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authorRahiel Kasim <rahielkasim@gmail.com>2016-02-25 20:28:34 +0100
committerRahiel Kasim <rahielkasim@gmail.com>2016-02-25 20:28:34 +0100
commit8fd80ce53bd641c6f9522d6811af2230217b5fed (patch)
treebbbfc0de152c9532044fed1615668392ba10dabb
parent39c0e5a13a65693b520e812639843194751f8d3d (diff)
add report
-rw-r--r--Report/Pictures/Euler_Sling.pngbin0 -> 43343 bytes
-rw-r--r--Report/Pictures/Flowchart.pngbin0 -> 78511 bytes
-rw-r--r--Report/Pictures/Flowchart.svg307
-rw-r--r--Report/Pictures/Leapfrog_Sling.pngbin0 -> 18259 bytes
-rw-r--r--Report/Pictures/errors_leap_symplectic.pngbin0 -> 212769 bytes
-rw-r--r--Report/Pictures/errors_leapfrog.pngbin0 -> 213262 bytes
-rw-r--r--Report/Pictures/errors_symplectic1.pngbin0 -> 266294 bytes
-rw-r--r--Report/Pictures/leapfrog.pngbin0 -> 1235 bytes
-rw-r--r--Report/Pictures/sunsistemoPrtSc.pngbin0 -> 539019 bytes
-rw-r--r--Report/Pictures/symplecticEuler_Sling.pngbin0 -> 18509 bytes
-rw-r--r--Report/references.bib150
-rw-r--r--Report/sunsistemo.pdfbin0 -> 1642469 bytes
-rw-r--r--Report/sunsistemo.tex651
13 files changed, 1108 insertions, 0 deletions
diff --git a/Report/Pictures/Euler_Sling.png b/Report/Pictures/Euler_Sling.png
new file mode 100644
index 0000000..c340853
--- /dev/null
+++ b/Report/Pictures/Euler_Sling.png
Binary files differ
diff --git a/Report/Pictures/Flowchart.png b/Report/Pictures/Flowchart.png
new file mode 100644
index 0000000..3eeffbf
--- /dev/null
+++ b/Report/Pictures/Flowchart.png
Binary files differ
diff --git a/Report/Pictures/Flowchart.svg b/Report/Pictures/Flowchart.svg
new file mode 100644
index 0000000..95433de
--- /dev/null
+++ b/Report/Pictures/Flowchart.svg
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diff --git a/Report/references.bib b/Report/references.bib
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+++ b/Report/references.bib
@@ -0,0 +1,150 @@
+@article{scholar:nbody,
+ author = {Trenti, M. and Hut, P. },
+ title = {{N}-body simulations (gravitational)},
+ year = {2008},
+ journal = {Scholarpedia},
+ volume = {3},
+ number = {5},
+ pages = {3930},
+ note = {{revision \#91544}}
+}
+
+@misc{wiki:n-body-problem,
+ author = "Wikipedia",
+ title = "N-body problem --- Wikipedia{,} The Free Encyclopedia",
+ year = "2015",
+ note = "[{\url{https://en.wikipedia.org/w/index.php?title=N-body_problem&oldid=685939410}}; accessed 29-October-2015]"
+}
+
+@INPROCEEDINGS{amuse,
+ author = {{McMillan}, S. and {Portegies Zwart}, S. and {van Elteren}, A. and
+ {Whitehead}, A.},
+ title = "{Simulations of Dense Stellar Systems with the AMUSE Software Toolkit}",
+booktitle = {Advances in Computational Astrophysics: Methods, Tools, and Outcome},
+ year = 2012,
+ series = {Astronomical Society of the Pacific Conference Series},
+ volume = 453,
+archivePrefix = "arXiv",
+ eprint = {1111.3987},
+ primaryClass = "astro-ph.IM",
+ editor = {{Capuzzo-Dolcetta}, R. and {Limongi}, M. and {Tornamb{\`e}}, A.
+ },
+ month = jul,
+ pages = {129},
+ adsurl = {http://adsabs.harvard.edu/abs/2012ASPC..453..129M},
+ adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
+@misc{pynbody,
+ author = {{Pontzen}, A. and {Ro{\v s}kar}, R. and {Stinson}, G.~S. and {Woods},
+ R. and {Reed}, D.~M. and {Coles}, J. and {Quinn}, T.~R.},
+ title = "{pynbody: Astrophysics Simulation Analysis for Python}",
+ note = {Astrophysics Source Code Library, ascl:1305.002},
+ year = 2013
+}
+
+@article{nemo,
+ author = {Teuben, P.J.},
+ title = {The Stellar Dynamics Toolbox NEMO},
+ journal = {Astronomical Data Analysis Software and Systems IV},
+ pages = {398},
+ year = {1995}
+}
+
+@misc{wiki:principia,
+ author = "Wikipedia",
+ title = "Philosophiæ Naturalis Principia Mathematica --- Wikipedia{,} The Free Encyclopedia",
+ year = "2015",
+ note = "[{\url{https://en.wikipedia.org/w/index.php?title=Philosophi%C3%A6_Naturalis_Principia_Mathematica&oldid=685592689}}; accessed 1-November-2015]"
+}
+
+@INBOOK{giancoli,
+ author = "Douglas C. Giancoli",
+ title = "{Physics for Scientists and Engineers with Modern Physics}",
+ publisher = "Pearson Education",
+ edition = "Fourth",
+ year = "2009",
+ type = "Chapter",
+ chapter = "6-2 Vector Form of Newton's Law of Universal Gravitation",
+ pages = "143"
+}
+
+@misc{wiki:euler,
+ author = "Wikipedia",
+ title = "Euler method --- Wikipedia{,} The Free Encyclopedia",
+ year = "2015",
+ note = "[{\url{https://en.wikipedia.org/w/index.php?title=Euler_method&oldid=687685307}};; accessed 1-November-2015]"
+}
+
+@misc{leapfrog,
+ author = "Steve McMillan",
