A Natural Symmetrization for the Plummer Potential
  We propose a symmetrized form of the softened gravitational potential which
is a natural extension of the Plummer potential. The gravitational potential at
the position of particle i (x_i,y_i,z_i), induced by particle j at
(x_j,y_j,z_j), is given by: phi_ij = -G m_j/|r_ij^2+e_i^2+e_j^2|^1/2, where G
is the gravitational constant, m_j is the mass of particle j, r_ij =
|(x_i-x_j)^2+(y_i-y_j)^2+(z_i-z_j)^2|^1/2 and e_i and e_j are the gravitational
softening lengths of particles i and j, respectively. This form is formally an
extension of the Newtonian potential to five dimensions. The derivative of this
equation in the x,y, and z directions correspond to the gravitational
accelerations in these directions and these are always symmetric between two
particles.
  When one applies this potential to a group of particles with different
softening lengths, as is the case with a tree code, an averaged gravitational
softening length for the group can be used. We find that the most suitable
averaged softening length for a group of particles is <e_j^2> = sum_j^N m_j
e_j^2 / M, where M = sum_j^N m_j and N are the mass and number of all particles
in the group, respectively. The leading error related to the softening length
is O(sum_j r_j d(e_j^2)/r_ij^3), where r_j is the distance between particle j
and the center of mass of the group and d(e_j^2) = e_j^2 - <e_j^2>. Using this
averaged gravitational softening length with the tree method, one can use a
single tree to evaluate the gravitational forces for a system of particles with
a wide variety of gravitational softening lengths. Consequently, this will
reduce the calculation cost of the gravitational force for such a system with
different softenings without the need for complicated forms of softening. We
present the result of simple numerical tests. We found that our modification of
the Plummer potential works well.
