The Cygnus region is a very bright and complex portion of the TeV sky, host to unidentified sources and a diffuse excess with respect to conventional cosmic-ray propagation models. Two of the brightest TeV sources, MGRO J2019+37 and MGRO J2031+41, are analyzed using Milagro data with a new technique, and their emission is tested under two different spectral assumptions: a power law and a power law with an exponential cutoff. The new analysis technique is based on an energy estimator that uses the fraction of photomultiplier tubes in the observatory that detect the extensive air shower. The photon spectrum is measured in the range 1-100 TeV using the last three years of Milagro data (2005-2008), with the detector in its final configuration. An F-test indicates that MGRO J2019+37 is better fit by a power law with an exponential cutoff than by a simple power law. The best-fitting parameters for the power law with exponential cutoff model are a normalization at 10 TeV of 7+5 –2 × 10–10 s–1 m–2 TeV–1, a spectral index of 2.0+0.5 –1.0, and a cutoff energy of 29+50 –16 TeV. MGRO J2031+41 shows no evidence of a cutoff. The best-fitting parameters for a power law are a normalization of 2.1+0.6 –0.6 × 10–10 s–1 m–2 TeV–1 and a spectral index of 3.22+0.23 –0.18. The overall flux is subject to a ~30% systematic uncertainty. The systematic uncertainty on the power-law indices is ~0.1. Both uncertainties have been verified with cosmic-ray data. A comparison with previous results from TeV J2032+4130, MGRO J2031+41, and MGRO J2019+37 is also presented.
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