# 1 Kepler s Laws of Planetary Motion

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1 1 Kepler s Laws of Planetary Motion 1.1 Introduction Johannes Kepler published three laws of planetary motion, the first two in 1609 and the third in The laws were made possible by planetary data of unprecedented accuracy collected by Tycho Brahe. The laws were both a radical departure from the astronomical prejudices of the time and profound tools for predicting planetary motion with great accuracy. Kepler, however, was not able to describe in a significant way why the laws worked. 1.2 First Law: Law of Ellipses The orbit of a planet is an ellipse where one focus of the ellipse is the sun. An ellipse is defined by two focii and all points for which the sum of the distances are the same. The semimajor axis (a) is the long distance from the center to edge of the ellipse. If r 1 and r 2 are the distances from the focii to any point on the ellipse, then r1 + r2 = 2a. The short axis is called the semiminor axis. How elliptical an orbit is can be described by the eccentricity (e). The eccentricity is equal to the distance between a focus and the center (c) of the ellipses divided by the semimajor axis (a). That is, e = c/a. The NAAP Eccentricity Demonstrator ( shows ellipses for different values of a and c. Note c is always less than a by definition. (a) The NAAP Eccentricity Demonstration can be accessed at http: //astro.unl.edu/naap/pos/animations/ellipsedemo.swf. (b) Equal areas over equal time. Figure 1: Kepler s First and Second Laws 1.3 Second Law: Law of Equal Areas A line from the planet to the sun sweeps out equal areas in equal amounts of time. With elliptical orbits a planet is sometimes closer to the sun than it is at other times. The point at which it is closest is called perihelion. The point at which a planet is farthest is called aphelion. Kepler s second law basically says that the planets speed is not constant moving slowest at aphelion and fastest at perihelion. The law allows an astronomer to calculate the orbital speed of a planet at any point. Figure 1(b) shows a diagram of this law. 1.4 Third Law: Law of Harmonies The period of a planet s orbit squared is proportional to its average distance from the sun cubed. The average distance of a planet from the sun is equal to its semimajor axis (a). If the period (P ) is measured in years and the semimajor axis (a) is given in astronomical units (the earth sun distance is 1 AU) then Kepler s Third Law can be written: P 2 = a 3 1 / 7

2 However, this equation is only good for our solar system. Isaac Newton was able to derive a more general form of the equation using his Law of Gravitation. 2 Newton and Planetary Motion 2.1 Introduction In 1687 Isaac Newton published Philosophiae Naturalis Principia Mathematica, a work of immense and profound impact. Newton s pronounced three laws of motion and a law of universal gravitation. They were a united set of principles which applied not only to the heavens but also to the earth in a uniform way. Their simplicity and extremely broad applicability forever changed astronomy. 2.2 First Law of Motion: Law of Inertia A body remains at rest, or moves in a straight line (at a constant velocity), unless acted upon by a net outside force. The law of inertia did not originate with Newton, nevertheless it is integral to his system of mechanics. An object in motion will remain in motion unless something acts upon it. Because a planet is moving in an ellipse (i.e. not a straight line) this law states that there must be some force acting upon the planet. If there were no force, the planet would fly off in a straight line. Figure 2: When the string is released, the ball will fly straight away, not along the curve. 2.3 Second Law of Motion: F = ma The acceleration of an object is proportional to the force acting upon it. The first law says that if no force is acting on an object, it will remain in motion. The second law tells how the motion will change when a force acts upon the object. Velocity is how fast an object is moving (speed or magnitude) and the direction it is moving. Acceleration is a change in velocity. An accelerating object can either change how fast it is moving, the direction it is moving, or both. 2.4 Third Law: Law of Reciprocal Actions For every action, there is an equal and opposite reaction. The law can be more fully stated as, Whenever one body exerts force upon a second body, the second body exerts an equal and opposite force upon the first body. That is, when the sun pulls on a planet with the force of gravity, the planet pulls on the sun with a force of equal magnitude. But, because the sun is so much more massive than the planet, Newton s second law says that the sun will experience much less acceleration. 2 / 7

