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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?  

Tweet Topic Started: Dec 4 2011, 03:29 AM (1,490 Views)  
Chris HoStuart  Jan 4 2012, 04:31 PM Post #46 
Administrator

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 nongrey 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 C_{p}/C_{v}: 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/T_{0} = (p/p_{0})^{R/Cp}. R, the gas constant, is C_{p}  C_{v}. Hence that exponent R/C_{p} works out to 11/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. 
Chris HoStuart  Jan 19 2012, 12:58 PM Post #47 
Administrator

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 Starshipv2.4.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 "earthlike" planet. (It isn't really very Earthlike, as the atmosphere is "grey"; but the optical depth has been chosen to give approximately an Earthlike surface temperature and emission to space.) Plot of energy flows, temperature and pressure vs altitude. Plot of energy flows, and temperature vs pressure. (Pressure as a percent of surface pressure) The tropopause given this grey atmosphere is lower than for the real Earth. Cheers  Chris 
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