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\section*{Format for delivery of report and programs}
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The format of the project is that of a printed file or hand-written report.
The programs should also be included with the report. Write only your candidate number on the first
page of the report and state clearly that this is your report for project 5 of FYS3150/FYS4150, fall 2012. 
There will be a box marked 'FYS3150/FYS4150' at the reception of the Department of Physics (room FV128). 

\section*{Project 5, The solar system, deadline   December 10, 12pm (noon)}

We study first a hypothetical solar system with one planet, say Earth, which orbits around the Sun.
The only force in the problem is gravity. Newton's law of gravitation  is given by a force $F_G$
\[
F_G=\frac{GM_{\mathrm{sun}}M_{\mathrm{Earth}}}{r^2},
\]
where $M_{\mathrm{sun}}$ is the mass of the Sun and $M_{\mathrm{Earth}}$ is the mass of Earth. The gravitational constant is $G$ and $r$ is the distance between Earth and the Sun.
We assume that the sun has a mass which is much larger 
than that of Earth. We can therefore safely neglect the 
motion of the sun in this problem.  In the first part of this project, your aim is to compute the motion
of the Earth using different methods for solving ordinary differential equations.

We assume that the orbit of Earth around the Sun 
is co-planar, and we take this to be the $xy$-plane.
Using Newton's second law of motion we get the following equations
\[
\frac{d^2x}{dt^2}=\frac{F_{G,x}}{M_{\mathrm{Earth}}},
\]
and 
\[
\frac{d^2y}{dt^2}=\frac{F_{G,y}}{M_{\mathrm{Earth}}},
\]
where $F_{G,x}$ and $F_{G,y}$ are the $x$ and $y$ components of the gravitational force. 
\begin{enumerate}
\item[a)]  Rewrite the above second-order ordinary differential equations as a set of coupled first order
differential equations. Write also these equations in terms of dimensionless variables.
As an alternative to the usage of dimensionless variables, you could also
use so-called  astronomical units (AU as abbreviation). If you choose the latter set of units, 
one astronomical unit of length, known as 1 AU, is the average distance between the Sun and Earth, that is
$1$ AU = $1.5\times 10^{11}$ m.  It can also be convenient to use years instead of seconds since years match
better the solar system. The mass of the Sun is $M_{\mathrm{sun}}=M_{\odot}=2\times 10^{30}$ kg. The mass of Earth is
$M_{\mathrm{Earth}}=6\times 10^{24}$ kg. The mass of other planets like Jupiter is 
$M_{\mathrm{Jupiter}}=1.9\times 10^{27}$ kg and its distance to the Sun is 5.20 AU. Similar numbers for Mars
are $M_{\mathrm{Mars}}=6.6\times 10^{23}$ kg and  1.52 AU, for Venus $M_{\mathrm{Venus}}=4.9\times 10^{24}$ kg and  0.72 AU, for Saturn are $M_{\mathrm{Saturn}}=5.5\times 10^{26}$ kg and  9.54 AU, for Mercury are $M_{\mathrm{Mercury}}=2.4\times 10^{23}$ kg and  0.39 AU, for Uranus are $M_{\mathrm{Uranus}}=8.8\times 10^{25}$ kg and  19.19 AU, for Neptun are $M_{\mathrm{Neptun}}=1.03\times 10^{26}$ kg and  30.06 AU and for Pluto are $M_{\mathrm{Pluto}}=1.31\times 10^{22}$ kg and  39.53 AU. Pluto is no longer considered  a planet, but we add it here for historical reasons.

Finally,  mass units can be obtained by using the fact that Earth's orbit is almost circular around the Sun.
For circular motion we know that the force must obey the following relation
\[
F_G= \frac{M_{\mathrm{Earth}}v^2}{r}=\frac{GM_{\odot}M_{\mathrm{Earth}}}{r^2},
\]
where $v$ is the velocity of Earth. 
The latter equation can be used to show that
\[
v^2r=GM_{\odot}=4\pi^2\mathrm{AU}^3/\mathrm{yr}^2.
\]
Discretize the above differential equations and set up an algorithm for solving these equations using the so-called Euler-Cromer method discussed in the lecture notes, chapter 8.
\item[b)]  Write then a program which solves the above differential equations for the Earth-Sun system
using the Euler-Cromer method. Find out which initial value for the velocity that gives a circular orbit
and test the stability of your algorithm as function of different time steps $\Delta t$. Find a possible maximum
value $\Delta t$ for which the Euler-Cromer method does not yield stable results. Make a plot of the results you obtain for the position of Earth (plot the $x$ and $y$ values) orbiting  the Sun.

Check also for the case of a circular orbit that both the kinetic and the potential energies are constants.
Check also that the angular momentum is a constant. Explain why these quantities
are conserved.
\item[c)] Modify your code by implementing the fourth-order Runge-Kutta method and compare the stability of your results by repeating the steps in b). Compare the stability of the two methods, in particular as functions
of the needed step length $\Delta t$. Comment your results.

\item[d)] Kepler's second law states that the line joining a planet to the Sun sweeps out equal areas in equal times. Modify your code so that you can verify Kepler's second law for the case of an elliptical orbit. 
Compare both the Runge-Kutta method and the Euler-Cromer method and check that the total energy and angular
momentum are conserved. Why are these quantities conserved?  
A convenient choice of starting values are an initial position of 1 AU and an initial velocity of 5 AU/yr.

