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Homework #3 Solutions
Questions due at the beginning of lecture on Mar 7, 2000.

Chapter 7 Question 3:  Pluto
Pluto does not fit the usual classification of terrestrial or Jovian because its average density is much lower than the terrestrials (more like the Jovians) but it does not have a huge large atmosphere like the Jovians.  Based on its density (1100 kg/m3) Pluto is probably composed of a mixture of rock (density ~3000 kg/m3) and ice (density ~1000 kg/m3).

Chapter 7 Question 4:  Largest satellites (moons).
The 7 largest moons are almost as large as the terrestrial planets.  Also, they have average densities that higher than all of the Jovian planets.  From the average density, spectroscopic studies, and spacecraft visits, we know that many of these moons have rocky surfaces, like the terrestrial planets

The 7 largest moons are the 4 Galilean moons of Jupiter:
          Io
          Europa
          Ganymede
          Callisto
     Our moon
     Titan (Saturn)
     Triton (Neptune)

Chapter 7 Question 7: Average density
Average density is defined as total mass divided by total volume (note typo in box 7-1).  If the average density of a planet is high (>3000 kg/m3), the planet is terrestrial.  If it is low (<2000 kg/m3), the planet is probably Jovian.  The average density can also gives an indication of the internal structure of a planet.  For instance, most rock on the surface of the Earth is a density of ~3000 kg/m3, but the Earth's average density is 5515 kg/m3.  Therefore, there is something much denser inside the earth (its iron core).

Chapter 7 Question 21: Spectral lines from a moon.
Looking at one spectrum, it is difficult to tell which lines come from which source.  However, if several spectra are recorded as the moon orbit its planet, the Doppler effect will shift the moon's lines back and forth (to the red and the blue) while the Earth's lines will stay in the same place.  The Sun's lines will be shifted even more than the moon's lines because of the combined motion of the Sun and the moon and the moon and the Earth.

Chapter 7 Question 22: Mass and average density of Mars
Using the orbital information of the Martian moon Phobos, we can  calculate the mass and average density of Mars.  Since we have a distance and a period and want a mass, we can use Newton's form of Kepler's 3rd law.

The period is given as in days, but needs to be converted to seconds for the units to work out:
We also know a, though it is given as bits and pieces.  We need to know the distance between the center of Mars and the moon, so we add up the altitude of the moon above Mars' surface and half the diameter of Mars to get the semimajor axis of the orbit (and convert to meters):

Like we did in the last homework, we can assume the satellite has a negligible mass compared to Mars, or m2 = 0.  Since 0 is a nice easy number to work with, lets plug that is first, solve the equation for m1 while it is still just variables and then plug in.

At this point, it is helpful to know that 1 N = 1 kg m/s2.  That means all of the units cancel out except kg, so m1 = 6.4 x 1023 kg.
    To calculate the density, you need volume as well.  You can assume Mars is a sphere, which has a volume of 4/3 p R3.  Remember to express R in meters: R = D/2 = 3.397 x 106 m, so the volume is 1.642 x 1020 m3.  Average density is this divided into m1, or 3900 kg/m3.

Chapter 8 Question 4: Center of the Earth
Scientists have studied the inner structure of the Earth using seismic waves.  Waves that originate on one side of the globe at the site of an earthquake can be detected over most of the Earth.  From the intensity of the waves in different places, scientists have deduced the density structure inside the Earth.

Chapter 8 Question 7: Plate tectonics
The Earth's crust is broken up into sections called plates.  These plates are floating on a layer of  semi-liquid (plastic) rock called the asthenosphere.  Convection in the asthenosphere (the lava lamp effect) disturbs the plates and causes them to move relative to each other.  Mountain ranges, such as the Andes, Rockies and the Himalayas have been pushed up along the boundaries of two colliding plates.  The Mid-Atlantic ridge is a range of mountains in the middle of the Atlantic Ocean that has formed as two plates separate and new crust is added volcanically from below.

