ASTR221 HW #2  due Sep 23rd 2005
1. The Roche Limit
In class we calculated the Roche limit, which showed
how
close a moon could get to its parent planet before it will be tidally
disrupted.
Saturn's moons are mostly icy bodies, with densities
of
about 1.2 g/cm^{3}. Calculate the Roche limit for these moons.
Compare
this to the size of Saturn's rings, most of which lie within 120,000 km
from
the planet's center. Noting that all of Saturn's moons lie outside of
130,000
km from the planet' center, what does this suggest as one possible
formation
mechanism for Saturn's rings?
Calculate the Roche limit for the Moon. Is it in any danger of being
disrupted
any time soon?
2. Atmospheric Escape
The Earth has an atmosphere, the Moon does not. Let's
try
and understand this. A planet's atmosphere will start to "leak away" if
the
typical thermal velocity
of the molecules in the atmosphere is greater than about (1/10)v_{escape}.
Remember in class, I said that temperature is a measurement of kinetic
energy of particles. The way we write this formally is to say that kT =
(1/3)mv^{2}, where k is Boltzmann's constant, T is
the temperature of the atmosphere, m is the mass of the molecule, and v
is the velocity of the typical molecule. (And no, that's not a typo, it
is 1/3 and not 1/2  this is due to the fact that all molecules dont
have the same velocity, they have a distribution of velocities and when
you integrate over that distribution to define the typical velocity,
the 1/2 becomes a 1/3 in the process. You'll see this down the road
when/if you take a statistical mechanics class in Physics...). Anyway,
this means that the the thermal velocity of the molecules is
given by v_{thermal} = sqrt(3kT/m).
Now, equate the thermal velocity to (1/10)v_{escape} to show
that the
temperature at which a planet's atmosphere will begin to leak away is
T>(1/150)(GMm/kR),
where M and R are the mass and radius of the planet, respectively.
The Earth's atmosphere is mostly (71%) N_{2}, a molecule with
mass
28 amu (1 amu = the proton mass = 1.66x10^{24} g).
 At what temperature will the Earth's nitrogen atmosphere
begin to leak away? The characteristic temperature of the Earth's upper
atmosphere
is about 1000 degrees Kelvin. Do we need to worry about losing our
atmosphere?
 At what temperature would the Moon lose a similar
atmosphere?
Compare your answer to the temperature on the Moon's surface which is
about
274 degrees Kelvin.
 What about the Earth's primordial atmosphere? Assume it
was all hydrogen, and calculate the escape temperature for that
atmosphere. Does this explain why we lost our primordial atmosphere?
3. Measuring the size of the Earth's core
If we ignore the Earth's crust, we
can treat the Earth's interior
as being composed of two zones  the mantle and the core. If the
typical
densities of the core and mantle are 10,900 kg/m^{3} and 4,500
kg/m^{3},
respectively, use the average density of the Earth to determine the
radius
of the core. Express your answer in units of Earth radii.
4. Fission origin of the Moon
Estimate the initial rotation
period of the Earth if the Moon had
been torn from it, as suggested by the fission theory. Compare your
answer
to the estimated rate of spin of the early Earth (5 hours or so). Is
the
fission theory a viable theory?
5. Radioactive dating
I mentioned in class that
radioactive dating of moon rocks set the
absolute age of the formation time of the lunar maria and the lunar
highlands.
Let's work this out in practice. First read this short tutorial about
how
radioactive dating works: Page 1, Page 2. Then go help out Homer:
NASA recently hired one Homer Simpson to be the curator
of their moon rock collection. On his very first day, Homer tripped
carrying the rock samples and mixed up the rocks from the lunar
highlands with the rocks taken from the maria. It is your job to sort
them out.
You measure the
abundances of isotopes at various points in two rocks and find the
following abundance patterns:
Rock A
^{147}Sm/^{144}Nd

^{143}Nd/^{144}Nd

0.1847

0.511721

0.1963

0.511998

0.1980

0.512035

0.2061

0.512238

0.2715

0.513788

0.2879

0.514154


Rock B
^{87}Rb/^{86}Sr

^{87}Sr/^{86}Sr

0.008

0.6994

0.025

0.7005

0.082

0.7043

0.090

0.7048

0.110

0.7050

0.155

0.7088

0.214

0.7122


You also know the following halflives for
radioactive
decay:
Decay

Half Life (tau)

^{147}Sm > ^{143}Nd

106 billion years

^{87}Rb > ^{87}Sr

47.5 billion years

 For Rock A, plot ^{143}Nd/^{144}Nd
vs ^{147}Sm/^{144}Nd and derive its age.
 For Rock B, plot ^{87}Sr/^{86}Sr
vs ^{87}Rb/^{86}Sr and derive its age.
 Which rock came from the highlands, and which
from a mare? Why?