3 2.5 Law of Universal Gravitation F = Gm 1m 2 r 2 Every object in the Universe attracts every other object with a force directed along the line of centers for the two objects that is proportional to the product of their masses and inversely proportional to the square of the separation between the two objects. While the law does not explain what gravity is, it does say how the force of gravity works. From this law and his laws of motion, Newton was able to derive all of Kepler s Laws of Planetary Motion. 3 Pre-Lab Questions 1. Match the following definitions with the proper law. C Kepler s First Law A. only a force acting on an object can change its motion B Kepler s Second Law B. planets move faster when close to the sun D Kepler s Third Law C. planets orbit the sun in elliptical paths A Newton s First Law D. planets with large orbits take a long time to complete an orbit 2. When written as P 2 = a 3, Kepler s Third Law (with P in years and a in AU) is applicable to... A. any object orbiting our sun. B. any object orbiting any star. C. any object orbiting any other object. 3. For a planet in an elliptical orbit to sweep out equal areas in equal amounts of time it must... A. move slowest when near the sun. B. move fastest when near the sun. C. move at the same speed at all times. D. have a perfectly circular orbit. 4. If a planet is twice as far from the sun at aphelion than at perihelion, then the strength of the gravitational force at aphelion will be as it is at perihelion. A. four times as much B. twice as much C. the same D. one half as much E. one quarter as much 4 Procedure Everything in this lab will require you to open the NAAP Planetary Orbit Simulator from this link: 3 / 7

4 4.1 Kepler s First Law Click on the Kepler s First Law tab if it is not already (it s open by default), and enable all 5 check boxes in the bottom center of the display. The white dot is the simulated planet. One can click on it and drag it around. Change the size of the orbit with the semimajor axis slider. Note how the background grid indicates change in scale while the displayed orbit size remains the same. (You can change the value of a slider by clicking on the slider bar or by entering a number in the value box.) Change the eccentricity and note how it affects the shape of the orbit. Be aware that the ranges of several parameters are limited by practical issues that occur when creating a simulator rather than any true physical limitations. The semi-major axis is limited to 50 AU since that covers most of the objects we are interested in our solar system and the eccentricity is limited to 0.7 since the ellipses would be hard to fit on the screen for larger values. Note that the semi-major axis is aligned horizontally for all elliptical orbits created in this simulator, where they are randomly aligned in our solar system. Animate the simulated planet. You may need to increase the animation rate for very large orbits or decrease it for small ones. The planetary presets set the simulated planets parameters to those like our solar system s planets. Explore these options. 1. For what eccentricity is the secondary focus (which is usually empty) located at the sun? What is the shape of this orbit? Solution: When e = 0. This is a circular orbit. 2. Create an orbit with a = 20 AU and e = 0. Drag the planet first to the far left of the ellipse and then to the far right. What are the values of r 1 and r 2 at these locations? r 1 (AU) r 2 (AU) Far Left Far Right Create an orbit with a = 20 AU and e = 0.5. Drag the planet first to the far left of the ellipse and then to the far right. What are the values of r 1 and r 2 at these locations? r 1 (AU) r 2 (AU) Far Left Far Right For the ellipse with a = 20 AU and e = 0.5, can you find a point in the orbit where r 1 and r 2 are equal? Sketch and label the ellipse, the location of this point, and r 1 and r 2 in the space below. 5. What is the value of the sum of r 1 and r 2 and how does it relate to the ellipse properties? Is this true for all ellipses? Solution: r 1 + r 2 = 2a = 40 AU. This is true for all ellipses regardless of eccentricity or semi-major axis. 4.2 Kepler s Second Law Use the clear optional features button to remove the First Law features. Open the Kepler s Second Law tab. 4 / 7

