Is there a package (or packages) for drawing the Sun–Earth–Moon system with either PSTricks, TikZ, or some other vector graphics language to illustrate Lunar and Solar eclipses, the phases of the Moon, and the seasons on the Earth?
I know of pst-solarsystem but it is not what I'm looking for; here, it is 'only' possible to draw the planets moving in circular orbits around the Sun.
A couple of months ago, I produced a beamer/tikz animation (available on texample.net) of the Earth's orbit around the Sun to illustrate the counterintuitive fact that Earth is farther from the Sun in Summer than it is in Winter. I used that example to demonstrate the power of inducing (and then resolving) cognitive dissonance in the classroom.
I've modified it to also show the Moon following an elliptical orbit around the Earth. (Of course, the Moon's orbit around the Earth is really only approximately elliptical, and does not lie in the Ecliptic Plane.) You can change the positions visited by the Earth and the Moon along their respective orbits (simply change how \Earthangle and \Moonangle are defined in the body of the \foreach, and/or modify the value of \N).
\documentclass{beamer}
\usepackage{lmodern}
\usepackage{tikz}
\setbeamertemplate{navigation symbols}{}
\begin{document}
\begin{frame}[fragile]
\frametitle{}
\begin{center}
\begin{tikzpicture}[scale=2.5]
\def\rS{0.3} % Sun radius
\def\rE{0.1} % Earth radius
% Major radius of Earth's elliptical orbit = 1
\def\eE{0.25} % Excentricity of Earth's elliptical orbit
\pgfmathsetmacro\bE{sqrt(1-\eE*\eE)} % Minor radius of Earth's elliptical orbit
\pgfmathsetmacro\rM{.7*\rE} % Moon radius
\pgfmathsetmacro\aM{2.5*\rE} % Major radius of the Moon's elliptical orbit
\def\eM{0.4} % Excentricity of Earth's elliptical orbit
\pgfmathsetmacro\bM{\aM*sqrt(1-\eM*\eM)} % Minor radius of the Moon's elliptical orbit
\def\offsetM{30} % angle offset between the major axes of Earth's and the Moon's orbits
% This function computes the direction in which light hits the Earth.
\pgfmathdeclarefunction{f}{1}{%
\pgfmathparse{
((-\eE+cos(#1))<0) * ( 180 + atan( \bE*sin(#1)/(-\eE+cos(#1)) ) )
+
((-\eE+cos(#1))>=0) * ( atan( \bE*sin(#1)/(-\eE+cos(#1)) ) )
}
}
% This function computes the distance between Earth and the Sun,
% which is used to calculate the varying radiation intensity on Earth.
\pgfmathdeclarefunction{d}{1}{%
\pgfmathparse{ sqrt((-\eE+cos(#1))*(-\eE+cos(#1))+\bE*sin(#1)*\bE*sin(#1)) }
}
% Draw the elliptical path of the Earth.
\draw[thin,color=gray] (0,0) ellipse (1 and \bE);
% Draw the Sun at the right-hand-side focus
\shade[
top color=yellow!70,
bottom color=red!70,
shading angle={45},
] ({sqrt(1-\bE*\bE)},0) circle (\rS);
%\draw ({sqrt(1-\b*\b)},-\rS) node[below] {Sun};
% Produces a series of frames showing one revolution
% (the total number of frames is controlled by macro \N)
\pgfmathtruncatemacro{\N}{12}
\foreach \k in {0,1,...,\N}{
\pgfmathsetmacro{\Earthangle}{360*\k/\N}
\pgfmathsetmacro{\Moonangle}{3*360*\k/\N} % <--- change the multiplying factor to suit your needs
% Draw the Earth at \Earthangle
\pgfmathsetmacro{\radiation}{100*(1-\eE)/(d(\Earthangle)*d(\Earthangle))}
\colorlet{Earthlight}{yellow!\radiation!blue}
\pgfmathparse{int(\k+1)}
\onslide<\pgfmathresult>{
\shade[
top color=Earthlight,
bottom color=blue,
shading angle={90+f(\Earthangle)},
] ({cos(\Earthangle)},{\bE*sin(\Earthangle)}) circle (\rE);
%\draw ({cos(\Earthangle)},{\bE*sin(\Earthangle)-\rE}) node[below] {Earth};
% Draw the Moon's (circular) orbit and the Moon at \Moonangle
\draw[thin,color=gray,rotate around={{\offsetM}:({cos(\Earthangle)},{\bE*sin(\Earthangle)})}]
({cos(\Earthangle)},{\bE*sin(\Earthangle)}) ellipse ({\aM} and {\bM});
\shade[
top color=black!70,
bottom color=black!30,
shading angle={45},
] ({cos(\Earthangle)+\aM*cos(\Moonangle)*cos(\offsetM)-\bM*sin(\Moonangle)*sin(\offsetM)},%
{\bE*sin(\Earthangle)+\aM*cos(\Moonangle)*sin(\offsetM)+\bM*sin(\Moonangle)*cos(\offsetM)}) circle (\rM);
}
}
\end{tikzpicture}
\end{center}
\end{frame}
\end{document}
Here is non-animated version in the article class.