+ title = "The Leapfrog Integrator",
+ note = "[{\url{http://einstein.drexel.edu/courses/Comp_Phys/Integrators/leapfrog/}}; accessed 2-November-2015]"
+}
+
+@techreport{stan,
+ author = {Standish, E.M.},
+ title = "{JPL Planetary and Lunar Ephimerides, DE403/LE403}",
+ institution = "JPL",
+ type = "Interoffice Memorandum",
+ number = " 314.10-127",
+ note = "Published at Auto-ID Center Board Meeting",
+ year = "1995",
+ month = may
+}
+
+@misc{planettexture,
+ author = "James Hastings-Trew",
+ title = "Planet Texture Maps",
+ note = "[{\url{http://planetpixelemporium.com/planets.html}}; accessed 7-October-2015]"
+}
+
+@misc{balltexture,
+ author = "Robin Wood",
+ title = "Texture Maps for Balls",
+ note = "[{\url{http://www.robinwood.com/Catalog/FreeStuff/Textures/TexturePages/BallMaps.html}}; accessed 13-October-2015]"
+}
+
+@misc{d3lib,
+ author = "Mike Bostock",
+ title = "D3",
+ note = "[{\url{http://d3js.org/}}; accessed 2-November-2015]"
+}
+
+@ARTICLE{three-body-planar,
+ author = {{{\v S}uvakov}, M. and {Dmitra{\v s}inovi{\'c}}, V.},
+ title = "{Three Classes of Newtonian Three-Body Planar Periodic Orbits}",
+ journal = {Physical Review Letters},
+archivePrefix = "arXiv",
+ eprint = {1303.0181},
+ primaryClass = "physics.class-ph",
+ keywords = {Few- and many-body systems, Nonlinear dynamics and chaos, Celestial mechanics},
+ year = 2013,
+ month = mar,
+ volume = 110,
+ number = 11,
+ eid = {114301},
+ pages = {114301},
+ doi = {10.1103/PhysRevLett.110.114301},
+ adsurl = {http://adsabs.harvard.edu/abs/2013PhRvL.110k4301S},
+ adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
+@misc{twoBodyAnalytical,
+ author = "MIT, Unknown author",
+ title = "The Kepler Problem: Planetary Mechanics and the
+Bohr Atom ",
+ note = "[{\url{http://web.mit.edu/8.01t/www/materials/modules/guide17.pdf}}; accessed 4-November-2015]"
+}
+
+@misc{sunsistemoGH,
+ author = "{Jaro Camphuijsen, Rahiel Kasim}",
+ title = "{Sunsistemo on GitHub}",
+ note = "[{\url{https://github.com/sunsistemo/sunsistemo}}; accessed 4-November-2015]"
+}
+
+@misc{sunsistemo,
+ author = "{Jaro Camphuijsen, Rahiel Kasim}",
+ title = "{Sunsistemo - The N-Body Simulator}",
+ note = "[{\url{https://sunsistemo.js.org}}; accessed 4-November-2015]"
+}
diff --git a/Report/sunsistemo.pdf b/Report/sunsistemo.pdf
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diff --git a/Report/sunsistemo.tex b/Report/sunsistemo.tex
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@@ -0,0 +1,651 @@
+\documentclass[a4paper]{article}
+\usepackage[english]{babel}
+\usepackage{pdfpages, titling}
+\usepackage{array, float}
+\usepackage{cite, tablefootnote}
+\usepackage{graphicx, caption, subfigure, wrapfig}
+\usepackage{listings}
+\usepackage{color}
+\usepackage{mathtools, braket}
+\usepackage{amssymb}
+\usepackage[nottoc]{tocbibind} % references in the toc
+\usepackage{textcomp}
+\usepackage{minted} % code formatting
+\usepackage[hidelinks]{hyperref}
+\graphicspath{{Pictures/}} % Specifies the directory where pictures are stored
+
+\newcommand{\vect}[1]{\boldsymbol{#1}}
+\newcommand{\subtitle}[1]{%
+ \posttitle{%
+ \par\end{center}
+ \begin{center}\large#1\end{center}
+ \vskip0.5em}%
+}
+\lstset{frame=tb,
+ language=Java,
+ aboveskip=3mm,
+ belowskip=3mm,
+ showstringspaces=false,
+ columns=flexible,
+ basicstyle={\small\ttfamily},
+ numbers=none,
+ breaklines=true,
+ breakatwhitespace=true,
+ tabsize=3
+}
+
+\author{Jaro Camphuijsen (6042473) and Rahiel Kasim (10447539)}
+\date{6 November 2015}
+\title{Sunsistemo}
+\subtitle{Visualizing the Few Body Problem}
+\begin{document}
+\maketitle
+
+\tableofcontents
+
+\section*{Abstract}
+The physics governing gravity between all bodies of mass in the universe is elegantly expressed by
+Newton in a single vector equation. However general analytic solutions are only available for
+systems with up to two bodies. In this paper we discuss the physics and algorithms required to build
+a numerical simulation of gravity for an arbitrary number of bodies. In addition we look at how to
+make a visualization of the few body simulator that is appealing to a large audience. We conclude by
+validating the simulation against observations of the Solar System by NASA.
+
+\newpage
+\section{Introduction}
+% Background, overview of relevant literature, structure of remainder of the report
+
+% Introduce the problem that you have studied
+% –Describe what others have done (related work)
+% –Describe the structure of your report
+% •Make sure that you include enough and appropriate references.
