Experiments on the Voyager 1 and 2 spacecraft and observations made by the International Ultraviolet Explorer (IUE) have provided evidence for the existence of energetic particle precipitation into the upper atmosphere of Jupiter from the magnetosphere. This auroral precipitation has been shown to generate large ionization and dissociation rates, to excite auroral emissions, and also to vibrationally excite molecular hydrogen. A theoretical model of vibrationally excited H2 in the upper atmosphere of Jupiter is presented in this paper. Models are considered for both the auroral region and also for lower latitudes, where H2 is vibrationally excited owing to processes associated with the absorption of solar ultraviolet radiation. The ground electronic state (X 1&Sgr;g +) of molecular hydrogen can be vibrationally excited in a variety of ways, such as by direct electron impact excitation, by dissociative recombination of H3 +, and by excitation of the Lyman and Werner bands. Vibrationally excited H2 can react with either H2 or H via vibration-translation (VT) interchange collisions, or via vibration-vibration (VV) interchange collisions with other H2 molecules. Reaction with H+ ions is possible for vibrational levels with v≥4. Molecular and eddy diffusion of the vibrationally excited H2 is also taken into account in these model calculations. The reaction of H+ ions with vibrationally excited molecular hydrogen is shown to be a very important loss process for this ion and reduces the electron densities theoretically calculated for the auroral region by about a factor of 10. The theoretical ionosphere calculations are coupled with the vibrationally excited H2 calculations in a self-consistent manner. ¿ 1987 American Geophysical Union