Space Mathematics by Bernice Kastner

Author:Bernice Kastner
Language: eng
Format: epub, azw3
Publisher: INscribe Digital

c. Find the altitude at which the pressure is 20 percent of standard atmospheric pressure.

Solution:

Substituting in the equation of part (a) gives (0.20) (1035) = (1035) (1/2)h/5 8, where h is in km, and so (0.2) = (1/2)h/5·8. Now, taking logarithms,

and

PROBLEM 2.

The rule for the variation of atmospheric pressure with height which was given in the previous problem can also be written

Atmospheric scientists often use this rule in one of its equivalent forms where the base is 10 or e, the base of the natural logarithms, instead of 2. Find k1 and k2 so that P = 1035 (2)-0·17h = 1035 (10)-k1h = 1035 (e)-k2h.

Solution:

We need to find k1 so that 20.17 = 10k1. Taking logarithms, 0.17 log 2 = k1 or k1 = (0.17) (0.301) - 0.051. For k2 we have 20.17 = ek2, or k2 = (0.17)loge2 -(0.17) (0.693) = 0.12.

PROBLEM 3.

Sometimes different bases are used together in the same application in atmospheric work. For example, atmospheric absorption of electromagnetic radiation from the Sun and other sources is dependent on the wavelengths of the incoming radiation. Instruments carried by rockets, balloons, and satellites have shown how far in the atmosphere such radiation penetrates before being reduced by a factor of 1/e, the conventional measure used in this work. The results are given in Fig. 6.1. Both the wavelength scale and the altitude scale are logarithmic, with the horizontal scale in base 10 and the vertical scale in base 2. (How much of this information could be displayed using linear scales even on a wall-sized chart?)

Fig. 6.1 shows that visible light and radio waves penetrate the atmosphere completely and reach Earth’s surface. However, gases such as oxygen, ozone, nitrogen, and water vapor absorb most of the infrared, ultraviolet, X-ray, and shorter wavelengths. At what altitude will solar infrared radiation of wavelength 10-4 m be reduced by a factor of 1/e?

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