We analyze the far-ultraviolet (FUV) spectra of 33 classical T Tauri stars (CTTS), including 20 new spectra obtained with the Advanced Camera for Surveys Solar Blind Channel (ACS/SBC) on the Hubble Space Telescope. Of the sources, 28 are in the ~1 Myr old Taurus-Auriga complex or Orion Molecular Cloud, 4 in the 8-10 Myr old Orion OB1a complex and one, TW Hya, in the 10 Myr old TW Hydrae Association. We also obtained FUV ACS/SBC spectra of 10 non-accreting sources surrounded by debris disks with ages between 10 and 125 Myr. We use a feature in the FUV spectra due mostly to electron impact excitation of h2 to study the evolution of the gas in the inner disk. We find that the h2 feature is absent in non-accreting sources, but is detected in the spectra of CTTS and correlates with accretion luminosity. Since all young stars have active chromospheres which produce strong X-ray and UV emission capable of exciting h2 in the disk, the fact that the non-accreting sources show no h2 emission implies that the h2 gas in the inner disk has dissipated in the non-accreting sources, although dust (and possibly gas) remains at larger radii. Using the flux at 1600 {AA}, we estimate that the column density of h2 left in the inner regions of the debris disks in our sample is less than ~ 3x10^-6 g cm^-2, nine orders of magnitude below the surface density of the minimum mass solar nebula at 1 AU.
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