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Venus Atmosphere Puzzle, one man's struggle with atmospheric physics
qraal
post Jun 5 2006, 12:15 PM
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Hi All

This might seem like a really dumb question, but what's the mass of the Cytherean atmosphere per unit area?

At first pass I thought it was easy - same as for an isothermal atmosphere, Po/g, where Po is surface pressure and g is surface gravity. Simple. Except Venus doesn't come close to approximating an isothermal atmosphere. From a graph in Mark Bullock's PhD thesis (Hi Mark if you're visiting) I pulled the figures for Po and To as 92 bar and 735 K, while the left-side of the temperature curve was 250 K at 0.1 bar and 63 km. At about 210 K the temperature drop with altitude stops, then slowly rises into the Cytherean stratosphere.

Ok. My atmospheric physics is pretty limited - I 'modelled' that lapse rate pressure curve as a power law:

P/Po = (T/To)^n

and likewise for density, d/do = (T/To)^n.

Temperature, T, as a function of altitude, Z, I computed as T(Z) = To*(1-Z/(n.Zo)).

Zo = (k.T/m.g), where k is Boltzmann's constant and m is the molecular mass of the atmosphere.

These equations I then integrated between 210 K and 0.033 bar, 70 km, and 735 K and 92 bar, zero altitude.

The resulting equation is m = (n/(n+1))*(do.Zo)*(1 - (T/To))^(n+1) - a bit of simple algebra and the Gas equation shows that do.Zo = Po/g.

Thus the mass is lower than for a simple isothermal atmosphere by roughly (n/(n+1)). In this case n = 6.33, higher than the dry adiabat for CO2 which gives n = 4.45.

Now an adiabatic or polytropic atmosphere is an idealisation, but it seems odd to me that whenever Venus' atmospheric mass is discussed people always use the higher isothermal value. Have I missed something important in the physics, or is Venus's atmospheric mass just 86.4% of the usually quoted value?
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ngunn
post Jun 10 2006, 10:39 AM
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Hi graal. I think that the messenger's messageless message is trying to say we have gone round in a circle . . . however I still need to understand things in my slow, hand-waving way.

First 2 questions. Are you now satisfied that you have found the reason why your integral doesn't go on to infinity, and are you now getting an answer for the atmospheric mass of Venus that is in line with what you expected intuitively?

For me there had to be a common sense reason why the integral terminates, and it had to be independent of external factors. A finite atmosphere must be able to restrain it's own urge to become infinite without being disciplined by the Van Allen police! So I did another thought experiment:

This time I provided my gravitating globe with an extremely thin atmosphere. In fact I gave it just one single molecule of gas. The molecule hops about receiving thermal kicks from the surface at each bounce. I quickly realised that this atmosphere could not be isothermal, because the molecule slows down as it rises and accelerates again as it falls. At the very top of the highest bounces (which must be vertical) the molecule stops altogether, thus momentarily reaching zero Kelvin. The atmosphere therefore has zero volume - and zero mass - from here on out.

So it is the fact that molecules are of finite size that causes the atmosphere to terminate. If the molecules were infinitely many and infinitesimally small we would never reach the point where the mean free paths become long and thermal collisions give way to ballistic trajectories. The atmosphere would indeed go on for ever whilst still having a finite total mass, a contradiction, therefore molecules are of finite size.

I don't know if this is of any use but thanks anyway for making me think.
Regards
Nigel
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qraal
post Jun 10 2006, 12:34 PM
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Hi Nigel

Ok. Let's try again. Everything you've said about the discreteness of the atmosphere is true and as I've already noted it's what really obtains - eventually the gases stop running into each other and get separated out by gravity. I know that, though your eloquent restatement was better than my clumsy exposition.

What I was discussing was an idealisation, the hydrostatic model of an isothermal atmosphere that's in a gravity field dropping off via the inverse square law. Assume the gas is infinitely divisible and you get an infinite mass at infinity - infinity BTW is the usual limit when integrating the equation (rho) = (rho)o*e^-([Z/Zo), which as you can see only achieves a zero value at Z/Zo = infinity. That's just the nature of the maths and I know its limitations. When molecules only collide every few metres or so things change drastically from the case of micrometre path lengths.

My quandary concerns the case of the adiabatic atmosphere - a whole different ballgame, to coin a phrase. An adiabatic atmosphere in its idealisation has a finite altitude at which temperature, pressure and density fall to zero. That's because of the temperature-altitude relation - i.e. T/To = [(n.Zo - Z)/(n.Zo)] so T/To = 0 at Z = n.Zo, and because (rho) = (rho)o*(T/To)^n it means it drops to zero too.

