From c1fbdf65c8b775a49c859351053c9c2e845cbe7b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?=D0=90=D0=BB=D0=B5=D0=BA=D1=81=D0=B5=D0=B9=20=D0=A8=D0=B5?= =?UTF-8?q?=D0=BF=D0=B5=D0=BB=D0=B5=D0=B2?= Date: Sun, 5 May 2024 06:54:42 +0200 Subject: [PATCH] edit EoT and sun movement, add maps of eclipses --- calc/analemma.nb | 42 +++-- calc/eclipses-map.nb | 152 ++++++++++++++++++ img/eclipse-map-1999-08-11.png | Bin 0 -> 110170 bytes img/eclipses-map-20.png | Bin 0 -> 100224 bytes img/m13.pdf | Bin 0 -> 419893 bytes img/m45.pdf | Bin 0 -> 242549 bytes img/m57.pdf | Bin 0 -> 253114 bytes img/optical-aberrations-coma.pdf | Bin 0 -> 154010 bytes img/sky-map.pdf | Bin 0 -> 762365 bytes sections/geometrical-astronomy/eclipses.tex | 16 ++ sections/objects/other.tex | 6 +- sections/optics/aberrations.tex | 2 +- .../spherical-astronomy/change-coordin.tex | 128 ++++++++++----- .../spherical-astronomy/spher-trigonom.tex | 2 + sections/spherical-astronomy/sun-time.tex | 95 ++++------- sys/spherical-astronomy.tex | 2 +- sys/tables.tex | 2 +- 17 files changed, 313 insertions(+), 134 deletions(-) create mode 100644 calc/eclipses-map.nb create mode 100644 img/eclipse-map-1999-08-11.png create mode 100644 img/eclipses-map-20.png create mode 100644 img/m13.pdf create mode 100644 img/m45.pdf create mode 100644 img/m57.pdf create mode 100644 img/optical-aberrations-coma.pdf create mode 100644 img/sky-map.pdf diff --git a/calc/analemma.nb b/calc/analemma.nb index 1baa327..0e23fbf 100644 --- a/calc/analemma.nb +++ b/calc/analemma.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 9329, 259] -NotebookOptionsPosition[ 8484, 238] -NotebookOutlinePosition[ 8876, 254] -CellTagsIndexPosition[ 8833, 251] +NotebookDataLength[ 9504, 257] +NotebookOptionsPosition[ 8658, 236] +NotebookOutlinePosition[ 9050, 252] +CellTagsIndexPosition[ 9007, 249] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ 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zJWNS%zNZXp;izA}B}D5LPq-lSt5>zGzQHZLYFXVkcVqkE=VucIcP32KIm-{8cqMw} z_73zWJSf1hqK`!nP5h-Rdr!w6OPj2X^GnIg*M*YDOec9-hyFjZa5?;#Mcd}Mg`|XC z@drJ4_?q&Or`aWCC4+4jGfRCJs=1j}Z2x>H-U)7Pf` zoR#{tnl`P>GtnU(xN*lutnOLu3lvG0rf>H7%O&44Q!iB!pc@(QN=Xe{5*K(M8TGw6UutDXQTAEhf|lXN|K_6#UkU8qb(H~@8K&{EP6QL4SFB4Y832Gn%L@-!~`mj8p2_Ite~lUVp9NaQ-BMX zLyw?=^4OG3rIFl#Qw(l6ixUdKQ4jAa)@XfJUv=t5wcJO~Nur3b$31)>t{fzOrR8|Os`{Cr`#yZCU_Xa>ua5y|b zGrtqye-j$W0W_o0_;%o=4dATV?Uoq^Z+bqYZF Y@vH-m?L$wo7z7N4)iE-%CpqZ+2jg6f2><{9 literal 0 HcmV?d00001 diff --git a/sections/geometrical-astronomy/eclipses.tex b/sections/geometrical-astronomy/eclipses.tex index 72c72e4..8abeddc 100644 --- a/sections/geometrical-astronomy/eclipses.tex +++ b/sections/geometrical-astronomy/eclipses.tex @@ -464,6 +464,22 @@ \subsection{Затмения} \end{wrapfigure} Найденное выше максимальное угловое расстояние центра ширмы от центра источника, при котором наблюдается частичное затмение, определяет ширину полосы вокруг эклиптики, \lookPicRef{pic:elipses-node-distance}, внутри которой должен оказаться центр Луны, чтобы на Земле была потенциальная возможность наблюдать солнечное затмение. Оказывается, что Луна должна располагаться