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I am trying to understand how to determine Mars' earth year of 686 days. through observation of 33 oppositions between mars and earth I have confirmed the average opposition time to be 780 days. From this I am having difficulty taking this information and determining an outcome of 686 days for Mars' year cycle once around the sun. thank for any help you can give me.

By a lucky chance, I just finished writing and posting a page about how the synodic period (which is the same as the average time between oppositions or any other specific aspect, such as Eastern or Western quadrature) can be determined from the orbital period of the planet. The equation involved can be worked in reverse to calculate the orbital period from the synodic period. (To refer to the web page in question, see http://cseligman.com/text/sky/synodicperiods.htm -- the page includes a link to a page about Planetary Aspects, where it mentions the fact that the average time from one aspect to the next recurrence of the same aspect is the same as the synodic period.)

For outer planets, the equation involved can be expressed as

1/S = 1/E - 1/P,

where E is the orbital period of the Earth, P is the orbital period of the other planet, and S is the synodic period. E is 365.256 days, S (as determined from your data) is 780 days (which is an exceptionally good average, as the exact value is 779.935 days), so moving the 1/P and 1/S to the opposite sides of the equation,

1/P = 1/E - 1/S, = 0.0027377 - 0.0012821 = 0.0014556

The reciprocal then yields almost exactly P = 687 days, which is very close to the correct orbital period for Mars, which is 686.98 days. So you are to be congratulated for obtaining an excellent result.

Astronomy

I can answer almost any question about astronomy and related sciences, such as physics and geology. I will not answer questions about astrology and similar pseudo-scientific rubbish.

I have been a professor of astronomy for over 40 years, and am working on an online text/encyclopedia of astronomy, and an online catalog of NGC/IC objects.**Publications**

Astronomical Journal, Publications of the Astronomical Society of the Pacific (too long ago to be really relevant, but you could search for Courtney Seligman on Google Scholar)**Education/Credentials**

I received a BA in astronomy and physics and a MA in astronomy, both from UCLA. I was working on my doctoral dissertation when I started teaching, and discovered that I preferred teaching to research.**Awards and Honors**

(too long ago to be relevant, but Phi Beta Kappa and Sigma Xi still keep trying to get me to become a paying member)