正四面体惑星の重力ポテンシャルの多重極展開

　\(\rm{Legendre}\)多項式の母関数展開 \[\dfrac{1}{|\vec{r}-\vec{r}'|}=\dfrac{1}{r}\dfrac{1}{\sqrt{1-2(r'/r)cos\theta '+(r'/r)^{2}}} =\dfrac{1}{r}\sum_{l=0}^{\infty}\Bigl(\dfrac{r'}{r}\Bigr)^{l}P_{l}(cos \theta '),\ (r' \lt r) \] 　を用いて密度\(\rho\)が一様な物体の重力ポテンシャルは以下のように\(\rm{Legendre}\)多項式を用いて展開できる。 \begin{align*} &\iiint_{V}\dfrac{-G\rho}{|\vec{r}-\vec{r}'|}dx'dy'dz'=\dfrac{-G\rho}{r}\iiint_{V}\sum_{l=0}^{\infty}\Bigl(\dfrac{r'}{r}\Bigr)^{l}P_{l}(cos \theta ')dx'dy'dz'\\ =& \sum_{l=0}^{\infty}\dfrac{1}{r^{2l+1}}\iiint_{V}(rr')^{l}P_{l}(cos \theta ')dx'dy'dz' = \sum_{l=0}^{\infty}\dfrac{1}{r^{2l+1}}\iiint_{V}(rr')^{l}P_{l}\Bigl(\dfrac{\vec{r}\cdot\vec{r}'}{rr'}\Bigr)dx'dy'dz' \end{align*} 　上記の積分を行うため、以下の座標変換を行う。 \[u=\frac{1}{\sqrt{2}}(x+y),\ v=\frac{1}{\sqrt{2}}(y-x),\ u'=\frac{1}{\sqrt{2}}(x'+y'),\ v'=\frac{1}{\sqrt{2}}(y'-x')\] 　正四面体の\(4\)つの頂点\((a,a,a),\ \)\((a,-a,-a),\ \)\((-a,a,-a),\ \)\((-a,-a,a)\)は上記の座標変換により、\((\sqrt{2}a,0,a),\ \)\((0,-\sqrt{2}a,-a),\ \)\((0,\sqrt{2}a,-a),\ \)\((-\sqrt{2}a,0,a)\)に移るので、関数\(f(x,y,z)\)の正四面体内での積分は、 \[ \iiint_{V}f(x',y',z')dx'dy'dz' =\int_{-a}^{a}\Big[\int_{-(1/\sqrt{2})(a-z)}^{(1/\sqrt{2})(a-z)}\Big[\int_{-(1/\sqrt{2})(a+z)}^{(1/\sqrt{2})(a+z)}f(u',v',z')du'\Big]dv'\Big]dz' \] 　と表される。各\(l\)について積分を行うと、 \[-\dfrac{G\rho}{r}\iiint_{V}P_{0}(cos \theta ')dx'dy'dz'=-\dfrac{8G\rho a^{3}}{3r}\] \[-\dfrac{G\rho}{r^{3}}\iiint_{V}(rr')P_{1}(cos \theta ')dx'dy'dz'=0\] \[-\dfrac{G\rho}{r^{5}}\iiint_{V}(rr')^{2}P_{2}(cos \theta ')dx'dy'dz'=0\] \[-\dfrac{G\rho}{r^{7}}\iiint_{V}(rr')^{3}P_{3}(cos \theta ')dx'dy'dz'=-\dfrac{8G\rho a^{6}}{3r^{7}}xyz\] \[-\dfrac{G\rho}{r^{9}}\iiint_{V}(rr')^{4}P_{4}(cos \theta ')dx'dy'dz'=\dfrac{G\rho a^{7}}{r^{9}}\cdot\dfrac{4}{15}[(x^{4}+y^{4}+z^{4})-3(x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2})]\] \[-\dfrac{G\rho}{r^{11}}\iiint_{V}(rr')^{5}P_{5}(cos \theta ')dx'dy'dz'=0\] \[-\dfrac{G\rho}{r^{13}}\iiint_{V}(rr')^{6}P_{6}(cos \theta ')dx'dy'dz'=-\dfrac{G\rho a^{9}}{r^{13}}\cdot\dfrac{16}{63}[(x^{6}+y^{6}+z^{6})-\frac{15}{2}(x^{4}y^{2}+x^{2}y^{4}+y^{4}z^{2}+y^{2}z^{4}+z^{4}x^{2}+z^{2}x^{4})+90x^{2}y^{2}z^{2}]\] \[-\dfrac{G\rho}{r^{15}}\iiint_{V}(rr')^{7}P_{7}(cos \theta ')dx'dy'dz'=\dfrac{G\rho a^{10}}{r^{15}}\cdot\dfrac{52}{5}xyz[(x^{4}+y^{4}+z^{4})-\frac{5}{3}(x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2})]\] \[-\dfrac{G\rho}{r^{17}}\iiint_{V}(rr')^{8}P_{8}(cos \theta ')dx'dy'dz'=-\dfrac{G\rho a^{11}}{r^{17}}\cdot\dfrac{1}{5}[(x^{8}+y^{8}+z^{8})-14(x^{6}y^{2}+x^{2}y^{6}+y^{6}z^{2}+y^{2}z^{6}+z^{6}x^{2}+z^{2}x^{6})+35(x^{4}y^{4}+y^{4}z^{4}+z^{4}x^{4})]\] \[-\dfrac{G\rho}{r^{19}}\iiint_{V}(rr')^{9}P_{9}(cos \theta ')dx'dy'dz'=-\dfrac{G\rho a^{12}}{r^{19}}\cdot 16xyz[(x^{6}+y^{6}+z^{6})-\frac{7}{2}(x^{4}y^{2}+x^{2}y^{4}+y^{4}z^{2}+y^{2}z^{4}+z^{4}x^{2}+z^{2}x^{4})+\frac{70}{3}x^{2}y^{2}z^{2}]\] \begin{align*} -\dfrac{G\rho}{r^{21}}\iiint_{V}(rr')^{10}P_{10}(cos \theta ')dx'dy'dz'&=\dfrac{G\rho a^{13}}{r^{21}}\cdot\dfrac{8}{11}[(x^{10}+y^{10}+z^{10})-\frac{45}{2}(x^{8}y^{2}+x^{2}y^{8}+y^{8}z^{2}+y^{2}z^{8}+z^{8}x^{2}+z^{2}x^{8}) \\ &+21(x^{6}y^{4}+x^{4}y^{6}+y^{6}z^{4}+y^{4}z^{6}+z^{6}x^{4}+z^{4}x^{6})+504(x^{6}y^{2}z^{2}+y^{6}z^{2}x^{2}+z^{6}x^{2}y^{2})\\ &-630(x^{4}y^{4}z^{2}+y^{4}z^{4}x^{2}+z^{4}x^{4}y^{2})] \end{align*} \[-\dfrac{G\rho}{r^{23}}\iiint_{V}(rr')^{11}P_{11}(cos \theta ')dx'dy'dz'=-\dfrac{G\rho a^{14}}{r^{23}}\cdot\dfrac{255}{7}xyz[(x^{8}+y^{8}+z^{8})-6(x^{6}y^{2}+x^{2}y^{6}+y^{6}z^{2}+y^{2}z^{6}+z^{6}x^{2}+z^{2}x^{6})+\dfrac{63}{5}(x^{4}y^{4}+y^{4}z^{4}+z^{4}x^{4})]\] \begin{align*} -\dfrac{G\rho}{r^{25}}\iiint_{V}(rr')^{12}P_{12}(cos \theta ')dx'dy'dz'&=-\dfrac{G\rho a^{15}}{r^{25}}\cdot\dfrac{829}{2730}[(x^{12}+y^{12}+z^{12})-33(x^{10}y^{2}+x^{2}y^{10}+y^{10}z^{2}+y^{2}z^{10}+z^{10}x^{2}+z^{2}x^{10}) \\ &-\frac{112690}{829}(x^{8}y^{4}+x^{4}y^{8}+y^{8}z^{4}+y^{4}z^{8}+z^{8}x^{4}+z^{4}x^{8}) \\ &+\frac{507031}{829}(x^{6}y^{6}+y^{6}z^{6}+z^{6}x^{6})+\frac{1907205}{829}(x^{8}y^{2}z^{2}+y^{8}z^{2}x^{2}+z^{8}x^{2}y^{2}) \\ &-\frac{4450145}{829}(x^{6}y^{4}z^{2}+x^{6}y^{2}z^{4}+y^{6}z^{4}x^{2}+y^{6}z^{2}x^{4}+z^{6}x^{4}y^{2}+z^{6}x^{2}y^{4})+\frac{22250725}{829}x^{4}y^{4}z^{4}] \end{align*} となる。