  The contribution of a chiral three-nucleon force to the strength of an
effective spin-orbit coupling is estimated. We first construct a reduced
two-body interaction by folding one-nucleon degrees of freedom of the
three-nucleon force in nuclear matter. The spin-orbit strength is evaluated by
a Scheerbaum factor obtained by the $G$-matrix calculation in nuclear matter
with the two-nucleon interaction plus the reduced two-nucleon interaction. The
problem of the insufficiency of modern realistic two-nucleon interactions to
account for the empirical spin-orbit strength is resolved. It is also indicated
that the spin-orbit coupling is weaker in the neutron-rich environment. Because
the spin-orbit component from the three-nucleon force is determined by the
low-energy constants fixed in the two-nucleon sector, there is little
uncertainty in the present estimation.
