Jupiter and Kepler's Laws

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University of Missouri, Columbia *

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1010

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Astronomy

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Oct 30, 2023

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Part 1: Measuring the Mass of Jupiter from Kepler’s 3 rd Law In this lab, you will be determining the mass of Jupiter by measuring the orbital parameters of its moons and applying them to Kepler’s 3 rd Law. All measurements will be taken though a simulated web interface. 1. To begin the lab, in your browser (either Chrome or Firefox), go to the following link: astro.sbaldridge.com/kepler.html 2. Familiarize yourself with the interface. There are 5 main panels. a. Viewpoint Window: This is the simulation of Jupiter with its 4 moons. Clicking in this space will plot the corresponding mouse location on the right. b. Time Frame: This window is directly below the Viewport window and controls the amount of time to advance in each interval as well as moving forward or backward in time. Clicking the right (left) arrow will advance (rewind) time by the interval you set. c. Moon Selection Tabs: At the upper right of the program, you can select which moon you are currently measuring data for. Changing moons will change the associated graph and orbital parameters. Notice each moon has a corresponding color. d. Graph Window: This is a graph of your measured moon positions and will update in real time (each time you click in the Viewpoint Window). You can also find buttons for undoing the last measurement and clearing all data as well as the print button. e. Orbital Parameters Frame: Once all your data for the corresponding moon is recorded, you will want to fit a sine curve which matches your data precisely. You change the parameters of the sine curve by changing the values in the appropriate boxes. The MSE (or Mean Square Error) tells you how close to the data your curve is. (A value of 0 means perfectly fit). 3. To begin recording the moon positions, you must first determine an appropriate interval by which to advance time. An interval set too small will make you need to collect lots of data in order to see an entire orbit. An interval set too large will advance time too fast and you will miss the important details of the orbit. Each moon will require a different optimal interval. 4. Once you determine an optimal interval, you may begin recording the moon’s positions as it changes with time. Start with Io (red moon). The steps are outlined below: a. Set interval to a reasonable value (for Io, you can use 2 hours) b. Click the right arrow to advance time 1 c. Click on Io to record its position (the precision of your click is important) d. Repeat steps 4b and 4c until you have recorded more than a complete orbit of Io. 5. Once your first moon is recorded, click on the next moon’s tab (Europa) to begin a new data set. Do not worry, your previous moon’s data will be saved. 1 You may alternatively use the keyboard for several actions for ease of access. The “s” key will record the position of the mouse cursor (replacing the mouseclick); the “d” key will advance time by the interval; the “a” key will reverse time by the interval.

6. Before recording data for a new moon, reset your time to 0 . This is done by clicking the curved arrow in the Time Frame. Determine a new time interval for the new moon. 7. Repeat the same measurement steps you did for Io. Be sure to have enough measurements for a complete wavelength to show. Once finished, move on to the next moon and repeat. 8. Once all 4 moons are recorded, we need to fit sine curves to the data. This is done in the orbital parameters frame. Descriptions of these sine curve parameters can be found in “The Sine Wave” supplemental sheet. 9. Match your T-zero first, do not worry about being exact just yet, we will fine tune things later. 10. Match the wavelength second, again a rough value is fine. 11. Match the amplitude last. 12. Once you have a rough sine curve to match your data, we need to fine tune our parameters until the curve is as close as possible. Do this by changing the 3 values by small increments until the MSE (mean square error) is as close to 0 as possible. The MSE number changes color for how far off your numbers are; red is bad, and green is good. 13. Once your curve is the best fit possible, repeat for the other moons. 14. Once the sine curves are fit for all 4 moons, you need to (1) save your data and (2) submit the file with your lab . Clicking the printer button will open a new window with formatted output ready to save. In printers, choose “Microsoft Print to PDF” to save as a PDF copy in your computer. 15. Fill out column 2 and 3 in the table below with the appropriate data from your measurements. 16. Next, we need to convert our period and amplitude into the correct units to use with Kepler’s 3 rd Law. That is, Years and Astronomical Units (AU). Fill in the corresponding values in column 4 and 5. a. Convert your Period (days) to Period (Years) by dividing by 365 b. Convert your Amplitude (JD) to Semi Major Axis (AU) by diving by 1050 c. All answers should be to 3 significant figures! Refer to the box at the bottom of the page. Period (Days) Amplitude (Jupiter Diameters) Period (Years) Semi Major Axis (AU) Io -3 2.94 -.00821 .0028 Europa -2.2 4.81 -.00602 .00458 ↓ Convert ↓