+% –Explain for each referenced paper why it is relevant for your paper.
+% •In some cases, depending on the size of each part, the problem statement and the related work should be made into separate sections or chapters.
+
+All interactions of gravity are described by Newton's law of universal gravitation. Given initial
+positions and velocities, there exists a unique solution for the dynamics of the gravitating bodies.
+Analytical solutions are only available for system with at most two bodies. \cite{scholar:nbody}.
+For systems with more bodies we have no choice but to resort to numerical methods. Solving this
+problem has a long history starting from the father of gravity himself.
+
+When Sir Isaac Newton tried to predict planetary motion around the sun from initial conditions and
+some basic geometry he found that over the years his curve deviated from the true motion. He even
+correctly blamed this on the multiple interactive gravitational forces exerted by the other bodies
+in the system. So the n-body problem was born \cite{wiki:n-body-problem}.
+
+Nowadays n-body simulation is a widely researched and used field. Simulations of galaxies with
+billions of stars are used to predict how colliding galaxies will interact with each other. There
+are many software libraries available for scientists to conduct these experiments, examples include
+AMUSE \cite{amuse}, Pynbody \cite{pynbody} and Nemo \cite{nemo}. These are all sophisticated
+interfaces that require knowledge of physics and programming experience. In contrast, our simulation
+environment will aim to be accessible and appealing to everyone with an interest in gravity.
+
+This would mean that running a simulation should be as easy as going to a website, clicking on a
+button and being in awe of gravity. This can currently only be achieved by programming the
+simulation in JavaScript, the only programming language available on the browser. Keeping in mind
+that the simulation should work on fast and slower machines, our research will concentrate on the
+n-body problem where n is small: the few body problem. To make the simulations appealing it will be
+visualized in real-time in 3D.
+
+The remaining structure of this paper is as follows. We start by elucidating the physics in the few
+body problem, then we will translate them to usable algorithms with pseudo-code. We continue by
+introducing our implementation, Sunsistemo, discuss the visual effects we have implemented to make
+it more aesthetically appealing and how we validated the simulation.
+
+% Your report must clearly explain what it is about, and should not just be a repository of facts.
+% •The structure of this part depends on the nature of your work. E.g.:
+% –Start by describing your methods and algorithms
+% •Use formulae, pseudo-code
+% –Describe the implementation
+% •Describe import choices, leave out what should be self-evident
+% •But (well documented) source code goes to appendices, if to be included at all
+% –Describe your experiment and the results
+% –Interpret your results
+% •As with the introduction, the core may be split into separate sections, like methods,
+% experimentation and interpretation of the results.
+
+\section{Physics}
+In 1687 Sir Isaac Newton published not only his famous three laws of motion, but also his law of
+universal gravitation in his book \textit{Philosophiae Naturalis Principia Mathematica}, a treatise
+that is regarded as the most important work in the history of science. \cite{wiki:principia} His law
+of universal gravitation quantifies the interaction between all particles with mass in the universe,
+so it explains not only how objects fall on Earth, but also how the planets in the solar system fall
+around the Sun. This law will be at the heart of our simulation.
+
+\subsection{Gravitation}
+In vector notation Newton's law of universal gravitation is written as follows:
+\begin{equation} \label{eq:newton}
+ \vect{F}_{12}=-G\frac{m_{1}m_{2}}{|\vect{r}_{21}|^{2}}\hat{\vect{r}}_{21}
+\end{equation}
+where $\vec{F}_{12}$ is the vector force on particle 1 (with mass $m_{1}$) exerted by particle 2
+(with mass $m_{2}$), $G$ is the gravitational constant, $\vect{r}_{21}$ is the vector pointing from
+particle 2 to particle 1 and $\hat{\vect{r}}_{21}$ is its unit vector. \cite{giancoli}
+
+For systems with more than two particles the total force on particle number 1 is:
+\begin{equation} \label{eq:multi}
+\vect{F}_{1}=\vect{F}_{12} + \vect{F}_{13} + ... + \vect{F}_{1n} = \sum_{i=2}^{n} \vect{F}_{1i}.
+\end{equation}
+
+Using Newton's second law, $\vect{F_{1}}=m_{1} \vect{a_{1}}$ and equation \ref{eq:newton} we can
+derive an expression for the acceleration on particle 1 due to particle 2:
+\begin{align} \label{eq:newton2}
+ \vect{F}_{12} &=-G\frac{m_{1}m_{2}}{|\vect{r}_{21}|^{2}}\hat{\vect{r}}_{21} = m_{1} \vect{a_{1}}
+ \implies \vect{a_{1}} = -G\frac{m_{2}}{|\vect{r}_{21}|^{2}}\hat{\vect{r}}_{21}
+ = G\frac{m_{2}}{|\vect{r}_{12}|^{2}}\hat{\vect{r}}_{12} \nonumber \\[+3mm]
+ &= G\frac{m_{2}}{|\vect{r}_{21}|^{3}}\vect{r}_{21}.
+\end{align}
+Now combining this with equation \ref{eq:multi} we find an expression for the acceleration on
+particle 1 due to an arbitrary number of particles:
+\begin{equation} \label{eq:nbody}
+\vect{a}_{1} = \sum_{i=2}^{n} \vect{a}_{1i} = \sum_{i=2}^{n} G\frac{m_{i}}{|\vect{r}_{i1}|^{3}}\vect{r}_{i1}.
+\end{equation}
+Given initial conditions for the positions and velocities of the bodies in a system, equation
+\ref{eq:nbody} gives the acceleration for a body at any time. In general there is no analytical
+solution for the positions of gravitating bodies over time for systems with more than two bodies
+\cite{scholar:nbody}. For this reason we have to use numerical methods to calculate the evolution of
+the system. How this is done will be discussed in section \ref{sec:numerical} on numerical
+integration.
