We present the first detailed quantitative study of the stellar populations of the Sagittarius (Sgr) streams within the Stripe 82 region, using photometric and spectroscopic observations from the Sloan Digital Sky Survey (SDSS). The star formation history (SFH) is determined separately for the bright and faint Sgr streams, to establish whether both components consist of a similar stellar population mix or have a distinct origin. Best fit SFH solutions are characterised by a well-defined, tight sequence in age-metallicity space, indicating that star formation occurred within a well-mixed, homogeneously enriched medium. Star formation rates dropped sharply at an age of ~5-7 Gyr, possibly related to the accretion of Sgr by the MW. Finally, the Sgr sequence displays a change of slope in age-metallicity space at an age between 11-13 Gyr consistent with the Sgr alpha-element knee, indicating that supernovae type Ia started contributing to the abundance pattern ~1-3 Gyr after the start of star formation. Results for both streams are consistent with being drawn from the parent Sgr population mix, but at different epochs. The SFH of the bright stream starts from old, metal-poor populations and extends to a metallicity of [Fe/H]~-0.7, with peaks at ~7 and 11 Gyr. The faint SFH samples the older, more metal-poor part of the Sgr sequence, with a peak at ancient ages and stars mostly with [Fe/H]<-1.3 and age>9 Gyr. Therefore, we argue in favour of a scenario where the faint stream consists of material stripped i) earlier, and ii) from the outskirts of the Sgr dwarf.
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