We present a new multi-component spectral energy distribution (SED) decomposition method and use it to analyze the ultraviolet to millimeter wavelength SEDs of a sample of dusty infrared-luminous galaxies. SEDs are constructed from spectroscopic and photometric data obtained with the Spitzer Space Telescope, in conjunction with photometry from the literature. Each SED is decomposed into emission from populations of stars, an AGN accretion disk, PAHs, atomic and molecular lines, and distributions of graphite and silicate grains. Decompositions of the SEDs of the template starburst galaxies NGC7714 and NGC2623 and the template AGNs PG0804+761 and Mrk463 provide baseline properties to aid in quantifying the strength of star-formation and accretion in the composite systems NGC6240 and Mrk1014. We find that obscured radiation from stars is capable of powering the total dust emission from NGC6240, although we cannot rule out a contribution from a deeply embedded AGN visible only in X-rays. The decomposition of Mrk1014 is consistent with ~65% of its power emerging from an AGN and ~35% from star-formation. We suggest that many of the variations in our template starburst SEDs may be explained in terms of the different mean optical depths through the clouds of dust surrounding the young stars within each galaxy. Prompted by the divergent far-IR properties of our template AGNs, we suggest that variations in the relative orientation of their AGN accretion disks with respect to the disks of the galaxies hosting them may result in different amounts of AGN-heated cold dust emission emerging from their host galaxies. We estimate that 30-50% of the far-IR and PAH emission from Mrk1014 may originate from such AGN-heated material in its host galaxy disk.
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