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% 2004/06/04  ��R ����   SSD�p�ɏC��
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\Dauthor{ ��R ���� } % �����̖��O
\Ddate{ 2003/10/23 } % �쐬��
\Dfile{ sec0403.tex } % �t�@�C���̖��O

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\subsection{�|�e���V�������_} % �߂̃^�C�g��
% ��������{���������ĉ�����.
�����Ǝ��]�ɂ��ό`�̌v�Z��i�߂�O�ɁC�|�e���V�������_�ɗR�����邢�����̌��ʂ��q�ׂĂ������D���x$\gamma$�C���a$C$�̋ώ��ȋ���̕��̂̏d�̓|�e���V�������l�����Ƃ��C���a$r$�C����$\delta r$�C����$\delta m$�C�ł��锖�����k�̓����̃|�e���V�����ƊO���̃|�e���V�������l����΂悢(Ramsey\ 1940)�D���k�̊O���̂���_�ɂ�����|�e���V�����́C���S����̋�����$r^{\prime}$�Ƃ����
\begin{equation}  %%%18%%%
V_{\mathrm{ ext } }(r^{\prime})  =  -  \frac{ \mathcal{G}\delta m}{r^{\prime}}
\end{equation}
�ƂȂ�C����͋��k�̒��S�ɑS�Ă̎��ʂ��W�܂����ƍl�����Ƃ��Ɠ����ł���D����ċψ�ȋ��̕\�ʂ̊O���̃|�e���V�����͈ȉ��̂悤�ɗ^������D
\begin{equation}  %%%19%%%
V_{\mathrm{ext}}(C)  =  -  \frac{ \mathcal{G} \sum \delta m}{C}  =  - \frac{4}{3}\pi \gamma  \mathcal{G}C^2.
\end{equation}
�d�͂͋t���̖@��\footnote{2�̕��̂ɓ����͂�����2�̕��̂����Ԓ����̕����ɂ���C�傫����������2��ɔ���Ⴗ��Ƃ����@���D}(inverse square law) �ɂ���ċL�q�����̂ŁC���k�̓����̓_�ɓ����͂̓[���ł���D���������āC���k�̓����̏d�̓|�e���V�����͈��ł���C�����̂ǂ̓_�ɂ����ă|�e���V�������v�Z���Ă����߂邱�Ƃ��ł���D���k�̒��S�ɂ�����|�e���V�������v�Z�����
\begin{equation} %%%20%%%
V_{\mathrm{int}}(r)  =  -  \frac{ \mathcal{G} \sum \delta m}{r}  =  - 4 \pi \gamma  \mathcal{G} r \delta r .
\end{equation}
�O���a$C$�C�����a$r$�̋ψ�ȋ��k�̓����̃|�e���V�����͈ȉ��̂悤�ɗ^������D
\begin{equation}  %%%21%%%
V_{\mathrm{int}}(C , r)  = - 4 \pi \gamma  \mathcal{G}  \int_{r}^{C} r dr   =  - 2 \pi \gamma  \mathcal{G} \bigl( C^2 - r^2 \bigr)    .
\end{equation}
���������āC�ώ��ȋ���̕��̂ɂ����钆�S����̋�����$r$�̓_�ł̓���$(r < C)$�̃|�e���V�����C�O��$(r > C )$�̃|�e���V���� �͈ȉ��̂悤�ɂȂ�D
\begin{eqnarray}  %%%22%%% %%%23%%%
   V_{\mathrm{int}}(r)  &=&  - \frac{4}{3}\pi \gamma  \mathcal{G}r^2  -  2 \pi \gamma  \mathcal{G} \bigl( C^2 - r^2 \bigr) = - \frac{2}{3} \pi \gamma  \mathcal{G} \bigl( 3 C^2 - r^2 \bigr), \\
   V_{\mathrm{ext}}(r) &=& - \frac{4}{3}\pi \gamma  \mathcal{G} \frac{C^3}{r} .
\end{eqnarray}

�ό`�������̂̓����̃|�e���V�����C�O���̃|�e���V�����͋��֐��ɂ���Đ����ł���D�|�e���V�������_�ɂ����邱���̊֐���p�����ȉ��̋c�_�́CRamsey(1940),\\
MacRobert(1967),Bullen(1975),and Blakely(1995)�Ɋ�Â��Ă���D