So the mass integral has finite limits rather than the unphysical "infinity". Integrated, the equation gives:
m = (n/(n+1))*(n.Zo) and n.Zo = Po/g. So as I've already noted the mass is lower than the isothermal case for the same surface pressure.

Why is this so? And is it correct? The maths says "yes" and it means that the atmospheric mass on Venus is less than the usually quoted figures. So 88.8 bar of CO2 is actually 76.8 'bar' in weight. And 865 tons/sq.m in mass not the usual 1,000.

Ok. So that's the state of play. Why is it so? My first thought is that because expansion cooling lowers the pressure as you climb in height the hydrostatic balance can be maintained without relying on just the weight of the gas column itself. The temperature gradient acts like an extra potential. Am I even close?

Adam

QUOTE (ngunn @ Jun 10 2006, 10:39 PM) *
Hi graal. I think that the messenger's messageless message is trying to say we have gone round in a circle . . . however I still need to understand things in my slow, hand-waving way.

First 2 questions. Are you now satisfied that you have found the reason why your integral doesn't go on to infinity, and are you now getting an answer for the atmospheric mass of Venus that is in line with what you expected intuitively?

For me there had to be a common sense reason why the integral terminates, and it had to be independent of external factors. A finite atmosphere must be able to restrain it's own urge to become infinite without being disciplined by the Van Allen police! So I did another thought experiment:

This time I provided my gravitating globe with an extremely thin atmosphere. In fact I gave it just one single molecule of gas. The molecule hops about receiving thermal kicks from the surface at each bounce. I quickly realised that this atmosphere could not be isothermal, because the molecule slows down as it rises and accelerates again as it falls. At the very top of the highest bounces (which must be vertical) the molecule stops altogether, thus momentarily reaching zero Kelvin. The atmosphere therefore has zero volume - and zero mass - from here on out.

So it is the fact that molecules are of finite size that causes the atmosphere to terminate. If the molecules were infinitely many and infinitesimally small we would never reach the point where the mean free paths become long and thermal collisions give way to ballistic trajectories. The atmosphere would indeed go on for ever whilst still having a finite total mass, a contradiction, therefore molecules are of finite size.

