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Mapping the Motion of a Planet in the Sky

The Wandering Stars

Summary: Track the position of one or several planets that is in the evening sky (could be Venus, Mars, Jupiter, or Saturn) against the starry background during the semester.

Needed Supplies: Logbook, pencils, compass (of the "pointy end and pencil end" variety, not of the "show me magnetic north" variety), crossbow, and a blown up photocopy of the region of the SC-001 star chart where the planet is located in the sky.

General Description:The word ``planet'' is derived from the Greek word for "wanderer,'' as ancient observers realized that several "stars'' moved against the background of "fixed'' stars. Because the planets orbit the Sun nearly all in the same plane (i.e. the solar system is "flat"), the planets always stay close to the ecliptic, something that you will observe for yourself. With the help of star charts and the crossbow, you will track the position of a planet among the fixed stars as it changes due to the orbital motion of both the Earth and the planet during a period of 2-3 months.

Procedure:

Note: The planets move relatively slowly in the sky; for some planets at some times, you may notice no positional change for several days or even weeks; for other planets or these same planets at different times, you may notice positional changes every day. For your chosen planet, note its position once every week (or more often if the planet moves rapidly, like Venus) throughout the entire semester. Start as soon as possible! These observations are simple and take little time once you know how to go about it. Do some observations outside of the lab period if the weather does not cooperate during scheduled labs.

1. Use a blown up photocopy of the star chart (200%) centered on the part of the sky where the planet is located. You will use this to chart its motion during the semester. Keep it in your logbook and always bring it with you at the lab!

2. Measure the angular distance of the planet from at least two, preferably three, stars. You will get better results if your stars are in different directions from the planet. (E.g. if one star is primarily north or south of the planet, be sure to choose another which is primarily east or west of the planet.) You must choose stars that are plotted on the 200% blow-up of the star chart. You will get better measurements if you correctly use the crosswbows during the lab, but can get adequate results using the fist-and-finger rules. See Measuring Angular Distances on the Sky for more information. If you are using the crossbow, try to measure to the nearest 0.25^o (that's 1/8" on the crossbow scale).

   Note the date, sky conditions (such as visibility of stars near the planet), and (importantly) the names of the stars and measurement method (fist-and-finger or crossbow) for each observation. You do not have to use the same offset stars every time! Stars closer to the planet will usually give you better results. Use the closest stars that you can see and that are plotted on the star maps. On worse nights, you may need to resort to more distant stars.

3. Plot the position of the planet on the map as accurately as you can. With a compass, you can triangulate the position of the planet given the two or three different angular distances to stars you've measured. Use the declination scale on the star map to set the width of your compass to the right size. Place the point of the compass on the offset star, and draw a small, light arc on the map around where you believe the planet to be. Repeat this for the other two stars. Where the arcs intersect (or as close as possible, as it is unlikely that all three arcs will perfectly intersect), draw a dot to indicate the position of the planet. Date this dot.

   Note: Star charts are like geographical maps. They are obtained by projecting a curved surface (a portion of a sphere) on a flat surface (sheet of paper). This inevitably leads to distortions, where the scale is not uniform across the entire map. This is why Greenland appears much larger than South America on many maps of the Earth, while in fact it is much smaller. For objects moving along the ecliptic (planets and the moon), the worst error you will get is about 10% (for example 2^o if the distance from the star to the planet is 20^o). For your SC-001 star chart, distances north-south are always accurate, while distances east-west are distorted. The closer your offset stars are to the planet you are measuring, the less you will be affected by this error. For stars predominantly to the east or west of the planet, you will do better if they are closer to the equator.

4. The ecliptic is shown on the star chart (wavy curve). Notice that it is marked with a scale in degrees and with the days of the year. The dates indicate the position of the Sun in the sky throughout the year. Use this to write down, for each day you measure the position of the planet, 1) The position of the planet along the ecliptic (in degrees) and 2) the position of the Sun. The difference between the two gives you the Sun-Earth-planet angle, as it would look it you were looking down on the solar system.

Example

Suppose that on April 20, 2001, you observed that the planet is located very near the star Regulus, in constellation Leo. Its position along the ecliptic (its ecliptic longitude) would be 150^o. On that date, the ecliptic longitude of the Sun is 30^o. The Sun-Earth-planet angle is 150^o-30^o=120^o.

Lab Write-up

Your write-up should consist of:

1. The star map with all recorded positions labeled by date.

2. An observation log showing dates, times, position measurements relative to stars, and personal comments as you see fit.

3. Make a table as follows:

   Date	Sun's Ecliptic Longitude	Planet's Ecliptic Longitude	Sun-Earth-Planet Angle	Earth's Ecliptic Longitude
   2001-Sep-11	168o	272o	104o	348o

   The S-E-P angle is given by the ecliptic longitude of the planet minus that of the Sun. The Earth's ecliptic longitude is opposite that of the Sun: E.E.L.=S.E.L. + 180^o.

4. Make a plot of the Sun-Earth-Planet angle as a function of the date (time in days).

5. Make a diagram of the solar system as seen from above, showing the orbit of the planet you have tracked during the semester as well as the orbit of the Earth. Your diagram must be drawn to scale. Assume that the orbits are circular and use the values given in the following table for the orbit dimensions. The semi-major axis is the radius of the orbit, given here in Astronomical Units (the Earth-Sun distance). Define a direction (from the Sun) to be 0^o of Ecliptic longitude. Using a protractor, plot the position of the Earth on its orbit (the Earth's ecliptic longitude) for each date of observation along its orbit (the ecliptic longitude increases counterclockwise). Then plot the position of the planet on its orbit for each date, using the Sun-Earth-Planet angle you have tabulated. Obviously, that last angle is centered on the Earth, not on the Sun. Note that in this diagram, planets orbit the Sun counterclockwise. The figure below shows an example using the entries for planet Mars in the table above.

Planet	Semi-Major Axis (A.U.)
Venus	0.72
Earth	1.00
Mars	1.52
Jupiter	5.20
Saturn	9.54






6. Answers to the following questions (HINT: Chapters 2 and 3 in the textbook may be useful in answering these questions):

   • a) In which direction is the planet moving against the background of stars?

   • b) Is the planet simply moving along one of the cardinal directions (say, east or west)? Explain why it is moving in the direction you observe.

   • c) Is there a relation between the direction of motion and the Sun-Earth-planet angle?

   • d) Do you notice any change in the motion (direction, speed) throughout the semester? If so, explain what is going on. Hint: use the plots you made in #4 and #5.

   • e) How does the planet's motion relate to the ecliptic? Is it above, below, moving closer or farther from the ecliptic. Explain what causes what you have observed.

   • f) If you observed more than one planet during the semester, compare their motions in light of your answers to the above questions. Which is moving faster? Why?

   • g) Using the diagram of the orbits (#5), determine the orbital period of the planet, i.e. how long it takes to complete one orbit around the Sun.