5 Press the start sweeping button. Adjust the semimajor axis and animation rate so that the planet moves at a reasonable speed. Adjust the size of the sweep using the adjust size slider. Click and drag the sweep segment around. Note how the shape of the sweep segment changes, but the area does not. Add more sweeps. Erase all sweeps with the erase sweeps button. The sweep continuously check box will cause sweeps to be created continuously when sweeping. Test this option. 1. Erase all sweeps and create an ellipse with a = 1 AU and e = 0. Set the fractional sweep size to one-twelfth of the period. Drag the sweep segment around. Does its size or shape change? Solution: It s a circle, so neither the shape nor the size change as the sweep segment is dragged around the orbit. 2. Leave the semi-major axis at a = 1 AU and change the eccentricity to e = 0.5. Drag the sweep segment around and note that its size and shape change. Where is the sweep segment the skinniest? Where is it the fattest? Where is the planet when it is sweeping out each of these segments? (What names do astronomers use for these positions?) Solution: Skinniest is when the planet is farthest from the star: aphelion. Fattest is when the planet is closest to the star: perihelion. 3. Halley s comet has a semimajor axis of about 18.5 AU, a period of 76 years, and an eccentricity of about 0.97 (so Halleys orbit cannot be shown in this simulator). The orbit of Halleys Comet, the Earths Orbit, and the Sun are shown in Figure 3 (not exactly to scale). Based upon what you know about Kepler s Second Law, explain why we can only see the comet for about 6 months every orbit (76 years)? Solution: For the six months when it s swinging through perihelion, it s moving at its fastest in the inner portion of the solar system. We can see it just fine. But when it moves out of the inner solar system, it gets harder to see, but it also slows down. So it spends the largest amount of its orbital period far away from us, moving very slowly, where we can t see it too well. Figure 3: The orbit of Halley s Comet, the Earth s orbit, and the sun. 4.3 Kepler s Third Law Use the clear optional features button to remove the Second Law features. Open the Kepler s Third Law tab. 1. Use the simulator to complete Table As the size of a planet s orbit increases, what happens to its period? Solution: Its period increases: a = 3 P Start with the Earth s orbit and change the eccentricity to 0.6. Does changing the eccentricity change the period of the planet? 5 / 7

6 Object P (years) a (AU) e P 2 a 3 Earth Mars Ceres Charon Table 1: Orbital period and distance for selected objects. Solution: No. 4.4 Newtonian Features Important: Use the clear optional features button to remove other features. Open the Newtonian features tab. Click both show vector boxes to show both the velocity and the acceleration of the planet. Observe the direction and length of the arrows. The length is proportional to the values of the vector in the plot. 1. The acceleration vector is always pointing towards what object in the simulator? Solution: The star. 2. Create an ellipse with a = 5 AU and e = 0.5. For each marked location on the plot in Figure 4 indicate a) whether the velocity is increasing or decreasing at the point in the orbit (by circling the appropriate arrow) and b) the angle θ between the velocity and acceleration vectors (you may have to eyeball the value). Note that one is completed for you. Figure 4: The orbit of Halley s Comet, the Earth s orbit, and the sun. 6 / 7

7 3. Where do the maximum and minimum values of velocity occur in the orbit? Solution: Maximum velocity is at perihelion. Minimum velocity is at aphelion. 4. Can you describe a general rule which identifies where in the orbit velocity is increasing and where it is decreasing? What is the angle between the velocity and acceleration vectors at these times? Solution: Orbital velocity increases when acceleration increases: from aphelion to perihelion, and the angle between the velocity and acceleration vectors is 45 < θ < 90. Orbital velocity decreases when acceleration decreases: from perihelion to aphelion, and the angle between the velocity and acceleration vectors is 90 < θ < 135. (The max value of θ depends on how eccentric the orbit is, but the rules above hold for all ellipses.) 7 / 7

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