|Name: ________________________||DUE 9/20||ID number: ________________________|
Predicting planetary positions was very difficult for ancient astronomers. With modern techniques, accurate predictions are possible -- if you can do the math. This assignment uses a simpler method which yields moderately accurate predictions -- good enough to find the planets in the sky.
Your job is to calculate the positions of the `classical' planets -- Mercury, Venus, Earth, Mars, Jupiter, and Saturn -- on the date of your choice. Feel free to pick any date you like -- a birthday, a holiday, or whatever. Instructions for calculating positions are given at the end of this handout. Use the work-sheet on the other side when doing your calculations, and plot your results on the diagrams below and on the other side.
Write the date you've chosen here: ____________________
Fig. 1. Orbits of Mercury, Venus, Earth, and Mars.
Fig. 2. Orbits of Earth, Jupiter, and Saturn.
Find the number of days D from January 1, 2000 to the date
Now fill in the values of r, s, f, and L for each planet.
The orbits shown above are marked in fractions of the orbital period. On each orbit, the point labeled `0.0' shows where the planet will be on January 1, 2000. Likewise, the point labeled `0.1' shows where the planet will be one-tenth of an orbital period later, and so on all the way around.
The first step is to find the number of days from January 1, 2000 to your chosen date. Call this number D; it should be negative if your chosen date is before January 1, 2000, and positive if it's after. To compute the number of days, multiply the number of years by 365.26 (the number of days in a year). Then use a calendar to figure out how many days to add.
An example may help make this clearer. Suppose the chosen
date is August 30, 1956. The next New Year's day, January 1, 1957, is
43 years before January 1, 2000. A calendar shows that there are 2
months of 30 days (September and November) and two months of 31 days
(October and December) between September 1, 1956 and January 1, 1957.
Finally, August 30, 1956 comes two days before September 1, 1956,
Now for each planet, divide D by the planet's orbital period P listed in the second column of the work-sheet, call the result r, and write it in the third column. If r is positive, take its integer part, multiply by -1, call the result s, and write it in the forth column. If r is negative, take the integer part of its absolute value, add 1, call the result s, and write it in the forth column. Now add r and s, call the result f, and write it in the fifth column. The number f should be between 0 and 1.
Again, an example may help. Let's say we're trying to
figure out the positions of Earth and Venus on August 30, 1956. We
already found that D = -15830 days. For Earth, we get
and for Venus,
For each planet, the f value you've just computed is the fraction of the planet's orbit it's completed by the chosen date. Using these values, you can mark the planet's position along its orbit. Listed by increasing distance from the Sun, Fig. 1 shows the orbits of Mercury, Venus, Earth, and Mars, while Fig. 2 shows the orbits of Earth, Jupiter, and Saturn. Find the orbit for Mercury, start at the point labeled `0.0', and scan counter-clockwise until you reach the f value you computed for Mercury. Do the same for all the planets; be sure to mark Earth on both Figs. 1 & 2!
To continue the example, we found f = 0.66 for Earth and f = 0.53 for Venus. Shown on the right is part of Fig. 1, with filled-in circles marking f = 0.66 on the orbit of Earth and f = 0.53 on the orbit of Venus.
Using a ruler, draw lines from Earth to each of the other planets in Figs. 1 & 2. These lines of sight show which direction you should look on the chosen date to see each planet. But to figure out just where in the ecliptic to look, one more step is needed. The outer circles in Figs. 1 & 2 should really be drawn with a nearly infinite diameter since they represent positions among the stars. It's impossible to draw the planetary orbits and the surrounding stars both to scale, so I've `cheated' by drawing the outer circles only a little bigger than the orbits they enclose. Because of this, a line drawn from Earth to another planet and beyond to the outer circle won't correctly indicate the planet's position along the ecliptic. So each time you draw a line from Earth to a planet, draw another line starting from the Sun and parallel to the line just drawn. The place where this second line reaches the outer circle shows where along the ecliptic you'll really find the planet.
The numbers along the outer circle are celestial longitudes -- they measure angles around the ecliptic. As you draw the two parallel lines for each planet, notice where the line from the Sun reaches the outer circle, and record the longitude L in the final column of the work-sheet.
Following the same example as above, the diagram on the right shows the line from Earth toward Venus and the parallel line from the Sun to the outer circle. It is this second line which shows where Venus appears in the sky as seen from Earth on August 30, 1956. Venus should have a celestial longitude L = 112° on August 30, 1956. I checked using the Your Sky web page. The result is shown below; the slanting line is the ecliptic, and Venus is shown by its usual symbol. As you can see, the results are in good agreement!