In a simple model of the atmosphere

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Foundations of mathematical language · Variational and asymptotic methods

Mathematical Methods for Physics, Ch Series, limits and controlled approximation

Problem

In a simple model of the atmosphere, the pressure at height hh relative to the surface pressure P0P_0 is P(h)/P0=e−h/HP(h)/P_0 = e^{-h/H}, where H=8.00 kmH = 8.00\,\mathrm{km} is the scale height. The exponential is approximated by its first three terms, P(h)P0≈1−hH+12(hH)2.\frac{P(h)}{P_0} \approx 1 - \frac{h}{H} + \frac{1}{2}\left(\frac{h}{H}\right)^2. Let x=h/Hx = h/H. ((a)) Write the Lagrange remainder for this truncation and use it to bracket the size of the error between two explicit bounds valid for x>0x > 0. ((b)) For h=2.40 kmh = 2.40\,\mathrm{km} (so x=0.300x = 0.300), evaluate both bounds and the true error, to three significant figures, and check the true error lies between them.

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