  The drag-free satellites of LISA will maintain the test masses in geodesic
motion over many years with residual accelerations at unprecedented small
levels and time delay interferometry (TDI) will keep track of their
differential positions at level of picometers. This may allow investigations of
fine details of the gravitational field in the Solar System previously
inaccessible. In this spirit, we present the concept of a method to measure
directly the gravitational effect of the density of diffuse Local Dark Matter
(LDM) with a constellation of a few drag-free satellites, by exploiting how
peculiarly it would affect their relative motion. Using as test bed an
idealized LISA with rigid arms, we find that the separation in time between the
test masses is uniquely perturbed by the LDM, so that they acquire a
differential breathing mode. Such a LDM signal is related to the LDM density
within the orbits and has characteristic spectral components, with amplitudes
increasing in time, at various frequencies of the dynamics of the
constellation. This is the relevant result, in that the LDM signal is brought
to non-zero frequencies.
