Question

6. [Sec. 12.7) Estimate the mass of the Earth's troposphere using the following assumptions, being mindful of units. (a) The Earth is a smooth sphere of radius 6000 km. (b) The troposphere's mass density varies linearly from 1 kg m-3 at the Earth's surface to 1/3 kg m-3 at the top of the tropo sphere and is not a function of latitude or longitude. (c) The troposphere occupies the first 11 km of the atmosphere above the Earth's surface.

Answer 1

Considering the spherical co - ordinates system, origin at the center of the Earth the density function of the Troposphere is only the function of radial distance and varies linearly delta = 1 - 1 - 1/2 /11 times (p - 6000) where 6000 lessthanorequalto P lessthanorequalto 6011 therefore delta = 4011/11 - 2P/33000where P is in meter mass of the Troposphere is given by integral_R dn = integral_R delta * dV M = integral^theta = 2n_theta = 0 integral^theta = phi_theta = 0 integral P = 6011 times 10^3_P = 6000 times 10^3 (4011/11 - 2p/33000) * P^* sin phi * dP * d phi * d theta = integral^theta = 2pi_theta = 0 integral^phi = pi_phi = 0 (4011/11 * P^3/3 - P^4/2 times 33000) Vertical-bar ^6011 times 10^3_6000 times 10^3 sin phi d phi d theta = integral^theta = 2 pi_theta = 0 integral^phi = pi_phi = 0

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