+
+A translation of equation \ref{eq:nbody} to python-esque pseudocode looks like:
+\begin{minted}[]{python}
+def accel(bodies, i):
+ """Calculate acceleration on body i"""
+ a = Vector3(0, 0, 0)
+ for (j != i) in bodies:
+ r = j.position - i.position
+ a += G * j.mass / (abs(r) ** 3) * r
+ return a
+\end{minted}
+Note that the acceleration is calculated for all bodies in the system, so we have a nested loop
+making this calculation of order $O(n^{2})$.
+
+In the preceding equations the bodies of mass are treated as point particles, i.e. spheres with zero
+radius. A more compelling visualization would have bodies with varying radii. To solve the then
+emerging problem of bodies being able to move through each other, we will look at the physics of
+collisions.
+
+\subsection{Collisions}
+\label{sec:collisions}
+Bodies with a well determined size can approach each other until their surfaces touch, when this
+happens there will be a moment of impact. What happens on such a collision depends on the material
+from which the bodies are made. Some options are:
+\begin{enumerate}
+\item The bodies do not collide but continue their movement. Which is the case for point particles
+ with a virtual texture shell.
+\item The two colliding bodies merge into one with combined masses. This is a simplification of what
+ happens to colliding blobs of fluid with high viscosity.
+\item The two bodies collide and change direction based on the initial velocities and body masses.
+ This can be a pure elastic collision where energy is conserved before and after the collision, or
+ an inelastic collision which does not conserve energy.
+\end{enumerate}
+For planets and other solid objects the third option makes the most sense, which is the one we will
+implement. For a particle of mass $m_1$ and velocity $v_1$ colliding with a particle of mass $m_2$,
+the velocity after an elastic collision $v_{1}^*$ is given by:
+
+\begin{equation}
+\label{eq:elColl}
+ \vect{v}_{1}^*=\vect{v}_{1} - \frac{2 m_2}{m_1 - m_2}\frac{\braket{\vect{v}_{12}|\vect{x}_{12}}}{\| \vect{r}_{12}\|^2}\cdot \vect{r}_{12}
+\end{equation}
+
+Where $v_{12} = (v_1 - v_2)$ and $x_{12} = (x_1 - x_2)$, with $v_1, v_2, x_1, x_2$ the velocities
+and positions of the particles before the collision took place. Although elastic collisions do not
+naturally occur in real world examples of the n-body problem, it was implemented to provide a more
+natural interaction between bodies, as before they would just pass through each other. Therefore
+when collisions are enabled, at each timestep every body is checked on whether it collides with
+another body. In the case of collision, the new velocity is calculated according to equation
+\ref{eq:elColl}:
+
+\begin{minted}[]{python}
+def collision(bodies, i):
+ """Check and calculate collision of body i"""
+ for (j != i) in bodies:
+ r12 = j.position - i.position
+ v12 = j.velocity - i.velocity
+
+ if r12 <= (j.radius + i.radius):
+ massFactor = 2 * j.mass / (i.mass + j.mass)
+ i.v = i.v - r12 * massFactor * (v12 * r12) / abs(r12)^2
+\end{minted}
+
+These were the necessary physics for the n-body problem. Given positions we can calculate the
+acceleration of the bodies and how they should interact when they collide. How they will actually
+move can be computed by numerical integration.
+
+\section{Numerical Integration}
+\label{sec:numerical}
+Numerical integration is used to evolve the initial positions and velocities of the n-body system
+over time. There exist several numerical methods to achieve this. Generally they are used to solve
+for solutions of ordinary differential equations. In our case the system to solve is:
+\begin{align} \label{eq:ode}
+ \frac{d \vect{v}(t)}{dt} &=a(\vect{x}_{1}(t), ..., \vect{x}_{n}(t)) \nonumber \\
+ \frac{d \vect{x}(t)}{dt} &= \vect{v}(t),
+\end{align}
+where the initial conditions $\vect{x}_{i}(0)$ and $\vect{v}_{i}(0)$ are given, and the solution is
+the set $(\vect{x}_{1}(t), ..., \vect{x}_{n}(t))$. Here we will review three of such methods to find
+these solutions and state which we've chosen to implement and why. Remember that this problem is
+three-dimensional and accordingly all the vectors have three dimensions as well.
+
+\subsection{Forward Euler Method}
+The simplest of such methods is the so called \textit{Forward Euler Method}. The idea is to take the
+initial condition of a variable and to continuously interpolate over a very small timestep $h$ times
+the derivative of that variable. These derivatives were defined by system \ref{eq:ode}. Notice that
+time is discretized in steps $h$. In general if we denote the \textit{n}th timestep by $t_{n}$ and
+the corresponding computed solution as $y_{n}$ the Euler method with initial conditions
+$(t_{n}, y_{n})$ is given by:
+\begin{equation}
+ y_{n+1} = y_n + h f(y_n, t_n),
+\end{equation}
+where $f$ is the differential equation of the system. \cite{wiki:euler} In our case the computation
+for $v_{n+1}$ and $x_{n+1}$ is as depicted in the following pseudocode:
+\begin{minted}[]{python}
+v[n + 1] = v[n] + h * a[n]
+x[n + 1] = x[n] + h * v[n]
+\end{minted}
+The forward Euler method is simple to implement but has a large error relative to the other methods.
