\documentclass[12pt, a4paper, english, singlespacing, parskip]{scrartcl}

%\documentclass[
%11pt, 				% The default document font size, options: 10pt, 11pt, 12pt
%oneside, 			% Two side (alternating margins) for binding by default, uncomment to switch to one side
%chapterinoneline,	% Have the chapter title next to the number in one single line
%english, 			% ngerman for German
%singlespacing, 	% Single line spacing, alternatives: onehalfspacing or doublespacing
%draft, 			% Uncomment to enable draft mode (no pictures, no links, overfull hboxes indicated)
%nolistspacing, 	% If the document is onehalfspacing or doublespacing, uncomment this to set spacing in lists to single
%liststotoc, 		% Uncomment to add the list of figures/tables/etc to the table of contents
%toctotoc, 			% Uncomment to add the main table of contents to the table of contents
%parskip, 			% Uncomment to add space between paragraphs
%nohyperref, 		% Uncomment to not load the hyperref package
%headsepline, 		% Uncomment to get a line under the header
%]{scrartcl or scrreprt or scrbook} % The class file specifying the document structure

\usepackage{lmodern} 		% Diese beiden packages sorgen für echte 
\usepackage[T1]{fontenc}	% Umlaute.

\usepackage{amssymb, amsmath, color, graphicx, float, setspace, tipa}
\usepackage[utf8]{inputenc} 
\usepackage[english]{babel}
\usepackage[pdfpagelabels,
        pdfstartview = FitH,
        bookmarksopen = true,
        bookmarksnumbered = true,
        linkcolor = black,
        plainpages = false,
        hypertexnames = false,
        citecolor = black, 
        breaklinks]{hyperref}
\usepackage{url}
\usepackage{tikz}
\usetikzlibrary{3d,arrows,calc,fadings,through,fit,shapes.geometric}
\tikzset{circle through 3 points/.style n args={3}{%
insert path={let    \p1=($(#1)!0.5!(#2)$),
                    \p2=($(#1)!0.5!(#3)$),
                    \p3=($(#1)!0.5!(#2)!1!-90:(#2)$),
                    \p4=($(#1)!0.5!(#3)!1!90:(#3)$),
                    \p5=(intersection of \p1--\p3 and \p2--\p4)
                    in },
at={(\p5)},
circle through= {(#1)}
}}

\usepackage{authblk} 		% titlepage stuff
\usepackage[titletoc, title]{appendix}

%===================
% BIBLIOGRAPHY
%===================
\usepackage[]{natbib}
\bibliographystyle{unsrtnat}
%DONT FORGET TO COMPILE THE BIBLIOGRAPHY WITH BIBTEX WHEN CHANGES ARE MADE.




%--------------------------------------------
% NEW COMMANDS
%--------------------------------------------

\newcommand{\corresponds}{\mathrel{\widehat{=}}}		% equals with hat

\newcommand {\arctanh}{\mathrm{arctanh}}				% Atanh
\newcommand{\arccot}{\mathrm{arccot }}					% Acotanh

\newcommand{\limz}[1]{\lim\limits_{#1 \rightarrow 0}}	% Limes of something towars zero

\newcommand{\bm}{\boldmath}								% Bold font in math
\newcommand{\dps}{\displaystyle}						

\newcommand{\e}{\mbox{e}}								% e noncursive in math mode

\newcommand{\del}{\partial}								% partial diff operator
\newcommand{\de}{\mathrm{d}}							% differential d
\newcommand{\D}{\mathrm{d}}								% differential d
\newcommand{\GRAD}{\mathrm{grad}\ }						% gradient
\newcommand{\DIV}{\mathrm{div}\ }						% divergence
\newcommand{\ROT}{\mathrm{rot}\ }						% rotation

\newcommand{\CONST}{\mathrm{const.\ }}					% constant
\newcommand{\var}{\mathrm{var}}							% variance

\newcommand{\g}{^\circ}									% degrees
\newcommand{\degr}{^\circ}								% degrees

\newcommand{\msol}{M_\odot}								% solar mass


\newcommand{\x}{\mathbf{x}}								% x vector
\newcommand{\xdot}{\dot{\mathbf{x}}}					% x dot vector
\newcommand{\xddot}{\ddot{\mathbf{x}}}					% x doubledot vector
\newcommand{\R}{\mathbf{r}}								% r vector
\newcommand{\rdot}{\dot{\mathbf{r}}}					% r dot vector
\newcommand{\rddot}{\ddot{\mathbf{r}}}					% r doubledot vector
\newcommand{\vel}{\mathbf{v}}							% v vector
\newcommand{\V}{\mathbf{v}}								% v vector
\newcommand{\vdot}{\dot{\mathbf{v}}}					% v dot vector
\newcommand{\vddot}{\ddot{\mathbf{v}}}					% v doubledot vector

\newcommand{\dete}{\mathrm{d}t}							% dt
\newcommand{\delte}{\del t}								% partial t
\newcommand{\dex}{\mathrm{d}x}							% dx
\newcommand{\delx}{\del x}								% partial x
\newcommand{\der}{\mathrm{d}r}							% dr
\newcommand{\delr}{\del r}								% partial r

\newcommand{\deldt}{\frac{\del}{\del t}}				% shortcut partial derivative, in line
\newcommand{\ddt}{\frac{\de}{\de t}}					% shortcut total derivative, in line
\newcommand{\DELDT}[1]{\frac{\del  #1}{\del t}}			% shortcut partial derivative, on fraction
\newcommand{\DDT}[1]{\frac{\de  #1}{\de t}}				% shortcut total derivative, on fraction

\newcommand{\deldx}{\frac{\del}{\del x}}				% shortcut partial derivative, in line
\newcommand{\ddx}{\frac{\de}{\de x}}					% shortcut total derivative, in line
\newcommand{\DELDX}[1]{\frac{\del  #1}{\del x}}			% shortcut partial derivative, on fraction
\newcommand{\DDX}[1]{\frac{\de  #1}{\de x}}				% shortcut total derivative, on fraction

\newcommand{\deldr}{\frac{\del}{\del r}}				% shortcut partial derivative, in line
\newcommand{\ddr}{\frac{\de}{\de r}}					% shortcut total derivative, in line
\newcommand{\DELDR}[1]{\frac{\del  #1}{\del r}}			% shortcut partial derivative, on fraction
\newcommand{\DDR}[1]{\frac{\de  #1}{\de r}}				% shortcut total derivative, on fraction

% replace \sum with \sum\limits
\let\oldsum\sum
\renewcommand{\sum}{\oldsum\limits}