��Ԃɂ�����d�̓|�e���V����$V$�͈ȉ��̃��v���X�������𖞂����D
\begin{equation}  %%%24%%%
  \nabla ^2 V = 0 .
\end{equation}
�ȉ���{\textbf{�I�C���[�̕�����}}(Euler's equation)�𖞂����Ȃ�΁C$V$��{\textbf{n��������}}(homogeneous of degree n)�ł���D
\begin{equation}  %%%25%%%
	x \frac{\partial V }{\partial x}  + y \frac{\partial V }{\partial y} + z \frac{\partial V }{\partial z} = n V  .
\end{equation}
���v���X�������𖞂��������̊֐����C{\textbf{�����a�֐�}}\footnote{���a�֐��́C�Ⴆ�ΎO�p�֐��̂悤�ɁC������������Ƃɖ߂�悤�Ȋ֐��D}(spherical solid harmonics)�Ƃ������D�����W�ŕ\���Ƃ��C�����͏d�v�Ȋ֌W�������C3�̊֐��ɂ킯�邱�Ƃ��ł��C���ꂼ��$r$,$\theta$,$\phi$��3�̕ϐ��̂���1�ɂ���ĕω�����D(���֐��ɂ��ẮCBlakely(1995)���ڂ����D)


�ɋ����W($r$,$\theta$,$\phi$)��p����ƁC$r$�͏d�S����̓��a�����̋����ł���C$\theta$�͋Ɏ�����Ƃ����]�ܓx�ł���C$\phi$�͔C�ӂ̌Œ肵����������Ƃ����o�x�ł���D���v���X�������͈ȉ��̂悤�ɂȂ�
\begin{equation}  %%%26%%%
\frac{\partial }{\partial  r }    \biggl( r^2  \frac{\partial V }{\partial r }  \biggr)   +    \frac{\partial }{\partial \mu } \biggl( ( 1  -  \mu^2   )     \frac{\partial V }{\partial \mu }  \biggr)  +  \frac{1}{(1  -  \mu^2 )}  \frac{\partial ^2 V }{\partial \phi ^2 }  =  0  ,
\end{equation}
������$ \mu = \cos \theta $�ł���D���v���X��������$ V = r^n S_n ( \mu , \phi ) $�ƒu���Ď�(4.26)�ɓ���邱�Ƃɂ���ĉ����C������$ S_n ( \mu , \phi ) $��$r$�ɂ��Ȃ��D���̂��Ƃ���
\begin{equation}  %%%27%%%
\frac{\partial }{\partial  r } \biggl( r^2  \frac{\partial V }{\partial r }  \biggr) =  n(n + 1) r^n  S_n   ,
\end{equation}
�ƂȂ�C��(4.26)��
\begin{equation}  %%%28%%%
\frac{\partial }{\partial \mu } \biggl( ( 1  -  \mu^2   )     \frac{\partial S_n }{\partial \mu }  \biggr)  +  \frac{1}{(1  -  \mu^2 )}  \frac{\partial ^2 S_n }{\partial \phi ^2 }  +  n(n + 1)  S_n   =  0  .
\end{equation}
�����{\textbf{���W�����h��������}}(Legendre's equation)�ƌ����C���̕������𖞂���$S_n$��{\textbf{���ʒ��a�֐�}}(spherical surface harmonic) �ƌĂ΂��֐��ł���D$n$��$-(n+1)$�Ə������Ƃ��C$n(n+1)$���ς�炸�c���Ă���̂ŁC���v���X�������̈�ʉ��͈ȉ��̂悤�ɏ�����D
\begin{equation}  %%%29%%%
  V  =  \sum_{n=0}^{\infty } \Bigl( A_n r^n  +  B_n r^{-(n + 1)}  \Bigr) S_n (\mu , \phi) .
\end{equation}
���̕������̊e���͂��ꂼ��$n$����$n+1$����{\textbf{�̋��֐�}}(solid harmonic)�ƌĂ΂��(Ramsey 1940)

