An Activity Guide to Accompany PLANET TRACKER Version 4.1 by David Chandler and Michael Zeilik (c) 1993 by David Chandler and Michael Zeilik This software and activity guide are specifically designed to be used with CONCEPTUAL ASTRONOMY (by M. Zeilik, 1st edition, 1993), ASTRONOMY: THE EVOLVING UNIVERSE (by M. Zeilik, 7th edition. 1994), and ASTRONOMY: THE COSMIC PERSPECTIVE (by M. Zeilik and J Gaustad, 2nd edition, 1990), all published by John Wiley & Sons. Those using other texts will find it equally useful. Directions for running the program and producing worksheets are provided within the program itself. Select "Help Topics / Notes" from the main menu to read selected information from the screen or print out all or part of the help file. All computations in the program are based on Keplerian orbits using mean orbital elements as described in Astronomical Algorithms by Jean Meeus. Given the small scale of the charts, the algorithms are perfectly adequate for their intended purposes. They are not suitable for critical uses such as navigation, but they have more than sufficient accuracy for general observational purposes well into the next century. For instructional purposes, to illustrate the patterns of planetary motion, they may be run for several thousand years. The algorithms lose validity and will eventually generate bizarre results if allowed to run tens of thousands of years into the future. Program Overview The authors of "Planet Tracker" are both teachers of astronomy, and the program was designed with the students in mind. "Planet Tracker" is a tool to investigate the laws of planetary motion create printed planet charts and worksheets in a variety of useful formats. The program is sharply focused on concepts of planetary motion that most of our students find difficult to grasp. You can think of the program as a simulation of the "raw" data from which models of planetary motion were derived. The key to the program's usefulness, however, is the way it displays this information. It allows us to see planetary motion from vantage points that were available to Copernicus and Kepler only in their imaginations. You will find that this program is very easy to learn as we have been very selective in its capabilities. All screens have instructions at the bottom. You can leave any screen by hitting the "ESC" key. The collection of exercises and activities that follow is presented only to suggest a few of the many possibilities. This file (ACTIVITY.TXT) is straight ASCII text. You are encouraged to import it into a word processor and modify it to meet your own needs. You may also print it out from the "Main Menu" of Planet Tracker. The exercises are divided into two categories: those using printouts only and those requiring direct student access to computers. We envision that this program will be used in a laboratory or small class setting. If you have enough computers for a small class, you will want to instruct the students in running the animations themselves. In that case, you will need a site license to run the program at the same time on multiple machines. For information on site licenses, which cost an additional $50, contact David Chandler Co., P. O. Box 309, La Verne, CA 91750. However, it is possible to develop "short" activities to use in larger classes. I (M.Z.) have done so with classes of 200 to 300 students at a time. Suggested Exercises (Student access to computers NOT required) 1. Planet Calendar Introduction The simplest use of the heliocentric planet charts is to pin them on the bulletin board and keep track of the positions of the planets from week to week. Text References Conceptual, Sections 1.2 and 1.3; Evolving, Sections 1.2 and 1.4; Cosmic, Sections 1.2 and 1.4. Procedure Use the heliocentric charts as a bulletin board display and update the planet positions daily or weekly. Using transparencies, the orbits can be projected onto butcher paper and traced at a larger scale. With the right adjustments, both the inner and outer planets can be combined into a single diagram. Key Concepts Through daily contact with this display, your students should learn to: => associate the length of a year with the orbit of the earth. => become familiar with the relative speed of the solar system orbits. => notice the high speed of the inner planets compared to the slow movement of the outer planets. => recognize that the planet orbits are not concentric circles. 