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Our first aim is to calculate the minimum mass for Primary Fragmentation in avariety of potential star-formation scenarios, i.e. (i) hierarchicalfragmentation of a 3-D medium; (ii) one-shot, 2-D fragmentation of ashock-compressed layer; (iii) fragmentation of a circumstellar disc. Our secondaim is to evaluate the role of H2 dissociation in facilitating SecondaryFragmentation and thereby producing close, low-mass binaries. Results: (i)Forcontemporary, local star formation, the minimum mass for Primary Fragmentationis in the range 0.001-0.004Msun, irrespective of the scenario considered.(ii)Circumstellar discs are only able to radiate fast enough to undergo PrimaryFragmentation in their cool outer parts (R>100AU). Therefore brown dwarfs (BDs)should have difficulty forming by Primary Fragmentation at R<30AU, explainingthe Brown Dwarf Desert.Conversely, Primary Fragmentation at R>100AU could bethe source of brown dwarfs in wide orbits, and could explain why massive discswith Rd>100AU are rarely seen.(iii)H2 dissociation can lead to collapse andSecondary Fragmentation, thereby converting primary fragments into close,low-mass binaries, with semi-major axes a~5AU(Msystem/0.1Msun), in goodagreement with observation; in this case, the minimum mass for PrimaryFragmentation becomes a minimum system mass, rather than a minimum stellarmass.(iv)Any primary fragment can undergo Secondary Fragmentation, producing aclose low-mass binary, provided only that the fragment is spinning. SecondaryFragmentation is therefore most likely in fragments formed in the outer partsof discs, and this could explain why a BD in a wide orbit about a Sun-like starhas a greater likelihood of having a BD companion than a BD in the field -asseems to be observed.Comment: 15 pages, A&A accepte

The minimum mass for star formation, and the origin of binary brown dwarfs