CESAR Science Case. Jupiter Mass. Calculating a planet s mass from the motion of its moons. Teacher

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1 Jupiter Mass Calculating a planet s mass from the motion of its moons Teacher

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3 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 Activity 2: Calculate the period of your favourite moon Activity 3: Calculate the orbital radius of your favourite moon Activity 4: Calculate the Mass of Jupiter Additional Activity: Predict a Transit Links

4 Fast Facts FAST FACTS Age range: Type: Guided investigation Complexity: Medium Teacher preparation time: 20 minutes Lesson time required: 1 hour 30 minutes Location: Indoors Includes use of: Computers, internet Outline In these activities students will apply their knowledge about the orbits of celestial bodies. Students will measure the main orbital parameters and use them to calculate new Students should already know 1. Orbital Mechanics (velocity, distance ) 2. Kepler s Laws 3. Secondary School Maths 4. Units conversion Curriculum relevance General Working scientifically. Use of ICT. Physics Kepler s Laws Circular motion Eclipses Space/Astronomy Research and exploration of the Universe. The Solar System Orbits You will also need Students will learn 1. How to apply theoretical knowledge to astronomical situations 2. Basics of astronomy software 3. How to make valid and scientific measurements 4. How to predict astronomical events Students will improve Their understanding of scientific thinking. Their strategies of working scientifically. Their teamwork and communication skills. Their evaluation skills. Their ability to apply theoretical knowledge to real-life situations. Their skills in the use of ICT. Paper, pencil, pen and computer with required software installed To know more CESAR Booklets: Planets Stellarium Cosmographia 4

5 Summary of activities Title Activity Outcomes Requirements Time 1. Properties of the Galilean Moons Students may choose their favourite Jupiter s moon by using Comographia Students improve: Their understanding of scientific thinking. Their strategies of working scientifically. Their skills in the use of ICT. Cosmographia installed Step by step Installation guide can be found in: Cosmographia Booklet 10 min 2. Calculate the period of your favourite moon Students inspect Stellarium software for making scientific measurements to obtain the orbital period of the moon Students improve: The first steps in the scientific method. Their strategies of working scientifically Their skills in the use of ICT. Completion of Activity 1. Stellarium installed Step by step Installation guide can be found in: Stellarium Booklet 10 min 3. Calculate the orbital radius of your favourite moon Students inspect Stellarium software for making scientific measurements to obtain the orbital distance of the moon and its velocity Students learn: How astronomers make calculus Students improve: The first steps in the scientific method. Their strategies of working scientifically Their skills in the use of ICT. Completion of Activity 1. Stellarium installed Step by step guide can be found in: Stellarium Booklet 15 min Their ability to apply theoretical knowledge 4. Calculate the Mass of Jupiter Students may use 3 rd Kepler s Law and the results previously obtained to calculate the mass of Jupiter Students learn: How astronomers make calculus Students improve: The final steps in the scientific method. Completion of Activities 1,2 and 3. Basic knowledge of stellar evolution and how the colour of a (massive) star relates to its age. 5 min 5

6 Title Activity Outcomes Requirements Time 5. Aditional Activity: Predict a Transit Students analyse the motion by another method, using uniformly accelerated motion equations. Students learn: How astronomers make calculus of real data. Basic properties of a star. What information can be seen and extracted from an astronomical image. Completion of all the previous Activities 15 min 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. 6

7 Background For this Science Case some software is required: - Cosmographia: - Stellarium: 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 7

8 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: F G = F C GMm R 2 = m a c and as a c = v2 R GMm R 2 = m v2 R again, as v = ω R = 2π T R Note that M is the mass of the main object and m is the mass of the orbiting one, v is the linear velocity of the moving body, R is the radius of the orbit, ω is the angular velocity of it, T is the period of the orbiting object (in seconds) and G is the gravitational constant, which value is G = m 3 kg 1 s 2 GM 4π 2 = R3 T 2 8

9 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 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. 9

