 公転軸系直交座標計算：太陽の高度)：
\[
cos( \alpha ) = \tau_s \ x - \tau_c \ y
\]
また緯度ａの正午の直交座標は，つぎの通り ( 公転軸系直交座標計算：正午の座標)：
よって，つぎのようになる： \[ \begin{align} \alpha_c &= \tau_s \ x - \tau_c \ y \\ &= \tau_s ( \frac{ a_s \ n_s \ \tau_s \ \tau_c + n_c \tau_s \sqrt{(a_c)^2 - (n_s)^2 (\tau_s)^2}}{1 - (n_s)^2 (\tau_s)^2} ) \\ & \quad - \tau_c ( \frac{ - a_s \ n_s \ (\tau_c)^2 - n_c \tau_c \sqrt{(a_c)^2 - (n_s)^2 (\tau_s)^2}}{1 - (n_s)^2 (\tau_s)^2} ) \\ \ \\ &= \frac{ a_s \ n_s \ (\tau_s)^2 \ \tau_c + n_c (\tau_s)^2 \sqrt{(a_c)^2 - (n_s)^2 (\tau_s)^2}}{1 - (n_s)^2 (\tau_s)^2} ) \\ & \quad + ( \frac{ - a_s \ n_s \ \tau_c (\tau_c)^2 +n_c (\tau_c)^2 \sqrt{(a_c)^2 - (n_s)^2 (\tau_s)^2}}{1 - (n_s)^2 (\tau_s)^2} ) \\ \ \\ &= \frac{ a_s \ n_s \ \tau_c + n_c \sqrt{(a_c)^2 - (n_s)^2 (\tau_s)^2} }{1 - (n_s)^2 (\tau_s)^2} ) \\ \\ \ \\ \end{align} \] |