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Home , Adrastea (moon), Amalthea (moon), Galilean moons, Jupiter mass, Metis (moon)

CESAR Science Case

Jupiter

Calculating a ’s mass from the motion of its

Teacher

The Mass of 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 . Choose your ...... 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 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… •

Space/Astronomy • Their understanding of scientific thinking. • Their strategies of working scientifically. • Research and exploration of the . • Their teamwork and communication skills. • The • 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: – – 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 and to calculate the Students improve: how the colour of a mass of Jupiter • The final steps in the (massive) 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

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Kepler’s Laws

The three Kepler’s Laws, published between 1609 and 1619, meant a huge revolution in the . With them scientists were able to make very accurate predictions of the motion of the planets, changing drastically the of Ptolomeo (who claimed that the was the centre of the Universe) and the heliocentric model of Copernicus ( where the 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 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 푎𝑔푎𝑖푛, 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 , which value is 퐺 = 6.674 ∙ 10−11 푚3 푘𝑔−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 , Thebes, and are the biggest inner moons known. They can be seen in Cosmographia and Stellarium too.

Figure 4: Inner (Credit: , NASA)

o Main Moons: These objects are bigger than the inner moons. The Jupiter main moons are called , and . They are in an of (1:2:4). 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.

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Another interesting exercise would be comparing the final results of the calculation of the 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/cm3)

Jupiter 1.8982 ∙ 1027 69 911 1.326

Io 8.9319 ∙ 1022 1 824 3.53

Europa 4.8000 ∙ 1022 1 563 3.01

Ganymede 1.4819 ∙ 1023 2 632 1.94

Callisto 1.07594 ∙ 1023 2 409 1.84

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)

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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 푑푎푦 18.45 ℎ표푢푟푠

Europa 3 푑푎푦푠 12.26 ℎ표푢푟푠

Ganymede 7 푑푎푦푠 3.71 ℎ표푢푟푠

Callisto 16 푑푎푦푠 16.53 ℎ표푢푟푠 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)

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An example is provided:

Your Moon Europa

Initial date (YYYY-MM-DD hh:mm:ss) Final date (YYYY-MM-DD hh:mm:ss)

2018-09-01 03:05:00 2018-09-04 15:25:00

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 12 . 3 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.

The Mass of Jupiter 12 CESAR Science Case

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 3rd 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 (휃) and the orbital distance of every moon (푅), can be calculated using basic trigonometry; and lastly(푑퐽퐸) 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):

푅 = 푑퐽퐸푠푒푛 휃 (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 information. Distance to earth in the right image, rounded

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Again, as an example, using the previous results:

Maximum Distance of your Moon to 0 ° 2 ‘ 40.31 ‘’ 0,0445 ° Jupiter

8 푑퐽퐸 = 5.718 퐴푈 8.55 · 10 푘푚

푅 = 푑퐽퐸 푠𝑖푛 휃 R = 8.55 · 108 sin ( 0.445 º) = 664 761 km

For meters, we multiply by 103

푅 = 664 761 푘푚 6.64 · 108 푚

2휋 푣 = 휔 ∙ 푅 = 푅 푇 3600 s T = 3 d 12.3h = 3·24+12.3 h = 84.3 h = 84.3 h· = 303 480 s 1 h

2π 8 3 v= 6.64·10 = 13.76·10 m/s = 13 760 m/s 303480

푣 = 13 763 푚/푠

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)

Orbital Radio (km) Orbital velocity (m/s) Moon (Semi-major Axis) Io 421 700 17 334

Europa 670 900 13 740

Ganymede 1 070 400 10 880

Callisto 1 882 700 8 204

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No solution is provided for the angular distance 휃 , 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 푅 (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:

| 푴풆풂풔풖풓풆풅 푽풂풍풖풆 − 푹풆풂풍 푽풂풍풖풆 | 푬 = · ퟏퟎퟎ (ퟐ) 푹 푹풆풂풍 푽풂풍풖풆 | 664 761-670 00 | 5 329 → E = · 100= · 100=0.78% R 670 00 670 000

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

푀 = 27 퐽 1.8982 · 10 푘𝑔

So then applying the third Kepler´s Law:

3 2 3 퐺푀퐽 푅 4휋 푅 = → 푀 = 4휋2 푇2 퐽 퐺 푇2

And for the previous example:

4π2 R3 4π2 (6.64·108 m )3 M = = · = 1.8867·1027 kg J G T2 6.674·10-11 m3 kg-1 s-2 (303480 s)2

The Mass of Jupiter 15 CESAR Science Case

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

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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.

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Lastly students are asked:

Answer to the questions of the Student’s Guide

Do you think it will be seen with 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 and night skies are changing their roles in the different seasons. The 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.

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Alternatively, a chart for future transits can be found here: https://www.skyandtelescope.com/wp-content/observing-tools/jupiter_moons/jupiter.html#

1 2

3 4

Figure 12: Sky& 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.

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Links

Software

• CESAR Booklet: Cosmographia

• Cosmographia Official Users guide https://cosmoguide.org/

• CESAR Booklet: Stellarium

• Stellarium Official Users Guide https://github.com/Stellarium/stellarium/releases/download/v0.18.1/stellarium_user_guide- 0.18.1-2.pdf

Planets

• CESAR Booklet: Planets

Kepler’s Laws

• CESAR Science Case: Orbits (Spanish only)

• Kepler’s Laws Animation http://astro.unl.edu/classaction/animations/renaissance/kepler.html

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