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(xi, yi, zi), induced by particle j at (xj, yj, zj), is given by:ϕij=-Gmj|rij2+ϵi2+ϵj2|1/2,where G is the gravitational constant, mj is the mass of particle j, rij = ∣(xi − xj)2 + (yi − yj)2 + (zi − zj)2∣1/2 and ϵi and ϵj 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 they are always symmetric between two particles.When one applies this potential to a group of particles with different softening lengths, as in 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 〈ϵj2〉=∑jNmjϵj2/M, where M=∑jNmj and N   are the mass and number of all particles in the group, respectively. The leading error related to the softening length is O∑jδrjδϵj2/rij3, where δrj is the distance between particle j   and the center of mass of the group and δϵj2=ϵj2-〈ϵj2〉. 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.

► We extend the Plummer potential to a symmetrized form for different softening lengths. ► We apply this symmetrized potential to a group of particles with different softenings. ► An averaged softening for the group can be used to evaluate the gravitational force. ► The use of this method leads to a large decrease in the force calculation with tree. ► The result of numerical tests shows that our symmetrized potential works well.