2. Identifying the Planets in the Sky Using the Zodiac Time Line Introduction The planets are easy to pick out because they are as bright or brighter than any of the stars. Once the planets are found the first time they are easy to pick out on subsequent nights and their motions can be followed. Text References Conceptual, Sections 1.2 and 1.3; Evolving, Sections 1.2 and 1.4; Cosmic, Sections 1.2 and 1.4. Procedure Start by locating the zodiac in the sky. One way to do this is to watch the moon. It follows the zodiac from one night to the next, although it strays farther from the ecliptic than most of the planets. Locate the sun on the zodiac chart. The half of the zodiac to the left of the sun (wrapping around if need be) is the portion visible in the evening. Identify the bright stars that lie along the zodiac. Any bright star out of place in this region of the sky is most likely a planet. Venus is the brightest starlike object in the sky, whenever it is visible. Jupiter is also brighter than any star. Saturn is about the brightness of the brightest stars and is a pale yellow color. Mars can be as bright as Jupiter or as faint as a second magnitude star depending on its distance from the earth. Mercury is as bright as the brightest stars, but it is usually only visible in the twilight. Checking the chart is critical to success in identifying Mercury. If several planets are visible in the evening at once, notice that they, along with the moon (if it is up), form a line close to the ecliptic. Once the planets are identified they are easy targets for small telescopes. Saturn shows its rings in even the smallest telescopes. Jupiter's moons are visible in small telescopes or even binoculars, changing positions from one night to the next. Mars looks like a tiny orange dot, but still clearly different from a star, which appears as a point of bright light. Venus and Mercury show phases but no surface details in a telescope. Crescent phases occur as they are moving retrograde (left to right on the chart), crossing between the earth and the sun. Key Concepts After completing this exercise, students should be able to => gain familiarity with the constellations and bright stars. => recognize planets as bright "starlike" objects that progress through the sky. => gain experience using a telescope. => recognize the difference between a star and a planet in a telescope. 3. Translating Heliocentric Orbit Diagrams to Earth's Perspective Introduction The heliocentric view of the solar system is the simplest way to describe planetary motion, but at first it seems unrelated to the complex way the planets move in the sky from earth's perspective. This exercise helps form a bridge between the two. Text References Conceptual, Sections 2.2 and 2.3; Evolving, Sections 2.4 and 3.2; Cosmic, Sections 2.4 and 3.2. Procedure Print out a heliocentric orbit chart for the current date. Draw an arrow from the earth, passing through the sun, all the way across the page. Mark this arrow with heavy lines. Now draw lines from the earth to each of the planets. Face south and point the sun arrow of the diagram to where the sun is at noon. Hold the paper roughly parallel to the ecliptic plane. Each of the other arrows should now be pointing at its respective planet, but the planets cannot be seen easily in daylight. Now rotate the sheet in the plane of the paper so that the sun arrow points toward the western horizon. Any planets to the left of the sun will still be up after the sun has set. Continuing to rotate the paper until the sun arrow points to the eastern horizon, you can see that any planet to the right of the sun should be up in the morning sky. Key Concepts Many students keep information about 3-dimensional "space" and their direct perceptions of the 2-dimensional "sky" in separate compartments in their minds. Any exercise that links the two will help clarify their understanding of both. 4. Relating the Heliocentric Orbit Diagrams to the Time Lines Introduction The previous exercise uses the heliocentric charts to relate the positions of the planets in their orbits with where they appear in the sky at night. The heliocentric charts can also illuminate the time lines, which in turn clarify how the planets move in the sky. Text References Conceptual, Sections 1.2, 1.3, and 2.3; Evolving, Sections 1.2, 1.4, and 3.2; Cosmic, Sections 1.2, 1.4, and 3.2. Procedure Draw arrows from the earth through the sun and from the earth through each of the planets on a heliocentric chart as in the previous exercise. Measure the angle of each planetary arrow from the arrow pointing at the sun. This tells how many degrees left or right of the sun to look to find each planet. Find the corresponding date on the solar time line and mark it with a dark horizontal line across the page. The total width of the chart represents 360 degrees wrapped around the sky, so each quarter of the chart represents 90 degrees. Mark the location of each planet for that date and compare with the angles measured from the heliocentric chart. They should agree. Now compare the positions with the same date on the zodiac time line. The view may be shifted, but the positions of the planets relative to the sun should be the same. From the heliocentric chart you can find the distance of a planet from the earth, whereas the time lines show the planet's position projected onto the sky. From the heliocentric charts, where would you expect Mars to appear it's brightest? What is the elongation angle at that point? Where would it be on the time lines at that point? Where should you look to find Mars in the sky when it is at its brightest? As a final check, print out a heliocentric chart and block out the data at the top of the page. From an extended zodiac or solar time line the student should be able to figure out the date of the chart to within a few days. Key Concepts The different charts present similar information in different forms. The heliocentric charts preserve the 3-dimensionality and indicate distances, whereas the time lines do not. On the other hand, the times lines better show how the planets move with time. These charts illustrate how similar information can be represented in very different formats. 