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

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\begin{document}
\chapter{Introduction}
\label{sec:introduction}
%<<<
% Intro Cosmic Ray
In the beginning of the $\mathrm{20^{th}}$~century, various types of radiation were discovered.
With the balloonflight of Victor Hess \Todo{ref} in 1912, one type was determined to come from beyond the atmosphere and therefore labelled ``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 (see Figure~\ref{fig:cr_flux}).
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, \glspl{GNSS} are used to synchronise the detectors.
However, advanced analyses require an even higher accuracy than currently achieved with these systems.
\\
This thesis investigates a relatively straightforward method (and its limits) to obtain this required timing accuracy for radio arrays.
\\

% >>>
\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.
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.
The various magnetic fields that they travel through deflect them due to their charge.
Because of this, they do not point back to their sources.
\\
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/expand}
\\
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 and flux as illustrated in Figure~\ref{fig:cr_flux}.
The acceleration of cosmic rays is thought to occur in highly energetic regions\Todo{expand}
\\

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 $10^{17}\eV$\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.
\Todo{rewrite paragraph}
\\
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 an energy above $1\;\TeV$ 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.
This happens until the energy is spread out\Todo{word} enough that the number of interactions decreases.
\\

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.
The atmospheric depth at which this number of particles reaches its maximum is called $\Xmax$.
\\
In Figure~\ref{fig:airshower:depth} the \Xmax is different for a photon, a proton and iron.
Typically, heavy nuclei have their first interaction higher up in the atmosphere than protons, with photons penetrating the atmosphere even further.
Therefore, measurements of $\Xmax$ allow to statistically discriminate between photons, protons and iron nuclei.
\\


The initial particle type also influences the particle content of an air shower.
Protons (and other nuclei) have access to hadronic interaction channels (such as 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}
\\
These different components have a different width.\Todo{rephrase}
The hadronic component is greatly collimated, while the electromagnetic component.
\\
\Todo{rewrite paragraph}

\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}%>>>

% Radio measurements
Processes in an air showers also generate radiation that can be picked up as coherent radio signals.
%% Geo Synchro
Due to the magnetic field of the Earth, the electrons in the air shower generate radiation.
Termed geomagnetic emission in Figure~\ref{fig:airshower:polarisation}, this has a polarisation that is dependent on the magnetic field vector $\vec{B}$ and the air shower velocity $\vec{v}$.
\Todo{expand?}
\\
%% Askaryan / Charge excess
An additional mechanism emitting radiation was first theorised by Askaryan\Todo{ref}.
Due to the large inertia of the positively charged ions with respect to their light, negatively charged electrons, a negative charge excess is created.
In turn, this generates radiation that is polarised radially towards the shower axis (see Figure~\ref{fig:airshower:polarisation}).
\\
%% Cherenkov ring
The relativistic speeds of the particles cause any radiation that is produced in the air shower to be forward beamed along the shower axis.
Additionally, the shower travels faster than the speed of light in the atmosphere.
This generates an
The detection of the radio signals is limited to an
This is limited by the so-called Cherenkov angle.
\Todo{finish paragraph}


\begin{figure}%<<< airshower:polarisation
	\centering
	\begin{subfigure}{0.47\textwidth}
		\includegraphics[width=\textwidth]{airshower/airshower_radio_polarisation_geomagnetic.png}%
		\caption{
			Geomagnetic emission
		}
		\label{fig:airshower:polarisation:geomagnetic}
	\end{subfigure}
	\hfill
	\begin{subfigure}{0.47\textwidth}
		\includegraphics[width=\textwidth]{airshower/airshower_radio_polarisation_askaryan.png}%
		\caption{
			Askaryan or charge-excess emission
		}
		\label{fig:airshower:polarisation:askaryan}
	\end{subfigure}
	\caption{
		From \protect \cite{Schoorlemmer:2012xpa, Huege:2017bqv}
		\protect \Todo{Krijn?}
		The Radio Emission mechanisms and the resulting polarisations of the radio signal: \subref{fig:airshower:polarisation:geomagnetic} geomagnetic and \subref{fig:airshower:polarisation:askaryan} charge-excess)
		See text for explanation.
	}
	\label{fig:airshower:polarisation}
\end{figure}%>>>>>>
%>>>>>>


%\subsection{Experiments}%<<<
%\label{sec:detectors}
\bigskip
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}).
Observatories therefore have to span huge areas to gather decent statistics at these highest energies on a practical timescale.
In recent and upcoming experiments, such as \gls{Auger} (and its upgrade \gls{AugerPrime}), \gls{GRAND} or \gls{LOFAR}, the approach is typically to instrument an area with a (sparse) grid of detectors to detect the generated air shower.\Todo{cite experiments here}
With distances up to $1.5\;\mathrm{km}$ (\gls{Auger}), the detectors therefore have to operate in a self-sufficient manner\Todo{word} with only wireless communication channels.
\\

These standalone detectors typically receive their timing from a \gls{GNSS}.
Previously, for timing of water-Cherenkov detectors, this timing accuracy was better than the resolved data.
Even for the first analyses of radio data, this was sufficient.
However, for advanced analyses such as radio interferometry, the timing accuracy must be improved.
\\

% Structure summary
In this thesis, a solution to enhance the timing accuracy of air shower radio detectors is demonstrated.
First, an introduction to radio interferometry is given in Chapter~\ref{sec:interferometry}.
This will be used later on and gives an insight into the timing accuracy requirements.
\\
Chapter~\ref{sec:waveform} reviews typical techniques to analyse waveforms to obtain timing information.
\\
Chapter~\ref{sec:disciplining} introduces the concept of a beacon transmitter to synchronise an array of radio antennas and constrains the achievable timing accuracy using the techniques described in the preceding chapter.
\\
Chapter~\ref{sec:single_sine_sync} establishes a method to synchronise an array using a single sine wave beacon while using the radio interferometric approach to resolve\Todo{word} an airshower.
\\
Finally, Chapter~\ref{sec:gnss_accuracy} investigates the limitations of the current hardware of \gls{GRAND} and its ability to record and reconstruct a beacon signal.
\end{document}
