1 座標・表記

図: Geometry for caluculation of the radiation field at $R\rt$ from the position of the radiating particle at the retarted time ( $t'=t_{{\rm ret}}$).

座標、表記は以下の通り：

$$t' = t_{{\rm ret}}$$ (1)

$${\bf R}(t') = \vr - \vr_0(t')$$ (2)

$$R(t') = \left\vert {\bf R}(t')\right\vert$$ (3)

$$t' = t - \frac{R(t')}{c}$$ (4)

$$\vn(t') = \frac{{\bf R}(t')}{R(t')}$$ (5)

$$\bm{\beta}(t') =\frac{\vv(t')}{c}$$ (6)

$$\gamma = \frac{1}{\sqrt{1-\bm{\beta}^2}}$$ (7)

$$\kappa(t') = 1-\vn(t')\cdot \bm{\beta}(t')$$ (8)

$$\del{f(t')}{t'} \equiv \dot{f}(t'),\quad \del{{\bf f}(t')}{t'} \equiv \dot{{\bf f}}(t') .$$ (9)

又、

$$R(t')= {\bf R}(t')\cdot \vn(t') = \left( \vr-\vr_0(t')\right)\cdot \vn(t')= \vn(t') \cdot \vr - \vn(t')\cdot \vr_0(t')$$

$$R^2(t') = {\bf R}(t')\cdot{\bf R}(t')... ...e \, dt' = dt -\frac{1}{c}\di{R(t')}{t'}dt' =dt - \vn\cdot \dot{{\bf R}}(t')dt'$$

であるから、以下を得る：

$$\therefore \quad R(t') = \vn(t')\cdot... ...appa(t')dt'; \quad t = \frac{\vr\cdot\vn}{c} +t' -\frac{\vn\cdot\vr_0(t')}{c} .$$ (10)

著者: 茅根裕司 chinone_at_astr.tohoku.ac.jp