In this exercise you will be using data from observations of the moons of Jupiter to plot the positions of the Galilean moons over a period of time. By plotting data on the positions of the moons at different times in their orbits, it is possible through the use of Kepler’s 3rd Law, to calculate the mass of Jupiter. The set of observations and calculations you will make in this exercise is very similar to the way astronomers originally calculated the mass of Jupiter. Introduction: About 400 years ago the great Italian scientist Galileo Galilei was the first person in human history to observe the four largest moons of Jupiter. With his homemade telescope, Galileo made regular observations of the moons Io, Europa, Ganymede, and Callisto for several decades. These 4 moons are known collectively as the ‘Galilean Moons’ of Jupiter. The order listed here, Io, Europa, Ganymede, and Callisto is the order from closest to furthest from the planet. (This order can be remembered by the well known acronym ‘I Eat Green Carrots’!) Galileo’s observations of these moons showed him that there were at least some objects in the universe which did not revolve around the Earth, and this helped convince Galileo that the heliocentric model of the Solar System (which had been proposed 60 years earlier by Copernicus) was indeed correct. Background: A smaller body in orbit around a larger central object will obey the Laws of Planetary Motion as originally stated by Johannes Kepler in the early 1600’s. In particular, Kepler’s 3rd Law of Planetary Motion describes how the orbital period, and semi-major axis (average distance from central object) of the orbiting body are related. In mathematical form Kepler’s 3rd Law says: M x T2 = a3 In this formula: M = Mass of central object, measured in Solar Mass Units. T = Period of orbiting body, measured in Earth years. a = Semi-major axis of orbiting body, measured in Astronomical Units (AU’s). By measuring the period ‘T’ and the semi-major axis ‘a’ of some of the moon’s in orbit around Jupiter, a calculation of the mass, ‘M’ of Jupiter can be made. Procedures: Use the data provided to plot the positions of the four Galilean moons over time as instructed in the steps below. 1. Plot the data points on the graph. For each moon, use the appropriate graph and carefully plot the moon’s position on the vertical axis versus its day number on the horizontal axis. If there were ‘cloudy skies’ for a day, then do not plot any data for that day number. Also, remember that we consider positions east of Jupiter to be negative, and positions west of Jupiter to be positive. For example, plot 4.36 E as – 4.36, and plot 2.94 W as +2.94. You will need to make four plots in total (one for each moon). You can use the provided plots or recreate these on your own graph paper. 2. Fit a curve to the data points. After plotting all the data points, it’s time to draw a curve through the points. The type of curve to draw is called a ‘sine wave’ or ‘sine curve’. The idea is to draw a smooth curve that passes through most of the dots you plotted. Do not simply draw straight lines to connect each adjacent dot! A sample sine curve is drawn below. Remember that a sine curve must reach the same value for each peak of the curve. Most peaks of the curve will not have data point right at the peak, so draw the peak of the curve at the same height even if there is no data point there. Likewise the curve must reach an equal negative value for each trough (or valley) the curve goes through. 3. Determine the period and semi-major axis. After drawing a smooth sine curve through the data points you can determine the period and semi-major axis of the moon’s orbit using your curve. Record the period and the semi-major axis in the spaces provided below each graph. • The period is the time it takes the moon to go through one complete orbit. To measure the period from the graph, make note of the day number for one peak of the graph, and then note the day number for an adjacent peak. Then subtract these two numbers to find the period. (A slightly more accurate way to find the period, is to count the number of days between 3 or 4 peaks, and divide by 3 or 4 to get the time for one period.) • The value of the semi-major axis is simply the maximum value the graph rises up to at each peak. The semi-major axis is the maximum height (measured in J.D.’s) that the graph reaches in the vertical direction. 4. Convert the units of the period and semi-major axis. In order to use Kepler’s 3rd Law to calculate the mass of Jupiter, it is necessary to first convert the units of the period from days to years, and the units of the semimajor axis from Jupiter Diameters to AU. • To convert the period to years, simply divide by 365, (since there are 365 days in one year). • The value of the semi-major axis is measured in Jupiter Diameters (J.D.’s), and there are 1050 Jupiter Diameters in one Astronomical Unit. To convert J.D.’s to AU’s, divide the value of the semi-major axis by 1050. 5. Calculate mass of Jupiter. We are now ready for the last step in the data analysis, which is to calculate of the mass of Jupiter. Kepler’s 3rd Law says M x T2 = a3 , and since we have values for ‘T’ and ‘a’, we can find M. • Start by solving the 3rd Law for M. We get, M=a3 /T2. • Now take the value of ‘a’ measured in AU’s and cube it. • Then take the value of ‘T’ measured in years, and square it. • Finally divide a3 by T2 , and we have our answer. 