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\begin{document}
%98���� 1998 0418, 0422

\baselineskip 14pt

\Large

\begin{flushleft}
{\LARGE \bf ��3���t�ˏ�̏���}
\end{flushleft}

\begin{flushleft}
{\Large \bf ��3.0����`}
\end{flushleft}

�t�ˋ��x�� $I$�A�����]���� $l^i$�A�������̊p�� $d\Omega$ �Ƃ���ƁA
\begin{eqnarray}
   �t�˃G�l���M�[���x \hs\hs\hs & E & \equiv \frac{1}{c} \int I d\Omega,
\\
   �t�˗����x�N�g�� \hs\hs\hs & F^i & \equiv \int I l^i d\Omega,
\\
   �t�˃X�g���X�e���\�� \hs\hs\hs & P^{ik} & \equiv \frac{1}{c} \int I l^i l^k d\Omega.
\end{eqnarray}
%\footnote
%{\large
%�����̗ʂ͂��Ƃ��Ƒ��Θ_�I�ȗʂȂ̂ŁA
%�t�ˏ�̃G�l���M�[�^���ʃe���\�� $\tilde{T}$ �Ƃ��āA
%�s��̌`�ɂ܂Ƃ߂邱�Ƃ��ł���B
%$$
%   \tilde{T}^{\mu\nu} = \left(
%   \begin{array}{cc}
%      E              & \frac{\ds 1}{\ds c}F^i \\
%      \frac{\ds 1}{\ds c}F^i & P^{ik}
%   \end{array}
%                  \right).
%$$
%}
�����������ŁA
\begin{eqnarray}
   l^i &=& \frac{�i�ϑ��_P�̍��W�j-�i�����\�ʂ̗v�fdS�̍��W�j}{�v�f�Ɗϑ��_�̊Ԃ̋���D}
\\
   d\Omega &=& �ϑ��_P��������\�ʂ̗v�fdS�������ޗ��̊p
\end{eqnarray}
�ŁA�ϕ��͊ϑ��_����݂�������̕\�ʑS�̂ɂ킽���Ď��s����B

%\newpage

%\large

\begin{flushleft}
{\Large \bf ��3.1���_�����̏ꍇ}
\end{flushleft}

\begin{eqnarray}
   �����]�� \hs & l^i &
   = \frac{(r, 0, 0)-(r-D\cos\theta, -D\sin\theta\sin\varphi, D\sin\theta\cos\varphi)}{D}
\nonumber \\
   &&= (\cos\theta, \sin\theta\sin\varphi, -\sin\theta\cos\varphi).
\\
   ���̊p \hs & d\Omega &
   = \sin\theta d\theta d\varphi.
\end{eqnarray}

�ϕ��̈�́A$0 \leq \theta \leq \theta_0$ ����� 
$0 \leq \varphi \leq 2\pi$ �ł���B

�܂� $\sin\theta_0 = R_*/r$ �ɂȂ�B

���̌��x $L$ �ւ̕ϊ��� $L=4 \pi R_*^2 \pi I$ ���g���B

\newpage

\large

\noi
\underline{�t�˃G�l���M�[���x}
\begin{eqnarray}
   E &=& \frac{1}{c}\int Id\Omega
     = \frac{1}{c} \int_0^{\theta_0} \int_0^{2\pi} I \sin\theta d\theta d\varphi
     = \frac{2\pi I}{c} \int_0^{\theta_0} \sin\theta d\theta
\nonumber \\
     &=& \frac{2\pi I}{c} \left( 1-\cos\theta_0 \right)
     = \frac{2}{c} \frac{L}{4\pi R_*^2} 
         \left[ 1 - \sqrt{1-\left(\frac{R_*}{r}\right)^2} \right]
\nonumber \\
     & \sim & \frac{1}{c} \frac{L}{4\pi r^2} \hs\hs �i�\�������j 
\end{eqnarray}

