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.
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?
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.
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?
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:
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.