Peak X-ray fluxes (in W m ) of solar flares above a detection threshold are modelled…

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

Peak X-ray fluxes

X

(in W m

^{-2}

) of

n = 200

solar flares above a detection threshold

x_{\min} = 10^{-6}\ \mathrm{W\,m^{-2}}

are modelled by the continuous power-law density

p(X) \propto X^{-\alpha}

for

X \ge x_{\min}

. The sum of log-ratios is

\sum_{i=1}^{200} \ln(X_i/x_{\min}) = 310

. ((a)) Find the maximum-likelihood estimate

\hat\alpha

and its standard error. ((b)) Estimate the fraction of flares with flux at least five times

x_{\min}

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