Estimating Atmospheric Masses Using the Barometric Equation

[Brewer]

I've been given some homework questions (about the solar system, but I'm sure that it fits in this section of the board) and its all fine until I get to the part asking about atmospheres. I have a feeling that I know what I'm supposed to do, but I feel that I'm missing something blindingly obvious, as I can't actually find an answer.

The question asks me to estimate the mass of carbon dioxide on Venus, Earth and Mars, given ground level pressure, surface temp, surface gravity and the fractional CO2 content.

My initial thought it to use the exponential atmosphere equation:

P(h) = P(0)*exp-(mgh/kT), and solve for m, and multiplying this by the fractional CO2 content. But I don't think that this is the way to go somehow, as I don't have a a value of P(h). And now I'm stumped. Can anyone help me at all (because my notes sure can't!)

Thanks guys

 

[Bystander]

Think about how you go about calculating the pressure of a mercury column.

 

[Brewer]

You mean along the lines of pressure = force/unit area?

 

[Bystander]

Yes. And --- force equals what?

 

[Brewer]

f = ma. so in this case is a = g?

If that's right, do I then equate that with the equation I put down earlier? If i do that I can't see how to solve for m, if there's an m in the exponential as well as the LHS.

 

[Brewer]

assuming unit area of course?

 

[Bystander]

f = ma. so in this case is a = g?

Yes.

If that's right, do I then equate that with the equation I put down earlier?

No.

If i do that I can't see how to solve for m, if there's an m in the exponential as well as the LHS.

You've been given P, g, and CO₂ content. "You mean along the lines of pressure = force/unit area?" What does the force equal?

 

[Brewer]

P=mg, therefore m=P/g right?

 

[Bystander]

Correct. Now what're the total areas?

 

[Brewer]

4*pi*R^2??

Does it matter the height of the atmosphere, so that it would be radius of planet and height above planet, or is it best to keep it all at the surface? It makes more sense to me to keep it at the surface, but doesn't the atmosphere not start until you get a certain distance up?

 

[Bystander]

You're asked to "estimate." g depends upon distance from COM, but, just for example, the Earth's atmosphere extends how far above the surface? That's the barometric equation you were playing with --- integrate to get however many "nines" of the total mass you'd like to play with. Height matters for low mass systems. Planets are in general not low mass. The "atmosphere" includes all gas phase mass associated with the planet. You can look at that any way you like: include all pore space in soils, cavern volumes; exclude gases dissolved in liquid or solid phases in the planetary structure. The question isn't asking for that kind of detail, so you are calculating the mass above the surface.