  An anomalous constant acceleration of (8.7 \pm 1.3) x 10^-8 cm/s^2 directed
toward the Sun has been discovered by Anderson et al. in the motion of the
Pioneer 10/11 and Galileo spacecrafts. In parallel, the WMAP results have
definitively established the existence of a cosmological constant Lambda=1/
L_U^2, and therefore of an invariant cosmic length-scale L_U=(2.72 \pm 0.10)
Gpc. We show that the existence of this invariant scale definitively implements
Mach's principle in Einstein's theory of general relativity. Then we
demonstrate, in the framework of an exact cosmological solution of Einstein's
field equations which is valid both locally and globally, that the definition
of inertial systems ultimately depends on this length-scale. As a consequence,
usual local coordinates are not inertial, so that the motion of a free body of
speed v is expected to contain an additional constant acceleration
a_P=v^2(\surd3 L_U), which is, using the WMAP five years results, (6.02 \pm
0.34) x 10^-8 cm/s^2 when v \approx c. Such an effect is too small to
contribute significantly to the Pioneer acceleration (since v_Pioneer \approx
12 km/s << c), but could be possibly observed in a dedicated space mission.
