\section{Thermal Analysis}
\label{sec:thermal-analysis}
The inital \acl{TMM} \cite{tmm} has been set up and analyzed at \acf{IDR} at \acf{UPM}. All figures presented in this chapter are taken from that report.\newline
\textit{ESATAN} has been used to perform the analysis. The current model is a low resolution nodal network with one node representing every main mechanical component with manually calculated thermal masses and conductances applied to it. For the moment just steady state cases have been studied, which end up in a thermal equilibrium of the structure depending on the chosen inputs. Figure \ref{fig:tmm-nodal-network} shows the details of it.

\begin{figure}[h!]
	\centering
	\includegraphics[width=1\linewidth]{cau-ath-djf-0007_i1-0/media/tmm_node-network.jpg}
	\caption[The schematic of the baselined node-network]{Schematic of the TMM as a nodal network.}
	\label{fig:tmm-nodal-network}
\end{figure}

\subsection{Design Drivers}
\label{sec:tmm-design-drivers}
The main thermal design drivers of \acs{AHEPaM} are the temperatures of the a) particle detectors and b) the \acf{PMT}. The electronics noise the silicon of the detectors produces is correlated to its temperature, with lower temperature producing less noise. Heritage shows that 0$^{\circ}$C is the ideal temperature for \acs{AHEPaM}'s detectors. That's even more true if little energies need to be detected.\newline
The second design driver is -30$^{\circ}$C for the \acs{PMT} in both the cold op- and non-op case. This is a value defined by the manufacturer, provided in the element's datasheet.\newline
Since this sets a limiting threshold at a comparatively high temperature, the instrument team is currently investigating the justification behind this value and if it can be relaxed towards colder values.\newline
Nevertheless, since at the time of writing mitigation is still under investigation, this value is part of the \acs{TMM}.

\subsection{Design Elements}
\label{sec:tmm-design-elements}
The different main design elements are described in \cite{ahepam-ddf} and are not repeated here.

\subsection{Boundary Conditions}
\label{sec:tmm-boundary-conditions}
As outlined in \cite{ahepam-ddf} key thermal design parameters which are not yet available have been chosen in order to conduct the simulation. More specific, a value for solar heating due to the sun's illumination has been applied. This limits the applicability of the results to a specific scenario, described in the following chapters. The benefit is that this approach provides information to put the units behaviour into perspective and allows to extrapolate from it.\\

The following general boundary conditions have been applied:

\begin{itemize}
    \item The unit is fully insulated, except for:
    \begin{itemize}
        \item its conductive interfaces.
        \item its radiative interfaces.
    \end{itemize}
    \item There are two external heat sources: 
    \begin{itemize}
        \item the sun at 1 AU.
        \item the conductive interface with the \acs{S/C}.
    \end{itemize}
    \item There are three potential heat sinks, depending on the scenario
    \begin{itemize}
    \item radiative coupling with space through the instruments surface.
    \item radiative coupling with space through external radiator 'applied' to the ebox' chassis frames.
    \item the S/C interface with an assumed $\infty$ thermal capacity.
    \end{itemize}
\end{itemize}

Consequently there is \textit{no} heat exchange with the surface of the \acs{S/C} or other (unknown) appendages, e.g. other payload.

\subsection{Studied Cases}
\label{sec:tmm-cases}
The following scenarios have been studied