\documentclass{article}
\usepackage{lmodern}
\usepackage{tikz}
\usepackage{kantlipsum}
\begin{document}
\section{Eclipses}
\kant[1]
\begin{figure}
\centering
\begin{tikzpicture}[scale=2.5]
\def\rS{0.3} % Sun radius
\def\Earthangle{30} % angle wrt to horizontal
\def\rE{0.1} % Earth radius
% Major radius of Earth's elliptical orbit = 1
\def\eE{0.25} % Excentricity of Earth's elliptical orbit
\pgfmathsetmacro\bE{sqrt(1-\eE*\eE)} % Minor radius of Earth's elliptical orbit
\def\Moonangle{-45} % angle wrt to horizontal
\pgfmathsetmacro\rM{.7*\rE} % Moon radius
\pgfmathsetmacro\aM{2.5*\rE} % Major radius of the Moon's elliptical orbit
\def\eM{0.4} % Excentricity of Earth's elliptical orbit
\pgfmathsetmacro\bM{\aM*sqrt(1-\eM*\eM)} % Minor radius of the Moon's elliptical orbit
\def\offsetM{30} % angle offset between the major axes of Earth's and the Moon's orbits
% This function computes the direction in which light hits the Earth.
\pgfmathdeclarefunction{f}{1}{%
\pgfmathparse{
((-\eE+cos(#1))<0) * ( 180 + atan( \bE*sin(#1)/(-\eE+cos(#1)) ) )
+
((-\eE+cos(#1))>=0) * ( atan( \bE*sin(#1)/(-\eE+cos(#1)) ) )
}
}
% This function computes the distance between Earth and the Sun,
% which is used to calculate the varying radiation intensity on Earth.
\pgfmathdeclarefunction{d}{1}{%
\pgfmathparse{ sqrt((-\eE+cos(#1))*(-\eE+cos(#1))+\bE*sin(#1)*\bE*sin(#1)) }
}
% Draw the elliptical path of the Earth.
\draw[thin,color=gray] (0,0) ellipse (1 and \bE);
% Draw the Sun at the right-hand-side focus
\shade[
top color=yellow!70,
bottom color=red!70,
shading angle={45},
] ({sqrt(1-\bE*\bE)},0) circle (\rS);
%\draw ({sqrt(1-\b*\b)},-\rS) node[below] {Sun};
% Draw the Earth at \Earthangle
\pgfmathsetmacro{\radiation}{100*(1-\eE)/(d(\Earthangle)*d(\Earthangle))}
\colorlet{Earthlight}{yellow!\radiation!blue}
\shade[%
top color=Earthlight,%
bottom color=blue,%
shading angle={90+f(\Earthangle)},%
] ({cos(\Earthangle)},{\bE*sin(\Earthangle)}) circle (\rE);
%\draw ({cos(\Earthangle)},{\bE*sin(\Earthangle)-\rE}) node[below] {Earth};
% Draw the Moon's (circular) orbit and the Moon at \Moonangle
\draw[thin,color=gray,rotate around={{\offsetM}:({cos(\Earthangle)},{\bE*sin(\Earthangle)})}]
({cos(\Earthangle)},{\bE*sin(\Earthangle)}) ellipse ({\aM} and {\bM});
\shade[
top color=black!70,
bottom color=black!30,
shading angle={45},
] ({cos(\Earthangle)+\aM*cos(\Moonangle)*cos(\offsetM)-\bM*sin(\Moonangle)*sin(\offsetM)},%
{\bE*sin(\Earthangle)+\aM*cos(\Moonangle)*sin(\offsetM)+\bM*sin(\Moonangle)*cos(\offsetM)}) circle (\rM);
\end{tikzpicture}
\caption{Sun, Earth and Moon}
\end{figure}
\end{document}
You can use the definition of pst-solarsystem.tex and remove all unwanted planets.
