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% 2004/06/04  ��R ����   SSD�p�ɏC��
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%%%%%%%%   Subtitle, Date, Filename Setting     %%%%%%%%
\Dauthor{ ��R ���� } % �����̖��O
\Ddate{ 2003/10/16 } % �쐬��
\Dfile{ sec0402.tex } % �t�@�C���̖��O

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\setcounter{section}{4}    % (�͔ԍ�) �������
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\subsection{�����o���W} % �߂̃^�C�g��
% ��������{���������ĉ�����.
����$m_p$�̘f���̂܂����܂�鎿��$m_s$�̉q���ɂ���ċN���钪�����l���Ă݂�D���̂����_�ƍl����ƁC���L���̖͂@���ɂ��C2�_�Ԃɓ����͂̑傫��$\langle$$F$$\rangle$��
\begin{equation}  %%%1%%%
\langle F \rangle =  \mathcal{G}\frac{m_pm_s}{r^2},
\end{equation}
������$r$�͒��S����̋����ł���D�f���Ɖq�������̋��ʂ̏d�S�̂܂��ɉ�]���Ă���Ƃ���Ɓi�}4.1�j�C2.7�߂Ō���ꂽ�悤�ɤ�������Ǝ��ʂ̊֌W��
\begin{equation}   %%%2%%%
 a_s/a_p=m_p/m_s,
\end{equation}
�ƂȂ�C�����ŏ��$a=a_p+a_s$�ł���D�f���̒��S�_$P_1$�̋O���͏d�S$C_1$�𒆐S�ɔ��a$a_p$�̉~�O���ł���D�����Ŏ��]�𖳎�����ƁC�܂葼�̓_�̓����𖳎�����ƁC�f���̒��̓_$P_2$�̕`���~�O���͓������a�������S�_�͈قȂ�C$C_1$�ƒu���������$C_2$�ƂȂ�D������$P_1$��$P_2$�ɂ��ꂽ�̂Ɠ����Ԃ񂾂��C$C_1$��$C_2$�ɂ����i�}4.2�j�D�܂�C�f�����`���S�Ă̓_�́C�傫���ƕ��������������S�͂ɂ���ĉ^�����Ă���̂ł���C�q������̓���������{\textbf{F}}�ɂ���ĉ^�����Ă���̂ł͂Ȃ��̂ł���\footnote{�܂�C�e�_�ɂ�����${\textbf{F}}$�͂��ꂼ��Ⴄ�Ƃ������ƁD}�D���ʂ̉��S�͖͂��L����{\textbf{$\langle$F$\rangle$}}�ɓ����� 
\begin{equation}   %%%3%%%
{\textbf{$\langle$F$\rangle$}}= ���S��(centrifugal\ force) \neq{\textbf{F}}.
\end{equation}
�ƂȂ�C�f����ό`������N����$\textbf{F}_{\mathrm{tidal}}$�͎��̂悤�ɒ�`�����D
\begin{equation}   %%%4%%%
{\textbf{F}}_{\mathrm{tidal}}  =   {\textbf{F}}-{\textbf{$\langle$F$\rangle$}}.
\end{equation}
���]�ɂ��͂����̂�ό`�����邪�i4.7�߁j�C�����ɂ��ό`�����]�ɂ��ό`����������΁C�ʁX�Ɍv�Z�ł��āC���`�I�ɑ������킹�邱�Ƃ��ł���D

%%%%%%%figure 4.1%%%%%%%
\begin{figure}[tbp]
\begin{center}
	\includegraphics[width=10cm,height=5.47cm]{./figure/4.1.bmp}
\end{center}	
	\caption{�f���Ɖq���͂��̋��ʏd�S�𒆐S�ɉ~�O�����Ƃ�D�d�S����Ƃ����������͂��ꂼ��$a_p$,$a_s$�ł���C�q���O���̘f������Ƃ�����������$a = a_p+a_s$�ł���D}
	\label{figure.4.1}
\end{figure}

%%%%%%%figure 4.2%%%%%%%
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=10cm,height=4.36cm]{./figure/4_2.jpg}
\end{center}
\caption{�f���̑S�Ă̓_�͓��������a$a_p$�̉~��`�����C���S�͈قȂ�D$P_1$,$P_2$�͂��ꂼ��$C_1$,$C_2$�𒆐S�Ƃ����~��`���D}
	\label{figure4.2}
\end{figure}