не дальше, чем $l = \gamma / \sin i \simeq 16^\circ$ от узла\footnote{в случае круговой орбиты Луны}, где $i$~--- угол наклона орбиты Луны к эклиптике. +\begin{figure}[h!] + \begin{subcaptionblock}[t]{0.63\tw} + \includegraphics[width=\tw]{eclipses-map-20.png} + \caption{Карта полных фаз затмений в 20-ом веке} + \label{pic:eclipse-20-century} + \end{subcaptionblock} + \hfill + \begin{subcaptionblock}[t]{0.32\tw} + \includegraphics[width=\tw]{eclipse-map-1999-08-11.png} + \caption{Карта затмения 11 августа 1999 года%\footnote{https://eclipse.gsfc.nasa.gov/SEplot/SEplot1951/SE1999Aug11T.GIF} + } + \label{pic:eclipse-1999-08-11} + \end{subcaptionblock} + \caption{} +\end{figure} + \begin{figure}[h!] \centering \vspace{-.5pc} diff --git a/sections/objects/other.tex b/sections/objects/other.tex index 2280f88..d5029bf 100644 --- a/sections/objects/other.tex +++ b/sections/objects/other.tex @@ -10,13 +10,13 @@ \subsection{Другие объекты} \begin{minipage}{0.32\tw} \centering \vspace{-1.2pc} - \includegraphics[width = .7\tw]{m13} + \includegraphics[width = .7\tw]{m13.pdf} \captionof{figure}{Шаровое скопление M13 (негатив)} \vspace{1pc} - \includegraphics[width = .7\tw]{m57} + \includegraphics[width = .7\tw]{m57.pdf} \captionof{figure}{Пла\-не\-тар\-ная туманность M57 (негатив)} \vspace{1.2pc} - \includegraphics[width = .7\tw]{m45} + \includegraphics[width = .7\tw]{m45.pdf} \captionof{figure}{Рассеянное звёздное скопление M45 (негатив)} \end{minipage} diff --git a/sections/optics/aberrations.tex b/sections/optics/aberrations.tex index 57afd67..626db6b 100644 --- a/sections/optics/aberrations.tex +++ b/sections/optics/aberrations.tex @@ -392,7 +392,7 @@ \subsection{Аберрации в оптике} \begin{wrapfigure}[13]{l}{0.4\tw} \centering \vspace{-1pc} - \includegraphics[width=0.4\tw]{optical-aberrations-coma.jpg} + \includegraphics[width=0.4\tw]{optical-aberrations-coma.pdf} \caption{Изображение <<хвоста>> Большой Медведицы, полученное с помощью широкоугольного объектива, страдающего ярко выраженной комой.} \label{pic:optics-aberrations-coma} \end{wrapfigure} diff --git a/sections/spherical-astronomy/change-coordin.tex b/sections/spherical-astronomy/change-coordin.tex index 3952810..56f57d6 100644 --- a/sections/spherical-astronomy/change-coordin.tex +++ b/sections/spherical-astronomy/change-coordin.tex @@ -37,52 +37,104 @@ \subsection{Годичное движение Солнца} \caption{Аналемма} \label{fig:analemma} \end{wrapfigure} -В течение сидерического года Земля совершает полный оборот вокруг Солнца. Вследствие этого Солнце движется относительно далёких звёзд для наблюдателя на Земле. Это движение совершается по большому кругу небесной сферы, называемому \term{эклиптикой} и совпадающему с плоскостью орбиты Земли. Однако, в силу прецессии земной оси с периодом около 25765~лет, период такого движения равен \imp{тропическому году}, который длиннее сидерического года примерно на 20~мин~25~сек. +В течение сидерического года Земля совершает полный оборот вокруг Солнца. Вследствие этого Солнце движется относительно далёких звёзд для наблюдателя на Земле. Это движение совершается по большому кругу