Galileo and the Moons of Jupiter: Exploring The

Science education projects Image courtesy of NASA Galileo and the moons of Jupiter: exploring the

night sky of 1610 Physics Mathematics ICT Simple harmonic motion Uniform circular motion Arcsine function Physics History of astronomy Ages 17+

Jupiter and the Galilean This article suggests a new moons were photographed in enquiry-based way of 1979 by spacecraft Voyager 1 teaching simple harmonic and assembled into this col- motion: students use their lage. They are in their relative knowledge of mathematics, positions but not to scale. physics and information and communication tech- nology to characterise the motion of Jupiter’s moons. They collect data from a Carla Isabel Ribeiro explains how software programme, process it and then plot graphs, mathematics can be used to study particularly of sine and arcsine functions, to calculate Jupiter’s moons – and how students the moons’ orbital periods. can do the same. The interdisciplinary nature of the article serves to make science more enjoyable. In addition, the activity devel- ops soft skills such as the n a January night in 1610, “… I should disclose and Galileo Galilei looked at Ju- presentation of results and piter through his telescope publish to the world the communication. By join- and saw what he thought were three occasion of discovering and ing an international project, O the students would have the stars near the planet. He continued his observations for about two months, observing four PLANETS, opportunity to share their and over this time realised that there never seen from the very results not only with other members of their class but were actually four ‘stars’, which beginning of the world up changed their position around Jupiter. with students from different Galileo concluded that the ‘stars’ were to our own times….” countries.

Corina Toma, Computer in fact planets orbiting Jupiter – the Galileo Galilei in Sidereus Medicean planets, as he named them Science High School at the time, but which are now known Nuncius (Starry Messenger; “Tiberiu Popoviciu” Cluj as the Galilean moons in honour of 1610) Napoca, Romania REVIEW their discoverer (figure 1, page 42).

www.scienceinschool.org Science in School I Issue 25 : Winter 2012 I 41 Figure 1: Jupiter and the Galilean moons Io, Europa, Ganymede, and Callisto (Jupiter is not on the same scale as the satellites.) Public domain image; image source: Wikimedia Commons Image courtesy of NASA Planetary Photojournal

This discovery of Galileo’s forms right when he claimed that the ‘stars’ the basis of a project that I devised near Jupiter were in fact the planet’s for my 12th-grade physics students satellites. To do this, students col- (17–18 years old) when teaching the lect data about the movement of the topic of simple harmonic motion. The moons using a computer simulation, project builds on a similar activity I and then show that this movement developed earlier, about the motion of has the characteristics of simple har- the Galilean moon Io (Ribeiro, 2012). monic motion, with Jupiter as the cen- Portrait of Galileo Galilei The current project is enquiry-based, tre. At the end of the project, students (Justus Sutermans, 1636) and aims to engage students with a produce a report (a document or pres- rich variety of scientific processes – entation) to describe their findings from exploring historical contexts to and the whole process – and, ideally, to approach it. I spent four months obtaining and analysing experimental share this with students from other working on the project with my stu- results and communicating their con- countries so they can learn to commu- dents, but if you do not have time clusions to others. nicate scientific work internationally. to run the whole project with your The aim of the project is for your The duration of the project will vary students, you could select individual students to prove that Galileo was depending on how the teacher decides activities from it.

Image courtesy of Nicola Graf

Displacement

Period A A

Amplitude Time B B

C C

Figure 2: Plotting the motion of an object attached to a spring against time produces a sine wave.

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Public domain image; image source: Wikimedia Commons

Figure 4: The 16th century geocentric model of the cosmos, depicted in Bartolomeu Velho’s Cosmographia (1568) Physics

Image courtesy of Carla Isabel Ribeiro Simple harmonic motion and A ment appears to be SHM, due to the uniform circular motion projection of their UCM around the planet onto our direction of vision Simple harmonic motion (SHM) is (figure 3). (The orbits are slightly ec- the term used to describe regular peri- centric, but only to a small degree, so Jupiter Jupiter’s odic motions such as the swinging of the moons can be considered to have a moon a pendulum or the oscillations of an circular orbit and to move at constant object attached to a spring. Plotting a speed.) graph of these motions (distance from the central point against time) pro- Cosmology in Galileo’s time duces the characteristic form of a sine Galileo’s observations of Jupiter’s wave (figure 2). moons were made during a time of SHM can be interpreted as a projec- scientific transition, from the geocen- tion onto one axis of an object that is tric (Earth-centred; figure 4) model of B moving with a uniform circular mo- Aristotle and the Catholic Church to tion (UCM). For example, imagine an the heliocentric (Sun-centred) model. object moving in a circle in a horizon- Galileo, of course, was to take up Time tal plane. If we view this from the side the heliocentric model explicitly some at ‘eye level’ (equivalent to projection years later, in a direct challenge to onto the x axis), we see a to-and-fro Catholic doctrine. However, his tel- motion exactly the same as that of an escopic observations, published in oscillating object attached to a spring. his book Sidereus Nuncius (Starry Mes- Figure 3: The movement of one of the Only when viewing the motion from senger) in 1610, were already causing Galilean moons around Jupiter: A) as seen from above the orbit plane and B) above would we see the movement as friction by contradicting the teachings as seen from Earth (viewed parallel to circular. of Aristotle. In Aristotle’s model, eve- the orbit plane). The black dots represent This relationship between SHM and rything in the cosmos orbits Earth – so the Galilean moon’s positions at equal UCM can be applied to the Galilean the idea of moons orbiting the planet intervals of time. moons as seen from Earth: their move- Jupiter was at odds with this.

