%
% TEX �{���p�t�H�[�}�b�g
%
% �E ���̃t�@�C���̓��C���t�@�C������ǂݍ��܂�邱�ƂŎg�p�����.
%
% �E ���̃t�@�C���P�̂ł̓R���p�C���ł��Ȃ��̂Œ���.
%
%
% 2002/10/09  �X�� ����   �쐬
% 2003/04/15  ���c ��h   ��b�����[�~�p�ɏC��
% 2003/06/05  ���c ��h   �J�E���^�[�� setting �ύX
% 2004/06/04  ��R ����   SSD�p�ɏC��
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%   Subtitle, Date, Filename Setting     %%%%%%%%
\Dauthor{ ��R ���� } % �����̖��O
\Ddate{ 2004/07/09 } % �쐬��
\Dfile{ sec0412.tex } % �t�@�C���̖��O

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%   Set Counter (chapter, section etc. ) %%%%%%%%

\setcounter{section}{4}    % (�͔ԍ�) �������
\setcounter{subsection}{11} % (�ߔԍ� -1) �������
\setcounter{footnote}{44}  % 
\setcounter{equation}{198}   % (���ԍ� -1) �������
\setcounter{page}{48}
\setcounter{figure}{15}     % (�}�ԍ� -1) �������

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%              Text Start                %%%%%%%%
%%%%%%%figure 4.16%%%%%%%
\subsection{Tides on Titan} % �߂̃^�C�g��
% ��������{���������ĉ�����.
4.11�߂ł́C$Q_s$�̐���l��p����$\dot{E}$���v�Z�����D
�������Ȃ���C
�����U��̃��J�j�Y�����킩���Ă���΁C
��ꌴ������$\dot{E}$�𒼐ڌ��ς��邱�Ƃ��ł��邱�Ƃ�����D
����͗Ⴆ�΁C
�C�̒����ɂ��U��̏ꍇ�ł���C
Titan�ɓK�p�ł���D
�������p���āC���̓y���̉q���̑�C���Ƀ��^�����������ꂽ(Kuiper 1944)�D
Vorager 1 �̊ϑ��ɂ����
�\�ʂ̑��C�� 1.6 bar ��
�\�ʉ��x�� �` 95 K �ł���C���^���̗Z�_(90.6 K)�ƕ��_(118 K)�̊Ԃł���
���Ƃ����炩�ɂȂ��Ă���(Hanel et al 1981)�D
����̂ɁC
���^���ƍ����̒Y�����f���t�̂̏�Ԃŉq���\�ʂɑ��݂��Ă��邩�����ꂸ�C
���̂��Ƃɂ����
�����̊C�̒��̒����̓����ɂ��Ă������̘_�_��������ł���(Lunine 1993 �Q��)�D
Titan�̋O�����S�����傫���Ƃ��������͖��ł���C
�Ȃ��Ȃ炱��ɂ��$Q_s > 200$�ƂȂ�(�\4.2�Q��)�C
���ɗB��킩���Ă���قڑS���𕢂��C�����V�̂ł���n����$Q_p \approx 12$�ł���(�\4.1�Q��)�D
�������Ȃ���C
�U��̊����͑S���𕢂��C�̐[���Ɉˑ����Ă���(Sagan �� Dermott 1982, Dermott �� Sagan 1995)�D

%%%%%%%figure 4.17%%%%%%%
\begin{figure}[bp]
\begin{center}
	\includegraphics[width=7.5cm,height=5.718cm]{./figure/4_17.jpg}
\end{center}	
	\caption{
	radial tide�ɂ�钪���̗���.
	�q�������_����ߓ_�ɓ����C�����ω��̐U������������̂ɔ����C
	����radial tide�݂̂ɂ�钪���̗���̕����������Ă���D
	�����̗���͉q���̒��S�Ƙf���O����guiding centre�����Ԑ��ɂ��đΏ̂ł���D	$P$�͔��p$\beta$�̉~���̕\�ʂ������D	
	���S���Ɛ����Ȏ���ʂ镽�ʂ����؂闬���
	�q���̐ԓ��ʂ����؂闬���
	��Ƀ[���ł���D
	}
	\label{figure.4.17}
\end{figure}