\noi
\underline{�t�˗���}
\begin{eqnarray}
   F^r &=& \int I l^r d\Omega
     = \int I\cos\theta d\Omega
%     = \int_0^{\theta_0} \int_0^{2\pi} I \sin\theta \cos\theta d\theta d\varphi
     = 2\pi I \int_0^{\theta_0} \sin\theta \cos\theta d\theta
\nonumber \\
     &=& \pi I \sin^2\theta_0
\nonumber \\
     &=& \frac{L}{4\pi r^2} \hs\hs �i�\�������ł����藧�j
\\
   F^\varphi &=& 0  \hs\hs\hs\hs �i\varphi �����̐ϕ���0�ɂȂ�j
\\
   F^z &=& 0        \hs\hs\hs\hs �i\varphi �����̐ϕ���0�ɂȂ�j
\end{eqnarray}

\noi
\underline{�t�˃X�g���X�e���\��}
\begin{eqnarray}
   P^{rr} &=& \frac{1}{c} \int I l^r l^r d\Omega
     = \frac{1}{c}\int I\cos^2\theta d\Omega
     = \frac{2\pi I}{c} \int_0^{\theta_0} \sin\theta \cos^2\theta d\theta
\nonumber \\
     &=& \frac{2\pi I}{c} \frac{ 1-\cos^3\theta_0 }{3}
     = \frac{2\pi I}{3c} \left( 1-\cos\theta_0 \right)
            \left( 1+\cos\theta_0+\cos^2\theta_0 \right)
\nonumber \\
     & \sim & E \hs\hs �i�\������; \theta_0 \sim 0�j 
\\
   P^{r\varphi} &=& 0  \hs\hs\hs\hs �i\varphi �����̐ϕ���0�ɂȂ�j
\\
   P^{rz} &=& 0        \hs\hs\hs\hs �i\varphi �����̐ϕ���0�ɂȂ�j
\\
   P^{\varphi\varphi} &=& \frac{1}{c}\int I l^\varphi l^\varphi d\Omega
     = \frac{1}{c}\int I \sin^2\theta\sin^2\varphi d\Omega
     = \frac{1}{c} \int_0^{\theta_0} \int_0^{2\pi} I \sin^3\theta \sin^2\varphi d\theta d\varphi
\nonumber \\
     &=& \frac{\pi I}{3c} 
         \left( 1-\cos\theta_0 \right)
         \left( 1-\cos\theta_0 + \sin^2 \theta_0 \right)
\nonumber \\
     &\sim& 0 \hs\hs �i�\�������j
\\
   P^{\varphi z} &=& 0  \hs\hs\hs\hs�i\varphi �����̐ϕ���0�ɂȂ�j
\\
   P^{zz} &=& \frac{1}{c}\int I l^z l^z d\Omega
     = \frac{1}{c}\int I \sin^2\theta\cos^2\varphi d\Omega
     = \frac{1}{c} \int_0^{\theta_0} \int_0^{2\pi} I \sin^3\theta \cos^2\varphi d\theta d\varphi
\nonumber \\
     &=& P^{\varphi\varphi}
\nonumber \\
     &\sim& 0 \hs\hs �i�\�������j
\end{eqnarray}

\newpage

\Large

\begin{flushleft}
{\Large \bf ��3.2�����ʌ����̏ꍇ}
\end{flushleft}

\begin{eqnarray}
   �����]�� \hs & l^i &
   = (\sin\theta\cos\varphi, \sin\theta\sin\varphi, \cos\theta).
\\
   ���̊p \hs  & d\Omega &
   = \sin\theta d\theta d\varphi.
\end{eqnarray}

�ϕ��̈�́A$0 \leq \theta \leq \pi/2$ ����� 
$0 \leq \varphi \leq 2\pi$ �ł���B

\noi
\underline{�t�˃G�l���M�[���x}
\begin{eqnarray}
   E &=& \frac{1}{c}\int Id\Omega
     = \frac{I}{c} \int \sin\theta d\theta d\varphi
     = \frac{2\pi I}{c} 
\end{eqnarray}

\noi
\underline{�t�˗���}
\begin{eqnarray}
   F^r &=& 0       \hs\hs\hs\hs �i\varphi �����̐ϕ���0�ɂȂ�j
\\
   F^\varphi &=& 0 \hs\hs\hs\hs �i\varphi �����̐ϕ���0�ɂȂ�j
\\
   F^z &=& \int I \cos\theta\sin\theta d\theta d\varphi
       = \pi I
\end{eqnarray}

