\section{Performance analysis}
\label{sec:performance-analysis}
\subsection{Measurement technique and its performance based on simulations}
\graphicspath {{../cau-ath-djf-0007_i1-0/media/figs_sim}}
The performance of AHEPaM is analysed using a GEANT4 simulation. For the majority of particles in the desired energy range the particles are expected to penetrate the entire instrument stack. Hence the analysis of the performance focuses on these particles. It has to be noted though that the particles stopping in the detector, i.e. the lower energy range (protons below 150~MeV), are evaluated as well by the instrument further improving its capabilities.

\subsection{Simulation setup}
\label{sec:sim_setup}
A model of the instrument was designed for GEANT4 studies as presented in fig.~\ref{fig:geometry_sketch} (left). It consists of
\begin{itemize}
    \item two \ac{BGO} scintillators, 2~cm thick each
    \item five double stack SSDs labeled SDA to SDE, 500~$\mu$m thick (for each double stack, the minimum of the energy deposition is used, hence, in the following SDA implies minimum(SDA1,SDA2)). The SSDs are also segmented:
    \begin{itemize}
        \item inner Segments
        \item 3 segments in a ring-like structure. Since the SSDs in a double stack are rotated with respect to each other this results in 6 virtual segments
        \item the inner three SSDs (SDB, SDC, SDD) have an additional outer ring that serves as an anti-coincidence.
    \end{itemize}
    \item two Aerogel Cherenkov detectors between SDA-SDB and SDD-SDE
    \item a simplified housing model
\end{itemize}


Fig. \ref{fig:geometry_sketch} (right) shows a visualisation of a test run with GEANT4 for an electron with an energy of 500~MeV. In the following analysis, however, particles are simulated in order to mimic the GCR fluxes. Therefore, an isotropic flux with GCR energy spectra as shown in fig. \ref{fig:adriani-e-p} was simulated. The resulting number of particles that traverse the detector stack is than used to calculate the expected count rates of the real instrument.\newline
In the following, particles that deposit their entire energy in the detector (stopping particles) and those that penetrate the entire detector stack (penetrating particles, sec. \ref{sec:sim_pene}) have been analysed separately due to different measurement techniques. For both, definitions for the selection of particle species have been designed in order to provide proton, electron and helium channels. For the proton channels, a quick estimate has been performed regarding the statistical requirements. The requirement analysis of electron channels also includes a quantitative analysis regarding the uncertainty introduced due to proton contamination. We are confident that based on the proton results the helium requirements do not require an individual analysis at this point and hence we only focus on protons and electrons here.

\begin{figure}
  \centering
  \includegraphics[width=7cm]{sim_model_invert.png}
  \includegraphics[width=7cm]{shower_el_500MeV}
  \caption{left: Sketch of the geometry as integrated in the GEANT4 simulation. Right: Visualisation of a simulation run with an electron  (500~MeV).}
  \label{fig:geometry_sketch}
\end{figure}


\begin{figure}
  \centering
  \includegraphics[width=8cm]{adriani-etal-combined-e-p.png}
  \caption{Proton and electron differential fluxes as measured by PAMELA \cite{picozza-etal-2007}. Proton data are taken from \cite{adriani-etal-2013}, electron data from \cite{adriani-etal-2015}. A force field solution (FFS) is superposed in blue using the same parameters for electrons as for protons and a fixed electron-to-proton ratio of $2\%$.}
  \label{fig:adriani-e-p}
\end{figure}


