Galilean Time Keepers

Experiment by Geoff Hitchcox, Christchurch, New Zealand ---- Submitted 24th February 2002

B efore the days of accurate chronometers and radio signals, the Jovian Galilean satellites were briefly used as a "reference" clock. In the book "Illustrated Longitude" by Dava Sobel, there is a map (page 35) showing how France was mapped more accurately using Jovian Satellites as a time reference. In my previous experiment I had concluded that it was the lack of isochronism (equal time) of IO that was the main error in the Roemer "Speed of Light" experiment.

Doing an extensive Internet search and a browse through the local library, I could not find any reference to the "performance" of the Jovian moons as timekeepers. I decided it was quicker to try my own experiment than keep searching! Not knowing the answer (as in the "Speed of Light" experiment) made it a more tantalising challenge.

The purpose of this experiment is to examine the Jovian moons purely in terms of a clock or timepiece. To place each moon "on the bench", and observe not only its rate, but also its "rate of rate". It is the latter specification that determines a good clock. A perfect clock would have a "rate of rate" of zero, as an atomic clock approximates. By placing Jovian moons on the "workbench", it also removes any "light delay" issues! The reason to conduct the experiment is to gain perspective on how accurate historical experiments (like Roemer's) were restricted by the Jovian moons lack of isochronism.

I decided to use the great historical record that is bound mathematically into the E5 ephemeris by Dr Jay Lieske. This is the current (circa 2002) most accurate model of the Galilean satellites. Using freely available source code, I was able to quickly optimise a "C" function to give the XYZ position of each moon (with respect to Jupiter) for any given time and date. Implementing a fast iterative routine to find the time that each moon reaches a "reference" RA (Right Ascension) followed. This enabled the software to produce a list of orbital times for each satellite from any given date. This highly intensive CPU task is but a breeze for a modern day Pentium!

I began by exploring the timekeeping of IO. Looking at the list of orbital times indicated an "orbital jitter" (more on that later) but no overall pattern. To extract the "rate of rate" for IO, I modified the software to "accumulate" the difference between the Mean Sidereal Rate of IO and the calculated orbital rate. This "accumulation" resulted in a very clean graph that shows a recurring pattern every 275 orbits of IO as per: [Cycle] If IO orbited at a constant rate, the above graph would be a straight line (no errors to accumulate). Not having any other data to compare this graph against, I began to wonder if it was just an artifact of my software. It then occurred to me, I could use the graph to "predict" the time IO would cross a reference RA that could be independently verified using the JPL HORIZONS online ephemeris. With that in mind I noted the following positions on the graph. [label] From A to B you can see that IO accumulates just under 200 seconds (over its average orbital rate). From B to D it loses just under 400 seconds, from A to E there should be no change!

Using the JPL ephemeris (see NOTE 1 for working) to determine the "accumulated" error between the above points, gave these very reassuring figures. A to B (194 secs), B to D (-393 secs) A to E (-0.37 secs). So the "sinewave" graph does seem to represent the behaviour of IO and is not just an artifact of the software. To further test the code, I used the software to calculate the "average" orbital period, using 27,500 orbits of IO (now that gave the Pentium a headache ;-) 
1.769137802 days = authors software (27,500 orbits)
1.769137786 days = AA value for IO Sidereal value

A "reasonable" agreement to 1 part per billion!
Being somewhat confident that the above "sinewave" graph is a how IO behaves - how does it come about?

To try and understand what was going on, I made a graph of the "sinewave" time error along with the distance IO was from its "mean" distance to Jupiter. This was suitably scaled to fit the graph, the "y" units do not apply to the RED curve. [Apsis] The change in orbital rate is thus explained by the slow rotation of the eccentric IO orbit, as it precesses its way around Jupiter in 275 orbits. At perijove (when IO is closest to Jupiter) its speed is greater than the mean - IO appears to run "fast". Conversely when IO is at the greatest distance (apojove) IO appears to run slow. Note, where the RED curve crosses Y=0, the "rate of rate" (tangent to the BLUE curve) becomes zero (but not for long!). The above graph is a nice example of Kepler's Law in action!

