Accretion-powered X-ray pulsars are among the most luminous X-ray sources in the Galaxy. However, despite decades of theoretical and observational work since their discovery, no satisfactory model for the formation of the observed X-ray spectra has emerged. In this paper, we report on a self-consistent calculation of the spectrum emerging from a pulsar accretion column that includes an explicit treatment of the bulk and thermal Comptonization occurring in the radiation-dominated shocks that form in the accretion flows. Using a rigorous eigenfunction expansion method, we obtain a closed-form expression for the Greens function describing the upscattering of monochromatic radiation injected into the column. The Greens function is convolved with bremsstrahlung, cyclotron, and blackbody source terms to calculate the emergent photon spectrum. We show that energization of photons in the shock naturally produces an X-ray spectrum with a relatively flat continuum and a high-energy exponential cutoff. Finally, we demonstrate that our model yields good agreement with the spectra of the bright pulsar Her X-1 and the low luminosity pulsar X Per.
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