Lab 3 Kepler 3rd law (1)

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College of the Desert *

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A003

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Astronomy

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Apr 29, 2024

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docx

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Kepler's Third Law Student Name: Objectives This exercise will help to demonstrate and verify Kepler's third law. As Kepler did, using Tycho Brahe's data, you will measure the periods (P) of the planets known in Kepler's time, and calculate their semimajor axis (a). You will then compare your calculations to current known values.  Introduction The early part of the 17th century was an extremely important time for science, particularly Astronomy. In 1609, Galileo Galilei was the first person to look through the newly invented telescope towards the heavens. His discoveries with this new piece of technology included, sunspots, lunar mountains, and of course, the four largest moons orbiting Jupiter.  At about the same time, Johannes Kepler, using the observational data of Tycho Brahe and their shared belief in the heliocentric model of the solar system, was the first to develop mathematical relationships describing planetary motion. These relationships are now known as Kepler's Laws of Planetary motion. Galileo was able to determine that the four moons he discovered around Jupiter obeyed Kepler's 3 rd  Law.  In this exercise, we will look at the naked eye planets and their periods to verify Kepler's 3 rd  Law.  Kepler's 3 rd  Law can be stated as:  o The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. P 2   ∝ a 3   (P=orbital period in years and a=semi-major axis of orbit in A.U.)  Equation 1. 1
Step 1. A. Start Stellarium. It should be in the default configuration you setup in the Using Stellarium exercise. You should be viewing to your South and set the program to full screen. B. From your home location, bring up the Date/Time Window and set the date to 2010/03/01 and time to 06:00:00 . C. Leave the Date/Time Window open and in the upper right hand corner. D. Turn off both the Ground button and the atmosphere button. Step 2. A. Click on the search button on the horizontal bar and type “solar system observer”, then press search. Press “Ctrl” and then the letter “G” B. Use the Search Window to locate the Sun. This will place you above the Sun's north pole, unlike Tycho and Kepler who had to observe from the Earth. C. Click on the “sky and viewing option” on the vertical bar, go to the SSO, check show planet orbits, show planet marker, and move the slider of labels and markers all the way to the right. D. Zoom in until the FOV is about 0.34°. The inferior planets and Earth should be visible (with white circles around them and their names visible to the upper right of each planet) Step 3. INFERIOR PLANETS A. Set the date to current date and time to current time. B. Zoom in until you see the whole orbit of Mercury. Watch Mercury as it orbit around the sun by pressing “L” a couple of times. C. Stop the animation. Record the Date/Time and place your finger on the screen where Mercury is. Increase the month value by pressing the up arrow on top of the hour section until Mercury has made one complete orbit. D. You can fine tune your answer by adjusting the day value in the Date/Time Window to get the planet closer to its original location and add/subtract the adjustment days. E. Wait until Mercury comes back to your finger position. 2
F. Record the number days for the complete orbit in Table 1. G. Reset the date to current date and time and repeat Step 3, sub steps A through C, for the planet Venus. Step 4. SUPERIOR PLANETS H. Zoom out until the orbit of Mars is visible (with a white circle around it and its name visible to the upper right) I. Reset the date and time to current date and time and repeat Step 3, sub steps A through C, for the planet Mars. Try to fine tune your calculation by adjusting the day section. J. Zoom out until the orbit of Jupiter, Saturn, Uranus, and Neptune are visible (with white circles around them and their names visible to the upper right of each planet) K. Reset the date and time to current date and time. While watching Jupiter and/or Saturn, use the Date/Time Window to increase the year value until each one has made one complete orbit. Try to fine tune your calculation by adjusting the day section. L. Record the number of years for one complete orbit in Table 1. Step 5. A. Convert P for Mercury and Venus into years (#days/365.25) and record in Table 1. B. The actual period of the planet is reported in Table 1. Determine how accurate your calculation is by determining the error % from the actual values of the orbital period. % Error = ( Measured value − Actualvalue Actualvalue ) × 100 C. Report your result in Table 1. D. Square the orbital period in years (P 2 ) for all of the planets and record in Table 2. E. Calculate the semimajor axis (a) for planets and record in Table 2. Recall that for planets orbiting the Sun, P 2  = a 3  or a = (P 2 ) 1/3 F. Calculate a 3 in table 2. 3
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