CESAR Science Case Jupiter Mass Calculating a planet’s mass from the motion of its moons Teacher The Mass of Jupiter 2 CESAR Science Case Table of Contents Fast Facts ...................................................................................................................................... 4 Summary of activities ................................................................................................................... 5 Background ................................................................................................................................... 7 Kepler’s Laws ................................................................................................................................................ 8 Activity description ....................................................................................................................... 9 Activity 1: Properties of the Galilean Moons. Choose your moon ............................................................... 10 Activity 2: Calculate the period of your favourite moon ............................................................................... 10 Activity 3: Calculate the orbital radius of your favourite moon .................................................................... 13 Activity 4: Calculate the Mass of Jupiter ..................................................................................................... 15 Additional Activity: Predict a Transit ............................................................................................................ 16 Links ............................................................................................................................................ 20 The Mass of Jupiter 3 CESAR Science Case Fast Facts FAST FACTS Outline Age range: 16-18 In these activities students will apply their knowledge about the orbits of celestial bodies. Type: Guided investigation Students will measure the main orbital parameters and use them to calculate new Complexity: Medium Teacher preparation time: 20 minutes Students should already know… Lesson time required: 1 hour 30 minutes 1. Orbital Mechanics (velocity, distance…) Location: Indoors 2. Kepler’s Laws 3. Secondary School Maths Includes use of: Computers, internet 4. Units conversion Curriculum relevance Students will learn… General 1. How to apply theoretical knowledge to astronomical situations • Working scientifically. 2. Basics of astronomy software • Use of ICT. 3. How to make valid and scientific measurements Physics 4. How to predict astronomical events • Kepler’s Laws • Circular motion Students will improve… • Eclipses Space/Astronomy • Their understanding of scientific thinking. • Their strategies of working scientifically. • Research and exploration of the Universe. • Their teamwork and communication skills. • The Solar System • Their evaluation skills. • Orbits • Their ability to apply theoretical knowledge to real-life situations. • Their skills in the use of ICT. You will also need… • Paper, pencil, pen and computer with required software installed To know more… • CESAR Booklets: – Planets – Stellarium – Cosmographia The Mass of Jupiter 4 CESAR Science Case Summary of activities Title Activity Outcomes Requirements Time 1. Properties of Students may Students improve: Cosmographia installed 10 min the Galilean choose their • Their understanding of Moons favourite Jupiter’s scientific thinking. Step by step Installation moon by using • Their strategies of guide can be found in: Comographia working scientifically. • Their skills in the use • Cosmographia of ICT. Booklet • Completion of Activity 1. 2. Calculate the Students inspect Students improve: 10 min period of your Stellarium software • The first steps in the • Stellarium installed favourite for making scientific scientific method. moon measurements to • Their strategies of Step by step Installation obtain the orbital working scientifically guide can be found in: period of the moon • Their skills in the use of ICT. • Stellarium Booklet • Completion of Activity 1. 