www.scienceinschool.org Science in School I Issue 25 : Winter 2012 I 43 Images courtesy of Stellarium The project step by step from Starry Messengerw1, in which Step 1: 17th century cosmology Galileo describes his observations and conclusions. Ask your students to research the cosmological ideas that were current Step 2: choose the planetarium A in early 17th century Europe. What software effect might they have had on Galileo, You will need to download the free- his investigations and conclusions? ware planetarium programme Stel- Students should also read excerpts lariumw2 or a similar simulation. Then

Images courtesy of Stellarium 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 B A 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 B intervals that differ between the four A 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 Figure 5: Simulation with the freeware of which should be A. programme Stellarium of Jupiter and its For example, Io is the moon that is moons Europa (A) and Io (B). closest to Jupiter, which means it orbits most quickly. If the students take their measurements at 1 h intervals, Step 4: testing Galileo’s the data will not be sufficient; 15 min conclusion intervals would be more appropriate. The fourth and most complicated For one of the more distant Galilean step is for students to show that Gali- moons, 15 min intervals would result leo was right when he concluded that in more data than is necessary, so a the SHM he observed is produced by longer interval should be used. a moon’s UCM around its planet. To find the centre of the planet on To find the orbital period T of their Stellarium, your students can use the moon, the students will need the red marks around Jupiter (figure 5) to mathematical equation for SHM: draw two lines that cross at its centre. x = A x sin(w t +j) (1) To find the distance from that point to where w is the angular frequency Images of Jupiter and the Galilean moons the moon being studied, they can ei- and j is a constant (the phase con- on 7, 8, 10 and 11 January 1610 as ther use an image programme or print stant), together with the equation link- simulated by the freeware programme off the screenshots and measure the w T Stellarium. ing and : distances with a ruler. w = 2p / T

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Image courtesy of Carla Isabel Ribeiro Figure 6: A graph of arcsin Graph of arcsin(x / A) = (2p / T)t for the moon Europa (x / A) = (2p / T)t for the 2 moon Europa, using observations made with the software Stellarium. The orbital period, T, of the moon can be calculated from the gradient, 2p / T, and the phase constant of 1 the moon, j, is the intercept on the y axis. x arcsin( / A ) (radians)

0 4 8 12 16 20

t (hours) Physics

where x is the displacement, A the which describes SHM – then it is rea- for the moon Europa. amplitude of the motion or maximum sonable to conclude that it is orbiting The graph above has a gradient of displacement, t is the time and j is a the planet as a moon. 0.0741. constant (the phase constant). Because equation (2) is linear, we Since the gradient is equal to 2p / T, Equation (1) can be transformed can see that if your students use their it follows that: into the linear equation (2), thus: data from step 3 to plot a graph of 2p / T = 0.0741 x = A x sin(w t +j) (1) arcsin(x / A) against t, the gradient ⇒ T =2p / 0.0741 ⇒ x / A = sin(w t +j) will be 2p / T, from which they can = 84.8 h ⇒ arcsin(x / A) = w t +j easily calculate the orbital period T. The more mathematically able stu- Since w = 2p / T The phase constant of the moon, j, dents could then carry out a regres- ⇒ arcsin(x / A) = (2p / T)t +j (2) is the intercept on the y axis. Figure sion analysis of the data to test the If your students observe an object 6 shows an example of a graph that ‘goodness of fit’ to equation (2). The behaving according to equation (2) – your students could plot using data value obtained in the case of Europa

Image courtesy of the Lunar and Planetary Institute Image courtesy of NASA Planetary Photojournal

An impact scar, 140 km wide, on Europa. It formed as the An active volcanic eruption at Tvashtar Catena, a chain of giant surface fractured minutes after a mountain-sized asteroid or volcanic calderas on Io, in November 1999. This image was comet slammed into the satellite. captured by NASA’s Galileo spacecraft. www.scienceinschool.org Science in School I Issue 25 : Winter 2012 I 45 Public domain image; image source: Wikimedia Commons

