LewisF_Lab_Kepler'sLaw_Minje Lab: Kepler’s Laws ASTR1120G Your Name goes here; keep the text red Introduction Throughout human history, the motion of the planets in the sky was a mystery: why did some planets move quickly across the sky, while other planets moved very slowly? Even two thousand years ago it was apparent that the motion of the planets was very complex. For example, Mercury and Venus never strayed very far from the Sun, while the Sun, the Moon, Mars, Jupiter and Saturn generally moved from the west to the east against the background stars (at this point in history, both the Moon and the Sun were considered “planets”). The Sun appeared to take one year to go around the Earth, while the Moon only took about 30 days. The other planets moved much more slowly. In addition to this rather slow movement against the background stars was, of course, the daily rising and setting of these objects. How could all of these motions occur? Because these objects were important to the cultures of the time—even foretelling the future using astrology. Being able to predict their motion was considered vital. The ancient Greeks had developed a model for the Universe in which all of the planets and the stars were embedded in perfect crystalline spheres that revolved around the Earth at uniform, but slightly different speeds. This is the “geocentric”, or Earth-centered model. But this model did not work very well–the speed of the planet across the sky changed. Sometimes, a planet even moved backwards! The Egyptian astronomer Ptolemy (85 − 165 AD) finally came up with a model for the motion of the planets that accounted for some of challenges. Ptolemy developed a complicated system to explain the motion of the planets, including “epicycles” and “equants”, that in the end worked reasonably well, and no other models for the motions of the planets were considered for 1500 years! While Ptolemy’s model worked well, the philosophers of the time did not like this model–their Universe was perfect, and Ptolemy’s model suggested that the planets moved in peculiar, imperfect ways. In the 1540’s Nicholas Copernicus (1473 − 1543) published his work suggesting that it was much easier to explain the complicated motion of the planets if the Earth revolved around the Sun, and that the orbits of the planets were circular. While Copernicus was not the first person to suggest this idea, the timing of his publication coincided with attempts to revise the calendar and to fix a large number of errors in Ptolemy’s model that had shown up over the 1500 years since the model was first introduced. But the “heliocentric” (Sun- centered) model of Copernicus was slow to win acceptance, since it did not work as well as the geocentric model of Ptolemy. Johannes LewisF_Lab_Kepler'sLaw_Minje Kepler (1571 − 1630) was the first person to truly understand how the planets in our solar system moved. Using the highly precise observations by Tycho Brahe (1546 − 1601) of the motions of the planets against the background stars, Kepler was able to formulate three laws that described how the planets moved. With these laws, he was able to predict the future motion of these planets to a higher precision than was previously possible. Materials Much of this lab involves using the UNL NAAP page (https://astro.unl.edu/downloads/ ) an and orbital simulator (http://www.astro.ucla.edu/undergrad/astro3/orbits.html) Lab Goals Many credit Kepler with the origin of modern physics, as his discoveries were what led Isaac Newton (1643 − 1727) to formulate the law of gravity. In this lab we will investigate Kepler’s laws. This assesses the following Course Goals (CGs): ● ● CG2: Apply the Scientific Method to solve problems in astrophysics. CG3: Demonstrate an understanding of math and physics principles to explain astrophysical phenomena. ● CG4: Adopt multiple perspectives to address modern astrophysics challenges As a lab class, this also address course goal 6 ● CG6: Discuss and solve astrophysics problems for a lab-based course And from Module 4 ● M6.2: Relate the need for Ptolemy's circles to 'backward' motion of a planet against the stars (CG4) ● M6.3: Explain Copernicus's arguments that the Earth is a planet orbiting the Sun, and how his reasoning accounts for Mar’s retrograde motion (CG4) ● M6.4: Describe the characteristics of planetary orbits discovered by Kepler as given by his 3 laws (CG3) ● M6.5: Describe Galileo's telescopic observations and discuss why there were so upsetting to ancient beliefs about the nature of the Universe (CG2) ● M6.7: Assess the role of Newton's theory of Gravity in the discovery of Neptune (CG1) ● M6.12: Explain how Pluto was expected to exist, in a manner similar to Neptune (CG1) ● M6.15: Explain how dwarf planets are distinguished from major planets (CG3) LewisF_Lab_Kepler'sLaw_Minje Lab Report Setup Before starting the lab, be sure to set up your lab report. To do this, first click “File” at the top left corner of this page and select “Make a copy…”. This will create your own personal version of this lab document which you can edit - this is now your lab report. Once you’ve made your copy, rename it with your last name and first initial, an underscore, and then “LabKepler” (for example, Nelson Muntz’s TA is called Ruth, Nelson’s report would be called “MuntzN_LabKepler_Ruth”). Next, share it with your TA so you can receive feedback as you go along by clicking the blue “Share” button in the upper right-hand corner