Wednesday, January 16, 2008
Terraforming Mars: Escape Velocity
So, the manual David gave me when he made me an Author said that I was supposed to alternate frivolous and serious posts*. CharlieMao was one of those; this is the other—it’s time for some physics.
One of the long-term goals of sending men to Mars has traditionally been to establish a permanent population on the planet. If that population is to be much more than hideously expensive spam in a can, they’ll need to be able to breathe something other than canned air and live outside of a pressurized environment. The process of turning Mars from its current status of a desert at the temperature of dry ice with a soft vacuum for an atmosphere to something vaguely tolerable is called terraforming.
The question of how one might go about terraforming the planet is an interesting one, but I’m not going to write about that. 8-) It’s been done, over, and over again. Instead, I will assume the success of terraforming and take a look at one complication that will arise afterwards.
(I’m just going to assert most of the physics here; if you want more information, it’s pretty easy to find.)
Temperature is defined as a measure of the mean (broadly, what most people think of as “the average") kinetic energy of a particle. Kinetic energy is proportional to the mass times the square of the velocity. That means that temperature is a measure of average particle velocity, with higher velocities for lower mass particles. (At any temperature, hydrogen will be moving much faster than oxygen at the same temperature, because the hydrogen is much less massive.)
Planetary escape velocity is proportional to the square root of (the planet’s mass divided by the planet’s radius). For Earth, this is a bit over 11 km/s; for Mars this is a bit over 5 km/s. Both of these are higher than the mean velocity of important components of the atmosphere at tolerable temperatures, but that turns out not to be enough.
Because temperature is only a measure of mean velocity, some particles are faster than the mean and some are slower. For our purposes (and for nearly any purposes, for that matter), we can treat the distribution of velocities as “normal”, which is commonly graphed as the sort of bell curve you have probably seen used for grading. Without going into the math, it turns out that if the escape velocity is greater than about 10 times the mean velocity, the particles will hang around for billions of years. If the escape velocity is about the same as the mean velocity, the time is only a few days.
So lets take a look at a few important gases and make some assumptions about a post-terraformed Mars:
My definition of “terraformed” includes liquid water and breathable air, so I will assume an average temperature of about 0 degrees C (32 degrees F or approximately 273 Kelvin) and an atmosphere with lots of free oxygen. It can’t be pure oxygen (which is quite dangerous), and most very heavy gases are either difficult to make or find or are toxic, so we’ll assume lots of free nitrogen as well. (In other words, we’ll assume something like the atmosphere of earth.) That means we need to find out the mean velocities of water vapor, oxygen, and nitrogen at 273 Kelvin.
Given these assumptions, the results are as follows:
Vrms O2 = 0.461 Km/s
Vrms N2 = 0.493 Km/s
Vrms H2O = 0.615 Km/s
Since the escape velocity for Mars is 5.03 Km/s, this means that the oxygen and nitrogen are close, but probably ok, but the water vapor is going to be bleeding off into space. (Remember that we need a velocity below 10% of the escape velocity for safety.) Even after terraforming, you will have to provide a constant source of new water to maintain a civilization.
By the way, this is why the current atmospheres of both Venus and Mars are predominantly CO2. Its molecular mass is enough higher than that of lighter gases (like methane and nitrogen) to keep it from escaping the planet in the high temperatures of Venus and the shallow gravity well of Mars.
* Actually, it was more like, “You’re an author; here’s a link; figure it out”, but I’m reading between the lines here.
Comments & Trackbacks
If we’ve already terraformed the planet, then we certainly would have put a space elevator up. If there’s an elevator, it needs to be bringing in new mass to replace what’s coming up from the surface (Martian mining products, I assume); might as well be water ice from the cometary belt.
But what does any of this have to do with American Idol?
You put math in the post, David is not going to like that.
It’s OK, David’s tiny journalist brain shut down as soon as he hit the word “kinetic”, and that was ahead of any maths. He’ll never know it was there.
