  We define, for smooth projective orbifold pairs $(X,D)$ notions of `slope
Rational connectedness', and of orbifold `slope Rational quotient' . These
notions extend to this larger context the classical notions of rationally
connected manifold and `rational quotient' (sometimes called `MRC fibration').
Our notions and proofs work entirely in characteristic zero, and are based on
the consideration of foliations with minimal positive slope with respect to
some suitable movable class. The existence of covering or connecting families
of `orbifold rational curves' is indeed presently unknown in the orbifold
context, in situations analogous to the classical case $D=0$. By contrast, the
notions we introduce here, are checkable in practice and can certainly be used
to show general properties expected from the existence of connecting families
of `orbifold rational curves'. The proofs given here in the orbifold context
provide new proofs in the classical case where $D=0$, since the classical
proofs did not seem to adapt, with the presently existing techniques, to this
broader context.