5. Planetary Aspects Introduction The terms conjunction, greatest elongation, quadrature, and opposition are called planetary aspects. They have to do with the angle between the sun and a planet along the ecliptic. Note: None of Zeilik's textbooks use the term "quadrature" in an effort to keep technical terms to a minimum. You may, however, want to introduce it in your class. Text References Conceptual, Sections 1.2, and 2.3; Evolving, Sections 1.4 and 3.2; Cosmic, Sections 1.4 and 3.2. Procedure On a heliocentric chart of the inner solar system, plot the two points where Venus would be in line with the sun. When Venus is between the earth and sun it is said to be at inferior conjunction. When it is behind the sun it is a superior conjunction. Is Mars ever in conjunction with the sun? Is Mars ever at inferior conjunction with the sun? Would you ever be able to see any planet at either kind of conjunction with the sun? Opposition occurs when a planet is opposite the sun in the sky. Locate the points on the heliocentric chart where Mars would be at opposition. How do you expect the viewing of Mars would be in a telescope when it is near opposition compared with elsewhere in its orbit? Is Venus ever at opposition? What would be the aspect of earth as seen from Venus (if you could see through the clouds on Venus) when Venus is at inferior conjunction? What would be the aspect of earth as seen from Mars when Mars is at opposition? Greatest elongation occurs when the elongation angle of an inferior planet is maximum. Locate the points of greatest elongation for Venus. Does the line from earth to Venus hit tangent to the orbit of Venus? If not, is there a greater angle of elongation for Venus than the one you chose? Quadrature takes place when the elongation of an outer planet is 90 degrees. (The elongations of the inner planets never reach 90 degrees. Why?) Find two places on the orbit of Mars when it is at quadrature. If you were on Mars looking at the earth, in what aspect would the earth be? In what aspect would you see earth from Venus when Venus is at greatest elongation? Finally, identify on the zodiac time line or the solar time line the points where Venus is at superior conjunction, inferior conjunction, and greatest elongation. Identify points where Mars is at quadrature and opposition. Key Concepts Understanding planetary aspects involves spatial perception skills. Some students may need coaching to answer how the earth would appear from some other point of view. Acting out these relationships with the aid of the heliocentric charts can be a significant aid to understanding. 6. Phases of the Planets Introduction Planets have phases just like the moon. Phases are caused by the varying angle of lighting from the sun. Many students misunderstand the phases of the moon; they believe that they are seeing the earth's shadow on the moon (as in a lunar eclipse). Introducing the phases of the planets may be an alternative approach to the same conceptual problem. Text References Conceptual, none; Evolving, Section 4.1 (see Figure 4.5); Cosmic, Section 4.1 (see Figure 4.5). Procedure Find a day on the solar time line (or any other resources of the program) when Venus is closer to the earth than the sun and Mercury is farther from the earth than the sun, or vice versa. Print out a heliocentric chart for that day. For each of the two planets, draw a line from the planet to the sun and from the planet to the earth and extend those lines across the page so they form an angle whose vertex is at the planet. One line represents our line of sight, the other represents the direction of lighting. In a darkened room with a bare light bulb and a tennis ball, place the ball in the position of the planet. Place the bulb so it illuminates the planet along the line from the sun to the planet. Place your eye at the level of the table so that you are looking along the line of sight from the earth toward the planet. Sketch the shape of the illuminated portion of the ball from this perspective. Repeat for the second planet. Move the ball representing the planet to various positions around the bulb representing the sun. When is it gibbous and when is it a crescent? Is it ever truly full when it can be seen from our point of view? Ptolemy thought Venus went in a circle centered at a point between us and the sun. How would the cycle of phases look if that were the case? Would Venus ever have a full phase? Galileo was the first one to observe the phases of Venus with a telescope. How could his observations tell if Ptolemy were correct? Or wrong! Does Mars have changing phases? (Yes!) Does it go through crescent