Context: The convective origin of supergranulation is highly controversial. Past measurements of intensity variations inside supergranules have often been influenced by the brightness enhancement at the cell boundaries due to the magnetic network. Aims: We conduct a precise determination of intensity variations inside supergranules. Methods: We determine the supergranule cell boundary from smoothed divergence maps derived from horizontal flow maps. We derive these flow maps from intensity maps obtained by MDI/SOHO in high resolution mode. We discuss the different possible approaches to take into account the influence of the magnetic field which can be used to determine the intensity variations inside supergranules. Results: We observe a significant decrease of the intensity from the center to the boundary of supergranules. We also obtain additional clues from the behavior of the maximum intensities and minimum intensities around each pixel, which are related to granules and intergranules: the maximum intensity decreases from center to boundary, while the minimum intensity is constant or increases depending how restrictive the selection is. The difference between intensity profiles versus divergence and relative distance to cell center also provides complementary information. The corresponding temperature differences between cell center and boundary are in the range 0.8-2.8 K. The intensity enhancement (for the magnetic network) or deficit (for intranetwork fields) depends on the localisation inside the cell. Conclusions: .It is the first time that such a detailed analysis of intensity variations inside supergranulation is performed. Our results are compatible with a convective origin of supergranulation, as the intensity decreases toward the boundary of the cells. However, new simulations of supergranulation are necessary to verify whether the compared behavior of granule and intergranule intensity variations is in close agreement with convection.
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