\section{Performance analysis}
\label{sec:performance-analysis}
\graphicspath {{media/figs_sim}}
The performance of AHEPaM is analysed using a GEANT4 simulation. Section \ref{sec:sim_setup} introduces the simulation setup while sections \ref{sec:sim_stop} and \ref{sec:sim_pene} evaluate the performance for particles that stop in the instruments or penetrate it. Section \ref{sec:sim_verdict} summarizes the results. The formula used for the uncertainty estimation that takes into account contamination of different particle types is derivated in section \ref{sec:sim_error}.


\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, sec. \ref{sec:sim_stop}) 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{Stopping particles}
\label{sec:sim_stop}

\begin{figure}
  \centering
  \includegraphics[width=16cm]{stop_A_inner-B_inner_counts.png}
  \caption{Energy deposition in SDA over the total deposited energy for particles stopping in the instrument. The columns refer to simulations with different particles, the second row shows results utilizing a Cherenkov detector which removes Ions with energies below 2~GeV/nuc. The colors represent expected count rates while the red, blue and cyan boxes represent proton, electron and helium boxes respectively.}
  \label{fig:stopping_ebox}
\end{figure}
Fig. \ref{fig:stopping_ebox} presents simulation results for the stopping coincidence (SDA, SDB, BGO1, SDC, BGO2 are triggered while the particles do not reach SDD and SDE). Since the idea is to utilize the dE/dx-E method, the energy loss in SDA ($\propto$dE/dx) is shown over the total deposited energy. The color code represents the expected count rates of these particles. The second row is using an additional Cherenkov detector as veto condition and hence no protons and helium particles with energies below 2~GeV/nuc are detected. The columns present expected count rates from i) protons, ii) electrons, iii) helium particles and iv) combined overall counts.\newline
Note that for the following study, only central segment combinations were investigated as a best case study regarding the systematical uncertainty. Including other segment combinations will increase statistics on the cost of a increased systematic uncertainty. Furthermore, the numbers only correspond to the count rates in one direction and hence, the expected count rate numbers have to be doubled.


\subsubsection{Proton requirement}
From fig.~\ref{fig:stopping_ebox} it is obvious, that proton and helium particles can be easily distinguished for stopping particles as the red box only contains entries from protons. The label for the shown boxes give the expected count rates inside the box. The expected count rate of 2.8 counts/ks is rather small. However, this is expected as protons can only stop in the second \ac{BGO} with energies of roughly 100-150~MeV and hence, the energy window of this channel is rather small. Furthermore, taking into account both sides of the detector, the count rate increases to 5.6 counts/ks. In a 10ks time interval this channel would have an stat. uncertainty of 
\begin{equation}
    \Delta p / p ~(10ks) = 1/(\sqrt{56}) = 0.14 = 14\%
\end{equation}
Note that these values are based on the requirement that both SDA and SDB are being hit in the central segment (to allow for the cleanest electron channels). Allowing i) hits in the same ring segments in SDA and SDB and ii) hits in the central segment of B independent of SDAs segment one can still ensure that the trajectories are going through the \acp{BGO} while increasing the geometry factor. Based on preliminary studies, it has been found that the expected count rate of protons increases by a factor of six to 34 counts/ks and hence an stat. uncertainty in a 10ks time interval of 
\begin{equation}
    \Delta p / p ~(10ks) = 1/(\sqrt{340}) = 0.055 = 5.5\%
\end{equation}
or in a 3ks time interval of
\begin{equation}
    \Delta p / p ~(3ks) = 1/(\sqrt{102}) = 0.099 = 9.9\%
\end{equation}
Furthermore, it has to be noted that the majority of protons in the required energy range will be detected in the penetrating channel (cf. \ref{sec:sim_pene_p}). 


 \subsubsection{Electron requirements}
% \begin{figure}
%   \centering
%   \includegraphics[width=16cm]{stop_A_inner-B_inner_cut-counts.png}
%   \caption{Energy deposition in SDA over the total deposited energy for particles stopping in the instrument with entries in a defined electron box. The columns refer to simulations with different particles, the second row shows results utilizing a Cherenkov detector which removes Ions with energies below 2~GeV/nuc. The colors represent expected count rates. The sum of the expected count rates is given in the title of each panel.}
%   \label{fig:stopping_ebox_counts}
% \end{figure}
Fig. \ref{fig:stopping_ebox} shows a possible definitions of electron channels. The labels for the shown boxes give the expected count rates inside the box. Protons (and to a much smaller extend helium particles) have a very small chance to deposit similar energies like electrons (in both detector SDA as in the entire instrument). Despite the small possibility for a single proton to do so (cf. fig. \ref{fig:stopping_ebox}), the much higher flux of protons compared to electrons (cf. fig. \ref{fig:adriani-e-p}) causes this contamination (i.e. the counts of protons in the blue electron box) to be significant compared to the expected counts of "real" electrons. \newline
In order to further improve our particle separation, several additional definitions have been introduced:
\begin{itemize}
    \item An upper and lower limit of the ratio of energy depositions in a \ac{BGO} over the energy loss in SDB has been defined for both \acp{BGO} %as seen in fig. \ref{fig:stopping_bgobgo}.
    %\item The ratio of the energy deposition in the \acp{BGO} has been limited to an interval %shown in fig. \ref{fig:stopping_bgoratio}.
    \item An upper limit for the accepted energy deposition in detector C has been defined %as shown in fig. \ref{fig:stopping_sdc}.
    \item An upper limit for the accepted energy deposition in detector B has been defined %as shown in fig. \ref{fig:stopping_sdc}.
    \item The requirement of a signal in the Cherenkov detector
\end{itemize}
%\begin{figure}
%  \centering
%  \includegraphics[width=12cm]{BGO1vsBGO2.png}
%  \caption{Explanation of addition stopping electron cut.}
%  \label{fig:stopping_bgobgo}
%\end{figure}

