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\section*{Introduction to numerical projects}

Here follows a brief recipe and recommendation on how to write a report for each
project.
\begin{itemize}
\item Give a short description of the nature of the problem and the eventual 
numerical methods you have used.
\item Describe the algorithm you have used and/or developed. Here you may find it convenient
to use pseudocoding. In many cases you can describe the algorithm
in the program itself.

\item Include the source code of your program. Comment your program properly.
\item If possible, try to find analytic solutions, or known limits
in order to test your program when developing the code.
\item Include your results either in figure form or in a table. Remember to
       label your results. All tables and figures should have relevant captions
       and labels on the axes.
\item Try to evaluate the reliabilty and numerical stability/precision
of your results. If possible, include a qualitative and/or quantitative
discussion of the numerical stability, eventual loss of precision etc. 

\item Try to give an interpretation of you results in your answers to 
the problems.
\item Critique: if possible include your comments and reflections about the 
exercise, whether you felt you learnt something, ideas for improvements and 
other thoughts you've made when solving the exercise.
We wish to keep this course at the interactive level and your comments can help
us improve it.
\item Try to establish a practice where you log your work at the 
computerlab. You may find such a logbook very handy at later stages
in your work, especially when you don't properly remember 
what a previous test version 
of your program did. Here you could also record 
the time spent on solving the exercise, various algorithms you may have tested
or other topics which you feel worthy of mentioning.
\end{itemize}



\section*{Format for electronic delivery of report and programs}
%
The preferred format for the report is a PDF file. You can also
use DOC or postscript formats. 
As programming language we prefer that you choose between C/C++, Fortran or Python.
The following prescription should be followed when preparing the report:
\begin{itemize}
\item Use Fronter to hand in your projects, log in  at 
blyant.uio.no and choose 'fellesrom fys3150'.
Thereafter you will see an icon to the left with 'hand in' or 'innlevering'.
Click on that icon and go to the given project. 
There you can load up the files within the deadline.
\item Upload {\bf only} the report file and the source code file(s) you have developed.
The report file should include all of your discussions and a list of the codes you have developed. 
Do not include library files which are available at the course homepage, unless you have
made specific changes to them.
\item Comments  from us on your projects, approval or not, corrections to be made 
etc can be found under
your Fronter domain and are only visible to you and the teachers of the course.

\end{itemize}
Finally, 
we do prefer that you work two and two together. Optimal working groups consist of 
2-3 students. You can then hand in a common report. 

\section*{Project 2, Structure of white dwarf stars, deadline   September  29  12am (midnight)}

This project contains a long description of the physics of compact objects such as white dwarfs.
It serves as a background for understanding the final differential equations you need to solve.

White dwarfs are cold objects which consist mainly of heavy nuclei such as $^{56}$Fe, with 26 protons, 30 neutrons
and  their respective electrons.  Charge equilibrium results in an equal quantity of electrons and protons.
You can read more about white dwarfs, neutron stars and black holes at the website of the
Joint Institute for Nuclear Astrophysics www.jinaweb.org or NASA's website www.nasa.org.
These stars are the endpoints of stars with masses of the size or smaller than our sun. 
They are the outcome of standard nuclear processes and end their lives as 
cold objects like white dwarfs when they have used up all their nuclear fuel.

Where a star ends up at the end of its life depends on the mass, or amount of matter, it was born with. 
Stars that have a lot of mass may end their lives as black holes or neutron stars. Low and medium mass stars 
will become something called a white dwarf. A typical white dwarf is half as massive as the Sun, yet 
only slightly bigger than the Earth. This makes white dwarfs one of the densest forms of matter, surpassed only by neutron stars.

Medium mass stars, like our Sun, live by burning the hydrogen that dwells within their cores, 
turning it into helium. This is what our Sun is doing now. The heat the Sun generates by its 
nuclear fusion of hydrogen into helium creates an outward pressure. In another 5 billion years, 
the Sun will have used up all the hydrogen in its core.

This situation in a star is similar to a pressure cooker. Heating something 
in a sealed container causes a build up in pressure. The same thing happens in the Sun. 
Although the Sun may not strictly be a sealed container, gravity causes it to act like one, pulling the 
star inward, while the pressure created by the hot gas in the core pushes to get out. 
The balance between pressure and gravity is very delicate. 

