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\begin{document}
\chapter{An Introduction to Cosmic Rays and Extensive Air Showers}
\label{sec:introduction}
%\section{Cosmic Particles}%<<<<<<
%<<<
\phantomsection
\label{sec:crs}
% Energy and flux
The Earth is bombarded with a variety of extra-terrestrial particles, with the energy of these particles extending over many orders of magnitude as depicted in Figure~\ref{fig:cr_flux}.
The flux of these particles decreases exponentially with increasing energy.
For \gls{UHE}, above $10^{6}\GeV$\Todo{limit}, it approaches one particle per~square~meter per~year, whereas for even higher energies the flux decreases to a particle per~square~kilometer per~year.
\\

\begin{figure}%<<< fig:cr_flux
	\centering
	\includegraphics[width=0.9\textwidth]{astroparticle/The_CR_spectrum_2023.pdf}
	\caption{
		From \protect \cite{The_CR_spectrum}.
		The diffuse cosmic ray spectrum (upper line) as measured by various experiments.
		The intensity and fluxes can generally be described by rapidly decreasing power laws.
		The grey shading indicates the order of magnitude of the particle flux, such that from the ankle onwards ($E>10^9\GeV$) the flux reaches $1$~particle per~square~kilometer per~year.
	}
	\label{fig:cr_flux}
\end{figure}%>>>

% CR: magnetic field
At \gls{UHE}, the incoming particles are primarily cosmic rays, atomic nuclei typically ranging from protons ($Z=1$) up to iron ($Z=26$).
Because these are charged, the various magnetic fields they passthrough will deflect and randomise their trajectories.
Ofcourse, this effect is dependent on the strength and size of the magnetic field and the speed of the particle.
It is therefore only at the very highest energies that the direction of an initial particle might be used to constrain the direction of its origin.
\\
% CR: galaxy / extra-galactic
The same argument (but in reverse) can be used to distinguish galactic and extra-galactic origins. 
The acceleration of these charged particles equally\Todo{word} requires strong and/or sizable magnetic fields.
Size constraints on our galaxy lead to a maximum energy for which a cosmic ray can still be contained in the galaxy.
This mechanism is expected to explain the steeper slope in Figure~\ref{fig:cr_flux} from the ``knee'' ($10^{6}\GeV$) onwards.
\\

% Photons and Neutrinos
Other particles at these energies include photons and neutrinos, which are not charged.
Therefore, these particle types do not suffer from magnetic deflections and have the potential to reveal their source regions.
Unfortunately, aside from both being much less frequent, photons can be absorbed and created by multiple mechanism, and neutrinos are notoriously hard to detect due to their weak interaction.
%\Todo{
%	$\gamma + \nu$ production by CR,
%	source / targets
%}
\\

%>>>
%\subsection{Air Showers}%<<<
\phantomsection
\label{sec:airshowers}
When a cosmic ray with an energy above $10^{3}\GeV$ 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 mean energy per particle is sufficiently lowered such that these particles are absorbed by the atmosphere.
\\

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, accurate measurements of $\Xmax$ allow to statistically discriminate between photons, protons and iron nuclei.
For example, the difference in $\langle\Xmax\rangle$ for iron and protons is roughly $100\;\mathrm{g/cm^2}$~\cite{Deligny:2023yms}.
\\


The initial particle type also influences the particle content of an air shower.
Depending on the available interaction channels we distinguish three components in air showers: the hadronic, electromagnetic and muonic components.
Each component shows particular development and can be related to different observables of the air shower.
\\
For example, detecting a large hadronic component means the initial particle has access to hadronic interactions (such as pions, kaons, etc.)\Todo{ref?} which is a typical sign for protons and other nuclei.
In contrast, for an initial photon, which cannot interact hadronicly, the energy will be 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, thus comprising the muonic component.
The lifetime, and ease of penetration of relativistic muons allow them to propagate to the Earth's surface, even if other particles have decayed or have been absorbed in the atmosphere.
\\

\begin{figure}%<<< airshower:depth
	\centering
	\includegraphics[width=0.5\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 ($\vec{v}$).
\\
%% Askaryan / Charge excess
An additional mechanism emitting radiation was 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
Due to the (varying) refractive index of the atmosphere, the produced radiation is concentrated on a ring-like structure called the Cherenkov-ring.
A peculiar time-inversion of the radiation from the whole air shower signals happens at this ring.
Outside this ring, radiation from the top of the air shower arrives earlier than radiation from the end of the air shower, whereas this is reversed inside thering.
\\
Consequently, all radiation from the whole air shower is concentrated in a small time-window at the Cherenkov-ring.
It is therefore important for radio detection to obtain measurements in this region.
\\

\begin{figure}%<<< airshower:polarisation
	\centering
	\begin{subfigure}{0.48\textwidth}
		\includegraphics[width=\textwidth]{airshower/airshower_radio_polarisation_geomagnetic.png}%
		\caption{
			Geomagnetic emission
		}
		\label{fig:airshower:polarisation:geomagnetic}
	\end{subfigure}
	\hfill
	\begin{subfigure}{0.48\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}
		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}%<<<
\phantomsection
\label{sec:detectors}
As mentioned, the flux at the very highest energy 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 the~\gls{Auger}\cite{Deligny:2023yms} and the~\gls{GRAND}\cite{GRAND:2018iaj}, the approach is typically to instrument a large area with a (sparse) grid of detectors to detect the generated air shower.
With distances up to $1.5\;\mathrm{km}$ (\gls{Auger}), the detectors therefore have to operate in a self-sufficient manner with only wireless communication channels and timing provided by \gls{GNSS}.
\\

In the last two decades, with the advent of advanced electronics, the detection using radio antennas has received significant attention.
A difficulty for radio detectors at these large distances.
\Todo{write paragraph}

For the detectors  (and its upgrade \acrlong{AugerPrime}\cite{Huege:2023pfb}),

Previously, for the timing of surface detectors such as water-Cherenkov detectors, this timing accuracy was better than the resolved data.
Even for the first analyses of radio data, this was sufficient.\Todo{ref or rm}
However, for advanced analyses such as radio interferometry, the timing accuracy must be improved.
\\
%%<<<
%% Radio
%In the last two decades, the detection using radio antennas has received significant attention \Todo{ref}, such that collaborations such as the~\gls{GRAND}\Todo{more?} 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.
%\\
%\Todo{remove - repeated at end of chapter}

% >>>

% 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 limitations of the current hardware of \gls{GRAND} and its ability to record and reconstruct a beacon signal.
\end{document}