���̏͂ŏq�ׂĂ���ό`�́C�����ɂ���Ă����]�ɂ���Ă����Ώ̂ł���C���W�����h����������
\begin{equation}  %%%30%%%
  ( 1  -  \mu^2   ) \frac{\partial ^2 \mathcal{P}_n(\mu ) }{\partial \mu ^2 } - 2 \mu \frac{\partial  \mathcal{P}_n(\mu) }{\partial \mu  }  + n( n + 1 ) \mathcal{P}_n(\mu)  = 0  ,
\end{equation}
�����Ń��W�����h��������$\mathcal{P}_n(\mu)$�́C
\begin{eqnarray}   %%%31%%%
	\mathcal{P}_n(\mu)  =  \frac{1\cdot 3 \cdot 5 \ldots (2n-1)}{n!}&& \!\!\!\!\!\!\!\Biggl[ \mu^n -\frac{n(n-1)}{2(2n-1)} \mu^{n-2} \nonumber  \\ 
	&& +  \frac{n(n-1)(n-2)(n-3)}{2 \cdot 4 \cdot (2n-1)(2n-3) } \mu^{n-4} + \cdots \Biggr]
\end{eqnarray}
�܂��́C���h���Q�X�̌���(Rodrigues's formula)
\begin{equation}  %%%32%%%
	\mathcal{P}_n(\mu)  =  \frac{1}{2^n n!}  \frac{d^n(\mu ^2 - 1 )^n}{d \mu ^n}
\end{equation}
�ɂ���ė^������D�]�ܓx$\theta$�͕ό`�̑Ώ̎����瑪���Ă���D
���W�����h����������{\textbf{�ы��֐�}}(zonal  harmonics)�ł���C�ŏ���5���͈ȉ��̂悤�ɗ^������D
\begin{eqnarray}   %%%33 to 37%%%
	\mathcal{P}_0(\mu)  &=&   1  ,				\\
	\mathcal{P}_1(\mu)  &=&   \mu  = \cos\theta  ,				\\
	\mathcal{P}_2(\mu)  &=&   \frac{1}{2}(3 \mu ^2 - 1) = \frac{1}{4}(3\cos2\theta +1),				\\
	\mathcal{P}_3(\mu)  &=&   \frac{1}{2}(5 \mu ^3 - 3\mu) = \frac{1}{8}(5\cos3\theta +3\cos\theta),								\\
	\mathcal{P}_4(\mu)  &=&	\frac{1}{8}(35 \mu ^4 - 30\mu^2 +3) = \frac{1}{64}(35\cos4\theta +20\cos2\theta +9 ).					
\end{eqnarray}


���ʒ��a�֐�(surface harmonics) �͒����֐��ł���C���̕\�ʂł����̎�̓�d�ϕ��́C�ȉ��̂悤�ȗL�p�Ȑ��������D�P�ʋ�($r=1$)�̖ʐϗv�f��$\sin\theta d \theta d \phi = -d \mu d \phi  $�D������$Y_m(\mu , \phi)$ , $S_n(\mu , \phi)$���C���ꂼ��$m$���C$n$���̋��ʒ��a�֐����Ƃ����$(m \neq n)$
\begin{equation}  %%%38%%%
\int_0^{2\pi} \int_{-1}^{+1} Y_m (\mu , \phi) S_n (\mu , \phi) d \mu d\phi = 0.
\end{equation}
������2�̋��ʒ��a�֐��ɂ����ė���$n$���ŁC������1���ы��֐�$\mathcal{P}_n(\mu)$�̂Ƃ��C
\begin{equation}  %%%39%%%
	\int_0^{2\pi} \int_{-1}^{+1} S_n (\mu , \phi) \mathcal{P}_n(\mu) d \mu d\phi = \frac{4\pi}{2n + 1} S_n(1) ,
\end{equation}
�ƂȂ�C$S_n(1)$��$\mathcal{P}_n(\mu)$�̋ɂł�$S_n(\mu , \phi)$�̒l�ł���D


