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
above.

*D* =

Now fill in the values of *r*, *s*, *f*, and
*L* for each planet.

PLANET | P (days) |
r | s | f | L |

Mercury | 87.97 | ____________ | ______ | ____________ | ______ |

Venus | 224.64 | ____________ | ______ | ____________ | ______ |

Earth | 365.26 | ____________ | ______ | ____________ | ______ |

Mars | 686.79 | ____________ | ______ | ____________ | ______ |

Jupiter | 4332.80 | ____________ | ______ | ____________ | ______ |

Saturn | 10759.7 | ____________ | ______ | ____________ | ______ |

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,
so

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 To continue the example, we found |

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 |

Joshua E. Barnes (barnes@ifa.hawaii.edu) Last modified: September 14, 1999