# Flexing Muscles and Moons

Target Level: Middle School

Description: What force is responsible for the tremendous volcanic activity on Jupiter's moon Io? How does the gravity 'tug-of-war' between Jupiter and its moons produce heat within the moons, and what do the interiors of Jupiter's inner moons Io and Europa look like as a result?

Objectives: Students calculate the difference in gravity across Io, and then model the tug-of-war heating (a process called 'tidal flexing') by repeatedly squeezing a rubber ball. Students then act out the inner structures of Io and Europa.

Vocabulary: Volcano, plate tectonics, gravity, orbit, ellipse, tidal flexing, friction, spherical

Materials: Each group will need two hand-sized flexible rubber balls (see Teacher Notes), scissors or X-acto knife, stopwatch or timer, and two thermometers

Introduction:

Jupiter's moon Io is the most volcanically active object in the solar system. On Earth, volcanoes are found mainly along the colliding boundaries of great crustal plates. From what we can tell, however, Io has no tectonic plates (plates in motion). What is the cause of this volcanic activity on Io? The answer is gravity. Gravity is a force of attraction between all objects in the Universe. The equation for gravity between any two objects is:

F = Gm1m2/R2

In this equation, F is the force of gravity, G is the gravitational constant (6.67x10-8 cm3/g-sec2), m1 is the mass of one object (such as a moon), m2 is the mass of a second object (such as another moon or planet), and R is the distance between the two objects.

That's a lot of numbers! To compare the gravity attraction between two moons when they are at different distances from each other, we can ignore G, m1, and m2, because they stay the same (the moons themselves aren't getting bigger or smaller!). When we do this, we see that the force of gravity between two objects is simply proportional to 1/R2. Stated another way, gravity is a force that weakens with the inverse square of the distance between two objects. When two values like gravity and distance are proportional in some way, it means that when one number changes, the other changes by a known factor. If we double the distance between two objects (2R), the gravitational force changes by a factor of 1/(2)2, or 1/4, i.e., when the objects are twice as far away, they exert 1/4 of the gravity pull on each other. If we halve the distance (1/2R), the gravitational force changes by a factor of 1/(1/2)2, or 4.

Now that you know how gravity works, let's take a closer look at Io. First, Io occupies an orbit very close to the largest planet in the solar system, Jupiter. Because it is so close, the pull of Jupiter is significantly higher on the side of Io that faces towards Jupiter than on the side that faces away from it. This gravity difference from one side of a moon to the other is called a gravity gradient. An example of a gravity gradient that you are familiar with is ocean tides on Earth. The rising of a high tide is caused by the gravity pull on the side of Earth that faces the Moon. The tidal bulge on the opposite side of Earth results from that side being attracted toward the Moon less strongly than is the central part of Earth. On planets or satellites without oceans, the same forces apply, but they cause stresses in the solid body.

Second, Io's orbit around Jupiter is an ellipse, which means that its distance from Jupiter changes during a complete orbit. When Io is close to Jupiter, the gravity of Jupiter tries to pull and stretch Io into the shape of an egg. When it is furthest away from Jupiter, Io relaxes to a more spherical shape. Finally, Jupiter has other large moons that exert their gravitational influence on Io, pulling it in other directions still. This is like a giant tug-of-war with Io stuck in the middle!

The rising and falling of Io's surface is caused by the same force (gravity) that causes the rise and fall of tides on Earth's oceans, so we call it 'tidal flexing'. The flexing of solid rock inside a moon produces a lot of friction and heat. Enough heat, in fact, to melt rock into magma and produce erupting volcanoes on Io. Tidal flexing also affects Europa, the next moon outward from Io, although the amount of energy produced is much less because of its greater distance from Jupiter (less pull of gravity leads to weaker flexing). Even so, enough heat may be produced in Europa to partially melt ice deep in the crust, which may have created an ocean under the surface! (See "Europa Geology Jigsaw Puzzle".)

Activity 1: Io's Gravity Gradient

Knowing how gravity is related to the distance between two points, we can calculate the relative gravity gradient across Io by comparing Jupiter's gravitational attraction at the center of Io with its gravitational attraction on both the near and far sides of Io. You may wish to draw a diagram showing the proper distances involved:

• radius of Jupiter: 71,000 km
• Mean Distance from Jupiter's surface to center of Io: 421,600 km
• radius of Io: 1,815 km
• Jupiter's gravitational force at center of Io
• Distance from center of Jupiter to center of Io = 492,600 km
• R = 492,600 km/492,600 km = 1
• F &181; 1/R2
• F &181; 1/(1)2
• F &181; 1, or 100% of the gravitational attraction of Jupiter

Now let's calculate the force of gravity from Jupiter on the side of Io that faces towards the planet (figure 1, point A). The radius of Io is 1,815 km, so we have to subtract this value from the distance between Jupiter and Io. Although we have not actually moved Io closer to Jupiter, this change in the value for R will allow us to calculate the approximate proportional change in F, or the Force of gravity, on the near side Io.