�����U��̊����́C
�C��̑S�Ă̓_�ł̒����̕��ϑ��x���v�Z���邱�Ƃɂ���Č��߂邱�Ƃ��ł���D
�����̍����Ɏ����I�ȕω������邱�Ƃɂ��C�������N����D
radial tide�݂̂��l����Ƃ���ƁC
�q���\�ʂ̂���_$P$�ł̒����̎��ԕω�(�}4.17�Q��)�͈ȉ��̂悤�ɗ^������
\begin{equation}   %%%199%%%
	\zeta = 3eh \mathcal{P}(\cos \beta) \cos nt
	\label{199}
\end{equation}
�ł���C������$h$�͊C�̒����̕��ϐU���ňȉ��̂悤�ɗ^�����
\begin{equation}   %%%200%%%
	h = \frac{\bigl( m_s/m_p \bigr)(C_s/a)^3 C_s}{[1-(3\sigma/5\rho)]+[1-\sigma / \rho] / \tilde{\mu _s}} \approx 120 \mathrm{m}
	\label{200}
\end{equation}
�ł���C�܂������ŊC�͎�ɖ��x$\sigma= 0.65$ g cm$^{-3}$��
�G�^�����琬���Ă���Ɖ��肷��(��(4.86)�Q��\footnote{
��(4.86)%%%86%%%
$$
	A (T_2 -  S_2 ) = \frac{  \zeta \tilde{\mu} }{1- \sigma / \rho + \tilde{\mu} ( 1 - 3\sigma /5 \rho )   }
$$
})�D
���S���ɂ���Ĉ����N�������q���̕��ό`��ɂ��Ă̌`�̐U��
�ɂ̂ݒ��ڂ��Ă���̂ŁC
���$h$�ɂ��Ă̎���
�ő̉q���ɂ����ĐÐ������t�����藧���Ă���Ƃ��ɂ��L���ł���D


