sign in sign up
<<
Home , Harmonices Mundi, Johannes Kepler

Exploring ’s Laws of Planetary : As Modified by Version: 1.0 Author: Sean S. Lindsay History: Written February 2020 Goal: To become familiar with Kepler’s Laws of Planetary Motion in the of Newton’s Laws of Motion and Newtonian . Specifically, this lab addresses: 1. What are Kepler’s Three Laws of Planetary Motion? 2. What are orbital and escape velocity? 3. How relates to the semi-major axis of an -shaped 4. How planetary connects Kelper’s First and Second Laws through an exploration of eccentricity, perihelion, and aphelion 5. How Newtonian modifies Kepler’s Laws of Planetary Motion

Tools used in this lab: • University of Colorado, Boulder’s PhET simulation: “My ” https://phet.colorado.edu/en/simulation/my-solar-system • Microsoft Excel 1. Kepler’s Three Laws of Planetary Motion In 1609, the German (27 December 1571 – 15 November 1630) published two of his three laws of planetary motion in his (A New ). Using the ’s incredibly comprehensive, accurate, and precise data set of planetary positions, Kepler was able to move past the limitation of circular embedded in the Copernican model of the solar system. From precise measurements of ’ position, Kepler deduced that planetary orbits are not circular in shape; they are instead elliptical in shape with the located at one of the two foci of the elliptical orbit. Mars was ideal for this task because other than , it has the highest eccentricity of any of the naked-eye visible . The non-circular, elliptical orbits explained the of Mars and allowed him to extend his ideas to how planets are moving along their orbits. Formulating their motion in terms of , he discovered that planets sweep out equal areas in equal . 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 sweeping out an area as it moves along its orbit. In equal intervals, that swept out area will always be equivalent meaning that the planet is orbiting the Sun at different 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 (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 “ 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 of the mean distance.” Putting this in the natural -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 , 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: ) 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 , 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 . 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 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 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. This law is sometimes called the period-radius law or the harmonic law. It connects the orbital period, and therefore how fast the planet is traveling on average to the average distance it is away from the Sun. While this equation may at first glance look relatively unremarkable, its determination by Johannes Kepler fundamentally changed humanity’s view of its place in the solar system and . For the first time, the distances to all of the known planets could be determined using their observed orbital periods. The distances to the planets were much greater than expected meaning that we live in a solar system and universe much larger than previously conceived. 1.4 – The Limitation of Kepler’s Three Laws of Planetary Motion While Kepler’s Three Laws of Planetary Motion fundamentally changed our understanding of our solar system, they were not without their flaws. Kepler provided a mathematical solution to the data set amassed by Tycho Brahe. This is what scientists call an empirical solution. It gives that can be highly predictive, but it does not give an explanation for how the natural phenomenon is operating. This means that Kepler’s Three Laws only applied to the conditions of the data, which were objects in orbit around the Sun. As determined, Kepler’s 3rd Law does not work for objects not in orbit around this Sun, including the: Earth- system; -Galilean system; planetary orbits in exoplanetary systems; orbits of binary ; rotations of spiral galaxies; etc. To extend the Kepler’s Laws to any system, a generalized and universal understanding of how objects move and the development of a theory of gravity (the making the celestial bodies move) need to be invented. For this, we need Isaac Newton. 2 – Understanding Kepler’s Laws Through Newtonian Isaac Newton (25 December 1642 – 20 March 1726) was a British mathematician, , and . He was fascinated by the works of Johannes Kepler and Italian astronomer and physicist, . Specifically, Newton wanted to understand the findings that Galileo made about how objects fall and what makes the planets orbit and follow Kepler’s Three Laws of Planetary Motion. To accomplish this, Newton developed a theory of universal motion, humanity’s first conception of gravity, and a new branch of mathematics, , which was required for his laws of motion and understanding of the force of gravity. These generalized physical explanations of how things move in the universe, and that all generates an attractive force called gravity allowed him to finally understand Kepler’s Laws of Planetary Motion, the of falling objects observed by Galileo, and apply them to the entire universe. Newton’s new physics finally unlocks what is making the planets orbit the Sun, and what makes all objects in the universe move in response to gravity. It was the dawn of a new understanding on the of how our universe works.

