A Likelihood Method for Measuring the Ultrahigh Energy Cosmic Ray Composition

Air fluorescence detectors traditionally determine the dominant chemical composit ion of the ultrahigh energy cosmic ray flux by comparing the averaged slant depth of the shower maximum, $X_{max}$, as a function of energy to the slant depths expect ed for various hypothesized primaries. In this paper, we present a method to make a direct measurement of the expected mean number of protons and iron by comparing the shap es of the expected $X_{max}$ distributions to the distribution for data. The advantages of this method includes the use of information of the full distribution and its ability to calculate a flux for various cosmic ray compositi ons. The same method can be expanded to marginalize uncertainties due to choice of spectra, hadronic models and atmospheric parameters. We demonstrate the technique with independent simulated data samples from a parent sample of protons and iron. We accurately predict the number of protons and iron in the parent sample and show that the uncertainties are meaningful.