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Grade Level: Middle School

Description: What force is responsible for the tremendous volcanic activity on Jupiter's moon Io? How does tidal flexing produce heat, and what do the interiors of Io and Europa look like as a result?

Objectives: Students calculate the gravity gradient across Io's surface, and then model the effects of tidal flexing by repeatedly squeezing a rubber ball (applying mechanical energy) and quantifying the result (thermal energy). Finally, students kinesthetically act out the dynamic inner structures of Io and Europa.

Materials: Each group will need two hand-sized flexible rubber balls (see Teacher Notes); scissors or x-acto knife; stopwatch or timer; thermometer; optional images for kinesthetic exercise (To print images click here)

Vocabulary: Volcano, plate tectonics, gravity, gravity gradient, orbit, ellipse, friction

With over sixty active volcanic regions located so far, Io is now considered the most volcanically active body in our 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. What could be the cause of the tremendous volcanic activity on Io? The answer is gravity.

Gravity is a force of attraction between all objects in the Universe. The equation for gravitational attraction between any two objects is given as

In this equation, F is the force of gravity, G is the Gravitational constant (6.67×10^-8 cm³/g-sec²), m is the mass of the first object, m' is the mass of the second object, and R is the distance between the two objects.

That's a lot of numbers! To compare the gravitational attraction between two objects when they are at different distances, we could ignore the values for G, m, and m', because they would stay the same. 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. 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 is called a gravity gradient. An example of a gravity gradient that you are familiar with is ocean tides on Earth. The tidal bulge on the side of Earth that faces the Moon is caused by the proximity of the Moon and its relatively stronger gravitational pull on that side. 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. Think of it as 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. This flexing produces a lot of friction and heat, and leads to 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. Even so, it may generate enough heat to partially melt ice deep in the crust, which may have resulted in an ocean under the surface! (See "Europa Geology Jigsaw Puzzle".)

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

Distance from center of Jupiter to center of Io = 492,600 km

R = 492,600 km/492,600 km = 1

F µ 1/R²

F µ 1/(1)²

F µ 1, or 100% of the gravitational attraction of Jupiter

Now let's calculate the gravitational force of Jupiter on the side of Io that faces towards the planet. The radius of Io is 1,815 km, so we have to subtract this value from the overall 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.

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 µ 1/R²

F µ 1/(.996)²

F µ 1.008, or 100.8% compared to gravitational attraction of Jupiter at the center of Io

1. Calculate the relative gravitational force of Jupiter on the side of Io that faces away from Jupiter.
2. What is the difference 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!

What happens when gravitational forces stretch and bend rock, just as Jupiter and its moons create stresses on Io? 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 on the data table provided.

Have someone hold the first ball but do not flex 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 the thermometer into each of the balls and record the temperatures on the data table.

Record the differences in initial and final temperatures, and then calculate and record the difference between these two values.

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

1. What was the purpose of measuring the temperature of a ball that was held in your hand, but not subjected to flexing?
2. Determine the rate of heating for the ball that was squeezed.
3. What are the theoretical temperature limits to heating the balls by these two methods?
4. How often is Jupiter's moon Io "stretched?"
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?

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 (download images here). You may wish to have your "student moons" orbit a central point (Jupiter), and respond to its tremendous gravitation at different positions and distances. 

• The Core of Io: madly 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! 

• The Core of Europa: dynamic, hot, under pressure
• The Rocky Mantle of Europa: several students standing around the Core of Europa, facing outward, arms interlocked, relatively immobile
• The Deep Ice/Ocean of Europa: first as ice, in a similar arrangement of the rocky mantle, several students encircling, arms interlocked, malleable
• The 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.

Teacher notes are available here.

Moons in Resonance

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This module was written by Brian Exton (National Optical Astronomy Observatories, Tucson AZ).

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Galileo Solid State Imaging Team Leader: Dr. Michael J. S. Belton

The SSI Education and Public Outreach webpages were originally created and managed by Matthew Fishburn and Elizabeth Alvarez with significant assistance from Kelly Bender, Ross Beyer, Detrick Branston, Stephanie Lyons, Eileen Ryan, and Nalin Samarasinha.

Last updated: September 17, 1999, by Matthew Fishburn

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