\noi
\underline{�t�˃X�g���X�e���\��}
\begin{eqnarray}
   P^{rr} &=& \frac{1}{c} \int I l^r l^r d\Omega
     = \frac{I}{c} \int \sin^3\theta \cos^2\varphi d\theta d\varphi
     = \frac{2\pi I}{3c} 
     = \frac{1}{3}E
\\
   P^{r\varphi} &=& 0  \hs\hs\hs\hs �i\varphi �����̐ϕ���0�ɂȂ�j
\\
   P^{rz} &=& 0  \hs\hs\hs\hs�i\varphi �����̐ϕ���0�ɂȂ�j
\\
   P^{\varphi\varphi} &=& \frac{1}{c}\int I l^\varphi l^\varphi d\Omega
     = \frac{I}{c}\int \sin^3\theta \sin^2\varphi d\theta d\varphi
     = \frac{2\pi I}{3c}
     = \frac{1}{3}E
\\
   P^{\varphi z} &=& 0  \hs\hs\hs\hs �i\varphi �����̐ϕ���0�ɂȂ�j
\\
   P^{zz} &=& \frac{1}{c}\int I l^z l^z d\Omega
     = \frac{I}{c}\int \cos^2\theta \sin\theta d\theta d\varphi
     = \frac{2\pi I}{3c}
     = \frac{1}{3}E
\end{eqnarray}

\vs

�Ȃ��\�ʂ����x $T$ �̍����t�˂���˂��Ă���Ȃ�΁A
�X�e�t�@���E�{���c�}���̖@�����A
$\pi I = \sigma T^4$ �ƂȂ�
�i$\sigma$ �̓X�e�t�@���E�{���c�}���̒萔�j�B

\newpage

\begin{flushleft}
{\Large \bf ��3.3���W���~���~�Ղ̏ꍇ}
\end{flushleft}

\begin{eqnarray}
   �t�ˋ��xI�ƕ\�ʉ��xT_{\rm d}
   \hs\hs\hs & \pi I & = \sigma T_{\rm d}^4
\\
   �\�ʉ��xT_{\rm d}�̔��a���z
   \hs\hs\hs & \sigma T_{\rm d}^4 &
        =  \frac{3GM\dot{M}}{8\pi r^3}
           \left( 1-\sqrt{\frac{r_{\rm in}}{r}} \right)
\\
   �~�ՑS�̂̌��xL_{\rm d}
   \hs\hs\hs & L_{\rm d} & = \frac{GM\dot{M}}{2\rin}
\end{eqnarray}

\noi
{\Large \bf ���~�ՋߖT�ߎ�}

\noi
\underline{�t�˃G�l���M�[���x}
\begin{equation}
   E = \frac{2}{c} \sigma T_{\rm d}^4 
     = \frac{2}{c} \frac{3GM\dot{M}}{8\pi r^3}
       \left( 1-\sqrt{\frac{r_{\rm in}}{r}} \right)
     = \frac{2}{c} \frac{L_{\rm d}}{4\pi} \frac{3r_{\rm in}}{r^3}
       \left( 1-\sqrt{\frac{r_{\rm in}}{r}} \right).
\label{nearE}
\end{equation}

\noi
\underline{�t�˗����� $r$ ������ $z$ ����}
\begin{eqnarray}
   F^r &=& 0,  \\
   F^z &=& \sigma T_{\rm d}^4
        =  \frac{3GM\dot{M}}{8\pi r^3}
           \left( 1-\sqrt{\frac{r_{\rm in}}{r}} \right)
        =  \frac{3L_{\rm d}r_{\rm in}}{4\pi r^3} 
           \left( 1-\sqrt{\frac{r_{\rm in}}{r}} \right).
\label{nearF}
\end{eqnarray}

\noi
\underline{�t�˃X�g���X�e���\��}
\begin{equation}
   P^{rr} = P^{\varphi \varphi}=P^{zz}=\frac{1}{3}E,
\hs\hs\hs\hs
   {\rm other} \quad P^{ij}=0.
\label{nearP}
\end{equation}

