Sidereal and Synodic Periods  (Math Alert ... some math is required)

At the very end of unit 1 you were introduced to two new terms, Sidereal and Synodic periods of a planet.  Briefly, the sidereal period is the time it takes for a planet to make one complete orbit around the sun.  We all know the Earth takes 1 year (or 365¼ days) to orbit the sun.  Inferior planets go around the sun in shorter times and superior planets take longer to complete a cycle.  The synodic period refers to the time it takes a planet to come the same alignment with respect to the Earth and Sun.  For example, if you were to observe Venus shining very brightly high over the western horizon at dusk, you would have to wait one synodic period of Venus before you were to observe this again.

Part 1 - It is obvious that the synodic period is easy to obtain .... just watch an alignment of a planet and count the number of days until you see it again.  Your job is to explain why it is impossible to obtain the sidereal period of a planet from observation alone.

If the Earth did not move in space, discovering the sidereal period would be a cakewalk.  However, the Earth is also moving along with all the other planets.  It would be like asking a runner in a track meet to call the winner (and all you can see is the person way ahead of you).

Part 2 - Copernicus derived formulas that allowed him to calculate the sidereal period of a planet if you have the synodic period.  Find these formula (it can be in several forms ... all of which are fine).  Define all terms and indicate the units used.  Why are there (usually) two formulas and not just one?

There are two formulas - one for inferior planets and a different one for superior planets.  You can combine them into one if you want to get fancy (not shown)

1/P = 1 + 1/S for inferior planet (Mercury, Venus)
1/P = 1 - 1/S for superior planet (Mars, Jupiter, etc.)

S = synodic period of planet (in years)
P = sidereal year of planet (in years)

Part 3 - Suppose there was a planet that took 4 years to go from opposition back to opposition.  Calculate the sidereal period for this hypothetical planet.  Show all your work.

Since only superior planets can go into opposition, you need to use the formula:  1/P = 1 - 1/S

If S = 4  we get:    

1/P = 1 - 1/S

1/P = 1 - 1/4

1/P = 3/4

P = 4/3           Meaning this planet would take 1.3333 year  ( or 487 days) to go around the sun.  No such planet exists

Part 4 - Look up the synodic period of Venus.  Calculate the sidereal period of Venus.  Show all your work.  How does your value compare to the actual value?

Venus is an inferior planet so you need to use the formula: 1/P = 1 + 1/S

If you look up the synodic period of Venus. you will get:  S = 583.92 days

Convert this to years to get S = 583.92/365.25 = 1.599 years

1/P = 1 + 1/S

1/P = 1 + 1/1.599

1/P  = 1 + .6254

1/P = 1.6254

P = .615 years   (compared to the book value of .615 years or 224.7 days)