  The presence of fallback disks around young neutron stars has been invoked
over the years to explain a large variety of phenomena. Here we perform a
numerical investigation of the formation of such disks during a supernova
explosion, considering both neutron star (NS) and black hole (BH) remnants.
Using the public code MESA, we compute the angular momentum distribution of the
pre-supernova material, for stars with initial masses M in the range 13 - 40
Msun, initial surface rotational velocities vsurf between 25% and 75% of the
critical velocity, and for metallicities Z of 1%, 10% and 100% of the solar
value. These pre SN models are exploded with energies E varying between 10^{50}
- 3x10^{52} ergs, and the amount of fallback material is computed. We find
that, if magnetic torques play an important role in angular momentum transport,
then fallback disks around NSs, even for low-metallicity main sequence stars,
are not an outcome of SN explosions. Formation of such disks around young NSs
can only happen under the condition of negligible magnetic torques and a
fine-tuned explosion energy. For those stars which leave behind BH remnants,
disk formation is ubiquitous if magnetic fields do not play a strong role;
however, unlike the NS case, even with strong magnetic coupling in the
interior, a disk can form in a large region of the {Z,M,vsurf,E} parameter
space. Together with the compact, hyperaccreting fallback disks widely
discussed in the literature, we identify regions in the above parameter space
which lead to extended, long-lived disks around BHs. We find that the physical
conditions in these disks may be conducive to planet formation, hence leading
to the possible existence of planets orbiting black holes.