一方、正四面体の対称性をもつ球面調和関数のうち正四面体群の\(A_{1}\)表現の基底となるものについては、 \begin{align*} Te_{3}(\theta,\varphi)=&\dfrac{1}{\sqrt{2}i}(Y_{3}^{2}(\theta,\varphi)-Y_{3}^{-2}(\theta,\varphi)) =\dfrac{1}{4}\sqrt{\dfrac{105}{2\pi}}\sin^{2}\theta\cos\theta\cdot\dfrac{e^{2i\varphi}-e^{-2i\varphi}}{\sqrt{2}i}\\ =&\dfrac{1}{8i}\sqrt{\dfrac{105}{2\pi}}r^{-3}z[(x+iy)^{2}-(x-iy)^{2}] =\dfrac{1}{2}\sqrt{\dfrac{105}{\pi}}r^{-3}xyz \end{align*} \begin{align*} Te_{4}(\theta,\varphi) =&\dfrac{\sqrt{2\cdot 3\cdot 5}}{12}(Y_{4}^{4}(\theta,\varphi)+Y_{4}^{-4}(\theta,\varphi)) +\dfrac{\sqrt{3\cdot 7}}{6}Y_{4}^{0}(\theta,\varphi)\\ =&\dfrac{5}{64}\sqrt{\dfrac{3\cdot 7}{\pi}}\sin^{2}\theta\cdot[e^{4i\varphi}+e^{-4i\varphi}] +\dfrac{1}{32}\sqrt{\dfrac{3\cdot 7}{\pi}}(35\cos^{4}\theta-30\cos^{2}\theta+3)\\ =&\dfrac{5}{64}\sqrt{\dfrac{21}{\pi}}r^{-4}[(x+iy)^{4}+(x-iy)^{4}] +\dfrac{1}{32}\sqrt{\dfrac{21}{\pi}}r^{-4}(35z^{4}-30z^{2}r^{2}+3r~{4})\\ =&\dfrac{1}{4}\sqrt{\dfrac{21}{\pi}}r^{-4}((x^{4}+y^{4}+z^{4})-3(x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2})) \end{align*} \begin{align*} Te_{6,o}(\theta,\varphi) =&\dfrac{\sqrt{7}}{4}(Y_{6}^{4}(\theta,\varphi)+Y_{6}^{-4}(\theta,\varphi)) -\dfrac{\sqrt{2}}{4}Y_{6}^{0}(\theta,\varphi)\\ =&\dfrac{21}{128}\sqrt{\dfrac{13}{2\pi}}\sin^{4}\theta(11\cos^{2}\theta-1)\cdot[e^{4i\varphi}+e^{-4i\varphi}] -\dfrac{1}{128}\sqrt{\dfrac{2\cdot 13}{\pi}}(231\cos^{6}\theta-315\cos^{4}\theta+105\cos^{2}\theta-5)\\ =&\dfrac{21}{128}\sqrt{\dfrac{13}{2\pi}}r^{-6}(11z^{2}-r^{2})[(x+iy)^{4}+(x-iy)^{4}] -\dfrac{1}{128}\sqrt{\dfrac{2\cdot 13}{\pi}}r^{-6}(231z^{6}-315z^{4}r^{2}+105z^{2}r^{4}-5r^{6})\\ =&\dfrac{1}{8}\sqrt{\dfrac{26}{\pi}}r^{-6}((x^{6}+y^{6}+z^{6})-\frac{15}{2}(x^{4}y^{2}+x^{2}y^{4}+y^{4}z^{2}+y^{2}z^{4}+z^{4}x^{2}+z^{2}x^{4})+90x^{2}y^{2}z^{2}) \end{align*} \begin{align*} Te_{7}(\theta,\varphi) =&\dfrac{\sqrt{3\cdot 11}}{12i}(Y_{7}^{6}(\theta,\varphi)-Y_{7}^{-6}(\theta,\varphi)) +\dfrac{\sqrt{3\cdot 13}}{12i}(Y_{7}^{2}(\theta,\varphi)-Y_{7}^{-2}(\theta,\varphi))\\ =&\dfrac{11}{256i}\sqrt{\dfrac{3\cdot 5\cdot 7\cdot 13}{\pi}}\sin^{6}\theta\cos\theta\cdot[e^{6i\varphi}-e^{-6i\varphi}] +\dfrac{1}{256i}\sqrt{\dfrac{3\cdot 5\cdot 7\cdot 