  Observations suggest that star formation in filamentary molecular clouds
occurs in a two-step process, with the formation of filaments preceding that of
prestellar cores and stars. Here, we apply the gravo-turbulent fragmentation
theory of Hennebelle & Chabrier 08, 09, 13 to a filamentary environment, taking
into account magnetic support. We discuss the induced geometrical effect on the
cores, with a transition from 3D geometry at small scales to 1D at large ones.
The model predicts the fragmentation behavior of a filament for a given mass
per unit length (MpL) and level of magnetization. This CMF for individual
filaments is then convolved with the distribution of filaments to obtain the
final system CMF. The model yields two major results: (i) the filamentary
geometry naturally induces a hierarchical fragmentation process, first into
groups of cores, separated by a length equal to a few filament Jeans lengths,
i.e. a few times the filament width. These groups then fragment into individual
cores. (ii) Non-magnetized filaments with high MpL are found to fragment
excessively, at odd with observations. This is resolved by taking into account
the magnetic field treated simply as additional pressure support). The present
theory suggests two complementary modes of star formation: while small
(spherical or filamentary) structures will collapse directly into prestellar
cores, according to the standard Hennebelle-Chabrier theory, the large
(filamentary) ones, the dominant population according to observations, will
follow the afore-described two-step process.
