We present deep spectroscopic observations of the classical T Tauri stars DF Tau and V4046?Sgr in order to better characterize two important sources of far-ultraviolet continuum emission in protoplanetary disks. These new Hubble Space Telescope-Cosmic Origins Spectrograph observations reveal a combination of line and continuum emission from collisionally excited H2 and emission from accretion shocks. H2 is the dominant emission in the 1400?? λ 1650?? band spectrum of V4046?Sgr, while an accretion continuum contributes strongly across the far-ultraviolet spectrum of DF Tau. We compare the spectrum of V4046?Sgr to models of electron-impact-induced H2 emission to constrain the physical properties of the emitting region, after making corrections for attenuation within the disk. We find reasonable agreement with the broad spectral characteristics of the H2 model, implying N(H2) ~ 1018?cm–2, T(H2) = 3000+1000 –500?K, and a characteristic electron energy in the range of ~50-100?eV. We propose that self-absorption and hydrocarbons provide the dominant attenuation for H2 line photons originating within the disk. For both DF Tau and V4046?Sgr, we find that a linear fit to the far-UV data can reproduce near-UV/optical accretion spectra. We discuss outstanding issues concerning how these processes operate in protostellar/protoplanetary disks, including the effective temperature and absolute strength of the radiation field in low-mass protoplanetary environments. We find that the 912-2000?? continuum in low-mass systems has an effective temperature of ~104?K with fluxes 105-107 times the interstellar level at 1?AU.
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