�f���ɂ����钪���o���W�̌`�����߂邽�߂ɤ�q�������f���̕\�ʂ̓_$P$�ɂ�����|�e���V����$V$�ɂ��čl���Ă݂悤�D(�q���͎��_�Ƃ��Ĉ���)
\begin{equation}     %%%5%%%
V=    - \mathcal{G}\frac{m_s}{\Delta},
\end{equation}
�����ŁC$\Delta$��$P$�Ɖq���̒��S�Ƃ̋����ł���i�}4.3�j�D�]���藝���
\begin{equation}    %%%6%%%
\Delta  =  a \Biggl[1-2 \biggl( \frac{R_p}{a}\biggr) \cos\psi + \biggl( \frac{R_p}{a} \biggr)^2  \Biggr]^{\frac{1}{2}},
\end{equation}
�ƂȂ�C������$\psi$�͂ӂ��̕��̂̒��S�����񂾐�����Ƃ����p�x�ł���D�قƂ�ǂ̏ꍇ $R_p/a$\ $\ll$$\ 1$ �ł���D�Ⴆ�΁C�n���̐ԓ����a��6,378 km�ł���C���O���̔�������384,400 km�ł���D���������āC��(4.6)��W�J��
	\footnote{�e�C���[�W�J����D$x = 1-2 \biggl( \frac{R_p}{a}\biggr) \cos\psi + 	\biggl( \frac{R_p}{a} \biggr)^2$ �Ƃ����āC
	\begin{eqnarray}   
 	V = - \mathcal{G}\frac{m_s}{\Delta} = - \mathcal{G}\frac{m_s}{a(1+x)^{\frac{1}{2}}} =  - \mathcal{G}\frac{m_s	}{a}(1+x)^{(-\frac{1}{2})} \nonumber
 	\end{eqnarray} 
	\begin{eqnarray}    
 	(1+x)^{(-\frac{1}{2})} = 1 - \frac{1}{2}x + \frac{3}{8}x^2 \cdots \nonumber
	\end{eqnarray} 
	$x$�����Ƃɖ߂��C�����i3���ȍ~�j�̍��𖳎�����Ǝ�(4.7)��������D
	}�C
\begin{eqnarray}   %%%7%%%
�@V\!\!\!&=&\!\!\!- \mathcal{G}\frac{m_s}{a}\Biggl[1+\biggl( \frac{R_p}{a}\biggr) \cos\psi + \biggl( \frac{R_p}{a} \biggr)^2\frac{1}{2}\bigl(3\cos^2\psi-1\bigr)+ \cdots \Biggr] \\
   &\approx&  V_1+V_2+V_3 \nonumber,
\end{eqnarray}
�ƂȂ�D�����ō����̍��͖��������D

%%%%%%figure4.3%%%%%%
\begin{figure}[tbp]
\begin{center}
	\includegraphics[width=10cm,height=3.55cm]{./figure/4_3.jpg}
\end{center}
	\caption{�f���̔��a$R_p$�C�q���̔�����$a$�C�f���̕\�ʏ�̓_$P$����q���܂ł̋���$\Delta$�Ƃ̊֌W�������D�_���͓��|�e���V������$V_2=- \mathcal{G}(m_s/a^2)R_p\cos\psi=constant�������D$}
	\label{figure4.3}
\end{figure}