\documentclass{article}
\usepackage{pstricks}
\usepackage{pst-solarsystem}
\makeatletter
\def\SunEarth{\pst@object{SunEarth}}
\def\SunEarth@i{{%
\pst@killglue%
\use@par%
\begin{pspicture}(-3.5,-3.5)(3.5,4.5)
\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt]%
\pstVerb{%
/JOUR \psk@SolarSystemD\space def
/MOIS \psk@SolarSystemM\space def
/AN \psk@SolarSystemY\space def
/HEURE \psk@SolarSystemH\space def
/MINUTE \psk@SolarSystemMi\space def
/SECONDE \psk@SolarSystemS\space def
%%%% Calcul du mill�naire Julien ---------------------
/lesMois [0 31 59 90 120 151 181 212 243 273 304 334] def
/EcartJours {lesMois MOIS 1 sub get JOUR add HEURE MINUTE 60 div add SECONDE 3600 div add 24 div add 1 sub} def
/EcartAn {AN 4 div AN 4 div floor sub cvi} bind def
EcartAn 0 eq {/EcartAn 1 def} if
EcartAn 1 eq {MOIS 2 gt {/EcartJours EcartJours 1 add def}if} if
/T {AN 2000 sub 365.25 mul 0.5 add EcartJours add EcartAn sub 365250 div} bind def
/T2 {T dup mul} bind def
/T3 {T2 T mul} bind def
}%
\rput(0,4){\psk@SolarSystemD/\psk@SolarSystemM/\psk@SolarSystemY}%
\ThreeDput{%
% \psframe[fillstyle=gradient,gradbegin=cyan,gradend=white](-7,-7)(7,7)
\multido{\r=22.5+45}{8}{\psline[linecolor=yellow](1;\r)}%
\psline[linestyle=dotted]{->}(-3,0)(3,0)
\uput[0](3,0){$\mathbf{\gamma}$}
\uput[90](3,0){0\textsuperscript{o}}
\uput[90](0,3){90\textsuperscript{o}}
\uput[180](-3,0){180\textsuperscript{o}}
\uput[270](0,-2.9){270\textsuperscript{o}}
\psline[linestyle=dotted](0,-3)(0,3)}%
\pscircle[linestyle=none,fillstyle=gradient,gradmidpoint=0,gradend=yellow,GradientCircle=true,gradbegin=gray]{0.5}%
{\psset{unit=2}
% Earth
\pstVerb{%
earLM earKA earHA earQ earP orbitalparameters
aear /radius exch 1 E dup mul sub mul
1 E LO LP sub cos mul add div def
}%
\ThreeDput{%
\psplot[polarplot=true,plotpoints=361,linecolor=red]{0}{360}{%
aear 1 E dup mul sub mul
1 E x LP sub cos mul add div}
\pnode(! radius LO cos mul radius LO sin mul){Terre}}
\rput(Terre){\psset{unit=2}%
\pscircle[style=planetes,gradbegin=blue,GradientPos={(0.013,0.03)}]{0.0536}%
\uput{0.08}[u](0,0){\footnotesize\textsf{Earth}}}%
\ifPst@values
\rput(-0.5,-4.25){\psPrintValue{LO}}
\rput(-0.5,-4.75){\psPrintValue{0.000}}
\rput(-0.5,-5.25){\psPrintValue{radius}}
\fi
}
\ifPst@values
\rput(-0.5,-7.75){Earth}
\rput(-6.5,-8.42){longitude at $^\mathrm{o}$}
\rput(-6.5,-9.42){latitude at $^\mathrm{o}$}
\rput(-6.5,-10.42){distance at U.A.}
\fi
\end{pspicture}}}
\makeatother
\begin{document}
\SunEarth[Day=30,Month=02,Year=2001,
Hour=23,Minute=59,Second=59,values=false]
\SunEarth[Day=30,Month=12,Year=2001,
Hour=23,Minute=59,Second=59,values=false]
\end{document}
pst-solarsystem package is aimed to an accurate representation.
