Using the energy-balance model with

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Uncertainty in measurement

Scientific Computing for Physicists, Ch The uncertainty of a simulated number: inputs, sensitivity and correlated output

Problem

Using the energy-balance model T=[S0(1−a)/(4εσSB)]1/4T = [S_0(1-a)/(4\varepsilon\sigma_{\rm SB})]^{1/4} with S0=1361±0.5 W m−2S_0 = 1361 \pm 0.5\,\mathrm{W\,m^{-2}}, a=0.30±0.02a = 0.30 \pm 0.02 and ε=0.612±0.02\varepsilon = 0.612 \pm 0.02, a Monte Carlo propagation with 10610^6 seeded draws gives T=287.9±3.13 KT = 287.9 \pm 3.13\,\mathrm{K}. Someone argues: "The Monte Carlo error scales as N−1/2N^{-1/2}, so if we redo this with 10810^8 draws instead of 10610^6, the sampling error shrinks by a factor of 1010. Since more draws means more information about ε\varepsilon, the reported uncertainty uT=3.13 Ku_T = 3.13\,\mathrm{K} should also shrink by close to that factor." ((a)) Find the error in this argument. ((b)) Classify each of S0S_0, aa and ε\varepsilon as aleatory or epistemic, giving a reason for each. ((c)) State what would actually reduce uTu_T, and why more Monte Carlo draws alone will not.

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