��(4.7)�̑�1���C$V_1=- \mathcal{G}(m_s/a)$�C�͒萔�ł���C${\textbf{F}}/m_p=- \nabla V$�C�ł��邱�Ƃ���C���̍��͘f���ɉ��̗͂��y�ڂ��Ȃ��D��(4.7)�̑�2���C$V_2=- \mathcal{G}(m_s/a^2)R_p\cos\psi$�C��$P$�_�ɂ��闱�q�ɉ~�O����`���̂ɕK�v�ȗ͂�^����D
�|�e���V�����ɂ���Đ�����͂͂ǂ̓_�ɂ����Ă��|�e���V�������z�̕����ɐ����C�����Ă��̓_��ʂ铙�|�e���V�����ʂɐ����ł���D�����ł͓��|�e���V�����ʂ͐����ʂł���C��������2���̂̒��S�����Ԑ��ɐ����ł���C�|�e���V�������z�͈ȉ��̂悤�ɗ^������D
\begin{equation}   %%%8%%%
- \frac{\partial V_2}{\partial (R_p\cos\psi)} =  \mathcal{G}\frac{m_s}{a^2} = \frac{\langle F \rangle}{m_p}.
 \end{equation}
��(4.7)�̑�3���ɂ��f���\�ʂ̃|�e���V�����͈ȉ��̂悤�ɏ���
\begin{equation}   %%%9%%%
V_3(\psi) = - \mathcal{G}\frac{m_s}{a^3}R_p^2\mathcal{P}_2(\cos\psi),
\end{equation}
������
\begin{equation}    %%%10%%%
 \mathcal{P}_2(\cos\psi) = \frac{1}{2}\bigl(3\cos^2\psi-1\bigr)
\end{equation}
�Ƃ́C���W�����h����������$n=2$�C$\cos$$\psi$�̂Ƃ��ł���i4.3�ߎQ�Ɓj�D������
\begin{equation}    %%%11%%%
\frac{{\textbf{F}}_{\mathrm{tidal}}}{m_p}  =   \frac{{\textbf{F}}}{m_p}-\frac{{\textbf{$\langle$F$\rangle$}}}{m_p} = - \nabla V - \frac{{\textbf{$\langle$F$\rangle$}}}{m_p} \approx - \nabla V_3(\psi),
\end{equation}
�ł���̂ŁC����͒����������N�����|�e���V�����ł���D
�����ŁC$V_3(\psi)$��
\begin{equation}    %%%12%%%
�@V_3(\psi) = \zeta g\mathcal{P}_2(\cos\psi),
\end{equation}
�Ə������Ƃ��ł���D������
\begin{equation}    %%%13%%%
 \zeta = \frac{m_s}{m_p}\Biggl( \frac{R_p}{a} \Biggr)^3R_p 
\end{equation}
�ł���D�܂��C
\begin{equation}   %%%14%%%
g =  \mathcal{G}\frac{m_p}{R_p^2}
\end{equation}
����͘f���ɂ�����C�d�͉����x�C�\�ʏd�͂ł���D�����ł́C$\zeta \mathcal{P}_2(\cos\psi$)�͘f���\�ʂɂ����Ă�����$\psi$�΂��āC{\bfseries ���t����}(equilibrium tide)�̑傫����\���D������$\mathcal{P}_2(\cos\psi)$��$ \psi =0,\pi$�̂Ƃ��ő�ł���C$\psi =\pi/2,3\pi/2$�̂Ƃ��ɍŏ��ɂȂ邱�Ƃɒ��ڂ��ꂽ���D�n����24���Ԃ�1�񎩓]����̂ŁC���̂��Ƃ��猎�ɂ���Ēn���ɂقږ��������Ɗ�����2��N���邱�Ƃ��킩��D

%%%%%%figure4.4%%%%%%
\begin{figure}[tbp]
\begin{center}
	\includegraphics[width=10cm,height=3.67cm]{./figure/4_4.jpg}
\end{center}
\caption{�d�̓|�e���V������$V_3(\psi)$�ɂ���Đ����钪���ό`�̌`�������������̐}�ƁC����Ɣ�r���ē_���Ŏ������͕̂ό`�̂Ȃ����̏ꍇ�̐}�ł���D(a)�����ό`�̑Ώ̎��͒n���ƌ��̒��S��ʂ�C�n����$z$�������Ɋp���x$\Omega$�Ŏ��]���C���̕��ω^����$n$�ł���C�n���̐ԓ��ɑ΂���O���X�Ίp��$I$�ł���D(b)���̗]�ܓx�ƌo�x��$(\theta_M,\phi_M)$�C�n���\�ʏ�̓_$P$�ł�$P(\theta_P,\phi_P)$�ł���D$\phi_P$,$\phi_M$�͋�Ԃɂ����ČŒ肳�ꂽ��������Ƃ��Ă���C�n���Ƌ��ɂ܂���Ă����������Ƃ��Ă���킯�ł͂Ȃ��Ƃ������Ƃɒ��ӂ��ꂽ���D}
	\label{figure4.4}
\end{figure}

