Determining the orbital period of Jupiter to a high level of accuracy

Updated March 11

This sketch represents a view from above of the Earth in its orbit (green), around the Sun (bright orange in the original sketch!), and an outer planet, Jupiter say, (pink/red).

The orbits are circular and are in the same plane.

At time 1, for an observer on the Earth, Jupiter is said to be at opposition, when at its closest to the Earth.

At time 2, 6 months later, the Earth has completed half an orbit. Being much further away from the Sun than is the Earth, Jupiter's angular velocity is much lower and it has completed only a small part of its orbit in that time.

At time 3, Jupiter is at opposition again and since it has moved along in its orbit, the Earth also has to move along its orbit for an extra distance (solid green line), in a time interval represented by the delta symbol, 𝛥, in green, between the points marked 1 and 3. Jupiter has now completed 𝛥/365 of an orbit in 365 + 𝛥 days.

In 2026, opposition was January 10th and in 2027 it will be February 11th. In that time, Jupiter will have completed 32/365 of an orbit in 397 days. 

From this, its orbital period in years is therefore 397/32 or 12.4.        (Method 1)

The accepted period is 11.86, so what happened? Mainly it's because I've assumed the orbits are circular and they are not - they are elliptical. Now, I'm going to try to eliminate that error, by considering a longer period of time when the eccentricities of the orbits of Jupiter and Earth will have been smoothed out, somewhat.

Let's look ahead to 2038, when opposition is on January 14th and Jupiter will have completed 369/365 ((365+4)/365) of an orbit in 12 and 4/365 years. That means it will have completed a full orbit in 365/369 of those 12 and 4/365 years, i.e. its orbital period is:

((12x365) + 4)/369 or 11.88 years.        (Method 2)

That's very close to the accepted figure. I've not taken into account the exact times of day of the consecutive oppositions (available here), nor that the Earth takes 365 and 1/4 days to orbit the Sun. If I do that, then I make it 11.85 years.        (Method 3)

A similar analysis using method 2 above for Saturn over July 15th 1990 to July 21st 2020 gives an orbital period of 29.45 years (actual=29.46) and for Mars over March 4th 2012 to April 9th 2014 gives 1.91 years (actual=1.88). Interestingly, the deviation from the accepted period is greater in the latter case (closer to us, shorter time interval) and smaller in the former (further from us, bigger time interval).