Jupiter Mass - Cesar ESAcesar.esa.int/upload/201809/jupiter_mass_teacher_guide.pdf · 2018. 9. 19. · 1. Open the console, by pressing F12, and paste the following script: 2. Their

CESAR Science Case

Jupiter Mass

Calculating a planet’s mass from the motion of its moons

Teacher

The Mass of Jupiter CESAR Science Case

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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


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Fast Facts

FAST FACTS

Age range: 16-18

Type: Guided investigation

Complexity: Medium

Teacher preparation time: 20 minutes

Lesson time required: 1 hour 30 minutes

Location: Indoors

Includes use of: Computers, internet

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…

• Paper, pencil, pen and computer with required software installed

To know more…

• CESAR Booklets: – Planets – Stellarium – Cosmographia

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

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.


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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:




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


• 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 ability to apply theoretical knowledge

• Completion of Activity 1.

• Stellarium installed

Step by step guide can be found in:

• Stellarium Booklet

15 min

4. Calculate the Mass of Jupiter

Students may use 3rd 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


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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.

Students improve:



• Their teamwork and communication skills.

• Their ability to apply theoretical knowledge to real-life situations.


• Completion of all the previous Activities

15 min


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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

https://www.cosmos.esa.int/web/spice/cosmographia

http://stellarium.org/


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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:

𝐹𝐺 = 𝐹𝐶 → 𝐺𝑀𝑚

𝑅2= 𝑚 𝑎𝑐

𝑎𝑛𝑑 𝑎𝑠 𝑎𝑐 =𝑣2

𝑅 →

𝐺𝑀𝑚

𝑅2= 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 = 6.674 ∙ 10−11 𝑚3 𝑘𝑔−1 𝑠−2

𝐺𝑀

4𝜋2=

𝑅3

𝑇2


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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.


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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/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:

2. Their view will be placed to Jupiter (similar as Figure 6)

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);


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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.

4. The real value of the period of the moons appears in Table 2

Figure 7: Jupiter Moons visualization (Credit: CESAR)

Moon Orbital Period

Io 1 𝑑𝑎𝑦 18.45 ℎ𝑜𝑢𝑟𝑠

Europa 3 𝑑𝑎𝑦𝑠 12.26 ℎ𝑜𝑢𝑟𝑠

Ganymede 7 𝑑𝑎𝑦𝑠 3.71 ℎ𝑜𝑢𝑟𝑠

Callisto 16 𝑑𝑎𝑦𝑠 16.53 ℎ𝑜𝑢𝑟𝑠

Table 2: Period of the galilean moons


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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.


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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 (𝜃) and the orbital distance of every moon (𝑅), can be calculated using basic trigonometry; and lastly(𝑑𝐽𝐸) 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 = 𝑑𝐽𝐸𝑠𝑒𝑛 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


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Again, as an example, using the previous results:

Maximum Distance of your Moon to Jupiter

0 ° 2 ‘ 40.31 ‘’ 0,0445 °

𝑑𝐽𝐸 = 5.718 𝐴𝑈 8.55 · 108 𝑘𝑚

𝑅 = 𝑑𝐽𝐸 𝑠𝑖𝑛 𝜃

R = 8.55 · 108 sin ( 0.445 º) = 664 761 km

For meters, we multiply by 103

𝑣 = 𝜔 ∙ 𝑅 =2𝜋

𝑇 𝑅

T = 3 d 12.3h = 3·24+12.3 h = 84.3 h = 84.3 h·3600 s

1 h = 303 480 s

v= 2π

303480 6.64·108 = 13.76·103 m/s = 13 760 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)

𝑅 = 664 761 𝑘𝑚 6.64 · 108 𝑚

𝑣 = 13 763 𝑚/𝑠

Moon Orbital Radio (km)

(Semi-major Axis)

Orbital velocity (m/s)

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 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:

𝑬𝑹 =| 𝑴𝒆𝒂𝒔𝒖𝒓𝒆𝒅 𝑽𝒂𝒍𝒖𝒆 − 𝑹𝒆𝒂𝒍 𝑽𝒂𝒍𝒖𝒆 |

𝑹𝒆𝒂𝒍 𝑽𝒂𝒍𝒖𝒆· 𝟏𝟎𝟎 (𝟐)

→ ER=| 664 761-670 00 |

670 00 · 100=

5 329

670 000· 100=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

So then applying the third Kepler´s Law:

𝐺𝑀𝐽

4𝜋2=

𝑅3

𝑇2 → 𝑀𝐽 =

4𝜋2

𝐺

𝑅3

𝑇2

And for the previous example:

MJ = 4π2

G

R3

T2 =

4π2

6.674·10-11 m3 kg-1 s-2·(6.64·108 m )3

(303480 s)2 = 1.8867·1027 kg

𝑀𝐽 = 1.8982 · 1027 𝑘𝑔


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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:

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


core.setDate("2018:08:17T00:20:50","utc");



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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:

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.

• By user interface: Use Figure 10 buttons and move to the predicted time and date.

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.


core.setDate("2018:08:17T00:20:50","utc");



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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#

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.

1

2

3

4

https://www.skyandtelescope.com/wp-content/observing-tools/jupiter_moons/jupiter.html


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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

https://cosmoguide.org/

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

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

http://astro.unl.edu/classaction/animations/renaissance/kepler.html

Documents

Jupiter Mass - Cesar ESAcesar.esa.int/upload/201809/jupiter_mass_teacher_guide.pdf · 2018. 9. 19. · 1. Open the console, by pressing F12, and paste the following script: 2. Their