参考資料
　１.日の出・日の入りの計算: 天体の出没時刻の求め方　長沢工（地人書館）
　２.NASA Eclips - POLYNOMIAL EXPRESSIONS FOR DELTA T (ΔT)


年　：　$year$
月　：　$month$
日　：　$day$
時　：　$hour$
分　：　$min$
秒　：　$sec$
経度：　$\lambda$（東経を正にとる）
緯度：　$\varphi$（北緯を正にとる）
標高：　$H　[m]$

$month \leqq 2$　のとき、
　$Y = year - 2001$
　$M = month + 12$
　$D = day$

$3 \leqq month \leqq 12$　のとき、
　$Y = year - 2000$
　$M = month$
　$D = day$


J2000.0（2000年1月1日力学時正午）からの経過日数の計算要素

$K' = 365 Y +30 M + D - 33.875 + \left[ \dfrac{3(M+1)}{5} \right] + \left[ \dfrac{Y}{4} \right]$

なお、$\left[ \dfrac{3(M+1)}{5} \right]$ と $\left[ \dfrac{Y}{4} \right]$ はガウス記号。

例えば、$\left[1.618\right]=1$、 $\left[3.1415\right]=3$、 $\left[-0.7\right]=-1$、 $\left[-13.2\right]=-14$


経過時間の日単位変換補整
$d = \biggl( hour + \dfrac{min}{60} + \dfrac{sec}{3600} \biggr) \times \dfrac{1}{24} $


地球の自転遅れ補正値　$DT$

計算要素
$y' = year + \bigl( month - 0.5 \bigr) \times \dfrac{1}{12} $


$y' \leqq -500$　のとき、
$DT_0 = -20 + 32 \biggl( \dfrac{y' - 1820}{100} \biggr)^2$

$-500 \leqq y' < 500$　のとき、
$\begin{equation} \begin{split} DT_0 = {} & 10583.6 - 1014.41 \biggl( \dfrac{y'}{100} \biggr) + 33.78311 \biggl( \dfrac{y'}{100} \biggr)^2 - 5.952053 \biggl( \dfrac{y'}{100} \biggr)^3 \\ & - 0.1798452 \biggl( \dfrac{y'}{100} \biggr)^4 + 0.022174192 \biggl( \dfrac{y'}{100} \biggr)^5 + 0.0090316521 \biggl( \dfrac{y'}{100} \biggr)^6 \end{split} \end{equation}$

$500 \leqq y' < 1600$　のとき、
$\begin{equation} \begin{split} DT_0 = {} & 1574.2 - 556.01 \biggl( \dfrac{y'-1000}{100} \biggr) + 71.23472 \biggl( \dfrac{y'-1000}{100} \biggr)^2 \\ & + 0.319781 \biggl( \dfrac{y'-1000}{100} \biggr)^3 - 0.8503463 \biggl( \dfrac{y'-1000}{100} \biggr)^4 \\ & - 0.005050998 \biggl( \dfrac{y'-1000}{100} \biggr)^5 + 0.0083572073 \biggl( \dfrac{y'-1000}{100} \biggr)^6 \end{split} \end{equation}$

$1600 \leqq y' < 1700$　のとき、
$DT_0 = 120 - 0.9808 \left( y' - 1600 \right) - 0.01532 \left( y' - 1600 \right)^2 + \dfrac{ \left( y' - 1600 \right)^3}{7129}$

$1700 \leqq y' < 1800$　のとき、
$\begin{equation} \begin{split} DT_0 = {} & 8.83 + 0.1603 \left( y' - 1700 \right) - 0.0059285 \left( y' - 1700 \right)^2 \\ & + 0.00013336 \left( y' - 1700 \right)^3 - \dfrac{ \left( y' - 1700 \right)^4}{1174000} \end{split} \end{equation}$

