\chapter{Results \label{sec:results}}

This chapter includes the main results of this study. The data for the plots were generated using the "New FC" Button from the main GUI described in section \ref{sec:gui:data_processing}.
The results contain plots for the different spectral types M, K, G, and F as well as combined results of every possible combination. Additionally for all these possible combinations, plots with data limits, e.g. only stars with a rotational period of less than 2 days or minimal/maximal normalized flare peak limits, were also generated. Additionally plots for stars with a more detailed spectral type like M0 or G5 were created. Furthermore there is a selection of individual star results. For every plot group a CSV file is generated, which contains information about every star and flare used to generate the plot.
The results also only contain the data of folded lightcurves which could be fitted with a sine function and less than 30 iterations of the fold optimization. The data which uses polynomial fits or both sine and polynomial fits as well as all plots and the accompanying CSV files are available at \href{https://drive.google.com/drive/folders/1L_F21W3WwZicVZg9052B12B2hJNuJBwK?usp=drive_link}{google drive}.

\section{M dwarfs \label{sec:results:m_dwarfs}}

This section shows the results for all 144 M dwarfs in this study. The list of stars can be found in table \ref{apA:list_of_m_stars}.\\
Figures \ref{fig:M-Flarecount-10_Bins} and \ref{fig:M-Flarecount-30_Bins} show histograms, with 10 and 30 bins respectively, of the amount of flares during the normalized phase.\\
Looking at figure \ref{fig:M-Flarecount-10_Bins} there is an even distribution within error of flares across the normalized phase, with the excepion of the bin at phase $0.5 \pi$. The bin at phase $0.5 \pi$ shows a significant dip of roughly twice the error below the surrounding bins.\\
Looking at the same data, just with 30 instead of 10 bins (figure \ref{fig:M-Flarecount-30_Bins}), the same dip is visible. In this figure the dip spans 3 bins. Additionally there are additional dips at around phase $0.7 \pi$, $1.3 \pi$ and $1.4 \pi$. Including the error, the major dip (which was already visible in figure \ref{fig:M-Flarecount-10_Bins}) is still below the average. Similar for the dips at phases $0.7 \pi$ and $1.3 \pi$. The dip at phase $1.4 \pi$ on the other hand overlaps with its error with the errorbars of the bins at phase \textasciitilde$1.7 \pi$ and and onward, which are good assumption for an average value. Due to the dips surrounding the center, it may look like there is an increased number of flares in the center. If we look at the errorbars, it is clear that only the bin at phase \textasciitilde$1.25 \pi$ is above the average.

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/M/M-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:M-Flarecount-10_Bins}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/M/M-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:M-Flarecount-30_Bins}
    \end{subfigure}
    \caption{Histogram showing the amount of flares per phase for all 144 M dwarfs used in this study. The x-axis represents the normalized phase of the folded lightcurves. There are 10 bins (\subref{fig:M-Flarecount-10_Bins})/30 bins (\subref{fig:M-Flarecount-30_Bins}) of the phase, showing the number of flares per bin. The error bar shows the standard deviation for the histogram. The blue line indicates an idialized phase (sine curve), with the maximum at phase $0 \pi$/$2 \pi$ and the minimum at phase $1 \pi$.}
    \label{fig:M-Flarecount}
\end{figure}

Filtering the data by the minimal flare peak (see figures \ref{fig:M-Flarecount-10_Bins_1.25_peak} to \ref{fig:M-Flarecount-30_Bins_1.5_peak}), makes the trend of figure \ref{fig:M-Flarecount-30_Bins} clearer. There are increasinly more flares in the minima at phase $1 \pi$ (more/bigger star spots) and maxima at phase $0$ and $2 \pi$ (less/smaller star spots) compared to the transitions.
Taking a closer look at figure \ref{fig:M-Flarecount-10_Bins_1.25_peak}, which accounts only for normalized flare peaks greater than 1.25 with 10 bins, the same dips (around phases $0.5 \pi$ and $1.5 \pi$) as in the previous figure (figure \ref{fig:M-Flarecount-30_Bins}) are visible. Comparing to the dip visible in figure \ref{fig:M-Flarecount-10_Bins}, it widened by 1 bin.
At phase \textasciitilde$1.7 \pi$ to $2 \pi$ there is a major peak. While the last bin overlaps slightly with the one at phase $0.9 \pi$ with accounting for error, the second to last does not.\\
The same data, just with 30 bins can be seen in figure \ref{fig:M-Flarecount-30_Bins_1.25_peak}. There are no continuous peaks like in the previous figure, but at the same phases there are bins with an increased number of flares. Additionally, at around phase $0.6 \pi$, there is a larger amount of flares compared to the bins next to it.\\
Increasing the normalized minimal flare peak further to 1.5 decreases the total amount of flares further, which results in a comparatively large error. This makes the error half the size or even larger than some of the bins in figure \ref{fig:M-Flarecount-10_Bins_1.5_peak}. Nontheless its visible that the largest flares seem to appear more often around phase $1 \pi$ and phase $0 \pi$/$2 \pi$. Including the error, those peaks are still higher than the flare count of the in previous figures mentioned dips.\\
Increasing the bin count to 30 (see figure \ref{fig:M-Flarecount-30_Bins_1.5_peak}) decreases the individual bin heights so far, that the errorbars start to explode in size, which causes all of them to overlap and not give any proper results.

