Tensor representation of quadrupole expansion for softened potentials in N-body simulations

A tensor representation for quadrupole expansion is proposed as a recipe for n-body simulations of collisionless stellar systems that do not perform the direct summation method, as is the case of TREECODE. The quadrupole expansion is almost straightforward in the non-softened case, but it is pretty complicated when some softening artifice modifies the potential. A shorthand formula of quadrupole correction for potential and gravity force is derived that can be easily adapted to computer codes like C and FORTRAN. A test is performed to show how relevant it is to incorporate quadrupole correction in n-body codes like TREECODE and particle-mesh methods.