of the page. A dialog box will appear; type “astr.1120.ta@gmail.com” into the field labeled “People,” checking to make sure that the “can edit” option is selected, and then click the blue “Send” button at the bottom left corner of the box. As you work on this lab, keep your answers in red text - this allows the TA to be able to easily find your answers to provide feedback and grades. Check Ins with your TA. You’ll find there are a few suggested check in points with your TA above and below. These are points for you to ask your TA to see if you are making progress down the right path, it is a chance for your TA to help you out if you are stuck, or if you accidentally make a common error. You’ll already have shared this document with your TA, so they can comment directly on your own google doc. Remember your TA has 1 work day to reply, so you will only be able to obtain useful help if you check in with your TA at least 24 hours before the deadline. Final submission statement. When you submit your labs each week you will sign a statement at the end of the lab each time. This statement means you confirm you have not copied work, you have not worked with another student outside any Lab lounge discussions, and you have not provided your work for anyone else to copy. Sanctions for any of these both fall under 'Plagiarism', and range from 0 on assignment (minor or 1st offense) to one letter grade reduction in course (minor, or 2nd offense) to F in course and dismissal from the university (major, or repeated offenses). LewisF_Lab_Kepler'sLaw_Minje Kepler’s Laws Before you begin the lab, let’s state Kepler’s three laws, the basic description of how the planets in our Solar System move. Kepler formulated his three laws in the early 1600’s, when he finally solved the mystery of how planets moved in our Solar System. These three (empirical) laws are: I. The orbits of the planets are ellipses with the Sun at one focus. II. A line from the planet to the Sun sweeps out equal areas in equal intervals of time. III. A planet’s orbital period squared is proportional to its average distance from the Sun cubed: P2 ∝ a3 In this lab, we will investigate these laws to develop your understanding of them. Simulator We will be using the NAAP simulators again, as we did for the Seasons lab and moon lab. If you have downloaded these already, then you already have the simulators you will need for this lab! You can just open the “NAAP Labs” application on your computer. If not, you can download these at the UNL NAAP page (https://astro.unl.edu/downloads/). You will want to download the NAAP Labs files. Once you’ve downloaded and installed the simulator, you should open the application. In the list of labs, we will be working with the Planetary Orbits lab (which is Lab Number 5 in this application). Once you’ve clicked on the lab, it will provide you some links with some useful and helpful information. Feel free to look at this information throughout the lab. Note, the simulator that we’ll use for this lab is the “Planetary Orbit Simulator.” Click on the “Planetary Orbit Simulator” in order to launch the simulator for this lab. Most of the text and questions for the following three sections are taken directly from the NAAP Planetary Orbits lab Student Guide. LewisF_Lab_Kepler'sLaw_Minje Kepler’s 1st Law Launch the NAAP Planetary Orbit Simulator. ● Open the Kepler’s 1st Law tab if it is not already (it’s open by default). ○ This can also be found here ○ https://astro.unl.edu/naap/pos/animations/kepler.html ● Enable all 5 check boxes. ● 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. ● Change the eccentricity and note how it affects the shape of the orbit. ● Tip: You can change the value of a slider by clicking on the slider bar or by entering a number in the value box. 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 simulator limits the semi-major axis to 50 AU since that covers most of the objects in which we are interested in our solar system and have limited eccentricity to 0.7 since the ellipses would be hard to fit on the screen for larger values. Note also 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 planet’s parameters to those like our solar system’s planets. Explore these options. We will now be using this simulator to answer some questions on Kepler’s 1st law. 1. For what eccentricity is the secondary focus (which is usually empty) located at the sun? What is the shape of this orbit? (3 points) replace this text LewisF_Lab_Kepler'sLaw_Minje 2. Create an orbit with a=20AU 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 r1 and r2 at these locations? (3 points) replace this text 3. 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 r1 and r2 at these locations? (3 points) replace this text 4. What is the value of the sum of r1 and r2 and how does it relate to the ellipse properties? Is this true for all ellipses? (4 points) replace this text It is easy to create an ellipse using a loop of string and two thumbtacks. The string is first stretched over the thumbtacks which act as foci. The string is then pulled tight using the pencil which can then trace out the ellipse. LewisF_Lab_Kepler'sLaw_Minje 5. Assume that you wish to draw an ellipse with a semi-major axis of a = 20 cm and an eccentricity of e = 0.5. How long would your string need to be? (Hint: think about the case where e = 0, i.e., a circle). Given that the eccentricity of an ellipse is c/a, where c is the distance of each focus from the center of the ellipse, how far apart would the thumbtacks (at the focii) need to be? (5 points) replace this text ****Time to Check In with your TA. Send them the document and ask them for feedback on your progress. Q5 , in particular, can be a puzzler, and so check with your TA to see if you are using the best method to work it out. If not, they can help guide you along to working this**** LewisF_Lab_Kepler'sLaw_Minje Kepler’s 2nd Law ● Use the “clear optional features” button to remove the 1st Law features. ● Open the Kepler’s 2nd Law tab. ● 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. 