Ganymede -9.54 7.63 -.0261 .00727 Callisto 13.9 13.48 .0381 .0128 All answers should be to 3 significant figures ! Examples: 6.44x10 -4 or 5.78 or 0.00752

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17. You now have P (period) and a (semi major axis) for each moon orbiting Jupiter. Using the following equation, calculate the mass of Jupiter as measured by each moon. Find the average. Since the mass is being compared to the mass of the Sun, your answers will be small and have a negative exponent. M ≈ a 3 P 2 (This is a modified Kepler’s 3 rd law which only works if units of AU, Years, and Solar mass are used) Mass of Jupiter as determined by Io: ___3.257 x 10^-4__________ Solar Masses Europa: _______2.651 x 10^-3__________ Solar Masses Ganymede: __-5.684 x 10^-4___________________ Solar Masses Callisto: ______1.445 x 10^-3_________ Solar Masses Average: _____.41725 x 10 ^-4______________________________ Solar Masses 1. Based on your results, imagine there was a moon orbiting Jupiter further than Callisto. Would this moon have a longer or a shorter period than Callisto? It would have a longer period than Callisto 2. Which would have a more detrimental impact on your final Jupiter mass: a 10% error in your measurements of a or P ? Why? ( Hint: look at the equation ) P because it is being multiplied to the third 3. Imagine you wish to now measure the mass of Saturn. Describe in your own words how you would proceed. What parameters would you need? How would you determine the mass? I would determine the mass with keplers 3 rd law and I would need AU and the time it takes for a full orbit

Part 2: Kepler's Laws Questions Figure 1 (below) shows a planet's orbit. The twelve positions shown are exactly one month apart. 1. Without measuring, are the shaded areas of the two triangles equal area? Explain your reasoning . Yes because the speed of the planet may increase but the time stays the same so when the sun orbits it covers the same amount areas in the same amount of time 2. Does the planet seem to be moving the same distance each month? yes 3. At which point will the planet be moving fastest? Slowest? Fasted at G and slowest at A 4. At point D, is the speed of the planet increasing or decreasing? Increasing 5. Provide a concise statement of the relationship between a planet's distance to the Sun and its orbital speed. As planets get closer to the sun the orbital speed increases. Figure 1

For the following 5 questions (6-10), consider an imaginary planetary system with 2 planets: Terra Small, Earth-like planet. Orbits close to the Star. Jove Large, Jupiter-like planet. Orbits further from the Star. For simplicity, assume the planets have orbital eccentricities of 0.0 (unless otherwise mentioned) and do not gravitationally interact with each other. The star has a mass very near the Sun's mass therefore we can invoke Kepler's version of his 3rd law: P 2 = a 3 . BIG HINT: You only need to use this relationship! 6. Which one of these planets (Terra or Jove) do you think will move around the central star in the least amount of time? Why? Terra because it is closer to the sun 7. If Terra and Jove were to switch positions, would your answer to the previous question change? If so, how? If not, why not? Yes, because then Jove would be closer to the sun therefore go faster 8. Do you think the orbital period for Terra would increase, decrease, or stay the same if its mass were increased? Why? I think it would stay the same because only the distance to the sun matters not the mass 9. Imagine both Terra and Jove were in orbit at the same distance from the star. Which of the two planets would orbit in the least amount of time? Terra 10. Imagine that Terra has a high eccentricity of 0.9 and Jove still has an eccentricity of 0.0. The semi- major axis of both planets is the same. Which planet orbits the star in the least amount of time? Jove as it has a rounder orbit

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11. Figure 2 (right) is a graphical representation of Kepler's 3rd law. According to the graph, how far from the central star does a planet orbit if it has an orbital period of 1 year? 1 AU 12. How long does it take a planet to complete one orbit if it is three times the distance as the planet in the previous question? 5 years 13. What is the name of the planet in Question 11? Earth The orbital distances, orbital periods, and masses for the 6 closest planets have been listed below. 14. Referencing the table above, which of the answers below best describe how a planet's mass will affect its orbital period? a. Planets that have small masses have longer orbital periods than planets with large masses. b. Planets with the same mass will have the same orbital period. c. Planets that have large masses have longer orbital periods than planets with small masses. d. A planet's mass does not affect its orbital period. Figure 2

15. Explain your reasoning (for #14) and cite a specific example from the table to support your choice. (Do you still stand by your answers for questions 6-10? If not, change them.) Mars(.11) is lighter than Venus (.82)however it has a longer orbital period than venus