\noi
\underline{�t�˗����� $\varphi$ ����}

\begin{equation}
   F^\varphi = \frac{4}{3}Ev_{\rm K}.
\end{equation}


\newpage

\noi
{\Large \bf ���ڑ�����}

\large
\Large

�����ł̕\���i�������A$R=\sqrt{r^2+z^2}$�A$\cos i =z/R$�j�F
\begin{eqnarray}
   E &=& \frac{1}{c} \frac{L_{\rm d}\cos i}{2\pi R^2}
     = \frac{2}{c} \frac{L_{\rm d}}{4\pi}\frac{z}{R^3}.
   \hs\hs\hs �i�����j
\label{farE} \\
   F^r &=& \frac{L_{\rm d}}{4\pi}\frac{2z}{R^3}\frac{r}{R},  
   \hs\hs\hs �i�����j
\label{farF1} \\
   F^z &=& \frac{L_{\rm d}}{4\pi}\frac{2z}{R^3}\frac{z}{R}.
   \hs\hs\hs �i�����j
\label{farF2}
\end{eqnarray}

\noi
\underline{�t�˃G�l���M�[���x�̐ڑ�����}

\begin{equation}
   E = \frac{2}{c} \frac{L_{\rm d}}{4\pi} \frac{3r_{\rm in}+z}{R^3} 
       \left( 1-\sqrt{\frac{r_{\rm in}}{R}} \right).
\label{E}
\end{equation}

\noi
\underline{�t�˗����x�N�g���̐ڑ�����}

\begin{eqnarray}
   F^r &=& \frac{L_{\rm d}}{4\pi}\frac{2rz}{R^4}
           \left(1-\frac{4r_{\rm g}}{R}\right),  
\\
   F^{\varphi}
   &=& \left( E + P^{\varphi\varphi} \right) \frac{r^2}{R^2} v_{\rm K}
    =  \frac{2}{c}\frac{L_{\rm d}}{4\pi}
       \frac{4\rin+z}{R^3}\frac{r^2}{R^2} v_{\rm K},
\label{Fp}
\\
   F^z &=& \frac{L_{\rm d}}{4\pi}\frac{3r_{\rm in}R+2z^2}{R^4}
           \left( 1-\sqrt{\frac{r_{\rm in}}{R}} \right).
\label{F}
\end{eqnarray}
 
\noi
\underline{�t�˃X�g���X�e���\���̐ڑ�����}

\begin{eqnarray}
   P^{rr} &=& \frac{2}{c}\frac{L_{\rm d}}{4\pi}
              \frac{r_{\rm in}R^2+zr^2}{R^5}
              \left( 1-\sqrt{\frac{r_{\rm in}}{R}} \right),  
\\
   P^{\varphi\varphi} &=& \frac{2}{c}\frac{L_{\rm d}}{4\pi}
              \frac{r_{\rm in}}{R^3}
              \left( 1-\sqrt{\frac{r_{\rm in}}{R}} \right), 
\\
   P^{zz} &=& \frac{2}{c}\frac{L_{\rm d}}{4\pi}
              \frac{r_{\rm in}R^2+z^3}{R^5}
              \left( 1-\sqrt{\frac{r_{\rm in}}{R}} \right).
\label{P} 
\end{eqnarray}

\newpage

\Large

\begin{flushleft}
{\LARGE \bf ��4���ᑬ�����D}
\end{flushleft}

���x�Ɋւ��āA$v/c$ �̈ꎟ�܂ł̋ߎ��B

\begin{flushleft}
{\Large \bf ��4.1���^��������}
\end{flushleft}

�����D�̑��x�x�N�g���� $v^i$�A���ʂ� $m$�A�L���f�ʐς� $S$ �Ƃ���ƁA
\begin{equation}
   \frac{dv^i}{dt} = -\bmnabla \phi
      + \frac{S}{mc} \left( F^i - Ev^i - v_k P^{ik} \right).
\end{equation}
�E�ӑ�ꍀ�͏d�́A
��񍀂��t�ˈ��i$F^i$�j���t�˒�R�i$ - Ev^i - v_k P^{ik}$�j�ł���B

\begin{flushleft}
{\Large \bf ��4.2���_�����̏ꍇ}
\end{flushleft}

���a $r$ �����̉^���������F
\begin{equation}
   \frac{dv^r}{dt} = -\frac{GM}{r^2}
      + \frac{S}{mc} \left( F^r - Ev^r - v^r P^{rr} \right).
\end{equation}

�_�����̏ꍇ�́A$E=P^{rr}=F^r/c$ �� $F^r=L/(4\pi r^2)$ �Ȃ̂ŁA
\begin{equation}
   \frac{dv^r}{dt} = -\frac{GM}{r^2}
      + \frac{S}{mc}\frac{L}{4\pi r^2} \left( 1 - \frac{2v^r}{c} \right).
\end{equation}