\subsection{Penetrating particles}
\label{sec:sim_pene}
The identification of particles that do not stop in the instrument (i.e. sum of all energy deposits is not equal the particles total energy any more) is not as straight forward as the dE/dx-E method for stopping ones. The idea of using the information that we have in the various detectors in order to find identification definitions for the different particle types remains the same.\newline
First, a set of cuts for all penetrating particles has been defined:
\begin{itemize}
    \item \textbf{min(SDA,SDB,SDC,SDD,SDE)$>$100~keV} - this ensures that we are indeed looking at penetrating particles. Note that this is the cleanest signal since the viewcone is restricted such that the particle is also hitting SDA and SDE. It is also possible to require just hits in SDB to SDD (including the \acp{BGO}). This will result in much higher statistics on the cost of worse particle separation (i.e. the Cherenkov detectors can not be used) which might be beneficial for proton channels (see discussion in \ref{sec:sim_pene_p})
    \item \textbf{SDB$<$SDD} - this serves as a proxy for the direction for non minimal ionizing particles. The analysis will be performed for SDD$<$SDB as well and hence, the statistics given in the following represent only half of the expected counts
\end{itemize}
These basic identification definitions have been applied to simulations of protons, electrons and helium particles up to the GeV range. Furthermore, i) a proton, ii) an electron and iii) a helium cut have been defined based on the energy losses in the SSDs before and after the \acp{BGO}, i.e.
\begin{itemize}
    \item proton identification: \textbf{max(SDB,SDD)$<$230~keV} - proton are expected to deposit 150~keV in 500$\mu$m silicon. Since they are less likely to create secondary particle shower in the \ac{BGO} than electrons, the energy loss in the SSD behind the \acp{BGO} is expected to remain below 230~keV as well.
    %\item electron cut: SDB$<$230~keV and SDD$>$230~keV - electrons at these high energies (they require $>$~250~MeV to penetrate the instrument) are also expected to deposit ~150~keV in 500$\mu$m silicon. However, they are producing a cascade of secondary particles in the \ac{BGO} and hence, the detector behind the \acp{BGO} is more likely to see several (secondary) electrons which in sum are causing higher energy depositions in the SSD.
    \item electron identification: \textbf{SDB$<$0.23~MeV \& SDC$<$7~MeV \& SDD$<$8~MeV} - electrons at these high energies (they require $>$~250~MeV to penetrate the instrument) are also expected to deposit 150~keV in SDB's 500$\mu$m silicon. The SDC and SDD are higher to allow electron showering while still cuting very high dE/dx caused by hadronic interactions (i.e. a secondary silicon particle caused by primary protons).
    %Cuts in SDC&SDD: Protonen können bei diesen Energien hadronische Schauer Produzieren, wodurch mehr Energie in SDC bzw SDD deponiert wird, als bei EM Schauer?
    \item helium identification: \textbf{min(SDB,SDD)$>$230~keV} - helium particles with energy sufficient enough to penetrate the instrument are expected to cause energy losses of 600~keV in 500$\mu$m silicon and hence, the SSDs before and behind the \ac{BGO} should see high energy depositions
\end{itemize}
Fig. \ref{fig:pene_supertrigger} presents the energy loss in the second \ac{BGO} as function of the energy loss in the first \ac{BGO} for the basic particle identification described above. Furthermore, the i) proton definition, ii) electron definition and iii) helium definition are applied in the i) left, ii) central and iii) right column, respectively. The different rows present results from 1) a proton, 2) a helium and 3) an electron simulation. The color code represents the expected count rates of these particles.\newline
From the figure, it can be concluded that 
\begin{itemize}
    \item a proton box with only neglectable contributions from electrons and helium particles can be defined (red box in left column)
    \item the majority of simulated helium particles (second row) is observed in the helium trigger (third column)
    \item the majority of simulated electrons (third row) is observed in the electron trigger (second column)
    \item the proton contribution to the electron channel (first row, second column) is significant while the helium contribution to the electron channel (second row, second column) is neglectable compared to the proton contamination
\end{itemize}
Fig. \ref{fig:pene_supertrigger_cherenkov} shows the same results as fig. \ref{fig:pene_supertrigger} with the additional requirement of a signal in the Cherenkov detector which removes the ion contributions below 2~GeV/nuc. % Vielleicht besser ~2GeV, die Schwelle liegt ja etwas über 2 GeV

