\documentclass[../thesis.tex]{subfiles}

\graphicspath{
	{.}
	{../../figures/}
	{../../../figures/}
}

\begin{document}
\chapter{Introduction}
\label{sec:introduction}


\section{Cosmic Particles}
\label{sec:crs}
Particles from outer space,
Particle type,
Energy,
magnetic fields -- origin,

\subsection{Air Showers}
\label{sec:airshowers}
Particle cascades,
Xmax?,
Radio emission,

\subsection{Experiments}
\label{sec:detectors}
Standalone devices,
\gls*{PA},
AugerPrime RD,
\gls*{GRAND},
\gls*{LOFAR}?,


\section{Interferometry}
\label{sec:interferometry}
Rough outline of Interferometry?
\\

Requires $\sigma_t \lesssim 1\ns$ \cite{Schoorlemmer:2020low}

\begin{figure}
	\includegraphics[width=0.5\textwidth]{radio_interferometry/Schematic_RIT_extracted.png}
	\caption{From H. Schoorlemmer}
\end{figure}


\begin{equation}
	\label{eq:propagation_delay}
	\Delta_i = \frac{ \left|{ \vec{x} - \vec{a_i} }\right| }{c} n_{eff}
\end{equation}


\begin{equation}
	\label{eq:interferometric_sum}
	S(\vec{x}, t) = \sum_i S_i(t + \Delta_i(\vec{x}))
\end{equation}

\begin{equation}
	\label{eq:coherence_condition}
	\Delta t \leq \frac{1}{f}
\end{equation}

\begin{figure}
	\begin{subfigure}[t]{0.3\textwidth}
		\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap_bad.png}
		\label{fig:trace_overlap:bad}
	\end{subfigure}
	\hfill
	\begin{subfigure}[t]{0.3\textwidth}
		\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap_medium.png}
		\label{fig:trace_overlap:medium}
	\end{subfigure}
	\hfill
	\begin{subfigure}[t]{0.3\textwidth}
		\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap_best.png}
		\label{fig:trace_overlap:best}
	\end{subfigure}
	\caption{Trace overlap due to wrong positions}
	\label{fig:trace_overlap}
\end{figure}

\section{Time Synchronisation}
\label{sec:timesynchro}
The main method of synchronising multiple stations is by employing a \gls{GNSS}.
This system should deliver timing with an accuracy in the order of $10\ns$ \cite{} (see Section~\ref{sec:grand:gnss}).
\\

Need reference system with better accuracy to constrain current mechanism (Figure~\ref{fig:reference-clock}).

\begin{figure}
	\centering
	\includegraphics[width=0.5\textwidth]{clocks/reference-clock.pdf}
	\caption{
		Using a reference clock to compare two other clocks.
	}
	\label{fig:reference-clock}
	\todo{
		redo figure with less margins,
		remove spines,
		rotate labels
	}
\end{figure}

\end{document}