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\Dauthor{ ��R ���� } % �����̖��O
\Ddate{ 2003/12/04 } % �쐬��
\Dfile{ sec0406.tex } % �t�@�C���̖��O

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\subsection{Darwin-Radau relation} % �߂̃^�C�g��
% ��������{���������ĉ�����.
Darwin-Radau relation(�Q�Ɨ� Cook 1980)�́C
�����\���̌W��$\mathcal{C}/mR^2$($m$�͕��̂̎��ʁC$R$�͕��ϔ��a)��
�f���܂��͉q���ɂ�����$q$�C$f$�C$J_2$�̒l��
�֘A�t����ߎ����ł���D
���̊֌W�́CClairaut(1743)�Ɋ�Â���Radau(1885)�ɂ���čŏ��Ɍ��o����C
Darwin(1899)�����̖��Ɋ�^���Ă���D
��b�ƂȂ鉼��́C�֘A���镨�̂͐Ð������t�ɂ���Ƃ������Ƃł���D
���̊֌W�͂������̈�����`�ŕ\���ł��邪�C�����ł�Cook(1980)�ɂ��`��K�p��
\begin{equation}  %%%112%%%
	\frac{\mathcal{C}}{mR^2} = \frac{2}{3} \Biggl[ 1 - \frac{2}{5}\biggl( \frac{5}{2}\frac{q}{f} - 1\biggr)^{1/2} \Biggr] .
\end{equation}
�����\���̌W��$\bar{\mathcal{C}}$�𖳎����̗ʂƂ��Ĉȉ��̂悤�ɒ�`��
\begin{equation}  %%%113%%%
	\bar{\mathcal{C}} = \frac{\mathcal{C}}{mR^2}
\end{equation}
��(4.110)�ŗ^����ꂽ$J_2$�C$q$�C$f$�̊֌W��p����ƁC��(4.112)��Darwin-Radau relation�͈ȉ��̂悤�ɏ������Ƃ��ł�

\begin{equation}  %%%114%%%
	\frac{J_2}{f} = -\frac{3}{10} + \frac{5}{2} \bar{\mathcal{C}} - \frac{15}{8}\bar{\mathcal{C}}^2.
\end{equation}


�������Ȃ���C���̌`��Darwin-Radau relation�ɂ���ė^������$\bar{\mathcal{C}}$��$J_2/f$�̊֌W�́C4.4�߂ōl�����C��茻���I�Șf���̕ό`���f����p���ē�����C����ʓI�Ȍ��ʂ̋Ɍ��̏ꍇ�ł���D
�j�|�}���g�����f���ɂ��CDermott(1979b)�͈ȉ��𓾂�
\begin{equation}  %%%115%%%
	\bar{\mathcal{C}} = \frac{2}{5} \Biggl[ \frac{\sigma}{\langle \rho \rangle} + \biggl(1 - \frac{\sigma}{\langle \rho \rangle} \biggr) \biggl(\frac{A}{B}\biggr)^{2} \Biggr] .
\end{equation}
�����\���̒�`�ƁC�m���Ă���j�ƃ}���g���̑傫���ƕό`��p���āC��ꌴ�����炱�̌��ʂ�������D
�q���̕\�ʂƊj�|�}���g�����E�ʂ͕��t�ł���Ƃ���ƁC��(4.79)
\footnote{��(4.79) 
$	A S_2  = F\frac{(5/2)\zeta _c}{1 + \tilde{\mu}} \ \ \  and\ \ \ B T_2 = H \frac{5}{2}\zeta  $
\begin{eqnarray}  
	H = \frac{2\langle \rho \rangle }{5\rho}\Biggl(  \frac{1 + \tilde{\mu} +(3/2) (A/B)^2 F \delta  }{ (1 + \tilde{\mu})( \delta + 2\sigma /5 \rho )  } \Biggr)
\nonumber
\end{eqnarray}
}
�ŗp����ꂽ�W��$H$�͎�(4.83)
\footnote{��(4.83)\begin{eqnarray}  
H_h = \frac{2\langle \rho \rangle }{5\rho} \Biggl(  \frac{1 + (3\delta / 5 \gamma) (A/B)^2  }{ \delta + 2\sigma /5 \rho - (9\delta \sigma / 25 \gamma \rho )(A/B)^2 } \Biggr)\nonumber
\end{eqnarray}}
�ŗ^������Ð��l$H_h$�����D
Dermott(1979b)��$J_2/f$�̒l��$H_h$���ȉ��̕������ɂ���Ċ֘A�t����
\begin{equation}  %%%116%%%
	\frac{J_2}{f} = \frac{2}{3} \biggl( 1 - \frac{2}{5H_h} \biggr) .
\end{equation}
$\delta$��$\gamma$
�̎��ɂ����
\footnote{��(4.82)
	$\delta = \bigl(\frac{A}{B}\bigr)^3 \bigl(1-\frac{\sigma}{\rho}\bigr)$�C
 ��(4.84)
	$\gamma = \frac{2}{5} + \frac{3\sigma}{5\rho}$}
�C�^����ꂽ�l$A/B$�ŁC$J_2/f$��$H_h$�Ɋ֘A�t���邱�Ƃ��ł�
\begin{equation}  %%%117%%%
	\frac{J_2}{f} = \frac{2}{3} + \frac{\bar{\mathcal{C}}-\frac{2}{5}(A/B)^2}{1-(A/B)^2} +  \frac{ 8 - 20 (A/B)^2 +10\bar{\mathcal{C}}\bigl[5(A/B)^3 - 2 \bigr] }{12\bigl[(A/B)^5 - 1 \bigr]  + 15\bar{\mathcal{C}}\bigl[2 - 5(A/B)^3 + 3(A/B)^5 \bigr]}  .
\end{equation}
���̎���2�̋Ɍ����l�����


