# Lab 6: The Planets and Kepler

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1 Lab 6: The Planets and Kepler The Motion of the Planets part I 1. Morning and Evening Stars. Start up Stellarium, and check to see if you have the Angle Tool installed it looks like a sideways A ( ) in the bottom menu. If not, go to the configuration window (F2), select the Plugins tab, choose Angle Measure and check the box to Load at startup, then restart Stellarium. Once Stellarium is running correctly, set your location to Amherst today before sunset, and turn off the atmosphere (A), but not the ground. a. Where is Venus today relative to the Sun? (If it s not labeled, hit P to turn on planet labels.) Hit the = key repeatedly to advance by a solar day ( key to go backward), and describe how Venus s position changes in the weeks and months ahead. A b. How long is it before Venus gets close to the Sun? Switch to just after sunrise and describe how Venus shifts again. How long is it in the morning sky? c. Now take a look at Mercury. When are the next two times it will be highest in the evening sky just after sunset? 2. The Sizes of Orbits. a. In Stellarium, go to the year Estimate when Venus is the farthest from the Sun when the Sun has just risen and estimate when Venus is the farthest from the Sun when the Sun has just set. Write the dates in the first two rows below. Then, use the angle tool to measure the angle between the planet and the Sun and determine the location of Venus relative to the Sun. Click on Venus and find its distance (from Earth NOT the Sun)

2 Date Elongation (deg) Location (right/left of Sun) Distance from Earth (AU) Phase January 31, 2004 October 15, 2004 b. What are the elongation, location, and distance of Venus on January 31, 2004 and October 15, 2004? c. Draw a diagram of Venus at each date and determine its phase as seen from Earth. Assume 1AU = 10 cm.

3 d. Measure each Sun-Venus and Earth-Venus distances. What is the difference between the two sets? What should the trajectory of Venus look like? e. Now find the elongation, phase, and distance for June 8, 2004 and March 29, How does this change the orbit? f. Estimate the size of Venus s orbit compared to Earth s orbit. 3. The Motion of Mars. Compared to Mercury and Venus, Mars has a very different set of motions on the sky. a. Set Stellarium to July 1, 2018 with the atmosphere (A) and ground (G) turned off. Switch your view to equatorial on the bottom menu (CTRL- M), and now follow Mars to see when it will next be bright. In the magnitude listing for Mars, the more negative the magnitude, the brighter it is. What date will this be? We ll discuss in class what this magnitude means. b. Center your view on the constellation Virgo as in the star chart shown here. Starting once a month in January 2016, plot the position of Mars in the star chart until October What is the motion of Mars during this time?

4 N E W S The Motion of the Planets part II Important as well are models of the solar system, which have developed considerably over time. Start at Review the Geocentric Model background material. The simulation of Ptolemy s model demonstrates the dominant model when Copernicus presented his heliocentric model. Thoroughly review the Heliocentric Model background material. Look at the Animation of the Copernican Solar System on the Heliocentricism page. Question 1: What relationship do you notice between how fast a planet moves in its orbit and its distance from the Sun?

5 Open the Elongations and Configurations page Question 2: The table below concerns various elongation configurations for a hypothetical superior planet. Complete any missing elongations, terminology, or lettered labels on the drawing where the Sun and Earth are shown. Use the following figures to check your answers. Location Elongation Term Earth A B C Superior Conjunction Inferior Conjunction Sun D West 20 XXX C Question 3: The table below concerns various elongation configurations for a hypothetical inferior planet. Complete any missing elongations, terminology, or lettered labels on the drawing where the Sun and Earth are shown. Use Figures 2 and 3 to check your answers. Location Elongation Term A 180 B C Western Quadrature Sun Earth D East 120 XXX C

6 Simulator Exercises Open up the Planetary Configurations Simulator and complete the following exercises. Question 4: In this exercise we will measure the synodic period of Mercury. Set the observer s planet to Earth and the target planet to Mercury. The synodic period of a planet is the time it takes to go from one elongation configuration to the next occurrence of that same configuration. However, it makes sense to use an easily recognized configuration like superior conjunction. Drag a planet (or the timeline) until Mercury is at superior conjunction. Now zero the counter, click start animation, and observe the counter. conjunction. A synodic period is that time until Mercury is once again at superior What is the synodic period of Mercury? Question 5: In the previous exercise superior conjunction was used as the reference configuration, but in practice it is not the best elongation configuration to use. Explain why. What is the best elongation configuration to use? (Hint: when is an inferior planet easiest to observe in the sky?) Do you get the same result for the synodic period you got in Question 4? Question 6: Use greatest elongation as the reference configuration to calculate the synodic period of Venus. (Be careful. There are two different occurrences of greatest elongation for an inferior planet: eastern and western.) Also, record the value of the greatest elongation of Venus Synodic period of Venus: Greatest elongation of Venus: What general trend do you notice between an inferior planet's distance from the Earth and its synodic period?

