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南中の直交座標計算

作成: 2020-09-28 更新: 2020-09-28

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問題 公転角がτのときの，緯度ａの南中の座標 (x, y, z) は？ 簡単のため \[ n_s = sin(n), \ \ n_c = cos(n) \\ a_s = sin(a), \ \ a_c = cos(a) \\ \tau_s = sin(\tau), \ \ \tau_c = cos(\tau) \] とおく。 さらにこの簡略記法 \[ \theta_s = sin(\theta) \\ \theta_c = cos(\theta) \] を，一般の \( \theta \) に対しても用いるとする。 南中の位置： 南中の位置ベクトルを ｘ とする。 ベクトルt, n をつぎのようにとる： このとき， \[ {\bf t} = \frac{ {\bf n} \times {\bf s} }{ |{\bf n} \times {\bf s}| } \\ \ \\ \begin{align} {\bf n} \times {\bf s} &= ( 0, - n_s, n_c ) \times ( \tau_s, - \tau_c, 0 ) \\ &= ( (- n_s) 0 - n_c (- \tau_c), n_c \tau_s - 0 0, 0 (- \tau_c) - (- n_s) \tau_s ) \\ &= ( n_c \tau_c, \ n_c \tau_s,\ n_s \tau_s ) \\ % \\ \ \\ % |{\bf n} \times {\bf s}| % &= \sqrt { (n_c \tau_c)^2 + (n_c \tau_s)^2 + (n_s \tau_s)^2 } \\ % &= \sqrt { (n_c)^2 + (n_s)^2 ( 1 - (\tau_c)^2 ) } \\ % &= \sqrt { 1 - (n_s)^2 (\tau_c)^2 ) } \end{align} \] さらに， \[ \begin{align} 0 &= {\bf x} \cdot {\bf t} \\ &= ( x, y, z ) \cdot (n_c \tau_c, \ n_c \tau_s, \ n_s \tau_s ) \\ &= (n_c \tau_c) x + (n_c \tau_s) y + (n_s \tau_s) z \quad \cdots \ (1)\\ \end{align} \\ \ \\ \] 緯度ａの点の座標 (x, y, z) では，つぎが必要条件になる ( 自転軸系経度緯度と公転軸系直交座標の変換式)： \[ (n_s) y + (- n_c) z + a_s = 0 \quad \cdots \ (2)\\ \] \[ (1) \times n_c + (2) \times (n_s \tau_s) \ : \\ 0 = ((n_c)^2 \tau_c x + (n_c)^2 \tau_s y + n_s n_c\tau_s z\\ \quad \quad + (n_s)^2 \tau_s y + (- n_c) n_s \tau_s z + n_s a_s \tau_s \\ \quad = (n_c)^2 \tau_c x + \tau_s y + n_s a_s \tau_s \\ \\ \ \\ \Longrightarrow \ \ y = \frac{ - (n_c)^2 \tau_c x - n_s a_s \tau_s }{ \tau_s } \\ \ \\ \ \\ (1) \times n_s - (2) \times (n_c \tau_s) \ : \\ 0 = n_s n_c \tau_c x + n_s n_c \tau_s y + (n_s)^2 \tau_s z \\ \quad \quad - ( n_s n_c \tau_s y - (n_c)^2 \tau_s z +n_c a_s \tau_s ) \\ \quad = n_s n_c \tau_c x + \tau_s z - n_c a_s \tau_s \\ \ \\ \Longrightarrow \ \ z = \frac{ - n_s n_c \tau_c x + n_c a_s \tau_s }{ \tau_s } \\ \\ \ \\ \] よって， \[ \begin{align} & x^2 + y^2 + z^2 = 1 \\ \Longrightarrow \ \ & x^2 + (\frac{ - (n_c)^2 \tau_c x - n_s a_s \tau_s }{ \tau_s })^2 + (\frac{ - n_s n_c \tau_c x + n_c a_s \tau_s }{ \tau_s })^2 = 1 \\ \ \\ \Longrightarrow \ \ & (\tau_s)^2 x^2 \\ & + ( (n_c)^2 \tau_c x )^2 + 2 ((n_c)^2 \tau_c x ) (n_s a_s \tau_s ) + ( n_s a_s \tau_s )^2 \\ & + ( n_s n_c \tau_c x )^2 - 2 ( n_s n_c \tau_c x ) ( n_c a_s \tau_s ) + ( n_c a_s \tau_s )^2 \\ & = (\tau_s)^2 \\ \ \\ \Longrightarrow \ \ & (\tau_s)^2 x^2 \\ & + (n_c)^2 (n_c)^2 ( \tau_c )^2 x^2 + 2 n_s (n_c)^2 a_s \tau_s \tau_c x + (n_s)^2 (a_s)^2 (\tau_s )^2 \\ & + (n_s)^2 (n_c)^2 (\tau_c)^2 x^2 - 2 n_s (n_c)^2 a_s \tau_s \tau_c x + (n_c)^2 (a_s)^2 (\tau_s)^2 \\ & = (\tau_s)^2 \\ \ \\ \Longrightarrow \ \ & ( (\tau_s)^2 + (n_c)^2 (n_c)^2 ( \tau_c )^2 + (n_s)^2 (n_c)^2 (\tau_c)^2 ) x^2 \\ & + 2 ( n_s (n_c)^2 a_s \tau_s \tau_c - n_s (n_c)^2 a_s \tau_s \tau_c ) x \\ & + (n_s)^2 (a_s)^2 (\tau_s )^2 + (n_c)^2 (a_s)^2 (\tau_s)^2 - (\tau_s)^2 = 0 \\ \ \\ \Longrightarrow \ \ & ( (\tau_s)^2 + (n_c)^2 ( \tau_c )^2 ) x^2 \\ & + (a_s)^2 (\tau_s )^2 - (\tau_s)^2 = 0 \\ \ \\ \Longrightarrow \ \ & ( 1 - (\tau_c)^2 + (n_c)^2 ( \tau_c )^2 ) x^2 \\ & - ( 1 - (a_s)^2 ) (\tau_s )^2 = 0 \\ \ \\ \Longrightarrow \ \ & ( 1 - (n_s)^2 ( \tau_c )^2 ) x^2 \\ & - (a_c)^2 (\tau_s )^2 = 0 \\ \ \\ \Longrightarrow \ \ & x = \pm \frac { a_c \tau_s} {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } \\ \ \\ \end{align} \] さらに \[ \begin{align} y &= - \frac{ (n_c)^2 \tau_c }{ \tau_s } x - n_s a_s \\ &= \mp \frac{ (n_c)^2 \tau_c }{ \tau_s } \frac { a_c \tau_s} {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } - n_s a_s \\ &= \mp \frac{ (n_c)^2 a_c \tau_c } {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } - n_s a_s \\ \ \\ z &= - \frac{ n_s n_c \tau_c }{ \tau_s } x + n_c a_s \\ &= \mp \frac{ n_s n_c \tau_c }{ \tau_s } \frac { a_c \tau_s} { \sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } + n_c a_s \\ &= \mp \frac{ n_s n_c a_c \tau_c } { \sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } + n_c a_s \\ \\ \ \\ \end{align} \] まとめ 南中 \[ x = \frac { a_c \tau_s} {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } \\ y = - \frac{ (n_c)^2 a_c \tau_c } {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } - n_s a_s \\ z = - \frac{ n_s n_c a_c \tau_c } { \sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } + n_c a_s \\ \] 北中 \[ x = - \frac { a_c \tau_s} {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } \\ y = + \frac{ (n_c)^2 a_c \tau_c } {\sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } - n_s a_s \\ z = + \frac{ n_s n_c a_c \tau_c } { \sqrt{ 1 - (n_s)^2 ( \tau_c )^2 } } + n_c a_s \\ \]