�P�ʋ��ɂ�����2�̓_�ɂ��čl���Ă݂�D�ω�����_$(\theta ^{\prime} , \phi ^{\prime})$�Œ肳�ꂽ�_$(\theta , \phi )$��2�_�ł���D���̒��S�ł����̓_�ɋ��܂ꂽ�p�� $\psi$ �Ƃ��C�ϕ����l�����
\begin{equation}  %%%40%%%
	\int_0^{2\pi} \int_{-1}^{+1} S_n (\theta ^{\prime} , \phi ^{\prime}) \mathcal{P}_n(\cos\psi) d \mu ^{\prime} d\phi ^{\prime} .
\end{equation}
�ω�����_�̍��W���𓮂����C�ω�����]�ܓx�����߂�V���������Œ�_$(\theta , \phi )$��ʂ�悤�ɂƂ�C�ω�����_�̐V�����p���W�� $(\Theta ^{\prime} , \Phi ^{\prime})$�C������ $\Theta ^{\prime} = \psi $�Ƃ���D���l��$ S_n (\theta ^{\prime} , \phi ^{\prime})$ �� $Y_n(\Theta ^{\prime} , \Phi ^{\prime})$ �Ƃ���D�����Ŏ�(4.39)���
\begin{equation}  %%%41%%%
	\int_0^{2\pi} \int_{-1}^{+1} Y_n (\Theta ^{\prime} , \Phi ^{\prime}) \mathcal{P}_n(\cos\Theta ^{\prime}) d (\cos\Theta ^{\prime}) d\Phi ^{\prime} = \frac{4\pi}{2n + 1} Y_n(1).
\end{equation}
�������Ȃ���C
\begin{equation}  %%%42%%%
Y_n(1) = S_n( \theta , \phi ).
\end{equation}
���̂��Ƃ���
\begin{equation}  %%%43%%%
	\int_0^{2\pi} \int_{-1}^{+1} S_n (\mu ^{\prime} , \phi ^{\prime}) \mathcal{P}_n(\cos\psi) d \mu ^{\prime} d\phi ^{\prime}  =  \frac{4\pi}{2n + 1} S_n(\mu  , \phi)
\end{equation}
�����ŁC$S_n(\mu , \phi )$  �� $(\mu , \phi )$�ɂ��āC$S_n(\mu ^{\prime}�� \phi ^{\prime})$��$(\mu ^{\prime} , \phi ^{\prime})$���ē����`�������֐��ł���(MacRobert 1967)�D


�Œ肳�ꂽ�_$P$�ł́C�ώ��ŋ��ɋ߂��`����������̂ɂ��|�e���V�����ɂ��čl����D���̕��̂̕\�ʂ͈ȉ��̂悤�ɒ�`����
\begin{equation}  %%%44%%%
	R(\theta ^{\prime})  =   C \Bigl[ 1 + \epsilon _2 \mathcal{P}_2(\cos\theta ^{\prime}) \Bigr],
\end{equation}
������$\epsilon _2 (\ll 1)$�͒萔�ł���C$C$�͕��ς̔��a��\���D$P$�_�͓���$(r < C)$�C�O��$(r > C )$�ǂ���ɂł��Ƃ�C�����W��$( r , \mu  , \phi )$�ł���C$ \mu = \cos\theta $�C�]�ܓx$\theta$�͒����o���W�̑Ώ̎�����Ƃ������̂ł���(�}4.5)�D$P$�_�ł̏d�̓|�e���V������2�̕����ɕ����邱�Ƃ��ł���D1�͋���̕��̂ɂ����̂ŁC��(4.22)�Ǝ�(4.23)�ɂ���ė^������D�����ЂƂ͎�Ȃ��̂ł͂Ȃ����C�ό`�������̂Ɨ��z�I�ȋ��̂Ƃ̊Ԃ̂킸���ȕ����̈ړ��ɂ����̂ł���D����_$P^{\prime}( r ^{\prime} , \mu ^{\prime}  , \phi ^{\prime} )$�ɂ����āC�����̔����w�̌�����$ \epsilon _2C \mathcal{P}_2(\mu ^{\prime})$�ł���C�̐ϗv�f��$\epsilon _2 C^3 \mathcal{P}_2(\mu ^{\prime}) d \mu ^{\prime} d \phi ^{\prime}  $�ł���D$P$�_�ł̎��ʗv�f�ɂ��|�e���V�����͋���$PP^{\prime} = \Delta $�ɂ���Č��܂�C������
\begin{equation}  %%%45%%%
	\frac{1}{\Delta}  =  \Bigl(  C^2 + r^2 - 2 C r \cos \psi  \Bigr)^{-1/2}.
\end{equation}
�ł���D$r < C$�Ȃ��
\begin{equation}  %%%46%%%
	\frac{1}{\Delta}  = \frac{1}{C} \biggl[ 1 + \Bigl( \frac{r}{C}\Bigr)^2 -2\mu  \Bigl( \frac{r}{C} \Bigr) \biggr]^{-1/2},
\end{equation}
�e�C���[�W�J��$r/C$�̎����ŏ��ׂ��̏��ɍ�����ׂ��
\begin{eqnarray}  %%%47%%%
	\frac{1}{\Delta}  = \frac{1}{C} \Biggl[ 1 +  \Bigl( \frac{r}{C} \Bigr)\mu + \Bigl( \frac{r}{C}\Bigr)^2 \Bigl(- \frac{1}{2} +\frac{3}{2} \mu ^2 \Bigr)  + \Bigl( \frac{r}{C}\Bigr)^3 \Bigl(- \frac{3}{2}\mu +\frac{5}{2} \mu ^3 \Bigr)  +  \cdots
	    \Biggr],
\end{eqnarray}
%%%%%%%figure 4.5%%%%%%%
\begin{figure}[tbp] 
\begin{center}
	\includegraphics[width=11cm,height=6.123cm]{./figure/4_5.jpg}
\end{center}
\caption{�_$P$�ɂ����ẮC���ϔ��a$C$�̋��̂��琶�����C�ό`�����f���ɂ��|�e���V�����ƁC�����o���W�ɂ�镨���̈ړ��D}
	\label{figure.4.5}
\end{figure}

