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, 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!)

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.

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).