6. 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? (3 points) replace this text 7. 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 “widest”? Where is it the “narrowest”? Where is the planet when it is sweeping out each of these segments? What names do astronomers use for these positions? (5 points) replace this text 8. What eccentricity in the simulator gives the greatest variation of sweep segment shape? (3 points) replace this text LewisF_Lab_Kepler'sLaw_Minje 9. 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 Halley’s orbit cannot be shown in this simulator.) The orbit of Halley’s Comet, the Earth’s Orbit, and the Sun are shown in the diagram below (not exactly to scale). Based upon what you know about Kepler’s 2nd Law, explain why we can only see the comet for about 6 months every orbit (76 years)? (5 points) replace this text LewisF_Lab_Kepler'sLaw_Minje Kepler’s 3rd Law Kepler’s third law is: P2 (years) = a3 (AU) Note the units. This works for the planet orbital period in years and the planet orbit semi-major axis in AU. It is really trivial for Earth, with an orbital period of 1 year and planet orbit semi-major axis of 1AU. P2 (years) = 12 = 1 X 1 = 1 a3 (AU) = 13 = 1 X 1 X 1= 1 Here is an example of how use this equation to make some other predictions. If the average distance of Jupiter from the Sun is about 5 AU, what is its orbital period? Setup the equation: P(Jupiter)2 = a(Jupiter)3 = 53 =5×5×5 = 125 So, for Jupiter, P2 = 125. How do we figure out what P is? We have to take the square root of both sides of the equation, which you can easily do with a calculator. P = √ P(Jupiter)2 = √ 125 =11.2 years The orbital period of Jupiter is approximately 11.2 years. Similarly, if you are given the period of an orbit, you can find the semimajor axis: just take the square of the period, and then you have to take the cube root of that number. Try this for Jupiter. Take the square of 11.2, then take the cube root of the result. Your answer should be 5. If you cannot do this on your calculator contact your TA so they can help you. Let’s investigate Kepler’s third law using the simulator LewisF_Lab_Kepler'sLaw_Minje ● Use the “clear optional features” button to remove the 2nd Law features. ● Open the Kepler’s 3rd Law tab. 10. Use the simulator to complete (replace the #) in the table below. (14 points) Object P (years) a (AU) e P2 a3 Earth # 1.00 # # # Mars # 1.52 # # # Ceres # 2.77 0.08 # # Chiron 50.7 # 0.38 # # 11. As the size of a planet’s orbit increases, what happens to its period? (3 points) replace this text 12. Start with the Earth’s orbit and change the eccentricity to 0.6. Does changing the eccentricity change the period of the planet? (3 point) replace this text 13. Kepler’s third law is P2 = a3 where P is measured in years, and a is measured in astronomical units. Using this relation, what would the period of an object be if it was an in orbit with a semi-major axis of 4 AU? Show your work. (4 points) replace this text 14. What would the orbital semimajor axis be for an object that had an orbital period of 10 years? (4 points) replace this text LewisF_Lab_Kepler'sLaw_Minje If you use units other than years for the period and AU for the semimajor axis, there would be some other numbers in the equation for Kepler’s third law, but the basic relation between the square of the period (P2) and the semimajor axies (a3 ) would still be the same. For example, say we measured the semimajor axis in kilometers (km) instead of in AU. We can do a unit conversion. Since 1 AU = 1.496 × 108 km, we have: You would get some different number if you used some different units for either the period or the semimajor axis, but you would always see a P2 on the left side and an a3 on the right. For this reason, scientist often represent the fundamentally important part of the relation as a proportionality rather than as an equality, in other words, they would say that P2 is proportional to a3, which is a statement that is true independent of the units used. This is often written as: P2 ∝ a 3 If you take the square root of both sides, this becomes: P ∝ a3/2 Using proportionalities often makes calculations easier, because you can use ratios of quantities from different objects. For example, if someone says that the semimajor axis of some object is twice that of Jupiter, you can tell them what the period of that object is relative to the period of Jupiter: without ever needing to know what the semimajor axis or the period of Jupiter is at all! 15. The proportionality part of Kepler’s third law holds for all orbiting objects, although the equality does not. Imagine we discovered another system of planets around another star, and found that a planet located at 1 AU from the star took 2 years to go around (this would happen if the star was less massive than our