6. Lastly, do the remaining calculations, and answer the remaining questions on the final page. Sample Data, Graph and Calculations for the a fake Moon Position The data shown in the table on the right has been plotted on the graph above with each data point marked by a red triangle. The data points were then connected with a sine curve (seen in the blue dots). The first peak is at 0.7 days and the next peak is at 4.3 days, so period (days between peaks) would be 4.3-0.7 = 3.6 Days. The semi-major axis would be the height of a peak, and the first peak reaches 5.0 J.D. so that is the semi-major axis of this moon’s orbit. T (Period) = 3.6 Days T (Period) = 3.6 / 365 = 0.009863 Years M = a3/T2 a (Semi-major axis) = 5.0 J.D. a (Semi-major axis) = 5.0 / 1050 = 0.004762 AU M = (0.004762)3/(0.009863)2 = (1.080 x 10-7)/(9.728 x 10-5) = 0.00111 Solar Masses Position Data for Io & Europa Data taken four times a day due to the close orbit of these two moons. Date 10/22 10/22 10/22 10/22 10/23 10/23 10/23 10/23 10/24 10/24 10/24 10/24 10/25 10/25 10/25 10/25 10/26 10/26 10/26 10/26 10/27 Time 0:00 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 Day Number 0 0.25 0.5 0.75 1.0 1.25 1.5 1.75 2.0 2.25 2.5 2.75 3.0 3.25 3.5 3.75 4.0 4.25 4.5 4.75 5.0 Io — 2.55 E 2.85 E 1.05 E 1.45 W 2.85 W (cloudy) — 2.45 E 2.90 E 1.30 E 1.20 W 2.80 W 2.25 W (cloudy) 2.35 E 3.00 E 1.42 E 1.13 W 2.80 W 2.35 W Europa 4.50 W 3.70 W 2.15 W — 1.90 E 3.65 E (cloudy) 4.65 E 4.00 E 2.40 E — 1.60 W 3.25 W 4.35 W (cloudy) 3.95 W 2.55 W 0.65 W 1.42 E 3.20 E 4.40 E Position Data for Ganymede & Callisto Data taken here just once a day at midnight. Date 10/22 10/23 10/24 10/25 10/26 10/27 10/28 10/29 10/30 10/31 11/1 11/2 11/3 11/4 11/5 11/6 11/7 11/8 11/9 11/10 11/11 11/12 Time 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 0:00 Day Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Ganymede 3.70 W 2.65 E 7.10 E 6.55 E 1.35 E 4.80 W 7.45 W 4.75 W — 6.55 E 7.05 E 2.45 E 3.88 W 7.35 W 5.55 W — 5.90 E 7.40 E (cloudy) 2.80 W 7.10 W 6.30 W Callisto 6.35 W 1.70 W 3.15 E 7.65 E 11.10 E 12.95 E 13.00 E 11.25 E 7.95 E 3.50 E 1.43 W 6.15 W 9.90 W 12.40 W 13.25 W 12.20 W 9.60 W 5.60 W (cloudy) 3.95 E 8.30 E 11.45 E Position versus Time Graph for Io T (Period) = _______________ Days a (Semi-major axis) = _________________ J.D. T (Period) = _______________ Years a (Semi-major axis) = _________________ A.U. Hints: • To determine the period in years, start with the value of ‘T’ measured in days and divide by 365, since there are 365 days in a year. • To determine the semi-major axis in AU’s, start with the value of ‘a’ measured in J.D’s, and divide by 1050 since there are 1050 Jupiter Diameters in 1 Astronomical Unit. • In the following calculation, be sure to use the value of ‘a’ measured in AU’s, and the value of ‘T’ measured in years. • See the sample calculations for the example moon on the second page. Calculation of the mass of Jupiter: MJ = a3 /T2 MJ = _____________ solar mass units Position versus Time Graph for Europa T (Period) = _______________ Days a (Semi-major axis) = _________________ J.D. T (Period) = _______________ Years a (Semi-major axis) = _________________ A.U. Calculation of the mass of Jupiter: MJ = a3 /T2 MJ = _____________ solar mass units Position versus Time Graph for Ganymede T (Period) = _______________ Days a (Semi-major axis) = _________________ J.D. T (Period) = _______________ Years a (Semi-major axis) = _________________ A.U. Calculation of the mass of Jupiter: MJ = a3 /T2 MJ = _____________ solar mass units Position versus Time Graph for Callisto T (Period) = _______________ Days a (Semi-major axis) = _________________ J.D. T (Period) = _______________ Years a (Semi-major axis) = _________________ A.U. Calculation of the mass of Jupiter: MJ = a3 /T2 MJ = _____________ solar mass units Further Data Analysis and Questions: 1. List the four values for the mass of Jupiter you calculated here: From Io’s data MJ = ___________________________ From Europa’s data MJ = ___________________________ From Ganymede’s data MJ = ___________________________ From Callisto’s data MJ = ___________________________ Use this data to calculate an average value for the mass of Jupiter. MJ = ___________________________ (To get the average of the four mass values, add all values and divide the sum by four.) 2. Now calculate the mass of Jupiter as measured in Earth mass units. The mass of the Earth in solar mass units is about 3.00×10-6, so take the average value of the mass of Jupiter from above, and divide by 3.00×10-6. MJ = ___________________________ Earth Masses 3. There are quite a few moons of Jupiter that are out beyond the orbit of Callisto. Will these moons have larger or smaller orbital periods than Callisto? Explain your answer. Conclusion: Write a short conclusion summarizing what you did and what you learned in this exercise. Were you able to graph the data effectively? Do you understand how the mass of Jupiter can be determined from observations of its moons? Include any difficulties you had in performing the exercise. Use a separate sheet of paper as necessary.
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