We report on the Fermi-LAT observations of the Geminga pulsar, the second brightest non-variable GeV source in the γ-ray sky and the first example of a radio-quiet γ-ray pulsar. The observations cover one year, from the launch of the Fermi satellite through 2009 June 15. A data sample of over 60,000 photons enabled us to build a timing solution based solely on γ-rays. Timing analysis shows two prominent peaks, separated by Δφ = 0.497 ± 0.004 in phase, which narrow with increasing energy. Pulsed γ -rays are observed beyond 18 GeV, precluding emission below 2.7 stellar radii because of magnetic absorption. The phase-averaged spectrum was fitted with a power law with exponential cutoff of spectral index Γ = (1.30 ± 0.01 ±0.04), cutoff energy E0 = (2.46 ± 0.04 ± 0.17) GeV, and an integral photon flux above 0.1 GeV of (4.14 ± 0.02 ± 0.32) × 10-6 cm-2 s-1. The first uncertainties are statistical and the second ones are systematic. The phase-resolved spectroscopy shows a clear evolution of the spectral parameters, with the spectral index reaching a minimum value just before the leading peak and the cutoff energy having maxima around the peaks. The phase-resolved spectroscopy reveals that pulsar emission is present at all rotational phases. The spectral shape, broad pulse profile, and maximum photon energy favor the outer magnetospheric emission scenarios.
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