I'm trying to replicate Rømer's experiment but can't seem to get even close to the correct value for the speed of light

Question

I am trying to replicate Rømer's experiment where he determined the speed of light by observing and measuring the eclipse timings of Io by Jupiter. I'm using Stellarium for this experiment and no matter how I try I can't seem to land on the correct value.

Here is my setup.

1. Using Stellarium I noted the date and time of when Io just begins to emerge from behind Jupiter and took this as a baseline.
2. I then predict the next eclipse by adding 42.5 h to the previous time.
3. I use Stellarium to see when the next eclipse will happen and note the date and time.
4. I then calculate the distance travelled by using data from NASA'a Horizon website for the respective distances of Io from Earth, and subtract those 2 values, which gives the distance.
5. The predicted and actual eclipse timings are different by 104.5 s
6. Using these values I then divide the distance by the time.

Here are my values:

• Baseline eclipse - 2023-11-01 07:00:46
• Next predicted eclipse - 2023-11-01 01:28:20
• Actual eclipse - 2023-11-01 01:26:35
• Time difference = 104.5 sec

Distance of Io from Earth:

• Baseline eclipse - 3.98457905366747 au = 596093026.42865 km (using 1 au=$1.496 \times 10^8$ km)
• Actual eclipse - 3.98258042749088 au = 595794031.95264 km

So distance travelled = 298994.47602 km

Then to calculate speed:

$$ s = d / t \\ s = \frac{298994.47602 \text{ km}}{104.5 \text{ s}} \\ s = 2861.06 \text{ km/s} $$

This is obviously incorrect and looks to be out by a factor of 100. I've double and triple checked all of my measurements and calculations, but I can't figure out where I've gone wrong.

Answer 1

In the current version of Stellarium, v24.2, there is a checkbox in the settings to turn "simulate light speed" on and off ("Sky and viewing options window" [F4] → "SSO" → "Simulate light speed", the same as in many older versions). It sounds like you might have it turned off. I think that's the default, in order to save a computational step and run the program a little faster.

If this is correct, another interpretation is that you have discovered, by using a version of Rømer's method, that the speed of light in your Stellarium is wonky. If the option is just disabled, however, I would have expected infinity rather than too slow by a factor of 100.

Another interpretation is that you have Stellarium correctly configured to simulate light speed, but the NASA database is also stimulating light speed, and your two data sources have some computational difference which means your two speed measurements are different at the 1% level.

Since you seem to be using consecutive eclipses, a third possibility is "catastrophic cancellation," as pointed out by another commenter. Consider that $$ 10^6+1-10^6=1 $$ is true in arithmetic, but $$ (10^6\pm10^3)+1-(10^6\pm10^3)=1\pm10^3 $$ means that any rounding or measurement uncertainty might obscure your effect.

The correct approach, as you have intuited in a comment, is to make the eclipse timing predictions yourself. Here's an approach that I would take:

1. Measure the time interval for twenty or so consecutive Io orbits, so that you have a good handle for the average value and also for the repeatability of your method.

2. Repeat these measurements, keeping track of both dates and times, when Jupiter is at conjunction (distance 6 a.u.), at opposition (distance 4 a.u.), and then at the next conjunction.

3. You should find that Io's orbital period is the same for all three sets of observations, and that you can use the observations of transit times at the first conjunction to predict the transit times at the second conjunction. (At conjunction and opposition, you won't get good eclipses.) However, your transits at opposition should arrive about twenty minutes earlier than your conjunction-based predictions.