Mapping the Motion of a Planet in the Sky

The Wandering Stars

Goals of the Lab

To follow the movement of one or several planets (Venus, Mars, Jupiter, or Saturn) in the night sky over the course of the semester.

Required Equipment: paper, pencils, compass (of the "pointy end and pencil end" variety, not of the "show me magnetic north" variety), crossbow, and a magnified photocopy of the region of the SC-001 star chart where the planet is located in the sky (you will be given this the first night you make an observation, bring it every following week).

Background: The word "planet" is derived from the Greek word for "wanderer." Ancient observers noticed that several "stars" moved against the background of "fixed" stars. Since the planets orbit the Sun more or less in the same plane (i.e., the solar system is "flat"), the planets always stay close to the ecliptic. You will observe this phenomenon. Using the star charts and the crossbow, you will track the position of a planet through the fixed background stars as it changes due to the orbital motion of both the Earth and the planet during a period of 2-3 months.

Part I: Measurements

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.

Use a blown up (200%) photocopy of the star chart centered on the part of the sky where the planet is located. You will use this map to chart the planet's motion during the semester. Always bring it with you to lab!
Measure the angular distance of the planet from at least 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. Use the crossbows in lab to make these measurements; see Measuring Angular Distances on the Sky for more information. Try to measure to the nearest 0.25° (that's 1/8 inch 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 each time! Stars closer to the planet will usually give the best results. Use the closest visible stars which are plotted on the star maps. On some partially cloudy nights, you may need to use more distant stars.
Plot the position of the planet on the 200% blowup of the SC-001 map. This is accomplished with a compass; triangulate the position of the planet from the measured angular distances between the planet and each of the three stars chosen above. Use the declination scale on the star map to calibrate your compass to the distance on your map. For example, if you measured a distance of 10 degrees between a star and the planet, equate 10 degrees as measured on the SC-001 map with the width of your compass. Then, place the point of the compass on the offset star, and draw a small, light arc on the map in the area 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.

Example

On March 30, 2004, at 8:00 PM, you see Jupiter in the constellation Leo. You measure that it is 12.5° (6.25" on the crossbow) from Regulus, 14.5° (7.25") from Denebola, and 28.0° (14") from Alphard. First, consider Regulus. On your blow-up map, use the declination scale to set the width of your compass to what corresponds to 12.5° on the map. (Note: if your compass has an angle measurement on it, do not use that; use the map!)

With that set, put the point of the compass at the center of Regulus, and draw a small arc on the map around the position where you saw Jupiter:

Next, set the compass to 14.5° using the declination scale, put the point of the compass on Denebloa, and draw another arc:

Next, set the compass to 28° using the declination scale, and draw an arc for the distance from Jupiter to Alphard. As you see in the picture, your three arcs won't always intersect. Mark a point which is as close as possible to all three arcs at once, much as is shown. Label that point with the date.

Note: Star charts are like geographical maps. They are created by projecting a curved surface (a portion of a sphere) onto a flat surface (sheet of paper). This process produces distortions, meaning that the scale is not uniform across the entire map. This same reason explains why Greenland appears much larger than South America on many maps of the Earth, while in reality 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° if the distance from the star to the planet is 20°. 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, 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.

What to turn in

What you turn in for this section should consist of:

The star map with all recorded positions labeled by date.
An observation log showing dates, times, position measurements relative to stars, and personal comments as you see fit.

Part II: Analysis

Make a table as follows:

Date Sun's Ecliptic Longitude Planet's Ecliptic Longitude Sun-Earth-Planet Angle Earth's Ecliptic Longitude

2001-Sep-11 168° 272° 104° 348°

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°.

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 if you were looking down on the solar system.

Example

Suppose that on April 20, 2001, you observed the planet to be located very near the star Regulus, in the constellation Leo. Its position along the ecliptic (its ecliptic longitude) would be 150°. On that date, the ecliptic longitude of the Sun is 30°. The Sun-Earth-planet angle is 150° -30° = 120°.
Make a plot of the Sun-Earth-Planet angle as a function of the date (time in days).
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° 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.

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

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

Answer the following questions:
- a) What 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 relationship 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 your observation.
- 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.

Astronomy 103, Summer, 2006