+The authors were unable to evolve a system with three bodies using this scheme without them going to
+infinity. The local truncation error, or the error in each step of the integrator is proportional to
+$h^{2}$ and the global truncation error, the cumulative effect of the local truncation error, is
+proportional to $h$. \cite{wiki:euler} Thus the Forward Euler Method is an order 1 integration
+method. An alternative way to think about the order of an integrator is as follows: a numerical
+integrator of order 5 means that if we shrink the stepsize with a factor of 10, we decrease the
+error with a factor of $10^5$.
+
+As a further demonstration we will show how each of the three methods performs on numerically
+solving the harmonic oscillator (simple pendulum): $0 = \ddot{x} + \dot{x} = \dot{v} + \dot{x}$,
+where $x$ is the position and $v$ the velocity. The performance of the forward Euler method can be
+seen in figure \ref{fig:euler}. The integration started from $x_{0}=5$ and $v_{0}=3$ with a stepsize
+of $h=0.2$. It is clear that the error of this method is quite large and the pendulum explodes to
+infinity. The next method largely improves on this.
+
+\begin{figure}
+\center{\includegraphics[height=7cm]{Euler_Sling}}
+\caption{A solution to the harmonic oscillator numerically integrated with the Forward Euler Method
+ starting from $x_{0}=5$ and $v_{0}=3$ with a stepsize of $h=0.2$. We see that the pendulum has an
+ unexpected explosive growth in both position and velocity.}
+\label{fig:euler}
+\end{figure}
+
+\subsection{Symplectic Euler}
+A small modification to the forward Euler Method leads to the \textit{Symplectic Euler Method}, the
+position is updated using the just calculated new velocity:
+\begin{minted}[]{python}
+v[n + 1] = v[n] + h * a[n]
+x[n + 1] = x[n] + h * v[n + 1]
+\end{minted}
+A numerical integrator is called \textit{symplectic} if an area of initial conditions in phase space
+stays constant under its iteration. What this means physically, is that the energy in the system
+stays constant. This difference was immediately apparent in the implementation: it was finally
+possible to simulate a stable system of three bodies. The method's local and global truncation error
+are the same as of the Euler method, only the order of energy conservation has improved.
+
+What this means for the harmonic oscillator is shown in figure \ref{fig:symplectic}. While all the
+parameters of the integration are the same, we see that the system did not undergo an exponential
+growth, the position and velocity stay bounded. The \textit{symplectic} part of the integrator
+caused this change. What rests for the next method is an improvement over the local and global
+truncation error.
+
+\begin{figure}
+\center{\includegraphics[height=7cm]{symplecticEuler_Sling}}
+\caption{A solution to the harmonic oscillator numerically integrated with the Symplectic Euler
+ Method starting from $x_{0}=5$ and $v_{0}=3$ with a stepsize of $h=0.2$. The pendulum returns to
+ its original position after a full period, as it is supposed to. There is however an asymmetry in
+ the solution, the pendulum reaches its highest point just after the velocity is zero, not at the
+ same time. This can be fixed by using either a smaller stepsize or using a higher order
+ integrator.}
+\label{fig:symplectic}
+\end{figure}
+
+\subsection{Leapfrog}
+\label{sec:leapfrog}
+The \textit{Leapfrog} integrator is a second order integration scheme. \cite{leapfrog} That makes
+its global truncation error proportional to $h^2$ (note that this is smaller than $h$ for $h<0$).
+Furthermore this integrator is time reversible: if you would replace the stepsize $h$ by $-h$, you
+would get back precisely the starting conditions (up to rounding error). Time reversibility thus
+guarantees conservation of energy.
+
+\begin{figure}
+\center{\includegraphics[height=4cm]{leapfrog}}
+\caption{An illustration of the leapfrog integration scheme. The velocity starts off at half a
+ timestep and the integration continues with the subsequent iterations ``leaping'' over each
+ other's time.}
+\label{fig:leapfrog}
+\end{figure}
+
+In figure \ref{fig:leapfrog} we see an illustration of the leapfrog integration scheme. The major
+difference with the previous methods is that the leapfrog integrator is not synchronous. The
+velocities and positions are calculated at different times. The integrator starts off at $x_{0}$ and
+$v_{1/2}$ after which they take full steps leaping over each other's time. In practice we don't have
+the initial velocity at half a timestep, so we have to kickoff the leapfrog integrator by doing half
+an integration step of the velocity using another integrator, for example using the Forward Euler
+Method:
+\begin{minted}[]{python}
+vHalf[0] = v0 + 0.5 * h * a[0]
+\end{minted}
+Then the positions and velocities can leap over each other:
+\begin{minted}[]{python}
+x[n + 1] = x[n] + h * vHalf[n]
+vHalf[n + 1] = vHalf[n] + * h * a[n + 1]
+\end{minted}
+Notice that here the velocity is calculated with the new position, while in the symplectic Euler
+method the position is calculated with the new velocity.
+
+This reduction in error also further improves the performance on the harmonic oscillator. In figure
+\ref{fig:leap} we see the simple pendulum now solved with the Leapfrog integrator. The difference
+with figure \ref{fig:symplectic} is in the general shape, it is now symmetric in the x and y axes
+instead of being a slanted ellipse. This is due to the Leapfrog integrator being of higher order.
+
+\begin{figure}
+\center{\includegraphics[height=7cm]{Leapfrog_Sling}}
+\caption{A solution to the harmonic oscillator numerically integrated with the Leapfrog integrator
+ starting from $x_{0}=5$ and $v_{0}=3$ with a stepsize of $h=0.2$. The pendulum returns to its
+ original position after a full period, .}
+\label{fig:leap}
+\end{figure}
+
+We've now seen three methods, each improving over the previous one. Our last method is of second
+order and conserves the energy in the sytem. An important observation to keep in mind is that these
+methods all require about the same number of floating-point operations. The Leapfrog scheme is
+computationally as intensive as the Forward Euler Method but is still of a higher order. We could
+look at integrators of higher order, but they will require more flops. Thus we consider the Leapfrog
+integrator as most fitting for our purposes: fast, energy conserving and of second order.