Jupiter's gravity force on near side of Io
Distance from Jupiter to near side of Io = 492,600 km - 1,815 km = 490,785 km
Change in Radius = 490,785 km/492,600 km = .996
F &181; 1/R2
F &181; 1/(.996)2 F &181; 1.008, or 100.8% compared to gravitational attraction of Jupiter at the center of Io

Questions:

1. Calculate the relative gravitational force of Jupiter on the side of Io that faces away from Jupiter (figure 1, point B).
2. What is the difference (in %) in Jupiter's gravitational attraction between the near and far sides of Io?

The values you come up with may not seem very impressive, but remember that Jupiter's mass is much greater than Io's. In terms of pulling and stretching, even a small gravity gradient turns out to be very significant. Remember also that we have not considered the effects of Io's eccentric orbit or the influence of the other Jovian moons. Tidal flexing of the Jovian moons is a complex phenomenon that we are just beginning to understand!

Activity 2: Great Balls of Fire!

What happens when gravity forces flex rock? Where does the heat actually come from? In this activity we will model the effects of tidal flexing on a solid body.

Procedure

Take two hand-sized foam rubber balls (or other flexible balls) and cut a single hole, just large enough to insert a thermometer, in each. Measure the starting/standing temperature of one of the balls. Record this in the table provided. Have someone hold the first ball but do not squeeze it. At the same time, have another person simulate the gravitational flexing of Io by alternately squeezing and relaxing the second ball for 5 minutes (or until they get too tired). As soon as time is up, insert a thermometer into each of the balls, one squeezed and the other not squeezed, and record the temperatures on the data table.

Record the initial and final temperatures, and then calculate and record the temperature change.

Finally, record the total squeezing time (this should be 5 minutes unless you stopped earlier).

 Starting temperature Ending temperature of ball NOT squeezed Ending temperature of ball squeezed Temperature change ball NOT squeezed Temperature change ball squeezed Temperature difference between the two balls Total flexing time (in minutes)

Questions:

1. Why did you measure the temperature of a ball that was held but not squeezed?
2. How fast did the squeezed ball heat up (calculate in degrees per minute).
3. What are the limits to heating the balls by holding and by squeezing?
4. How often is Jupiter's moon Io 'flexed?'
5. What are some other ways of heating up a planet or moon?
6. How could scientists get a direct measure of the amount of tidal flexing on Io or Europa?

Teacher Notes

Answers to Gravity Questions:

1. Jupiter's gravitational force on far side of Io
- Distance from Jupiter to far side of Io = 492,600 km + 1,815 km = 494,415 km
- Change in Radius = 494,415 km/492,600 km = 1.004
- F &181; 1/R2
- F &181; 1/(1.004) 2
- F &181; .992, or 99.2% compared to gravitational attraction of Jupiter at the center of Io
2. The difference in Jupiter's gravitational attraction between the near and far sides of Io is about 1.5%.

Great Balls of Fire!

Ball selection. The best rubber balls for this activity are made of thick foam, solid throughout, but 'squishable'. You will need to pre-test the balls you select to see if they can be heated by a measurable amount.

Answers to Review Questions:

1. Measuring the temperature of a ball that was held but not squeezed serves as a 'control'. Heat from your hand may contributeto warming the ball.
2. Rates will vary. Typical results for solid foam rubber balls are about 1 degree F per minute.
3. You could heat the ball held (but not squeezed) to body temperature, 98.6 degrees F. Theoretically, you could heat the ball by squeezing it until it melted and flowed out of your hand. Of course you would have to squeeze it VERY fast and for a LONG time, and your hand would be badly burned in the process!
4. Once during each orbit around Jupiter, which is approximately 1.77 Earth days.
5. Solar energy; impacts by meteors from space; radioactive decay from within.
6. Scientists would need to precisely measure the diameter of each moon through a complete orbit to see the changes. A laser beam bounced off the surface (similar to the echo location used by bats, except using laser light rather than sound waves) could accurately measure the changes in the moon's surface.

Physical Demonstration Using Students

Another way to illustrate the effects of tidal flexing is to select small groups of students to act out the dynamic interior structures of Io and Europa. You may wish to have your "student moons" orbit a central point (Jupiter), and respond to its tremendous gravitation at different positions and distances.

Io Group

Core of Io: super hot and wild

Molten Interior of Io: several students standing around the core of Io, facing outward, arms loosely interlocked, relatively mobile

Sulfur Volcanoes: responding to the gyrations of molten sulfur compounds bursting forth

The tidal flexing is so dramatic that the Molten Interior constantly creates new volcanic activity!

Europa Group

Core of Europa: dynamic, hot, under pressure

Rocky Mantle of Europa: several students standing around Core of Europa, facing outward, arms interlocked, relatively immobile

Deep Ice/Ocean of Europa: first as ice, in a similar arrangement to rocky mantle, several students encircling, arms interlocked, malleable

Surface Ice of Europa: several students with hands placed outward, flat, cold, showing an icy surface

Tidal flexing stretches and unstretches the whole group. As long as the whole structure (surface ice, deep ice and rock) is solid, the tidal flexing is a relatively small effect. If heat from the core emanates through the rock enough to sustain a melting of the deep ice, creating an ocean, then the ocean sloshes around inside, creating a more pronounced distortion of the shape of Europa.