\noindent$\bullet $�j���_�̏ꍇ$A/B \to 0$��
\begin{equation}  %%%118%%%
	J_2/f \to \bar{\mathcal{C}}.
\end{equation}
$\bullet $ Darwin-Radau relation�̏ꍇ$A/B \to 1$��
\begin{equation}  %%%119%%%
	J_2/f \to -\frac{3}{10} + \frac{5}{2} \bar{\mathcal{C}} - \frac{15}{8}\bar{\mathcal{C}}^2 .
\end{equation}

%%%%%%figure 4.9%%%%%%%
\begin{figure}[tbp] 
\begin{center}
	\includegraphics[width=4cm,clip] %width=10cm,height=10cm
	{./figure/4_9.eps} %pdf
\end{center}

\caption{�����\��$\bar{\mathcal{C}}$�̊֐��Ƃ��Ă�$J_2/f$�C���ꂼ��(i)�j���_�̃��f��(�_��), (ii)$A/B=0.5$�̃��f��(�j��)�C(iii) Darwin-Radau relation�̏ꍇ(����)�D�n���Ƌ���f����$J_2/f$�̒l�������Ă���D}
	\label{figure.4.9}
\end{figure}

�}4.9�́C�j���_�̃��f���CDarwin-Radau ���f���C$A/B = 0.5$�̈�ʓI�Ȋj�|�}���g���̃��f���ɂ�����C$\bar{\mathcal{C}}$�̊֐��Ƃ��Ă�$J_2/f$���v���b�g�������̂������Ă���D
�n���Ƌ���f���ɂ�����$J_2/f$�̒m���Ă���l�������Ŏ����Ă���C���ꂼ��̘f���ɂ�����$\bar{\mathcal{C}}$�̋Ɍ��l�������Ă���D
$J_2/f$�̒l�ŗ^������$\bar{\mathcal{C}}$�̒l�̕����C$J_2/f$���傫���Ȃ�ɂ�ď������Ȃ��Ă������Ƃɒ��ڂ��ꂽ���D
�L�[�|�C���g�́C$J_2/f$�̒l�ɂ��C�����\���ɐ�����u�����Ƃ��ł���Ƃ������Ƃł���D
��(4.105)����\footnote{��(4.105) \begin{eqnarray}  
	J_2 = \frac{ \mathcal{C} - \frac{1}{2}(\mathcal{A+B}) }{ma^2} \approx \frac{\mathcal{C-A}}{ma^2} \nonumber
\end{eqnarray}}
$\mathcal{C-A}$�̒l�ƁC���̏������ѕt����ƁC$\mathcal{C}$��$\mathcal{A}$�̐���l��������D
���̂悤�Ȑ���l�͘f�������̏ڍׂȃ��f���ɐ�����^����D



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