�q�������_����ߓ_�ɓ����Ƃ��̒����̍����̕ω��ɂ��čl���悤(�}4.17)�D
$P$�_�ł̎���$dt$�Ԃɂ����钪���̍����̑�����
\begin{equation}   %%%201%%%
	d\zeta = -3eh \mathcal{P}(\cos \beta) n \sin nt dt
	\label{201}
\end{equation}
�ƂȂ�D�ʐς̑������ȉ��̂悤�ɗ^����
\begin{equation}   %%%202%%%
	dA = 2 \pi C_s^2\sin \beta d \beta
	\label{202}
\end{equation}
���p$\beta$�̉~���̒���
�\�ʂ̂����̕����𕢂��Ă���C�̑̐ς̑����́C�ȉ��̂悤�ɗ^������D
\begin{eqnarray}   %%%203%%%
	\Delta V &=& -3ehn \sin nt dt 2\pi C_s^2 \int _0^{\beta} \mathcal{P}(\cos \beta) \sin \beta d\beta \nonumber \\
	&=& 3eh\pi C_s^2 n \sin nt \cos \beta \sin ^2 \beta dt
	\label{203}
\end{eqnarray}
�����̗��ꂪ�[��$D$�̊C�ŋN�������Ƃ���ƁC���̂Ƃ�
\begin{eqnarray}   %%%204%%%
	\Delta V = 2 \pi C_s \sin \beta D v dt
	\label{204}
\end{eqnarray}
�ƂȂ�C������$v$��$P$�_�ł̒����̗���̑��x�ł���D
���������āC��(\ref{203})�Ǝ�(\ref{204})����ȉ��𓾂�D
\begin{eqnarray}   %%%205%%%
	v = \frac{1}{4}nC_s \biggl( \frac{3eh}{D} \biggr) \sin 2 \beta \sin nt
	\label{205}
\end{eqnarray}
���E�w�̗����ɂ��P�ʖʐς�����̃G�l���M�[�U��̊�����
\begin{eqnarray}   %%%206%%%
	\dot{E} = f\sigma v^3
	\label{206}
\end{eqnarray}
�ł���C�����Ŗ������̒萔$f \approx 0.003$(Sears 1995)��
�\�ʂ̖��C�W��(skin friction)�ł���(Goldreich �� Soter 1966)�D
$\dot{E}$�̕��ϒl�𓾂邽�߂ɁC
$v$�̕\����(\ref{205})�͎��Ԃ�(�ʐςɂ���ďd�݂�����)��Ԃ�
���ς����ׂ��ł���D
�ȉ��̂悤�ɗ^��
\footnote
{
�{�� $	\langle  \sin ^3 nt  \rangle = 4 \pi / 3 
$ �͌�A�D
}
\begin{eqnarray}   %%%207%%%
	\langle  \sin ^3 nt  \rangle = 4 / 3 \pi
	\label{207}
\end{eqnarray}
�܂�\footnote
{
�{�� �E��$ = 32\pi / 35
$ �͌�A�D
}
\begin{eqnarray}   %%%208%%%
	\langle  \sin ^3 2 \beta  \rangle =  \frac{2}{\pi} \int _0^{\pi/2} \sin ^3 2\beta \sin \beta d \beta = 32 / 35\pi
	\label{208}
\end{eqnarray}
�ł���C�v�Z����
\begin{eqnarray}   %%%209%%%
	\langle  v ^3  \rangle = \frac{2}{105 \pi} \biggl[ nC_s \Bigl( \frac{3eh}{D} \Bigr) \biggr]^3
	\label{209}
\end{eqnarray}
�ł���C�܂��Cradial tide�݂̂ɂ��G�l���M�[�U��̊����͈ȉ��̂悤�ɗ^������D
\begin{eqnarray}   %%%210%%%
	\dot{E}_{\mathrm{radial}} = 4\pi C_s^2 f \sigma \langle  v ^3  \rangle 
	\label{210}
\end{eqnarray}
���v�̃G�l���M�[�U��̊����𓾂�G
����͈ȉ��̂悤�ɏ�����
\begin{eqnarray}   %%%211%%%
	\dot{E}_{\mathrm{total}} = x \dot{E}_{\mathrm{radial}}
	\label{211}
\end{eqnarray}
������$x$�͒萔�Ō��߂邱�Ƃ��ł���D
�����U��̃��J�j�Y�������`�ߒ��ł���ƁC���̂Ƃ��ȉ��̂悤�ɏ�����
\begin{eqnarray}   %%%212%%%
	\dot{E}_{\mathrm{total}} = \dot{E}_{\mathrm{radial}} + \dot{E}_{\mathrm{librational}}
	\label{212}
\end{eqnarray}
���̏ꍇ�C��(4.191)
\footnote
{��(4.191)
\begin{eqnarray}  %%%191%%%
	E_{\mathrm{radial}} = \frac{19}{5} \mu \epsilon _{\mathrm{max}}^2 \pi C_s^3 \int _0 ^{\pi} [\mathcal{P}_2(\cos \beta)]^2 \sin \beta d \beta = \frac{57}{50} \mu \epsilon _{\mathrm{max}}^2  \mathcal{V}_s \nonumber
	\label{191}
\end{eqnarray}
}
��
��(4.195)
\footnote
{��(4.195)
\begin{eqnarray}  %%%195%%%
	E_{\mathrm{librational}} &=& \frac{19}{10} \mu \epsilon _{\mathrm{max}}^2 C_s^3 \int \bigl[\sin ^2 \theta \sin 2\phi \bigr]^2 \sin\theta d\theta d\phi \nonumber \\
	&=& \frac{19}{10} \mu \epsilon _{\mathrm{max}}^2 2\pi C_s^3 \int _0^{\pi/2} \sin ^5\theta d\theta  \nonumber \\
	&=&  \frac{4}{3}\frac{57}{50} \mu \epsilon _{\mathrm{max}}^2 \mathcal{V}_s�@\nonumber
	\label{195}
\end{eqnarray}
}
����$x = 7/3$�𓾂�D
�������Ȃ���C
�����U��̃��J�j�Y�������E�w�̗����ɂ����̂ŁC$v^3$�ɔ�Ⴗ��ƁC
�����U��̃��J�j�Y���͊m���ɔ���`�ɂȂ�C
��(\ref{212})�͖��炩�ɓK�p�ł��Ȃ��D
numerical hydrodynamical code��p���āC
Sears(1995)��$x \approx 6$�����߁C
�܂�Detmott �� Sagan(1995)�͂��̐���l��p����
���݂�Titan�̍����O�����S��($e = 0.029$)��
�[���S���𕢂��[��0.6 km �ȏ�̒Y�����f�̊C�����邱�ƁC
�܂���
�C���S���𕢂��ĂȂ��Ƃ���ƁC
�t�̒Y�����f�͐��X�̂Ȃ����Ă��Ȃ��C��N���[�^�[�΂Ɍ����đ��݂��邱��
�ɂ���Ă̂ݐ����ł��邱��
���������D

\clearpage



%%%%%%%%              Text End                  %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
