calculate the mass of the Moon in kg. Find your percentage error.
Prelab Sheet for “Kepler’s Laws”
SectionName
1. Kepler’s three laws deal with the motion of
2. Earth and the other planets travel around the Sun in what shape orbits?
3. The point in the orbit where Earth is farthest from the Sun is called its
4. How many foci does an ellipse have?
5. Consider the ellipse at right.
What does line a represent?
Twice line a is the length of the
Calculate the eccentricity. (You will need to
Make several length measurements.)
6. Referring to Figure 4 on previous page, what is the relationship between the
area of triangle ANSun and that of triangle GFSun?
7. A planet moves fastest when it is at what point in its orbit? (one word)
8. How long is Earth’s period?
9. AU stands for
10. What is the average distance in miles from Earth to the Sun?
11. The average distance of Uranus from the Sun is 19.2 AU. How many Earth
years does it take for Uranus to make one revolution around the Sun? Show setup and answer.
Setup: Answer ______
12. The small moon names Phobos takes 7.7 hours to orbit Mars once. The semimajor axis of Phobo’s orbit is 9.4 x 103 km in length. Find the mass of Mars. (Use equation 5.)
Setup:Answer ______
13. In Part A, the line lengths are measured to the nearest
14. Find the area in mm2of the right triangle shown.
(the side of each square is 1 mm long.)
Data Sheet for “Kepler’s Laws”
Section # Partner’s Last Name Your Name
A.Drawing and Measuring an Ellipse
1. Setup for the eccentricity (from data below):Answer: ______
DistanceBetween foci / Major axis / Semimajor
axis / Minor axis / Semiminor
axis
Length (cm)
2. Explain how the data in the table below prove that your drawing is an ellipse.
Distance from Focus 1(cm) / Distance from Focus 2
(cm) / Total of the two
Distances (cm)
Point A
Point B
Point C