## Moment of Inertia

The moment of inertia of a planet is a measure of how much force is required to increase the spin of the planet. (In more technical terms, the angular acceleration of an object is proportional to the torque acting on the object, and the proportional constant is called the moment of inertia of the object.)

The moment of inertia depends on the mass of the planet and on how this mass is distributed around the planet's center. The farther the bulk of the mass is from the center of the planet, the greater the moment of inertia. In other words, if all the mass is at the outside, it takes more force to spin the planet than if all the mass is at the center. This is similar to an example of two wheels with the same mass: one is a solid plate and the other is a bicycle wheel, with almost all the mass at the rim. The bicycle wheel has the greater moment of inertia and takes more force to create the same angular acceleration. The units of the moment of inertia are units of mass times distance squared; for example, lb X ft2 or kg X m2.

By definition, the moment of inertia I is defined as the sum of mr2 for every piece of mass m of the object, where r is the radius for that mass m. In a planet, the density changes with radius, and so the moment of inertia needs to be calculated with an integral:

where rg is the center of the planet and r is the total radius of the planet, p(r) is the change of density with radius in the planet, and r is the radius of the planet and the variable of integration. To compare moments of inertia among planets, scientists calculate what is called the moment of inertia factor. By dividing the moment of inertia by the total mass of the planet M and the total radius squared R2, the result is the part of the moment of inertia that is due entirely to radial changes in density in the planet.

lie slightly below the level of the surrounding plains. In contrast, Thrace (right) is longer, shows a hummocky texture, and appears to stand at or slightly above the older surrounding bright plains.

Europa is thought to have an iron core, a silicate mantle, a layer of liquid water, and a crust of ice.The liquid ocean under the crust of ice

This division also produces a non-dimensional number because all the units cancel. The equation for the moment of inertia factor, K, is as follows:

The issue with calculating the moment of inertia factor for a planet is that, aside from the Earth, there is no really specific information on the density gradients inside the planet. There is another equation, the rotation equation, that allows the calculation of moment of inertia factor by using parameters that can be measured. This equation gives a relationship between T, the rotation period of the planet; K, the moment of inertia factor of the planet; M, the mass of the planet; G, the gravitational constant; R, the planet's polar radius; D, the density for the large body; a, the planet's semimajor axis; i, the orbital inclination of the planet; m, the total mass of all satellites that orbit the large body; d, the mean density for the total satellites that orbit large body; and r,the mean polar radius of all satellites that orbit the large body:

By getting more and more accurate measures of the moment of inertia factor of Mars, for example, from these external measurements, scientists can test their models for the interior of Mars. By integrating their modeled density structures, they can see whether the model creates a moment of inertia factor close to what is actually measured for Mars. On the Earth, the moment of inertia factor can be used to test for core densities, helping constrain the percentage of light elements that have to be mixed into the iron and nickel composition.

could explain the lineations in the crust: Movement of the fluid beneath created tension and compression in the crust, making it buckle or fault. Europa's surface is covered with long curved features called arcuate ridges, unique in the solar system.The longest arcuate ridge is called Astypalaea Linea, and it is over 560 miles (900 km)

This chaos region on Europa is thought to have been made by movement in a partly solid, partly liquid water crust. (NASA/JPL/Galileo)

This chaos region on Europa is thought to have been made by movement in a partly solid, partly liquid water crust. (NASA/JPL/Galileo)

long.Wide, overlapping cracklike features also may indicate a kind of ice plate tectonics, in which certain areas spread and new ice forms in the cracks, while in other areas the icy plates press together and make long compressional ridges.

These long cracks may be caused by gravitational pulls from Jupiter and from the other Galilean satellites. Intriguingly, this particular process would only work if the icy shell were floating on a liquid ocean. Since water flows so much more easily than solid ice, it can respond to tidal forces more quickly and it can move farther than solid materials would. If Europa has a subsurface sea, then its ice crust should rise and fall by 100 feet (30 m) during each 3.6-day orbit around Jupiter. Shown in the lower color insert on page C-7, these wild fluctuations are sufficient to cause the cracking and ice resurfacing seen on the planet. If there is no liquid ocean, tides will only cause the surface to rise and fall by three feet (1 m), insufficient to cause the lineations.