\noindent
��(4.34)���玮(4.37)���悭���Ă݂�ƁC��(4.47)�͈ȉ��̂悤�ɏ���
\begin{equation}  %%%48%%%
	\frac{1}{\Delta}  =  \frac{1}{C} \sum_{n=0}^{\infty} \Bigl( \frac{r}{C}\Bigr)^n \mathcal{P}_n(\cos\psi)  + \mathcal{O}(\epsilon_2).
\end{equation}
���������āC$P$�_�ɂ����鑍�|�e���V�����ւ̔񒆐S�̃|�e���V�����̑傫����
\begin{equation}  %%%49%%%
	V_{\mathrm{nc,int}} = - \gamma  \mathcal{G} C^2 \epsilon_2 \sum_{n=0}^{\infty} \Bigl( \frac{r}{C}\Bigr)^2 \int \int \mathcal{P}_2(\mu ^{\prime}) \mathcal{P}_n(\cos\psi)d \mu ^{\prime} d\phi ^{\prime} .
\end{equation}
��(4.43)���C
\begin{equation}  %%%50%%%
	\sum_{n=0}^{\infty} \Bigl( \frac{r}{C}\Bigr)^n \int \int \mathcal{P}_2(\mu ^{\prime}) \mathcal{P}_n(\cos\psi)d \mu ^{\prime} d\phi ^{\prime}  = \frac{4\pi}{5} \Bigl( \frac{r}{C}\Bigr)^2 \mathcal{P}_2(\cos\theta)
\end{equation}
���������āC�����ɂ�����$P$�_�ɂ����鑍�|�e���V�����ւ̔񒆐S�̃|�e���V�����̑傫����
\begin{equation}  %%%51%%%
	V_{\mathrm{nc,int}} = - \frac{4\pi}{5} \gamma  \mathcal{G} r^2 \epsilon_2 \mathcal{P}_2 (\cos \theta),
\end{equation}
�����āC$P$�_�ɂ���������̑��|�e���V�����͂��̍��Ǝ�(4.22)�ɂ���ē����錋�ʂ̘a�ł���C
\begin{equation}  %%%52%%%
	V_{\mathrm{int}}(r , \theta) =  - \frac{4}{3} \pi C^3 \gamma  \mathcal{G} 	\Biggl[ \frac{3C^2 -r^2}{2C^3} +  \frac{3}{5} \frac{r^2}{C^3} \epsilon _2 \mathcal{P}_2 (\cos\theta)\Biggr].
\end{equation}
���l�ɁC$(r > C)$�Ȃ��
\begin{equation}  %%%53%%%
	\frac{1}{\Delta}  =  \frac{1}{r} \sum_{n=0}^{\infty} \Bigl( \frac{C}{r}\Bigr)^n \mathcal{P}_n(\cos\psi)  + \mathcal{O}(\epsilon_2)
\end{equation}
�ł���C�O���̑��|�e���V������
\begin{equation}  %%%54%%%
	V_{\mathrm{ext}}(r , \theta) =  - \frac{4}{3} \pi C^3 \gamma  \mathcal{G} 	\Biggl[ \frac{1}{r} +  \frac{3}{5} \frac{C^2}{r^3} \epsilon _2 \mathcal{P}_2 (\cos\theta)\Biggr].
\end{equation}

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