Physics 281: Homework #5: due May 9

By the end of the week, there will be 7 problems (6 from lectures, plus a bonus).

1. One problem astronomical observatories face is water vapor—especially relevant to infrared and sub-millimeter work, where the absorption/emission of light by water vapor veils the sky. The water content of the atmosphere is characterized in millimeters of precipitable water. That is, how deep the water column would be if you condensed it all out into a sheet. How many millimeters of water (overburden) do you expect as a function of altitude for air saturated with non-condensing water (no clouds)? Produce numbers for sea level through 5 km, at some reasonable cadence. Try both a dry lapse rate and a wet one, to bound the problem. Use the top of the troposphere as the upper bound of where water can be. For all cases, use 17 g/m³ as the density of water vapor at sea level (at 20°C), and the scaling with temperature we covered in class.
   
   
2. Use the fact that water has a surface tension of γ = 0.07 N/m to compute the largest a falling raindrop (in terminal velocity) may be without the ram pressure from air ripping it apart.
   
   
3. If we say that pre-industrial CO₂ was at 280 ppmv, and at this time we enjoyed a greenhouse effect of 33°C (21°C from H₂O; 7°C from CO₂; and 5°C from methane and other gases), we can estimate the equilibrium warming accompanying an increase in CO₂ concentration to 400 ppmv. But if it is warmer (including the ocean surface), there will be a higher equilibrium density of water in the atmosphere as well, which is a more potent greenhouse gas.
   
   1. If the pre-industrial average temperature is 288 K, what does it become from just the added CO₂ component, using a simple linear scaling?
   2. Using this number, by what factor does the concentration of water increase, using the scaling discussed in Murphy Lecture 7? What additional temperature increase might we expect from the extra water—again using a simple linear scaling based on water concentration?
   3. If the increase from added water vapor is at all significant compared to that from CO₂, you might consider iterating. What happens when you do this, and what is the implication?
   4. Practically speaking, more water in the air probably means more clouds. If we assume that clouds alone are responsible for the 30% reflection of sunlight from the earth, what will be the temperature effect of more clouds in the proportion suggested by the water increase calculated for part (b)? For simplicity, just calculate the the change to the effective IR temperature of earth (255 K under 30% reflection), and assume this difference applies to ground level as well. Who wins? Does this problem make you want to stay in physics rather than chase down the complexities of weather?
   
   
4. Let's say we wanted to sequester atmospheric CO₂, so that we could continue our present rate of fossil fuel usage without increasing concentrations of atmospheric CO₂ beyond their current (dangerous?) levels. For simplicity, let's say that we sequester into chunks of concrete (effectively calcium carbonate, CaCO₃).
   1. At a density of about 2.5 times that of water, how much physical volume of concrete would a typical American need to "generate" to account for their lifetime of fossil fuel use? Assume each American uses energy at a rate of 10,000 W, and that 85% of this comes from fossil fuels. Put the answer in a linear dimension and compare to something familiar.
   2. If you did a reasonable job on the first part, you'll realize that we may have trouble finding space for it all. So let's dump it in the ocean! How much does sea level rise if we made enough concrete to bring the atmospheric CO₂ concentration back to pre-industrial levels? Finally, we catch a break!
   
   
   
5. The sun beats down on dark pavement depositing 1000 W/m² of energy. Let's say that radiation and convection together account for about 25 W/m²/K of transfer out, and that air in contact with the road acquires the resulting road surface temperature, relaxing to the ambient level approximately one meter above the surface.
   1. What is the steepest angle that will produce a mirage in these conditions? How far ahead is this apparent "puddle" when you are sitting in a car driving down the road?
   2. If it becomes difficult to see/notice mirages beyond about 400 m from your position in a car (because the road is not planar and straight, etc.), how large must the temperature difference be in order to get a mirage? What does this suggest in terms of limits in solar input or convective/radiative coupling?
   
   
   
6. Atmospheric refraction poses a challenge to astronomical observations not only because the pointing/tracking must account for the deflection, but because the dispersion of the atmosphere bends different colors differently. An image therefore risks having elongated stars. An unfortunately-oriented (i.e., horizontal) spectrograph slit will have a wavelength-dependent acceptance for an object on the sky, which will imprint onto the spectrum and may lead to false claims or at least bad data. We can explore the problem in two ways:
   1. If you operate a CCD camera wide open with no filters, it will cover wavelengths from roughly 350–1000 nm. How close to the zenith must you stay if you do not want the resulting chromatic smear to compete with the atmospheric "seeing" (blur size imposed by atmospheric turbulence—nothing to do with transparency), which for good sites is in the neighborhood of one arcescond (5 microradians)?
   2. If you want to operate down to elevations 20 degrees above the horizon, how narrow must the pass-band of a blue-ish filter be (in nanometers) in order to avoid smearing the image?
   
   
   
7. Invent a problem of your own in the spirit of this week's class topics. You do not need to provide a complete solution, though you might outline how it may go. The art is to strike a balance between trivial and hopelessly complex.
   

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