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The giant space ship example; Can you calculate surface temperatures given a moon sized interstellar space ship with its own atmosphere?
Topic Started: Dec 4 2011, 03:29 AM (1,509 Views)
Chris Ho-Stuart
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Jan 4 2012, 03:16 PM
"Try this. Alter the value for gamma on the first sheet, using the Dwarf example. By default, it is 1.4. Try entering 1.32, and then 1.34, and observe the sudden change in the height of the tropopause. This is another indication I am on about the right track; this is a theoretically expected behaviour. "

Ok, so atmosphere lowers and requires more energy input.
I have no idea why. Can't remember what gamma is. This isn't heated by sunlight, but bond albedo- what surface is made of- should still a have affect- is that what gamma is.
It's not particularly important. Those interested can find more in Principles of Planetary Climate by R. Pierrehumbert. The relation falls apart as we go on to consider real non-grey gases, or pressure variations of k. I just found it interesting, and a useful confirmation that I am getting sensible results in the spreadsheet so far.

A quick review for the record: gamma is the ratio Cp/Cv: specific heat at constant pressure to specific heat at constant volume. For diatomic gases, it is generally about 7/5 (or 1.4). It has nothing to do with albedo. (One nice thing about the spaceship problem is that albedo doesn't show up at all; it is an elegant problem in which the greenhouse effect is cleanly isolated from other effects.)

The relation between temperature and pressure under a dry adiabat is T/T0 = (p/p0)R/Cp. R, the gas constant, is Cp - Cv. Hence that exponent R/Cp works out to 1-1/gamma.

Now power varies as the fourth power of temperature, so in the mathematics you tend to see an exponent for pressure that is 4R/Cp, And it's just a mathematical curiosity than in the simple case of constant k which we've been using, whether that exponent is greater than or less than 1 makes the difference between shallow and deep tropopause. A bit of algebra means
[indentblock]4R/Cp = 1 when 4 - 4/gamma = 1, or gamma = 4/3[/indentblock]

Since the atmosphere thins much more gradually above the tropopause, a lower tropopause actually gives a higher atmosphere in general.
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Chris Ho-Stuart
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Sorry for the delay... I have been a bit busy. I still have not done the rather difficult conversion to a real gas, but I have updated the grey gas spreadsheet.

Here is the download link for Starship-v2.4.xls (version 2.4). Fixed up a bug or two, and added plots of the atmospheric profile. Here, for example, are the plots for the "earth-like" planet. (It isn't really very Earth-like, as the atmosphere is "grey"; but the optical depth has been chosen to give approximately an Earth-like surface temperature and emission to space.)

Plot of energy flows, temperature and pressure vs altitude.
Posted Image

Plot of energy flows, and temperature vs pressure. (Pressure as a percent of surface pressure)
Posted Image

The tropopause given this grey atmosphere is lower than for the real Earth.

Cheers -- Chris
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