# 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/cm3. 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)vescape. 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)mv2, 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 vthermal = sqrt(3kT/m).

Now, equate the thermal velocity to (1/10)vescape 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%) N2, 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/m3 and 4,500 kg/m3, 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?

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:

 147Sm/144Nd 143Nd/144Nd 0.1847 0.511721 0.1963 0.511998 0.1980 0.512035 0.2061 0.512238 0.2715 0.513788 0.2879 0.514154
 87Rb/86Sr 87Sr/86Sr 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) 147Sm --> 143Nd 106 billion years 87Rb --> 87Sr 47.5 billion years
• For Rock A, plot 143Nd/144Nd vs 147Sm/144Nd and derive its age.
• For Rock B, plot 87Sr/86Sr vs 87Rb/86Sr and derive its age.
• Which rock came from the highlands, and which from a mare? Why?