Because a white dwarf is no longer able to create internal pressure, gravity unopposedly crushes 
it down until even the very electrons that make up a white dwarf's atoms are mashed together. 
In normal circumstances, identical electrons (those with the same "spin") are not allowed to occupy the same energy level. 
Since there are only two ways an electron can spin, only two electrons can occupy a single energy level. 
This is what's know in physics as the Pauli Exclusion Principle. And in a normal gas, this isn't a problem; 
there aren't enough electrons floating around to completely fill up all the energy levels. But in a white dwarf, 
all of its electrons are forced close together; soon all the energy levels in its atoms are filled up with electrons. 
If all the energy levels are filled, and it is impossible to put more than two electrons in each level, 
then our white dwarf has become degenerate. For gravity to compress the white dwarf anymore, 
it must force electrons where they cannot go. Once a star is degenerate, gravity cannot compress it any more 
because quantum mechanics tells us there is no more available space to be taken up. So our white dwarf survives, 
not by internal combustion, but by quantum mechanical principles that prevent its complete collapse.



With a surface gravity of 100,000 times that of the earth, the atmosphere of 
a white dwarf is very strange. The heavier atoms in its atmosphere sink and the lighter ones remain at the surface. 
Some white dwarfs have almost pure hydrogen or helium atmospheres, the lightest of elements. 
Also, the very strong gravity pulls the atmosphere close around it in a very thin layer, that, 
if were it on earth, would be lower than the tops of our skyscrapers!

\subsection*{Equilibrium equations}
We assume that the star is in thermal equilibrium. It exhibits also charge equilibrium, meaning the number of electrons has to balance the number of protons. The gravitational pull 
on every element of volume is balanced by the pressure set up by a degenerate gas of electrons
at $T=0$, since the temperature of the star is well below the so-called Fermi temperature of the 
electrons. 
The electrons are assumed to be relativistic and since the protons and neutrons have much lower
kinetic energy, we assume that the pressure which balances the gravitational force is mainly
set up by the relativistic electrons. The kinetic energy of the electrons is also much larger
than the electron-electron repulsion or the attraction from the nuclei. This means that we can treat 
the system as a gas of free degenerate electrons at $T=0$ moving in between a lattice of nuclei like  
iron. This is our ansatz. 
Based on this we can derive the pressure which counterbalances the gravitational force given by
(for every element of volume in a distance $r$ from the center of the star)
\[
F_{Grav}=-\frac{Gm(r)}{r^{2}}\rho(r),
\]
with $G$ being the  gravitational constant, $\rho (r)$ the mass density  (mass per volume) of a volume
element a distance $r$ from the center of the star,
and  $m(r)$ is the integrated mass within a radius $r$. The latter reads
\[
m(r)=4\pi\int_{0}^{r}\rho (r')r'^{2}dr'
\]
which yields a differential equation between the total mass and the mass density
\[
\frac{dm}{dr}=4\pi r^{2}\rho (r).
\]
In equilibrium, the pressure $P$ balances the gravitational force
\[
\frac{dP}{dr}=-\frac{Gm(r)}{r^{2}}\rho (r),
\]
and using 
\[
\frac{dP}{dr} = \frac{d\rho}{dr}\frac{dP}{d\rho}
\]
 we obtain
\[
\frac{d\rho}{dr}=-\left(\frac{dP}{d\rho}\right)^{-1}\frac{Gm}{r^{2}}\rho.
\]
Together with $\frac{dm}{dr}=4\pi r^{2}\rho (r)$
we have now two coupled first-order ordinary differential equations
which determine the structure of the white dwarf given an equation of state
$P(\rho )$.
The total radius is given by the condition 
$\rho (R)=0$. 
Similarly, the mass for $r=0$
is $m=0$.  
The density at $r=0$ is given by the central density $\rho_{c}$, a parameter you will have to play with 
as input parameter.

By integrating the last equation, we find the density profile of the star.
The radius $R$ is determined by the point where 
the density distribution
is $\rho =0$. 
The mass is then given by $M=m(R)$.
Since both the total mass and the radius $R$ will depend on the central density
$\rho_{c}$, a variation of this parameter will allow us to study stars 
with different masses.
However, before we can proceed, we need the pressure for a relativistic gas of electrons.

