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緯度ａ経度ｂでの＜南中の方角＞

作成: 2020-09-29 更新: 2020-09-29

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公転角度τ，緯度a，経度ｂでは，つぎのベクトル \( ( d_x, d_y, d_y ) \) が太陽方角ベクトルになる ( 緯度ａ経度ｂの点で見る＜太陽の方角＞)： \[ \begin{align} d_x &= \tau_s a_c^2 b_s^2 \\ & \quad + \tau_c n_c a_c^2 b_s b_c \\ & \quad - \tau_c n_s a_s a_c b_c \\ & \quad + \tau_s a_s^2 \\ \ \\ d_y &= - \tau_c n_s^2 a_c^2 b_s^2 \\ & \quad - \tau_c a_c^2 b_c^2 \\ & \quad - \tau_s n_c a_c^2 b_s b_c \\ & \quad - 2 \tau_c n_s n_c a_s a_c b_s \\ & \quad + \tau_s n_s a_s a_c b_c \\ & \quad - \tau_c n_c^2 a_s^2 \\ \ \\ d_z &= \tau_c n_s n_c a_c^2 b_s^2 \\ & \quad - \tau_s n_s a_c^2 b_s b_c \\ & \quad + ( \tau_c n_c^2 a_s a_c - \tau_c n_s^2 a_s a_c ) b_s \\ & \quad - \tau_s n_c a_s a_c b_c \\ & \quad - \tau_c n_s n_c a_s^2 \\ \ \\ | {\bf d} | &= \sqrt { 1 - ( a_c ( n_c \tau_c b_s - \tau_s b_c ) - n_s a_s \tau_c )^2 } \end{align} \\ \ \\ \] また, 「南中の経度計算式」より： \[ b_c = \frac { \tau_s} {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } \\ b_s = \frac { - n_c \tau_c }{ \sqrt{1 - (n_s)^2 ( \tau_c )^2} } \\　 \] 以上を合わせて： \[ \begin{align} d_x &= \tau_s a_c^2 b_s^2 \\ & \quad + \tau_c n_c a_c^2 b_s b_c \\ & \quad - \tau_c n_s a_s a_c b_c \\ & \quad + \tau_s a_s^2 \\ \ \\ &= \tau_s a_c^2 (\frac { - n_c \tau_c }{ \sqrt{1 - (n_s)^2 ( \tau_c )^2} })^2 \\ & \quad + \tau_c n_c a_c^2 \frac { - n_c \tau_c }{ \sqrt{1 - (n_s)^2 ( \tau_c )^2} } \frac { \tau_s} {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } \\ & \quad - \tau_c n_s a_s a_c \frac { \tau_s} {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } \\ & \quad + a_s^2 \tau_s \\ \ \\ &= \tau_s a_c^2 \frac { (n_c)^2 (\tau_c)^2 }{ 1 - (n_s)^2 ( \tau_c )^2 } \\ & \quad + \tau_c n_c a_c^2 \frac { - n_c \tau_s \tau_c }{ 1 - (n_s)^2 ( \tau_c )^2 } \\ & \quad - \tau_c n_s a_s a_c \frac { \tau_s} {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } + a_s^2 \tau_s \\ \ \\ &= \frac { \tau_s (a_c)^2 (n_c)^2 (\tau_c)^2 - \tau_c n_c a_c^2 n_c \tau_s \tau_c }{ 1 - (n_s)^2 ( \tau_c )^2 } \\ & \quad - \frac { \tau_c n_s a_s a_c \tau_s} {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } + a_s^2 \tau_s \\ \ \\ &= \frac {(n_c)^2 (a_c)^2 \tau_s (\tau_c)^2 - (n_c)^2 (a_c)^2 \tau_s (\tau_c)^2 }{ 1 - (n_s)^2 ( \tau_c )^2 } \\ & \quad - \frac {n_s a_s a_c \tau_s \tau_c } {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } + a_s^2 \tau_s \\ \ \\ &= - \frac { n_s a_s a_c \tau_s \tau_c } {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } + a_s^2 \tau_s \\ \ \\ \ \\ \end{align} \] \[ \begin{align} d_y &= - \tau_c n_s^2 a_c^2 ( \frac { - n_c \tau_c }{ \sqrt{1 - (n_s)^2 ( \tau_c )^2} } )^2 \\ & \quad - \tau_c a_c^2 ( \frac { \tau_s} {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } )^2 \\ & \quad - \tau_s n_c a_c^2 \frac { - n_c \tau_c }{ \sqrt{1 - (n_s)^2 ( \tau_c )^2} } \frac { \tau_s} {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } \\ & \quad - 2 \tau_c n_s n_c a_s a_c \frac { - n_c \tau_c }{ \sqrt{1 - (n_s)^2 ( \tau_c )^2} } \\ & \quad + \tau_s n_s a_s a_c \frac { \tau_s} {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } \\ & \quad - \tau_c n_c^2 a_s^2 \\ \ \\ &= - \tau_c n_s^2 a_c^2 \frac { (n_c)^2 (\tau_c)^2 }{ 1 - (n_s)^2 ( \tau_c )^2 } \\ & \quad - \tau_c a_c^2 \frac { (\tau_s)^2 } {1 - (n_s)^2 ( \tau_c )^2 } \\ & \quad - \tau_s n_c a_c^2 \frac { - n_c \tau_s \tau_c }{ 1 - (n_s)^2 ( \tau_c )^2 } \\ & \quad + \frac { 2 \tau_c n_s n_c a_s a_c n_c \tau_c + \tau_s n_s a_s a_c \tau_s} {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } \\ & \quad - n_c^2 a_s^2 \tau_c \\ \ \\ &= \frac { - \tau_c n_s^2 a_c^2 (n_c)^2 (\tau_c)^2 - \tau_c (a_c)^2 (\tau_s)^2 + \tau_s n_c (a_c)^2 n_c \tau_s \tau_c } { 1 - (n_s)^2 ( \tau_c )^2 } \\ & \quad + \frac { 2 n_s (n_c)^2 a_s a_c (\tau_c)^2 + n_s a_s a_c (\tau_s)^2 } {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } \\ & \quad - (n_c)^2 (a_s)^2 \tau_c \\ \ \\ \ \\ & - \tau_c n_s^2 a_c^2 (n_c)^2 (\tau_c)^2 - \tau_c (a_c)^2 (\tau_s)^2 + \tau_s n_c (a_c)^2 n_c \tau_s \tau_c \\ \ \\ &= - (n_s)^2 (n_c)^2 a_c^2 \tau_c (\tau_c)^2 - (a_c)^2 (\tau_s)^2 \tau_c + (n_c)^2 (a_c)^2 (\tau_s)^2 \tau_c \\ \ \\ &= ( - (n_s)^2 (n_c)^2 (\tau_c)^2 - (\tau_s)^2 + (n_c)^2 (\tau_s)^2 ) (a_c)^2 \tau_c \\ \ \\ &= ( - (n_s)^2 (n_c)^2 (\tau_c)^2 - (n_s)^2 (\tau_s)^2 ) (a_c)^2 \tau_c \\ \ \\ &= ( - (n_c)^2 (\tau_c)^2 - (\tau_s)^2 ) (n_s)^2 (a_c)^2 \tau_c \\ \ \\ &= ( - (n_c)^2 (\tau_c)^2 - ( 1 - (\tau_c)^2 ) ) (n_s)^2 (a_c)^2 \tau_c \\ \ \\ &= ( (\tau_c)^2 - (n_c)^2 (\tau_c)^2 - 1 ) ) (n_s)^2 (a_c)^2 \tau_c \\ \ \\ &= - ( 1 - (n_s)^2 (\tau_c)^2 ) (a_c)^2 (n_s)^2 \tau_c \\ \ \\ \ \\ & 2 n_s (n_c)^2 a_s a_c (\tau_c)^2 + n_s a_s a_c (\tau_s)^2 \\ \ \\ &= ( 2 (n_c)^2 (\tau_c)^2 + (\tau_s)^2 ) n_s a_s a_c \\ \ \\ &= ( (n_c)^2 (\tau_c)^2 + (n_c)^2 (\tau_c)^2 + ( 1 - (\tau_c)^2 ) ) n_s a_s a_c \\ \ \\ &= ( (n_c)^2 (\tau_c)^2 + 1 - (n_s)^2 (\tau_c)^2 ) ) n_s a_s a_c \\ \ \\ \ \\ d_y &= \frac { - ( 1 - (n_s)^2 (\tau_c)^2 ) (n_s)^2 (a_c)^2 \tau_c } { 1 - (n_s)^2 ( \tau_c )^2 } \\ & \quad + \frac {( (n_c)^2 (\tau_c)^2 + 1 - (n_s)^2 (\tau_c)^2 ) ) n_s a_s a_c } {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } \\ & \quad - (n_c)^2 (a_s)^2 \tau_c \\ \ \\ &= - ( (n_s)^2 (a_c)^2 + (n_c)^2 (a_s)^2 ) \tau_c \\ & \quad + \frac {( (n_c)^2 (\tau_c)^2 + ( 1 - (n_s)^2 (\tau_c)^2 ) ) n_s a_s a_c } {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } \\ \ \\ \end{align} \] \[ \begin{align} 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 a_s a_c - \tau_c (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 \\ \ \\ &= \tau_c n_s n_c (a_c)^2 ( \frac { - n_c \tau_c }{ \sqrt{1 - (n_s)^2 ( \tau_c )^2} } )^2 \\ & \quad - \tau_s n_s (a_c)^2 \frac { - n_c \tau_c }{ \sqrt{1 - (n_s)^2 ( \tau_c )^2} } \frac { \tau_s} {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } \\ & \quad + ( \tau_c (n_c)^2 a_s a_c - \tau_c (n_s)^2 a_s a_c ) \frac { - n_c \tau_c }{ \sqrt{1 - (n_s)^2 ( \tau_c )^2} } \\ & \quad - \tau_s n_c a_s a_c \frac { \tau_s} {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } \\ & \quad - n_s n_c (a_s)^2 \tau_c \\ \ \\ &= ( \frac { \tau_c n_s n_c (a_c)^2 (n_c)^2 (\tau_c)^2 + \tau_s n_s (a_c)^2 n_c \tau_c \tau_s } {1 - (n_s)^2 ( \tau_c )^2} \\ & \quad + \frac { - ( \tau_c (n_c)^2 a_s a_c - \tau_c (n_s)^2 a_s a_c ) n_c \tau_c - \tau_s n_c a_s a_c \tau_s } { \sqrt{1 - (n_s)^2 ( \tau_c )^2} } \\ & \quad - n_s n_c (a_s)^2 \tau_c \\ \ \\ \ \\ & \tau_c n_s n_c (a_c)^2 (n_c)^2 (\tau_c)^2 + \tau_s n_s (a_c)^2 n_c \tau_c \tau_s \\ \ \\ &= n_s n_c (n_c)^2 (a_c)^2 (\tau_c)^2 \tau_c + n_s n_c (a_c)^2 (\tau_s)^2 \tau_c \\ \ \\ &= ( (n_c)^2 (\tau_c)^2 + (\tau_s)^2 ) n_s n_c (a_c)^2 \tau_c \\ \ \\ &= ( (n_c)^2 (\tau_c)^2 + 1 - (\tau_c)^2 ) n_s n_c (a_c)^2 \tau_c \\ \ \\ &= ( 1 - (n_s)^2 (\tau_c)^2 ) n_s n_c (a_c)^2 \tau_c \\ \ \\ \ \\ & - ( \tau_c (n_c)^2 a_s a_c - \tau_c (n_s)^2 a_s a_c ) n_c \tau_c - \tau_s n_c a_s a_c \tau_s \\ \ \\ &= - \tau_c (n_c)^2 a_s a_c n_c \tau_c + \tau_c (n_s)^2 a_s a_c n_c \tau_c - \tau_s n_c a_s a_c \tau_s \\ \ \\ &= - (n_c)^2 n_c a_s a_c (\tau_c)^2 + (n_s)^2 n_c a_s a_c (\tau_c)^2 - n_c a_s a_c (\tau_s)^2 \\ \ \\ &= ( - (n_c)^2 (\tau_c)^2 + (n_s)^2 (\tau_c)^2 - (\tau_s)^2 ) n_c a_s a_c \\ \ \\ &= ( - (n_c)^2 (\tau_c)^2 + (n_s)^2 (\tau_c)^2 - 1 + (\tau_c)^2 ) n_c a_s a_c \\ \ \\ &= ( (n_s)^2 (\tau_c)^2 + (n_s)^2 (\tau_c)^2 - 1 ) n_c a_s a_c \\ \ \\ &= ( (n_s)^2 (\tau_c)^2 - (1 - (n_s)^2 (\tau_c)^2 ) ) n_c a_s a_c \\ \ \\ \ \\ d_z &= ( \frac { ( 1 - (n_s)^2 (\tau_c)^2 ) n_s n_c (a_c)^2 \tau_c } {1 - (n_s)^2 ( \tau_c )^2} \\ & \quad + \frac { ( (n_s)^2 (\tau_c)^2 - (1 - (n_s)^2 (\tau_c)^2 ) ) n_c a_s a_c } { \sqrt{1 - (n_s)^2 ( \tau_c )^2} } \\ & \quad - n_s n_c (a_s)^2 \tau_c \\ \ \\ &= n_s n_c (a_c)^2 \tau_c - n_s n_c (a_s)^2 \tau_c \\ & \quad + \frac { ( (n_s)^2 (\tau_c)^2 - (1 - (n_s)^2 (\tau_c)^2 ) ) n_c a_s a_c } { \sqrt{1 - (n_s)^2 ( \tau_c )^2} } \\ \ \\ &= n_s n_c ( (a_c)^2 - (a_s)^2 ) \tau_c \\ & \quad + \frac { ( (n_s)^2 (\tau_c)^2 - (1 - (n_s)^2 (\tau_c)^2 ) ) n_c a_s a_c } { \sqrt{1 - (n_s)^2 ( \tau_c )^2} } \end{align} \\ \ \\ \] \[ \begin{align} | {\bf d} |^2 &= 1 - ( a_c ( n_c \tau_c b_s - \tau_s b_c ) - n_s a_s \tau_c )^2 \\ \ \\ &= 1 - ( a_c ( n_c \tau_c \frac { - n_c \tau_c }{ \sqrt{1 - (n_s)^2 ( \tau_c )^2} } - \tau_s \frac { \tau_s} {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } ) - n_s a_s \tau_c )^2 \\ \ \\ &= 1 - ( - a_c \frac { (n_c)^2 (\tau_c)^2 + (\tau_s)^2 }{ \sqrt{1 - (n_s)^2 ( \tau_c )^2} } - n_s a_s \tau_c )^2 \\ \ \\ &= 1 - ( - a_c \frac { (n_c)^2 (\tau_c)^2 + 1 - (\tau_c)^2 }{ \sqrt{1 - (n_s)^2 ( \tau_c )^2} } - n_s a_s \tau_c )^2 \\ \ \\ &= 1 - ( - a_c \frac { 1 - (n_s)^2 (\tau_c)^2 }{ \sqrt{1 - (n_s)^2 ( \tau_c )^2} } - n_s a_s \tau_c )^2 \\ \ \\ &= 1 - ( - a_c \sqrt{1 - (n_s)^2 ( \tau_c )^2} - n_s a_s \tau_c )^2 \\ \ \\ &= 1 - ( (a_c)^2 (1 - (n_s)^2 ( \tau_c )^2) + 2 a_c \sqrt{1 - (n_s)^2 ( \tau_c )^2} n_s a_s \tau_c + (n_s)^2 (a_s)^2 (\tau_c)^2 ) \\ \ \\ &= 1 - (a_c)^2 + (n_s)^2 (a_c)^2 ( \tau_c )^2 - (n_s)^2 (a_s)^2 (\tau_c)^2 - 2 n_s a_s a_c \tau_c \sqrt{ 1 - (n_s)^2 ( \tau_c )^2} \\ \ \\ &= (a_s)^2 + (n_s)^2 (a_c)^2 ( \tau_c )^2 - (n_s)^2 (a_s)^2 (\tau_c)^2 - 2 n_s a_s a_c \tau_c \sqrt{ 1 - (n_s)^2 ( \tau_c )^2} \\ \ \\ &= (a_s)^2 + ( (a_c)^2 - (a_s)^2 ) (n_s)^2 (\tau_c)^2 - 2 n_s a_s a_c \tau_c \sqrt{ 1 - (n_s)^2 ( \tau_c )^2} \\ \ \\ \end{align} \] まとめ 太陽方角ベクトル \( ( d_x, d_y, d_y ) \) ： \[ \begin{align} d_x &= - \frac { n_s a_s a_c \tau_s \tau_c } {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } + a_s^2 \tau_s \\ \ \\ d_y &= - ( (n_s)^2 (a_c)^2 + (n_c)^2 (a_s)^2 ) \tau_c \\ & \quad + \frac {( (n_c)^2 (\tau_c)^2 + ( 1 - (n_s)^2 (\tau_c)^2 ) ) n_s a_s a_c } {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } \\ \ \\ d_z &= n_s n_c ( (a_c)^2 - (a_s)^2 ) \tau_c \\ & \quad + \frac { ( (n_s)^2 (\tau_c)^2 - (1 - (n_s)^2 (\tau_c)^2 ) ) n_c a_s a_c } { \sqrt{1 - (n_s)^2 ( \tau_c )^2} } \\ \ \\ | {\bf d} |^2 &= (a_s)^2 + ( (a_c)^2 - (a_s)^2 ) (n_s)^2 (\tau_c)^2 - 2 n_s a_s a_c \tau_c \sqrt{ 1 - (n_s)^2 ( \tau_c )^2} \end{align} \]