We analyse a comprehensive set of MIR/FIR observations of Stephans Quintet (SQ), taken with the Spitzer Space Observatory. Our study reveals the presence of a luminous (L_{IR}approx 4.6x10^43 erg/s) and extended component of infrared dust emission, not connected with the main bodies of the galaxies, but roughly coincident with the X-ray halo of the group. We fitted the inferred dust emission spectral energy distribution of this extended source and the other main infrared emission components of SQ, including the intergalactic shock, to elucidate the mechanisms powering the dust and PAH emission, taking into account collisional heating by the plasma and heating through UV and optical photons. Combining the inferred direct and dust-processed UV emission to estimate the star formation rate (SFR) for each source we obtain a total SFR for SQ of 7.5 M(sun)/yr, similar to that expected for non-interacting galaxies with stellar mass comparable to the SQ galaxies. Although star formation in SQ is mainly occurring at, or external to the periphery of the galaxies, the relation of SFR per unit physical area to gas column density for the brightest sources is similar to that seen for star-formation regions in galactic disks. We also show that available sources of dust in the group halo can provide enough dust to produce up to L_{IR}approx 10^42 erg/s powered by collisional heating. Though a minority of the total infrared emission (which we infer to trace distributed star-formation), this is several times higher than the X-ray luminosity of the halo, so could indicate an important cooling mechanism for the hot IGM and account for the overall correspondence between FIR and X-ray emission.
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