  Context: Potentially hazardous asteroids (PHA) in Earth-crossing orbits pose
a constant threat to life on Earth. Several mitigation methods have been
proposed, and the most feasible technique appears to be the disintegration of
the impactor and the generation of a fragment cloud by explosive penetrators at
interception. However, mitigation analyses tend to neglect the effect of
orbital dynamics on the trajectory of fragments.
  Aims: We aim to study the effect of orbital dynamics of the impactor's cloud
on the number of fragments that hit the Earth, assuming different interception
dates. We investigate the effect of self-gravitational cohesion and the axial
rotation of the impactor.
  Methods: We computed the orbits of 10^5 fragments with a high-precision
direct N-body integrator of the eighth order, running on GPUs. We considered
orbital perturbations from all large bodies in the Solar System and the
self-gravity of the cloud fragments.
  Results: Using a series of numerical experiments, we show that orbital shear
causes the fragment cloud to adopt the shape of a triaxial ellipsoid. The shape
and alignment of the triaxial ellipsoid are strongly modulated by the cloud's
orbital trajectory and, hence, the impact cross-section of the cloud with
respect to the Earth. Therefore, the number of fragments hitting the Earth is
strongly influenced by the orbit of the impactor and the time of interception.
A minimum number of impacts occur for a well-defined orientation of the
impactor rotational axis, depending on the date of interception.
  Conclusions: To minimise the lethal consequences of an PHA's impact, a
well-constrained interception timing is necessary. A too-early interception may
not be ideal for PHAs in the Apollo or Aten groups. Thus, we find that the best
time to intercept PHA is when it is at the pericentre of its orbit.
