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\Dauthor{ �Έ� �q } % �����̖��O
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\subsection{��ʉ����ꂽ�i�N�ۓ�}

��������͔񋅑̂Ȓ��S���̎�����^�����Ă���N�̖��ɂ�����A
�i�N���_�̂���ʉ����ꂽ�`�ɂ��čl����B
���̌`�̍��������邱�Ƃ�$j$�̐ۓ��|�e���V�������l�����Ƃ��A
�Ε��������S�V�̂̌��ʂ��l���ł���B
\begin{equation}
\begin{split}
 \langle \mathcal{R} _j^{(ob1)} \rangle 
 & = \\
 & \frac{1}{2} n_j a_j^2 n_j
   \left[ \frac{3}{2} J_2 \left( \frac{R_c}{a_j} \right) ^2
          -\frac{9}{8} J_2^2 \left( \frac{R_c}{a_j} \right) ^4
          -\frac{15}{4} J_4 \left( \frac{R_c}{a_j} \right) ^4
   \right] e_j^4 \\
 & -\frac{1}{2} n_j a_j^2 n_j
   \left[ \frac{3}{2} J_2 \left( \frac{R_c}{a_j} \right) ^2
          -\frac{27}{8} J_2^2 \left( \frac{R_c}{a_j} \right) ^4
          -\frac{15}{4} J_4 \left( \frac{R_c}{a_j} \right) ^4
   \right] I_j^2
\end{split}
\end{equation}
�����ł�$R_c$�͒��S�V�̂̔��a�ł���B
$J_2 , J_4$�͏����̓�̒n��̏d�͌W���ł���i��6.255 �Q�Ɓj�B

���̕\���͐ۓ���^���Ă���V�̂�ۓ����Ă���V�̂𕪗��ł���A
�܂肻�ꂼ��̓V�̂����m�ɐڐG�v�f�������Ă���A
�Ƃ����󋵂ɂ̂ݓ��Ă͂܂�Ƃ������Ƃ𒍈ӂ��Ă����B
�f���̃����O�̏ꍇ�ϑ����ꂽ�ʂ͐���Ƀ����O�̊􉽔������ł���B
�i���Ȃ킿�ł��K���ȑȉ~�̓����O�̗��q�����ǂ������̂ł���B�j
�����ł̓����O�̕��ω^���̒�`��ς���K�v������B
�Ȃ��Ȃ炱�̏ꍇ�������͎��i6.244�j�ŕ��ω^���Ƃ��Ďg��ꂽ$a$��
�����ł͂Ȃ�����ł���B