�n���̒����ό`�͎��ۂ͈ȉ��̗��R�ł����ƕ��G�ł���D(�@)���z�ƌ��C�����ɂ���Ē������N����C(�A)�n���𒆐S�ɍl����ƁC���z�������ΐS�I�ŁC�n���̐ԓ��ɑ΂��ČX�����O�����Ƃ��Ă܂���Ă���D�����O���̗��S���𖳎����čl����ƁC���z�ƌ��͂��ꂼ��O�̑傫�Ȓ�����n���ɂ����炷�D�}4.4a�ɂ����āC�����ό`�̑Ώ̎��͒n���ƌ��̒��S��ʂ�C�n����$z$�������Ɋp���x$\Omega$�Ŏ��]���Ă���C���͕��ω^��$n$�������C�n���̐ԓ��ɑ΂���O���X�Ίp��$I$�ł���D�}4.4b�ɂ����āC���̗]�ܓx�ƌo�x��$(\theta_M,\phi_M)$�C�n���\�ʏ�̓_$P$�ł�$P(\theta_P,\phi_P)$�ł���D$\phi_P$,$\phi_M$�͋�Ԃɂ����ČŒ肳�ꂽ��������Ƃ��Ă���C�n���Ƌ��ɂ܂���Ă����������Ƃ��Ă���킯�ł͂Ȃ��Ƃ������Ƃɒ��ӂ��ꂽ���D


�]���藝����C�x�N�g��$OP$��$OM$�̊p�x$\psi$�� \footnote{�ʒu�x�N�g���𐬕��\�����C���ς̊֌W���瓱��.  $\mid OM \mid \mid OP \mid \cos\psi = x_Px_M + y_Py_M + z_Pz_M $}
\begin{equation}   %%%15%%%
\cos\psi = \cos\theta_P\cos\theta_M + \sin\theta_P\sin\theta_M\cos(\phi_P - \phi_M).
\end{equation}
�ƂȂ�C����������
\begin{eqnarray}    %%%16%%%
\frac{1}{2}(3\cos^2\psi-1) &=& \frac{1}{2}(3\cos^2\theta_P-1)\frac{1}{2}(3\cos^2\theta_M-1) \nonumber \\
& & +\frac{3}{4}\sin^2\theta_P\sin^2\theta_M\cos2(\phi_P - \phi_M) \nonumber \\ 
& & +\frac{3}{4}\sin2\theta_P\sin2\theta_M\cos(\phi_P - \phi_M).
\end{eqnarray}


�n����ɌŒ肳�ꂽ�_�ɂ�����ܓx$\theta_P$���^�����C$P$�_�ł̒����̑傫����$\phi_P$�C$\theta_M$�C$\phi_M$�ɂ���ĈقȂ�D��(4.16)�ɂ������1���̕ω��́C$\cos^2\theta_M = (1/2)(1+\cos2\theta_M)$�̎��Ԃɂ��ω��ɂ���Č��܂�D���������āC���̍���$2n$�̐U���ɂ���ĕω����C{\bfseries �������� }(fortnightly tide)���N�����D��2����$2(\Omega-n)\approx 2\Omega$�̐U���ɂ���ĕω����C{\bfseries �������� }(semidiurnal tide)���N�����D��3���́C$(\Omega-n)\approx \Omega$�̐U���ɂ���ĕω����C{\bfseries ������ }(diurnal tide)���N�����D��3����$\sin2\theta_M$���܂ނ��߁C�������͊u�T�ŋ����}�g������D���̑��̒����Ɋւ��鍀�͌��̋O�����S���Ɋ֌W���Ă���D

���z�ɂ�锼�N�����C���������C�������ɑΉ����āC�U��$2n_{\mathrm{Sun}}$�C$2(\Omega-n_{\mathrm{Sun}})\approx 2 \Omega$�C$(\Omega-n_{\mathrm{Sun}})\approx \Omega$������D�����ŁC$n_{\mathrm{Sun}}$�͑��z�́i�n���́j���ω^���ł���D��(4.13)���C���z�ƌ��̒����̑傫���̔䗦�͂ǂ�ȏꍇ�ł�
\begin{equation}   %%%17%%%
\frac{m_{\mathrm{Sun}}}{m_{\mathrm{Moon}}}\Biggl(\frac{a_{\mathrm{Moon}}}{a_{\mathrm{Sun}}}\Biggr)^3 \approx 0.46.
\end{equation}
�ƂȂ�D���ɂ���Ēn���ɋN���钪����$\zeta$ = 0.36 m�C������z�ɂ���ċN���钪����$\zeta$ = 0.16 m�ł���

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