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Homework #6 Solutions
Questions due at the beginning of lecture on April 25, 2000.



From Chapter 17.

Question 1.  Discovery of asteroids
Astronomers were looking for a planet between Mars and Jupiter and found many small objects (diameters less than 500 km) orbiting the Sun.  These are now called asteroids or minor planets.

Question 2.  The asteriod belt.
Most asteriods have orbits that lie within the plane of the ecliptic and at distances from the between 2 and 3.5 AU.  These are the "belt" asteroids.  Several asteroids, such as Apollo and Icarus have more elliptical and/or inclined orbits.

Question 3.  Kirkwood gaps
Kirkwood gaps are gaps in the distribution of asteroid orbital periods.  These gaps occur and simple fractions (e.g. 1/2, 1/3, etc) of Jupiter's orbital period.  Kirkwood gaps are caused by the gravitational interaction between Jupiter and asteriods with these periods.  The orbits of asteriods that fall into the Kirkwood gaps are regularly disturbed every few orbits (2, 3, etc.)  The orbits are drawn into progressively more and more ellipitcal shapes until they come close enough to Jupiter for Jupiter's gravitational attraction to have more effect than the Sun's.  At this point, the asteroid is usually  flung completely out of the solar system.  This same effect explains why Saturn's moon Mimas causes the Cassini division in Saturn's rings.

Question 5.  The Trojan asteroids
The Trojan asteroids are groups of asteroids that share Jupiter's orbit.  One group orbits ahead of Jupiter by 60 degrees, the other behind by 60 degrees.  The stability of these orbits are made possible by a symmetry between the gravity of the Sun and Jupiter called Lagrange points.  A similar arangement is found in Saturn's moons Tethys, Calypso and Telesto.

From Chapter 13.

Question 2.  Jupiter's chemical composition compared to the Sun
The chemical compositions of the Sun and Jupiter are very similar:
 
Element Sun Jupiter
Hydrogen 74% 71%
Helium 25% 24%
Other 1% 5%

This similarity suggests that the Sun and Jupiter formed from the same source of material (the solar nebula) and probably at the same time.

Question 3.  Jupiter's excess heat
Jupiter is large enough so that it is probably still cooling down from its initial formation 5 billion years ago.  This means its interior is still warm and heat is continually leaking out.

Question 4.  Belts and zones
Belts and zones are the dark and light bands (respectively) at the top of Jupiter's atmosphere.  They are formed by convection in the atmosphere (the lava lamp effect) where belts are cool material that is sinking and zones are warmer material that is rising. The Great Red Spot is more like a zone than a belt because it is made up of warmer material that is rising up from below.

Question 5.  Jupiter is oblate
Jupiter is oblate (its equitorial diameter is larger than its polar diameter) because Jupiter is mostly a rotating ball of gas.  As it rotates, the material at the equator gets flung out by centrifugal acceleration with greater for than the material at the poles.  The amount of oblateness indicates the size and density structure of the outer layers of the atmosphere.

Question 6.  Internal rotation
Jupiter's internal rotation rate is probably about the same as the rotation rate of its magnetic field, which is generated deep inside the planet.

From Chapter 14.

Question 1.  Compare and contrast Galilean satellites to Solar System
Jupiter's Galilean satellites all appear to have formed at the same time as Jupiter, just as the planets probably formed at about the same time as the Sun.  Our evidence for this is that the Galilean satellites all orbit roughly in the plane of Jupiter's equator and their compositions and internal structure are a function of distance away from Jupiter.  The evidence indicates that heat from Jupiter helped keep water and other light elements in vapor form on Io and Europa, which is why they are somewhat more dense than Ganymede and Callisto.  On the other hand, Io, the innermost Gallilean satelllite is very geologically active because of tidal heating and heating from Jupiter's strong magnetic field.  Mercury, the innermost planet, on the other hand, is geologically dead: it is too small to have retained its heat of formation and it does not generate much internal heat from tital interactions with the Sun.

Question 2.  Source of energy for Io's volcanos.
Io's volcanos are powered by heat that is generated from within Io.  Where does this heat come from?  Most of it probably comes from tidal forces; some might be caused by Jupiter's strong magnetic field.  Io's orbit it distorted by its periodic proximity to Europa and Ganymede.  The resulting elliptical orbit brings Io closer to and farther away from Jupiter, which changes the internal stresses on Io, resulting in frictional heating.

Question 5.  The Io torus
The Io torus is a doughut-shaped cloud of plasma (super hot gas)  that is locked into Jupiter's magnetic field by the interaction of the charged particals and the magnetic field.  The material in the torus is probably volcanicly ejected from Io, which would explain why the torus is close to Io's orbit.

Question 9.  Evidence of liquid water under Europa's surface.
First of all, the surface of Europa is mostly ice.  The picture below shows  that there is something under this ice that is either soft and slippery or all together liquid.  Pieces of the ice surface have clearly broken apart from eachother and floated or slipped  to new positions.  New material has frozen in between the

Question 10.  Life on Europa?
Europa may have liquid water, an ingredient essential for life as we know it.

Question 11.  Ganymede's magnetic field
Magnetic field is generated by moving charges.  Since Ganymede has a magnetic field, that means there must be charges moving inside of it, which implies some sort of liquid.  However, Ganymede's current orbit is too far from Jupiter to have enough tidal heating to keep its interior warm enough to support liquid.  One possibility is that Ganymede previously had a more ellipitical orbit that brought it closer to Jupiter, where tidal heating is greater.  Then something happened to circularize Ganymede's orbit and it has yet to cool off.

Question 20.  Kepler's 3rd law

Let's make a table with the book's P and a values and then calculate P2/a3 for all of the satellites.  This should work out to be the quantity in the square brakets in all cases:


Moon P (days) P (seconds) a (km) a (m) P2/a3 (sec2/m3)
Io 1.769 152840 421600 4.216 x 108 3.12 x 10-16
Europa 3.551 306810 670900 6.709 x 108 3.12 x 10-16
Ganymede 7.155 618190 1.070 x 106 1.070 x 109 3.12 x 10-16
Callisto 16.689 1441900 1.883 x 106 1.883 x 109 3.11 x 10-16

 
 


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