phase? (No!) Key Concepts => Phases are not logically limited to the moon, but rather apply to any object illuminated by an outside source where the direction of lighting changes over time. => Inferior planets 7. Inferior and Superior Planets Introduction Inferior planets are defined as the planets whose orbits lie within the earth's orbit. Superior planets lie outside the earth's orbit. This is the determining factor in how the planets appear to move in the sky. Note: None of Zeilik's textbooks use the terms "inferior" and "superior" in an effort to keep technical terms to a minimum. You may, however, want to introduce them in your class. Text References Conceptual, Section 1.2; Evolving, Section 1.4; Cosmic, Section 1.4. Procedure On the heliocentric charts find the largest elongation angle possible for Mercury and again for Venus. When a planet is to the left of the sun it remains in the evening sky after sunset. Will either Mercury or Venus ever be found in the eastern evening sky or in the western morning sky? Will either planet ever be visible at midnight? Why are Mercury and Venus referred to as morning and evening stars? Does either planet ever go through opposition? Do any of the superior planets ever pass in front of the sun? On the solar time line the paths of the inferior planets behave very differently from the paths of the superior planets. The superior planets move continuously from left to right, pass behind the sun, and wrap around at the opposition point behind the earth. The inferior planets, on the other hand, move right to left behind the sun and left to right as they cross back in front of it from our point of view on earth. Can you explain why they behave oppositely? (Hint: who passes whom in the race among then planets to get around the sun?) Key Concepts => How a planet appears to move depends on whether we view it from outside or inside its orbit. => The earth moves faster than the superior planets and slower than the inferior planets. This affects their apparent direction of motion through the sky. => Superior planets never pass between the earth and the sun and inferior planets never circle to the far side of the earth from the sun. 8. Regularities of Planetary Motion Introduction Planetary motion as observed from the earth is well represented by the zodiac time line. The regularities in the motions on this chart are clues to the motions we would observe from space. Text References Conceptual, Sections 1.2 and 1.3; Evolving, Sections 1.4 and 3.2; Cosmic, Sections 1.4 and 3.2. Procedure Print out a Zodiac Time Line covering at least three years. Find a date when one of the superior planets is at opposition and another is behind the sun. Print out an outer solar system geocentric and matching heliocentric plot for that data. (Use a multiple of ten days as the plotting interval, since the zodiac time line is plotted in multiples of ten days.) When does retrograde motion occur on the zodiac time line? What is occurring on the heliocentric and geocentric charts at that time? When does maximum prograde motion occur? Note the orbit of Mars in particular. How long does it take to move from behind the sun to quadrature (half way to opposition)? How long does it take to move from quadrature to opposition? Compare these times for Jupiter and Saturn as well. What progression do you observe? Plot the points of quadrature for Mars, Jupiter, and Saturn on the heliocentric chart. Are we as off center with respect to Jupiter or Saturn as we are for Mars? Speaking of asymmetries, can you explain why the inferior planets take longer to go from right to left than from left to right across the sun on the time lines? (Consider where the points of greatest elongation are on the heliocentric chart.) Why is Venus' path more asymmetric than Mercury's? 9. Sidereal versus Synodic Periods Introduction How long does it take the hands of a clock to move from one alignment to the next? Consider their angular rates (the reciprocals of their periods). The fast hand moves at 1 rev/h, the slow hand moves at 1/12 rev/h, therefore the fast hand gains on the slow hand at a rate of 11/12 rev/h. Taking the reciprocal of the relative rate gives 12/11 h/rev, or 65.4545 minutes as the time between alignments. If the clock hands were the earth and another planet, the absolute rates would be the sidereal rates and the relative rate would be the synodic rate. The relation in terms of the periods is 1/Te - 1/Tp = 1/Tsyn for superior planets, or 1/Tp - 1/Te = 1/Tsyn for inferior planets. Since our observation point is on the earth, one of the planets participating in the race, it is only the synodic periods of the planets that are directly observable. Given the synodic period of a planet and the sidereal period of the earth (365.25 days), the problem is to find the sidereal period of the planet. Text References Conceptual, Section 2.3; Evolving, Section 3.2 and Focus 3.1; Cosmic, Section 3.2. Procedure Print out either the zodiac or solar time line for several years. Measure synodic periods of each planet by counting the number of days from one alignment with