LAB 2 ASTRO KEPLERS LAWS

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Red Rocks Community College *

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101

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Astronomy

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Oct 30, 2023

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LAB 2 KEPLER’S LAWS* 80pts. OBJECTIVE: Students will use interactive web simulations to explore Kepler’s three laws of planetary motion. BACKGROUND Johannes Kepler was a German mathematician and astronomer who worked with the famous observer Tycho Brahe. Using much of Brahe’s observational data for his work on planetary motion, particularly the motion of Mars, Kepler developed three simple laws which describe motion of planets not only in our solar system but in star systems all over the universe! SKILLS/COMPETENCIES:  Define each of the related vocabulary words.  Recognize the appropriate symbols used.  State the concepts introduced.  Distinguish between different concepts within a topic.  Interpret tables or graphs.  Evaluate the relevancy of data.  Set up and solve problems using geometry, algebra, and trigonometry as required.  Apply concepts to new situations.  Demonstrate the ability to select and apply contemporary forms of technology to solve problems or compile information. MATERIALS:  Computer with internet access and flash capability PRE-LAB

There is no formal pre-lab for this lab but your instructor may have special instructions for you. LAB TASK 1: BACKGROUND Before Kepler, many astronomers were working with the Copernican model of the solar system. Please explain the Copernican model by answering the following questions: 1. (4) What did the Copernican solar system look like (you must draw a simple sketch along with your written answer )? Be specific. In the Copernican solar system model, the sun is at the center of the solar system and the orbit of every planet, including the orbit of the moon around the earth, moved in a perfectly circular rhythm. It is also worth mentioning that the starts beyond out solar system are fixed and do not move.

2. (2) In the Copernican solar system, what were the shapes of the planets’ orbits? In a Copernican solar system, the shapes of the planets’ orbits moved in a perfectly circular orbit. 3. (2) In the Copernican solar system , what caused the observed retrograde motion of Mars and the other planets? The observed retrograde motion of Mars and the other planets in the Copernican solar system model was described as a perspective effect. More specifically, this is caused when Earth is passing a slower moving outer planet. 4. (2) Why did astronomers suspect the Copernican model was either incorrect or incomplete? Astronomers suspected the Copernican model was incorrect because the orbits of planets moved in elliptical patterns rather than in the pattern of a perfect circle. This meant that all the details of the planets movement in the celestial sphere could not be explained easily, and usually involved great exceptions.

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5. (6) Please state each of Kepler’s three laws in your own words : Kepler’s 1 st Law: Planets that orbit around a star follow an elliptical path with the star at a set focus. Kepler’s 2 nd Law: A planet on an orbit around a star move faster at its perihelion and move slower at its aphelion. Also, a planet orbiting a star covers equal areas over equal times. Kepler’s 3 rd Law: Far away planets that orbit a star move slower relative to close planets that orbit the same star. This is calculated using the equation, (Time of the orbit) squared is equal to (Average orbital radius) cubed or p^2=a^3. TASK 2: KEPLER’S FIRST LAW Kepler’s first law describes the shape of a planet’s orbit. Let’s start by making sure you know what an ellipse is! You will need to make sure that your browser has pop-ups turned on before opening the interactive software. GO to the website below and download and install the NAAP Labs for your platform system. This section of the lab will be using the Planetary Orbit Simulator which can be found under the Planetary Orbits link: https://astro.unl.edu/nativeapps/ 1. (2) For the ellipse below, please label the approximate location of the: foci, semi- major axis and semi-minor axis.