4 2.1 – Newton’s Three Laws of Motion & Newton’s Universal Law of Gravity Before we get to what Newton’s three laws of motion are, we need to define a few terms required to understand Newton’s ideas. Since, we are talking about how things move, we will start with the difference between speed and velocity. Figure 3 shows the relationship between velocity and speed. Velocity is a speed plus information on the direction of travel. Speed is just the magnitude (how fast) the object is traveling. For example, a car driving at 60 mph is a speed, but a car driving at 60 mph heading due North is a velocity. In , however, we do not use miles per hour (mph) as our unit for speed and velocity; we use meters per second (m/s) as our standard. A force is any influence that changes the motion of an object. Force is measured in Newtons (N), which in base units is (1 = 1 N). So, 1 Newton of force is required to move 1 kilogram of mass at an acceleration Figure 3. The relationship of speed and 2 of 1 m/s . The change in motion in response to a force velocity. Velocity is a speed plus a could be a change in speed and/or a change in direction. direction. In both cases, we say that the object experienced acceleration. An acceleration is simply a change in motion, or more formally, the rate of change of the object’s velocity. That is, the acceleration is how the velocity is changing as a function of time, and that change can be a change in direction and/or a change in speed. An object’s resistance to change its motion, i.e., accelerate, is called . The higher the inertia, the more resistant to a change in motion a mass is. The more massive the object, the higher the inertia, so inertia is really a concept saying that it requires more force to accelerate more massive objects. Another way to put that is that given the same force acting on two unequal , the more massive object will accelerate less because it has higher inertia. With our basic concepts defined, we can now understand Newton’s Three Laws of Motion. Newton’s First Law of Motion: The Inertial Law An object at rest will remain at rest unless acted upon by and outside force. An object in motion with a constant velocity (same speed and direction) will continue to move at the same constant velocity unless acted upon by and outside force. Newton’s First Law of Motion essentially states that an object will not change its motion unless it is forced to. Newton’s Second Law of Motion: The Force Law When a force F acts on a body of mass m, it produces an acceleration a equal to the force divided by the mass, i.e., a = F/m. Equivalently, the force that caused an acceleration a for mass m is equal to the mass times the acceleration, i.e., F = ma. Newton’s Second Law of Motion is the law that explains how an object will change its motion, i.e., accelerate, when an outside force acts on it. It gives a mathematical relationship for what the acceleration is given the amount of force and the amount of mass. Newton’s First Third of Motion: The Reaction Law When one body exerts a force on a second body, the second body exerts a force equal in magnitude, but opposite in direction, on the first body.

5 Newton’s Third Law of Motion explains that come in equal and opposite pairs. Any force acting on a body will also have an accompanying force that has the same strength (magnitude), but opposite in direction. An example would be standing on a scale. The force of gravity between you and the Earth pulls you down on the scale. In response, the springs of the scale push back up on you with an equal, but opposite direction (upward) force. This depresses the springs, which can be translated into your weight. Another example is how rockets fly at launch. When the rocket fuel is ignited, it is directed out of nozzles toward the ground. The force created by this rapidly escaping gas acts downward. By Newton’s Third Law, there is an equal and opposite, upward force acting on the rocket causing it to lift off the ground. Newton’s Universal Law of Gravity Everything that has mass produces an attractive force acting on all other objects that have mass. The strength of this force is directly proportional to the product of the masses and inversely proportional to the square of the distance between the masses. This force is called the gravitational force, or gravity for short.

In equation form, the force of gravity, Fg between two masses, m1 and m2, separated by a distance r, is � � � = � � � (5) � �� , where G is the Universal Gravitational Constant equal to 6.67 * 10-11 m3/(kg s2). Newton wanted to explain the results of experiments conducted by Galileo Galilei, as well as, understand what was causing the motion of the planets. Galileo conducted experiments to determine that when objects fall, they fall with a constant acceleration equal to 9.8 m/s2 that is independent of the mass of the object. By Newton’s Three Laws of Motion, there must be some force that is casing this constant acceleration. What is this force? Why, it is gravity, of course! Living on a massive planet, the Earth’s mass of 5.97*1024 kg produces a large gravitational force acting on other masses. Applying Newton’s Second Law, this force of gravity must be equal to the mass times the acceleration. We can calculate the 24 force of gravity between a mass m, the Earth with mass, ME = 5.97 * 10 kg at a distance of the center of 6 the Earth to the surface of the Earth, or rather one Earth radius, RE = 6,378 km = 6.378 * 10 m. We set that equal to Newton’s second law to see how much acceleration the object of mass m feels. Since we are talking about the acceleration due to gravity, let’s call that g instead of a. Now, we have