\begin{figure}
  \centering
  \includegraphics[width=16cm]{pene_A-E.png}
  \caption{Expected count rates as function of the energy losses in BGO2 and BGO1 for penetrating particles. The rows correspond to 1) proton, 2) helium and 3) electron simulations. The columns are based on 1) the proton, 2) the electron and 3) the helium trigger.}
  \label{fig:pene_supertrigger}
\end{figure}

\begin{figure}
  \centering
  \includegraphics[width=16cm]{pene_A-E_cher.png}
  \caption{Expected count rates as function of the energy losses in BGO2 and BGO1 for penetrating particles utilizing a Cherenkov detector which removes input from ions below 2~GeV/nuc. The rows correspond to 1) proton, 2) helium and 3) electron simulations. The columns are based on 1) the proton, 2) the electron and 3) the helium trigger.}
  \label{fig:pene_supertrigger_cherenkov}
\end{figure}

\subsubsection{Proton requirement}
\label{sec:sim_pene_p}
The sum of all counts in the red boxes in fig. \ref{fig:pene_supertrigger} is given in the legend. For protons, the expected count rate is 0.41 counts/s in the first column (i.e. in the proton cut). 
%Since neither electrons nor helium contribute significantly to the box in any column a proton channel can be based on all three triggers (columns) and hence the expected count rate is 0.41+0.5+0.026=0.936 counts /s.  
Due to the second basic cut (SDB$<$SDD) these counts have to be doubled for a bi-directional analysis and hence, the total count rate is expected to be 0.82 counts/s for protons. Based on fig. \ref{fig:pene_supertrigger_cherenkov} (which shows the result including a Cherenkov detector) 0.0586 counts/s of these counts are caused by protons above 2~GeV (0.117 counts/s if considering both directions). Hence, protons in the penetrating channel (roughly above 150~MeV) and below 2~GeV are expecting to cause count rates of 0.7 counts/s. \newline
In a 10ks time frame this would result in 7000 counts. Equally distributed over five channels this would result in 
\begin{equation}
    \Delta p / p ~(5ch., t=10ks) = 1/(\sqrt{7000/5}) = 0.027 = 2.7\%
\end{equation}
utilizing just the penetrating protons.
For 3ks and two channels this will result in
\begin{equation}
    \Delta p / p ~(2ch., t=3ks) = 1/(\sqrt{2096/2}) = 0.031 = 3.1\%.
\end{equation}

Requiring only the detectors from SDB to SDD to be hit, the statistics can be increased (up to count rates of 3.8 counts/s). However, this includes trajectories that do not hit any of the Cherenkov detectors. For protons this would mean no clear separation at 2~GeV. 
In a 10ks time frame this would result in 38000 counts. Equally distributed over five channels this would result in 
\begin{equation}
    \Delta p / p ~(5ch., t=10ks) = 1/(\sqrt{38000/5}) = 0.012 = 1.2\%
\end{equation}
utilizing just the penetrating protons.
For 3ks and two channels this will result in
\begin{equation}
    \Delta p / p ~(2ch., t=3ks) = 1/(\sqrt{11400/2}) = 0.014 = 1.4\%.
\end{equation}

\subsubsection{Electron requirements}
\label{sec:sim_pene_e}
The blue box presented in the figure presents a simple approach to reduce and to quantify the proton contamination of the electron channels. The number given in the legend represent the expected count rate in the box.\newline
Utilizing the Cherenkov, the expected count rate in the electron box caused by protons is reduced. The expected counts as well as the contribution of protons to the electron channel are summarized in table \ref{tab:pene_elec}.


