The previously unidentified very high-energy (VHE; E > 100 GeV) gamma-ray source HESS J1303-631, discovered in 2004, is re-examined including new data from the H.E.S.S. Cherenkov telescope array. Archival data from the XMM-Newton X-ray satellite and from the PMN radio survey are also examined. Detailed morphological and spectral studies of VHE gamma-ray emission as well as of the XMM-Newton X-ray data are performed. Significant energy-dependent morphology of the gamma-ray source is detected with high-energy emission (E > 10 TeV) positionally coincident with the pulsar PSR J1301-6305 and lower energy emission (E <2 TeV) extending sim 0.4^{circ} to the South-East of the pulsar. The spectrum of the VHE source can be described with a power-law with an exponential cut-off N_{0} = (5.6 pm 0.5) X 10^{-12} TeV^-1 cm^-2 s^-1, Gamma = 1.5 pm 0.2) and E_{rm cut} = (7.7 pm 2.2) TeV. The PWN is also detected in X-rays, extending sim 2-3 from the pulsar position towards the center of the gamma-ray emission region. The spectral energy distribution (SED) is well described by a one zone leptonic scenario which, with its associated caveats, predicts a very low average magnetic field for this source. Significant energy-dependent morphology of this source, as well as the identification of an associated X-ray PWN from XMM-Newton observations enable identification of the VHE source as an evolved PWN associated to the pulsar PSR J1303-6305. However, the large discrepancy in emission region sizes and the low level of synchrotron radiation suggest a multi-population leptonic nature. The low implied magnetic field suggests that the PWN has undergone significant expansion. This would explain the low level of synchrotron radiation and the difficulty in detecting counterparts at lower energies, the reason this source was originally classified as a dark VHE gamma-ray source.
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