We present the first-ever rotationally resolved spectroscopic investigation of (624) Hektor and (911) Agamemnon, the two largest Jupiter Trojans. The visible and near-infrared spectra that we have obtained at the TNG telescope (La Palma, Spain) do not show any feature or hints of heterogeneity. In particular we found no hints of water-related absorptions. No cometary activity was detected down to ~23.5 R-mag/arcsec2 based on the complementary photometric data. We estimated upper limits on the dust production rates of Hektor and Agamemnon to be ~30 kg/s and ~24 kg/s, respectively. We modelled complete visible and near-infrared spectra of our targets using the Shkuratov formalism, to define the upper limit to the presence of water ice and more in general to constrain their surface composition. For both objects, successful models include amorphous carbon, magnesium-rich pyroxene and kerogen, with an upper limit to the amount of water ice of a few percent.
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