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CESAR Science Case

Calculating a ’s mass from the motion of its

Student’s Guide

Mass of Jupiter 2 CESAR Science Case

Table of Contents

The Mass of Jupiter ...... ¡Error! Marcador no definido. Kepler’s Three Laws ...... 4 Activity 1: Properties of the ...... 6 Activity 2: Calculate the period of your favourite ...... 9 Activity 3: Calculate the orbital radius of your favourite moon ...... 12 Activity 4: Calculate the Mass of Jupiter ...... 15 Additional Activity: Predict a Transit ...... 16 To know more… ...... 19 Links ...... 19

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Background

Kepler’s Three 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 , changing drastically the geocentric model 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 1: 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 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 푘푔−1 푠−2

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Therefore, the quotation previously mentioned is achieved: 푇2 ∝ 푅3

퐺푀 푅3 = 4휋2 푇2

In this Science Case we are going to make use of two of the most practical astronomical tools: Cosmographia and Stellarium. By applying the third Law of Kepler we could calculate the mass of Jupiter.

Please check CESAR Booklets (see links) to know how to install and configure these software packages.

Did you know?

Jupiter is the largest planet in the : more than 11 times bigger than the Earth and around of 320 times heavier than it. Jupiter is also the planet with more moons orbiting around it: 79 moons have been discovered up to 2018.

However these moons are not equal in size. The biggest ones (, , and ) were the first discovered by Galileo Galilei in 17th century. It was in Galileo’s honour why they are called the Galilean Moons.

Figure 2: The Galilean moons (Credits:NASA)

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Activity 1: Properties of the Galilean Moons

For this first activity Cosmographia will be used.

1. Double click on “Cosmographia.app” and it will pop up the application as Figure 3.

Figure 3: Cosmographia starting view

Note: The menu of Cosmographia is at the left part of the window.

2. Select the first option (the white circles). The image of the solar system bodies will be displayed. See Figure 4

Figure 4: Solar System bodies as seen in Cosmographia

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3. Find out Jupiter and click on it. Cosmographia will drive you to Jupiter (Figure 5)

Figure 5: Solar System bodies

4. Scroll out and turn the mouse until you find the Galilean moons (Io, Europa, Ganymede and Callisto), as in Figure 6

Figure 6: Solar System bodies as seen in Cosmographia

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5. Display the trajectory for the four moons by doing right click on each one (see Figure 7)

Figure 7: Trajectory toggle menu as seen in Cosmographia

6. Display the properties of Jupiter and every Galilean moons and will Table 1

Figure 8: Right click on Jupiter (left image); Jupiter properties (right image) as seen in Cosmographia

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Now is your turn to write down the most important properties of them on the following table:

Object Mass (kg) Radius (km) Density (g/cm3)

Jupiter

Io

Europa

Ganymede

Callisto Table 1: Chart of properties of Jupiter and the Galilean Moons

Now that you are familiar with the Galilean moons, choose your favourite one for the following activities.

Your Moon

Activity 2: Calculate the period of your favourite moon

The third Kepler’s Law involves several orbital parameters, and the periodicity of the motion (period or T term in the equations) is one of them. You will calculate the period using “Stellarium”.

Open Stellarium

Open the console, by pressing F12, and paste the following script on it:

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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Once you are done click on the play button ( ) and a figure similar to Figure 9 will be displayed

Figure 9: Stellarium view, after running the script

The name of the Galilean moons will be displayed. Check how your favourite moon moves around Jupiter. It is following a periodic motion! However, as you may guess the real motion is slower (indeed 3000 times slower). Slow down the motion at your convenience.

Calculate how much time your moon spends in doing a complete loop around Jupiter.

For this pay attention to the parameters “date” and “time” at the lower part of the display. Select one point in time and write down the value. See the evolution of the movement and note down the time when the moon is again at the same position. The difference between those 2 points is the period.

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

Calculate the time difference here

Table 2: Moon period calculation

Write down the result of your moon here:

Period days hours

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Figure 10: Jupiter Moons visualization (Credit: CESAR) As shown in Figure 10, be aware that in Stellarium we are seeing Jupiter as it is seen from Earth. However, the motion of the moons is circular, but we are just watching a projection in 2 dimensions.

Did you know?

Jupiter has always been a very interesting to study. Since the first observation from Galileo a lot of improvements have been achieved.

Figure 11: Artist's concept of the proposed JUICE mission to the Jupiter system (Credit: Wikipedia)

Several space missions have passed very close to this planet: like and 11, both and 2, Galileo, Cassini-Huygens and Ulysses has made some fly-bys around Jupiter. And other missions like JUNO (from NASA) have been developed for this planet in particular. ESA is currently working in JUICE, which will be launched in 2022.

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Activity 3: Calculate the orbital radius of your favourite moon

The orbital radius of a moon, assuming a circular movement, can be defined as the maximum distance between Jupiter and your moon. As you can see in Figure 13 , this value can be obtained by using trigonometry basis.

To calculate this you will make use of Stellarium and the plugin Angle Measure.

Note: Make sure the plugin “Angle Measure” is active in your configuration. Otherwise do the following:

• Move your mouse to the left part of the screen • Open the configuration menu (or F2 in your keyboard) • Select Angle Measure > Load at startup • Restart Stellarium.