the sun to the next, preferably averaging over several cycles. Calculate the sidereal period for each planet using the formulas above. As a complementary exercise, use a heliocentric chart generated for the day of an alignment between the earth, some other planet, and the sun. Count the days until the next alignment by stepping each planet forward the same number of days. Compute the synodic period from the two sidereal periods using the formula and compare your answer with the direct count. Key Concepts => The sidereal period of a planet is a fundamental astronomical measurement, but it must be found indirectly from earth-based observations. => The measurement of the sidereal period of Mars is the first step in Kepler's method for determining the orbit of Mars. 10. Determining the Orbit of Mars Introduction Determining the orbit of Mars is the key in Kepler's discovery of elliptical orbits. The method consists of finding the position of Mars on pairs of dates 687 days apart, preferably close to opposition. Since the sidereal period of Mars is about 687 days, Mars would be at the same point in its orbit on both days. The earth, however, would be at different locations on those two dates. Constructing the line of sight from the earth to Mars on both dates locates the actual position of Mars in space by triangulation. Text References Conceptual, Section 2.5; Evolving, Section 3.2 (see Figure 3.7); Cosmic, Section 3.2 (see Figure 3.9). Procedure Prepare several time lines (either kind) covering at least two years each starting shortly before an opposition of Mars. Pick pairs of dates 687 days apart and measure the elongation of Mars on each date. If the first date is about 20 to 30 days after an opposition, the other date will be shortly before the next opposition. (Use a scale factor of 360 degrees = 6", which converts to 2.36 degrees per mm. to measure angles horizontally across the time lines.) Make a heliocentric plot of the earth's orbit at five day intervals starting January 1. Use a protractor with the earth orbit plot to lay off the elongation of Mars for each date. Where the lines intersect is one point on Mars' orbit. Repeat for other pairs of dates to find more points on the orbit. Finally, compare the plot with another heliocentric chart that contains the orbit of Mars. Key Concepts => This lab exercise uses the time lines to simulate the work of Tycho Brahe, but it presents the information in more concrete visual form. => The information that is so neatly summarized in tables of planetary data is often not directly observable. It must be deduced by observations from our earth-based perspective. 11. Determining the Orbit of Mercury Introduction Not all greatest elongations of Mercury are created equal. By scanning down the time lines (the solar time line is best for this exercise) it becomes apparent that evening elongations are smallest in the spring and largest in the fall. When we are looking at Mercury at greatest elongation we are looking tangent to its orbit. If we construct enough tangents to Mercury's orbit, the shape of the orbit is revealed. Text References Conceptual, Section 2.3; Evolving, Section 3.2 (see Figure 3.6); Cosmic, Section 3.2 (see Figure 3.8). Procedure Print out several years worth of solar time line and an orbit of the earth at 5 or 10 day intervals starting January 1 without the planets. Using the same angle measuring techniques as in the Mars orbit lab, find the elongations of Mercury at each greatest elongation and plot the lines of sight on the orbit diagram. Compare the results with a heliocentric inner planet chart. Key Concepts This exercise illustrates another example of using indirect methods to determine the nature of our solar system from our limited vantage point on earth. 12. Kepler's Laws Introduction Kepler introduced three laws of planetary motion: 1) Planets move in elliptical orbits with the sun at one focus. 2) A planets sweeps out equal areas in equal times. 3) The square of the period of an orbit is proportional to the cube of its semi-major axis. Text References Conceptual, Section 2.5; Evolving, Section 3.5; Cosmic, Section 3.4. Procedure Before Kepler, the orbits of the planets were thought to be off-centered, or eccentric, circles. Kepler showed that they are actually ellipses, although they are close to circular. See how close the circular approximation actually is by finding the centers of the orbits and drawing the circular approximations to the orbits. To the accuracy of simple drafting tools can you notice a discrepancy between the circles and the orbits? Which planets have the worst fit? If a is the semi-major axis of the ellipse and c is the distance from the center to the focus, the eccentricity is defined by e = c/a. Measure the eccentricities of each of the orbits. To test the equal area law, use pairs of days close enough together that the curvature of the orbit between them is not significant. Use the radius to one point as the base, measure the altitude, and compute the area. Compare for different parts of the orbit. How do the "area sweep rates" compare for different planets?Kepler's third law can be verified directly using a numerical table of orbital elements. An interesting consequence of the third law, assuming circular orbits, is that the orbital speed of a planet is inversely proportional to its orbital radius. The spacing of the dots lessens dramatically with radius in the heliocentric plots. It is simple to check that the speeds drop off inversely with the radius. 