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Home , Jupiter mass, Moons of Jupiter

PHYS133 – Lab 3 The Revolution of the of

Goals:

• Use simulated magnification in Stellarium-web to observe Jupiter and the position of its four largest ("Galilean") moons.

• Measure the time between maximum distances of each from Jupiter to determine their orbital characteristics (i.e., period and semi-major axis).

• Employ Newton’s form of Kepler’s third law to determine the of Jupiter.

What You Turn In:

• Calculations of the mass of Jupiter for each moon’s .

• Question answer sheet, including determining also the rotation period of Jupiter.

Background Reading: Background reading for this lab can be found in your text book (specifically, Chapters 3 and 4 and especially section 4.4) and the notes for the course.

Equipment needed for the lab: • Computer with Internet Connection • Web browser on site: https://stellarium-web.org/ Equipment provided by the student: • Computer • Pen • Calculator

Background: Astronomers cannot directly measure many of the things they study, such as the and distances of the and their moons. Nevertheless, we can deduce some properties of celestial bodies from their motions despite the fact that we cannot directly measure them. In 1543, Nicolaus Copernicus hypothesized that the planets revolve in circular around the . Tycho Brahe (1546-1601) carefully observed the locations of the planets and 777 over a period of 20 years using a sextant and compass. These observations were used by Johannes Kepler, to deduce three empirical mathematical laws governing the orbit of one object around another.

UDel Physics 1 of 9 Spring 2020 PHYS133 Lab 3 The Revolution of the Kepler’s Third Law is the one that applies to this lab. For a moon orbiting a much more massive parent body, it states the following: a3 M = p2

Where • M is the mass of the parent body in units of the mass of the sun (i.e., solar masses) • a is the length of the semi-major axis in units of the mean -Sun distance, 1 A.U. (astronomical unit). If the orbit is circular (as we assume in this lab), the semi-major axis is equal to the radius of the orbit. • p is the period of the orbit in Earth years. The period is the amount of time required for the moon to orbit the parent body once.

In 1609, the telescope was invented, allowing the observation of objects not visible to the . used a telescope to discover that Jupiter had four moons orbiting it and made exhaustive studies of this system, which was especially remarkable because the Jupiter system is like a miniature version of the . Studying such a system could open a way to understand the motions of the solar system as a whole. Indeed, the Jupiter system provided clear evidence that Copernicus’ heliocentric model of the solar system was physically possible. Unfortunately for Galileo, the inquisition took issue with his findings; he was put on trial and forced to recant.

We will observe the four moons of Jupiter that Galileo saw through his telescope, known today as the . They are named , , and , in order of distance from Jupiter. You can remember the order by the mnemonic “I Eat Green Carrots.”

Looking at Jupiter through a small telescope, you might see the following:

The moons appear to be lined up because we are looking more-or-less edge-on at the orbital plane of the moons of Jupiter. If we watched, as Galileo did, over a succession of clear nights, we would see the moons shuttle back and forth, more or less in a line. While the moons actually move in roughly circular orbits, you can only see the perpendicular distance of the moon to the line of sight between Jupiter and Earth. If you could view Jupiter from “above” (see Figure to the right), you would see the moons traveling in apparent circles.

As you can see from graph on the next page, the perpendicular distance of the moon should be a sinusoidal curve if one plots it versus time. By taking enough measurements of the position of a moon, one can fit a sine curve to the data and determine the radius of the orbit (the amplitude of the sine curve) and the period of the orbit (the period of the sine curve). Once we know the radius and period of the orbit of that moon and convert them into appropriate units, we can determine the mass of Jupiter by using Kepler’s Third Law.

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UDel Physics 2 of 9 Fall 2020 PHYS133 Lab 3 The Revolution of the Moons of Jupiter

GRAPH OF APPARENT POSITION OF A MOON

The apparent position of a moon varies sinusoidally with the changing angle from the line of sight, θ, as it orbits Jupiter. Here the apparent position is measured in units of the radius of the moon’s orbit, R, and the angle measured in degrees.

The in-person version of this lab has a program that allows you to make measurements of the time variation of each moon's position and then fit a Sine curve to the data.

For the present online version of the lab, based on the Stellarium-web software, you will instead just measure: 1. the maximum apparent distance a from Jupiter (in Jupiter diameters DJ) of each of the 4 Galilean moons; 2. the time P between successive maximum separations to the same side of Jupiter.

Procedure 1. As done in the Lab02 for last week, launch the Stellarium-web site on your favorite web browser. 2. Feel free to play around to familiarize yourself again with the site.