%\begin{figure}
%  \centering
%  \includegraphics[width=12cm]{BGO1-BGO1_1D.png}
%  \caption{Explanation of addition stopping electron cut.}
%  \label{fig:stopping_bgoratio}
%\end{figure}

%\begin{figure}
%  \centering
%  \includegraphics[width=12cm]{SDC_1D.png}
%  \caption{Explanation of addition stopping electron cut.}
%  \label{fig:stopping_sdc}
%\end{figure}

\begin{figure}
  \centering
  \includegraphics[width=16cm]{stop_A_inner-B_inner_cut-counts.png}
  \caption{Energy deposition in SDA over the total deposited energy for particles stopping in the instrument with entries in a defined electron box. All cuts as discussed above have been applied to the simulation results as well in order to suppress the proton contamination. The columns refer to simulations with different particles, the second column shows the result utilizing a Cherenkov detector which removes Ions with energies below 2~GeV/nuc. The colors represent expected count rates. The sum of the expected count rates is given in the legend of each panel.}
  \label{fig:stopping_allcuts}
\end{figure}
Fig. \ref{fig:stopping_allcuts} shows the contribution of protons to the electron box without (left panel) and with the Cherenkov detector required (middle panel) as well as the electron contribution (since electrons at these energies always trigger the Cherenkov we don't distinguish between both cases here). In comparison to fig. \ref{fig:stopping_ebox} the efficiency of these cuts and especially the benefit of the Cherenkov detector is evidently. A critical quantity in the systematic uncertainty is the ratio of proton counts to proton plus electron counts ($r$).
The expected counts as well as this contribution of protons to the stopping electron channel are given in table \ref{tab:stopping_elec}.\newline % stopping electrons?
\begin{table}
\centering
\begin{tabular}{ || c | c | c | c | c | } 
  \hline
  & ebox & ebox+cuts & ebox+ch & ebox+cuts+ch \\ 
  \hline
  p counts / ks  & 0.98 & 0.183 & $50.4\cdot 10^{-3}$ & $6.58\cdot 10^{-3}$ \\ 
  \hline
  e counts / ks & 0.170 & 0.119 & 0.170 & 0.119 \\ 
  \hline
  C$_a$= e+p counts / ks & 1.151 & 0.302 & 0.220 & 0.126  \\ 
  \hline
  r=p/(p+e)  & 0.85 & 0.61 & 0.23 & 0.05 \\ 
  \hline
  $\Delta C_e / C_e$ (t=100ks) & 0.85 & 0.59 & 0.29 & 0.26 \\
  \hline
  $\Delta C_e / C_e$ (t=3000ks) & 0.16  & 0.11 & 0.053 & 0.048 \\
  \hline
\end{tabular}
  \caption{Expected count rates in the electron box (column 1), including the improved cuts (2), utilizing the Cherenkov detector (3) and with cuts and the Cherenkov (4) based on protons and electrons.}
  \label{tab:stopping_elec}
\end{table}
Utilizing the equation in section \ref{sec:error_estimation_eq}, the relative uncertainty of the electron has been calculated for all scenarios for different time resolutions (for more details see sec. \ref{sec:sim_pene_p}) and considering both directions (i.e. by doubling the expected count rates).\newline
Note that these numbers only reflect coincidences requiring the inner segments of SDA and SDB. Early studies suggest that by utilizing different segment combinations the count rate can be increased by a factor of six with r$\approx$0.16 which would result in $\Delta C_e / C_e$=0.15 for 100ks (daily) and $\Delta C_e / C_e$=0.05 for 800ks (nine days). Furthermore, it has to be noted that this accounts only for stopping electrons (energies up to $\approx$200~MeV), the penetrating electrons further improve the statistics (cf. \ref{sec:sim_pene_e}).