13}{\pi}}\sin^{2}\theta(143\cos^{5}\theta-110\cos^{3}\theta+15\cos\theta)\cdot[e^{2i\varphi}-e^{-2i\varphi}]\\ =&\dfrac{11}{256i}\sqrt{\dfrac{1365}{\pi}}r^{-7}z[(x+iy)^{6}-(x-iy)^{6}] +\dfrac{1}{256i}\sqrt{\dfrac{1365}{\pi}}r^{-7}(143z^{5}-110z^{3}r^{2}+15zr^{4})\cdot[(x+iy)^{2}-(x-iy)^{2}]\\ =&\dfrac{3}{4}\sqrt{\dfrac{1365}{\pi}}r^{-7}xyz[(x^{4}+y^{4}+z^{4})-\frac{5}{3}(x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2})] \end{align*} \begin{align*} Te_{8}(\theta,\varphi) =&\dfrac{\sqrt{2\cdot 3\cdot 5\cdot 13}}{48}(Y_{8}^{8}(\theta,\varphi)+Y_{8}^{-8}(\theta,\varphi)) +\dfrac{\sqrt{2\cdot 3\cdot 7}}{24}(Y_{8}^{4}(\theta,\varphi)+Y_{8}^{-4}(\theta,\varphi)) +\dfrac{\sqrt{3\cdot 11}}{8}Y_{8}^{0}(\theta,\varphi)\\ =&\dfrac{65}{4096}\sqrt{\dfrac{3\cdot 11\cdot 17}{\pi}}\sin^{8}\theta\cdot[e^{8i\varphi}+e^{-8i\varphi}] +\dfrac{7}{1024}\sqrt{\dfrac{3\cdot 11\cdot 17}{\pi}}\sin^{4}\theta(65\cos^{4}\theta-26\cos^{2}\theta+1)\cdot[e^{4i\varphi}+e^{-4i\varphi}] \\ &+\dfrac{1}{2048}\sqrt{\dfrac{3\cdot 11\cdot 17}{\pi}}(6435\cos^{8}\theta-12012\cos^{6}\theta+6930\cos^{4}\theta-1260\cos^{2}\theta+35)\\ =&\dfrac{65}{4096}\sqrt{\dfrac{561}{\pi}}r^{-8}[(x+iy)^{8}+(x-iy)^{8}] +\dfrac{7}{1024}\sqrt{\dfrac{561}{\pi}}r^{-8}(65z^{4}-26z^{2}+1)[(x+iy)^{4}+(x-iy)^{4}] \\ &+\dfrac{1}{2048}\sqrt{\dfrac{561}{\pi}}r^{-8}(6435z^{8}-12012z^{6}r^{2}+6930z^{4}r^{4}-1260z^{2}r^{6}+35r^{8})\\ =&\dfrac{1}{16}\sqrt{\dfrac{561}{\pi}}r^{-8}((x^{8}+y^{8}+z^{8})-14(x^{6}y^{2}+x^{2}y^{6}+y^{6}z^{2}+y^{2}z^{6}+z^{6}x^{2}+z^{2}x^{6})+35(x^{4}y^{4}+y^{4}z^{4}+z^{4}x^{4})) \end{align*} \begin{align*} Te_{9,t}(\theta,\varphi) =&-\dfrac{\sqrt{2\cdot 13}}{8i}(Y_{9}^{6}(\theta,\varphi)-Y_{9}^{-6}(\theta,\varphi)) +\dfrac{\sqrt{2\cdot 3}}{8i}(Y_{9}^{2}(\theta,\varphi)-Y_{9}^{-2}(\theta,\varphi))\\ =&-\dfrac{13}{1024i}\sqrt{2\cdot \dfrac{3\cdot 5\cdot 11\cdot 19}{\pi}}\sin^{6}\theta(17\cos^{3}\theta-3\cos\theta)\cdot[e^{6i\varphi}-e^{-6i\varphi}] \\ &+\dfrac{3}{1024i}\sqrt{\dfrac{2\cdot 3\cdot 5\cdot 11\cdot 19}{\pi}}\sin^{2}\theta(221\cos^{7}\theta-273\cos^{5}\theta+91\cos^{3}\theta-7\cos\theta)\cdot[e^{2i\varphi}-e^{-2i\varphi}]\\ =&-\dfrac{13}{1024i}\sqrt{\dfrac{6270}{\pi}}r^{-9}(17z^{3}-3zr^{2})[(x+iy)^{6}-(x-iy)^{6}] \\ &+\dfrac{3}{1024i}\sqrt{\dfrac{6270}{\pi}}r^{-9}(221z^{7}-273z^{5}r^{2}+91z^{3}r^{4}-7zr^{6})[(x+iy)^{2}-(x-iy)^{2}]\\ =&-\dfrac{3}{8}\sqrt{\dfrac{6270}{\pi}}r^{-9}xyz((x^{6}+y^{6}+z^{6})-\frac{7}{2}(x^{4}y^{2}+x^{2}y^{4}+y^{4}z^{2}+y^{2}z^{4}+z^{4}x^{2}+z^{2}x^{4})+\frac{70}{3}x^{2}y^{2}z^{2}) \end{align*} \begin{align*} Te_{10,o}(\theta,\varphi) =&\dfrac{\sqrt{3\cdot 11\cdot 17}}{48}(Y_{10}^{8}(\theta,\varphi)+Y_{10}^{-8}(\theta,\varphi)) +\dfrac{\sqrt{11}}{8}(Y_{10}^{4}(\theta,\varphi)+Y_{10}^{-4}(\theta,\varphi)) -\dfrac{\sqrt{2\cdot 3\cdot 5\cdot 13}}{48}Y_{10}^{0}(\theta,\varphi)\\ =&\dfrac{561}{1024}\sqrt{\dfrac{2\cdot 5\cdot 7\cdot 13}{\pi}}\sin^{8}\theta(19\cos^{2}\theta)-1)\cdot[e^{8i\varphi}+e^{-8i\varphi}] +\dfrac{33}{256}\sqrt{\dfrac{2\cdot 5\cdot 7\cdot 13}{\pi}}(969\cos^{6}\theta-765\cos^{4}\theta+135\cos^{2}\theta-3)\cdot[e^{4i\varphi}+e^{-4i\varphi}] \\ &-\dfrac{3}{512}\sqrt{\dfrac{2\cdot 5\cdot 7\cdot 13}{\pi}}(46189\cos^{10}\theta-109395\cos^{8}\theta+90090\cos^{6}\theta-30030\cos^{4}\theta+3465\cos^{2}\theta-63)\\ =&\dfrac{561}{1024}\sqrt{\dfrac{910}{\pi}}r^{-10}(19z^{2}-r^{2})[(x+iy)^{8}+(x-iy)^{8}] +\dfrac{33}{256}\sqrt{\dfrac{910}{\pi}}r^{-10}(969z^{6}-765z^{4}r^{2}+135z^{2}r^{4}-3r^{6})[(x+iy)^{4}+(x-iy)^{4}] \\ &-\dfrac{3}{512}\sqrt{\dfrac{910}{\pi}}r^{-10}(46189z^{10}-109395z^{8}r^{2}+90090z^{6}r^{4}-30030z^{4}r^{6}+3465z^{2}r^{8}-63r^{10})\\ =&-\dfrac{3}{2}\sqrt{\dfrac{910}{\pi}}r^{-10}[(x^{10}+y^{10}+z^{10})-\frac{45}{2}(x^{8}y^{2}+x^{2}y^{8}+y^{8}z^{2}+y^{2}z^{8}+z^{8}x^{2}+z^{2}x^{8}) \\ &+21(x^{6}y^{4}+x^{4}y^{6}+y^{6}z^{4}+y^{4}z^{6}+z^{6}x^{4}+z^{4}x^{6})+504(x^{6}y^{2}z^{2}+y^{6}z^{2}x^{2}+z^{6}x^{2}y^{2})\\ &-630(x^{4}y^{4}z^{2}+y^{4}z^{4}x^{2}+z^{4}x^{4}y^{2})] \end{align*} \begin{align*} Te_{11}(\theta,\varphi) =&\dfrac{\sqrt{3\cdot 7\cdot 19}}{48}(Y_{11}^{10}(\theta,\varphi)-Y_{11}^{-10}(\theta,\varphi)) +\dfrac{3\sqrt{3}}{16}(Y_{11}^{6}(\theta,\varphi)-Y_{11}^{-6}(\theta,\varphi)) +\dfrac{\sqrt{2\cdot 3\cdot 5 \cdot 17}}{48}(Y_{11}^{2}(\theta,\varphi)-Y_{11}^{-2}(\theta,\varphi)) \\ =&\dfrac{133}{16384i}\sqrt{\dfrac{11\cdot 13\cdot 17\cdot 23}{\pi}}\sin^{10}\theta\cos\theta\cdot[e^{10i\varphi}-e^{-10i\varphi}] +\dfrac{9}{16384i}\sqrt{\dfrac{11\cdot 13\cdot 17\cdot 23}{\pi}}\sin^{6}\theta(399\cos^{5}\theta-190\cos^{3}\theta+15\cos\theta)\cdot[e^{6i\varphi}-e^{-6i\varphi}] \\ &+\dfrac{5}{8192i}\sqrt{\dfrac{11\cdot 13\cdot 17\cdot 23}{\pi}}\sin^{2}\theta(2261\cos^{9}\theta-3876\cos^{7}\theta+2142\cos^{5}\theta-420\cos^{3}\theta+21\cos\theta)\cdot[e^{2i\varphi}-e^{-2i\varphi}]\\ =&\dfrac{133}{16384i}\sqrt{\dfrac{55913}{\pi}}r^{-11}z[(x+iy)^{10}-(x-iy)^{10}] +\dfrac{9}{16384i}\sqrt{\dfrac{55913}{\pi}}r^{-11}(399z^{5}-190z^{3}r^{2}+15zr^{4})[(x+iy)^{6}-(x-iy)^{6}] \\ &+\dfrac{5}{8192i}\sqrt{\dfrac{55913}{\pi}}r^{-11}(2261z^{9}-3876z^{7}r^{2}+2142z^{5}r^{4}-420z^{3}r^{6}+21zr^{8})[(x+iy)^{2}-(x-iy)^{2}] \\ =&\dfrac{5}{16}\sqrt{\dfrac{55913}{\pi}}r^{-11}xyz((x^{8}+y^{8}+z^{8})-6(x^{6}y^{2}+x^{2}y^{6}+y^{6}z^{2}+y^{2}z^{6}+z^{6}x^{2}+z^{2}x^{6})+\dfrac{63}{5}(x^{4}y^{4}+y^{4}z^{4}+z^{4}x^{4})) \end{align*} \begin{align*} Te_{12,o8}(\theta,\varphi) =&\dfrac{1}{20\sqrt{82}}(\sqrt{3\cdot 13\cdot 17\cdot 19}(Y_{12}^{8}(\theta,\varphi)+Y_{12}^{-8}(\theta,\varphi)) -4\sqrt{2\cdot 7\cdot 13}(Y_{12}^{4}(\theta,\varphi)+Y_{12}^{-4}(\theta,\varphi)) +9\sqrt{2\cdot 11}Y_{12}^{0}(\theta,\varphi))\\ =&\dfrac{1}{20\sqrt{82}}(\dfrac{62985}{4096}\sqrt{\dfrac{2\cdot 11}{\pi}}\sin^{8}\theta(161\cos^{4}\theta-42\cos^{2}\theta+1)\cdot[e^{8i\varphi}+e^{-8i\varphi}] \\ &-\dfrac{1365}{1024}\sqrt{\dfrac{2\cdot 11}{\pi}}\sin^{4}\theta(7429\cos^{8}\theta-9044\cos^{6}\theta+3230\cos^{4}\theta-340\cos^{2}\theta+5)\cdot[e^{4i\varphi}+e^{-4i\varphi}] \\ &+\dfrac{45}{2048}\sqrt{\dfrac{2\cdot 11}{\pi}}(676039\cos^{12}\theta-1939938\cos^{10}\theta+2078505\cos^{8}\theta-1021020\cos^{6}\theta+225225\cos^{4}\theta-18018\cos^{2}\theta+231))\\ =&\sqrt{\dfrac{11}{41\pi}}(\dfrac{12597}{16384}r^{-12}(161z^{4}-42z^{2}r^{2}+r^{4})[(x+iy)^{8}+(x-iy)^{8}] \\ &-\dfrac{273}{4096}r^{-12}(7429z^{8}-9044z^{6}r^{2}+3230z^{4}r^4-340z^{2}r^{6}+5r^{8})[(x+iy)^{4}+(x-iy)^{4}] \\ &+\dfrac{9}{8192}r^{-12}(676039z^{12}-1939938z^{10}r^{2}+2078505z^{8}r^{4}-1021020z^{6}r^{6}+225225z^{4}r^{8}-18018z^{2}r^{10}+231r^{12}))\\ =&\dfrac{9}{8}\sqrt{\dfrac{11}{41\pi}}r^{-12}((x^{12}+y^{12}+z^{12})-33(x^{10}y^{2}+x^{2}y^{10}+y^{10}z^{2}+y^{2}z^{10}+z^{10}x^{2}+z^{2}x^{10}) \\ &+\frac{815}{24}(x^{8}y^{4}+x^{4}y^{8}+y^{8}z^{4}+y^{4}z^{8}+z^{8}x^{4}+z^{4}x^{8}) +\frac{1631}{12}(x^{6}y^{6}+y^{6}z^{6}+z^{6}x^{6})+\frac{5125}{4}(x^{8}y^{2}z^{2}+y^{8}z^{2}x^{2}+z^{8}x^{2}y^{2}) \\ &-\frac{35875}{12}(x^{6}y^{4}z^{2}+x^{6}y^{2}z^{4}+y^{6}z^{4}x^{2}+y^{6}z^{2}x^{4}+z^{6}x^{4}y^{2}+z^{6}x^{2}y^{4})+\frac{179375}{12}x^{4}y^{4}z^{4}) \end{align*} \begin{align*} Te_{12,o12}(\theta,\varphi) =&\dfrac{1}{160\sqrt{246}}( 1025(Y_{12}^{12}(\theta,\varphi)+Y_{12}^{-12}(\theta,\varphi)) +\sqrt{2\cdot 3\cdot 7\cdot 11\cdot 23}(Y_{12}^{8}(\theta,\varphi)+Y_{12}^{-8}(\theta,\varphi))\\ &+3\sqrt{11\cdot 17\cdot 19\cdot 23}(Y_{12}^{4}(\theta,\varphi)+Y_{12}^{-4}(\theta,\varphi)) +2\sqrt{7\cdot 13\cdot 17\cdot 19\cdot 23}Y_{12}^{0}(\theta,\varphi))\\ =&\dfrac{1}{160\sqrt{246}}( \dfrac{5125}{4096}\sqrt{\dfrac{7\cdot 13\cdot 17\cdot 19\cdot 23}{\pi}}\sin^{12}\theta\cdot[e^{12i\varphi}+e^{-12i\varphi}] +\dfrac{165}{2048}\sqrt{\dfrac{7\cdot 13\cdot 17\cdot 19\cdot 23}{\pi}}\sin^{8}\theta(161\cos^{4}\theta-42\cos^{2}\theta+1)\cdot[e^{8i\varphi}+e^{-8i\varphi}] \\ &+\dfrac{495}{4096}\sqrt{\dfrac{7\cdot 13\cdot 17\cdot 19\cdot 23}{\pi}}\sin^{4}\theta(7429\cos^{8}\theta-9044\cos^{6}\theta+3230\cos^{4}\theta-340\cos^{2}\theta+5)\cdot[e^{4i\varphi}+e^{-4i\varphi}] \\ &+\dfrac{5}{1024}\sqrt{\dfrac{7\cdot 13\cdot 17\cdot 19\cdot 23}{\pi}}(676039\cos^{12}\theta-1939938\cos^{10}\theta+2078505\cos^{8}\theta-1021020\cos^{6}\theta+225225\cos^{4}\theta-18018\cos^{2}\theta+231))\\ =&\sqrt{\dfrac{676039}{246\pi}}( \dfrac{1025}{131072}r^{-12}[(x+iy)^{12}+(x-iy)^{12}] +\dfrac{33}{65536}r^{-12}(161z^{4}-42z^{2}r^{2}+r^{4})[(x+iy)^{8}+(x-iy)^{8}] \\ &+\dfrac{99}{131072}r^{-12}(7429z^{8}-9044z^{6}r^{2}+3230z^{4}r^4-340z^{2}r^{6}+5r^{8})[(x+iy)^{4}+(x-iy)^{4}] \\ &+\dfrac{1}{32768}r^{-12}(676039z^{12}-1939938z^{10}r^{2}+2078505z^{8}r^{4}-1021020z^{6}r^{6}+225225z^{4}r^{8}-18018z^{2}r^{10}+231r^{12}))\\ =&\dfrac{1}{32}\sqrt{\dfrac{676039}{246\pi}}r^{-12} ((x^{12}+y^{12}+z^{12})-33(x^{10}y^{2}+x^{2}y^{10}+y^{10}z^{2}+y^{2}z^{10}+z^{10}x^{2}+z^{2}x^{10}) \\ &+\frac{495}{2}(x^{8}y^{4}+x^{4}y^{8}+y^{8}z^{4}+y^{4}z^{8}+z^{8}x^{4}+z^{4}x^{8}) -462(x^{6}y^{6}+y^{6}z^{6}+z^{6}x^{6})) \end{align*} 　なお、以上の球面調和関数は全て規格化済みである。
　正四面体惑星の総質量は\(M=\frac{8}{3}\rho a^{3}\)なので、重力ポテンシャルは、 \begin{align*} U(r,\theta,\varphi)=&-\dfrac{GM}{r}[1 +\Bigl(\dfrac{a}{r}\Bigr)^{3}\cdot 2\sqrt{\dfrac{\pi}{105}}Te_{3}(\theta,\varphi) -\Bigl(\dfrac{a}{r}\Bigr)^{4}\cdot\dfrac{2}{5}\sqrt{\dfrac{\pi}{21}}Te_{4}(\theta,\varphi) +\Bigl(\dfrac{a}{r}\Bigr)^{6}\cdot\dfrac{16}{21}\sqrt{\dfrac{\pi}{26}}Te_{6,o}(\theta,\varphi)\\ &-\Bigl(\dfrac{a}{r}\Bigr)^{7}\cdot\dfrac{26}{5}\sqrt{\dfrac{\pi}{1365}}Te_{7}(\theta,\varphi) +\Bigl(\dfrac{a}{r}\Bigr)^{8}\cdot\dfrac{6}{5}\sqrt{\dfrac{\pi}{561}}Te_{8}(\theta,\varphi) -\Bigl(\dfrac{a}{r}\Bigr)^{9}\cdot 16\sqrt{\dfrac{\pi}{6270}}Te_{9,t}(\theta,\varphi)\\ &+\Bigl(\dfrac{a}{r}\Bigr)^{10}\cdot\dfrac{2}{11}\sqrt{\dfrac{\pi}{910}}Te_{10,o}(\theta,\varphi) +\Bigl(\dfrac{a}{r}\Bigr)^{11}\cdot\dfrac{306}{7}\sqrt{\dfrac{\pi}{55913}}Te_{11}(\theta,\varphi)\\ &+\Bigl(\dfrac{a}{r}\Bigr)^{12}\cdot (\dfrac{254294}{1399125}\sqrt{\dfrac{41\pi}{11}}Te_{12,o8}(\theta,\varphi) + -\dfrac{14858}{5125}\sqrt{\dfrac{246\pi}{676039}}Te_{12,o12}(\theta,\varphi)) +\cdots] \end{align*} と表される。最低次の非球対称の項は\(3\)次となる。各球面調和関数の前に現れる係数が力学的形状係数に相当するものであるが、それぞれ、 \[2\sqrt{\dfrac{\pi}{105}}=0.34594756\cdots, \ -\dfrac{2}{5}\sqrt{\dfrac{\pi}{21}}=-0.15471245\cdots, \ \dfrac{16}{21}\sqrt{\dfrac{\pi}{26}}=0.264843275\cdots, \ -\dfrac{26}{5}\sqrt{\dfrac{\pi}{1365}}=-0.24046633\cdots,\] \[\dfrac{6}{5}\sqrt{\dfrac{\pi}{561}}=0.08979967\cdots, \ -16\sqrt{\dfrac{\pi}{6270}}=0.35814686\cdots, \ \dfrac{2}{11}\sqrt{\dfrac{\pi}{910}}=0.01068295\cdots, \ \dfrac{306}{7}\sqrt{\dfrac{\pi}{55913}}=0.32767395\cdots, \] \[\dfrac{254294}{1399125}\sqrt{\dfrac{41\pi}{11}}=0.62194233\cdots, \ -\dfrac{14858}{5125}\sqrt{\dfrac{246\pi}{676039}}=-0.09802189\cdots, \] となる。