I don't know if this is of any use but thanks anyway for making me think.
Regards
Nigel
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Posts in this topic
- qraal   Venus Atmosphere Puzzle   Jun 5 2006, 12:15 PM
- - remcook   QUOTE do.Zo = Po/g This is the equation for hydro...   Jun 5 2006, 02:15 PM
|- - qraal   Hi rem That's exactly what I did and I still ...   Jun 5 2006, 11:45 PM
|- - The Messenger   QUOTE (qraal @ Jun 5 2006, 05:45 PM) That...   Jun 9 2006, 01:57 PM
- - ngunn   In an atmosphere in which temperature decreases ra...   Jun 6 2006, 11:37 AM
|- - qraal   Hi ngunn Thanks for the reply. You know I wondere...   Jun 7 2006, 12:14 PM
- - ngunn   Well you've done the detailed calculations, no...   Jun 7 2006, 12:46 PM
|- - qraal   Hi ngunn I've read it before, but replicated ...   Jun 9 2006, 12:43 PM
- - Phil Stooke   There's some great Venus atmosphere stuff (and...   Jun 7 2006, 01:50 PM
|- - DonPMitchell   QUOTE (Phil Stooke @ Jun 7 2006, 06:50 AM...   Jun 7 2006, 08:33 PM
- - ngunn   Very interesting, and I follow the reasoning you g...   Jun 9 2006, 12:57 PM
- - ngunn   Hi graal. I think that the messenger's messag...   Jun 10 2006, 10:39 AM
|- - qraal   Hi Nigel Ok. Let's try again. Everything you...   Jun 10 2006, 12:34 PM
- - ngunn   Fine. I agree the adiabatic model is probably a b...   Jun 12 2006, 11:23 AM
|- - The Messenger   QUOTE (ngunn @ Jun 12 2006, 05:23 AM) Fin...   Jun 12 2006, 08:56 PM
- - ngunn   Hi Messenger. You found my post confusing??!...   Jun 14 2006, 09:04 AM
- - remcook   just a note: Titan's temperature does vary sig...   Jun 14 2006, 11:36 AM
- - qraal   Hi All Well I decided to do a numerical experimen...   Jun 23 2006, 12:42 PM
|- - The Messenger   [quote name='qraal' date='Jun 23 2006,...   Jun 23 2006, 04:44 PM
- - qraal   Hi again And the average temperature is 630 K. H...   Jun 23 2006, 01:11 PM
|- - ngunn   QUOTE (qraal @ Jun 23 2006, 02:11 PM) Hmm...   Jun 23 2006, 01:51 PM
- - qraal   Hi Messenger & ngunn Hey thanks for the nice ...   Jun 25 2006, 12:50 AM
- - RNeuhaus   Venus' Double Vortex Confirmed in New Animatio...   Jun 27 2006, 10:21 PM
|- - DonPMitchell   At last, some pictures from VEX. These are fascin...   Jun 28 2006, 01:33 AM
||- - qraal   Thanks Don Man that's so bizarre. Venus is a...   Jun 28 2006, 10:21 AM
||- - qraal   Hi All Updated the gravity - now it's a linea...   Jun 30 2006, 02:11 PM
|- - JRehling   QUOTE (RNeuhaus @ Jun 27 2006, 03:21 PM) ...   Jul 3 2006, 10:02 PM
- - edstrick   The soviet venus descent probes measured light lev...   Jul 1 2006, 07:43 AM
- - qraal   Hi ed Thanks for the heads up on that reference, ...   Jul 1 2006, 12:38 PM
- - edstrick   The Univ of Arizona has had a method of producing ...   Jul 2 2006, 08:41 AM
- - DonPMitchell   The U of Arizona books are essential. There are a...   Jul 2 2006, 05:56 PM
|- - qraal   Hi Don Hey thanks for the insights - Venus is tru...   Jul 2 2006, 11:47 PM
|- - rlorenz   QUOTE (DonPMitchell @ Jul 2 2006, 01:56 P...   Jul 13 2007, 12:08 PM
- - DonPMitchell   That is a mystery. Many believe there are no crys...   Jul 3 2006, 01:01 AM
|- - qraal   Hi Don The exobiological theory is the most excit...   Jul 3 2006, 08:52 AM
- - edstrick   The Pioneer Large Probe Cloud Particle Size Spectr...   Jul 3 2006, 09:18 AM
- - DonPMitchell   Vega-1 and Vega-2 performed similar experiments. ...   Jul 3 2006, 04:16 PM
|- - qraal   Thanks Don & Ed More data to cram into my bra...   Jul 4 2006, 01:16 AM
- - DonPMitchell   Planetary circulation is fascinating. Rotating pa...   Jul 4 2006, 01:26 AM
|- - Bob Shaw   QUOTE (DonPMitchell @ Jul 4 2006, 02:26 A...   Jul 4 2006, 02:31 PM
- - MichaelT   QUOTE (qraal @ Jun 5 2006, 12:15 PM) Hi A...   Jul 4 2006, 09:25 AM
|- - ngunn   QUOTE (MichaelT @ Jul 4 2006, 10:25 AM) I...   Jul 4 2006, 03:51 PM
|- - MichaelT   QUOTE (ngunn @ Jul 4 2006, 03:51 PM) meas...   Jul 4 2006, 04:40 PM
|- - The Messenger   QUOTE (MichaelT @ Jul 4 2006, 10:40 AM) O...   Jul 4 2006, 07:22 PM
- - edstrick   I'd forgotten the instruments on the Vega prob...   Jul 4 2006, 10:17 AM
- - ngunn   Which is what I still don't understand (or bel...   Jul 5 2006, 09:19 AM
|- - qraal   Hi ngunn For the first 200 km of altitude Venus...   Jul 5 2006, 11:18 AM
- - ngunn   Ah! qraal with a Q - my apologies. It's th...   Jul 5 2006, 01:08 PM
- - qraal   Hi ngunn & MichaelT As you might've guess...   Jul 5 2006, 10:32 PM
|- - MichaelT   QUOTE (qraal @ Jul 5 2006, 10:32 PM) If s...   Jul 6 2006, 06:27 PM
|- - qraal   Hi Michael There's quite a lot of literature,...   Jul 10 2006, 07:56 AM
- - DonPMitchell   You're looking at Mark Bullock's thesis I ...   Jul 5 2006, 10:49 PM
- - ljk4-1   Venus Atmosphere Profile from a Maximum Entropy Pr...   Sep 26 2006, 02:36 PM
|- - rlorenz   QUOTE (ljk4-1 @ Sep 26 2006, 10:36 A...   Jul 13 2007, 12:16 PM
- - qraal   Hi All I haven't bothered with this for some ...   Jul 12 2007, 11:05 PM
- - edstrick   "....Sadly, it isnt cheap..." I'd sa...   Jul 14 2007, 06:03 AM


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