Learn how you and your students can use mathematics to study Jupiter’s moons.
On a January night in 1610, Galileo Galilei looked at Jupiter through his telescope and saw what he thought were three stars near the planet. He continued his observations for about two months, and over this time realised that there were actually four ‘stars’, which changed their position around Jupiter. Galileo concluded that the ‘stars’ were in fact planets orbiting Jupiter – the Medicean planets, as he named them at the time, but which are now known as the Galilean moons in honour of their discoverer (figure 1).
‘… I should disclose and publish to the world the occasion of discovering and observing four PLANETS, never seen from the very beginning of the world up to our own times….”
Galileo Galilei in Sidereus Nuncius (Starry Messenger; 1610)
This discovery of Galileo’s forms the basis of a project that I devised for my 12th-grade physics students (17–18 years old) when teaching the topic of simple harmonic motion. The project builds on a similar activity I developed earlier, about the motion of the Galilean moon Io (Ribeiro, 2012). The current project is enquiry-based, and aims to engage students with a rich variety of scientific processes – from exploring historical contexts to obtaining and analysing experimental results and communicating their conclusions to others.
The aim of the project is for your students to prove that Galileo was right when he claimed that the ‘stars’ near Jupiter were in fact the planet’s satellites. To do this, students collect data about the movement of the moons using a computer simulation, and then show that this movement has the characteristics of simple harmonic motion, with Jupiter as the centre. At the end of the project, students produce a report (a document or presentation) to describe their findings and the whole process – and, ideally, share this with students from other countries so they can learn to communicate scientific work internationally.
The duration of the project will vary depending on how the teacher decides to approach it. I spent four months working on the project with my students, but if you do not have time to run the whole project with your students, you could select individual activities from it.
Simple harmonic motion (SHM) is the term used to describe regular periodic motions such as the swinging of a pendulum or the oscillations of an object attached to a spring. Plotting a graph of these motions (distance from the central point against time) produces the characteristic form of a sine wave (figure 2).
SHM can be interpreted as a projection onto one axis of an object that is moving with a uniform circular motion (UCM). For example, imagine an object moving in a circle in a horizontal plane. If we view this from the side at ‘eye level’ (equivalent to projection onto the x axis), we see a to-and-fro motion exactly the same as that of an oscillating object attached to a spring. Only when viewing the motion from above would we see the movement as circular.
This relationship between SHM and UCM can be applied to the Galilean moons as seen from Earth: their movement appears to be SHM, due to the projection of their UCM around the planet onto our direction of vision (figure 3). (The orbits are slightly eccentric, but only to a small degree, so the moons can be considered to have a circular orbit and to move at constant speed.)
Galileo’s observations of Jupiter’s moons were made during a time of scientific transition, from the geocentric (Earth-centred; figure 4) model of Aristotle and the Catholic Church to the heliocentric (Sun-centred) model.
Galileo, of course, was to take up the heliocentric model explicitly some years later, in a direct challenge to Catholic doctrine. However, his telescopic observations, published in his book Sidereus Nuncius (Starry Messenger) in 1610, were already causing friction by contradicting the teachings of Aristotle. In Aristotle’s model, everything in the cosmos orbits Earth – so the idea of moons orbiting the planet Jupiter was at odds with this.
Ask your students to research the cosmological ideas that were current in early 17th century Europe. What effect might they have had on Galileo, his investigations and conclusions? Students should also read excerpts from Starry Messenger^{w1}, in which Galileo describes his observations and conclusions.
You will need to download the freeware planetarium programme Stellarium^{w2} or a similar simulation. Then divide your students into four groups, and assign each group to one of the four Galilean moons (Io, Callisto, Ganymede or Europa).
Ask your students to use Stellarium to investigate the position of their moon over time: in a table, each group should record the displacement (x) – the distance of their moon from the centre of Jupiter – at a succession of times (t). They will also need to find the maximum displacement (A) of the moon from the planet’s centre.
Measurements should be taken at intervals that differ between the four moons, depending on their distance to the planet and, therefore, their orbital period. The students should use trial and error to find the most appropriate interval for their moon. Their aim is to obtain at least 10 measurements, one of which should be A.
For example, Io is the moon that is closest to Jupiter, which means it orbits most quickly. If the students take their measurements at 1 h intervals, the data will not be sufficient; 15 min intervals would be more appropriate. For one of the more distant Galilean moons, 15 min intervals would result in more data than is necessary, so a longer interval should be used.
To find the centre of the planet on Stellarium, your students can use the red marks around Jupiter (figure 5) to draw two lines that cross at its centre. To find the distance from that point to the moon being studied, they can either use an image programme or print off the screenshots and measure the distances with a ruler.
The fourth and most complicated step is for students to show that Galileo was right when he concluded that the SHM he observed is produced by a moon’s UCM around its planet.
To find the orbital period T of their moon, the students will need the mathematical equation for SHM:
x = A × sin(ω t + φ) (1)
where w is the angular frequency and j is a constant (the phase constant), together with the equation linking w and T:
ω = 2π / T
where x is the displacement, A the amplitude of the motion or maximum displacement, t is the time and φ is a constant (the phase constant).
Equation (1) can be transformed into the linear equation (2), thus:
x = A × sin(ω t + φ) (1)
→ x / A = sin(ω t + φ)
→ arcsin(x / A) = ω t + φ
Since ω = 2π / T
→ arcsin(x / A) = (2π/ T)t + φ (2)
If your students observe an object behaving according to equation (2) – which describes SHM – then it is reasonable to conclude that it is orbiting the planet as a moon.
Because equation (2) is linear, we can see that if your students use their data from step 3 to plot a graph of arcsin(x / A) against t, the gradient will be 2π / T, from which they can easily calculate the orbital period T. The phase constant of the moon, φ, is the intercept on the y axis. Figure 6 shows an example of a graph that your students could plot using data for the moon Europa.
The graph above has a gradient of 0.0741.
Since the gradient is equal to 2π/ T, it follows that:
2π / T = 0.0741
→ T = 2π / 0.0741
= 84.8 h
The more mathematically able students could then carry out a regression analysis of the data to test the ‘goodness of fit’ to equation (2). The value obtained in the case of Europa (R^{2} = 0.998) shows that the data comply closely with the equation, and thus confirms that this object behaves like a moon in orbit.
The accepted value for Europa is about 3.55 days (85.2 h), which is quite similar to the value calculated above. The difference between the two values can be a good starting point for a discussion about the accuracy of experimental results. What might have gone wrong? Were any errors random or systematic?
In this case, the error may have its origin in the measurement of A, since the moon’s positions were simulated 1 h apart, and the maximum displacement could have been reached between two measurements. You could ask your students to suggest ways to minimise this error.
The last step is for each group of students to present their work, describing the whole investigation and showing their results. In science, it is important to communicate, so you could get your students to make their presentations to students in another class, or perhaps as part of a school fair or open day. They should think about how best to communicate their work. How can they make it simple for others to understand? What images could they use to help explain what they did?
More ambitiously, you could even set up a collaboration with a school in another country. Scientists sometimes have to communicate in foreign languages – often, but not always, English. Based on this project, I am hoping to set up an international collaboration through the eTwinning network^{w3}. On a smaller scale, I would also be happy to hear from other schools that would like to work together on the project.
The original edition is also available online.
de Moraes IG, Pereira JAM (2009) Using simple harmonic motion to follow the Galilean moons – testing Kepler’s third law on a small system. Physics Education 44: 241. doi: 10.1088/0031-9120/44/3/002
The article can be downloaded free of charge from the website of the University of Picardie, France.