Impact crater diameters (in km) on a planetary surface are measured for craters above…

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Data analysis and model fitting in the laboratory

Many Parts, One Behaviour, Ch Heavy tails: power laws and how to measure them

Problem

Impact crater diameters DD (in km) on a planetary surface are measured for n=120n = 120 craters above the lower cutoff xmin⁡=1 kmx_{\min} = 1\ \mathrm{km}, and modelled by the continuous power-law density p(D)∝D−αp(D) \propto D^{-\alpha} for D≥xmin⁡D \ge x_{\min}. The sum of log-ratios is ∑i=1120ln⁡(Di/xmin⁡)=180.5\sum_{i=1}^{120} \ln(D_i/x_{\min}) = 180.5. ((a)) Find the maximum-likelihood estimate α^\hat\alpha and its standard error. ((b)) Estimate the fraction of craters with diameter at least 8 km8\ \mathrm{km}.

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