� = � � � � = �� �

�� � = � � (6) ��

2 Plugging in the values for G, ME, and RE, gives the acceleration due to gravity g = 9.8 m/s . This value for g is called the surface gravity of Earth. 2.2 – Connecting Newton’s Laws to Kepler’s Laws Clearly, Newton’s Laws of Motion and his Universal Force of Gravity are powerful tools to understand how things move in response to the force of gravity from massive objects. In the previous example of Earth’s surface gravity, we found the acceleration all objects near the surface of Earth experience, which dictates the motion of objects moving around on Earth. What about how the planets move in response to the Sun’s gravity? The Sun has about a million times more mass the Earth with a mass of 1 MSun = 1.99 * 1030 kg. That is going to generate a seriously strong gravitational force pulling the planets toward the Sun.

6 What keeps the planets from falling into the Sun? The answer to this question is same answer to the question, “what causes the planets to orbit the Sun?”. The graphic in Figure 4 demonstrates the following narrative explaining how Newton’s laws and Gravity explain planetary orbits. Consider a planet moving with a velocity, v, in the presence of the Sun’s gravitational pull. By Newton’s First Law of Motion, this force of gravity acting on the planet will change the object’s velocity. In fact, Newton’s Second Law of Motion tells us that the planet will experience an acceleration towards the Sun, which will constantly change the motion of the planet trying to make it move toward the Sun. Figure 4. A graphical depiction demonstrating how Newton’s Laws of Motion and the Universal Law of Gravity explain planetary orbits. The leftmost image of the planet shows that according to Newton’s 1st Law, the planet wants to move straight down at a speed of vorb. This speed is the orbital velocity necessary to create , Fc to balance the force of gravity Fg. Gravity deflects the motion of the planet toward the Sun changing the velocity. At the orbital velocity, this deflection makes a circular path around the Sun causing the planet to forever fall towards the planet while always missing it. This is the secret to orbiting. The light image of the planet at a second point in its orbit shows how the velocity, force of gravity, and centripetal force vectors change to create orbital motion. Orbital Velocity So, what is the secret to orbiting? In order to offset the pull of gravity toward the Sun, the planet needs to move at just the right velocity. To understand this, think about spinning a mass around on a string. The mass has , and you can feel the tension in the string as it spins around. This tension in the string is the string exerting a force on the mass pulling it toward the center point of motion. This force that you feel pulling toward the center of the circle is called centripetal force, or “center-seeking” force a that body experiences to have it move on a curved path The centripetal force is given by the equation Fc = (mv2)/r, where m is the mass, v is the velocity and r is the distance from the center of the circle, i.e., the radius. For an object in orbit around the Sun, this centripetal force is gravity. This gives us a way to determine the velocity, v, to keep the object in orbit. To find it, we use the fact that here the centripetal force equal to the force of gravity. We call this velocity the orbital velocity, vorb. The value of vorb can be found by setting the force of gravity equal to the centripetal force. Consider a planet with mass, Mp, orbiting the Sun with mass, MSun, at a distance a. Then, �� �� � = � ⇒ � = � � � � = � �

�� � = ��� (7) ��� �

7 So, for a planet to orbit the Sun at a distance a away from the Sun, it simply needs to be doing so at the orbital velocity given in Equation 7. Notice that as the distance increases, the velocity decreases indicating that objects orbiting farther from the Sun orbit slower, just as Copernicus originally stated in his original heliocentric model. Given that the total distance traveled for one complete orbit is the circumference of a circle with radius, a, and that to orbit, an object needs to move along this distance at the orbital velocity, you can calculate how long it would take the planet to complete exactly one orbit, i.e., the orbital period by dividing the circumference of the circle (2��) by the orbital velocity. This is the connection Newton made to explain Kepler’s Third Law of Planetary Motion. The orbital period of a planet can be determined using Newton’s Laws of Motion and Gravity. Escape Velocity If there is a velocity required to have a planet, or other object, orbit, then what happens when that velocity is increased? Is it possible to get an object moving fast enough to escape the gravitational pull of the Sun altogether? The answer is yes, and the minimum velocity required to do so is called the escape velocity. The derivation of escape velocity requires calculus (remember Newton needed to invent calculus), and I won’t belabor you with those details, but if you are curious and know the maths, you simply need to integrate the gravitational potential from a distance R away from the source of gravity to infinity. That gives you the total change in gravitational potential required to escape the gravity of the massive object (e.g., the Sun). Set that change in gravitational to the kenetic energy (1/2 mv2), and solve for the that velocity, and you get the minimum velocity to escape. Nicely done, us. That velocity comes out to simply be