7 Question 7: Now use the simulator to find the value of Mercury's greatest elongation. Greatest elongation of Mercury: Compare the values of greatest elongation for Mercury and Venus. What relationship do you notice between the value of greatest elongation of a planet and its distance from the Sun? Can you create a hypothetical 3 rd inferior planet in the simulator to check your reasoning? Question 8: Now we will measure the synodic period of Mars. As before, set Mars up in a particular elongation configuration, zero the counter, and then animate the simulator again to see how long it takes Mars to return to the same configuration. Synodic period of Mars: Question 9: Just as with superior conjunction in Question 5, conjunction is not the best configuration to observe a superior planet in the sky. Explain why this is and explain which configuration is best for observing a superior planet. Measure the synodic periods of Jupiter and Saturn. Synodic period of Jupiter: Synodic period of Saturn:

8 Question 10: Look over the synodic periods of the superior planets. Is there a trend? What value does the synodic period of a superior planet approach as we consider planets farther and farther away from Earth? Explain this trend. Question 11: Compare your answer above and your answer to the last part of Question 6, and then state a relationship between a planet s synodic period and its distance from Earth that is valid for both inferior and superior planets.

9 Kepler s Laws The first part of the lab focuses on making the great laws of orbits come to life, and the role of energy in that. Start at Answer the following questions after reviewing the Kepler's Laws and Planetary Motion background page. Question 12: Draw a line connecting each law on the left with a description of it on the right. Kepler s 1 st Law only a force acting on an object can change its motion Kepler s 2 nd Law planets move faster when close to the sun Kepler s 3 rd Law planets orbit the sun in elliptical paths Newton s 1 st Law planets with large orbits take a long time to complete an orbit Question 13: When written as P 2 = a 3 Kepler's 3rd Law (with P in years and a in AU) is applicable to a) any object orbiting our sun. b) any object orbiting any star. c) any object orbiting any other object. Question 14: The ellipse to the right has an eccentricity of about a) 0.25 b) 0.5 c) 0.75 d) 0.9

10 Question 15: For a planet in an elliptical orbit to sweep out equal areas in equal amounts of time it must a) move slowest when near the sun. b) move fastest when near the sun. c) move at the same speed at all times. d) have a perfectly circular orbit. Answer the following question after reviewing the Newton and Planetary Motion background page. Question 16: If a planet is twice as far from the sun at aphelion (farthest point) than at perihelion (nearest point), then the strength of the gravitational force at aphelion will be as it is at perihelion. a) four times as much b) twice as much c) the same d) one half as much e) one quarter as much Kepler s 1st Law Go back to the main page and launch the NAAP Planetary Orbit Simulator. Open the Kepler s 1 st Law tab if it is not already (it s open by default). Enable all 5 check boxes. The white dot is the simulated planet. One can click on it and drag it around. Change the size of the orbit with the semimajor axis slider. Note how the background grid indicates change in scale while the displayed orbit size remains the same. Change the eccentricity and note how it affects the shape of the orbit. Tip: You can change the value of a slider by clicking on the slider bar or by entering a number in the value box. Be aware that the ranges of several parameters are limited by practical issues that occur when creating a simulator rather than any true physical limitations. We have limited the semi-major axis to 50 AU since that covers most of the objects in which we are interested in our solar system and have limited eccentricity to 0.7 since the ellipses would be hard to fit on the screen for larger values. Note that the semi-major axis is

11 aligned horizontally for all elliptical orbits created in this simulator, where they are randomly aligned in our solar system. Animate the simulated planet. You may need to increase the animation rate for very large orbits or decrease it for small ones. The planetary presets set the simulated planet s parameters to those like our solar system s planets. Explore these options. Question 17: For what eccentricity is the secondary focus (which is usually empty) located at the sun? What is the shape of this orbit? Question 18: Create an orbit with a = 20 AU and e = 0. Drag the planet first to the far left of the ellipse and then to the far right. What are the values of r 1 and r 2 at these locations? r 1 (AU) r 2 (AU) Far Left Far Right Question 19: Create an orbit with a = 20 AU and e = 0.5. Drag the planet first to the far left of the ellipse and then to the far right. What are the values of r 1 and r 2 at these locations? r 1 (AU) r 2 (AU) Far Left Far Right Question 20: For the ellipse with a = 20 AU and e = 0.5, can you find a point in the orbit where r 1 and r 2 are equal? Sketch the ellipse, the location of this point, and r 1 and r 2 in the space below.