�����̃��C�g�l�X����
\begin{equation}
   \Gamma \equiv \frac{LS}{4\pi cGMm}
\end{equation}
�Œ�`����΁A�^���������́A
\begin{equation}
   \frac{dv^r}{dt} = -\frac{GM}{r^2}
             \left[ 1 - \Gamma \left(1-\frac{2v^r}{c}\right)\right].
\end{equation}

\underline{�I�[���x}
\begin{eqnarray}
   v^r_\infty &=& \frac{\Gamma-1}{2\Gamma}c
\nonumber \\
              &=& \frac{1}{2}c \hs\hs\hs �i\Gamma �������̂Ƃ��j.
\end{eqnarray}

\newpage

\begin{flushleft}
{\Large \bf ��4.3�����ʌ����̏ꍇ}
\end{flushleft}

���� $z$ �����̉^���������F
\begin{equation}
   \frac{dv^z}{dt}
      = \frac{S}{mc} \left( F^z - Ev^z - v^z P^{zz} \right).
\end{equation}

���ʌ����ł́A$E=(2/c)F^z$�A$F^z=\sigma T^4$�A$P^{zz}=E/3$ �Ȃ̂ŁA
\begin{equation}
   \frac{dv^z}{dt}
      = \frac{S}{mc} \left( F^z - \frac{4}{3}Ev^z \right)
      = \frac{S}{mc} F^z \left( 1 - \frac{8}{3} \frac{v^z}{c} \right).
\end{equation}

\underline{�I�[���x}
\begin{equation}
   v^z_\infty = \frac{3}{8}c.
\end{equation}

\begin{flushleft}
{\Large \bf ��4.4���W���~���~�Ղ̏ꍇ�i����j}
\end{flushleft}

���� $z$ �����̉^���������F
\begin{equation}
   \frac{dv^z}{dt}
      = -\frac{GM}{z^2}
        + \frac{S}{mc} \left( F^z - Ev^z - v^z P^{zz} \right).
\end{equation}

���㉓���ł́A
\begin{equation}
   E = \frac{2}{c} \frac{L_{\rm d}}{4\pi z^2},
\hs\hs\hs
   F^z = \frac{2L_{\rm d}}{4\pi z^2},
\hs\hs\hs
   P^{zz} = E
\end{equation}
�ƂȂ�̂ŁA
\begin{equation}
   \frac{dv^z}{dt}
      = -\frac{GM}{z^2}
        +\frac{S}{mc} \frac{2L_{\rm d}}{4\pi z^2}
         \left( 1 - \frac{2v^z}{c} \right).
\end{equation}

�����̗L�����C�g�l�X����
\begin{equation}
   \Gamma_{\rm d} \equiv \frac{2L_{\rm d}S}{4\pi cGMm}
\end{equation}
�Œ�`����΁A�^���������́A
\begin{equation}
   \frac{dv^z}{dt} = -\frac{GM}{z^2}
             \left[ 1 - \Gamma_{\rm d} \left(1-\frac{2v^z}{c}\right)\right].
\end{equation}

\underline{�I�[���x}
\begin{eqnarray}
   v^z_\infty &=& \frac{\Gamma_{\rm d}-1}{2\Gamma_{\rm d}}c
\nonumber \\
              &=& \frac{1}{2}c \hs\hs\hs �i\Gamma_{\rm d} �������̂Ƃ��j.
\end{eqnarray}

\newpage

\begin{flushleft}
{\LARGE \bf ��5�������������D}
\end{flushleft}

���ꑊ�Θ_�I�̘g���ł͊��S�Ȍ`�B

\begin{flushleft}
{\Large \bf ��5.1���^��������}
\end{flushleft}

�����D��4�����x�� $u^i$�A���ʂ� $m$�A�L���ʐς� $S$ �Ƃ���ƁA
\begin{eqnarray}
   c\frac{\ds du^i}{\ds d\tau} &+& c^2\Gamma^i_{\mu\nu}u^\mu u^\nu  
\nonumber \\
   &=&     \frac{\ds S}{\ds mc} \gamma 
        \left[ F^i - \gamma^2Ev^i - v_k P^{ik}
        +\frac{\ds \gamma^2}{\ds c^2}v^i
         \left( 2v_k F^k -v_j v_k P^{jk} \right) \right].
\nonumber \\
{}^{}
\end{eqnarray}
������ $\tau$ �͌ŗL���ԂŁA
$\Gamma^i_{\mu\nu}$ �̓��[�}���N���X�g�b�t�F���̋L���B