+
+\section{Implementation}
+We've studied the necessary mathematics and algorithms to implement the few body problem. We've
+implemented these in JavaScript so aspiring gravity simulators only need a web browser with internet
+access. Our implementation is called Sunsistemo, available on \url{https://sunsistemo.js.org}. The
+source code is available at \url{https://github.com/sunsistemo/sunsistemo}.
+
+We made use of the JavaScript library three.js which enables users to make GPU accelerated 3D
+graphics in the browser through the use of WebGL. WebGL in itself is cumbersome to use as it is a
+low level language. A simple 3D object would require many lines of control and shader code, however
+three.js offers an higher level API for the creation and manipulation of 3D objects which are then
+rendered using WebGL.
+
+In figure \ref{fig:flowchart} the flowchart of Sunsistemo is shown, a high level overview of
+Sunsistemo's design. We can divide Sunsistemo in two parts, simulation and visualization. Body
+objects with positions and velocities live in the simulation part and they are represented by
+three.js spheres in the visualization part.
+
+\begin{figure}
+\center{\includegraphics[height=9cm]{Flowchart}}
+\caption{Highlevel flowchart of Sunsistemo. Red arrows represent the processes that are executed
+ every frame of the visualization}
+\label{fig:flowchart}
+\end{figure}
+
+
+\subsection{Simulation}
+The simulation starts with initiation of a system containing an array of body objects. Body objects
+have some static parameters (mass, radius, texture, rotation) and some dynamic variables (position,
+velocity) which can be changed during the simulation. The system also defines some simulation and
+visualization parameters (stepsize, steps per animation frame, etc.). Every timestep new body
+positions and velocities are calculated from the values of the previous timestep using the leapfrog
+algorithm from section \ref{sec:leapfrog}. After every step, each body is checked for collisions
+which are performed using the algorithm described in section \ref{sec:collisions}. The integration
+and collision algorithms update the velocity and position vectors of each body so these new
+variables can be used to update the visualization and are used as input for the next simulation
+step.
+
+\subsection{Systems}
+Sunsistemo is built in a modular fashion so it can simulate many different systems. The systems
+currently defined are:
+\begin{itemize}
+\item \textbf{Two Bodies:} two bodies in a Keplerian orbit.
+\item \textbf{Three Bodies:} two planets orbiting a star. This system isn't stable, after some time
+ one of the planets leaves.
+\item \textbf{The Solar System:} the Sun, Mercury, Venus, Earth, the Moon, Mars, Jupiter, Saturn,
+ Uranus, Neptune and Pluto. The initial conditions for this system are from NASA's HORIZONS system.
+ \cite{stan}
+\item \textbf{Random bodies:} a collection of many bodies each with random positions and velocity.
+\item \textbf{Angular Momentum:} many planets orbiting around the Sun. There is no interaction when
+ bodies collide: they pass through each other.
+\item \textbf{Angular with Bounce:} same as the previous but now the planets are balls and they
+ experience elastic collisions when they touch.
+\item \textbf{N-body choreographies:} there are some analytic solutions for the N-body problem where
+ $n>2$. These are special cases where often the bodies are of equal mass. We've implemented all
+ systems reported in the paper by {{\v S}uvakov} and {Dmitra{\v s}inovi{\'c}}.
+ \cite{three-body-planar} These systems are in theory stable, but in our simulation the numerical
+ errors build up and some bodies escape.
+\end{itemize}
+
+\subsection{Visualization}
+The initial idea of the project was to make a visualization of the few-body problem that is
+appealing to the eye and pleasant to use. Using three.js the following features were added to make
+the visualization aesthetically attractive:
+\begin{itemize}
+\item \emph{Spheres}\\
+ To add to the visual experience of the end user, instead of point particles the visualization uses
+ sphere geometries of variable size to visualize the bodies. This decision made it necessary to
+ implement close range interaction, as described in chapter \ref{sec:collisions}, as the spheres
+ would pass through each others radius without it. Sphere size can be coupled to the body radius,
+ for example to show the relative physical size of the planets in the solar system. The number of
+ polygons of which the sphere is build up can be adapted to alleviate the computational cost in
+ case of systems with many bodies. After each simulation cycle the new body position values are
+ used to update sphere positions and the new simulation frame is rendered.
+
+\item \emph{Lighting}\\
+ To achieve better 3D experience, proper lighting is necessary. A combination of ambient lighting
+ and a point source is used. In the solar system the point source can be coupled to the sphere
+ representing the sun. To the sun we've also added a halo sprite around the sphere, which mimics
+ the solar atmosphere. Phong shading is used to determine the light level on the geometries. Phong
+ shading interpolates the surface normals across a geometry to show a smoother surface, in contrast
+ to flat shading which calculates the light level from the actual polygon surface normal.
+
+
+\item \emph{Textures}\\
+ For visual appeal and to be able to discern the different bodies, a body texture is applied to the
+ sphere geometries. Textures of the eight solar system planets, the Sun and Pluto were used for the
+ solar system and other random systems. \cite{planettexture} Different textures can be used to
+ mimic other physical systems, for example the random bodies system where softball and tennisball
+ textures are used. \cite{balltexture}
+
+\item \emph{Bumpmap}\\
+ The use of the point light source, which causes directed lighting, enables us to assign a bump map
+ to the spheres. The bump map adjusts the surface normals which are used for calculating the
+ surface lighting. There is no use in this for simulation purposes but from an aesthetic point of
+ view it is interesting to have a detailed surface instead of a smooth one. Since we already use
+ Phong shading, the implementation of a bump map does not require much more computational power.
