diff --git a/examples/theory/multipoles.tex b/examples/theory/multipoles.tex index edae2f85018c87526c254f4996d8e7f72bfc5406..bfb2f7f97a86a066fddebe49319d40fbdfd4a94f 100644 --- a/examples/theory/multipoles.tex +++ b/examples/theory/multipoles.tex @@ -127,42 +127,37 @@ In the B-H approximation, the potential at position $\p{i}$ due to a set of particles $A$ is given by \begin{equation} - \phi(\p{i}) = -\sum_{\bf n} M_{A, \bf n} D_{\bf n}(\p{i} - \muu_A) + \phi(\p{i}) = -\sum_{\bf n} M_{A, \bf n} D_{\bf n}(\muu_A - \p{i}) \end{equation} Keeping only the terms up to second order (i.e. letting the sum run over -all vectors with $|{\bf n}|\leq2$) and writing $\rr = \p{i} - \muu_A$, we get: +all vectors with $|{\bf n}|\leq2$) and writing $\rr_{i,A} = \muu_A - \p{i}$, we +get: \begin{eqnarray} - \phi(\p{i}) &=& -M_{{\rm tot},A} \frac{G}{|\rr|} -\frac{1}{2}\sum_\alpha -I_{A,\alpha} \left(\frac{3Gr_\alpha^2}{|\rr|^5} - -\frac{G}{|\rr|^3}\right) \\ -& &- \frac{1}{2}\sum_{\alpha,\beta} \delta_{\alpha\beta}I_{A,\alpha\beta} -\frac{3Gr_\alpha r_\beta}{|\rr|^5} \\ - &=& -M_{{\rm tot},A} \frac{G}{|\rr|} + -\frac{1}{2}\left( I_{A,xx} + I_{A,yy} +I_{A,zz}\right)\frac{G}{|\rr|^3} \\ + \phi(\p{i}) &=& -M_{{\rm tot},A} \frac{G}{|\rr_{i,A}|} -\frac{1}{2}\sum_\alpha +I_{A,\alpha} \left(\frac{3Gr_\alpha^2}{|\rr_{i,A}|^5} - +\frac{G}{|\rr_{i,A}|^3}\right) \\ +& &+ \frac{1}{2}\sum_{\alpha,\beta} (1-\delta_{\alpha\beta})I_{A,\alpha\beta} +\frac{3Gr_\alpha r_\beta}{|\rr_{i,A}|^5} \\ + &=& -M_{{\rm tot},A} \frac{G}{|\rr_{i,A}|} + +\frac{1}{2}\left( I_{A,xx} + I_{A,yy} +I_{A,zz}\right)\frac{G}{|\rr_{i,A}|^3} +\\ & & - \frac{1}{2}\sum_{\alpha,\beta} I_{A,\alpha\beta} \frac{3Gr_\alpha -r_\beta}{|\rr|^5} \\ -&=& -M_{{\rm tot},A} \frac{G}{|\rr|} + \frac{G}{2} \frac{{\rm -tr}(\underline{I_A})}{|\rr|^3} - \frac{3G}{2}\frac{\rr^T \cdot -\underline{I_A} \cdot \rr}{|\rr|^5} +r_\beta}{|\rr_{i,A}|^5} \\ +&=& -G \left[ \frac{M_{{\rm tot},A}}{|\rr_{i,A}|} - \frac{1}{2} \frac{{\rm +tr}(\underline{I_A})}{|\rr_{i,A}|^3} + \frac{3}{2}\frac{\rr_{i,A}^T \cdot +\underline{I_A} \cdot \rr_{i,A}}{|\rr_{i,A}|^5}\right] \end{eqnarray} -The accelerations $\acc{i} = -\nabla_{\rr}\phi(\p{i})$ are then given by: +The accelerations $\acc{i} = -\nabla_{\rr_{i,A}}\phi(\p{i})$ are then given by: \begin{eqnarray} - a_{i,x} &=& \left[M_{{\rm tot},A} \frac{G}{|\rr|^3} - \frac{3G}{2} -\frac{{\rm tr}(\underline{I_A})}{|\rr|^5} - \frac{3G\underline{I_A}}{|\rr|^5} + -\frac{15G}{2}\frac{(\rr^T \cdot -\underline{I_A} \cdot \rr)}{|\rr|^7}\right] r_x \\ - a_{i,y} &=& \left[M_{{\rm tot},A} \frac{G}{|\rr|^3} - \frac{3G}{2} -\frac{{\rm tr}(\underline{I_A})}{|\rr|^5} - \frac{3G\underline{I_A}}{|\rr|^5} + -\frac{15G}{2}\frac{(\rr^T \cdot -\underline{I_A} \cdot \rr)}{|\rr|^7}\right] r_y\\ - a_{i,z} &=& \left[M_{{\rm tot},A} \frac{G}{|\rr|^3} - \frac{3G}{2} -\frac{{\rm tr}(\underline{I_A})}{|\rr|^5} - \frac{3G\underline{I_A}}{|\rr|^5} + -\frac{15G}{2}\frac{(\rr^T \cdot -\underline{I_A} \cdot \rr)}{|\rr|^7}\right] r_z +\acc{i} = G\left[\frac{M_{{\rm tot},A} \rr_{i,A}}{|\rr_{i,A}|} +-\frac{3}{2}\frac{{\rm tr}(\underline{I_A})\rr_{i,A}}{|\rr_{i,A}|^5} +-\frac{3\underline{I_A}\cdot\rr_{i,A}}{|\rr_{i,A}|^5} ++\frac{15}{2}\frac{(\rr_{i,A}^T \cdot \underline{I_A} \cdot +\rr_{i,A})\rr_{i,A}}{|\rr_{i,A}|^7}\right] \end{eqnarray}