�܂�
\begin{eqnarray}
   4�����x        \hs\hs\hs & u^\mu & = \frac{dx^\mu}{cd\tau}
                    = \left( \gamma, \gamma \frac{v^i}{c} \right).
\\
   ���[�����c���q \hs\hs\hs & \gamma & = \frac{1}{\sqrt{1-v^2/c^2}};
                    \hs\hs v^2 = v_k v^k.
\end{eqnarray}
���ꑊ�Θ_�͍l���邪�A��ʑ��Θ_�͍l���Ȃ��B
���Ȃ킿����͕��R�Ƃ���i$v_k = v^k$�j�B

\newpage

\begin{flushleft}
{\Large \bf ��5.2���_�����̏ꍇ}
\end{flushleft}

���a $r$ �����̉^���������i$v_r=v^r=v$�A$\gamma=1/\sqrt{1-v^2/c^2}$�j�F
\begin{eqnarray}
   \frac{\ds d}{\ds d\tau} \left( \gamma v \right) 
   &=& -\frac{\ds GM}{\ds r^2}
\nonumber \\
   &+& \frac{\ds S}{\ds mc} \gamma 
        \left[ F^r - \gamma^2 Ev - v P^{rr}
               +\gamma^2 v \left( \frac{\ds 2v F^r}{\ds c^2}
                   - \frac{\ds v^2}{\ds c^2}P^{rr} \right) \right].
\nonumber \\
{}^{}
\end{eqnarray}

�_�����̏ꍇ�́A$E=P^{rr}=F^r/c$ �� $F^r=L/(4\pi r^2)$ �Ȃ̂ŁA
\begin{equation}
   \frac{d}{d\tau} \left( \gamma v \right) 
      = -\frac{GM}{r^2}
      + \frac{S}{mc} F^r \gamma^3 \left( 1 - \frac{v}{c} \right)^2.
\end{equation}

�����̃��C�g�l�X�� $\Gamma$ �i$=LS/4\pi cGMm$�j�𓱓�����ƁA
\begin{equation}
   \frac{d}{d\tau} \left( \gamma v \right) 
      = -\frac{GM}{r^2}
         \left[ 1 - \Gamma \gamma^3 \left(1-\frac{v}{c}\right)^2 \right].
\end{equation}

\underline{�I�[���x}
\begin{eqnarray}
   \Gamma &=& \frac{1}{\gamma^3 (1-v_\infty/c)^2}
          = \frac{(1+v_\infty/c)^{3/2}}{(1-v_\infty/c)^{1/2}}.
\\
   v_\infty &=& c \hs\hs\hs �i\Gamma �������̂Ƃ��j.
\end{eqnarray}

\begin{flushleft}
{\Large \bf ��5.3�����ʌ����̏ꍇ}
\end{flushleft}

���� $z$ �����̉^���������i$v_z=v^z=v$�A$\gamma=1/\sqrt{1-v^2/c^2}$�j�F
\begin{equation}
   \frac{d}{d\tau} \left( \gamma v \right)
      = \frac{S}{mc} \gamma 
        \left[ F^z - \gamma^2 Ev - v P^{zz}
               +\gamma^2 v \left( \frac{2v F^z}{c^2}
                   - \frac{v^2}{c^2}P^{zz} \right) \right].
\end{equation}

���ʌ����ł́A$E=(2/c)F^z$�A$F^z=\sigma T^4$�A$P^{zz}=E/3$ �Ȃ̂ŁA
\begin{equation}
   \frac{d}{d\tau} \left( \gamma v \right)
      = \frac{S}{mc} F^z \gamma^3
        \left[ \left(\frac{v}{c}\right)^2 - \frac{8}{3} \frac{v}{c} + 1 \right].
\end{equation}

\underline{�I�[���x}
\begin{equation}
   v_\infty = \frac{4-\sqrt{7}}{3}c \sim 0.45c.
\end{equation}


\end {document}