�O���^�����Ă��鎿�ʂ͑��̋O���^�����Ă���V�̂���ʏ�̉i�N�ۓ��������邾�낤�B
����$m_j$�ł̎���$m_k$�ɂ��i�N���ʂ��l���悤�B
�������̂悤�ɏ����A
\begin{equation}
\begin{split}
 \langle \mathcal{R} _D (\alpha _{jk}) \rangle
 = & \frac{1}{8} \alpha _{jk} b_{3/2}^{(1)} e_j^2
     -\frac{1}{4} \alpha _{jk} b_{3/2}^{(2)} e_j e_k
     \cos(\varpi _j -\varpi _k) \\
   & -\frac{1}{8} \alpha _{jk} b_{3/2}^{(1)} I_j^2
     +\frac{1}{4} \alpha_{jk} b_{3/2}^{(2)} I_j I_k
     \cos(\Omega _J -\Omega _k)
\end{split}
\end{equation}
$\alpha _{jk}$�͓�̓V�̂̔������̔�Ƃ���ƁA
�O���̓V�̂ɂ��ۓ���
\begin{equation}
 \langle \mathcal{R} _j^{(\mathrm{sec})} \rangle
 = \frac{\mathcal{G} m_k}{a_k}
   \langle \mathcal{R} _D (a_j / a_k) \rangle
   \qquad (a_j �� a_k)
\end{equation}
�ƂȂ�A���
\begin{equation}
 \langle \mathcal{R} _j^{(\mathrm{sec})} \rangle
 = \frac{\mathcal{G} m_k}{a_k} \frac{a_k}{a_j}
   \langle \mathcal{R} _D (a_k / a_j) \rangle
   \qquad (a_j �� a_k)
\end{equation}
�ƂȂ�B
�ۓ����O�����炩�������炩���L���邱�Ƃ̕s�ւ��́A
$\alpha _{jk}$��$\bar{\alpha} _{jk}$�����̂悤�ɒ�`���邱�Ƃŉ����ł���B
\begin{equation}
 \alpha _{jk} = \begin{cases}
                 a_k / a_j &
                 \text{$a_j �� a_k$�̏ꍇ�i��������̐ۓ��j} \\
                 a_j / a_k &
                 \text{$a_j �� a_k$�̏ꍇ�i�O������̐ۓ��j}
                \end{cases}
\end{equation}
������
\begin{equation}
 \bar{\alpha} _{jk} = \begin{cases}
                       1 &
                       \text{$a_j �� a_k$�̏ꍇ�i��������̐ۓ��j} \\
                       a_j / a_k &
                       \text{$a_j �� a_k$�̏ꍇ�i�O������̐ۓ��j}
                      \end{cases}
\end{equation}
���̕\�L���@�͍ŏ�Dermott �� Nicholson(1986)��
�V�����̉q���Ɋւ���ۓ����_�Ŏg��ꂽ�B
�ʏ��$\mathcal{G} \approx n_j^2 a_j^3 / m_c$�Ƃ����ߎ����g����
$m_j$�����̑S�Ă�$N-1$�̎��ʂ���󂯂�ۓ��֐��̉i�N�������������Ƃ��ł���B
\begin{equation}
\begin{split}
 \langle \mathcal{R} _j^{(\mathrm{sec})} \rangle
 = & n_j a_j^2 n_j \sum_{k=1,k \neq j}^N \frac{m_k}{m_c +m_j}
     \bigg[ \frac{1}{8} \alpha _{jk} \bar{\alpha} _{jk}
            b_{3/2}^{(1)} e_j^2 -\frac{1}{8} \alpha _{jk}
            \bar{\alpha} _{jk} b_{3/2}^{(1)} I_j^2 \\
   & -\frac{1}{4} \alpha_{jk} \bar{\alpha} _{jk} b_{3/2}^{(2)}
     e_j e_k \cos(\varpi _j -\varpi _k)
     +\frac{1}{4} \alpha _{jk} \bar{\alpha} _{jk} b_{3/2}^{(1)}
     I_j I_k \cos(\Omega _j - \Omega _k)
     \bigg]
\end{split}
\end{equation}
�𓾂�B

$\mathcal{R} ^{(ob1)}$��$\mathcal{R}^{(sec)}$�̍���g�ݍ��킹�A
����$m_j$�Ɋւ���ۓ��֐����ȉ��̂悤�ɏ�����F
\begin{equation}
\begin{split}
 \mathcal{R} _j = &n_j a_j^2 \bigg[
                   \frac{1}{2} A_{jj} e_j^2
                   +\frac{1}{2} B_{jj} e_j^2 \\
                  &+\sum_{k=1, k\neq j}^N A_{jk} e_j e_k
                    \cos(\varpi _j -\varpi _k)
                   +\sum_{k=1, k\neq j}^N B_{jk} I_j I_k
                    \cos(\Omega _j -\Omega _k) \bigg]
\end{split}
\end{equation}
������
\begin{gather}
 \begin{split}
  A_{jj} =+n_j \bigg[ \frac{3}{2}
          &J_2 \left( \frac{R_c}{a_j} \right) ^2
           -\frac{9}{8} J_2^2 \left( \frac{R_c}{a_j} \right) ^4
           -\frac{15}{4} J_4 \left( \frac{R_c}{a_j} \right) ^4 \\
          &+\frac{1}{4} \sum_{k=1, k\neq j}^N \frac{m_k}{m_c +m_j}
            \alpha_{jk} \bar{\alpha}_{jk}
            b_{3/2}^{(1)} (\alpha _{jk}) \bigg]
 \end{split} \\
  A_{jk} =-\frac{1}{4} \frac{m_k}{m_c +m_j} n_j \alpha _{jk}
          \bar{\alpha} _{jk} b_{3/2}^{(2)} (\alpha _{jk})
          \quad (j \neq k) \\
 \begin{split}
  B_{jj} =-n_j \bigg[ \frac{3}{2}
          &J_2 \left( \frac{R_c}{a_j} \right) ^2
           -\frac{27}{8} J_2^2 \left( \frac{R_c}{a_j} \right) ^4
           -\frac{15}{4} J_4 \left( \frac{R_c}{a_j} \right) ^4 \\
          &+\frac{1}{4} \sum_{k=1, k\neq j}^N \frac{m_k}{m_c +m_j}
            \alpha_{jk} \bar{\alpha}_{jk}
            b_{3/2}^{(1)} (\alpha _{jk}) \bigg]
 \end{split} \\
  B_{jk} =+\frac{1}{4} \frac{m_k}{m_c +m_j} n_j \alpha _{jk}
          \bar{\alpha} _{jk} b_{3/2}^{(1)} (\alpha _{jk})
          \quad (j \neq k) \\
\end{gather}
�ł���B
$A_{jj} , A_{jk} , B_{jj} , B_{jk}$�͓��$N�~N$�s��A
$\mathbf{A}$��$\mathbf{B}$�̒萔�v�f�ł���ƍl���邱�Ƃ��ł���B

