Add theory of the star formation

theory/Star_Formation/bibliography.bib

+@ARTICLE{schaye2008,
+   author = {{Schaye}, J. and {Dalla Vecchia}, C.},
+    title = "{On the relation between the Schmidt and Kennicutt-Schmidt star formation laws and its implications for numerical simulations}",
+  journal = {\mnras},
+archivePrefix = "arXiv",
+   eprint = {0709.0292},
+ keywords = {stars: formation , galaxies: evolution , galaxies: formation , galaxies: ISM},
+     year = 2008,
+    month = jan,
+   volume = 383,
+    pages = {1210-1222},
+      doi = {10.1111/j.1365-2966.2007.12639.x},
+   adsurl = {http://adsabs.harvard.edu/abs/2008MNRAS.383.1210S},
+  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
+@ARTICLE{schaye2004,
+   author = {{Schaye}, J.},
+    title = "{Star Formation Thresholds and Galaxy Edges: Why and Where}",
+  journal = {\apj},
+   eprint = {astro-ph/0205125},
+ keywords = {Galaxies: Evolution, Galaxies: Formation, Galaxies: ISM, ISM: Clouds, Stars: Formation},
+     year = 2004,
+    month = jul,
+   volume = 609,
+    pages = {667-682},
+      doi = {10.1086/421232},
+   adsurl = {http://adsabs.harvard.edu/abs/2004ApJ...609..667S},
+  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
+@ARTICLE{kennicutt1998,
+   author = {{Kennicutt}, Jr., R.~C.},
+    title = "{The Global Schmidt Law in Star-forming Galaxies}",
+  journal = {\apj},
+   eprint = {astro-ph/9712213},
+ keywords = {GALAXIES: EVOLUTION, GALAXIES: ISM, GALAXIES: SPIRAL, GALAXIES: STELLAR CONTENT, GALAXIES: STARBURST, STARS: FORMATION, Galaxies: Evolution, Galaxies: ISM, Galaxies: Spiral, Galaxies: Starburst, Galaxies: Stellar Content, Stars: Formation},
+     year = 1998,
+    month = may,
+   volume = 498,
+    pages = {541-552},
+      doi = {10.1086/305588},
+   adsurl = {http://adsabs.harvard.edu/abs/1998ApJ...498..541K},
+  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+

theory/Star_Formation/run.sh

+#!/bin/bash
+echo "Generating PDF..."
+pdflatex -jobname=starform starformation_standalone.tex
+bibtex starform.aux
+pdflatex -jobname=starform starformation_standalone.tex
+pdflatex -jobname=starform starformation_standalone.tex