\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:stopping_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}

Utilizing the equation in section \ref{sec:error_estimation_eq}, 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}
\label{sec:sim_verdict}
Based on the analysis above we have decided that \textbf{a Cherenkov is recommended in order to fulfill the requirements}. 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. A detailed mathematical description of the systematic uncertainties introduced by this method is given in section \ref{sec:error_estimation_eq}.\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.}

\subsection{Assessment of measurement uncertainties}
\label{sec:sim_error}
The estimation of the uncertainties for the electron channels has to include the uncertainties introduced by the correction for the proton contamination. This section explains the calculation of these uncertainties.
\subsubsection{Assumptions and variables}

Assumption: 
\begin{itemize}
    \item The proton flux $f_p$ has already been determined by unfolding the proton countrates 
    \item $f_p$ leads to background counts $C_{pb}$ in a given electron box, computed by folding $f_p$ with the given response and integrating over time
    \item the electron counts $C_e$ in a given box are given as $C_e=C_a - C_{pb}$, where $C_a$ are the total counts in the box
    \item the error of $C_{pb}$ is given as $\Delta C_{pb} = \sqrt{ \sqrt{C_{pb}}^2+\Delta C_{pbsys}^2}$, where $\Delta C_{pbsys}$ is the systematic uncertainty for $C_{pbsys}$, mostly resulting from unfolding
    \item the ratio is defined as $r= C_{pb} / C_a$ 
\end{itemize}



\subsubsection{Estimation of uncertainty}
\label{sec:error_estimation_eq}
Since the uncertainty of $C_e$ and $ C_{pb}$ are not uncorrelated, they are just added together:
$$ \Delta C_e = \Delta C_{pb} + \Delta C_a$$

$$ \frac{\Delta C_e}{C_e} = \frac{\Delta C_{pb} + \Delta C_a}{C_a - C_{pb}}$$

$$ \frac{\Delta C_e}{C_e} = \frac{\sqrt{ C_{pb}+\Delta C_{pbsys}^2} + \sqrt{C_a}}{C_a - C_{pb}}$$

$$ \frac{\Delta C_e}{C_e} = \frac{\sqrt{ C_a \cdot r+\Delta C_{pbsys}^2} + \sqrt{C_a}}{C_a - C_a \cdot r}$$

This serves as the uncertainty of the counts and therefore is the lower boundary for the uncertainty of the countrates and fluxes of the electrons in a given box. 

\subsubsection{Examples for the error estimation}
In the following, $\Delta C_{pbsys}$ has been chosen to be the result of an unfolded proton flux uncertainty of $2.25 \%$. 

Figure \ref{fig:both_error_plots} (left) shows the relative uncertainty for the measured electron counts assuming 400 real electrons have been measured for a proton contamination between 0 and 0.9. As seen, a proton contamination of 0.5 (1:1 electrons and protons in electron box) leads to more than a factor of two in the resulting uncertainty.

Figure \ref{fig:both_error_plots} (right) shows the same behaviour for varying numbers of real measured electrons. The contour lines mark constant uncertainties. Hence, to achieve the same uncertainty for different proton contamination, one has to follow the corresponding contour lines. For example, a proton contamination of 0 and 400 real measured electrons lead to an uncertainty of 5 \% (see also previous figure). To achieve the same uncertainty of 5 \% with a contamination of 0.5, the number of real electrons needed increases to roughly 2400, by a factor of 6. This significantly impacts the necessary integration time for our measurement. 

Table \ref{tab:error} shows selected values from the conducted Monte-Carlo simulation with realistic cosmic ray flux values and detector setup.

\begin{table}
\centering
\begin{tabular}{c|c|c|c}
$\Delta C_{pbsys}$ & $r$ & $C_e$ & $\frac{\Delta C_e}{C_e}$ \\\hline
0 & 0 & 400 & 0.05 \\
0 & 0.2 & 500 & 0.08 \\
0 & 0.5 & 800 & 0.12 \\
10 & 0.2 & 500 & 0.082 \\
10 & 0.2 & 1300 & 0.05 \\
\end{tabular}
\caption{Assumed proton uncertainty ($\Delta C_{pbsys}$), relative proton contamination "r", number of real electrons counted $C_e$ and the overall uncertainty derived for various combination of the former values.}
\label{tab:error}
\end{table}

\begin{figure}
  \centering
  \includegraphics[width=7cm]{error.pdf}
  \includegraphics[width=7cm]{error_2D.pdf}
  \caption{Left: Relative uncertainty from proton contamination for 400 real measured electrons (left) and for varying numbers of real measured electrons (right). Proton flux is assumed to have an uncertainty of 2.25 \%}
  \label{fig:both_error_plots}
\end{figure}


%\begin{figure}
%  \centering
%  \includegraphics{error.pdf}
%  \caption{Relative uncertainty from proton contamination for 400 real measured electrons. Proton flux is assumed to have an uncertainty of 2.25 \%}
%  \label{fig:error}
%\end{figure}

%\begin{figure}
%  \centering
%  \includegraphics{error_2D.pdf}
%  \caption{Relative uncertainty from proton contamination for varying numbers of real measured electrons. Proton flux is assumed to have an uncertainty of 2.25 \%}
%  \label{fig:error2D}
%\end{figure}

\subsubsection{Result of the error estimation}

The background protons have a severe impact on the uncertainty of the electron countrates. Even if the response of electrons is higher by two orders of magnitude, the higher relative abundance of protons leads to need for significantly higher integration times to achieve the 5\% goal, even before considering unfolding uncertainty.
