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\Dauthor{ ��R ���� } % �����̖��O
\Ddate{ 2004/07/23 } % �쐬��
\Dfile{ sec0414.tex } % �t�@�C���̖��O

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\subsection{The Double Synchronous State} % �߂̃^�C�g��
% ��������{���������ĉ�����.
���̋O���ƒn���̎��]���C
���Ɍ��ɂ���Ēn���ɋN���锼�������̓�����
���z�n�̔N���ʂ��Ă��Ȃ�ω����邱�Ƃ�
�傢�ɍl������(Burns 1986)�D
���݂��̌n�̊p�^���ʂ̂قƂ�ǂ����̋O���ɑ��݂��邪�C
���`�����ɂ͂����炭�t�̏󋵂ɂȂ��Ă���C
�p�^���ʂ̂قƂ�ǂ��n���̎��]�ɑ��݂��Ă����ł��낤�D
�n���������Ƃ��ł���ő厩�]�U����$\Omega_{\mathrm{max}}$
�𐄒肷�邱�Ƃ��ł��C�ȉ��̂悤�ɗ^������
\begin{equation}   %%%233%%%
	\alpha m_p C_p^2 \Omega _{\mathrm{max}} \approx m_s a^2 n + \alpha m_p C_p^2 \Omega
	\label{233}
\end{equation}
������$\Omega$�͌��݂̒n���̎��]�U�����ł���C�܂�
�Ή�����(corresponding)�n���̍ŏ����]������ �` 4 h �ɂȂ邾�낤�D
�������Ȃ���C
�n�̍ŏ��̈ʒu�͂킩���Ă��炸�C
����̑��肩��͌��߂邱�Ƃ��ł��Ȃ��D



%%%%%%%figure 4.19%%%%%%%
\begin{figure}[bp]
\begin{center}
	\includegraphics[width=10cm,height=6.091cm]{./figure/4_19.jpg}
\end{center}	
	\caption{
	���̋O��$L_{\mathrm{orb}}$(�j��)��
	�n���̎��]$L_{\mathrm{spin}}$(����)�̊Ԃ̊p�^���ʂ̕ω��D
	���ۂ�$L_{\mathrm{orb}}$�̋Ȑ���̐�����
	���̒n������̋�����n�����a�Ŏ����Ă���D
	���ۂ�$L_{\mathrm{spin}}$�̋Ȑ���̐�����
	�n���̎��]������\���Ă���D
	���ꂼ��̐����̑g�ݍ��킹�ɑ΂��ĔN�ŏ����ꂽ���Ԃ́C
	���݂̔z�u��4.5$\times$10$^9$�N�̐i���̌��ʂł���Ɖ��肵�Čv�Z���ꂽ
	�O���i���̎��Ԃł���D
	�������̉���̉��C����5$\times$10$^{10}$�N�o��
	�n��-���n����d������ԂɂȂ�C���̂Ƃ�
	�n���̎��]�����ƌ��̌��]������47���ɂȂ�C
	���̔�������$\approx 87$�n�����a�ɂȂ�ł��낤�D
	}
	\label{figure.4.19}
\end{figure}


�����I�ړI�̂��߁C
���̋O����ԓ��ɕ��s�Ƃ��C
���z�����̉e���𖳎�����ƁC
���̂Ƃ�
�n�̑��p�^����$L_{\mathrm{tot}}$��
�O���p�^����$L_{\mathrm{orb}}$��
�n���̎��]�p�^����$L_{\mathrm{spin}}$�Ƃ̘a�ƂȂ�C
$L_{\mathrm{tot}}$�͕ۑ������Ƃ���ƁC
$L_{\mathrm{orb}}$��$L_{\mathrm{spin}}$�̘a�̎���
$L_{\mathrm{tot}}$�Ŋ��邱�Ƃ��o���āC
\begin{equation}   %%%234%%%
	1 = 0.832 \left( \frac{a}{60.4} \right)^{1/2} + 0.168 \left( \frac{24}{T} \right)
	\label{234}
\end{equation}
�𓾂�D
������
$a$�͒n�����a��P�ʂƂ������̋O���̔������ł���C
$T$�͎��Ԃ�P�ʂƂ����n���̎��]�����ł���D
���ς���
�n���̒����U��֐���$Q_{\mathrm{Earth}} \approx 34$�Ɖ��肷��ƁC
���̂Ƃ�
���̔������ƑΉ�����n���̎��]������
�n�̐i���̒��̗l�X�Ȏ��Ԃɂ���
�v���b�g���邽�߂�
��(\ref{234})��p���邱�Ƃ��ł���(�}4.19)�D
�����ŁC
�������݂̋O���ɂȂ�܂�$4.5\times 10^9$ y �����Đi�����Ă����Ɖ��肷��D
�����U��֐���$Q_{\mathrm{Earth}}$
 �` 34 �ɕۂ����Ƃ���ƁC
���̂Ƃ� �` $5 \times 10^{10}$ y �̐i���̌�ɂ�
�n���̎��]���猎�̋O���ւ̊p�^���ʂ̗A���͂قڊ�����
�n��
�n���ƌ��̎��]���������������Ō��̋O�������Ɠ�����
\textbf{��d����(double synchronous)}��ԂɂȂ�ł��낤�D 
�����
���̔������� �` 87 �n�����a�Œn���̎��]������ �` 47 d �ł���Ƃ���
�B�������D
���̒i�K�ł̑��z�����͒n���̎��]��}������悤�ɓ����C
�����ł��Ȃ��D�ǂ��spin-orbit couplings(5�͎Q��)�̔�����
�U����������
���ׂĂ̋��Œ�͉�������Ă��܂����낤(Peale 1986)�D
�n���̓����O���͌��̋O�����z���Ċg�����Ă����C
�����Ă��̂Ƃ����ɂ���Ēn���ɋN���锼��������
���̔�����������������悤�ɓ����ł��낤�D
$5 \times 10^{10}$ y ��肳��ɐi��������C
���͒n���ɋ߂Â��C
�������͂ɂ��j�󂳂�C
�傫�ȃ����O�n���`������ł��낤(Jeffreys 1970)�D