The above graph is the timekeeping as viewed from a constant RA wrt Jupiter. At any other RA, the graph has the same pattern just phase shifted along the X axis. The graph can be likened to the "Equation of Time", used to show how far our sun is ahead or behind "mean" earth time. This was often pasted as a table inside old Grandfather clocks, so you could use your sundial to correct the clock!

Although there are many ways we can examine IO as a clock (short term, long term, Fourier, Allan variance etc) the above graph is the most important aspect in terms of Roemer's experiment. Putting the IO graph into more common units, gives a +/- 18 part per million accuracy. This equates IO to a clock having an accuracy of +/- 1.6 seconds per day!

Historicity

At first glance it would seem that because IO has a +/- 18 part per million clock error, that we could get the same accuracy for the "Speed of Light" in the Roemer experiment! However it is the way the timing error accumulates and the mathematical method used in the Roemer experiment, that turns this "average clock performance" into a large percentage error for the value of the "Speed of Light".

The Roemer experiment is based on the assumption that IO orbits at a constant rate. From the above "sinewave" graph, we can see that IO can run fast or slow and accumulate an error. Also, because the experiment usually involves "appearance and disappearance", this forces one set of readings to be "fast" and the other "slow". This compounds the error because the 2 data sets are subtracted from each other, NOT added!

In the Roemer experiment, 4 sightings are usually made creating two data sets. Although the data sets may be many weeks or months in duration (therefore the IO time error is small), the 2 (equal orbit number) data sets are subtracted from each other. If everything was perfect, the result of the subtraction would be the time difference due to the speed of light. However our above graph will throw in a hundred or more seconds of error to the subtraction, magnifying the timing error (parts per million) into parts per ten! This is what turns our subtle timing error of IO into a major problem for Roemer and his followers!

Each time the Roemer experiment is performed, it is likely to be on a different part of the graph, this is somewhat "forced" by the following 2 cycles having a sliding phase shift. 
Interval of above IO Timing Cycle = 1.769137786 x 275 = 486.513 days = 1.332 Years
Interval Earth-Jovian conjunction = Jovian "viewing" cycle 396 days  = 1.084 Years
Historically the IO clock, and earthly measurement errors, were overcome by doing many experiments and averaging the results (as did Delambre (1749-1822) in 1809).

For a single Roemer experiment, the above IO clock error can be corrected using the "Revised Roemer method" I proposed in my first experiment.
A closer look at the graph

It was mentioned above that the data had an observed "jitter".
To take a closer look at the graph, lets zoom in on C. [zoom C] Notice how every 2nd pair of orbits have the same "y" value and then "leap" about 8 seconds to the next pair! Whether this graph is what IO gets up to, or whether it is an artifact of the E5 model, I do not know. It is interesting to compare the 8 second "leap" and the 1.3 yr cycle, to the actual Observed - Calculated values (referenced to the E2 ephemeris) by Anthony Mallama, et al. Furthermore, examining the IO actual timings for 2000 and 2001, shows that the more negative the O-C difference is, the closer IO was to apojove (where "C" is on the graph). This may indicate that apojove is the least accurate part of the IO E5 orbit. This association may not have been made before.

Comparing the zoom on C to that of E: [zoom E] Notice how the vertical intervals between each value are more consistent, a "straight line" with a small sinewave running along it (as the "rate of rate" changes due to Europa). The graph does not show the "steps" that the zoom in on "C" shows. This graph "E" is when IO is at perijove, it is here that the conjunction of Europa (at apojove) occurs every second orbit of IO. Running an orbital simulator of IO and Europa did not show any alignment of the moons that (I could see) would account for the "steps" found on the graph at "C".

However such "zooming" has its limits. Although there is now a long history of observations and continued excellent work by dedicated amateurs, it is only recently using CCD precision that the "residual error" in measurement has begun to reduce. Anyone who has tried to time a Galilean eclipse would agree that it is not an easy task due to the slow change in light intensity.
The Future!