theory/Star_Formation/starformation.tex

+\section{Star Formation in EAGLE}
+
+In this section we will shortly explain how the star formation in EAGLE works.
+The implemented star formation is based on the \citet{schaye2008}, instead of 
+the constant density threshold used by \citet{schaye2008}, a metallicity 
+dependent density threshold is used, following \citet{schaye2004}. An important 
+property of the implemented star formation law is the explicit reproducability 
+of the Kennicutt-Schmidt star formation law \citep{kennicutt1998}:
+\begin{align}
+ \dot{\Sigma}_\star &= A \left( \frac{\Sigma}{1 ~\text{M}_\odot ~\text{pc}^{-2}} \right)
+\end{align}
+
+\noindent In which $A$ is the normalization of the Kennicutt-Schmidt, $\dot{\Sigma}_\star$
+is the surface density of newly formed stars, $\Sigma$ is the gas surface 
+density and $n$ is the power law index. In the case of the star formation 
+implementation of \citet{schaye2008}, the star formation law is given by 
+a pressure law:
+
+\begin{align}
+\dot{m}_\star &= m_g A ( 1~\text{M}_\odot~\text{pc}^{-2})^{-n} \left( 
+\frac{\gamma}{G} f_g P \right)^{(n-1)/2}.
+\end{align}
+
+\noindent In which $m_g$ is the gas particle mass, $\gamma$ is the ratio of specific heats,
+$G$ is the gravitational constant, $f_g$ is the mass fraction of gas (unity in 
+EAGLE), and $P$ is the total pressure of the gas particle. In this equation
+$A$ and $n$ are directly constrainted from the observations of the Kennicutt-
+Schmidt law so both variables do not require tuning. Further there are 
+two constrains on the over density which should be $\Delta > 57.7$ (why this 
+specific number? \citet{schaye2008} says $\Delta \approx 60$), and the
+temperature of the gas should be atleast $T_\text{crit}<10^5 ~\text{K}$
+
+Besides this it is required that there is an effective equation of state. 
+Specifically we could take this to be equal to:
+\begin{align}
+ P &= P_\text{eos} (\rho) = \left( \frac{\rho_\text{g}}{\rho_\text{g,c}} \right)^{\gamma_\text{eff}}.
+\end{align}
+\noindent In which $\gamma_\text{eff}$ is the polytropic index. But the EAGLE 
+code just uses the EOS of the gas?
+
+\noindent Using this it is possible to calculate the propability that a gas particle is 
+converted to a star particle:
+\begin{align}
+ \text{Prob.} = \text{min} \left( \frac{\dot{m}_\star \Delta t}{m_g}, 1 \right) 
+ = \text{min} \left( A \left( 1 ~\text{M}_\odot ~\text{pc}^{-2} \right)^{-n} \left( \frac{\gamma}{G} f_g P_\text{tot} \right)^{(n-1)/2}, 1 \right).
+\end{align}
+
+\noindent In general we use $A=1.515 \cdot 10^{-4}~\text{M}_\odot ~\text{yr}^{-1} ~\text{kpc}^{-2}$ 
+and $n=1.4$. In the case of high densities ($n_\text{H} > 10^3 ~\text{cm}^{-3}$),
+the power law will be steaper and have a value of $n=2$. This will also adjust
+the normalization of the star formation law, both need to be equal at the 
+pressure with a corresponding density. This means we have:
+\begin{align}
+\begin{split}
+ A \left( 1 ~\text{M}_\odot ~\text{pc}^{-2} \right)^{-n} \left( \frac{\gamma}{G} f_g P_\text{tot} \right)^{(n-1)/2} \\
+ = A_\text{high} \left( 1 ~\text{M}_\odot ~\text{pc}^{-2} \right)^{-n_\text{high}} \left( \frac{\gamma}{G} f_g P_\text{tot} \right)^{(n_\text{high}-1)/2}. 
+\end{split}
+\end{align}
+\begin{align}
+A_\text{high} = A \left( 1 ~\text{M}_\odot ~\text{pc}^{-2} \right)^{n_\text{high}-n} \left( \frac{\gamma}{G} f_g P_\text{tot}(\rho_{hd}) \right)^{(n-n_\text{high})/2}. 
+\end{align}
+
+This is differently from the EAGLE code ($f_g=1$) which uses:
+\begin{align}
+A_\text{high} = A \left( \frac{\gamma}{G} P_\text{tot} (\rho_{hd}) \right)^{(n-n_\text{high})/2} . 
+\end{align}
+
+Besides this we also use the metallicity dependent density threshold given by \citep{schaye2004}:
+\begin{align}
+n^*_\text{H} (Z) &= n_\text{H,norm} \left( \frac{Z}{Z_0} \right)^{n_z}.
+\end{align}
+In which $n_\text{H,norm}$ is the normalization of the metallicity dependent 
+star formation law, $Z$ the metallicity, $Z_0$ the normalization metallicity,
+and $n_Z$ the power law of the metallicity dependence on density. standard 
+values we take for the EAGLE are $n_\text{H,norm} = 0.1 ~\text{cm}^{-3}$, 
+$n_Z=-0.64$ and $Z_0 = 0.002$.
+
+For the initial pressure determination the EAGLE code uses (Explanation needed):
+\begin{align}
+ P_\text{cgs} &= (\gamma -1) \frac{n_\text{EOS, norm} \cdot m_H}{X} T_{EOS,jeans} \cdot \frac{k_B}{1.22 \cdot (\gamma -1) m_H }.
+\end{align}
+
+To determine the pressure for the star formation law the EAGLE code uses the 
+physical pressure? Is this the effective EOS of the real EOS of the gas?
+
+Compared to the EAGLE code we can calculate a fraction of the calculations already 
+in the struct which are not depending on time, this may save some calculations. 
+
+
+
+
+
+
+

theory/Star_Formation/starformation_standalone.tex

+\documentclass[fleqn, usenatbib, useAMS, a4paper]{mnras}
+\usepackage{graphicx}
+\usepackage{amsmath,paralist,xcolor,xspace,amssymb}
+\usepackage{times}
+\usepackage{comment}
+\usepackage[super]{nth}
+
+\newcommand{\todo}[1]{{\textcolor{red}{#1}}}
+\newcommand{\gadget}{{\sc Gadget}\xspace}
+\newcommand{\swift}{{\sc Swift}\xspace}
+\newcommand{\nbody}{$N$-body\xspace}
+\newcommand{\Lag}{\mathcal{L}}
+
+%opening
+\title{Star formation equations in SWIFT}
+\author{Folkert Nobels}
+\begin{document}
+
+\date{\today}
+
+\pagerange{\pageref{firstpage}--\pageref{lastpage}} \pubyear{2018}
+
+\maketitle
+
+\label{firstpage}
+
+\begin{abstract}
+Making stars all over again.
+\end{abstract}
+
+\begin{keywords}
+\end{keywords}
+
+\input{starformation}
+
+
+
+\bibliographystyle{mnras}
+\bibliography{./bibliography.bib}
+
+\label{lastpage}
+
+\end{document}