2. (6) Now use the simulator to investigate Kepler’s First Law. Use the orbit settings and other options to demonstrate how Kepler’s First Law models the motions of planets in our Solar System. Then, use the chart below to create four ellipses (the first will be a special case of ellipse). Use the center of point of the plot for the focus of the first shape and the pairs of foci (squares, circles and triangles) to draw the other 3 shapes. Each of the 4 ellipses should have a semi-major axis of 10 graph-units (measured from the center point of the horizontal axis) and the arc of ALL the shapes should pass through those two points. Please use different colored pencils or pens to distinguish the different ellipses and label them to show which foci go with which color. SKIP

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3. (8) Use the Planetary Orbit Simulator to find the semimajor axis and eccentricity for all eight of the planets in our Solar System and Pluto. Click on the “Kepler’s 1 st Law” tab at the lower left of the screen and check the “show center” box to see how the position of the Sun is offset from the center of an ellipse. (You may click on or off any of the other features – feel free to explore and experiment!) Planet Name Semimajor Axis (in AU) Eccentricity Mercury 0.387 AU 0.206 Venus 0.723 AU 0.007 Earth 1 AU 0.017 Mars 1.52 AU 0.093 Jupiter 5.20 AU 0.048 Saturn 9.54 AU 0.056 Uranus 19.2 AU 0.047 Neptune 30.1 AU 0.009 Pluto 39.4 AU 0.249 4. (2) Which two objects from the table have the most eccentric orbits? Mercury and Pluto have the most eccentric orbits.

5. (2) Which two planets in our Solar System have the least eccentric orbits? Neptune and Venus have the least eccentric orbits. 6. (2) Now experiment with the eccentricity slider bar to find what eccentricity puts the Sun exactly at the center of the planet’s orbit. What is the eccentricity and what is the name of the shape of this special ellipse? For any planet, the eccentricity to put the Sun at the center of the planets orbit is 0. The name of the shape of this special ellipse is a perfect circle.

7. (4) There are a couple of special points along a planet’s orbit that you need to know about. Look up the definitions of these two special locations and write your own version (paraphrased) here: Perihelion: This is the point where a planet is closest to the star it orbits. Aphelion: This is the point where a planet is farthest to the star it orbits. 8. (4) The equation for the distance from the Sun at perihelion is: Perihelion = a ( 1 − ε ) . The equation for the distance from the Sun at aphelion is: Aphelion = a ( 1 + ε ) . a = average distance from the sun ( the semi-major axis ) ε = eccentricity of the orbit Use these equations to calculate the distance in kilometers of Earth from the Sun at aphelion and at perihelion. Give your final answer in S.N. Show your work. 150,000,000km (Rounded AU). 150,000,000(1-0.017) = 147450000km or 1.4545*10^8km is the distance of the Earth from the sun at perihelion. 150,000,000(1+0.017) = 152,550,000km or 1.5255*10^8km is the distance of the Earth from the sun at perihelion.

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9. (4) What percentage of the Earth’s average distance from the Sun is the difference from perihelion to aphelion? (This requires two calculations. Do the first in S.N. and give your final answer in percent.) Show your work. (1.4545*10^8 – 1.5255*10^8) or (7.1*10^6). This is the difference of perihelion and aphelion. (7.1*10^6) /(1.5*10^8) = (0.047)*(100), (4.7*10^-2) or 4.7%. The percentage of AU that is equal to the difference of perihelion and aphelion is equal to 4.7%.

TASK 3: KEPLER’S 2 ND LAW Kepler’s 2 nd law is about how the speed of a planet changes as its distance from the Sun changes. Toggle to the 2 nd Law on the menu in the lower left of the simulator page. 1. (4) Calculate Earth’s average orbital speed in kilometers per hour and in miles per hour. To make this calculation a bit easier, simplify Earth’s orbit to a perfect circle with a radius of one AU. Show your work. C=2(pi)(radius) C=2(pi)(150,000,000) C=942477796.08km/1yr * (1yr/365days) * (1day/24hr) = 107588.79km/1hr The earths average orbital speed is 107588.79km/1hr or 66852.575mi/1hr Now you’re ready to observe a simulation showing Kepler’s 2 nd Law:  Make sure the simulation is on the “Kepler’s 2 nd Law” tab at the lower left of the Planetary Orbit simulator.  Then, check the “sweep continuously” option 2. (3) Select Mercury from the Orbit Settings pull-down menu and then click “start sweeping”. What do the pie-pieces represent and describe their shapes? Each pie piece represents the same amount of area of distance every constant amount of time. The shapes of these pie pieces range from long and thin to short and fat. The long and thin pieces represent Mercury’s aphelion while the short and fat pieces represent Mercury perihelion.

