ASTR330 Homework #4


Homework hint: All problems in the text have answers at the back of the book! Some problems require just a written response, while others ask you to calculate something. Please write up all answers clearly, completely, and as succinctly as possible. You can work with others, but your final answers must be written up on your own.

Chapter 7

Question 1.
a) Kepler's First Law: The offset between an orbit's focus (where the Sun is) and its center is given by the equation offset = a e, where a is the orbit's semimajor axis and e is its eccentricity (see Fig. 7.1). The semiminor axis is given by b = a sqrt(1-e2). Make a table with the following columns: e=eccentricity, ae=offset, q=perihelion distance, Q=aphelion distance, and b=semiminor axis. Put in the following values for eccentricity: 0.0, 0.1, 0.2, 0.4, 0.6, 0.8, 0.9, 0.99 and work out each of the other entries. Assume a=1AU. Draw a few pictures with carefully measured axes on scratch paper or on the computer (you do not need to turn these in). How good an approximation is the statement "Ellipses look like circles offset from the center"?
b) Kepler's Second Law: For eccentricities e small enough that b ~ 1 in Table 1, the speed at perihelion of an elliptical orbit is 1+e times the average speed along the orbit while the speed at aphelion is 1-e times the average. Find the eccentricity at which a planet would move twice as fast at perihelion as at apohelion. What is the ratio of Mercury's highest to lowest speeds?
c) Kepler's Third Law: Work out how long it takes Venus to lap the Earth. Start by determining how many degrees per Earth day each planet moves along its orbit. Subtract the two rates and figure out how long it takes the difference to go through a full 360 degrees. Give you answer in both days and years. Finally find the inegers X and Y: "In X Earth years Venus will circle the Sun almost exactly Y times." How does this illustrate Kepler's third law?

Rings and Satellites

Question 2. One way to estimate the Roche Limit is to calculate when the tidal force of the planet is strong enough to lift pebbles off the surface of a satellite. This is close to the location where tides on a satellite are strong enough to actually rip the satellite apart. Inside the Roche Limit we expect rings, and outside we expect satellites. The Roche radius is given by: rRoche = Rsat (3Mplanet / Msat)0.333, where "M" stands for mass, "R" for radius, and "sat" for satellite.
a) Rewrite the equation in terms of densities by assuming that the planet and satellites are spheres and substituting in the equation for their masses. Solve for rRoche/Rplanet, the Roche Limit measured in planetary radii.
b) Calculate the Roche limit for all of the planets by using the Planetary Calculator at http://janus.astro.umd.edu/astro/calculators/pcalc.html . Plug in your expression from a), assuming a satellite made from fluffy snow with density 500 kg/m3. Be sure to put a "*" symbol for multiplication in your expression.
c) How do the terrestrial planets differ from the giant ones in terms of their Roche limits? Notice that the Roche Limit is not a precise concept - it differs for satellites with different densities.

Chapter 8

Question 3.

a) Describe how the Earth and Jupiter formed, identifying key stages in their growth from kilometer-sized planetesimals to their current masses. Make a single timeline for the first 1,000,000 years of Solar System history showing the how the masses of Earth and Jupiter grew during their formation. Plot time on the horizontal axis and mass on the vertical axis and sketch a growth curve for each planet. Use info from the plots in section 8.2.7 on runaway growth, and statements in the text about what happens after that. Can you explain why there is not a planet in the asteroid belt?
b) If the temperature of the solar nebula were hotter everywhere, how might this have affected things? What if it were cooler everywhere?

Chapter 9

Question 4.

One way in which the age of meteorites is determined is by looking for atoms that are in the wrong place. For instance, a given mineral which usually has Rubidium (Rb) atoms in it sometimes has Strontium (Sr) ones substituted in instead. In most cases, Strontium, which has a different chemical behavior than Rubidium, cannot form naturally in the same minerals that need Rubidium. So how did the Strontium get there? What has happened is that some isotopes of Rubidium are radioactive - these form into the mineral normally which then solidifies. Then the radioactive Rubidium atoms start to decay one by one, each leaving a single Strontium atom behind. Since the mineral is already solidified, the Strontium cannot leave the mineral and hence is trapped in the wrong place. Measuring what fraction of the Rubidium that has decayed tells us the age of the meteorite.
The equations that govern how much Rubidium has decayed to Strontium and how much of each we expect to find after a time, t, are:

Here T = 48.8 billion years is the halflife of Rubidium, and "87" refers to the number of nucleons (protons + neutrons) in each element, "87Sr" and "87Rb" are the number of Strontium and Rubidium atoms now, and "87Sr(t=0)" and "87Rb(t=0)" are the number of Strontium and Rubidium atoms when the meteorite first formed (at time t=0).
a) Test the equations to see if they make sense. Check the limits of a short passage of time (t = 0) and a long passage of time (t >> T). What do the equations predict in these limits? Discuss whether these predictions are reasonable.
b) One book that discusses Strontium and Rubidium gives this equation: Apply your tests from part a) and discuss whether this equation makes sense.
c) Using the first set of equations, what percentage of the radioactive Rubidium should have decayed by now if the meteorites is 4.46 billion years old? Nearly all meteorites show this same percentage, indicating that they all have similar ages.

Question 5.

a) Describe "Calcium-Aluminum rich inclusions". Describe how and when they form (if known), any interesting properties that they have, and why they are interesting.
b) Describe "Chondrules". Describe how and when they form (if known), any interesting properties that they have.

Question 6.

Take a look at figure 9.13 on page 339. a) Can you explain how it is that the relative ages along the x-axis are known so precisely while the absolute ages are more uncertain?

Fusion in the Sun: Extra Credit.

Still time to fuse your way to an iron nucleus in the Fe26 game: http://dimit.me/Fe26/. Include a screenshot of your best Fe26 game for up to three points extra credit (Full credit if you can form a 56Iron tile. The highest score posted to the class blog gets an additional 3 points.

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