\begin{itemize}
    \item[] A.1) Hot-operational.
    \begin{itemize}
        \item[] {The sun at a distance at 1 \acs{AU} illuminates 1 face of the pyramid-shaped instrument cover. The cover has the optical properties of \acs{OSR}s, means a high IR emissivity and a low solar absorption. The opposite side of the instrument cover is used as a radiator.\newline \newline
        Heat sources: I/F at +40$^{\circ}$C, instrument dissipates 8.86W in total, sun at 1 AU through instrument cover.\newline
        Heat sinks: Instrument cover.}
    \end{itemize}
    \item[] A.2) Hot-operational with additional external Radiator.
    \begin{itemize}
        \item[] {Same as before, just with an external radiator of approx. 10cm x 10cm applied to each of the three ebox chassis frames.\newline \newline
        Heat sources: I/F at +40$^{\circ}$C, instrument dissipates 8.86W in total, sun at 1 AU through instrument cover.\newline
        Heat sinks: Instrument cover, external radiator with a total area of rounded 0.03m$^2$.}
    \end{itemize}
    \item[] B.1) Cold Operational.
    \begin{itemize}
        \item[] {Now, as the other extreme, the I/F temperature is changed to -40$^{\circ}C$ and the additional heat intake from the sun is switched off.\newline \newline
        Heat sources: Instrument dissipates 8.86W in total.\newline
        Heat sinks: I/F at -40$^{\circ}$C, 1 side of the instrument cover.}
    \end{itemize}
    \item[] {B.2) Side case: Spot heating at the cherenkov frames in order to lift the \acs{PMT} temperature.}
    \begin{itemize}
        \item[] {Here, with the same interface setting as B.1 before, spot heating of 0.328W per cherenkov chassis has been applied. The goal is to see the effect on the PMT, since it has a cold temperature threshold of -30$^{\circ}$C.\newline \newline
        Heat sources: Instrument dissipates 8.86W in total, 2x 0.328W spot heating.\newline
        Heat sinks: I/F at -40$^{\circ}$C, instrument cover.}
    \end{itemize}
    \item[] C.1) Cold Non-operational.
    \begin{itemize}
        \item[] {In the non-op case no power is applied or dissipated by the unit.\newline \newline
        Heat sources: None.\newline
        Heat sinks: I/F at -40$^{\circ}$C, instrument cover.}
    \end{itemize}
    \item[] C.2) Cold Non-operational with survival heating power.
    \begin{itemize}
        \item[] {In order to prevent the minimum threshold of -30$^{\circ}$C at the \acs{PMT}s 5.265W is applied at the base frame. A safety margin of 5$^{\circ}$C for the \acs{PMT} is included.\newline \newline
        Heat sources: 5.265W of survival heating power at the base-frame.\newline
        Heat sinks: I/F at -40$^{\circ}$C, instrument cover.}
    \end{itemize}
\end{itemize}

\newpage

\subsection{Results}
\label{sec:tmm-results}
The results are verified against the design temperature of the particle detectors A to E (\textit{SDA .. SDE} and the \acl{PMT} of the cherenkov detectors. Their specific temperature are listed at the right in table \ref{tab:tmm-results}.

\begin{table}[h!]
	\caption[Results List.]{TMM Results}
    \centering
	\includegraphics[width=1\linewidth]{cau-ath-djf-0007_i1-0/media/tmm_2nd_results-table.pdf}
	\label{tab:tmm-results}
\end{table}

\subsection{Discussion and Next Steps}
\label{sec:tmm-next-steps}
The results clearly indicate that in case the instrument receives external heat for such a long time, that it reaches its equilibrium temperature, it is going to need a radiator. Looking at the effect of the analyzed radiator area and taking the shape of the instrument into account, the surface of the instrument cover could  calculatively be sufficient to lower the temperature to the design temperature.\newline However depending on the location on the S/C the accommodation of \acs{AHEPaM} might become very complex, since the instruments performance would probably be strongly depending on external illumination, e.g. light reflections of payload or S/C appendages.\newline An external radiator might be beneficial, because (given appropriate thermal straps or conductive connection in general) such an element could be accommodated in permanent shadow and thus would, together with \acs{MLI}, that could be used in that scenario, lead to a significantly more robust instrument design.\\

Additionally the results show, that if a radiator is required and it can (at the first assumption) not be switched on or off, it would reduce the instrument temperature well below the cold threshold of the PMT. Case B.1 gives a hint on what might happen. Unfortunately B.1 is overly optimistic, since it does not include the cooling effect of the external radiator, analyzed in A.2. Consequently it can be assumed, that, in case of a radiator, OP-heating in the range of 5W would be required.\\

Finally the need for survival heating power is obvious, see C.2, if the minimum threshold of -30$^{\circ}$C can not be mitigated, means shifted towards colder temperatures.\\

So it can be concluded, that the accommodation of \acs{AHEPaM} on the \acs{ATHENA} \acs{S/C} has a driving influence on adapting the thermal design and the need of heating power. Furthermore it should be thoroughly investigated, if the \acs{PMT} can be operated and stored in colder conditions.

% \begin{table}[h!]
% \centering
% \caption{Telescope FM/ DM design differences.}
% \label{tab:telescope-diffs}
% \begin{tabular}{lll}
% \hline
%  & \multicolumn{1}{c}{\textit{\textbf{FM}}} & \multicolumn{1}{c}{\textit{\textbf{DM}}}                                     \\
%  \hline
% \textit{Positioning} & $\vert50\vert30\vert30\vert50\vert$                            & \begin{tabular}[c]{@{}l@{}}$\vert51.4\vert27.3\vert30\vert51.4\vert$ \end{tabular} \\
% \hline
% \end{tabular}
% \end{table}