+ For even more realistic lighting a specular map can be added which adjusts the amount of specular
+ reflection as opposed to diffuse reflection. For example, the water covered parts of the earth
+ surface have a higher specular reflection while the reflection of land covered parts is mainly
+ diffuse. Bump and specular maps where obtained from the same source as body textures.
+
+
+\item \emph{Scaling}\\
+ To make an appealing visualization of the solar system it is inevitable to scale the objects radii
+ with respect to the orbit radii. Without this scaling the ratio of orbit radius to object radius
+ for the earth would be:
+\begin{equation}
+q = \frac{1.5 *10^{11} \mbox{ m}}{6.4*10^6 \mbox{ m}} = 2.3 *10^4
+\end{equation}
+Consequently we would need a screen of approximately $10^4$ pixels wide to see its full orbit and
+have the earth visualized as a single pixel. Additionally the ratio between the radius of the Sun
+and the radius of the inner planets is in the order of $10^2$ and $10^1$ for the radii of inner and
+outer planets. This implies we need a non-linear scaling for the size of the celestial objects. We
+decided on a scaling of:
+\begin{equation}
+s = \sqrt[5]{r_{\mbox{planet}}} \cdot 2 \cdot 10^9
+\end{equation}
+
+This scaling works as an equalizer for the object radii because of the fifth root and blows up the
+size of the objects to be of approximately the same order as the orbit radius.
+
+\item \emph{GUI}\\
+ To let users access and interact with the visualization from the browser a graphical user
+ interface is needed. While this may change in the future, for now a simple button menu was chosen
+ which lets the user select predefined systems and choreographies of which one is the solar system.
+ The interface was built with the Javascript data binding library D3.js. \cite{d3lib}
+
+\item \emph{Interaction}\\
+ Users can also interact in real-time with the simulation. Using their mouse they can manipulate
+ the camera: left-clicking and dragging will change the perspective of the camera, right-clicking
+ and dragging will change the position of the camera and scrolling will zoom in or out.
+
+\end{itemize}
+
+
+\section{Validation}
+We've created a model of gravity using Newton and the Leapfrog integrator. Everyone can create a
+model, but few can create a good model. That is why it's required to validate the model. We will
+perform an error analysis for the solar system simulation.
+
+We can define the error of the simulation at time $t$ as:
+\begin{equation} \label{eq:error}
+E = \frac{1}{n} \sqrt{ \sum_{i=0}^{n} \|\hat{\vect{x}}_i - \vect{x}_i\|^2},
+\end{equation}
+where $n$ is the number of bodies, $\hat{\vect{x}}_i$ the computed position of body $i$ and
+$\vect{x}_i$ its real position. A validation of the model would then look like a plot of the error
+as defined in equation \ref{eq:error} over time.
+
+\subsection{The Solar System}
+The solar system is constantly being observed by astronomers. NASA's HORIZONS system provides highly
+accurate data for objects in the solar system. They report having data on ``(696540 asteroids, 3366
+comets, 178 planetary satellites, 8 planets, the Sun, L1, L2, select spacecraft, and system
+barycenters)'' \cite{stan} We took the initial conditions for the solar system from NASA for the
+simulation. For the validation we compare the computed data with data from NASA over a range of
+years
+
+In figure \ref{fig:err_leap} we see the error of the leapfrog method with stepsizes of 24 hours
+(blue) and 12 hours (green). The error increases linearly over time. Lowering the stepsize naturally
+decreases the error. The red line has errors with exactly half of those of stepsize 24h. For a
+second order integrator halving the stepsize should lead to a $2^2$ factor of decrease in error, but
+this was not observed. The distance between the red and green lines show that halving the stepsize
+roughly also halves the error.
+\begin{figure}
+\center{\includegraphics[height=9cm]{errors_leapfrog}}
+\caption{The error over a range of years in the integration of the Solar System using the Leapfrog
+ integration method.}
+\label{fig:err_leap}
+\end{figure}
+
+In figure \ref{fig:err_symp} we see the performance of the symplectic Euler method. There is a clear
+difference in how the error evolves: it is periodic instead of linear. Another clear difference is
+in the effectiveness of reducing the error in decreasing the stepsize: halving gives an error of
+roughly the same magnitude. This is expected as the leapfrog integrator was of higher order.
+\begin{figure}
+\center{\includegraphics[height=9cm]{errors_symplectic1}}
+\caption{The error over a range of years in the integration of the Solar System using the Symplectic
+ Euler integration method.}
+\label{fig:err_symp}
+\end{figure}
+
+Finally in figure \ref{fig:err_leap_sym} we see a comparison between performance of the leapfrog and
+symplectic Euler method. This is clear evidence confirming that the Leapfrog integrator has an
+overall smaller error than the Symplectic Euler method.
+\begin{figure}
+\center{\includegraphics[height=9cm]{errors_leap_symplectic}}
+\caption{The error over time for the Leapfrog and Symplectic Euler methods. The Leapfrog method
+ performs better with the same stepsize.}
+\label{fig:err_leap_sym}
+\end{figure}
+
+This concludes the validation. Of course we can decrease the stepsize to get lower and lower errors
+but that is not the point of the validation. We've confirmed that:
+\begin{enumerate}
+\item The simulation ``works''. The magnitude of the error is sensible, it is on the order of the
+ position of Mercury (1E10). The gradual increase in error is also as expected.
+\item We've demonstrated that the Leapfrog method does indeed have smaller errors than the
+ Symplectic Euler method.