небесной сферы~--- \term{эклиптике}~--- проекции орбиты Земли на небесную сферу. -\begin{wrapfigure}[12]{r}{0.5\tw} +Однако, в силу прецессии земной оси, период такого движения равен \term{тропическому году}, который короче сидерического года примерно на 20~мин~25~сек. + +%\begin{wrapfigure}[12]{r}{0.5\tw} +% \centering +% \vspace{-.9pc} +% \tikzsetnextfilename{sun-path} +% \begin{tikzpicture} +% \begin{axis}[ +% width = .5\tw, +% height = 4.5cm, +% xlabel = {Прямое восхождение $\alpha^h$}, +% ylabel = {Склонение $\delta^{\circ}$}, +% extra y ticks = {23.44, -23.44}, +% ytick = {-20, -10, 0, 10, 20}, +% ymax = 25, +% ymin = -25, +% xmax = 24, +% xmin = 0, +% xtick = {0, 4, 8, 12, 16, 20, 24}, +% x dir = reverse +% ] +% \addplot [domain=0:24, samples=100] {atan(sin(x*15)*tan(23.44))}; +% \end{axis} +% \end{tikzpicture} +% \caption{График зависимости склонения Солнца от его прямого восхождения} +%\end{wrapfigure} +В моменты, когда Солнце находится в \imp{точке весеннего равноденствия}~$\aries$ (20~марта, реже~21) его координаты: $\alpha=0^h$, $\delta=0^{\circ}$. Во время прохождения этой точки обе координаты Солнца растут. Так происходит до момента, пока Солнце не пройдет \imp{точку летнего солнцестояния} (21~июня, реже~20), координаты которой~--- $\alpha=6^h$ и $\delta=\varepsilon$. После этого склонение Солнца начинает убывать. В момент прохождения \imp{точки осеннего равноденствия}~$\libra$ (22~или 23~сентября), координаты Солнца составляют $\alpha=12^h$, $\delta=0^{\circ}$. После прохождения \imp{точки зимнего солнцестояния} с координатами $\alpha=18^h$, $\delta=-\varepsilon$ (22~или 21~декабря) склонение Солнца начинает увеличиваться. + +\begin{wrapfigure}[5]{r}{0.4\tw} \centering - \vspace{-.9pc} - \tikzsetnextfilename{sun-path} - \begin{tikzpicture} - \begin{axis}[ - width = .5\tw, - height = 4.5cm, - xlabel = {Прямое восхождение $\alpha^h$}, - ylabel = {Склонение $\delta^{\circ}$}, - extra y ticks = {23.44, -23.44}, - ytick = {-20, -10, 0, 10, 20}, - ymax = 25, - ymin = -25, - xmax = 24, - xmin = 0, - xtick = {0, 4, 8, 12, 16, 20, 24}, - x dir = reverse - ] - \addplot [domain=0:24, samples=100] {atan(sin(x*15)*tan(23.44))}; - \end{axis} + \vspace{-1pc} + \tikzsetnextfilename{sun-lambda-alpha} + \tdplotsetmaincoords{70}{-70} + \begin{tikzpicture}[tdplot_main_coords] + \footnotesize + + \def\R{5} + \def\EPS{23.5} + \def\ALPHA{45} + \def\LAMBDA{atan(tan(\ALPHA)/cos(\EPS))} + \def\DELTA{asin(sin(\LAMBDA)*sin(\EPS))} + + % Draw triangle + \tdplotsetrotatedcoords{0}{0}{0}; + \draw[tdplot_rotated_coords, semithick] (\R,0,0) arc (0:{\ALPHA}:\R); + \tdplotsetrotatedcoords{90}{-\EPS}{-90}; + \draw[tdplot_rotated_coords, semithick] (\R,0,0) arc (0:\LAMBDA:\R); + \tdplotsetrotatedcoords{90+\ALPHA}{-90}{-90}; + \draw[tdplot_rotated_coords, semithick] (\R,0,0) arc (0:\DELTA:\R); + + % Draw points + \tdplotCsDrawPoint{\R}{180}{-90} + \tdplotCsDrawPoint{\R}{180 + \ALPHA}{-90 + \DELTA} + \tdplotCsDrawPoint{\R}{180 + \ALPHA}{-90} + + % Label arcs + \tdplotCsLabelPoint{\R}{180 + \ALPHA / 2}{-90}{$\alpha$}{below} + \tdplotCsLabelPoint{\R}{180 + \ALPHA / 2}{-90 + \DELTA / 