3. (3) Now select “Neptune” from the pull-down menu and describe how and why the “pie pieces” appearance differs from Mercury’s. Compared to Mercury, Neptune has a very constant shape of pie pieces. This is because Neptune has a very low eccentricity compared to Mercury. A low eccentricity means that the pie shapes will be more uniform. A high eccentricity means that the pie shapes will vary from long and thin to short and fat. TASK 4: KEPLER’S 3 RD LAW Kepler’s 3 rd law is often confused with Kepler’s 2 nd law. The 2 nd Law describes the orbit of just one planet while the 3 rd law compares the orbits of different planets and gives a simple and useful mathematical relationship between orbital period and orbital distance for our solar system. 1. (6) Math is not just for calculating numbers; math is also a way of communicating how properties and quantities are related. See for yourself by answering these questions: According to Kepler’s 3 rd Law, how would a planet’s orbital period change if the planet’s… a. eccentricity doubled? (What does the equation say?) If the eccentricity of a planet is doubled then the planet’s orbital period would not change. b. mass halved? (What does the equation say?)

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If the mass of a planet is halved then the planet’s orbital period would not change. c. distance from the Sun quadrupled? (What does the equation say?) If the distance from the Sun is quadrupled then the planets orbital period increases exponentially.

2. (4) Use Kepler’s 3 rd Law to find out the orbital period of Vesta, an asteroid orbiting our Sun with a semimajor axis of 2.36 AU. Show your work. Vesta has an eccentricity of 0.09. p^2=a^3, so Vesta has a semimajor axis of 2.36AU. This means that p^2=2.36^3. 2.36^3 is 13.14. This means that p^2=13.14. The square root of 13.14 is 3.62. This means that the orbital period of Vesta is 3.62 earth years. TASK 5: PULLING IT ALL TOGETHER 1. (4) How did Kepler’s three laws of planetary motion change human understanding of the Solar System? (Be specific and thorough.) Kepler’s three laws of planetary motion has changed the human understanding of the solar system because it made the conceptualization of planetary movement more specific, and therefore more logical. For example, it was common belief that planets had orbits of perfect circles. Tycho believed this especially because he saw the stars as heavenly objects. Kepler’s first law objects this theory by stating that a planets orbit is an ellipse with its star, or sun, at a specific focus point. This understanding has then allowed humans to predict planetary movements with much greater accuracy. Overall, Kepler’s laws have allowed humans to predict, explain and understand planetary movement with great accuracy as it has provided us specific information, rules and guidelines regarding how planets move.

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2. (2) Which of Kepler’s laws explains that more distant planets orbit more slowly than planets closer to the Sun? Kepler’s third law states that more distant planets orbit more slowly than planets closer to the sun through the equation of p^2=a^3 with p standing for the orbital period of the planet in earth years and a standing for the average distance of the planet and the sun in AU. 3. (2) Which of Kepler’s laws explains that planets move faster when they are at perihelion than when they are at aphelion? Kepler’s second law explains that planets move faster at perihelion and slower at aphelion as the area measured of a planets orbit while at perihelion is equal to the area measured of a planet on at aphelion during the same amount. 4. (2) Which of Kepler’s laws describes the shape of orbits? Kepler’s first law describes the shape of orbits by describing them as ellipses with the star that the planet orbits at a specific focus point. These ellipses have an eccentricity ratio, being 0 out of 1, that describe how much the ellipse deviates from a perfect circle. With 0 being a perfect circle and 1 being a straight line, every number in between result in varying semimajor axis. The semimajor axis describes the distance from the star that a planet is when at perihelion and aphelion in AU.