+\end{enumerate}
+
+\section{Conclusions}
+% In this section you give a clear resume of the conclusions from your work and discuss its relation to previous research.
+% –For that reason, some authors will actually put the section on related work after the core section(s)
+% –The conclusions should reflect back on the original problem statement in your introduction.
+% •When applicable, your work may result in suggestions on how to further validate or extend your research.
+We set out to make a few body simulator that is both accessible and visually pleasing. We started
+with an overview of the physics and continued with numerical integration. Introducing our
+implementation, Sunsistemo \cite{sunsistemo}, we discussed the visualization techniques that were
+applied to make it appealing. We have also validated the numerical model and shown that the choice
+for implementing the Leapfrog integrator was not in vain. In figure \ref{fig:sunsistemoPrtSc} we see
+a screenshot of Sunsistemo that can be reproduced by anyone within seconds by going to the website.
+With that we conclude that we've largely accomplished our initial goals.
+
+This is however not the end, there are still many aspects of the model that can be improved.
+
+\begin{figure}
+\center{\includegraphics[height=9cm]{sunsistemoPrtSc}}
+\caption{A screenshot of Sunsistemo in action. Seen are the Sun and its five inner planets orbiting
+ around it.}
+\label{fig:sunsistemoPrtSc}
+\end{figure}
+
+\section{Recommendations}
+In further development of Sunsistemo much can be done to improve the application. Both on the
+simulation side and on the graphical visualization side. These major improvements are more or less
+ordered by priority and can also be found on the GitHub page \cite{sunsistemoGH} amongst other minor
+issues.
+
+\begin{itemize}
+\item \emph{Adaptive stepsize}\\
+ The closer two bodies approach each other, the higher the gravitational acceleration. With a
+ faster changing velocity, the error in the simulation also grows because the linear distance
+ covered per step becomes bigger. In Sunsistemo this can most prominently be seen when collisions
+ are turned off, bodies approach each other to very small distance and tend to fly out of orbit
+ after these close approaches because they maintain the high velocities obtained at perihelion for
+ too long due to a large step size. A solution would be to change the step size to the order where
+ this effect does not occur, however this would cost a lot of computational power and is not
+ necessary for most part of the orbit since acceleration and velocities are not that large. A
+ variable step size, proportional to the minimum separation between the body and any other body,
+ could solve this problem.
+
+\item \emph{DIY simulations}\\
+ The goal of Sunsistemo was to encourage people to start experimenting with the n-body simulator,
+ at the moment however the interaction with the end user is limited to the simple GUI which only
+ offers a wide range of predefined systems. It would be better if users could define their own
+ system by selecting the amount of bodies and whether they are distributed randomly or at specified
+ initial positions and velocities. The API of Sunsistemo is suitable for this, but the GUI still
+ has to be build.
+
+\item \emph{Validation with the analytical 2-body problem}\\
+ Besides the validation using NASA data of the solar system there is another way validating the
+ model using the analytic solution of the 2-body problem. The differential equation for a system of
+ two bodies of mass $m_1$ and $m_2$:
+\begin{equation}
+\label{eq:dtTwoBody}
+\frac{dr}{dt} = \sqrt{\frac{2}{\mu}}\left(E-\frac{1}{2}\frac{L^2}{\mu r^2} + \frac{G m_1 m_2}{r}\right)^{\frac{1}{2}}
+\end{equation}
+
+with $\mu = \frac{m_1 m_2}{m_1 + m_2}$ the reduced mass and $r = |{\vect{r}_1 - \vect{r}_2}|$ the
+distance between the two bodies. It could be integrated directly, however no explicit solution for
+$r(t)$ can be obtained. So a different approach should be used converting the Cartesian simulation
+coordinates to cylindrical reduced mass coordinates and compare them with solutions of the form
+$r(\theta)$ which can be obtained.
+
+
+\item \emph{Higher order (symplectic) integrators}\\
+ The currently used leapfrog algorithm is a second-order symplectic integrator. To achieve a more
+ stable simulation we can implement integration algorithms of higher order. This will result in
+ smaller errors, however higher order integrators in general require more function evaluations per
+ simulation step. Nevertheless it would be interesting to test several integrators like Verlet
+ integration or the Runge-Kutta method.
+
+\item \emph{Approximation scheme}\\
+ When using these higher order integrators it might be necessary to implement a more sophisticated
+ algorithm to add all interactions on a body and approximate this for far away bodies to decrease
+ computational cost. For example the Barnes-Hut method which divides space in cubic cells and lets
+ the body of interest interact with the center of mass of each cell instead of every single body.
+ This would also possibly allow Sunsistemo to scale up to be an n-body simulator instead of the
+ few-body simulator it is now with decreasing performance at around 400 bodies.
+
+\item \emph{Better physics for collisions}\\
+ At the moment the physics of collisions between bodies is described by the perfect elastic
+ collision. Perfect elastic collisions do not exist in nature, so to improve the model the close
+ range interaction should be changed. Merging is an interesting option for a colliding planet and
+ star or black hole, semi-elastic collisions could also be implemented for smaller celestial
+ objects, which bounce off each other but lose energy in the process. It must be said however that
+ for simulation of systems like the solar system, collisions are not of importance because of the
+ difference in scale between orbit and object radii.
+
+\item \emph{Stereoscopy}\\
+ To make the visualization even more appealing the 3D vision could be enhanced by adding
+ stereoscopy. This is easy to implement by rendering the same simulation from two different angles
+ and overlay the rendered frames with different hue filters for anaglyph or stack them side-by-side
+ for display on a 3D television, projector or oculus rift.
+\end{itemize}
+
+
+\bibliography{references}
+\bibliographystyle{unsrt}
+
+\end{document}