2}{$\lambda$}{above right} + \tdplotCsLabelPoint{\R}{180 + \ALPHA}{-90 + \DELTA / 2}{$\delta$}{left} + + % Mark right angle + \tdplotsetrotatedcoords{\ALPHA}{0}{0}; + \draw [tdplot_rotated_coords](\R,-0.2,0.2) coordinate (c) (\R,0,0.2) coordinate (a1) -- (c) (\R,-0.2,0) coordinate (a2) -- (c) pic [angle radius=0.2cm] {right angle=a1--c--a2}; + + % Mark and label angle t near P + \def\angleRadius{0.6} + \tdplotsetrotatedcoords{0}{0}{0}; + \draw [tdplot_rotated_coords, canvas is yz plane at x = \R]({\angleRadius * cos(\EPS)},0) arc (0:{\EPS - 3}:\angleRadius); + \tdplotCsLabelPoint{\R}{6}{88.3}{$\varepsilon$}{left} \end{tikzpicture} - \caption{График зависимости склонения Солнца от его прямого восхождения} + \caption{} + \label{pic:sun-lambda-alpha} \end{wrapfigure} -В моменты, когда Солнце находится в \imp{точке весеннего равноденствия}~$\aries$ (20~марта, реже~21) его координаты: $\alpha=0^h$, $\delta=0^{\circ}$. Во время прохождения этой точки обе координаты Солнца растут. Так происходит до момента, пока Солнце не пройдет \imp{точку летнего солнцестояния} (21~июня, реже~20), координаты которой~--- $\alpha=6^h$ и $\delta=\varepsilon$. После этого склонение Солнца начинает убывать. В момент прохождения \imp{точки осеннего равноденствия}~$\libra$ (22~или 23~сентября), координаты Солнца составляют $\alpha=12^h$, $\delta=0^{\circ}$. После прохождения \imp{точки зимнего солнцестояния} с координатами $\alpha=18^h$, $\delta=-\varepsilon$ (22~или 21~декабря) склонение Солнца начинает увеличиваться. +Найдём выражение для прямого восхождения Солнца $\alpha$ через переменные $\lambda$ и $\varepsilon$, где $\varepsilon = 23.44^\circ$~--- угол наклона экватора Земли к эклиптике. Для этого рассмотрим формулу пяти элементов~\eqref{eq:formula-5-elem} для прямоугольного сферического треугольника, представленного на рисунке~\picRef{pic:sun-lambda-alpha}: +\begin{gather} + \sin \delta \cos 90^\circ = \cos \lambda \sin \alpha - \sin \lambda \cos \alpha \cos \varepsilon,\notag\\ + \cos \lambda \sin \alpha = \sin \lambda \cos \alpha \cos \varepsilon,\notag\\ + \frac{\tg \alpha }{\tg \lambda} = \cos \varepsilon. + \label{eq:tgAlpha/tgLambda} +\end{gather} -Пренебрегая сферическими искажениями, годичный путь Солнца по небесной сфере можно считать синусоидой, откуда +Из аналогичных рассуждений несложно получить, что прямое восхождение Солнца связано со склонением формулой. Рассмотрим формулу пяти элементов~\eqref{eq:formula-5-elem}, +\begin{gather*} + \cos \delta \sin \alpha = \sin \delta \cos \alpha \cos 90^\circ + \sin \lambda \cos \varepsilon,\\ + \cos \delta \sin \alpha = \sin \lambda \cos \varepsilon. +\end{gather*} +Выразим $\sin \lambda$ из теоремы синусов~\eqref{eq:sphere-th-sinus}: +\begin{equation*} + \sin \lambda = \frac{\sin \delta}{\sin \varepsilon}. +\end{equation*} +и подставим в полученное выше равенство: \begin{equation} - \delta=\varepsilon\cdot\sin \frac{2 \pi t}{T}, + \sin\alpha = \frac{\tg\delta}{\tg\varepsilon}. + \label{eq:sin-alpha} \end{equation} -где $t$~--- время, прошедшее с момента весеннего равноденствия, $T$~--- тропический год. -Более точная формула следует из