1. What frac-tion of the beam is absorbed in passing through a layer containing 1 kgm−2of the gas?What mass of gas would the layer have to contain onorder to absorb half the incident radiation?
per unit mass(and consequentlythe heating rate) is strongest, not at the level of unit optical depth butnear the top of the atmosphere, where the incident radiation is virtuallyundepleted. 65
4.46Consider a hypothetical planetary atmosphere comprised entirely of thegas in Exercise 4.43. The atmospheric pressure at the surface of the planetis 1000 hPa, the lapse-rate is isothermal, the scale height is 10 km and thegravitational acceleration is 10 m s−2.Estimate the height and pressureof the level of unit normal optical depth.At the level of unit normal optical depthτλ=kλZ∞zρdz= 1Hence,kλgZ∞zρgdz=kλpg= 1It follows thatp=gkλ=100.01= 103PaFrom the hypsometric equationz=Hlnμp0p¶=10 ln 100=46km4.47(a) What percentage of the incident monochromatic intensity with wave-lengthλand zero zenith angle is absorbed in passing through the layerof the atmosphere extending from an optical depthτλ=0.2 toτλ=4.0?(b) What percentage of the outgoing monochromatic intensity to spacewith wavelengthλand zero zenith angle is emitted from the layer of theatmosphere extending from an optical depthτλ=0.2 toτλ=4.0? (c)In an isothermal atmosphere, through how many scale heights would thelayer in (a) and (b) extend?
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