ASTR221 HW #3 - due Oct 7

1. Ancient Martian Greenhouse

Okay, we know that at one time Mars had liquid water on its surface. While there is still some debate over whether or not this water was transient or long-lived (ie in oceans), let's take the idea that there were long-lived bodies of water on Mars long ago and ask how we could change Mars' climate to permit that. To support liquid water, Mars must have once had a thick enough CO₂ atmosphere to have a greenhouse effect warm its surface. Here we are going to use some simple arguments to estimate how thick that atmosphere would have been.

First, we need to calculate the infrared (IR) optical depth of the early Martian CO₂ atmosphere (remember that surface IR radiation -- heat -- is trapped on Mars when the IR optical depth is high). A simple estimate of the surface temperature of a planet blanketed by an atmosphere is Tsurface = (1+tau)^(1/4) Tequil, where Tequil is the equilibrium temperature of the planet (without factoring in any greenhouse effect) and tau is the optical depth of the atmosphere.

First, calculate the equilibrium temperature of early Mars. There's one complication here: in the early history of the solar system the Sun was not as bright as it is now-- it was actually a bit cooler and smaller. To correct for this, calculate the equilibrium temperature of Mars today, then reduce that temperature by a factor of 0.9 (recent estimates have the luminosity of the early Sun at about 70% of its current luminosity, and 0.7^(1/4) ~ 0.9). Now, calculate the value of tau needed to have early Mars warm enough for liquid water.
(Hint: Make sure to do everything in units of degrees Kelvin!)

Now, the IR cross section of CO₂ is very roughly 5x10^-26 cm². Given this cross section and the value of tau you just calculated, what is the total column density of CO₂ molecules (in molecules per square meter) needed to provide this optical depth? (Hint: Remember how column density, optical depth, and cross section are all related. Look at these notes for a refresher.)

To go from column density (molecules per square meter) to volume density (molecules per cubic meter) we need to use the thickness of the atmosphere. A reasonable estimate of the thickness of the atmosphere is the scale height, H. What would the scale height of CO₂ have been on early Mars? (Careful: Think about what temperature you should use in the scale height equation.) Use this scale height to convert your column density to a volume density.

So we have the volume density of CO₂ needed to warm up Mars. Last step -- we want the surface pressure of CO₂. Use the ideal gas law to convert density to pressure, then convert your pressure to Earth atmospheres (1 atmosphere =10⁶ dyne/cm². Again, make sure you are using an intelligent temperature.)

2. Hydrostatic Equilibrium

Starting with the equation of hydrostatic equilibrium, calculate how pressure changes with depth in the ocean, i.e., P(z). Look at your notes for how we calculated the pressure in the atmosphere of the Earth, and remember that water, unlike a gas, is uncompressible.

The atmospheric pressure on the surface of Venus is 9x10⁶ N/m², or 90 times that of the Earth's surface atmospheric pressure. How far down in the Earth's ocean do you have to go to experience this pressure?

3. Jupiter's Heat

Jupiter is giving off heat through gravitational contraction. We're going to calculate how much, and how this has changed throughout the history of the solar system. When an gravitating object contracts, half of the lost gravitational energy is radiated away (the other half goes into heating up the object). For simplicity, assume that throughout the collapse, Jupiter can always be considered a sphere of uniform density, for which the gravitation energy can be written U=-(3/5)GM²/R.

First, estimate the total amount of energy radiated by Jupiter by gravitational contraction over the last 4.5 billion years. Assume Jupiter started out as a ball of gas with its current mass, except much, much bigger in size.

Now estimate the average rate of energy output from Jupiter over the last 4.5 billion years from gravitational contraction alone.

Calculate the current rate Jupiter is giving off energy due to gravitational contraction. Remember that if Jupiter was at its equilibrium temperature, it would simply be radiating back the energy it got from the Sun. Since Jupiter is hotter than this, it must be emitting more energy than what it receives from the Sun. Calculate how much more.

Compare the current rate (in part 3) with the average rate (in part 2). What does this tell you about how the energy output from Jupiter has changed over time? How might this affect the conditions in which Jupiter's moons formed?

4. Planet Jumble!

In an alternate universe, the great donkey-god Asinus visited our solar system 4.5 billion years ago. He decided he don't like the way the terrestrial planets are arranged (there's no pleasing some people), so he changed it around. Earth and Mercury got swapped (in terms of their distance from the Sun), as did Mars and Venus. Everything else about the planets stayed the same (ie mass, rotation speed, etc). Remember that the planets start out with their original (ie not yet evolved) atmospheres.

Write a 2-3 page essay describing what the terrestrial planets would be like today in that alternate universe. You may need to calculate some things (temperatures, etc), but the calculations are not part of the essay (ie I don't want a page and a half of equations, followed by three sentences).