Appendix A: Quadrupole expansion for the softened potential

The correct quadrupole expansion for the softened potential may be derived by considering the potential contribution from the enclosed mass in a cubic cell of the octal-tree. By Taylor expanding the softened potential, $\tilde\Phi$, around the cell's mass-center, we have:

$$\tilde\Phi=\tilde\Phi_{\rm o}+\tilde\Phi_2+...,$$ (A1)

where $\Phi_{\rm o}$ and $\Phi_2$ are respectively the monopole and the quadrupole contributions. Additional terms are suppressed since only quadrupole approximation is required. The dipole term $\Phi_1$ is zero since the expansion is being taken around the cell's mass-center.

The product-of-inertia tensor, with respect to the cell's mass-center, O, is defined as

$${\vec J}_{_O}=\sum_jm_j{\vec r}_{_Oj}{\vec r}_{_Oj}.$$ (A2)

where ${\vec r}_{_Oj}$ is the relative position of a particle j with respect to the cell's mass-center.

Let ${\vec R}$ be the position vector of the cell's mass-center. For notation saving, we introduce the softened mass-center distance:

$$R_{\epsilon}=\bigl(R^2+\epsilon^2\bigr)^{1/2},$$ (A3)

where $R=\vert{\vec R}\vert$.

The softened monopole-contribution to the cell's potential is written as

$$\tilde\Phi_{\rm o}={-{G}M_{\rm o}\over R_{\epsilon}},$$ (A4)

where, $M_{\rm o}$ is the cell mass:

$$M_{\rm o}=\sum_jm_j.$$ (A5)

By analogy to the definition of unity vector, we conveniently define the following vector:

$${\bf\hat R}_{\epsilon}={{\vec R}\over R_{\epsilon}},$$ (A6)

observing that ${\bf\hat R}_{\epsilon}{\vec\cdot}{\bf\hat R}_{\epsilon}\leq1$.

For convenience, we introduce the softened quadrupole mass,

$$M_2= (2R_{\epsilon}^2)^{-1} \biggl( {3}{\bf\hat R}_{\epsilon}... ...c\cdot}{\bf\hat R}_{\epsilon} -{\rm tr} {\vec J}_{_O} \biggr),$$ (A7)

to give the quadrupole contribution to the potential in the form:

$$\tilde\Phi_2={-{G}M_2\over R_{\epsilon}}\cdot$$ (A8)

In the non-softened case, $\epsilon\rightarrow0$, the quadrupole mass becomes the conventional form

$$M_2= {1\over 2}{{\bf\hat R}\over R} {\vec\cdot}\biggl[3{\vec ... ...{\rm tr} {\vec J}_{_O})\biggr]{\vec\cdot}{{\bf\hat R}\over R},$$ (A9)

where ${\vec 1}$ is the unity tensor, and the expression inside brackets is the non-softened case for the cell's quadrupole tensor.

The quadrupole correction is performed by applying Eqs. (A2), (A7), (A8), and extracting the gradient, to obtain the quadrupole contribution to gravity-acceleration. The six Cartesian components of the tensor ${\vec J}_{_O}$ are stored in the tree data-structure along with a tree-descent, visiting all the cubic cells.