Ready for calculating the radius? Follow the next steps to make the measurement:

• Stop the motion of the moons around Jupiter by pressing in the lower menu (or K in your keyboard) • Move your mouse again to the lower menu and press (you can also press Ctrl + A) in order to enable the Angle Measuring plugin. • Click on the centre of Jupiter and later the centre of your favourite moon.

WARNING: The Kepler’s Laws are just valid if the measurement is done from the centre of the astronomical objects, so be sure that you click on the centre of the moon and the centre Jupiter.

Figure 12: Using Angle Measure plugin

• Write down your measurement here, in the column 2 in the units given by the program. Convert these units to degrees (with decimals) and write down the distance in degrees. Note: We are considering angular distance, that is why the units are given in degrees.

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Maximum Distance ° ‘ ‘’ ° of your Moon to Jupiter Write your calculations here Remember: 1° = 60’ and 1’= 60’’ → 1°= 3600 ’’

Did you know?

The relationship between the angular distance (휃) and the orbital distance of every moon (푅) to Jupiter, 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 Figure 13 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 = 푑퐽퐸 · 푠푖푛 휃 (1)

Figure 13: Sketch for calculating moon-Jupiter distance

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• Note down the distance from Earth to Jupiter (푑퐽퐸), which is shown in the left side of Stellarium. This value appears in AU, Astronomical Units, so convert it to km, considering the proper conversion:

Note: 1 AU is the mean distance from the Earth to the Sun, and 1 퐴푈 = 149 584 372 푘푚,

푑퐽퐸 = 퐴푈 푘푚

• Using the equation 푅 = 푑퐽퐸푠푖푛 휃, we can obtain the distance from Jupiter to your moon R (see Figure 13 for further explanation):

푅 = 푑퐽퐸 푠푖푛 휃

푅 = 푘푚 푚

And finally, once we have the radius of the orbit we can obtain the velocity of the moon, where v is the lineal velocity, 휔 the angular velocity, 푇 is the period of the moon (in seconds) and 푅 the maximum distance previously calculated (but now in metres). 2휋 푣 = 휔 ∙ 푅 = 푅 푇

푣 = 푚/푠

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Activity 4: Calculate the Mass of Jupiter

Using the Kepler’s Laws, as we see on Background Section we can obtain the mass of the planet:

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

Remember that 푅 is the radius of the moon’s orbit (in metres), 푇 is the period of your moon (in seconds) and 퐺 is the gravitational constant, which value is 퐺 = 6.674 ∙ 10−11 푚3 푘푔−1 푠−2

Remember to use scientific notation!

푀 = 푘푔 퐽

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Additional Activity: Predict a Transit

Did you know?

Astronomers call transit to the pass of an astronomical object in front of another. They are very similar to eclipses, but they just differ on the apparent size of these two objects: if one is apparently largest than the other it is called a transit; but if their apparent size is similar it is an eclipse.

For example, here on Earth we can see some eclipses during every year. A solar eclipse occurs when the Moon is between the Earth and the Sun, this way in several places of Earth it will be dark, even it was daytime.

Figure 14: Phenomena concept (left), solar eclipse (centre) and Mercury transit (right) Credits: CESAR/NASA

The same happens in other planets of the Solar System and other .

In Jupiter we can see these transits from Earth using ground-based telescopes and satellites.

Figure 15: Io and Europa transit, using Stellarium

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Figure 16 shows the trajectory of the moons (X axis) over time (Y axis), and we will obtain a sinusoidal movement, as seen on Figure 16. But remember, this movement is a straight line as seen from Earth, we saw in Figure 10. In the graph below, the black curve of the graph means the separation from the moon to Jupiter, given in .

Every time the moon is on the “dark part” (called umbra) of the graph it will be transiting Jupiter whether behind it or a background pass. The grey part is assigned to the penumbra.

Figure 16: Jupiter Moon Tracker Results

Predict the transit generated by your favourite moon on Jupiter surface:

WARNING: If your moon is Callisto you may change to another for this activity. A transit also depends on the angle of inclination of the orbit and as Callisto is so far from Jupiter this transit is not visible from the Earth.

• In the case of Io and Europa the date and time of one transit is already in the script. But try to find another, press for moving to your current date and time. • If you have Ganymede, you have to look for it. One way you can do it is moving forward on time, for example at x300 time rate, and stopping the animation when you find it.

It is recommended to use the same moon as chosen before as favourite as we are going to use some of the already calculated values (T, the period).

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1. Open Stellarium 2. Open a console and add these lines to your previous script

StelMovementMgr.zoomTo(0.0167, 5); core.setDate("2018:08:17T00:20:50","utc"); core.setTimeRate(300);

3. Write down the date (start and end times) of a transit of your chosen as favourite moon. Note: the format (YYYY-MM-DD hh:mm:ss)

Beginning of the 1st transit End of the 1st transit

4. Now your job is to predict the next transit. When is going to be the next transit?

WARNING: Remember from previous activities that the movement of the moons is periodic, and that you have already calculated the period.

Beginning of the 2nd transit End of the 2nd transit

Do you think it will be seen with telescopes on Earth? And with space telescopes? Explain the reasoning behind your answers

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To know more…

Links

Software

• CESAR Booklets: Cosmographia, Stellarium

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

• 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: General Understanding of the solar System

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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