% vim: fdm=marker fmr=<<<,>>>
\documentclass[../thesis.tex]{subfiles}

\graphicspath{
	{.}
	{../../figures/}
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\begin{document}
\chapter{Introduction}
\label{sec:introduction}

% Intro Cosmic Ray
In the beginning of the 20th century, various types of radiation were discovered.
With the balloonflight of Victor Hess \Todo{ref} in \Todo{year}, one type was determined to come from beyond the atmosphere and named ``Cosmic Rays''.
With many discoveries following, the field of (astro-)particle physics evolved.
\\
% Current state, (nudge to radio)
Large collaborations are now detecting cosmic rays with a variety of methods over a large range of energy\Todo{ref figure}.
Still, questions on their origin remain.\Todo{list questions or remove}
\\
% Radio
In the last decade, the detection using radio antennas has received significant attention \Todo{ref}, such that collaborations such as \gls{GRAND} are building observatoria that fully rely on radio measurements.
%
For such radio arrays, the analyses require an accurate timing of signals within the array.
Generally, \gls{GNSS} is used to synchronise the detectors.
However, advanced analyses require an even higher accuracy.
\\

In this thesis, methods (and their limits) to obtain this accuracy for radio arrays are investigated.


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

\hrule

% Cosmic Particles = CR + Photon + Neutrino
There is a variety of extra terrestrial particles with which the Earth is bombarded.\Todo{rephrase}
These can be classified into three main types: charged nuclei (typically protons $Z=1$ up to iron $Z=26$), photons and neutrinos, each with different propagation effects.
\\
The charged nuclei are the bulk of the measured particles.
They do not point back to their sources because they are deflected by magnetic fields due to being charged.
\\
Photons do not suffer from being charged, and thus have the potential to identify their sources.
However, they can be absorbed and created by multiple mechanisms.\Todo{rephrase}
\\
Finally, neutrino's interact weakly, thus pointing back to their sources as well.
Unfortunately, this weak interaction also troubles the detection of the neutrino's.\Todo{rephrase}
\\
Note that cosmic rays are deemed\Todo{rephrase} to be charged nuclei.
\\

\begin{figure}%<<< cr_flux
	\centering
	\includegraphics[width=0.8\textwidth]{astroparticle/The_CR_spectrum_2023.pdf}
	\caption{
		From \protect \cite{The_CR_spectrum}.
		Cosmic Ray flux as a function of energy-per-nucleon.
	}
	\label{fig:cr_flux}
\end{figure}%>>>


% Energy
Cosmic rays span a large range of energy as illustrated in Figure~\ref{fig:cr_flux}.
The acceleration of cosmic rays is thought to occur in highly energetic regions
\\

Using the charged nuclei, an argument can be made to distinguish two types of sources.
\\
Being charged, the nuclei will gyrate in magnetic fields.
With an approximate size of $ $\Todo{size} and an average magnetic field of $5\mathrm{\;\mu G}$\Todo{}, the Milky Way can only contain particles up to an energy of about $10^{16}\eV$\Todo{fill}.
Still, particles with higher energies have been observed (see Figure~\ref{fig:}).
These higher energy particles must thus come from beyond our galaxy.
\\
Likewise, with an rapidly increasing flux for lower energies, one component can be assorted\Todo{rephrase} as coming from within the galaxy.
\\



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

\hrule
When a particle with a high enough energy comes into contact with the atmosphere, secondary particles are generated, forming an air shower.
This air shower consists of a cascade of interactions producing more particles that subsequently undergo further interactions.
Thus, the number of particles rapidly increases further down the air shower.
Figure~\ref{fig:airshower:depth} shows the number of particles as a function of atmospheric depth where $0\mathrm{\; g/cm^2}$ corresponds with the top of the atmosphere.
\\
An important feature that allows to statistically discriminate photons from protons and iron nuclei is the atmospheric depth at which this number of particles reaches its maximum, called $\Xmax$.
Part of this is explained by the depth of first interaction.
Due to the higher charge of heavy nuclei, they interact earlier in the atmosphere.
\\

The particle content of an air shower is dependent on the initial particle type.
Protons (and other nuclei) have access to hadronic interaction channels (pions, kaons, etc.)\Todo{ref?} through which most energy is passed.
In turn, the resulting air showers contain a large hadronic component.\Todo{check wording}
\\
In contrast, an initial photon cannot interact hadronicly, meaning its energy is dumped into the electromagnetic part of the air shower.
\\
Finally, any charged pions created in the air shower will decay into muons while still in the atmosphere.
This muonic component is a reliable part to measure.\Todo{rephrase}
\\


\begin{figure}%<<< airshower:depth
	\centering
	\includegraphics[width=0.3\textwidth]{airshower/shower_development_depth_iron_proton_photon.pdf}
	\caption{
		From H. Schoorlemmer.
		Shower development as a function of atmospheric depth for an energy of $10^{19}\eV$.
	}
	\label{fig:airshower:depth}
\end{figure}%>>>


\begin{figure}%<<< airshower:polarisation
	\centering
	\begin{subfigure}{0.47\textwidth}
		\includegraphics[width=\textwidth]{airshower/airshower_radio_polarisation_geomagnetic.png}%
	\end{subfigure}
	\hfill
	\begin{subfigure}{0.47\textwidth}
		\includegraphics[width=\textwidth]{airshower/airshower_radio_polarisation_askaryan.png}%
	\end{subfigure}
	\caption{
		From \protect \cite{Schoorlemmer:2012xpa} \protect\cite{Huege:2017bqv}
		\protect \Todo{Krijn?}
		Radio Emission mechanisms (left: geomagnetic, right: charge-excess)
	}
	\label{fig:airshower:polarisation}
\end{figure}%>>>>>>
\subsection{Experiments}%<<<
\label{sec:detectors}
At the very highest energy, the flux is in the order of one particle per square kilometer per century (see Figure~\ref{fig:cr_flux}).
To gather decent statistics at these highest energies on a practical timescale, observatories therefore have to span huge areas.
\\

The earliest 

\hrule
Standalone devices,
\gls*{Auger},
AugerPrime RD,
\gls*{GRAND},
\gls*{LOFAR}?,

%>>>>>>
\section{Radio Interferometry}
\label{sec:interferometry}
The radio signals emitted from the air shower can be recorded by radio antennas.

Unlike, astronomical interferometry, the source of the signal is closeby, therefore


\begin{figure}
	\centering
	\includegraphics[width=0.5\textwidth]{radio_interferometry/rit_schematic_true.pdf}%
%	\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(\vec{x}) = \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{figure}
	\centering
	\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}

\begin{figure}
	\centering
	\includegraphics[width=0.7\textwidth]{2006.10348/fig03_b.png}%
	\caption{
		From \protect \cite{Schoorlemmer:2020low}.
		$\Xmax$ resolution as a function of detector-to-detector synchronisation.
	}
	\label{fig:xmax_synchronise}
\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.
%		\protect \todo{
%			redo figure with less margins,
%			remove spines,
%			rotate labels
%		}
%	}
%	\label{fig:reference-clock}
%\end{figure}

\end{document}