3. When you're ready, toggle off the "Atmosphere" icon along the bottom menu, and the upper-right icon to close the side bar. 4. Then either navigate on your own to a south facing view towards Jupiter, or just use the Search box at the top to automatically center on Jupiter. Upon clicking on Jupiter, your screen should then look like the "observation screen" given below.

5. Next toggle the icon to convert to the button that allow one to zoom in (+) or out (-).

6. With Jupiter thus selected, click repeatedly on the (+) button to zoom into a resolved image of Jupiter with its various moons:

7. Of course, depending on the time set, the positions of the 4 Galilean moons shown here will vary.

UDel Physics 3 of 9 Fall 2020 PHYS133 Lab 3 The Revolution of the Moons of Jupiter

OBSERVATION SCREEN

8. Next go to the bottom right to toggle on the time controller window

9. Now do something you can’t do with the real sky, namely click on the up/down arrows for minutes, hours and days to magically move time forward/back. While you do this watch the moons zip back and forth as the time and even change. With this play, it’s fairly easy to see that what the moons are really doing is circling the while you view their orbits edge-on. 10. When you are satisfied that you understand the motions of Jupiter’s moons and why they appear the way they do, you are ready to start the lab.

UDel Physics 4 of 9 Fall 2020 PHYS133 Lab 3 The Revolution of the Moons of Jupiter

Procedure

Data Collection

1. First, for each of the 4 Galilean moons, you will find on the data sheet below, space to write down: a. The name of the moon.

b. The maximum separation, a, of that moon from Jupiter, measured in Jupiter diameters DJ. c. The date and time of at least 2 successive moments of such maximum separation to the same side of Jupiter. 2. Starting first with the innermost moon Io, adjust the time and day to when Io is furthest from Jupiter. (Hint: you may find it useful to toggle on the grid icons to provide a better reference background for adjusting to this maximum) 3. Once you have adjusted time to find this maximum, you can measure the separation, a, by placing a ruler (or perhaps a string) across your computer screen. Record this in units of Jupiter's diameter, DJ. 4. Next, run the clock backward or forward to find the previous or next maximum distance. Record the time it took to get from one maximum distance to the next. 5. Continue this process for all four moons. 6. Once you have the separation and time difference, calculate the mass of Jupiter using the data sheet below.

UDel Physics 5 of 9 Fall 2020 PHYS133 Lab 3 The Revolution of the Moons of Jupiter

Names: ______Section: ______

______Date: ______

Data Sheet

Initial Time Final Time Semi-major Axis Moon Period (days) (days) (days) (Jup Diameters)

Io

Europa

Ganymede

Callisto

Semi-major Axis Period (years) (AU)

UDel Physics 6 of 9 Fall 2020 PHYS133 Lab 3 The Revolution of the Moons of Jupiter

Table 1 – Mass of Jupiter in Solar Masses

where M is the mass of Jupiter in solar masses Msun a3 M = a is the orbital radius in A.U. (astronomical units) 2 p is the period of the orbit in Earth years. p

From Io MJI =

From Europa MJE=

From Ganymede MJG=

From Callisto MJC=

Average: MAVG =

Show your work below:

UDel Physics 7 of 9 Fall 2020 PHYS133 Lab 3 The Revolution of the Moons of Jupiter

Questions

1. There are moons beyond the orbit of Callisto. Will they have larger or smaller periods than Callisto? Why?

2. What are the sources of error when it comes to the calculation? Which measurement is more likely to contribute to the error? Why?

3. How does your result compare to the actual mass of Jupiter? Answer in terms of percent. (Google the currently accepted value of Jupiter’s mass)

4. The orbit of earth’s moon has a period of 27.3 days and a radius (semi-major axis) of 2.56 x 10-3 A.U. (= 3.84 x 105 km). What is the mass of the Earth? How does this compare to the mass of Jupiter?

UDel Physics 8 of 9 Fall 2020 PHYS133 Lab 3 The Revolution of the Moons of Jupiter 5. Now consider the other possibility, that Jupiter is being gravitationally affected by Ganymede. Using Kepler’s third law, find what the radius of Jupiter’s “orbit” would be approximately. Use the period of Ganymede you calculated previously, and the mass of Ganymede, which is 1.48 x 1028 kg. This calculation will show that while Jupiter’s movement is real, it is impossible to detect this via the naked eye. This orbit is more akin to a wobble.

6. Using Stellarium, find the on Jupiter. Use the red spot as a reference point in order to observe Jupiter’s rotation. Scroll through time in order to find out how long it takes for the red spot to return to the same spot. How long is a day in Jupiter? How does it compare to Earth’s day? What is Jupiter's equatorial rotation speed Vrot, in km/hr How does it compare to Earth's equatorial rotation speed?

UDel Physics 9 of 9 Fall 2020