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