��� � = = √� ∗ � (8) ��� � ��� which is about 1.414 ∗ �. So, 41.4% increase in the orbital velocity will cause an object to escape the gravity of the thing it is orbiting. If that object is the Sun, then it will leave our Solar System. If that object is the Earth, you have just determined the velocity required to send a spaceship away from the Earth to another part of our solar system, say Mars, , Jupiter, or beyond. Modifying Kepler’s Laws Newton’s insights led to a scientific understanding of Kepler’s Three Laws of Planetary Motion. The natural processes that govern the orbits of planets, and any other objects in orbit, were now accessible, understandable, and mathematically predictable. What changes and explanations needed to be added to turn Kepler’s Laws, which only apply to objects in orbit around the Sun, into ones that apply for any orbital system of objects (i.e., The Moon orbiting the Earth, the Jupiter- system and moons of other planets in general, planets orbiting other stars, stars orbiting within a galaxy, etc.) Newton’s Modification to Kepler’s First Law of Planetary Motion Orbits are ellipses with the center mass of the system located at one focus that is common to both ellipses (the other focus is empty). Newton explained orbital motion understanding how the force of gravity of the Sun acting on a planet requires a planet to move at the orbital velocity to remain in orbit. However, what does Newton’s Third Law of Motion tell us? If the Sun is exerting a force of gravity on the planet causing it to orbit, then there must be an equal in magnitude, but opposite in direction, force of gravity of the planet acting on the Sun. If there is a force acting on the Sun, then by Newton’s Second Law, it must change its motion. What keeps the Sun from heading directly toward the planet and them colliding? The answer is the same as before – The Sun must have its own orbit where it moves at the correct orbital velocity! With the Sun

8 being much more massive than a planet, and the force of gravity being the same strength, the acceleration of the Sun is much less (again, by Newton’s Second Law), so its orbit is extraordinarily small. Regardless, we must drop the notion from Kepler’s First Law that orbits are elliptical in shape (still true) with the Sun located at one of the two foci. If the Sun is moving, it cannot just sit at one of the two foci. If the Sun is orbiting as well, then we must ask the question of what is it orbiting? It turns out that both the planet and the Sun are orbiting their common center of mass. In simple terms, the center of mass is the balance point between the two masses. If you were to put them on a giant cosmic seesaw, then to have both sides balance one another, the fulcrum would have to be moved to the center of mass. Figure 5 shows how the location of the center of mass changes with different masses on the balance. Figure 5. A graphical depection of center of mass for (top) two equal masses; (middle) two unequal mass where m1 = 2m2; and (bottom) two very unequal masses where m1 = 10m2. With increasing unequalness in the masses, the center of mass moves toward the more massive object.

The realization that both the planet and Sun are on orbits around their center of mass gives us Newton’s modification to Kepler’s First Law of Planetary Motion. The change is that the center of mass is now at the foci of the two elliptical orbits (the orbit of the planet and the orbit of the Sun). Newton’s Modification to Kepler’s Second Law of Planetary Motion What about Kepler’s Second Law of Planetary Motion? This one is a bit trickier to deal with in sweeping out equal areas in equal times, but that does hold for both elliptical orbits. What does that practically mean? Consider the velocity of objects on elliptical orbits. For low eccentricities, the average velocity of an object in orbit is approximately equal to the circular orbital velocity given in Equation 7. For higher eccentricities, the average velocity is more complicated, and we won’t consider that case here. As seen in Fig. 1b, objects orbiting the Sun on elliptical orbits have the perihelion (closest to the Sun) and aphelion (farthest from the Sun) points. Our alternative to Kepler’s Second Law of Planetary Motion indicates that objects are traveling faster nearer the Sun and slower farther from the Sun. The fastest speed is at perihelion, and the slowest speed is at aphelion. So, the speed of an object orbiting the Sun at perihelion will be faster than the average speed, vorb, but to remain in orbit, it must also be less than the escape speed, vesc. Any faster, and it would break orbit and leave the solar system. Now considering aphelion, we can see that speed of the object must be less than the orbital velocity, vorb. The speed of an object in an elliptical orbit is not constant. It is always changing, such that it gets faster as it nearer to the massive object it is orbiting, and it gets slower as it moves away from the massive object it is orbiting. We can understand this in terms of Newton. For ease of communication, let’s consider a planet orbiting the Sun. As the planet moves away from the Sun, the Sun’s gravity is constantly pulling on it trying to change its motion (accelerate) it in toward the Sun. Hence, the velocity of the planet is getting smaller as gravity pulls it back toward the Sun. It loops around aphelion where it is travelling the slowest, and then begins to “fall” toward the Sun. The Sun’s gravity is now accelerating the planet in the