13. Distances to the Planets Introduction The heliocentric charts are scale drawings of the solar system. Distances measured on the charts in centimeters are proportional to actual distances. Knowing one distance, the distance from the earth to the sun, any other distance can be determined. Text References Conceptual, Section 10.1; Evolving, Section 13.1; Cosmic, Section 8.2 (see Figure 8.5). Procedure Taking the distance from the earth to the sun to be 93 million miles or 150 million km, find the: a) distances from the sun to the other planets. b) closest approach of Venus and Mars to earth. c) range of distances between Mars and earth. d) ranges of opposition distances of Mars from the earth. Key Concepts This kind of exercise reinforces math skills in proportional thinking, scientific notation, and metric units. 14. How Big Would it Look in a Telescope? Introduction This question arose in class when I (DC) announced an opportunity to view the 1988 opposition of Mars through the telescopes of a local astronomy club. Rather than give them the answer, we used the class time to figure it out together. Text References Conceptual, Section 2.3; Evolving, Sections 3.2, 4.1, and Focus 6.1; Cosmic, Sections 4.1 and 7.1. Procedure All you need to know is the distance to the planet and the size of the planet. The diameter of a circle (measured in cm) representing a planer compared to the distance of the paper it is drawn on (45 cm is typical reading distance) is proportional to the actual size of the planet compared to its distance, as long as the size and distance of the planet are measured in the same units. The dot size computed in this way will be very tiny, as it would appear to the naked eye. Multiply by 100 or so to get an idea of the size of a telescopic image. Different groups in the class could compute the sizes of various planets under different circumstances. A revealing comparison is to calculate the apparent size of a star. Assume the diameter of Alpha Centauri is comparable to that of the sun, about 1.4 million km. It's distance is about 4.5 light years, and light travels 300,000 km/s. From this data, you can be show that the image of such a star on a sheet of paper would be about 1 millionth of a centimeter. Even at a few hundred power the disk would not be discernible. Key Concepts Many students who ask simple quantitative questions do not realize that the answers are within their own power to compute. Helping a student answer their own questions with mathematics they already know can help motivate further study of mathematics. Suggested Exercises for a Laboratory Class (Where students have direct access to computers) Motion Along the Ecliptic 1. Select Mercury and the sun only. Note the dates of several inferior conjunctions (left to right crossing of the sun). What is Mercury's motion at these times relative to the stars? 2. Select Venus and the sun only. Set the vertical exaggeration to 5. At what point in its orbit does Venus stray farthest from the ecliptic? Why? Allow the program to run through many years. (You may have to set the interval to 5 days or more to speed up the program.) What evidence is there for resonances between the orbits of Venus and the earth? Do other planets show similar resonances? Hint: try Mercury! 3. Select Mars and the sun only. Watch Mars goes through a retrograde loop and note the date when it is in the middle of a loop. What is happening in the top view relative to the sun? 4. Starting with Mars, plot one planet at a time. Note how their positions relative to the stars correlates with the position of the sun (compare top and lower views). Note the sizes, elapsed time intervals, and the amount of drift of the retrograde loops across the zodiac. Also note that the shape of a retrograde loop is directly related to its position among the background stars. 5. Find the synodic period of Mercury (or any other planet) in as many different ways as possible. The Julian date makes finding the time interval easy. 6. What happens when the calculation interval is set to 365.25 days? To 365 days? Split Screen Animations 1. Apparent Dome of the Sky versus Zodiac as Seen from Earth Note that in the left display, the sun is fixed. The little white dots traveling around the dome represent stars in their seasonal motion with respect to the sun. On the right, the stars are fixed. This pairing is almost identical to the ecliptic views, but now the sky is wrapped around in a circle. Follow a planet such as Mars and note where the sun is when Mars undergoes retrograde motion in both views. 