(1) 太陽の方向
太陽方向ベクトルは，つぎの通り (  太陽の方向)：
\[
{\bf s} = (\tau_s, \ - \tau_c, \ 0)
\\
\]
秋分は，\( \tau = \pi / 2 \) なので，
\[
{\bf s} = ( 1, \ 0, \ 0)
\\
\]
(2) 太陽の高度
公転角度がτのとき，緯度ａ, 経度ｂの点での太陽の仰角余角αは，つぎの通り ( 太陽の高度)：
\[
cos( \alpha ) = a_c \tau_s \ b_c - n_c a_c \tau_c \ b_s + n_s a_s \tau_c
\\ \ \\
\]
秋分は，\( \tau = \pi / 2 \) ── \( \tau_s =1, \ \tau_c =0 \) ──なので，
\[
cos( \alpha ) = a_c \ b_c
\\ \ \\
\]
(3) 太陽の方角
公転角度τ，緯度a，経度ｂでは，つぎのベクトル \( {\bf d} = ( d_x, d_y, d_y ) \) が太陽方角ベクトルになる ( 太陽の方角)：
\[
\begin{align}
d_x &=
- \tau_s (a_c)^2 (b_c)^2 \\
& + \tau_c n_c (a_c)^2 b_s b_c \\
& - \tau_c n_s a_s a_c b_c
+ \tau_s
\\ \ \\
d_y &=
\tau_c (n_c)^2 (a_c)^2 (b_s)^2 \\
&- \tau_s n_c (a_c)^2 b_s b_c \\
& - 2 \tau_c n_s n_c a_s a_c b_s \\
&+ \tau_s n_s a_s a_c b_c \\
&+ \tau_c (n_s)^2 (a_s)^2 - \tau_c
\\ \ \\
d_z &=
\tau_c n_s n_c a_c^2 b_s^2 \\
& \quad - \tau_s n_s a_c^2 b_s b_c \\
& \quad + \tau_c ( n_c^2 - n_s^2 ) a_s a_c b_s \\
& \quad - \tau_s n_c a_s a_c b_c \\
& \quad - \tau_c n_s n_c a_s^2
\\ \ \\
| {\bf d} | &= \sqrt { 1 - ( a_c ( n_c \tau_c b_s - \tau_s b_c ) - n_s a_s \tau_c )^2 }
\end{align}
\\ \ \\
\]
秋分は，\( \tau = \pi / 2 \) ── \( \tau_s =1, \ \tau_c =0 \) ──なので，
\[
d_x = - a_c^2 b_c^2 + 1 \\
d_y = -n_c a_c^2 b_s b_c + n_s a_s a_c b_c \\
d_z = n_s a_c^2 b_s b_c - n_c a_s a_c b_c \\
| {\bf d} | = \sqrt { 1 - ( a_c ( - b_c ) )^2 } \\
\quad = \sqrt { 1 - ( a_c)^2 (b_c)^2 }
\\
\]
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