\subsection*{Equation of state for a white dwarf}
We will treat the electrons as a relativistic gas of fermions at $T=0$.
From statistical physics we can then obtain the particle density as
(we set $\hbar = c = 1$)
\[
n=N/V=\frac{1}{\pi^{2}}\int_{0}^{k_{F}}k^{2}dk=\frac{k_{F}^{3}}{3\pi^{2}},
\]
where $k_F$ is the Fermi momentum, here represented by the wave number $k_F$.
The wave number is connected to the momentum via $k_F=p_F/\hbar$.
The energy density is given by
\[
\varepsilon=E/V=\frac{1}{\pi^{2}}\int_{0}^{k_{F}}k^{2}dk
\sqrt{(k)^{2}+m_{e}^{2}}.
\]
This expression is of the form $\int y^2\sqrt{y^2+a^2}$. Performing the integration we obtain
\[
E/V=n_0m_ex^3\epsilon (x),
\]
where we have defined
\[
\epsilon (x) =
\frac{3}{8x^3}\left( x(1+2x^2)\sqrt{1+x^2}-ln(x+\sqrt{1+x^2})\right),
\]
with the dimensionless variable $x$ defined as
\[
x=\frac{k_F}{m_e}=\left(\frac{n}{n_{0}}\right)^{1/3}, 
\]
with
$n_{0}=\frac{(m_{e})^{3}}{3\pi^2}$ with $m_{e}$ the electron mass.
Inserting numerical values
for the electron mass we get
\[
n_{0}=5.89\times 10^{35} \mathrm{m}^{-3}.
\]

Since the mass of the protons and neutrons are larger by a factor $10^3$ than the mass
of the electrons $m_e$, 
we can approximate the total mass of the star by the mass density of the nucleons (protons and neutrons).
This mass density is given by
\[
\rho = M_p n_p,
\]
with $M_{p}$ being the  mass of the proton and $n_p$ the particle density of the nucleons.
The mass of the proton and the neutron are almost equal and we have set them equal here.
The particle density  $n_p$ can be related to the electron density $n$, which is the quantity
we can calculate. The relation is simple,
\[
n_p = n/Y_e ,
\]
where $Y_{e}$ is the number of electrons per nucleon.  For $^{56}$Fe we get $Y_{e}=\frac{26}{56}=0.464$, since we 
need to have as many electrons as protons in order to obtain a total charge of zero. 
The mass density is now
\[
\rho = M_p n/Y_e,
\]
and with
\[
x=\left(\frac{n}{n_{0}}\right)^{1/3}=\left(\frac{\rho}{\rho_{0}}\right)^{1/3},
\]
and inserting the numerical value for the proton mass we obtain
\[
\rho_{0}=\frac{M_{p}n_{0}}{Y_{e}}=9.79\times 10^{8}Y_{e}^{-1} \mathrm{kg}\hspace{1mm}
\mathrm{m}^{-3}.
\]
Using the parameter $Y_{e}$ we can then study stars with different compositions.
The only input parameters to your code are then $\rho_c$ and $Y_e$.

Now we want the equation for the pressure, based on the energy density.
Using the thermodynamical relation
\[
P=-\frac{\partial E}{\partial V}=-\frac{\partial E}{\partial x}
\frac{\partial x}{\partial V},
\]
we can find the pressure as a function of the mass density $\rho$.
Thereafter we can find $\frac{dP}{d\rho}$, which allows us to 
determine the mass and the radius of the star. 

The term
\[
\frac{\partial x}{\partial V},
\]
can be found using the fact that  $x\propto n^{1/3} \propto V^{-3}$. This results in
\[
\frac{\partial x}{\partial V}= -\frac{x}{3V}.
\]
Taking the derivative with respect to $x$ we obtain
\[
P=\frac{1}{3}n_0m_e x^4\frac{d\epsilon}{dx}.
\]
We want the derivative of  $P$ in terms of the mass density $\rho$. Using
$x=\left(\frac{\rho}{\rho_{0}}\right)^{1/3}$, we obtain
\[
\frac{dP}{d\rho}=\frac{dP}{dx}\frac{dx}{d\rho}.
\]
With
\[
\frac{dP}{dx}=\frac{1}{3}n_0m_e
\frac{d}{dx} \left(x^4\frac{d\epsilon}{dx}\right),
\]
and
\[
\frac{dx}{d\rho}=\frac{1\rho_0^{2/3}}{3\rho_0\rho^{2/3}}
=\frac{1}{3\rho_0 x^2},
\]
we find
\[
\frac{dP}{d\rho}=Y_{e}\frac{m_{e}}{M_{p}}\gamma (x),
\]
where we defined
\[
\gamma (x)=\frac{x^{2}}{3\sqrt{1+x^{2}}}.
\]
This is the equation for the derivative of the pressure to be used to find
\[
\frac{d\rho}{dr}=-\left(\frac{dP}{d\rho}\right)^{-1}\frac{Gm}{r^{2}}\rho.
\]

Note that $x$ and $\gamma(x)$ are dimensionless quantities.

