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Galileo and the moons of Jupiter: exploring the night sky of 1610Submitted by brown on 07 November 2012
By Carla Isabel Ribeiro
‘… 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….”
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 and uniform circular motion
Cosmology in Galileo’s time
The project step by step Step 1: 17th century cosmology 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. Step 2: choose the planetarium software 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). Step 3: tracking Jupiter’s moons Step 4: testing Galileo’s conclusion 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:
where w is the angular frequency and j is a constant (the phase constant), together with the equation linking w and 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:
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:
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. Step 5: presenting the results
Reference Ribeiro CI (2012) Io and its simple harmonic motion. Physics Education 47: 268270. doi: 10.1088/00319120/47/3/F04 Web references w1 – Download a recent English translation of Sidereus Nuncius (Starry Messenger), Galileo’s famous early work describing discoveries made with the telescope. Pages 17 and 18 contain his observations of the moons of Jupiter for the dates featured in this project.
w2 – Stellarium, the planetarium simulation used in the project, can be downloaded free of charge. w3 – The eTwinning website promotes school collaboration in Europe through the use of information and communication technologies (ICT). Available in 23 languages, it has nearly 50 000 members and more than 4000 registered projects between two or more schools across Europe. Resources The Physclips website offers a clear explanation of simple harmonic motion, with video and animation. NASA’s Solar System Exploration website offers uptodate information about Jupiter and its moons, including space missions. For an article describing a similar project exploring the Galilean moons, using a telescope equipped with a chargecoupled device (CCD) camera, see:
Project CLEA (Contemporary Laboratory Experiences in Astronomy), hosted by Gettysburg College, USA, offers laboratory exercises that illustrate modern astronomical techniques. Each exercise consists of a dedicated computer programme, a student manual, and a technical guide for the teacher. Several of the exercises involve Jupiter’s moons. If you found this article inspiring, you may like to browse the other astronomyrelated articles in Science in School. Carla Isabel Ribeiro teaches chemistry and physics at a public Portuguese school, and is particularly interested in astronomy. For the past 12 years, she has taught students ranging in age from 13 to 18. Review This article suggests a new enquirybased way of teaching simple harmonic motion: students use their knowledge of mathematics, physics and information and communication technology to characterise the motion of Jupiter’s moons. They collect data from a software programme, process it and then plot graphs, particularly of sine and arcsine functions, to calculate the moons’ orbital periods. The interdisciplinary nature of the article serves to make science more enjoyable. In addition, the activity develops soft skills such as the presentation of results and communication. By joining an international project, the students would have the opportunity to share their results not only with other members of their class but with students from different countries. Corina Toma, Computer Science High School “Tiberiu Popoviciu” Cluj Napoca, Romania
