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Exploring Kepler's Laws of Planetary Motion

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Exploring Kepler's Laws of Planetary Motion Exploring Kepler’s Laws of Planetary Motion: As Modified by Isaac Newton Version: 1.0 Author: Sean S. Lindsay History: Written February 2020 Goal: To become familiar with Kepler’s Laws of Planetary Motion in the light of Newton’s Laws of Motion and Newtonian Gravity. Specifically, this lab addresses: 1. What are Kepler’s Three Laws of Planetary Motion? 2. What are orbital velocity and escape velocity? 3. How orbital period relates to the semi-major axis of an ellipse-shaped orbit 4. How planetary speed connects Kelper’s First and Second Laws through an exploration of eccentricity, perihelion, and aphelion 5. How Newtonian physics modifies Kepler’s Laws of Planetary Motion Tools used in this lab: • University of Colorado, Boulder’s PhET simulation: “My Solar System” https://phet.colorado.edu/en/simulation/my-solar-system • Microsoft Excel 1. Kepler’s Three Laws of Planetary Motion In 1609, the German mathematician Johannes Kepler (27 December 1571 – 15 November 1630) published two of his three laws of planetary motion in his work Astronomia nova (A New Astronomy). Using the Tycho Brahe’s incredibly comprehensive, accurate, and precise data set of planetary positions, Kepler was able to move past the limitation of circular orbits embedded in the Copernican model of the solar system. From precise measurements of Mars’ position, Kepler deduced that planetary orbits are not circular in shape; they are instead elliptical in shape with the Sun located at one of the two foci of the elliptical orbit. Mars was ideal for this task because other than Mercury, it has the highest eccentricity of any of the naked-eye visible planets. The non-circular, elliptical orbits explained the motions of Mars and allowed him to extend his ideas to how planets are moving along their orbits. Formulating their motion in terms of geometry, he discovered that planets sweep out equal areas in equal times. Here, the area swept out is defined at a segment of the ellipse carved out by imagining the area a line between the Sun and planet sweeping out an area as it moves along its orbit. In equal time intervals, that swept out area will always be equivalent meaning that the planet is orbiting the Sun at different speeds in different parts of its orbit: faster when closer to the Sun, and slower when farther away. In 1619, Kepler published his third law of planetary motion in Harmonices Mundi (The Harmony of the World). Using orbital period and semi-major axis distance (the average distance away from the Sun a planet is) data, he discovered what he called a “music of the spheres.” He viewed the relationship between the periods and semi-major axis distances as a harmony of the worlds, and therefore third law is often referred to as the harmonic law. In Harmonices Mundi, he penned (translated from Latin), “I first believed I was dreaming… But it is absolutely certain and exact that the ratio which exists between the period times of any two planets is precisely the ratio of the 3/2th power of the mean distance.” Putting this in the natural Earth-based solar system units with orbital period, P, measured in Earth-years and semi-major axis distance, a, measured in Earth-Sun distances, or rather astronomical units, AU, this becomes P2 = a3; Kepler’s 3rd Law of planetary motion. 1 1.1 – Kepler’s First Law of Planetary Motion The orbit of a planet is an ellipse with the Sun at one of the two foci. Figure 1 shows the geometry of an ellipse and a planet on an elliptical orbit according to Kepler’s First Law. While mathematically, an ellipse is defined as a conic section, it is easiest to think of an ellipse a “squashed” circle. A circle can be defined by its center and radius. The radius is distance from the center of the circle to any point on the edge of the circle. As a circle is deformed into an ellipse, it will have a long cut through the center and a short cut through the center. The longest cut through the ellipse is called the major axis, and the short cut is called the minor axis. Half the major axis, from the center of the ellipse to the edge of the ellipse along the major axis, defines the semi-major axis, a. Half the minor axis, from the center of the ellipse to the edge of the ellipse along the minor axis, defines the semi-minor axis, b. Notice that the major axis goes through both foci, while the minor axis goes through neither. As the ellipse becomes less circular, the foci (singular: focus) of the ellipse move farther from the center of the ellipse and