$1800 \leqq y' < 1860$　のとき、
$\begin{equation} \begin{split} DT_0 = {} & 13.72 - 0.332447 \left( y' - 1800 \right) + 0.0068612 \left( y' - 1800 \right)^2 \\ & + 0.0041116 \left( y' - 1800 \right)^3 - 0.00037436 \left( y' - 1800 \right)^4 \\ & + 0.0000121272 \left( y' - 1800 \right)^5 - 0.0000001699 \left( y' - 1800 \right)^6 \\ & + 0.000000000875 \left( y' - 1800 \right)^7 \end{split} \end{equation}$

$1860 \leqq y' < 1900$　のとき、
$\begin{equation} \begin{split} DT_0 = {} & 7.62 + 0.5737 \left( y' - 1860 \right) - 0.251754 \left( y' - 1860 \right)^2 \\ & + 0.01680668 \left( y' - 1860 \right)^3 - 0.0004473624 \left( y' - 1860 \right)^4 + \dfrac{\left( y' - 1860 \right)^5}{233174} \end{split} \end{equation}$

$1900 \leqq y' < 1920$　のとき、
$\begin{equation} \begin{split} DT_0 = {} & -2.79 + 1.494119 \left( y' - 1900 \right) - 0.0598939 \left( y' - 1900 \right)^2 \\ & + 0.0061966 \left( y' - 1900 \right)^3 - 0.000197 \left( y' - 1900 \right)^4 \end{split} \end{equation}$

$1920 \leqq y' < 1941$　のとき、
$DT_0 = 21.2 + 0.84493 \left( y' - 1920 \right) - 0.0761 \left( y' - 1920 \right)^2 + 0.0020936 \left( y' - 1920 \right)^3$

$1941 \leqq y' < 1961$　のとき、
$DT_0 = 29.07 + 0.407 \left( y' - 1950 \right) - \dfrac{\left( y' - 1950 \right)^2}{233} + \dfrac{\left( y' - 1950 \right)^3}{2547}$

$1961 \leqq y' < 1986$　のとき、
$DT_0 = 45.45 + 1.067 \left( y' - 1975 \right) - \dfrac{\left( y' - 1975 \right)^2}{260} - \dfrac{\left( y' - 1975 \right)^3}{718}$

$1986 \leqq y' < 2005$　のとき、
$\begin{equation} \begin{split} DT_0 = {} & 63.86 + 0.3345 \left( y' - 2000 \right) - 0.060374 \left( y' - 2000 \right)^2 \\ & + 0.0017275 \left( y' - 2000 \right)^3 + 0.000651814 \left( y' - 2000 \right)^4 + 0.00002373599 \left( y' - 2000 \right)^5 \end{split} \end{equation}$

$2005 \leqq y' < 2050$　のとき、
$DT_0 = 62.92 + 0.32217 \left( y' - 2000 \right) + 0.005589 \left( y' - 2000 \right)^2$

$2050 \leqq y' < 2150$　のとき、
$DT_0 = -20 + 32 \left( \dfrac{ y' - 1820 }{100} \right)^2 - 0.5628 \left( 2150 - y' \right)$

$2150 \leqq y'$　のとき、
$DT_0 = -20 + 32 \left( \dfrac{ y' - 1820 }{100} \right)^2$



$1955 \leqq y' \leqq 2005$のとき、

$DT = DT_0$

$y' < 1955$ または $2005 < y'$のとき、
$DT = DT_0 - 0.000012935 \left( y' - 1955 \right)^2$



