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Viewing Single Post From: The giant space ship example  

Chris HoStuart  Jan 4 2012, 04:31 PM 
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. 
The giant space ship example · Physical theory for climate 