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

Context.The minimum mass for star formation is a critical parameter with profound astrophysical, cosmological and anthropic consequences. Aims.Our first aim is to calculate the minimum mass for $Primary\, F\!ragmentation$ in a variety of potential star-formation scenarios, i.e. (a) hierarchical fragmentation of a 3D medium; (b) one-shot, 2D fragmentation of a shock-compressed layer; (c) fragmentation of a circumstellar disc. By Primary Fragmentation we mean fragmentation facilitated by efficient radiative cooling. Our second aim is to evaluate the role of H2 dissociation in facilitating $S\!econdary\, F\!ragmentation$ and thereby producing close, low-mass binaries. Methods.We use power-law fits to the constitutive physics, a one-zone model for condensing fragments, and the diffusion approximation for radiative transport in the optically thick limit, in order to formulate simple analytic estimates. Results.(i) For contemporary, local star formation, the minimum mass for Primary Fragmentation is in the range $0.001\,{\rm to}\;0.004\,{M}_{\odot}$, irrespective of the star-formation scenario considered. This result is remarkable since, both the condition for gravitational instability, and the radiation transport regime operating in a minimum-mass fragment, are different in the different scenarios. (ii) Circumstellar discs are only able to radiate fast enough to undergo Primary Fragmentation in their cool outer parts ($R \ga 100\,{\rm AU}$). Therefore brown dwarfs should have difficulty forming by Primary Fragmentation at $R \la 30\,{\rm AU}$, explaining the Brown Dwarf Desert. Conversely, Primary Fragmentation at $R \ga 100\,{\rm AU}$ could be the source of brown dwarfs in wide orbits about Sun-like stars, and could explain why massive discs extending beyond this radius 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 $a \sim 5\,{\rm AU}\,(m_{_{\rm SYSTEM}}/0.1\,{M}_{\odot})$, in good agreement with observation; in this circumstance, 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 primary fragment is spinning. Secondary Fragmentation is therefore most likely in primary fragments formed in the outer parts of circumstellar discs (since such fragments inevitably spin), and this could explain why a brown dwarf in a wide orbit about a Sun-like star has a greater likelihood of having a brown-dwarf companion than a brown dwarf in the field – as seems to be observed. Moreover, we show that binary brown dwarfs formed in this way can sometimes be ejected into the field without breaking up