I have grouped Saturn and Uranus together because, from my viewpoint, they present a similar problem. Their sidereal periods are longer than the time interval that I have been making measurements for, so that the methods that I used to find the periods of the other planets don’t work so well. For Uranus especially I am extrapolating its behavior over quite a short fraction of an orbit. For both planets, I did pull out from my old notebooks a more-or-less casual observation from several years earlier, which I finished up making a lot of use of, but the values I calculate come out rather higher than the handbook values. It is just my bad luck that both planets have been moving through aphelion. I have made some new observations beyond the results in my book, but the gain in precision is very slow.

Here is a summary of my measurements of the oppositions of both planets:

Saturn

Date Julian day (DJD) Longitude at oppositon

1/27/2006, 14:23 UT 38743.1 127.5°

3/22/2010, 0:29 hrs 40257.52 181.2707°

4/3/2011, 23:12 hrs 40635.47 193.8506°

5/23/2015, 1:31 hrs 42145.56 241.6343°

6/3/2016, 7:29 hrs 42522.81 253.1207°

The first result is a bit less accurate than the others.

For Uranus

Date Julian day (DJD) Longitude at opposition

5/15/1980, 12:hrs UT 29355 233°

9/23/2010, 17:05 hrs 40441,21 358.606°

9/26/2011, 0:37 hrs 40810,53 2.59°

10/15/2016, 11:37 hrs 42656.98 22.51°

The first observation is very approximate. The date in uncertain by ±30 days and the longitude by ±7.5°

One way to use these numbers, very similar to what I did in several places in my book, is to plot graphs in Excel and ask for linear trendlines. For example, I can plot the longitude at opposition versus the Julian day, and from the slope I can calculate the sidereal year. The values that come out from the slopes of the lines are not significantly different from the values that I calculated by hand, given in chapters 5 and 7 of the book.

More or less as an experiment, I tried another approach. It has the advantage that it actually needs only the measurements taken during one retrograde passage. For each passage, my least-squares fitting program gives values of the distance of the planet from the Sun at opposition, and a number that I called the instantaneous period. This is actually just a way of parameterizing the angular velocity. For uniform circular motion, the angular velocity is related to the period by ω = 2π/T and it was easier for me to continue using T as the fitting parameter than to bring in ω as a new parameter. Kepler’s second law is that the planet sweeps out equal areas in equal times and that is equivalent to saying that r²ω or r²/T is constant. The fitting program also prints out a value of r²/T. Here are the results for the two planets.

Saturn

Year distance time r²/T

2010 9.58±0.04 29.5±0.2 3.11±0.04

2011 9.72±0.07 30.0±0.4 3.15±0.06

2015 9.97±0.02 32.22±0.09 3.085±0.012

2016 10.04±0.02 32.65±0.15 3.086±0.019

For Uranus

2010 20.1±0.2 93.0±2.0 4.33±0.12

2011 20.0±0.1 93.1±1.3 4.30±0.08

2016 19.86±0.05 90.9±0.5 4.34±0.03

As I pointed out in my book, the values of r²/T for each planet are equal within the uncertainties, and this does verify Kepler’s second law. They should be approximately equal to a²/P, where a is the semi-major axis and P is the sidereal period. There is also a factor of √(1 – e²). This is close enough to 1 compared with the precision of my values, but I do try to take it into account in my analysis of results for Mars. The new step I am taking now is to apply Kepler’s third law. If a is measured in astronomical units and P is measured in Earth years, which are the units used in the tables here, Kepler’s third law is that P² = a³. Then the ratio a²/P is just equal to the square root of a or the cube root of P. That is

a = (a²/P)², P = (a²/P)³.

For Saturn, the best value in the table is 3.085±0.012, and then

a = 9.52 ± 0.07, P = 29.4 ± 0.3

For Uranus, the best value is 4.34±0.03, so that

a = 18.8 ± 0.03, P = 82 ± 2.

Some of these are better than I arrived at earlier and some are worse. The value of P for Uranus is by far the best estimate I have made. As I said, all this method of analysis needs is data from one retrograde passage so you could get these numbers in just a few months observations.