  Recent works have proposed the idea of a tidal screening scenario, in which
tidal forces determine the mass that a protostar can accrete to explain the
IMF. In this scenario, gravitationally unstable fragments will compete for the
gas reservoir in a star-forming clump. In this contribution, we propose to
properly include the action of an external gravitational potential in the Jeans
linear instability analysis as previously proposed by Jog. We have found that
an external gravitational potential can reduce the critical mass required for
the perturbation to collapse if the tidal force produced is compressive or
increase it if it is disruptive. Our analytical treatment provides (a) new mass
and length collapse conditions; (b) a simple equation for observers to check
whether their observed fragments can collapse; and (c) a simple equation to
compute whether collapse-induced turbulence can produce the levels of observed
fragmentation. Our results suggest that, given envelopes with similar mass and
density, the flatter ones should produce more stars than the steeper ones. If
the density profile is a power-law, the corresponding power-law index
separating these two regimes should be about 1.5. We finally applied our
formalism to 160 fragments identified within 18 massive star-forming cores of
previous works. We found that considering tides, 49% of the sample may be
gravitationally unstable and that it is unlikely that turbulence acting at the
moment of collapse has produced the fragmentation of these cores. Instead,
these fragments should have formed earlier when the parent core was
substantially flatter.
