  In turbulent fragmentation, star formation occurs in condensations created by
converging flows. The condensations must be sufficiently massive, dense and
cool to be gravitationally unstable, so that they start to contract; {\it and}
they must then radiate away thermal energy fast enough for self-gravity to
remain dominant, so that they continue to contract. For the metallicities and
temperatures in local star forming clouds, this second requirement is only met
robustly when the gas couples thermally to the dust, because this delivers the
capacity to radiate across the full bandwidth of the continuum, rather than
just in a few discrete spectral lines. This translates into a threshold for
vigorous star formation, which can be written as a minimum ram-pressure Pcrit ~
4 10^-11 dyne. Pcrit is independent of temperature, and corresponds to flows
with molecular hydrogen number-density nH2 and velocity v satisfying nH2 v^2 >
800 cm^-3 (km/s)^2. This in turn corresponds to a minimum molecular hydrogen
column-density for vigorous star formation, NH2crit ~ 4 10^21 cm^-2 (SIGMAcrit
~ 100 MSun pc^-2), and a minimum visual extinction AVcrit ~ 9. The
characteristic diameter and line-density for a star-forming filament when this
threshold is just exceeded -- a sweet spot for local star formation regions --
are 2Rfil ~ 0.1 pc and mufil ~ 13 MSun pc^-2. The characteristic diameter and
mass for a prestellar core condensing out of such a filament are 2Rcore ~ 0.1
pc, and Mcore ~ MSun. We also show that fragmentation of a shock-compressed
layer is likely to commence while the convergent flows creating the layer are
still ongoing, and we stress that, under this circumstance, the phenomenology
and characteristic scales for fragmentation of the layer are fundamentally
different from those derived traditionally for pre-existing layers.
