Characterizing the atmospheres of planets orbiting M dwarfs requires understanding the spectral energy distributions of M dwarfs over planetary lifetimes. Surveys like MUSCLES, HAZMAT, and FUMES have collected multiwavelength spectra across the spectral types range of Teff and activity, but the extreme ultraviolet flux (EUV, 100 to 912 Angstroms) of most of these stars remains unobserved because of obscuration by the interstellar medium compounded with limited detector sensitivity. While targets with observable EUV flux exist, there is no currently operational facility observing between 150 and 912 Angstroms. Inferring the spectra of exoplanet hosts in this regime is critical to studying the evolution of planetary atmospheres because the EUV heats the top of the thermosphere and drives atmospheric escape. This paper presents our implementation of the differential emission measure technique to reconstruct the EUV spectra of cool dwarfs. We characterize our methods accuracy and precision by applying it to the Sun and AU Mic. We then apply it to three fainter M dwarfs: GJ 832, Barnards Star, and TRAPPIST-1. We demonstrate that with the strongest far ultraviolet (FUV, 912 to 1700 Angstroms) emission lines, observed with Hubble Space Telescope and/or Far Ultraviolet Spectroscopic Explorer, and a coarse X-ray spectrum from either Chandra X-ray Observatory or XMM-Newton, we can reconstruct the Suns EUV spectrum to within a factor of 1.8, with our models formal uncertainties encompassing the data. We report the integrated EUV flux of our M dwarf sample with uncertainties between a factor of 2 to 7 depending on available data quality.
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