Hey!
You can say a lot of things about David, but “journalist” is over the line. Sure, there’s the technical meaning of “one who writes a journal”, but that’s about as current as using “gay” for “carefree”. We all know that “journalist” is just code for the R word.
Let’s have a little civility here.
8-)
You’re right. My comment was way out of line. I owe David an apology.
David, I’m sorry that you have a tiny journalist brain.
Man, I’m probably gonna have to buy my own drinks this year. But if I can’t make fun of David, then who can I make fun of?
No, seriously, send me a list of people. I got a lot of stuff to work out.
Hey!
If you’re not nice, I’m going to take my blog and go home.
I told you he would not like the math.
They were going to have him as a “citizen judge” on American Idol, but then he saw that you have to add up numbers for the scorecard. “Pass, man - I didn’t flunk out of J-school to do arithmetic!”
Actually, the main challenge to terraforming Mars is the lack of an molten core. Without that, and the resulting magnetosphere, any atmosphere we were able to import—including the oxygen and nitrogen, would have to be constantly replenished as the solar wind eroded it.
On the bright side, assuming it were possible to come up with enough additional iron to somehow melt down and inject into Mars to re-establish its magnetosphere, the added mass would have to be enough to alter the gravity figures.
Keeping the molten core molten would probably require bringing in a large asteroid to serve as a moon, to exert constant tidal stress on the planet. That too would alter the gravity figures, albeit slightly in the other direction.
Do the shallower slope of the gravitational potential between the planet and the moon plus the increased atmospheric tides more than compensate for the increased total depth of the gravity well? Interesting question, but I suspect the modelling is beyond what I’m capable of without far more time than I’m willing to spend.
BTW, the usual solution for providing a core for the planet that I’ve seen involves a really big asteroidal impact. I’d hate to have to write that EIS.
8-)
"Anybody not wearing 2 million sunblock is gonna have a real bad day.”
And you just know Mars First! would sue…
Somehow my comment didn’t take.
My question: You say:
Isn’t that off by an order of 10? It seems like even water vapor doesn’t have a 9th of the energy it needs to escape?
Did you misplace a decimal, or am I missing something?
Missing something, but I could have written it a bit more clearly.
The particle velocity is a mean, with the individual particle velocities distributed over a bell curve. The problem is those particles near the higher end of the bell curve are going fast enough to escape the planet. When the mean velocity is greater than 10% of the escape velocity, a substantial portion of those particles will escape. The closer the mean is to the escape velocity, the faster the particles will escape.
Roger.
And I assume that the mass of increased gasses present after terraforming won’t be enough to raise the escape velocity any appreciable degree.
So Heinlein was full of crap in both Red Planet and Farmer in the Sky</i>. Dang.
I’m guessing that just bringing in the relatively heavy new moon would be enough to cause the core to liquefy naturally as tidal forces caused seismic rubbing. Might take a while, though. The attendant volcanism would contribute to the atmospheric thickening too.
I thought of that, but I think we do also want to deepen Mars’ gravity well, which will require more mass.
The volcanic gases that would be released are also among the most effective greenhouse gases there are, thus contributing to an increase in global temps. Too bad they also range from unbreathable to outright toxic.
Toxicity of atmosphere is merely an annoyance. People live in LA, Houston, Hong Kong, and Beijing, don’t they?
I can see the letters going out now:
CONGRATULATIONS CITIZEN! Your demonstrated tolerance for less-than-perfect air has moved you, (and everybody else in your precinct) into the category of PREFERRED PERSONS in the ongoing effort to establish a viable colony on Mars. Everything will be provided, just go quietly with the nice men delivering this letter.
Thank you
The Govt.
Oh well, I guess somebody has to be first.
Personally, I’m holding out for the time when a bunch of like-minded folks can throw in together on a FTL freighter and go find someplace that’s less of a ‘fixer-upper’.
Amen to that.