2. Apparent Dome of the Sky versus Earth and Sun Held Stationary Note the basic similarity of the two views. When you go out every evening at sunset, the earth is beneath you and the sun is at approximately the same position relative to the horizon. The planetary orbits do NOT retrace themselves exactly in the right view. (Why? Because the elliptical orbits are rotating as seen from the earth.) The lack of depth in the left view is dramatic in this comparison. But note that the angular speeds of the planets on the left correlate with their distance from the earth on the right. For the outer planets, note the large variation of the distance of Mars from the earth compared to the relatively smaller variations of Jupiter and Saturn. That is why Mars varies so much in brightness in comparison. 3. Heliocentric versus Geocentric Run first without trails and note the similarity of the motions. Add the trails and note the great difference in the paths. Run Mars alone. Note that the loops on the right correspond to oppositions on the left. 4. Zodiac versus Geocentric Run Mars with a large calculation interval over hundreds of years. Not all the oppositions are the same. Find the "most favorable" oppositions in terms of time of the year and location among the constellations. Determine the next opposition of Mars and the most favorable one within your lifetime. 5. Heliocentric versus Earth and Sun Held Stationary The display on the right is a rotating frame of reference. Add trails. The inner planets move at slower rate compared to the left view. But the outer planets reverse their direction of motion! (Why? Because the earth's angular speed is subtracted from that of the other planets. Since the outer planets move slower than the earth, as viewed from the sun, in the right view they appear to go backwards.) 6. Heliocentric versus Zodiac Note that the "sky dome" appears to rotate synchronously in these views. The stars are "fixed", and all local motions are relative to them. Find a time when Mars and Mercury are in conjunction with each other. What is happening with these planets in the heliocentric view? (Specific example for in-class or lab activities) Note to the Instructor: For this activity, you will need to run a Zodiacal Time Line and a Solar Time Line selecting Mercury and Mars for a period of about two years each. (To get a continuous chart, note the last date printed at the end of the first page and select it as the starting date for the second year. Trim the heading from the second chart and fit them together.) Start with the date of the class or some other convenient starting date. Be sure that the time span includes one retrograde motion of Mars. Astronomy 101 Major Motions of the Planets (c) 1993 M. Zeilik Name:_______________________________Class/section:______________ Purpose: To describe the positions of the sun and selected, naked-eye planets along the ecliptic and infer general patterns in their motions from graphs. Predictions: Write your responses to these questions on this page before you look at the graphs: 1) In what one way does the motion of the sun along the ecliptic resemble that of the planets? 2) In what one way does the motion of the sun along the ecliptic differ from that of the planets? Materials: Pencil, ruler, calculators, and graphs of planetary positions. Text references: Section 1.2, Conceptual (1/e); Section 1.4, Evolving (7/e) Procedure: 1. Examine carefully the graphs of the positions of the planets. They show time running down the page, and the dates at the left are given in 10-day intervals (the year is given on the right). Their vertical extent covers a little more than a year, and their horizontal span is 360 degrees. The bolder line is the sun. The angular distance between the sun and a planet on any given date is called the planet's elongation. The planetary positions run across the page horizontally through the constellations of the zodiac (see the diagram at the top of the page; each constellation of the zodiac is drawn and labeled). Once around the zodiac is a complete circle of 360 degrees. The solid line through the center of the constellations is the path of the sun in the sky relative to the star--the ecliptic. The legend is in the upper right corner for the planets plotted and the year. Note that east is to the left, and west is to the right. Seek out patterns in the motions you see for Mercury and Mars. Answer the following questions on this page: 3) Which planet always appears close to the sun in angle (elongation)? 4) How frequently does Mercury move from one side of the sun to the other? Is this change more or less frequent for Mars? 5) For Mercury, does its maximum angular distance (maximum elongations) from the sun remain constant or does it vary? 6) How does the sun's motion differ in this plot? In what direction does the sun move relative to the stars? How long does it take to move 360 degrees? 7) How does the sun's resemble in general that of Mercury and Mars 8) How does the sun's motion differ from that of Mercury and Mars? 9) Note the cases when the planet's motion is toward the right (west) rather than to the east. What is happening to the planets in the sky during these times?