9 direction of its motion, so the planet increases its velocity. Because the average velocity is the orbital velocity, the planet misses the Sun, and swings around perihelion where it is travelling fastest and then starts to move away from the Sun again slowing down until it reaches aphelion. And so on, and so forth, over and over again as the planet moves in an elliptical orbit. The greater the eccentricity, the bigger the distance difference between aphelion and perihelion; the bigger the difference in the strength of gravity at aphelion compared to perihelion; and the bigger the difference in the speed at aphelion versus perihelion. There substantial difference in how Newton modified Kepler’s Second Law of Planetary Motion. Instead, it is more about understanding the velocity of the orbiting objects on elliptical orbits. This is summarized by what is known as the Vis viva (Latin for “living force”) equation, which gives the velocity of an object with a semi-major axis distance a on an elliptical orbit at a distance r away from the second object with mass M. The Vis viva Equation: �� = �� � − � (9) ������ � � Newton’s Modification to Kepler’s Third Law of Planetary Motion Finally, we get to heart of the matter with Newton’s modification of Kepler’s Third Law of Planetary Motion. For simplicity, we will assume circular orbits since the end result is the same. Consider two objects with masses, m1 and m2 in circular orbits around their common center of mass. Let’s consider the orbital period of m1 orbiting at a semi-major axis distance a at its orbital velocity. The time for the planet to complete one full orbit is the time it would take to travel the entire circumference of its orbit. The circumference of a circle is the distance 2��. Here we have to consider the combined mass of the objects (m1+m2 = MTotal) This gives the orbital period as ������������� � = = 2�� � �� �

� ∗ = 2��, squaring both sides gives � ∗ = 4��, solving for P gives ��� �� = �� (10) ������� Equation 10 is the full form of Kepler’s Third Law of Planetary Motion as modified by Newton. To use this version, you need all the units to be SI units. That means the period is measured in seconds, the mass in kilograms (kg), and the semi-major axis in meters. Notice that it still has the period squared is proportional to the semi-major axis cubed. The part the Kepler missed was that it also depends on the total mass of the system, Mtotal. We can recover some of the simplicity of the unmodified version if we choose the correct units and relate everything to our solar system. If we return to having the period in year and the semi-major axis in astronomical units (AU), and put the total mass of the system in solar masses, where the Sun has a mass of 1 solar mass, or 1 MSun, then we have �� �� = (11) ������ In-Lab Activities Name: Lab Instructor: Lab Meeting Day/Time:

10 Activity 1: Exploring Kepler’s Third Law • This activity requires you to use UC Boulder’s PhET My Solar System simulation. Link: PhET My Solar System (https://phet.colorado.edu/en/simulation/my-solar-system) o You will need to enable Flash for this simulation to work. If you are having trouble, please ask your instructor for assistance. • This activity also uses the Excel Spreadsheet for this lab available on the lab website.

The start of our in-lab activities is for you to discover Kepler’s 3rd Law of Planetary Motion for yourself. For this task, you will use the University of Colorado, Boulder’s PhET online simulation “My Solar System.” This simulation operates by simply putting in Newton’s Laws of Motion and Newtonian Gravity and letting things move accordingly. No other physics or programming is put into the simulation other than the graphics. As such, you should be able to discover Kepler’s Laws (as modified by Newton) by playing with the simulation. We will start with a guided exercise for you to take data and empirically discover Kepler’s Third Law: P2 = a3.