���������Ŏ��i7.18�j�Ɓi7.19�j�ŗ^������V�����ϐ�
$h_j , k_j , p_j , q_j$���g���ƁA���̂悤�Ȏ��i7.20�j�̈�ʌ^�𓾂�F
\begin{equation}
 \begin{split}
  \langle \mathcal{R} \rangle 
  = &n_j a_j^2 \bigg[ \frac{1}{2} A_{jj} (h_j^2 +k_j^2) 
                     +\frac{1}{2} B_{jj} (p_j^2 +q_j^2) \\
    &+\sum_{k=1, k\neq j}^N A_{jk} (h_j h_k +k_j k_k)
     +\sum_{k=1, k\neq j}^N B_{jk} (p_j p_k +q_j q_k) \bigg]
 \end{split}
\end{equation}
�^���������͎��i7.25�j�Ɓi7.26�j�̂悤�ɗ^�����A����
\begin{gather}
 h_j =\sum_{i=1}^N e_{ji} \sin(g_i t +\beta _i) ,\quad
 k_j =\sum_{i=1}^N e_{ji} \cos(g_i t +\beta _i) ,\quad \\
 p_j =\sum_{i=1}^N I_{ji} \sin(f_i t +\gamma _i) ,\quad
 q_j =\sum_{i=1}^N I_{ji} \cos(f_i t +\gamma _i)
\end{gather}
������$g_i , f_i$�͓�̍s��
$\mathbf{A} , \mathbf{B}$��$N$�ŗL�l�ꎮ�ł���B
����܂łǂ���ŗL�x�N�g��$e_{ji} , I_{ji}$�̑傫���Ɠ����x��
$\beta _i ,\gamma _i$�ʂ͋��E�l���猈�܂�B
���鎞��$t$�ɂ����ė��S����2��Ǝ���$m_j$�̋O���X�Ίp��
\begin{gather}
 e_j^2 =\left[ \sum_{i=1}^N e_{ji} \sin(g_i t+\beta _i) \right] ^2
  +\left[ \sum_{i=1}^N e_{ji} \cos(g_i t+\beta _i) \right] ^2 \\
 I_j^2 =\left[ \sum_{i=1}^N I_{ji} \sin(f_i t+\gamma _i) \right] ^2
  +\left[ \sum_{i=1}^N I_{ji} \cos(f_i t+\gamma _i) \right] ^2 \\
\end{gather}
���l�ɁA���鎞�Ԃł̋ߓ_�o�x�Ə���_�o�x��
$h_j , k_j , p_j , q_j$����킩��B

�G�������̓����ɂ��ЂƂd�v�ȈႢ��
�O���X�Ό�_�̌ŗL�l�ɂ��͂�k�d�����݂��Ȃ��Ƃ����_�ł���B
���w�I�ɂ͂����$J_2 , J_4$�����s��$\mathbf{B}$��
���͂���`�ɂ͈ˑ����Ă��Ȃ��s������ÂɊ܂�ł���Ƃ������Ƃɂ��G
����������$\mathbf{B}$��$N$�Ɠ��������N�A�܂�s����̐��������Ă���B
�����I�ɂ͘f���̔񋅌`�Ȍ`�ɗR�����邤�܂���`���ꂽ�Q�ƕ��ʂ����邱�Ƃɂ����
�k�d�͏������ꂽ�Ƃ������Ƃł���B

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