Galileo Galilei shows the Doge of Venice how to use his telescope (HJ Detouche, 1754)

Public domain image; image source: Wikimedia Commons

A replica of the earliest surviving telescope attributed to Galileo Galilei, on display at the Griffith Observatory, Los Angeles, CA, USA

(R2 = 0.998) shows that the data com- In this case, the error may have its showing their results. In science, it ply closely with the equation, and origin in the measurement of A, since is important to communicate, so you thus confirms that this object behaves the moon’s positions were simulated could get your students to make their like a moon in orbit. 1 h apart, and the maximum displace- presentations to students in another The accepted value for Europa is ment could have been reached be- class, or perhaps as part of a school about 3.55 days (85.2 h), which is quite tween two measurements. You could fair or open day. They should think similar to the value calculated above. ask your students to suggest ways to about how best to communicate their The difference between the two val- minimise this error. work. How can they make it simple ues can be a good starting point for a for others to understand? What im- discussion about the accuracy of ex- Step 5: presenting the results ages could they use to help explain perimental results. What might have The last step is for each group of what they did? gone wrong? Were any errors random students to present their work, de- More ambitiously, you could even or systematic? scribing the whole investigation and set up a collaboration with a school

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Image courtesy of NASA / JPL / NASA of courtesy Image Issue 25 : Winter 2012 Winter Issue 25 : I To learn how to use learn how To this code, see page 65. you may like to browse the other you may like to browse articles in Science astronomy-related in School. See: www.scienceinschool. org/astronomy teaches chemis Carla Isabel Ribeiro Science in School If you found this article inspiring, this article inspiring, If you found try and physics at a public Portuguese try and physics at a interested school, and is particularly For the past 12 years, in astronomy. she has taught students ranging in age 13 to 18. from Science educationScience projects -

Europa photographed by by Europa photographed website offers up-to-date informa- website offers its moons, tion about Jupiter and See: including space missions. tem. Physics Education 44: 241. doi: 10.1088/0031-9120/44/3/002 The article can be downloaded the website of from of charge free France, the University of Picardie, (www.u-picardie.fr/~dellis/ Documents/PhysicsEducation/ Not_Experimental) or via the direct link: http://tinyurl.com/blfc4k6 Astronomy), tory Experiences in College, hosted by Gettysburg laboratory exercises USA, offers that illustrate modern astronomical consists of techniques. Each exercise a dedicated computer programme, a student manual, and a techni- Several cal guide for the teacher. involve Jupiter’sof the exercises moons. See: www3.gettysburg. edu/~marschal/clea/ CLEAhome.html http://solarsystem.nasa.gov/ planets the Galilean exploring project equipped moons, using a telescope device (CCD) with a charge-coupled camera, see: (2009) JAM de Moraes IG, Pereira Using simple harmonic motion to follow the Galilean moons – testing Kepler’s law on a small sys third encounter on 9 July 1996 encounter on 9 July Voyager 2 during its close 2 during its close Voyager NASA’s Solar System Exploration System Exploration Solar NASA’s (Contemporary Labora- CLEA Project For an article describing a similar For an article describing

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Physics Education ), Galileo’s famous . On a smaller scale, I would . On a smaller scale, w3 original edition is also available explanation of simple harmonic mo tion, with video and animation. See: www.animations.physics.unsw.edu. au Sidereus Nuncius translation of Sidereus online. See: www.rarebookroom.org/Control/ galsid can simulation used in the project, See: of charge. be downloaded free www.stellarium.org school collaboration in Europe the use of information through and communication technologies in 23 languages, Available (ICT). it has nearly 50 000 members and projects than 4000 registered more schools across between two or more See: www.etwinning.net Europe. (Starry Messenger early work describing discoveries made with the telescope. Pages 17 and 18 contain his observations of the moons of Jupiter for the dates See: in this project. featured http://homepages.wmich. edu/~mcgrew/Siderius.pdf harmonic motion. 47: 268-270. doi: 10.1088/ 0031-9120/47/3/F04 Resources a clear The Physclips website offers w2 – Stellarium, the planetarium website promotes w3 – The eTwinning Web references Web English w1 – Download a recent The Reference its simple CI (2012) Io and Ribeiro in another country. Scientists some country. in another times have to communicate in foreign in foreign times have to communicate but not always, languages – often, I am project, English. Based on this international col hoping to set up an the eTwinning laboration through network also be happy to hear from other from also be happy to hear like to work to- schools that would gether on the project. www.scienceinschool.org