Instructions Setting up the simulation • Start the PhET My Solar System Simulation • In the green outlined control box on the right, change the accuracy to meter to be in the middle between “accurate” and “fast” For each simulation, you are going to change the x position and the y-velocity of the planet (purple boxes in simulation). Once you set up the first simulation, these are the only two values you will change between simulations. To help draw that to your attention, the planet’s x-positions in Table 1 have been shaded green, and the y-velocity (vy) has been shaded blue. Body 1 will be a star and Body 2 will be a massive planet (1/50th the mass of the star) For all simulations set the following: • Set the mass of “body 1” equal to 500 by clicking on the yellow box and changing he number to 500. • Keep the mass of “body 2” at 10. Change it to 10 if it is set to something else. • Set the y and x position of the star to be 0. • Set the x velocity of the star to be 0. Set the y velocity of the star to be -1. • Set the y-position and x-velocity of the planet (pink boxes) to be 0. These values will be the same for all simulations. You will only be changing the x-position and the y velocity. For each simulation set the x-position and y- equal to the values indicated in Table 1: Kepler’s 3rd Law Simulations. For example, your first simulation, “Simulation 1” has the x-position of the planet (pink boxes) set to 50 and the y-velocity set to 320. Your task is to measure the orbital period and the semi-major axis of the planet’s orbit.

11 Instructions to measure P and a • To measure the period, observe 5 orbits and then press STOP at the very end of the 5th orbit. o If the planet isn’t exactly at the end of the 5th orbit, that is okay. A small difference divided by the 5 orbits will only be a small error. We counted 5 orbits to reduce the error in measuring the orbital period. • To measure the semi-major axis, click the “Tape Measure” option in the control box. Measure the major axis of the planet’s orbit and divide that number by 2.

1. Set the x-position and y-velocity of the planet for the conditions of Simulation 1 and press START. Record your measured orbital period and semi-major axis distance (both unitless quantities) in Table 1 and on you Excel Spreadsheet. a. Repeat the process for Simulations 2 through 9. Table 1: Kepler’s 3rd Law Simulations x y semi-major Simulation M_Sun M_planet x y Period velocity velocity axis 1 500 10 50 0 0 320 2 500 10 75 0 0 260 3 500 10 100 0 0 220 4 500 10 125 0 0 200 5 500 10 150 0 0 185 6 500 10 175 0 0 170 7 500 10 200 0 0 160 8 500 10 225 0 0 150 9 500 10 250 0 0 143

As you enter your measurements into the Excel Spreadsheet, the spreadsheet will create a graph of orbital period (y-axis) versus semi-major axis (x-axis). Notice that data points are not fit by a line.

2. Right-click a data point on your P vs. a graph. Select “Add Trendline” Fit the data with a power law and display the equation on the graph. You may want to increase the font size of the equation. Record your equation, where y is P and x is a.

Equation: P = ______a

3. Rewriting Kepler’s 3rd Law: P2 = a3 by taking the square-root of both sides gives P = a1.5 where P is in years and a is in AU. This simulation has unitless periods and distances, so you will have a coefficient before a. If the simulation matches Kepler’s 3rd Law, then you should have an exponent very near to 1.5. What is the absolute error between the expected exponent of 1.5 and your determined exponent.

|| �������� ����� = 100 ∗ =______

12 Activity 2: Connecting Orbital Velocity to Orbital Period & Kepler’s 3rd Law In this activity, you will calculate the orbital velocity for four objects in the solar system that are orbiting the Sun to determine their orbital periods assuming circular orbits. You will then compare your determined orbital period to the orbital period you determine by using Kepler’s 3rd Law. Record your answers in Tables 2 and 3. Table 2. Period from Orbital Velocity

Object a [AU] a [m] Circumference [m] vorb [km/s] P [years] Venus 0.72 Earth 1.0 Asteroid 2.77 1 Neptune 30.1

4. Table 2 Calculations. Feel free to create an Excel Spreadsheet to do the calculations for you. a. Convert the semi-major axis, a, distances from AU to meters [m] for all 4 objects listed in Table 2. Use conversion factor: 1 AU = 1.496 * 1011 m Work

b. Assuming that the orbits are circular, calculate the circumference in meters of the orbits for all 4 objects listed in Table 2. ������������ = 2�� Work

13 c. Calculate the orbital velocity, vorb, in km/s for all 4 objects in Table 2. 30 MSun = 1.99 * 10 kg G = 6.67 * 10-11 m3/(kg s2) In meters per second [m/s], � = Convert your orbital velocity from m/s to km/s by dividing by 1,000. Work

d. Calculate the orbital period in years for all 4 objects listed in Table 2. [] Period in seconds: � = [ ] Convert your orbital period to years, and record that value in Table 2.