the ratio of the major axis length to minor axis length increases. This deviation from circular is quantified by the ellipse’s eccentricity, e, such that (�� − ��) � = $ (1) �� Notice that for ellipses, 0 ≤ � < 1, where e = 0 makes it so a = b, and you have circle; and e = 1 means b = 0, and the shape is no longer an ellipse. As e increases toward 1, the foci move farther from the center of the ellipse and the major axis become increasingly longer than the minor axis making the ellipse appear more and more “squashed.” Figure 1a: The geometry of an ellipse. The major, semi- Figure 1b: An elliptical orbit according to Kepler’s first Law major, minor, and semi-minor axes are labeled. The two foci of Planetary Motion. The green circle represents a planet of an ellipse are equidistant from the center of the ellipse and moving on an elliptical orbit. The Sun is placed at one of the fall along the major axis. These points determine the shape two foci, while the other one remains empty. The point the of the ellipse including the ratio of the major to minor axis planet is closest to the Sun is called perihelion. The point the lengths and the eccentricity. planet is farthest from the Sun is called aphelion. Figure 1b shows what an elliptical orbit for a planet looks like according to Kepler’s First Law of Planetary Motion. The Sun is at one of the two foci of the ellipse; the other focus is at point of empty space. A little bit of geometry and calculation can show that the average distance between the edge of the ellipse and one of the foci is equal to the semi-major axis length, a. This means that the semi-major axis length is the mean (average) distance a planet is away from the Sun. The Sun being placed at one of the two foci also creates two special points in the planet’s elliptical orbit around the Sun: 2 • Perihelion: The point where the planet is closest to the Sun. The distance away from the Sun is defined as ��������� ��������, � = �(� − �), (2) where a is the semi-major axis distance and e is the eccentricity of the ellipse. • Aphelion (Pronounced “Ap – Helion”): The point where the planet is farthest from the Sun. The distance away from the Sun is defined as �������� ��������, � = �(� + �), (3) where a is the semi-major axis distance and e is the eccentricity of the ellipse. Notice that as the eccentricity increase, the perihelion distance gets smaller and the aphelion distance gets larger. 1.2 – Kepler’s Second Law of Planetary Motion A line segment connecting the planet and the Sun sweeps out equal areas of its elliptical orbit in equal intervals of time. The geometry phrasing of Kepler’s Second Law of Planetary Motion makes it rather difficult to internalize what is communicating about how planets move on their orbits. Figure 2 shows two equal areas of an elliptical orbit that a planet as swept out and the length of the arcs around the elliptical orbit the planet is taking. Area 1 is equal to Area 2, and therefore the time for the planet to travel the distance of Arc 1 and Arc 2 must be the same. Arc 1 is a longer distance, so the planet must be orbiting faster than it is as it moves along Arc 2. This insight gives an alternative, and more intuitive way to state Kepler’s Second Law. Alternative to Kepler’s Second Law: A planet moves faster near perihelion and slower near aphelion. It orbital speed is fastest at perihelion and slowest at aphelion. Figure 2. Kepler’s 2nd Law states that a line segment connecting the Sun to a planet will sweep out equal areas in equal amounts of time. Area 1 and Area 2 are equal, therefore, the time for the planet to travel Arc 1 is the same as Arc 2. Arc 1 is longer, so the planet must be moving faster near perihelion. Arc 2 is shorter, so the planet must be moving slower near aphelion. Exploring the alternative to Kepler’s Second Law further, we can start comparing the orbital speed at aphelion and perihelion compared to what the orbital speed would be if it were a circular orbit. Again, some calculation can show that the speed a planet is moving at perihelion is faster than what it would be for the orbital velocity of a circular orbit with radius equal to the semi-major axis length. The planet will be moving slower than the circular orbital velocity at perihelion. 3 1.3 – Kepler’s Third Law of Planetary Motion The square of the orbital period is directly proportional to the cube of the semi-major axis of its orbit. If the orbital period is measured in years and the semi-major axis is measured in astronomical units (AU), then this becomes �� = �� (4) Here the orbital MUST be in years and the semi-major axis MUST be in AU.
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