# 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. Go to the Inner Solar System Viewer at http://janus.astro.umd.edu/java/ISSV/innerSS.html .
a) Kepler's Third Law: Look at Mercury, Venus, and Earth. Earth (the blue dot) goes around the Sun in one year. Venus and the Earth start out together near the right - about how many years does it take Venus to lap the Earth? About how many times does Mercury lap Earth in one year? How does this illustrate Kepler's third law? Hit "Reload" to restart the simulation.
b) Kepler's First Law: Toggle the asteroid orbit to a=1AU, e=0.0, and w=0 degrees. Now increase the eccentricity (e) of the asteroid to 0.3 in increments of 0.05. What do elliptical orbits with low eccentricity look like?
c) Kepler's Second Law: Now toggle the asteroid orbit to a=1AU, e=0.9, and w=40 degrees. Watch the asteroid for a full orbit; where is it fastest? Draw a picture showing how this relates to Kepler's second law. Finally, make a prediction for the period of the asteroid's orbit using Kepler's third law. Now check your prediction by waiting until the asteroid goes through its closest approach to the Sun (its pericenter) and note Earth's position - where is Earth when the asteroid returns to pericenter?

## 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. 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:

87Sr = 87Sr(t=0) + 87Rb(t=0) (1 - 2 -t/T)
87Rb = 87Rb(t=0) 2 -t/T
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:
87Sr = 87Sr(t=0) + 87Rb(t=0) (2 t/T - 1).
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 329. 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.

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.