3. Calculate the Students inspect Students learn: 15 min orbital radius Stellarium software • How astronomers • Stellarium installed of your for making scientific make calculus favourite measurements to Step by step guide can be moon obtain the orbital Students improve: found in: distance of the • The first steps in the moon and its scientific method. • Stellarium velocity • Their strategies of Booklet working scientifically • Their skills in the use of ICT. • Their ability to apply theoretical knowledge 4. Calculate the Students may use Students learn: • Completion of Activities 5 min Mass of 3rd Kepler’s Law • How astronomers 1,2 and 3. Jupiter and the results make calculus • Basic knowledge of previously obtained stellar evolution and to calculate the Students improve: how the colour of a mass of Jupiter • The final steps in the (massive) star relates scientific method. to its age. The Mass of Jupiter 5 CESAR Science Case Title Activity Outcomes Requirements Time 5. Aditional Students analyse Students learn: • Completion of all the 15 min Activity: the motion by • How astronomers previous Activities Predict a another method, make calculus of real Transit using uniformly data. accelerated motion • Basic properties of a equations. star. • What information can be seen and extracted from an astronomical image. Students improve: • Their understanding of scientific thinking. • Their strategies of working scientifically. • Their teamwork and communication skills. • Their ability to apply theoretical knowledge to real-life situations. • Their skills in the use of ICT. The Mass of Jupiter 6 CESAR Science Case Background For this Science Case some software is required: - Cosmographia: https://www.cosmos.esa.int/web/spice/cosmographia - Stellarium: http://stellarium.org/ Booklet’s on how to install and configure them for this specific Case are available to download, and can be found here: Link 1 Link2 Figure 1: Cosmographia Figure 2: Stellarium The Mass of Jupiter 7 CESAR Science Case Kepler’s Laws The three Kepler’s Laws, published between 1609 and 1619, meant a huge revolution in the 17th century. With them scientists were able to make very accurate predictions of the motion of the planets, changing drastically the geocentric model of Ptolomeo (who claimed that the Earth was the centre of the Universe) and the heliocentric model of Copernicus ( where the Sun was the centre but the orbits were perfectly circular). These laws can be summarised as follows: 1. First Law: The orbit of every planet is an ellipse, with the Sun at one of the two foci. 2. Second Law: A line joining a planet and the Sun sweeps out equal areas during equal intervals of time. Figure 3: Second Law of Kepler (Credit: Wikipedia) 3. Third Law: The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. Considering that the planet moves in a circular orbit with no friction, the gravitational force equalizes the centrifugal force. Therefore, the third Kepler’s law can be express as: 퐺푀푚 퐹 = 퐹 → = 푚 푎 퐺 퐶 푅2 푐 푣2 퐺푀푚 푣2 푎푛푑 푎푠 푎 = → = 푚 푐 푅 푅2 푅 2휋 푎푎푛, 푎푠 푣 = 휔 ∙ 푅 = 푅 푇 Note that 푀 is the mass of the main object and 푚 is the mass of the orbiting one, 푣 is the linear velocity of the moving body, 푅 is the radius of the orbit, 휔 is the angular velocity of it, 푇 is the period of the orbiting object (in seconds) and 퐺 is the gravitational constant, which value is 퐺 = 6.674 ∙ 10−11 푚3 푘−1 푠−2 퐺푀 푅3 = 4휋2 푇2 The Mass of Jupiter 8 CESAR Science Case Activity description During these activities, students will make use of two of the most used software for astronomical purposes. Their goal is to obtain the Jupiter’s mass by applying the Kepler’s Laws and basic maths based on measurements done with Cosmographia and Stellarium. The mass can be obtained measuring the period and the radio of the orbit of one moon. Jupiter has 79 moons (up to 2018), which can be divided into 2 groups: - Irregular moons: small objects with very distant and eccentric orbits - Regular moons: bigger objects with nearly-circular orbits o Inner Moons: These objects orbit around the planet in very close orbits. The Jupiter inner moons are called Amalthea, Thebes, Metis and Adrastea are the biggest inner moons known. They can be seen in Cosmographia and Stellarium too. Figure 4: Inner Moons of Jupiter (Credit: Galileo spacecraft, NASA) o Main Moons: These objects are bigger than the inner moons. The Jupiter main moons are called Io, Europa and Ganymede. They are in an orbital resonance of (1:2:4). Callisto is the furthest one. They are also known as Galilean moons, as Galileo discovered them in 1610. Figure 5: The Galilean moons (Credit: NASA) For this Science Case students are asked to choose one of the four Galilean moons and execute measurements with it. The Mass of Jupiter 9 CESAR Science Case Another interesting exercise would be comparing the final results of the calculation of the Jupiter mass obtained by the different students (groups), as there would be students who will choose different moons. Activity 1: Properties of the Galilean Moons. Choose your moon Students will use Cosmographia for this activity. As it appears in Cosmographia booklet students may enable, by right clicking: - The trajectory of the four moons (step 5 of the student´s guide) - The properties of each moon (step 6 of the student´s guide) The solution to the chart asked is: Table 1: Chart of properties of Galilean Moons with key Object Mass