Running the software for 2,750 IO orbits into the "future" gives the following graph (for a 13.3 year period from Jan 2002). It is reasonably consistent, although you will notice the top of the graph is increasing toward the end! Considering the Callisto orbit takes 560 years to precess around Jupiter, there must be many long term changes that influence the IO orbital rate. [13yr]
Europa, Ganymede and Callisto:

Applying the same technique to the other 3 Jovian moons, produced 3 quite different graphs (see NOTE 2). When comparing all four moons, it is clear that Callisto has the best "timekeeping" stability followed by Ganymede, IO and then Europa. This would seem to indicate the order to choose which moon to use for "time referencing experiments" like that of Roemer. However the slower orbital rate of the outer moons, make it more difficult to judge when the eclipse has occurred (unless advanced CCD methods are used).

Summary:

This experiment reveals the magnitude of the timing error of the Galilean moons when used as a reference clock. It shows how the timing error accumulates, and an explanation as to why this happens.

An explanation is given as to why the small timing error of IO, creates a large error in the Roemer "Speed of Light" experiment.

The experiment provides the background information needed to understand the "Modified Roemer Method" proposed in the first experiment.

As an aid to modern research, perhaps viewing the E5 ephemeris in the manner of this paper, may help with some of the few remaining minor "issues" with the Galilean mathematical model.

Information links used in the experiment:

Source code used as the basis for my software. This is the E5 ephemeris as implemented by Meeus and coded by Kerry Shetline.

E5 Ephemerides (PDF document) by Dr Jay Lieske.

Galilean Eclipse timings by Anthony Mallama.

For those who enjoyed "Longitude" by Dava Sobel, this thesis (PDF document) is a sort of mathematical supplement to the book.

Note 1:

To find the IO "accumulated" timing error at the 3 points on the graph using the JPL Ephemeris.

Using the time and date at "A" from my data, I used the JPL ephemeris to give the Right Ascension of IO (wrt Jupiter). This angle (231.6 degrees) was the reference angle to measure the "time and date" at the other orbits at B, D and E.

Using the AA IO sidereal value as reference, the following data shows how fast (or slow) IO is running at the orbits "predicted" by the graph.

The "macro" and method used to access the JPL ephemeris are found at the end of the first experiment. 
AA Sidereal value = 1.769137786 days

------------------------------------------------------------------
A to B

Date__(UT)__HR:MN:SC.fff Date________JDUT     R.A.
2002-Jan-02 04:20:39.333 2452276.68101080     231.6044552
2002-May-04 05:58:57.500 2452398.74927662     231.6049165
                      diff = 122.06826582 days
69 orbits using AA value   = 122.070507234
                      diff =   0.002241414 days = 193.66 secs
------------------------------------------------------------------
B to D

Date__(UT)__HR:MN:SC.fff Date________JDUT     R.A.
2002-May-04 05:58:57.500 2452398.74927662     231.6049165
2002-Dec-29 02:05:54.000 2452637.58743056     231.6044360
                      diff = 238.83815394 days
135 orbits using AA value  = 238.83360111
                      diff =  -0.00455283 days = -393.36 secs
------------------------------------------------------------------
A to E

Date__(UT)__HR:MN:SC.fff Date________JDUT     R.A.
2002-Jan-02 04:20:39.333 2452276.68101080     231.6044552
2003-May-03 16:39:13.500 2452763.19390625     231.6047571
                      diff = 486.51289545 days
275 orbits using AA value  = 486.51289115
                      diff =  -0.0000043 days = -0.37 secs
Note 2:

Graphs of the other Galilean Satellites. [Europa] From the above we can see that Europa has 5 times the "timing error" of IO. This means its use in the Roemer "Speed of Light" experiment, will produce 5 times the error of IO. These Europa timing errors can still be "fixed" using the method I proposed in the first experiment. The Europa graph (BLUE) has the same unusual behaviour around orbit 75 that was observed on the IO graph at "C". [Ganymede] With Ganymede it is still possible to see the relationship with the rotating eccentric orbit and the "time error". There are now other factors that are perturbing the orbit, notice the smaller sinewave superimposed on both the BLUE and the RED. [Callisto] The outer moon Callisto (16.7 day orbit) has the most stable timekeeping of the four. It can be seen that Ganymede perturbs the orbit and there are "slow" long term precessional effects.