\item[e)] Consider then a planet which begins at a distance of 1 AU from the sun. Find out by trial and error
what the initial velocity must be in order for the planet to escape from the sun.  Can you find an exact answer?

\item[f)]  We will now study the three-body problem, still with the Sun kept fixed at the center but 
including Jupiter (the most massive planet in the solar system, having a mass that is approximately 1000 times
smaller than that of the Sun) together with Earth. This leads us to a three-body problem. Without Jupiter, Earth's motion is stable and unchanging with time. The aim here is to find out how much Jupiter alters Earth's motion.

The program you have developed can easily be modified by simply adding the magnitude of the force betweem Earth and Jupiter.

This force is given again by 
\[
F_{\mathrm{Earth-Jupiter}}=\frac{GM_{\mathrm{Jupiter}}M_{\mathrm{Earth}}}{r_{\mathrm{Earth-Jupiter}}^2},
\]
where $M_{\mathrm{Jupiter}}$ is the mass of the sun and $M_{\mathrm{Earth}}$ is the mass of Earth. 
The gravitational constant is $G$ and $r_{\mathrm{Earth-Jupiter}}$ is the distance between Earth and Jupiter.

We assume again that the orbits of the two planets are co-planar, and we take this to be the $xy$-plane. 
Modify your first-order differential equations in order to accomodate both the
motion of Earth and Jupiter by taking into account the distance in $x$ and
$y$ between Earth and Jupiter. Set up the algorithm and plot the positions of Earth and Jupiter using the fourth-order Runge-Kutta method.  Include an adaptive solver to your Runge-Kutta method, using for example Richardson's extrapolation scheme
discussed in chapter three of the lecture notes. Alternative you can look up the adaptive scheme proposed by Fehlberg,
see for example \url{http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta%E2%80%93Fehlberg_method}.

Discuss the stability of the solutions using the standard Runge-Kutta4 solver and the adaptive scheme.

Repeat 
the calculations by increasing the mass of Jupiter by a factor of 10 and 1000
 and plot the position of Earth.  Study again the stability of the standard and the adaptive Runge-Kutta solvers.

\item[g)] Finally, using your optimal Runge-Kutta solver, we carry out a real three-body calculation where all three systems, 
Earth, Jupiter and the Sun are in motion. To do this, choose the center-of-mass position of the three-body system as 
the origin rather than the position of the sun. Give the sun an initial velocity which makes the total momentum of the system exactly zero (the center-of-mass will remain fixed). Compare these results with those from the previous exercise and comment your results. Extend your program to include all planets in the solar system (if you have time, you can also include the various moons, but it is not required) and discuss your results. Try to find data for the initial positions and velocities for all planets. 


\item[h)] The perihelion precession of Mercury. This part is optional but gives you an additional 30\% on the final score!  

An important test of the general theory of relativity was comparing its prediction for the
perihelion precession of Mercury to the observed value. The observed value of the perihelion precession, when
all classical effects (such as the perturbation of the orbit due to gravitational attraction from the other planets) are
subtracted, is $43''$ ($43$ arc seconds) per century.

Closed elliptical orbits are a special feature of the Newtonian $1/r^2$ force. In general, any correction to the
pure $1/r^2$ behaviour will lead to an orbit which is not closed, i.e. after one complete orbit around the Sun, the
planet will not be at exactly the same position as it started. If the correction is small, then each orbit around
the Sun will be almost the same as the classical ellipse, and the orbit can be thought of as an ellipse whose 
orientation in space slowly rotates. In other words, the perihelion of the ellipse slowly precesses around the Sun.

You will now study the orbit of Mercury around the Sun, adding a general relativistic correction to the Newtonian
gravitational force, so that the force becomes
\[
F_G = \frac{GM_\mathrm{Sun}M_\mathrm{Mercury}}{r^2}\left[1 + \frac{3l^2}{r^2c^2}\right]
\]
where $M_\mathrm{Mercury}$ is the mass of Mercury, $r$ is the distance between Mercury and the Sun, $l=|\vec{r}\times\vec{v}|$ is the magnitude of Mercury's orbital angular momentum per unit mass, 
and $c$ is the speed of light in vacuum. Run a simulation 
over one century of Mercury's orbit around the Sun with no other planets present, starting with Mercury at perihelion on the $x$ axis.
Check then the value of the perihelion angle $\theta_\mathrm{p}$, using
\[
\tan \theta_\mathrm{p} = \frac{y_\mathrm{p}}{x_\mathrm{p}}
\]
where $x_\mathrm{p}$ ($y_\mathrm{p}$) is the $x$ ($y$) position of Mercury at perihelion, i.e. at the point
where Mercury is at its closest to the Sun. You may use that the speed of Mercury at perihelion is $12.44\,\mathrm{AU}/\mathrm{yr}$, and that the distance to the Sun
at perihelion is $0.3075\,\mathrm{AU}$.
You need to make sure that the time resolution used in your simulation
is sufficient, for example by checking that the perihelion precession you get with a pure Newtonian force is at least
a few orders of magnitude smaller than the observed perihelion precession of Mercury. Can the observed perihelion 
precession of Mercury be explained by the general theory of relativity?














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%\title{\bf{Project 1, FYS4560}}
%\author{Magnar K. Bugge \\
%\texttt\small{m.k.bugge@fys.uio.no}}
%\maketitle

\section*{The perihelion precession of Mercury}


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