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Viewing Single Post From: The giant space ship example  

Chris HoStuart  Dec 10 2011, 10:43 PM 
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In my earlier post #22, I got a little bit ahead of myself. I'm going to rewrite it entirely here, in a slightly simpler form, without any mention of the backradiation. We don't need it (yet). Backradiation exists when you have nonzero k; and the numbers allow you to infer it must be there. But as long as we assume a fixed lapse rate, we can calculate power requirements on the fusion heaters without calculating the backradiation. The Grey Gas approximation A grey gas is one for which the interaction with radiation is the same at all frequencies. (It has no colour.) This is unlike any real gas, but it is a useful stepping stone towards solving the problem given for this topic. I’ll consider the starship with a grey atmosphere and a fixed lapse rate. 1. BeerLambert law The specific transparency of a gas can be described with an absorption coefficient k. In general, this will depend on temperature, pressure, and frequency. In this post, I consider a simpler case in which it is constant. The units of k are area/mass. For a given mass of the gas, k gives a cross section area which absorbs incident radiation. The earlier post gives a bit more background derivation, ending up as follows.
2. Absorption and emission in the grey atmosphere Let us divide the atmosphere into N layers, all of equal mass per unit area. Hence, each layer has the same drop in pressure dp, and dp = p_{0}/N, where p_{0} is the pressure at the bottom of the atmosphere. The mass per unit area of the layer is dp/g. Hence radiation passing through any one layer has a fraction 1e^{k.dp/g} which is absorbed. We can apply Kirchoff's law, and conclude that the emissivity matches absorptivity; so this layer is radiating E.s.T(p+f)^{4}, where s is the StefanBoltzmann constant, 5.67e8 in SI units, and where T(p+f) is a representative temperature for the entire layer; somewhere between T(p) and T(p+dp), the temperatures at the top and bottom of the layer. When everything is all at the same temperature and at thermal equilibrium, the layer is absorbing energy from both top and bottom, and is emitting also out the top and the bottom. We're only going to worry about radiation propagating upwards (for now). Let Q(p) denote the radiation propagating upwards through the atmospheric level with pressure equal to p. Removing the absorbed radiation and adding the emitted radiation over a single layer we have: [indentblock]Q(p+dp) = (1  E).Q(p) + E.s.T(p+f)^{4}[/indentblock] We know the temperature profile T(z) already, because the lapse rate is fixed by assumption. All that is needed now is to iterate up the atmosphere from Q(p_{0}) at the surface, to Q(0) at the top of the atmosphere. That will be the energy lost to space, and which needed to be supplied by the onboard fusion heaters of the starship. 3. Temperature and lapse rate in the atmosphere The original post at post #22 describes how to calculate a lapse rate, based on a neutral dry atmosphere. A good online reference for this subsection is Thermodynamics & Statistical Mechanics, lecture notes by Richard Fitzpatrick of the University of Texas. (Also available as a bookl.) In particular, see this page: "The adiabatic atmosphere". We've already picked on the DALR as a basis for temperature in the atmosphere. Convenient formulae for altitude and temperature in terms of pressure are as follows:
4. Convection and the stratosphere As gbaikie has noted, the term "stratosphere" is ambiguous when we look at planets in general. Different authorities use different definitions of the word. Suffice to say… forget this section. Let's just proceed with the calculation of Q(0) exactly as given above, and then look at the results to see what implications follow, and where we might need to improve the model further. I'll get to that! 5. The spreadsheet. Version 1.1 Nerds and scientists and programmers will sneer… but I'm going to give you an Excel spreadsheet. Real experts would do this in R, at the very least. The spreadsheet has a huge advantage  a much wider audience will be able to grab it and experiment with various parameters. I'll use Excel 97, so old windows platforms can use it; and if you use Open Office or some such, it should work there also. Please let me know of any problems running the spreadsheet! I have taken a bit of time to clean it up. The spreadsheet is "protected"; a standard feature in Excel. It's easy to remove protection if you want to modify more than the intended user inputs. But for simply using it to calculate, enter information in the green cells. These cells have editing enabled. Your spreadsheet program should protect you from making changes to other more important cells. The spreadsheet is attached as a zip folder (to keep within the size limits on attached files). Try it out, tell me what you think and what you get. Especially tell me if you find bugs, or results that seem to be outside what you should expect. I'll probably be able to explain the result  or fix a bug which gave spurious numbers! The "k" value is entered in mm^2/kg, rather than m^2/kg. This means values are not tiny. The "k" value chosen for Earth is selected simply to give the same emission to space as we have in reality; the value is 97. With this, the emission to space for surface temperature of 15C and a dry adiabatic lapse rate is 241.2 W/m^2, which is about the same as what we have in reality. When this same value is applied to the dwarf planet starship, with temperature 10C, the emission to space is 105.54 W/m^2. That's actually a stronger greenhouse impact than for Earth, which surprised me at first. I had expected a weaker impact given the weak lapse rate. However, it seems that the more important fact is that the atmosphere ends up being much higher than on Earth, and so the effective emission layers are much colder. This gives a very strong greenhouse impact. The fusion generators now need 1.9 million GWatts, rather than the 6.6 million we had previously. Woohoo. The greenhouse effect in this case means only 30% of the original requirement is needed. I can play around with other parameters as I choose. I've entered a new example (and there's space for anyone to provide four new templates of their own choosing. My new example has:
With this, my spreadsheet gives a mere 3.4 W/m^2 trickling out the top of the atmosphere, a whopping decrease from the 272 W/m^2 at the surface. With the smaller surface area, my fusion generators now look a heck of a lot more reasonable. 6.8 thousand GWatts only. (10 times less area, 0.75 reduction in surface emission with the lower temperature, and power scaled down by 0.012 because of the massive greenhouse effect.) Play around, see what you get, and bug reports welcome! Cheers  Chris PS. Edited this post for corrected the emissivity formula for a single layer above. The spreadsheet looks to be doing it correctly, so no change there. Edited by Chris HoStuart, Dec 10 2011, 11:53 PM.

The giant space ship example · Physical theory for climate 