– egreg
Jun 2 '13 at 13:52
If you want to be really accurate, you should use a programming language to simulate the trajectories. From the examples, the eccentricities are off in the solutions above. Here is Python code I use all the time to create this kind of stuff. Note: I work in orbital mechanics.
Here are some notes I am working on that may help you if you want to code it up. This notes discuss orbital mechanics. They aren't done yet but with the link you can always check to see when it is complete.
Here is the Earth and Mars in astronomical units. You can add_patches for the spheres for the Sun, Earth, and Moon. The code just needs to be changed to fit your problem.
#!/usr/bin/env ipython
# This program solves the 3 Body Problem numerically and plots the
# trajectories
import numpy as np
from scipy.integrate import odeint
from mpl_toolkits.mplot3d import Axes3D
import pylab
mu = 1.0
# r0 = [-149.6 * 10 ** 6, 0.0, 0.0] # Initial position
# v0 = [-5.04769, -29.9652, 0.0] # Initial velocity
# u0 = [-149.6 * 10 ** 6, 0.0, 0.0, 29.9652, -5.04769, 0.0]
u0 = [-1.0, 0.0, 0.0, 0.169474, -1.0067, 0.0]
e0 = [-1.0, 0.0, 0.0, 0.0, -1.0, 0.0]
m0 = [1.53, 0.0, 0.0, 0.0, 1.23152, 0.0]
def deriv2(e, dt):
n = -mu / np.sqrt(e[0] ** 2 + e[1] ** 2 + e[2] ** 2)
return [e[3], # dotu[0] = u[3]'
e[4], # dotu[1] = u[4]'
e[5], # dotu[2] = u[5]'
e[0] * n, # dotu[3] = u[0] * n
e[1] * n, # dotu[4] = u[1] * n
e[2] * n] # dotu[5] = u[2] * n
def deriv(u, b):
n = -mu / np.sqrt(u[0] ** 2 + u[1] ** 2 + u[2] ** 2)
return [u[3], # dotu[0] = u[3]'
u[4], # dotu[1] = u[4]'
u[5], # dotu[2] = u[5]'
u[0] * n, # dotu[3] = u[0] * n
u[1] * n, # dotu[4] = u[1] * n
u[2] * n] # dotu[5] = u[2] * n
def deriv3(m, t):
n = -mu / np.sqrt(m[0] ** 2 + m[1] ** 2 + m[2] ** 2)
return [m[3], # dotu[0] = u[3]'
m[4], # dotu[1] = u[4]'
m[5], # dotu[2] = u[5]'
m[0] * n, # dotu[3] = u[0] * n
m[1] * n, # dotu[4] = u[1] * n
m[2] * n] # dotu[5] = u[2] * n
b = np.arange(0.0, 2 * np.pi, .01)
dt = np.arange(0.0, 2 * np.pi, .01)
t = np.arange(0.0, 2.5 * np.pi, .01)
u = odeint(deriv, u0, b)
e = odeint(deriv2, e0, dt)
m = odeint(deriv3, m0, t)
x, y, z, x2, y2, z2 = u.T
x3, y3, z3, x4, y4, z5 = e.T
x6, y6, z6, x7, y7, z7 = m.T
fig = pylab.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot(x3, y3, z3)
ax.plot(x, y, z)
ax.plot(x6, y6, z6)
pylab.axis((-2, 2, -2, 2))
pylab.show()
Ignore the u trajectory. The part that will aid you is the e for earth and m for mars. But as I said, you will need to make the appropriate adjustments.
Ignore the green path.
Here is the image this creates:
Here is a plot with a 3d sphere for you to look at too:
