Stellarium – Kepler’s Third Law

Name: ___________________________ Date: ____________________________ Kepler’s Third Law (Stellarium Exercise #8) Tycho Brahe and Johannes Kepler were the first astronomers to study the motions of objects in the sky in a modern way. Tycho, a Dane, kept careful measurements of the locations of the planets in the sky. He then gave his data to Kepler, a German mathematician and astronomer, who, around the year 1600, tried to come up with a simple mathematical formula that would explain how the planets moved in the sky. This was a huge task, since at that time astronomers weren’t even sure whether the planets orbited the Sun or vice versa! But Kepler found a solution. He came up with his famous three laws of planetary motion. Kepler’s laws are: 1. All planets move in elliptical (NOT circular) orbits around the Sun, with the Sun at one of the ellipse’s foci. 2. The nearer the planet is to the Sun, the faster it moves. 3. The average radius of the planet’s orbit (R) and the period of the planet’s orbit (T), are related by the equation: 𝑹𝟑 𝑻𝟐 = 𝑴, where M is the mass of the thing the planet is orbiting around, measured in Solar masses. Of course for any Solar System planet the thing that everything orbits around is the Sun! So for any planet in the Solar System, if we measure its R (how far the planet is from the Sun) in Astronomical Units, and its period T (how long it takes to go around the Sun once) T in years, then M in Kepler’s third law=1. In other words, the mass of the Sun is 1 Solar Mass (obviously!) 𝑹𝟑 Kepler’s Third Law is amazingly powerful! Look at the equation in it. It tells us that 𝟐 is the same for any 𝑻 planet or comet or asteroid or spacecraft that orbits the Sun. In other words, if we know a planet’s distance from the Sun, we can use Kepler’s Third Law to give us its period, or vice versa! Kepler’s laws apply not just to objects circling the Sun, of course, but to any object orbiting any other object due to the force of gravity. The constant C in Kepler’s Third Law may change for different orbital systems, but Kepler’s laws are true anywhere! Kepler’s Third Law is used over and over again in astronomy. Astronomers use it to measure the mass of the central object around which an object orbits. In this assignment we’ll put it to the test! We will measure the planets’ orbital periods and radii, and see if Kepler’s Third Law holds true. PART A We need to measure the periods of some of the planets, just as Tycho did. The easiest way to do this would be to look down on the Solar System from high above it and time how long it takes a planet to go around the Sun once. Of course, this is impossible to do in the real world, but it is easy to do in Stellarium! Certainly, Tycho couldn’t have looked down on the Solar System from above – he was stuck here on Earth! Tycho used a slightly more complex, but very clever way of measuring the planets’ periods. Since we have Stellarium, we’ll just pretend we have a spaceship that allows us to speed out above the Solar System and look down on it. Start Stellarium. If they are on, turn off the Ground, the Atmosphere and the Fog by pressing G, A and F. Open the Search window and search for Solar System Observer. Press Enter. You are now looking at a point high above the Solar System, directly above the Ecliptic plane. To go to that point, or to any point in the Solar System that you have selected, press CTRL-G. Do that now. Now you are standing at that point high above the Solar System! Now look back at the Sun by opening the Search window again, entering Sun and pressing Enter. You are now looking down at the Sun from your position high above it. 4/8/2024 1 Let’s display the Planets and their orbits. Do this by opening the Sky and Viewing Options window and, in the Planets and Satellites section, checking the three boxes next to Show Solar System Objects, Show Planet Markers and Show Planet Orbits. Also put a check in the box next to Labels and Markers, and drag the slider next to Labels and Markers all the way to the right! Finally, in the Stars section, un-check the Dynamic Eye Adaptation box. Close the Sky and Viewing Options window and zoom in until the orbit of Mars fills the screen. You should see the first four planets, Mercury, Venus, Earth and Mars, each with their orbit around the Sun displayed in red! You may also see the orbits of several of the largest Asteroids, for example Vesta, also marked in red. Press L several times, making time run faster and faster, until you can see the planets move around in their orbits. Note how the planets move. You will have to zoom in and out to see all the planets – not just the first four, and you may have to press L a few more times to let time go by faster. • Which way do the planets move around the Sun – clockwise or counter-clockwise? __________________ • Are there any exceptions? ________________________________________________________________ • Does Kepler’s Second law seem to hold true, at least qualitatively, from the way the planets move? In other words, which planets would you expect to move fastest or slowest? What is the general rule about how fast planets move around the Sun compared to how far they are from the Sun? Is this what you see? ___________________________________________________________________________________ Press the 7 key. This key stops time. Now zoom in until the orbit of Mercury fills the screen. Mark the position of the planet Mercury on your computer screen with a small piece of scotch tape or a Post-It note. • What is the date? (Look in the Information bar at the bottom of the screen) ______________________ Enter this date as the Start Date for Mercury in Table 1 on the next page. Now press L about six times to speed up time. You should see Mercury moving in its orbit, one day at a time, as it goes around the Sun. Press 7 to stop Mercury when it returns to its original position, which you marked on screen. You may have to step time forward and backward by one day at a time (using the – and = keys), to get the planet back exactly to where it started. • Now what is the date? __________________________________ Enter this date as the End Date for Mercury in Table 1. Now calculate how many days passed between Mercury’s Start Date and Mercury’s End Date? The result is Mercury’s Orbital Period, T, measured in days. Enter it in Table 1. Now we need to do the same for all the other planets. Stop time by pressing 7. Zoom out until Venus’ orbit just fits in your screen. Again mark Venus’ position on the screen and note the date in Table 1. Let time pass as fast as you want by pressing the Increase Time Speed button (or by pressing L) a few times – but be prepared to hit 7 when you reach the end of the orbit! Stop when Venus has gone through one full orbit. Enter the End Date in Table 1, just as you did for Mercury. Follow the same procedure for the other planets. NOTE: as you can see, the further you get from the Sun, the slower the speed of the planets – that’s Kepler’s 2nd Law!. You may have to press L a few more times with each planet to make time run faster, in order to get through a complete orbit in a reasonable time! Enter all your data in Table 1. 4/8/2024 2 For each planet, calculate its orbital period in days, just as you did for Mercury. You need to be careful, though, to count the days correctly! For example, if a planet takes from September 5, 2008 until January 21, 2011 to go around the Sun once, that would be two years, four months (to make things easier, just assume that each month has 30 days) and 16 days, or 365+365+30+30+30+30+16 = 866 days. Enter the results for Orbital Periods in Table 1. Table 1 Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Start Date End Date Orbital Period, T (days) Now we need to measure each planet’s orbital radius. To find each planet’s orbital radius, we’ll travel to the Sun, and then measure the distance to each of the planets from the Sun’s point of view. Stellarium will tell us the distance to the planet – it’s orbital radius at that moment. Click on the Sun, and then press CTRL-G to travel to it. Now open the Search window and type in Mercury and press Enter. Mercury is centered and selected. Look for the distance from the Sun to Mercury (in AU’s) – it’s listed in the information at the upper left corner of the screen, in the line that says “DISTANCE.” Enter the distance to Mercury in the second column of Table 2 below. Look at each planet in turn, and enter each planet’s orbital radius in the 2nd column of Table 2 below. KEEP ONLY TWO DECIMAL PLACES FOR EACH MEASUREMENT! Table 2 Planet Orbital Radius (R) (AU) Orbital Period (T) (years) R3 (AU3) T2 (years2) Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Now convert each planet’s Orbital Period from days into years, by dividing the Orbital Period from Table 1 by 365. Enter the result for each planet in the 3rd column of Table 2 – Orbital Period in years. Now calculate R3 for each planet, by cubing the Orbital Radius (R3 = R x R x R). Enter this result for each planet in the 4th column of Table 2. Finally, calculate T2 for each planet by squaring the Orbital Period (in years) that you entered for each planet in Table 2. Put the result for each planet in the last column in Table 2. 4/8/2024 3 PART B 𝑅3 Now, according to Kepler, 2 = 1 for any planet, if we measure R in astronomical units and T in years. 𝑇 Let’s see if this is true. Use the data from Table 2 to fill in Table 3 below. For each planet, divide its value of R3 from Table 2 by its value of T2 in Table 2 and call that value “M” for each planet. Enter the value of M for each planet in Table 3. Keep two decimal places in each calculated value of M. Table 3 M (=R3/T2) Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune 𝑅3 If Kepler was right, all these values of M (remember, 𝑀 = 𝑇 2 ) should be the same – they should all roughly equal 1. • Are they all close to 1? _____________________________________ PART C This part involves a little math, but don’t panic! Often in science we do experiments over and over again and measure some quantity many times, and then take the average value, hoping to average out any random errors. In Table 3 we’ve calculated M eight different times. Unless you made perfect observations, your values of M will not all be exactly the same. What we can do is calculate a precise number that tells us how close to each other all our values of M are. This number is called the standard deviation. The lower the standard deviation, the closer all your answers are to each other. For example, if all our values of M were exactly the same, the standard deviation would be zero. That would mean we were very careful experimenters! If the values of M were all the same and were all equal to 1, that would also be a very good indication that Kepler was right! To calculate the standard deviation, first we need to calculate your average value of M. Add up all the values of M in Table 3, and divide the result by 8 (since we measured eight planets). This is the average value ̅. of M. We’ll call it 𝑀 ̅ =_______________________________. 𝑀 ̅ . For each value of M, we call this Now we need to calculate how far each value of M is from 𝑀 difference “𝛥𝑀 .” Calculate this number for each planet and fill in Table 4 on the next page. 4/8/2024 4 Table 4 Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune ̅ 𝜟𝑴 = 𝑴 − 𝑴 Now we’re ready to calculate the standard deviation. The standard deviation is given by 𝟏 𝝈 = √ 𝜮(∆𝐌𝟐 ) 𝑵 Where N is the number of times we made the measurement of M – in this case, eight. What this equation says is that we SQUARE each 𝛥𝑀 for each planet, then ADD UP all those squared numbers (the Σ means “add up”), then divide by our value of N, which is 8, and then take the square root of the result. • What is your standard deviation, σ? ________________________________. Remember, the closer the standard deviation is to zero, the more all your values are M are nearly the same. If the expected value of M (in this case, 1), is within one standard deviation of your average value, then ̅ − 1 is less than σ, then you’ve we can say that they are, for all intents and purposes, equal. In other words, if 𝑀 proved Kepler correct! • Was Kepler correct, according to your data? _____________________________________ PART D For each of these questions, use Kepler’s Third Law, as we did above, to get the answer. REMEMBER, CONVERT ALL ORBITAL RADII TO AU’S AND ALL ORBITAL PERIODS TO YEARS! • Suppose you wanted to look for an asteroid. Asteroids are clustered in a band between Mars and Jupiter, at around 4 AU. How long would it take an asteroid at this location to circle the Sun? ________________. • If we found a planet that took 1,000 Earth years to go around the Sun, how far would it be from the Sun? ______________________________________________. 4/8/2024 5 Write a brief conclusion summarizing your results and what you learned in this assignment. Use the back of this page if you need more room. ___________________________________________________________________________________ ___________________________________________________________________________________ ___________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ 4/8/2024 6

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