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 Fragmentation 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 H-2 dissociation in facilitating Secondary Fragmentation 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 to 0.004 M-circle dot, 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 greater than or similar to 100 AU). Therefore brown dwarfs should have difficulty forming by Primary Fragmentation at R less than or similar to 30AU, explaining the Brown Dwarf Desert. Conversely, Primary Fragmentation at R greater than or similar to 100 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) H-2 dissociation can lead to collapse and Secondary Fragmentation, thereby converting primary fragments into close, low-mass binaries, with semi-major axes a similar to 5AU( m(SYSTEM)/0.1 M-circle dot), 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.