J2000.0(2000年1月1日力学時正午からの経過日数)：　$K$

$K = K' + d + \dfrac{DT}{86400}$


時刻変数：　$T$

$T = \dfrac{K}{365.25}$


黄道傾角：　$e$

$e = 23.439291 - 0.000130042 T$


太陽の視黄経：　$\lambda_{S2}$

$\begin{equation} \begin{split} \lambda_S = {} & 280.4603 + 360.00769 T \\ & + (1.9146 - 0.00005 T) \sin \left( (357.538 + 359.991 T) \times \dfrac{\pi}{180} \right) \\ & + 0.02 \sin \left( (355.05 + 719.981 T) \times \dfrac{\pi}{180} \right) \\ & + 0.0048 \sin \left( (234.95 + 19.341 T) \times \dfrac{\pi}{180} \right) \\ & + 0.002 \sin \left( (247.1 + 329.64 T) \times \dfrac{\pi}{180} \right) \\ & + 0.0018 \sin \left( (297.8 + 4452.67 T) \times \dfrac{\pi}{180} \right) \\ & + 0.0018 \sin \left( (251.3 + 0.2 T) \times \dfrac{\pi}{180} \right) \\ & + 0.0015 \sin \left( (343.2 + 450.37 T) \times \dfrac{\pi}{180} \right) \\ & + 0.0013 \sin \left( (81.4 + 225.18 T) \times \dfrac{\pi}{180} \right) \\ & + 0.0008 \sin \left( (132.5 + 659.29 T) \times \dfrac{\pi}{180} \right) \\ & + 0.0007 \sin \left( (153.3 + 90.38 T) \times \dfrac{\pi}{180} \right) \\ & + 0.0007 \sin \left( (206.8 + 30.35 T) \times \dfrac{\pi}{180} \right) \\ & + 0.0006 \sin \left( (29.8 + 337.18 T) \times \dfrac{\pi}{180} \right) \\ & + 0.0005 \sin \left( (207.4 + 1.5 T) \times \dfrac{\pi}{180} \right) \\ & + 0.0005 \sin \left( (291.2 + 22.81 T) \times \dfrac{\pi}{180} \right) \\ & + 0.0004 \sin \left( (234.9 + 315.56 T) \times \dfrac{\pi}{180} \right) \\ & + 0.0004 \sin \left( (157.3 + 299.3 T) \times \dfrac{\pi}{180} \right) \\ & + 0.0004 \sin \left( (21.1 + 720.02 T) \times \dfrac{\pi}{180} \right) \\ & + 0.0003 \sin \left( (352.5 + 1079.97 T) \times \dfrac{\pi}{180} \right) \\ & + 0.0003 \sin \left( (329.7 + 44.43 T) \times \dfrac{\pi}{180} \right) \end{split} \end{equation}$

$0 \leqq \lambda_S$　のとき、
　$ \lambda_{S2} = \left( \lambda_S \quad mod \quad 360 \right)$

$\lambda_S < 0$　のとき、
　$ \lambda_{S2} = \left( \lambda_S \quad mod \quad 360 \right) + 360$

ただし、　$mod$　は余りを表す記号。
例えば、　$(5 \quad mod \quad 2) = 1$、　$(17 \quad mod \quad 3) = 2$


恒星時： $\theta_2$
$\theta = 325.4606 + 360.007700536 T + 0.00000003879 T ^ 2 + 360 d + \lambda$

$0 \leqq \theta$　のとき、
$\theta_2 = \left( \theta \quad mod \quad 360 \right)$

$\theta < 0$　のとき、
$\theta_2 = \left( \theta \quad mod \quad 360 \right) + 360$


太陽の赤経　$\alpha_2$
$\alpha = \arctan \left( \tan \left( \lambda_{S2} \dfrac{\pi}{180} \right) \times \cos \left( e \dfrac{\pi}{180} \right) \right) \times \dfrac{180}{\pi}$

$0 \leqq \alpha$　のとき、
$\alpha_1 = \left( \alpha \quad mod \quad 180 \right)$

$ \alpha < 0$　のとき、
$\alpha_1 = \left( \alpha \quad mod \quad 180 \right) + 180$

$0 \leqq \lambda_{s2} < 180$　のとき、
$\alpha_2 = \alpha_1$

$180 \leqq \lambda_{s2} < 360$　のとき、
$\alpha_2 = \alpha_1 + 180$



赤緯：　$\delta$
$\delta = \arcsin \left( \sin \left( \lambda_{S2} \dfrac{\pi}{180} \right) \sin \left( e \dfrac{\pi}{180} \right) \right) \times \dfrac{180}{\pi}$