LewisF_Lab_Kepler'sLaw_Minje Sun). How long would it take a planet that was located at 4 AU from that star to orbit the star? Use equation 9 and explain your reasoning. (5 points) replace this text ****Time to Check In with your TA. Send them the document and ask them for feedback on your progress. Some student may have problems with completing the Table above, and Q15, in particular, can be a puzzler, so check with your TA to see if you are using the best method to work it out **** LewisF_Lab_Kepler'sLaw_Minje Going Beyond the Solar System It turns out that Kepler’s law can be explained by understanding more fundamental laws of motion and the law of gravity. Because of this, it turns out that the proportionality of Kepler’s laws hold for all orbiting objects. However, there is a simplification made in the simulator that you have been using, in that it assumes that the planets are much less massive than the Sun, which is generally true, but not perfectly so. As you will learn when you study gravity (if you haven’t already), when one object pulls on the other by the force of gravity, the pull is mutual : both objects pull on each other. As a result, both objects actually orbit around a spot called the center of mass of the system. If one object is much more massive than the other, then the center of mass is very close to the center of the massive object, and the massive object (like the Sun) barely moves, while the less massive object (like the planets) do almost all of the orbiting. But if the masses of two objects are more equal, then you can easily see the orbits of both objects. You can investigate this using a binary star simulator. A binary star system is a system where two stars orbit each other. You can find such a simulator at: http://www.astro.ucla.edu/undergrad/astro3/orbits.html In this simulation there are two stars in orbit around each other, a blue one and a red one. You can change the masses of the stars, as well as the eccentricity of the orbit and the inclination. Start by setting the first mass to 100, the second mass to 1, the eccentricity to 0, and the inclination to 90. Here you should be able to see a situation like you got with the previous simulator: the red object orbits around the blue object. You can plan with the eccentricity and you should see Kepler’s laws: as the object becomes more eccentric, the blue object is located at one of the focii of the ellipse (Kepler’s first law), the speed of the red orbiting object is much faster when near to the blue object than when it is far away. However, you can see another additional feature: the blue object is actually making a tiny orbit itself! Investigate what happens as you increase the mass of the blue object, and also when you change the eccentricity. LewisF_Lab_Kepler'sLaw_Minje 16. Keeping the eccentricity at zero, describe the orbits when the masses of the two objects are equal. Where is the center of mass in relation to the two orbits? (5 points) replace this text 17. Describe the shape and size of the orbits as the masses become more unequal. (5 points) replace this text 18. Describe the orbits (shape, size, location, and speed) as you change the eccentricity but keep the masses of the two objects equal. (5 points) replace this text 19. Describe the orbits as you change the eccentricity and the relative masses of the objects. (5 points) replace this text 20. Finally, go back to the original configuration, with the blue object much more massive than the red one. This starts to be close to a situation where you have a (big) planet orbiting a star. However, a star is also very different from a planet because it shines with its own light, while a planet does not. So if such a system was out there (and they are!), we can’t generally see the planet directly, because it is lost in the light coming from the star. However, this simulation suggests a way we might tell that the planet is there, just by looking at the star. From looking at the simulation, can you explain what that might be? (5 points) replace this text ***Time to Check in with your TA. Understanding the simulator, and understanding how to use this to answer Q16-20 is critical to this lab. Ask your TA for help in making sure you have done this correctly*** LewisF_Lab_Kepler'sLaw_Minje End of Lab Question 21. Do your research on google to find one object in the solar system with an extremely circular orbit and one with an extremely eccentric orbit. Which objects did you find and which websites did use (2 points) replace this text 22. Describe the differences in these two orbits by discussing the differences in the minimum and maximum distances from the Sun, their orbital periods, and their masses. (6 points) replace this text ***End of Lab *** Now sign this assignment by entering your name and date in the statement below, and submit this lab using the instructions on the Course Canvas page I, (replace this text with your name), submit this lab project (replace this text with todays date dd/mm/yy) as part of ASTR1120G as purely my own work. I have not copied from anyone, nor have I allowed my work to be copied. I have not worked with another student outside the Lab Lounge on the canvas page. I have read the NMSU student handbook policy on plagiarism, both intentional and unintentional, and I realize that if I copy or allowed myself to be copied, my behavior will result in dismissal from the course and NMSU. 
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