experimental physics - How did Rømer measure the speed of light by observing Jupiter's moons, centuries ago?

I am interested in the practical method and I like to discover if it is cheap enough to be done as an experiment in a high school.

Answer

Method

The method is based on measuring variations in perceived revolution time of Io around Jupiter. Io is the innermost of the four Galilean moons of Jupiter and it takes around 42.5 hours to orbit Jupiter.

The revolution time can be measured by calculating the time interval between the moments Io enters or leaves Jupiter's shadow. Depending on the relative position of Earth and Jupiter, you will either be able to see Io entering the shadow but not leaving it or you will be able to see it leaving the shadow, but not entering. This is because Jupiter will obstruct the view in one of the cases.

You might expect that if you keep looking at Io for a few weeks or months you will see it enter/leave Jupiter's shadow at roughly regular intervals matching Io's revolution around Jupiter.

However, even after introducing corrections for Earth's and Jupiter's orbit eccentricity, you still notice that for a few weeks as Earth moves away from Jupiter the time between observations becomes longer (eventually by a few minutes). At other time of year, you notice that for a few weeks as Earth moves towards Jupiter the time between observations becomes shorter (again, eventually by a few minutes). This few minutes difference comes from the fact that when Earth is further away from Jupiter it takes light more time to reach you than when Earth is closer to Jupiter.

Say you have made two consecutive observations of Io entering Jupiter's shadow at t₀ and t₁ separated by n Io's revolutions about Jupiter T. If the speed of light was infinite, one would expect

\begin{equation} t_1 = t_0 + nT \end{equation}

This is however not the case and the difference

\begin{equation} \Delta t = t_1 - t_0 - nT \end{equation}

can be used to measure the speed of light since it is the extra time that light needs to travel the distance equal to the difference in the separation of Earth and Jupiter at t₁ and t₀:

\begin{equation} c = \frac{\Delta d}{\Delta t} = \frac{d_{EJ}(t_1)-d_{EJ}(t_0)}{\Delta t} \end{equation}

(both numerator and denominator can be negative representing Earth approaching or receding from Jupiter)

In reality more than two observations are needed since T isn't known. It can be approximated by averaging observations equally distributed around Earth's orbit accounting for eccentricity or simply solved for as another variable.

Practical considerations

Note that you will not manage to see Io enter/leave Jupiter's shadow every Io's orbit (i.e. roughly every 42.5 hours) since some of your observation times will fall on a day or will be made impossible by weather conditions. This is of no concern however. You should simply number all Io's revolutions around Jupiter (timed by Io entering/leaving Jupiter's shadow) and note which ones you managed to observe. For successful observations you should record precise time. It might be good to use UTC to avoid problems with daylight saving time changes. After a few weeks you will notice cumulative effect of the speed of light in that the average intervals between Io entering/leaving Jupiter's shadow will become longer or shorter. Cumulative effect is easier to notice. At minimum you should try to make two observations relatively close to each other (separated by just a few Io revolutions) and then at least one more observation a few weeks or months later (a few dozens of Io revolutions). This will let you calculate the average time interval between observations within a short and long time period by dividing the length of the time period by the number of revolutions Io has made around Jupiter in that period. The average computed over the long time period will exhibit cumulative effect of the speed of light by being noticeably longer or shorter than the average computed over the short time period. More observations will help you make a more accurate determination of the speed of light. You must plan all of the observations ahead since you can't make the observations when Earth and Jupiter are close to conjunction or opposition.

Calculations

Once you collected the observations you should determine the position of Earth and Jupiter at the times of the observations (for example using JPL's Horizons system). You can then use the positions to determine the distance between the planets at the time the observations were made. Finally, you can use the distance and the variation in Io's perceived revolution period to compute the speed of light.

You will notice that roughly every 18 millions kms change in the distance of Earth and Jupiter makes an observation happen 1 minute earlier or later.

Cost

The cost of the experiment is largely the cost of buying a telescope that allows you to see Io. Note that the experiment takes a few months and requires measuring time of the observations with the accuracy of seconds.

History

See this wikipedia article for historical account of the determination of the speed of light by Rømer using Io.