�n�����z�n�̔N����Z�����Ԃœ�d����Ԃɐi�����������
�ȉ��̂悤�ɉ��肷�邱�Ƃɂ���Č��߂邱�Ƃ��ł��C
�ŏ���$L_{\mathrm{tot}} = L_{\mathrm{spin}}$��
�ŏI�I��$L_{\mathrm{tot}} = L_{\mathrm{orb}}$�ƂȂ�Ƃ����
�ŏI�I�Ȕ�����$a_f$�͍ŏ��̎��]�U����$\Omega _i(=2\pi /T_i)$
�ƈȉ��ɂ���Ċ֘A�t������D
\begin{equation}   %%%235%%%
	m_s (\mathcal{G}m_p )^{1/2} a_f^{1/2} =  \alpha m_p C_p^2 \Omega _i
	\label{235}
\end{equation}
�]���āC
\begin{equation}   %%%236%%%
	 \left(\frac{a_f}{C_p}\right)^{1/2} =  \alpha \frac{m_p}{m_s} \left(\frac{3 \Omega _i ^2}{4 \pi \mathcal{G} \rho _p}\right)^{1/2}
	\label{236}
\end{equation}
�ƂȂ�D
��(4.214)
\footnote
{��(4.214)
$$
\frac{2}{13}a_0 ^{13/2} = \frac{3k_{2p}}{Q_p} \left( \frac{\mathcal{G}}{m_p} \right) ^{1/2} C_p^5 m_s \Delta t
$$
}
�ɑ�����āC
���B�����d������Ԃɂ���
�q���Ƙf���̎��ʂ̔��
�ȉ��̗ՊE�l
\footnote{
�e�L�X�g����
$
	\left(\frac{m_p}{m_s}\right)_{\mathrm{crit}}
$
�͌�A�D
}
\begin{equation}   %%%237%%%
	 \left(\frac{m_s}{m_p}\right)_{\mathrm{crit}} = \left(\frac{2 Q_p}{39 \alpha k_{2p} \Omega _i \Delta t}\right)^{1/14} \left(\frac{3 \alpha ^2}{4 \pi \mathcal{G} \rho _p}\right)^{1/2} \frac{2 \pi}{T_i}
	\label{237}
\end{equation}
�𒴂��Ȃ��Ă͂Ȃ�Ȃ��Ɛ���ł���D
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��ɍŏ��̎��]�����ɂ���Č��߂��C
�f���܂蒆�S�V�̂̑傫�����܂ތn�̂��̑��̌W���ւ̈ˑ��͔��ɏ������D
�n��-���n�ɂ���
$T_i$ �` 4 h �Ɖ��肷��ƁC
���̂Ƃ�$(m_s /m_p) \approx 0.0147$�ł���C
�ϑ����ꂽ�� 0.0123 ���킸���ɑ傫���D


���z�n���ōł��������ʔ�����n��
��������Charon�̌n�ł���D
�n�b�u���F���]�����ɂ�閻�����̏d�S�܂��̂�炬�̊ϑ��ɂ��ƁC
���ʔ��0.0837$\pm$0.0147�ł���(Null et al 1993)�C
����͗ՊE�l�����Ȃ�����Ă���D
���݂ł͖�������Charon��
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���邱�Ƃ̏o���Ȃ����f���̑g���������]�x�����󂯂Ă��邾�낤�D
���x 3 g cm$^{-3}$ ��
������ $\mu$ �` $5 \times 10^{10}$ N m$^{-2}$ ��$Q_p$ �` 100 
(�S�ēK�����l)��
�����Ȋ�ΓV�̂Ɏ�(4.237)��K�p���C
�ŏ��̎��]������ �` 8 h �Ɖ��肷���(Alfv$\mathrm{\acute{e}}$n 1964)�C
���̂Ƃ����a$\sim 100$ km �̏��f���̗ՊE���ʔ�� 0.034 ��
10 m �̏����ȏ��f���ł� 0.065 �܂ł����������Ȃ��D
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