14 Table 3: Period from Kepler’s 3rd Law Object a [AU] P [years] Absolute Error Venus 0.72 Earth 1.0 Asteroid 1 Ceres 2.77 Neptune 30.1

5. Table 3 Calculations. You are welcome to use an Excel spreadsheet to do the calculations for you. a. Use the semi-major axis values in Table 2 or 3 and Kepler’s Third Law P2 = a3 to calculate the orbital period of the 4 objects in years. Work

b. Calculate the absolute error between the orbital period determined by Kepler’s 3rd Law (Table 3) and the orbital period determined from orbital velocity (Table 2). Use Kepler’s 3rd Law orbital period as the expected value.

15 Activity 3: Exploring Newton’s Modifications In this final activity, you will explore Newton’s Modifications to Kepler’s Three Laws of Planetary Motion. Specifically, you will observe two objects orbiting one another to see that both objects orbit a common center of mass. You will confirm that they are both orbiting a common center of mass by measuring the ratio of their semi-major axes and comparing that to the ratio of their semi-major axes calculated by their center of mass location. To spare you from having you going through the arduous mathematics, Figure 6 shows the connection between semi-major axis lengths and center of mass for two objects in orbit around one another. In the figure, the two masses are m1 and m2. The equation for how to determine the center of mass between two masses is given, but you do not have to work with that. Instead, the maths to calculate the related semi- major axis distances, a1 for the orbit of m1, and a2 for the orbit of m2 are show in purple text. Figure 6. The maths for the semi-major axis distances for two objects, mass 1 (m1) and mass 2 (m2), in orbit around one another. The total distance between the two objects is d. The semi-major axis distances are a1 for mass 1’s orbit and a2 for mass 2’s orbit. The equations in purple at the bottom relate the semi-major axes to the masses and the total distance.

A small bit of algebra reveals that the ratio between the semi-major axis distances is � � � �� �� + �� �� = � = (12) �� � � �� �� + ��

PhET My Solar System Simulations for two objects in orbit around one another. For these simulations, you are only going to change the masses of body 1 and body 2, and you are going to record the mass ratio, the expected semi-major axis ratio, and then measure the two semi-major axes for the two orbits to determine an observes semi-major axis ratio. Set the following initial conditions for the simulation Position Velocity Mass x y x y Body 1 Changes -80 0 0 -125 Body 2 Changes 80 0 0 125

16 6. Simulation 1: Set M1 = M2 = 300. a. Calculate the mass ratio: M1/M2

� = �

b. Calculate the expected semimajor axis ratio using Equation 12.

� �������� = �

c. Use the “Tape Measure” to measure a1 and a2. Calculate the measured semi-major axis ratio

a1 = ______a2 = ______

� �������� = � Do you answers match?

d. Describe the size and shape of Orbit 1 and Orbit 2

7. Simulation 2: Set M1 = 500 M2 = 300 a. Calculate the mass ratio: M1/M2

� = �

b. Calculate the expected semimajor axis ratio using Equation 12.

� �������� = �

c. Use the “Tape Measure” to measure a1 and a2. Calculate the measured semi-major axis ratio

a1 = ______a2 = ______

� �������� = � Do you answers match?

d. Describe the size and shape of Orbit 1 and Orbit 2

17 8. Simulation 3: Set M1 = 1500 M2 = 300 a. Calculate the mass ratio: M1/M2

� = �

b. Calculate the expected semimajor axis ratio using Equation 12.

� �������� = �

c. Use the “Tape Measure” to measure a1 and a2. Calculate the measured semi-major axis ratio

a1 = ______a2 = ______

� �������� = � Do you answers match?

d. Describe the size and shape of Orbit 1 and Orbit 2

9. Simulation 4: Set M1 = 1000 M2 = 10 a. Calculate the mass ratio: M1/M2

� = �

b. Calculate the expected semimajor axis ratio using Equation 12.

� �������� = �

c. The semi-major axis a1 for M1 is too small for you to measure using the “Tape Measure.” Measure the semi-major axis a2 and use it to predict the size of of a1

a2 = ______

Predicted a1 = ______

18