4-1

$\beta$を$\gamma$で書き換えて$\kappa$を$\theta$の関数として表し、図示せよ。但し

$$\theta\ll 1 \cos\theta \sim 1 -\frac{\theta^2}{2}\, ; \gamma\gg 1 \,\,\, \beta \sim 1-\frac{1}{2\gamma^2} \, .$$ (16)

4-1解答

$$\kappa(\theta) = 1-\vn\cdot \bm{\beta} =1-\beta \cos\theta =1-\left(1-\frac{1}{2... ...ma^2}-\frac{\theta^2}{2}\right)+ {\cal O}\left(\frac{\theta^2}{\gamma^2}\right)$$

$$=\frac{1}{2} \left(\theta^2 +\frac{1}{\gamma^2}\right)+ {\cal O}\left(\frac{\theta^2}{\gamma^2}\right)$$ (17)

図 3: $\gamma \gg 1,\theta \ll 1$の場合の $\kappa (\theta )$

著者: 茅根裕司 chinone_at_astr.tohoku.ac.jp