時角：　$t$
$t = \theta_2 - \alpha_2$

高度：　$h$
$h = \arcsin \left( \sin \left( \delta \dfrac{\pi}{180} \right) \sin \left( \varphi \dfrac{\pi}{180} \right) + \cos \left( \delta \dfrac{\pi}{180} \right) \cos \left(\varphi \dfrac{\pi}{180} \right) \cos \left( t \dfrac{\pi}{180} \right) \right) \times \dfrac{180}{\pi}$

大気差：　$R$
$R = 0.0167 \div \left( \tan \left( \dfrac{h + 8.6}{h + 4.4} \times \dfrac{\pi}{180} \right) \right)$


${\bf SunAngle} = h + R$




$0 \leqq \left(\sin \left(\delta \dfrac{\pi}{180} \right) \cos \left(\varphi \dfrac{\pi}{180} \right) - \cos \left(\delta \dfrac{\pi}{180} \right) \sin \left(\varphi \dfrac{\pi}{180} \right) \cos \left(t \dfrac{\pi}{180} \right) \right)$　のとき、

$A = \arctan \left( \left(-\cos \left(\delta \dfrac{\pi}{180} \right) \sin \left(t \dfrac{\pi}{180} \right) \right) \div \left( \sin \left(\delta \dfrac{\pi}{180} \right) \cos \left(\varphi \dfrac{\pi}{180} \right) - \cos \left(\delta \dfrac{\pi}{180} \right) \sin \left(\varphi \dfrac{\pi}{180} \right) \cos \left(t \dfrac{\pi}{180} \right) \right) \right) \times \dfrac{180}{\pi}$


$ \left( \sin \left( \delta \dfrac{\pi}{180} \right) \cos \left(\varphi \dfrac{\pi}{180} \right) - \cos \left(\delta \dfrac{\pi}{180} \right) \sin \left(\varphi \dfrac{\pi}{180} \right) \cos \left(t \dfrac{\pi}{180} \right) \right) < 0$　のとき、

$A = \arctan \left( \left(-\cos \left(\delta \dfrac{\pi}{180} \right) \sin \left(t \dfrac{\pi}{180} \right) \right) \div \left(\sin \left(\delta \dfrac{\pi}{180} \right) \cos \left(\varphi \dfrac{\pi}{180} \right) - \cos \left(\delta \dfrac{\pi}{180} \right) \sin \left(\varphi \dfrac{\pi}{180} \right) \cos \left(t \dfrac{\pi}{180} \right) \right) \right) \times \dfrac{180}{\pi} + 180$


$0 \leqq A$　のとき、
　${\bf Houikaku } = A$

$ A < 0 $　のとき、
　${\bf Houikaku }= A+360$



また、太陽出没高度の算出のため、

太陽距離　$r$
$\begin{equation} \begin{split} \log_{10}r = {} & \left( 0.007256 - 0.0000002 T \right) \sin \left( \left(267.54 + 359.991 T \right) \times \dfrac{\pi}{180} \right) \\ & + 0.000091 \sin \left( \left( 265.1 + 719.98 T \right) \times \dfrac{\pi}{180} \right) \\ & + 0.000030 \\ & + 0.000013 \sin \left( \left( 27.8 + 4452.67 T \right) \times \dfrac{\pi}{180} \right) \\ & + 0.000007 \sin \left( \left( 254 + 450.4 T \right) \times \dfrac{\pi}{180} \right) \\ & + 0.000007 \sin \left( \left( 156 + 329.6 T \right) \times \dfrac{\pi}{180} \right) \\ \end{split} \end{equation}$


視半径　$S = \left( \dfrac{16}{60} + \dfrac{1.18}{3600} \right) \times \dfrac{1}{r} $


大気差　$R = \dfrac{35}{60} + \dfrac{8}{3600}$


視差　$ \Pi = \dfrac{8.794148}{3600} \times \dfrac{1}{r}$


地平線の伏角　$E = \dfrac{2.12}{60} \sqrt{H}$


出没高度
$ {\bf k = \Pi - S - E - R}$