\subsection*{Dimensionless form of the differential equations}

In the numerical treatment of the two differential equations we need
to rescale our equations in terms of dimensionless quantities, since several of the
involved constants are either extremely large or very small.
Furthermore, the total mass is of the order of the mass of the sun, approximately
$2\times 10^{30}$kg while the mass of the electron is $9\times 10^{-31}$ kg.

We introduce therefore a dimensionless radius $\overline{r}=r/R_{0}$, a dimensionless density
$\overline{\rho}=\rho /\rho_{0}$ (recall that $x^{3}=\rho /\rho_{0}$)
and a dimensionless mass $\overline{m}=m/M_{0}$.

We determine below the constants  $M_{0}$ and $R_{0}$ by requiring that the equations
for $\frac{dm}{dr}$ and $\frac{d\rho}{dr}$ have to be dimensionless.  
We get then
\[
\frac{dM_0\overline{m}}{dR_0\overline{r}}=
4\pi R_0^2\overline{r}^{2}\rho_0\overline{\rho},
\]
resulting in
\[
\frac{d\overline{m}}{d\overline{r}}=
4\pi R_0^3\overline{r}^{2}\rho_0\overline{\rho}/M_0.
\]
If we want this equation to be dimensionless, we must require
\[
4\pi R_{0}^{3}\rho_{0}/M_0=1.
\]
Correspondingly, we have
\[
\frac{d\rho_0\overline{\rho}}{dR_0\overline{r}}=
-\left(\frac{GM_0M_p}{Y_em_ec^2 }\right)\frac{\overline{m}}
{\gamma R_0^2\overline{r}^{2}}
\rho_0\overline{\rho},
\]
with $R_0$
\[
R_{0}=\left(\frac{Y_{e}m_{e}c^2}{4\pi\rho_{0} GM_{p}}\right)^{1/2}
=7.72\times 10^{6} Y_{e}\hspace{2mm} \mathrm{m}.
\]
in order to yield a dimensionless equation. This results in 
\[
M_{0}=4\pi R_{0}^{3}\rho_{0}=5.67 \times 10^{30}Y_{e}^{2}\hspace{2mm} \mathrm{kg}.
\]
The radius of the sun is $R_{\odot}=6.95\times 10^{8}$ m 
and the mass of the sun is 
$M_{\odot}=1.99\times 10^{30}$ kg.

Our final differential equations 
$\overline{\rho}$ and
$\overline{m}$ read 
\[
\frac{d\overline{\rho}}{d\overline{r}}=-\frac{\overline{m}}{\gamma}\frac{\overline{\rho}}
{\overline{r}^{2}},
\hspace{5mm}\frac{d\overline{m}}{d\overline{r}}=
\overline{r}^{2}\overline{\rho}.
\]
These are the equations you need to code.

\begin{enumerate}
\item[a)] Verify the steps in the above derivations. Write a program which solves the two coupled
differential equations
\[
\frac{d\overline{\rho}}{d\overline{r}}=-\frac{\overline{m}}{\gamma}\frac{\overline{\rho}}
{\overline{r}^{2}},
\]
and
\[
\frac{d\overline{m}}{d\overline{r}}=
\overline{r}^{2}\overline{\rho},
\]
using the fourth order Runge-Kutta method (write your own code for the fourth order Runge-Kutta method) 
by integrating outward from $\overline{r}=0$.
Choose $Y_e=1$ and calculate the mass and radius of the star by varying the central density $\overline{\rho}_c$
ranging from $10^{-1}$ to $10^6$. 
Check the stability of your solutions by varying the radial step $h$. 
Discuss your results.  Your Runge-Kutta code should take as input the name of the function which sets up the derivatives.


Note that your initial value of $\overline{r}$ should not be zero, else you will get a division by zero.  Using a small value for the initial radius you should also figure out the initial values for the mass and the initial density. Hint, use a Taylor expansion to obtain the initial conditions.

\item[b)] 
Scale the mass-radius relation you found in a) to the cases corresponding to $^{56}$Fe and
$^{12}$C. Three white dwarf stars, Sirius B, 40 Eri B and Stein 2051, have masses and radii
in units of solar values determined from observations to be
$(1.053\pm0.028 M_{\odot},0.0074\pm 0.0006 R_{\odot})$, $(0.48\pm0.02 M_{\odot},0.0124\pm 0.0005 R_{\odot})$,
and $(0.72\pm0.08 M_{\odot},0.0115\pm 0.0012 R_{\odot})$, respectively.
Verify that these values are consistent with the model you have developed. Can you say something about the
compositions of these stars?
\end{enumerate}


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