\begin{table}
\centering
\begin{tabular}{ || c | c | c | c | c | } 
  \hline
  & etrigger+ebox (blue box) &   etrigger+ebox+ch (blue box) \\ 
  \hline
  p counts / ks  & 4.03 & 0.490  \\ 
  \hline
  e counts / ks & 10.3 & 10.2  \\ 
  \hline
  C$_a$= e+p counts / ks & 14.33 & 10.69  \\ 
  \hline
  r=p/(p+e) counts & 0.282 & 0.042  \\ 
  \hline
  $\Delta C_e / C_e$ (t=50ks) & 0.080 & 0.055 \\
  \hline
  $\Delta C_e / C_e$ (t=100ks) & 0.057  & 0.039  \\
  \hline
\end{tabular}
  \caption{Expected count rates in the electron channel without (first column) and with Cherenkov detector (second column) based on penetrating protons and electrons. Due to the second basic cut (SDB$<$SDD) these numbers have to be doubled for a bi-directional analysis.}
  \label{tab:pene_elec}
\end{table}

The relative uncertainty of the electron has been calculated for all scenarios for different time resolutions (cf. sec. \ref{sec:sim_pene_p}) and considering both directions (i.e. SDB$<$SDD and SDB$>$SDD by doubling C$_a$).\newline
Splitting the counts equally in two electron channels would result in
$$ \Delta C_e / C_e (2 ch., t=50ks)=5.5\% $$ 
i.e. two electron channels with 5.5\% accuracy on a 13 hours time resolution, or 
$$ \Delta C_e / C_e (2 ch., t=100ks)=3.9\% $$ 
i.e. two electron channels with 3.9\% accuracy on a daily basis.\newline
For five channels the statistical uncertainty increases
$$ \Delta C_e / C_e (5 ch., t=100ks)=6.1\% $$
on a daily basis, or the time resolution has to be reduced to i.e. 5 day averages
$$ \Delta C_e / C_e (5 ch., t=500ks)=2.8\% $$ 
It has to be noted that this accounts only for penetrating electrons (energies above $\approx$200~MeV), the stopping electrons additionally improve the statistics.

Neglecting the outer rings of SDB-SDD as veto-counter, the statistics can be increased by a factor of three. The disadvantages are an increased proton contamination and a reduced energy resolution. The resulting uncertainties for a 50ks or 100ks time frame are as follows
$$ \Delta C_e / C_e (2 ch., t=50ks)=3.5\% $$
i.e. two electron channels with 3.5\% accuracy on a 13 hours time resolution, or
$$ \Delta C_e / C_e (2 ch., t=100ks)=2.5\% $$ 
i.e. two electron channels with 2.5\% accuracy on a daily basis.\newline
$$ \Delta C_e / C_e (5 ch., t=50ks)=5.5\% $$
for five electron channels on a 13 hours time resolution, or
$$ \Delta C_e / C_e (5 ch., t=100ks)=3.9\% $$ 
for five electron channels on a daily basis.



\subsection{Verdict of the performance analysis}

Based on the analysis above we have decided that a Cherenkov is recommended in order to fulfill the requirements. However, a de-scoped version of \ac{AHEPaM} without the Cherenkov detectors has been proven to provide the capabilities of separating electrons from protons utilizing the methods described in \cite{ahepam-djf} based on sufficient statistics which could be achieved by integrating over longer time periods. Given the small temporal variations of electrons in the energy range above 50~MeV this de-scoped version is expected to provide electron fluxes within the required systematical and statistical uncertainty range if the requirements on the integration period for electrons would be relaxed. Furthermore, the performance analysis for protons and electrons have been done with a high-accuracy and a high-statistics data product for both species. \ac{AHEPaM} will be able to produce all these data products simultaneously. The design philosophy behind these data product is as follows: 

\begin{itemize}
    \item The high-accuracy data products limit the opening angle by requiring all detectors to be hit by a particle. This reduces the geometric factor leading to lower statistics compared to the high-statistics channels. The benefit, however, is a reduced contamination (i.e. less protons in the electron channel) leading to lower systematic uncertainties.
    \item The high-statistics data products do not require all detectors to be hit, improving the opening angle and thus the statistics on the cost of a higher contamination (i.e. the Cherenkov detectors can not be utilized since they are missed by particles with oblique trajectories).
\end{itemize}
Since these data products will be produced in parallel, the high-statistics data products can be used to monitor temporal variations over a time period of interest. If no variations are observed, this information allows the usage of the high-accuracy mode in that given time period by accumulating statistics over that given time period.\newline
In addition, measurements over a prolonged time period can be utilized to validate and/or improve the high-statistics channel by comparing the fluxes to the high-accuracy mode. This is especially important for the electron measurements due to the proton contamination. This contamination will be corrected for by using the measured proton spectrum and the simulated response (i.e. the likely-hood of a proton to end up in the electron channel) in order to estimate the number of protons in the electron channels and subtract this from the measured electron channel count rate.\newline

