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4. Fitting a Spectral Line 30 points A spectrometer is used to record X-ray photons with energies between 0 and 5 keV. The spec- trometer has 50 channels with a resolution of 0.1 keV per channel. An X-ray spectrum with counts ni in channel i is available in the file xray _data. txt. 50 40 30 20 10 2 3 5 energy [keV] You are searching for the presence of a spectral line in the data. Due to the energy resolution of your X-ray telescope, the spectral line is broadened and can be modeled as a Gaussian of width 0 = 0.2 keV. In addition, your instrument has a nonzero background rate B which is the same in each channel. Thus, if the amplitude of the spectral line is S and its location is E0, the expected count in channel i is A = B + S where Ei is the central energy of channel i. The flat background B and spectral line amplitude S are unknown. The line energy E0 may or may not be known. (a) Assuming the counts nj obey Poisson statistics, write the likelihood of the data {ni] given S, B, Eo, and 0. (b) Suppose you know E0 = 2.5 keV. Maximize the likelihood (minimize - In C) from part (a) to estimate S and B and the uncertainties on the estimates. Note that if you use a min- imization algorithm like BFGS you will be able to access the inverse of the Hessian matrix of the likelihood. Plot your best fit on top of the data and express your best estimates in the form B I UB and S US.

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