Different approaches to calculating the thickness of the icy shells and oceans on Europa bring different predictions. Using gravity measurements from the Galileo mission, a team of scientists lead by Frank Sohl at the University of Münster concluded that Europa has an ice crust between 75 and 105 miles (120 and 170 km) thick. Its core is constrained to be between 10 and 45 percent of the its radius, and its mantle is thought to be silicate and dominated by olivine, as Io's mantle is. A separate team also at the University of Münster, led by Hauke Hussmann, created computer models of the tidal heating created in Europa by Jupiter's giant gravity field and determined that Europa's ice shell can be only a few tens of kilometers thick, with underlying water oceans about 60 miles (100 km) deep. Hussman's estimate, then, gives Europa an ice shell only about half, or less, than the one Sohl's estimate gives. Either of these estimates indicates that Europa has more water than the total on the planet Earth!

Some scientists suggest that heat from below breaks up blocks of ice at the surface, which drift around and agglomerate to form Europa's surface features. The heat from below comes from radioactivity and tidal heating, while the surface is cooling everywhere. The oceans are thought to be convecting fiercely enough to have no stratification, in contrast to Earth's oceans, which are stratified. Other scientists believe that Europa's oceans are maintained as liquid and not allowed to freeze by tidal friction heating from Jupiter. These 15 images show the

Chaos regions, such as the Thera region and the Thrace region, may range of surface images on also be caused by dynamics of a liquid ocean under an ice crust. Europa. (NASA/JPL/Galileo)

Heating in the solid interior of the planet from tides and radioactive elements may cause icy blobs buoyant enough from warming to rise through the liquid ocean. These blobs, called diapers, would cause disruption of the crust when they strike it from below, creating chaos regions. Smaller diapers may cause other small circular features, called lenticulae. Lenticulae are named after the Latin word for freckles, and may be domes, pits, or dark spots. In the image presented on page 83, a wide variety of surface feature examples from Europa are shown, including chaos regions, lineations, lenticulae, and craters modified by flowing ice.

The discovery of a liquid water ocean on Europa is immensely exciting, since the combination of liquid water and organic molecules are what made life possible on Earth, so perhaps Europa, with those same elements, may also produce life. As Richard Terrile, a NASA scientist, has pointed out, "How often is an ocean discovered? The last one was the Pacific, by Balboa, and that was five hundred years ago!" Two experts in cratering, Elisabetta Pierazzo at the University of Arizona and Christopher Chyba of Stanford University, have calculated that at low impact velocities (10 mi/sec, or 16 km/sec), the carbon and carbon-based molecules held in comets are retained on Europa and not expelled back into space. Over solar system history, large comets may have delivered 1 to 10 billions of tons of carbon to Europa. This is a few times more carbon than is contained in the upper 60 feet (200 m) of Earth's oceans, where most life exists, but it is about 100 times less than all the carbon in all the oceans when the entire ocean depth is considered. Another researcher, Paul M. Schenk at the Lunar and Planetary Institute, measured the depths of impact craters on Europa, Ganymede, and Callisto, and found that Europa's crater shapes indicate an especially dense and cold ice shell, indicating that exchange of organic materials between the surface and the liquid ocean beneath could be slow or even impossible.

Because Europa is a possible place to find life, planetary scientists are especially careful not to allow any spacecraft from Earth to contaminate the moon (no matter how clean the spacecraft is when it leaves Earth, and no matter how long its passage in space, a spacecraft is almost certain to carry microscopic life with it from Earth). Europa is the main reason that NASA crashed the Galileo mission into Jupiter; the scientists wanted to leave no chance that the spacecraft might fall into Europa as its communications and steering failed at the end of its lifetime.

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