The derived uncertainties for the protons and electrons in their corresponding high-accuracy and high-statistics data products are summarized below. Note that while the proton uncertainty is derived from the expected counting statistics, the electron uncertainties also include the systematic uncertainties caused by proton contamination.



\subsubsection*{High-accuracy proton channels}
It has been shown that requiring a coincidence from SDA up to SDE for protons allows for utilizing the Cherenkov in order to separate at 2~GeV. Using this coincidence, protons from 150~MeV up to 2~GeV can be detected in
\begin{itemize}
    \item five channels on 10ks time resolution with 2.7\% stat. uncertainty
    \item two channels on 3ks time resolution with 3.1\% stat. uncertainty
\end{itemize}
This would also provide an integral channel for protons above 2~GeV.
Note that the protons up to 150~MeV can also improve the statistics utilizing the stopping channels. \newline

\subsubsection*{High-statistic proton channels}
In addition, it is possible to enhance statistics by only requiring the detectors SDB to SDD to be hit (with the disadvantage that no Cherenkov signal is ensured). This would result in
\begin{itemize} 
    \item five channels on 10ks time resolution with 1.2\% stat. uncertainty
    \item two channels on 3ks time resolution with 1.4\% stat. uncertainty
\end{itemize}
\textbf{Note that the instruments hard- and software can be designed such that both, the detailed and high statistic proton analysis are performed in parallel.} %An additional study with coincidences neglecting SDA and SDE entirely (i.e. SDB to SDD are triggered) is currently ongoing and we expect such coincidence to further increase statistics for the high statistic proton channel. 
\newline

\subsubsection*{High-accuracy electron channels}
Regarding electrons we have found that
\begin{itemize} 
    \item two channels on 50ks (13 hours) time resolution with 5.5\% uncertainty (including both the statistical as well as the systematical uncertainty due to proton contamination)
    \item two channels on 100ks (1 day) time resolution with 3.9\% uncertainty (including both the statistical as well as the systematical uncertainty due to proton contamination)
    \item five channels on 100ks (1 day) time resolution with 6.1\% uncertainty (including both the statistical as well as the systematical uncertainty due to proton contamination)
    \item five channels on 500ks (5 days) time resolution with 2.8\% uncertainty (including both the statistical as well as the systematical uncertainty due to proton contamination)
\end{itemize}
Furthermore, no large flux variation at these energies are expected for electrons on short time scales.\newline
Note that the electrons up to 200~MeV can also improve the statistics utilizing the stopping channels.


\subsubsection*{High-statistic electron channels}
Similar to the high statistic proton channel, it is possible to enhance statistics by neglecting the outer rings of SDB-SDD as veto-counter. This leads to
\begin{itemize} 
    \item two channels on 50ks (13 hours) time resolution with 3.5\% uncertainty (including both the statistical as well as the systematical uncertainty due to proton contamination)
    \item two channels on 100ks (1 day) time resolution with 2.5\% uncertainty (including both the statistical as well as the systematical uncertainty due to proton contamination)
    \item five channels on 50ks (13 hours) time resolution with 5.5\% uncertainty (including both the statistical as well as the systematical uncertainty due to proton contamination)
    \item five channels on 100ks (1 day) time resolution with 3.9\% uncertainty (including both the statistical as well as the systematical uncertainty due to proton contamination)
\end{itemize}
\textbf{Note that the instruments hard- and software can be designed such that both, the detailed and high statistic electron analysis are performed in parallel.}