10 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 (kg) Radius (km) Density (g/cm 3 ) Jupiter Io Europa Ganymede Callisto Activity 2: Calculate the period of your favourite moon For this activity Stellarium is used. Students have to calculate the period of their moon by playing around with Stellarium and the time. With Stellarium open students may 1. Open the console, by pressing F12, and paste the following script: core.setobserverlocation("madrid, Spain"); LandscapeMgr.setFlagLandscape(false); LandscapeMgr.setFlagAtmosphere(false); LandscapeMgr.setFlagFog(false); core.selectobjectbyname("jupiter", true); core.setmountmode("equatorial"); core.settimerate(3000); StelMovementMgr.setFlagTracking(true); StelMovementMgr.zoomTo(0.167, 5); 2. Their view will be placed to Jupiter (similar as Figure 6) 10

11 Figure 6: Stellarium view, after running the script 3. Now they have to calculate the period of the moon. It s quite simple, they may register a position of the moon and write down the first date. Then wait for the moon to reach the same position and write down the second date. Students may remember that the motion of the moons is circular, but from Earth we are just watching a projection in 2 dimensions, as it appears in the Figure 7. Moon Orbital Period Io 1 day hours Europa 3 days hours Ganymede 7 days 3.71 hours Callisto 16 days hours Table 2: Period of the galilean moons 4. The real value of the period of the moons appears in Table 2 Figure 7: Jupiter Moons visualization (Credit: CESAR) 11

12 An example is provided: Your Moon Europa Initial date (YYYY-MM-DD hh:mm:ss) Final date (YYYY-MM-DD hh:mm:ss) :05: :25:00 Calculate the time difference here Same year and same month 4th 1st = 3 days 15h 3h = 12 h 25 min 05 min = 20 min And as 1h = 60 min ; 20 min = 0.3 h Period 3 days hours Students can also play with the time rate in Cosmographia and check their result of the period of their moon by visualizing the motion in 3D. 12

13 Activity 3: Calculate the orbital radius of your favourite moon For this activity students have to calculate the radio of the orbit of their moon, as the 3 rd Law of Kepler involves this term. And, as explained in Stellarium booklet, the plugin Angle Measure plugin needs to be enabled. The relationship between angular distance (θ) and the orbital distance of every moon (R), can be calculated using basic trigonometry; and lastly(d JE ) is the distance from Jupiter to the Earth, which is obtained with Stellarium. As you can see in Error! No se encuentra el origen de la referencia. we can use the definition of the sine, which states that: in a rectangular triangle, the ratio between the length of the opposite side of an angle and the length of the hypotenuse is the sine of that angle. Which can also be expressed mathematically with the equation (1): R = d JE sen θ (1) The distance from Earth to Jupiter can be obtained with Stellarium. When you select one object, at the left side of the screen a bunch of information is displayed. Figure 8: Stellarium view with astronomical object information. Distance to earth in the right image, rounded 13

14 Again, as an example, using the previous results: Maximum Distance of your Moon to Jupiter ,0445 d JE = AU km R = d JE sin θ R = sin ( º) = km For meters, we multiply by 10 3 R = km m v = ω R = 2π T R T = 3 d 12.3h = h = 84.3 h = 84.3 h 3600 s = s 1 h v= 2π = m/s = m/s v = m/s With this information both orbital radio and velocity can be calculated Table 3: Chart with orbital radio and velocity for each Galilean moon (Credit: Wikipedia) Moon Orbital Radio (km) (Semi-major Axis) Orbital velocity (m/s) Io Europa Ganymede Callisto

15 No solution is provided for the angular distance θ, since it will depend on the distance from Earth to Jupiter, which is not always the same. To know if the measurement is done correctly students must calculate R (distance from Jupiter to the moon) and then this result must be compared to the real values in Table 3. It won t be the same value, but it may differ a bit due to errors in the measurements. An error less than or equal to 5% might be acceptable. The same goes for the value of the velocity. To calculate the relative error for any measurement: Measured Value Real Value E R = 100 (2) Real Value E R = = =0.78% Note: A negative value for the relative error will probably mean that the absolute value of equation (2) has not been applied Activity 4: Calculate the Mass of Jupiter The most accurate value for the mass of Jupiter is M J = kg So then applying the third Kepler s Law: GM J 4π 2 = R3 T 2 M J = 4π2 G R 3 T 2 And for the previous example: M J = 4π2 R 3 G T 2 = 4π m 3 kg -1 s -2 ( m ) 3 ( s) 2 = kg 15

