The Earth's moon has a mass approximately 1/81 that of the Earth, and the moon's center of mass is on average approximately 60 Earth radii away from the center of mass of the Earth.

Q1: Sketch this situation as two circles, including a line from center to center from Earth to moon, extending the line on through the Earth to both surfaces of the Earth. Label objects and the points on the Earth's surface closest and farthest from the moon.

Q2: The Earth has an average radius of 6.37 x 10⁶ m and a mass of 5.98 x 10²⁴ kg. Calculate the gravitational field strength of the moon at these points on the Earth:

a. the nearest point on the Earth's surface to the moon g_CloseMoon

mg = GMm/r²   or   g = GM/r²

g = (6.67 x 10^-11Nm²/kg²)(7.38 x 10^22kg)/(3.76 x 10⁸m)²

g = 3.49 x 10^-5N/kg

b. the center of the Earth g_AvgMoon

g = (6.67 x 10^-11Nm²/kg²)(7.38 x 10²²kg)/(3.82 x 10⁸m)²

g = 3.37 x 10^-5N/kg

c. the point on the Earth's surface farthest from the moon g_FarthestMoon

g = (6.67 x 10^-11Nm²/kg²)(7.38 x 10^22kg)/(3.89 x 10⁸m)²

g = 3.26 x 10^-5N/kg

Q3: What is the percentage difference between g_CloseMoon and g_FarthestMoon?

(g_close - g_{far)/(gfar) x 100 = percent difference}

(3.49 x 10^-5N/kg - 3.26 x 10^-5N/kg)/(3.26 x 10^-5N/kg) x 100 = 7.05%

Q4: Imagine our model Earth is a perfectly smooth sphere initially covered with a uniform, thin layer of water. How would this water eventually be distributed due to the differing values of g_CloseMoon, g_AvgMoon and g_FarthestMoon? Sketch the water distribution and explain this in your own words. Hint: where does water fall fastest, average and slowest towards the moon? Where does it pile up??

The oceans of the earth closest to the moon experience a stronger gravitational field strength than those on the far side because gravitational field strength is dependent upon the radius between two objects. For this reason, the oceans closest to the moon fall toward the moon with a greater velocity than the center of gravity of the earth. This causes the water to bulge out from the side of the earth closest to the moon forming what we call a high tide. The center of gravity of the earth falls toward the moon with a greater velocity than the oceans on the far side of the earth because the radius between the center of gravity of the earth and the moon is smaller than the radius between the oceans on the far side of the earth and the moon. For this reason, the earth falls away from the oceans on the far side of the earth causing the water to bulge out from the far side of the earth. This is another high tide phase that the oceans cycle through. The oceans of the earth at 90 degrees to the direction of the moon experience low tides because much of the ocean's volume is welling up on the close and far side of the earth.

Q5: Imagine our model Earth is now rotating about its center of mass once every 24 hours. Use this to explain in your own words the water tides experienced by a stick person rotating with the earth at a point on the Earth's surface. How many high and low tides each should the person see in 24 h?

The oceans on the side of the earth closest to the moon fall toward the moon with a higher velocity than the center of gravity of the earth because the gravitational field strength is dependent on the radius between two objects; the oceans closest to the moon have a radius with the moon that is slightly less than the center of gravity of the earth, and thus, experience a slightly higher gravitational attraction and are pulled away from the earth. This is what causes a high tide. At the same time, the center of mass of earth is falling toward the moon slightly faster than the oceans on the far side of the earth because the radius between the center of mass of the earth and the moon is smaller than the radius between the oceans on the far side of the earth and the moon. In effect, this means that the oceans on the far side of the earth are actually being left behind as the center of mass of the earth falls toward the moon. This causes a second high tide on the earth at any given moment. The oceans that fall 90 degrees on either side of the earth with respect to the moon experience a low tide because the water is bulged out on the earth at the closest and furthest point from the moon.

The earth rotates one full revolution about it's axis about once every 24 hours. This means that after 12 hours, the earth has made a 180 degree turn. The oceans that were experiencing a high tide on the side of the earth closest to the moon are now experiencing a high tide on the far side of the earth and vice versa. This means that in each 24 hour period, the earth experiences 2 high tides and 2 low tides.

Q6: From elementary school astronomy examples, describe a situation where the Roche limit plays a role in preventing a moon from forming.

The Roche limit is a situation where an object's tensile strength is too weak to withstand the gravitational tidal force of a larger object. If the earth's moon were too close to the earth, the difference between the acceleration with which the outside of the moon falls toward the earth and the acceleration with which the inside of the moon falls toward the earth would exceed the tensile strength of the moon, and it would be pulled apart. Similarly, accretion of a moon around a planet is impossible inside of the Roche limit. The gravitational tidal force exerted on the material within the Roche limit would be too great to allow it to form a moon because it would be pulled apart due to the difference in internal and external accelerations.

I was expecting Saturn's rings to be the 'obvious' described case here - DMI

Q7: Hypothesize why the comet Shoemaker-Levy 9 did not strike the surface of Jupiter intact.

When the comet Shoemaker-Levy 9 approached Jupiter, the gravitational force exerted on the comet by the planet exceeded the tensile strength of the comet. Again, the difference in the internal and external accelerations of the comet falling toward Jupiter became great enough that the comet's tensile strength was exceeded. This is why Shoemaker-Levy 9 was torn into several smaller comets and finally into debris that collided with Jupiter.

For more information, visit: http://www.jpl.nasa.gov/sl9.html, especially the background information section.