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When we look at a rich galaxy cluster, why is it that elliptical galaxies tend to live near the center while spiral galaxies tend to live at the outskirt?

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This is just a reflection a the way elliptical galaxies are formed: in general there are older objects than spiral galaxies. So they have had more time to sink into the potential of the cluster via dynamical friction. During this orbital decay the galaxies are stripped out of their gas content and end up being gas-depleted, slow star-forming, elliptical galaxies.

The trend you mention also follows the Morphology-Environment relation, although much debate is still going on in that direction

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