12 Question 21: What is the value of the sum of r 1 and r 2 and how does it relate to the ellipse properties? Is this true for all ellipses? Question 22: It is easy to create an ellipse using a loop of string and two thumbtacks. The string is first stretched over the thumbtacks which act as foci. The string is then pulled tight using the pencil which can then trace out the ellipse. Assume that you wish to draw an ellipse with a semi-major axis of a = 20 cm and e = 0.5. Using what you have learned earlier in this lab, what would be the appropriate distances for a) the separation of the thumbtacks and b) the length of the string? Please fully explain how you determine these values. Hint: Using the equation for eccentricity in the Kepler's Laws of Planetary Motion section may help.

13 Kepler s 2nd Law Use the clear optional features button to remove the 1st Law features. Open the Kepler's 2nd Law tab. Press the start sweeping button. Adjust the semimajor axis and animation rate so that the planet moves at a reasonable speed. Adjust the size of the sweep using the adjust size slider. Click and drag the sweep segment around. Note how the shape of the sweep segment changes, but the area does not. Add more sweeps. Erase all sweeps with the erase sweeps button. The sweep continuously check box will cause sweeps to be created continuously when sweeping. Test this option. Question 23: Erase all sweeps and create an ellipse with a = 1 AU and e = 0. Set the fractional sweep size to one-twelfth of the period. Drag the sweep segment around. Does its size or shape change? Question 24: Leave the semi-major axis at a = 1 AU and change the eccentricity to e = 0.5. Drag the sweep segment around and note that its size and shape change. Where is the sweep segment the skinniest? Where is it the fattest? Where is the planet when it is sweeping out each of these segments? (What names do astronomers use for these positions?) Question 25: What eccentricity in the simulator gives the greatest variation of sweep segment shape? Question 26: Halley s comet has a semimajor axis of about 18.5 AU, a period of 76 years, and an eccentricity of about 0.97 (so Halley s orbit cannot be shown in this

14 Object P (years) a (AU) e P 2 a 3 Earth 1.00 Mars 1.52 Ceres Chiron simulator.) The orbit of Halley s Comet, the Earth s Orbit, and the Sun are shown in the diagram below (not exactly to scale). Based upon what you know about Kepler s 2 nd Law, explain why we can only see the comet for about 6 months every orbit (76 years)? Kepler s 3 rd Law Use the clear optional features button to remove the 2nd Law features. Open the Kepler's 3rd Law tab. Question 27: Use the simulator to complete the table below. Question 28: As the size of a planet s orbit increases, what happens to its period? Question 29: Start with the Earth s orbit and change the eccentricity to 0.6. Does changing the eccentricity change the period of the planet?

15 Newtonian Features Important: Use the clear optional features button to remove other features. Open the Newtonian features tab. Click both show vector boxes to show both the velocity and the acceleration of the planet. Observe the direction and length of the arrows. The length is proportional to the values of the vector in the plot. Question 30: The acceleration vector is always pointing towards what object in the simulator? Question 31: Create an ellipse with a = 5 AU and e = 0.5. For each marked location on the plot below indicate a) whether the velocity is increasing or decreasing at the point in the orbit (by circling the appropriate arrow) and b) the angle θ between the velocity and acceleration vectors. Note that one is completed for you. θ = 61º 6161 θ = θ = θ = θ = θ = θ = θ =

16 Question 32: Where do the maximum and minimum values of velocity occur in the orbit? Can you describe a general rule which identifies where in the orbit velocity is increasing and where it is decreasing? What is the angle between the velocity and acceleration vectors at these times? Astronomers refer to planets in their orbits as forever falling into the sun. There is an attractive gravitational force between the sun and a planet. By Newton s 3 rd law it is equal in magnitude for both objects. However, because the planet is so much less massive than the sun, the resulting acceleration (from Newton s 2 nd law) is much larger. Acceleration is defined as the change in velocity both of which are vector quantities. Thus, acceleration continually changes the magnitude and direction of velocity. As long as the angle between acceleration and velocity is less than 90, the magnitude of velocity will increase. While Kepler s laws are largely descriptive of what planet s do, Newton s laws allow us to describe the nature of an orbit in fundamental physical laws!

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