сферической тригонометрии и имеет вид -\begin{equation} - \delta=\arcsin\left(\sin\varepsilon\cdot\sin \frac{2 \pi t}{T}\right). - \label{eq:delta-sun} -\end{equation} -Известно, что движение Солнца по эклиптике происходит неравномерно, поэтому данные формулы не являются абсолютно точными. -Прямое восхождение Солнца связано со склонением формулой -\begin{equation} - \sin\alpha=\frac{\tg\delta}{\tg\varepsilon}. - \label{eq:sin-alpha} -\end{equation} -Выражения \eqref{eq:delta-sun} и \eqref{eq:sin-alpha} следуют из формул перехода между экваториальной и эклиптической системами координат, получаемых из сферической тригонометрии. diff --git a/sections/spherical-astronomy/spher-trigonom.tex b/sections/spherical-astronomy/spher-trigonom.tex index 85c8142..70839a6 100644 --- a/sections/spherical-astronomy/spher-trigonom.tex +++ b/sections/spherical-astronomy/spher-trigonom.tex @@ -59,6 +59,7 @@ \subsection{Сферическая тригонометрия} Заметим, что правая часть равенства циклична по переменным $a$, $b$ и $c$, следовательно, \term{сферическая теорема синусов} имеет вид \begin{equation} \frac{\sin A}{\sin a} = \frac{\sin B}{\sin b} = \frac{\sin C}{\sin c}. + \label{eq:sphere-th-sinus} \end{equation} Далее получим \term{формулу пяти элементов}. Для этого запишем теорему косинусов в выразим в ней один из косинусов, применяя ее же: @@ -68,6 +69,7 @@ \subsection{Сферическая тригонометрия} \cos a - \cos a \cos^2 b = \sin c \sin b \cos A + \sin a \sin b \cos b \cos C,\nonumber\\ \cos a \sin^2 b = \sin c \sin b \cos A + \sin a \sin b \cos b \cos C,\nonumber\\ \cos a \sin b = \sin a \cos b \cos C + \sin c \cos A. + \label{eq:formula-5-elem} \end{gather} diff --git a/sections/spherical-astronomy/sun-time.tex b/sections/spherical-astronomy/sun-time.tex index 3194ce1..55d47f0 100644 --- a/sections/spherical-astronomy/sun-time.tex +++ b/sections/spherical-astronomy/sun-time.tex @@ -1,7 +1,5 @@ \subsection{Солнечное время. Уравнение времени} - - \term{Истинные солнечные сутки}~--- промежуток времени между двумя последовательными одноимёнными кульминациями Солнца. \term{Истинное солнечное время}~--- промежуток времени между нижней кульминацией Солнца и его текущим положением. Рассчитывается по формуле @@ -10,13 +8,13 @@ \subsection{Солнечное время. Уравнение времени} \end{equation} где $t_{\text{сол}}$~--- часовой угол Солнца. -\term{Среднее Солнце}~--- точка небесной сферы, которая равномерно движется по небесному экватору, с угловой скоростью, равной средней скорости изменения прям +\term{Среднее Солнце}~--- точка небесной сферы, которая равномерно движется по небесному экватору с угловой скоростью, равной средней скорости изменения прямого восхождения Солнца. -\term{Среднее солнечное время} ($T_\text{ср}$)~--- промежуток времени между нижней кульминацией \imp{среднего Солнца} и текущим его положением. Зная долготу наблюдателя, нетрудно вычислить среднее солнечное время: +\term{Среднее солнечное время} ($T_\text{ср}$)~--- время, прошедшее с последней нижней кульминации \imp{среднего Солнца}. Зная долготу наблюдателя, нетрудно вычислить среднее солнечное время: \begin{equation*} T_\text{ср} = \text{UTC} + \frac{\lambda}{15^\circ/\text{час}}, \end{equation*} -где UTC~--- \imp{всемирное время}~--- среднее (или, что тоже самое, истинное) солнечное время на Гринвиче (меридиан с долготой $\lambda = 0^\circ$). +где UTC~--- \imp{всемирное время}~--- среднее солнечное время на нулевом меридиане (меридиан с долготой $\lambda = 0^\circ$). \term{Поясное} или \term{гражданское время}~--- среднее солнечное время на срединном меридиане географического часового пояса. В России также установлено декретное время, которое на 1 час больше поясного. @@ -41,26 +39,30 @@ \subsection{Солнечное время. Уравнение времени} \caption{График уравнения времени} \label{pic:time-eq} \end{wrapfigure} -\term{Уравнение времени}~--- разница между истинным солнечным временем и средним солнечным временем, возникающая по причине неравномерности движения Земли по орбите и наклона земного экватора к плоскости эклиптики (см.~Рис.\,\ref{pic:time-eq}). Получим приближенное выражение для него. Для это этого вспомним уравнение Кеплера~\eqref{eq:kepler-eq} и величину эксцентриситета орбиты Земли $e_\oplus = 0.017 \ll 1$, откуда следует, что можно применять формулы приближенного вычисления, см.~Раздел~\ref{sec:form}. Рассмотрим выражение -\begin{multline*} +\term{Уравнение времени}~--- разница между истинным солнечным временем и средним солнечным временем, возникающая по причине неравномерности движения Земли по орбите и наклона земного экватора к плоскости эклиптики (см.~Рис.\,\ref{pic:time-eq}). + +Получим приближенное выражение для величины уравнения времени. Для этого вспомним величину эксцентриситета орбиты Земли $e_\oplus = 0.017 \ll 1$, и рассмотрим выражение +\begin{multline} \sin E = \sin(E - M + M) =\\ = \underbrace{\sin (E - M)}_{\simeq E - M} \cos M + \underbrace{\cos(E - M)}_{\simeq 1} \sin M \simeq\\ \simeq (E - M) \cos M + \sin M, -\end{multline*} -так как $E - M \sim e$. Отсюда можно сделать первое приближение + \label{eq:sun-time-sinE} +\end{multline} +так как $E - M \sim e$. Применим метод последовательных приближений, чтобы получить зависимость $E(M)$. Используем уравнение Кеплера~\eqref{eq:kepler-eq} и полученное выражение для $\sin E$~\eqref{eq:sun-time-sinE} для первого приближения: \begin{equation*} E_1 = M + e \sin E = M + e \big( (E - M) \cos M + \sin M \big) \simeq M + e \sin M. \end{equation*} -Воспользуемся полученным приближением, чтобы точнее оценить $E$. -\begin{equation*} +Воспользуемся полученным приближением, чтобы точнее оценить $E$, +\begin{equation} E = M + e \big( (E_1 - M) \cos M + \sin M \big) = M + e \sin M + \frac{e^2}{2} \sin 2M. -\end{equation*} + \label{eq:sun-time-E(M)} +\end{equation} Теперь запишем первые три члена многочлена Тейлора формулы перехода от истинной аномалии к эксцентрической~\eqref{eq:kepler-eq-E-nu-2}: \begin{equation*} @@ -68,70 +70,26 @@ \subsection{Солнечное время. Уравнение времени} = 2 \arctg \left(\sqrt{\frac{1+e}{1-e}} \tg \frac{E}{2} \right) \simeq E + e \sin E + \frac{e^2}{4} \sin 2E. \end{equation*} -Подставим сюда выражение для $E(M)$ и воспользуемся формулой для разложения синуса суммы: +Подставим сюда выражение~\eqref{eq:sun-time-E(M)} для $E(M)$ и воспользуемся формулой для разложения синуса суммы: \begin{multline*} \nu \simeq M + e \sin M + \frac{e^2}{2} \sin 2M + \\ - + e \bigg( \sin M \cdot \underbrace{\cos (e \sin M + \ldots)}_{\simeq 1} + \cos M \cdot \underbrace{\sin ( e \sin M + \ldots )}_{\simeq e \sin M} \bigg) + \\ - + \frac{e^2}{4} \bigg( \sin 2M \cdot \underbrace{\cos (2e \sin M + \ldots)}_{\simeq 1} + \cos 2M \cdot \underbrace{\sin (2e \sin M + \ldots)}_{\ll 1}\bigg) \simeq \\ + + e \bigg( + \sin M \cdot \underbrace{\cos (e \sin M + \ldots)}_{\simeq 1} + + \cos M \cdot \underbrace{\sin ( e \sin M + \ldots )}_{\simeq e \sin M} + \bigg) + \\ + + \frac{e^2}{4} \bigg( + \sin 2M \cdot \underbrace{\cos (2e \sin M + \ldots)}_{\simeq 1} + + \cos 2M \cdot \underbrace{\sin (2e \sin M + \ldots)}_{\ll 1,~\text{с уч-м коэф-та}} + \bigg) \simeq \\ \simeq M + 2e \sin M + \frac{5e^2}{4} \sin 2M. \end{multline*} Обозначим как $\omega = 103^\circ$ эклиптическую долготу перицентра, тогда эклиптическая долгота Солнца $\lambda = \nu + \omega$. -\begin{wrapfigure}[5]{r}{0.4\tw} - \centering - \vspace{-1pc} - \tikzsetnextfilename{sun-lambda-alpha} - \tdplotsetmaincoords{70}{-70} - \begin{tikzpicture}[tdplot_main_coords] - \footnotesize - - \def\R{5} - \def\EPS{23.5} - \def\ALPHA{45} - \def\LAMBDA{atan(tan(\ALPHA)/cos(\EPS))} - \def\DELTA{asin(sin(\LAMBDA)*sin(\EPS))} - - % Draw triangle - \tdplotsetrotatedcoords{0}{0}{0}; - \draw[tdplot_rotated_coords, semithick] (\R,0,0) arc (0:{\ALPHA}:\R); - \tdplotsetrotatedcoords{90}{-\EPS}{-90}; - \draw[tdplot_rotated_coords, semithick] (\R,0,0) arc (0:\LAMBDA:\R); - \tdplotsetrotatedcoords{90+\ALPHA}{-90}{-90}; - \draw[tdplot_rotated_coords, semithick] (\R,0,0) arc (0:\DELTA:\R); - % Draw points - \tdplotCsDrawPoint{\R}{180}{-90} - \tdplotCsDrawPoint{\R}{180 + \ALPHA}{-90 + \DELTA} - \tdplotCsDrawPoint{\R}{180 + \ALPHA}{-90} - % Label arcs - \tdplotCsLabelPoint{\R}{180 + \ALPHA / 2}{-90}{$\alpha$}{below} - \tdplotCsLabelPoint{\R}{180 + \ALPHA / 2}{-90 + \DELTA / 2}{$\lambda$}{above right} - \tdplotCsLabelPoint{\R}{180 + \ALPHA}{-90 + \DELTA / 2}{$\delta$}{left} - - % Mark right angle - \tdplotsetrotatedcoords{\ALPHA}{0}{0}; - \draw [tdplot_rotated_coords](\R,-0.2,0.2) coordinate (c) (\R,0,0.2) coordinate (a1) -- (c) (\R,-0.2,0) coordinate (a2) -- (c) pic [angle radius=0.2cm] {right angle=a1--c--a2}; - - % Mark and label angle t near P - \def\angleRadius{0.6} - \tdplotsetrotatedcoords{0}{0}{0}; - \draw [tdplot_rotated_coords, canvas is yz plane at x = \R]({\angleRadius * cos(\EPS)},0) arc (0:{\EPS - 3}:\angleRadius); - \tdplotCsLabelPoint{\R}{6}{88.3}{$\varepsilon$}{left} - \end{tikzpicture} - \caption{} - \label{pic:sun-lambda-alpha} -\end{wrapfigure} - -Далее найдём выражение для прямого восхождения Солнца $\alpha$ через переменные $\lambda$ и $\varepsilon$, где $\varepsilon = 23.44^\circ$~--- угол наклона экватора Земли к эклиптике. Для этого рассмотрим формулу пяти элементов для прямоугольного сферического треугольника, представленного на рисунке~\picRef{pic:sun-lambda-alpha}: -\begin{gather} - \sin \delta \cos 90^\circ = \cos \lambda \sin \alpha - \sin \lambda \cos \alpha \cos \varepsilon,\notag\\ - \cos \lambda \sin \alpha = \sin \lambda \cos \alpha \cos \varepsilon,\notag\\ - \frac{\tg \alpha }{\tg \lambda} = \cos \varepsilon. -\end{gather} -Теперь запишем формулу тангенса половинного угла для $\tg \sfrac{\varepsilon}{2}$ и воспользуемся полученным выражением для $\cos \varepsilon$: +Теперь запишем формулу тангенса половинного угла для $\tg \frac{\varepsilon}{2}$ и воспользуемся выражением~\eqref{eq:tgAlpha/tgLambda} для $\cos \varepsilon$: \begin{gather*} \tg^2 \frac{\varepsilon}{2} = \frac{1 - \cos \varepsilon}{1 + \cos \varepsilon} \equiv y ,\\ 1 - \frac{\tg \alpha}{\tg \lambda} = y \cdot \left( 1 + \frac{\tg \alpha}{\tg \lambda} \right),\\ @@ -144,8 +102,9 @@ \subsection{Солнечное время. Уравнение времени} \alpha_1 = \lambda - \arcsin \left( y \sin (\alpha_0 + \lambda) \right) \overset{\varepsilon \ll 1}{\simeq} \lambda - y \sin 2 \lambda,\nonumber\\ \alpha_2 = \lambda - \arcsin \left( y \sin (\alpha_1 + \lambda) \right) - \overset{\varepsilon \ll 1}{\simeq} \lambda - y \sin 2 \lambda + \frac{y^2}{2} \sin 4 \lambda. \label{eq:second-approx-alpha-lambda} + \overset{\varepsilon \ll 1}{\simeq} \lambda - y \sin 2 \lambda + \frac{y^2}{2} \sin 4 \lambda. \label{eq:second-approx-alpha-lambda}. \end{gather} + Используем~\eqref{eq:second-approx-alpha-lambda} для записи уравнения времени: \begin{multline} \eta @@ -164,4 +123,4 @@ \subsection{Солнечное время. Уравнение времени} \begin{equation} \eta = t_\text{ист} - t_\text{ср} = -7.65 \sin \frac{2\pi d}{P} + 9.86 \sin 2 \left( 1.80 + \frac{2\pi d}{P} \right)~\text{мин}, \end{equation} -где $T_\text{сид}$~--- сидерический год (здесь не учитываются поправки, связанные с прецессией Земной оси), а $d$~--- время от момента прохождения точки перицентра (в современную эпоху это происходит в период со 2 по 5 января). +где $P = T_\text{сид}$~--- сидерический год (здесь не учитываются поправки, связанные с прецессией Земной оси), а $d$~--- время от момента прохождения точки перицентра (в современную эпоху это происходит в период со 2 по 5 января). diff --git a/sys/spherical-astronomy.tex b/sys/spherical-astronomy.tex index 2ff5a2f..1ef37eb 100644 --- a/sys/spherical-astronomy.tex +++ b/sys/spherical-astronomy.tex @@ -40,7 +40,7 @@ \section{Сферическая астрономия} \input{sections/spherical-astronomy/coordin-sys.tex} \input{sections/spherical-astronomy/sky-phenom.tex} \input{sections/spherical-astronomy/spher-trigonom.tex} -\input{sections/spherical-astronomy/sun-time.tex} \input{sections/spherical-astronomy/change-coordin.tex} +\input{sections/spherical-astronomy/sun-time.tex} \input{sections/spherical-astronomy/refrac.tex} \input{sections/spherical-astronomy/dusk.tex} diff --git a/sys/tables.tex b/sys/tables.tex index e7608ae..ec67474 100644 --- a/sys/tables.tex +++ b/sys/tables.tex @@ -7,7 +7,7 @@ \section*{Приложение} \begin{figure}[h!] \centering \vspace{-.5pc} - \includegraphics[width=\tw]{sky-map} + \includegraphics[width=\tw]{sky-map.pdf} \caption{Карта звёздного неба до $-45^\circ$ по склонению} \vspace{-1cm} \end{figure}

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