Chapter 8 Question 11: Structure of Earth's atmosphere.
The Earth's atmosphere is divided into four layers by differences in the behavior of the temperature in these regions (see figure).  The Earth's atmosphere is mostly nitrogen, but life, specifically plant life, has produced large amounts of oxygen.  Now the Earth's atmosphere is almost 1/4 oxygen.   The lower region of the atmosphere (the troposphere), is not efficient at absorbing energy directly from the Sun (unlike the Ozone layer in the stratosphere).  Instead, sunlight falls on the ground which then heats the atmosphere from the bottom.  The hot air rises and moves toward the colder parts of the Earth's surface.  The resulting circulation (called convection) would tend to move air from the equator to the poles, but the rotation of the Earth breaks this pattern up (see figure).


Chapter 8 Question 15: Earth's surface relatively crater free
First of all, the moon and the Earth have been bombarded at the same rate since they started orbiting each other.  Mars and Mercury also have had similar bombardment rates.  The Earth's surface, however,  is being continually renewed by the processes of plate tectonics and volcanism.  Wind and rain also act to wear mountains down even as they are formed.  As a result, impact craters from meteors are erased within a few hundred million years.  The moon and Mercury and Mars do not have any plate tectonics nor (at present) surface water.  As a result, there are fewer processes that can erase the impact craters.

Chapter 8 Question 18: The ages of Hawaii and Kauai islands.
The Hawaiian islands are over a hot spot in the Earth's mantle.  The hot spot stays in the same place, but the Pacific plate moves at the rate of 5 cm per year northwest.  Volcanoes that form over the hot spot eventually become extinct as the plate moves too far from the hot spot.  When a new path for the magma to reach the surface forms, a new volcano is born, to the southeast of the older volcanos.  If two islands are separated by 300 miles we can calculate their relative ages using the speed of pacific plate.  It is easiest to do this problem if you just write down the numbers and their units to see how to multiply or divide them to get years in the final answer.  But first lets convert miles to centimeters so we have consistent length units:

Now we experiment with multiplying and dividing until the units cancel to give us years:

Note that the answer in the back of the book is 1 x 107 years, which is just our answer rounded to one significant digit.  Since we only have input values with one significant digit (300 mi and 5 cm/year), one significant digit is all we should have in our answer as well.

Chapter 9 Question 3:  Temperature on the moon
The temperature swings on the moon are more severe than on the Earth because the moon has no atmosphere to store heat.   Our atmosphere, particularly the CO2 in our atmosphere acts like a blanket.  This is why the increase in the CO2 content in the atmosphere shown on page 214 of the text is potentially alarming.

Chapter 9 Question 4: Moon rocks are so old...
The moon and the Earth probably formed at about the same time.  The moon, however, being smaller, cooled faster and long ago ceased to have processes like volcanism and plate tectonics, which can bring recently molten material to the surface.

Chapter 9 Question 6:  Few craters on the Maria
Shortly after the planets formed, there were still a lot of planetesimals floating around.  These planetesimals continued to rain down for about a billion years.  Toward the end of this intense period of bombardment, a few relatively big objects hit the moon.  The craters were deep enough to weaken the crust and allow lava to pour out onto the moon's surface (the moon bled).  These lava beds are the Maria.  Since there weren't many planetesimals left after this point, there haven't been very many impacts since then.  Thus, the Maria have relatively few craters.

Chapter 9 Question 18: How much does an 80 kg person weigh on the moon?
Weight is the effect of the force of gravity on a mass.  That is, in the "F = ma" equation, mass is m and weight is F.  What is a?  The problem-solving tips box above this problem gives the "acceleration due to gravity on the Earth's surface" as 9.8 m/s2.  That is, a = 9.8 m/s2.  So a person on the Earth whose mass is 80 kg weighs;

F = ma = 80 kg x 9.8 m/s2 = 784 N

Table 9-1 gives the surface gravity of the moon compared to Earth as 0.17.  So an 80 kg person would weigh

F = ma = 80 kg x 9.8 m/s2 x 0.17 = 133 N

Another way to approach this problem is to use Newton's universal law of gravitation:

where m1 is the mass of the Earth (5.974 x 1024 kg) and R is the radius of the Earth (6.378 x 106 m).  Plugging in just G, m1, and R, we get

F = (9.795 m/s2)(m2)

In other words, we just derived the acceleration of gravity on the Earth's surface.  The same thing works for the moon.  Try it!


 
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