16 Additional Activity: Predict a Transit For predicting a future transit student must first find a previous one. Stellarium is the most recommended software for this purpose. Adding the following code to the previous script run: StelMovementMgr.zoomTo(0.0167, 5); core.setdate("2018:08:17t00:20:50","utc"); core.settimerate(300); Figure 9: Io and Europa transit, using Stellarium Jupiter will fill the screen (Figure 9), and the script is already programmed for visualizing the Europa s and Io s transit. In order to visualize new transits students must press on, or pressing number 8 in their keyboard, to adjust the date of Stellarium to the current time and date. Later, with the button, the time rate can be changed. Each time they press the time rate the speed is multiplied by 10, therefore just touching this button two or three times the motion will be adequate for this activity. Press to stop the motion. Figure 10 shows the menu for changing the time rate, which is in the lower and left part of the screen. Figure 10: Time rate menu 16

17 To predict transits students must have in mind the already calculated period of its favourite moon. Just adding the period to the initial time/end time they will predict when the next transit will start/end. For this activity it is not recommended to choose Callisto, as it is the furthest moon of the Science Case. The reason why this happens is because the moon s orbit is not always parallel to the equator, they usually have some inclination; and the transit, which is the projection of the satellite in the planet, is more prompted at further distances. Figure 11 shows a sketch for orbit inclination. Figure 11: Inclination sketch of an orbit (not at a real scale). Blue line represents Earth s direction Yellow moon is close to Jupiter, so the transit could be seen. Green mon has the same inclination, but as it is further away the transit could not be seen. 17

18 Lastly students are asked: Answer to the questions of the Student s Guide Do you think it will be seen with telescopes on earth? And with space telescopes? Why? The transits of Jupiter s Galilean Moons can always be seen with space telescopes. But there are two main reasons why some transits cannot be seen from Earth: Optical telescopes on Earth depend on light conditions. That s why they just operate in night conditions. So only the transits that can be seen are those at night. Also their seeing depend on the position of the Earth. The constellations that can be seen in Summer are not the same constellations visible at Winter. That is because the Earth is moving around the Sun and the axis of rotation is tilted 23.4º, so the day and night skies are changing their roles in the different seasons. The stars and constellations that can be seen during the whole year (in a defined latitude) are called circumpolar. In conclusion, the orbit of the Earth and the orbit of Jupiter are also factors to take into account. Teachers and students can check if their prediction is correct by entering that date and time into Stellarium software and checking if the shadow of the moon appears in Jupiter. This can be achieved by two different ways: By console: Open the console by pressing F12 and add the following lines to the code. Change the second line by entering the predicted date and time. Run the script. StelMovementMgr.zoomTo(0.0167, 5); core.setdate("2018:08:17t00:20:50","utc"); core.settimerate(0); By user interface: Use Figure 10 buttons and move to the predicted time and date. 18

19 Alternatively, a chart for future transits can be found here: Figure 12: Sky&Telescope Jupiter s transits predictor Looking at Figure 12 teachers can check if students have predicted correctly the transit. In order to do that: Enter the predicted date and time for the transit in circled number 1 textboxes. Click on Recalculate using entered date and time in the 2 nd circle, to have a representation of the moons position on that time, Hit Display satellite events on date above in the 3 rd circle and all the information will be displayed on 4 th textbox. 19

20 Links Software CESAR Booklet: Cosmographia Cosmographia Official Users guide CESAR Booklet: Stellarium Stellarium Official Users Guide pdf Planets CESAR Booklet: Planets Kepler s Laws : Orbits (Spanish only) Kepler s Laws Animation 20

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