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/M/M_minFlarePeak_1.25-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:M-Flarecount-10_Bins_1.25_peak}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/M/M_minFlarePeak_1.25-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:M-Flarecount-30_Bins_1.25_peak}
    \end{subfigure}
    \caption{Histograms showing the amount of flares with a normalized peak of greater than 1.25 per phase for all 144 M dwarfs used in this study. The x-axis represents the normalized phase of the folded lightcurves. There are 10 (\subref{fig:M-Flarecount-10_Bins_1.25_peak}) /30 (\subref{fig:M-Flarecount-30_Bins_1.25_peak}) bins of the phase, showing the number of flares per bin. The error bar shows the standard deviation for the histogram. The blue line indicates an idialized phase (sine curve), with the maximum at phase $0 \pi$/$2 \pi$ and the minimum at phase $1 \pi$.}
    \label{fig:M-Flarecount-peaks_1.25_peak}
\end{figure}
\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/M/M_minFlarePeak_1.5-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:M-Flarecount-10_Bins_1.5_peak}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/M/M_minFlarePeak_1.5-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:M-Flarecount-30_Bins_1.5_peak}
    \end{subfigure}
    \caption{Histograms showing the amount of flares with a normalized peak of greater than 1.5 per phase for all 144 M dwarfs used in this study. The x-axis represents the normalized phase of the folded lightcurves. There are 10 (\subref{fig:M-Flarecount-10_Bins_1.5_peak})/30 (\subref{fig:M-Flarecount-30_Bins_1.5_peak}) bins of the phase, showing the number of flares per bin. The error bar shows the standard deviation for the histogram. The blue line indicates an idialized phase (sine curve), with the maximum at phase $0 \pi$/$2 \pi$ and the minimum at phase $1 \pi$.}
    \label{fig:M-Flarecount-peaks_1.5_peak}
\end{figure}

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/M/M_maxFlarePeak_1.01-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:M-Flarecount-10_Bins_1.01_maxpeak}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/M/M_maxFlarePeak_1.01-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:M-Flarecount-30_Bins_1.01_maxpeak}
    \end{subfigure}
    \caption{Histograms showing the amount of flares with a normalized peak of less than 1.01 per phase for all 144 M dwarfs used in this study. The x-axis represents the normalized phase of the folded lightcurves. There are 10 (\subref{fig:M-Flarecount-10_Bins_1.01_maxpeak})/30 (\subref{fig:M-Flarecount-30_Bins_1.01_maxpeak}) bins of the phase, showing the number of flares per bin. The error bar shows the standard deviation for the histogram. The blue line indicates an idialized phase (sine curve), with the maximum at phase $0 \pi$/$2 \pi$ and the minimum at phase $1 \pi$.}
    \label{fig:M-Flarecount-peaks_1.01_maxpeak}
\end{figure}

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/M/M_maxFlarePeak_1.05-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:M-Flarecount-10_Bins_1.05_maxpeak}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/M/M_maxFlarePeak_1.05-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:M-Flarecount-30_Bins_1.05_maxpeak}
    \end{subfigure}
    \caption{Histograms showing the amount of flares with a normalized peak of less than 1.05 per phase for all 144 M dwarfs used in this study. The x-axis represents the normalized phase of the folded lightcurves. There are 10 (\subref{fig:M-Flarecount-10_Bins_1.05_maxpeak})/30 (\subref{fig:M-Flarecount-30_Bins_1.05_maxpeak}) bins of the phase, showing the number of flares per bin. The error bar shows the standard deviation for the histogram. The blue line indicates an idialized phase (sine curve), with the maximum at phase $0 \pi$/$2 \pi$ and the minimum at phase $1 \pi$.}
    \label{fig:M-Flarecount-peaks_1.05_maxpeak}
\end{figure}

\FloatBarrier
\section{K dwarfs \label{sec:results:k_dwarfs}}

This section shows the results for all 37 K dwarfs in this study. A full list of the stars used can be found in table \ref{apA:list_of_k_stars}.\\
Figure \ref{fig:K-Flarecount-10_Bins} shows the amount of flares per phase with 10 bins of 37 K type dwarfs in used in this study. Overall the distribution is even within error, with a slight increase in flares when going from the maximum to the minimum of the lightcurve at phase $0.5 \pi$, and a slight decrease when going from minimum to maximum at phase $1.5 \pi$. There is also a peak at around $1 \pi$ with around 95 flares compared to the 60-80 flares per bin in the rest of the histogram. This peak is dominated by $V*~V471~Tau$. The individual results for this star are visible in section \ref{sec:results:individual}.\\
Looking at the same dataset with 30 bins for the histogram (figure \ref{fig:K-Flarecount-30_Bins}), the peak in the phase minimum at $1 \pi$ is still visible. Additionally there appear more peaks at phase $>1.3 \pi$ ($1.3,~1.6,~1.9 \pi$), while the amount of flares between phase $0 \pi$ and $1 \pi$ shows a slight trend to more flares with dips inbetween and a larger dip right before and after the big peak at phase $1 \pi$ which was also visible in figure \ref{fig:K-Flarecount-10_Bins} with 10 bins. The major dips at phase $0.7 \pi$, $1.3 \pi$, $1.6 \pi$ and $1.9 \pi$ are all well outside the errorbars of the surrounding peak bins, while the smaller ones between phase $0 \pi$ and $1 \pi$ overlap with their errorbars with their surrounding bins.

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/K/K-Flarecount-10_Bins.png}
    \caption{10 bins}
    \label{fig:K-Flarecount-10_Bins}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/K/K-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:K-Flarecount-30_Bins}
    \end{subfigure}
    \caption{Histogram showing the amount of flares per phase for all 37 K dwarfs used in this study. The x-axis represents the normalized phase of the folded lightcurves. There are 10 bins (\subref{fig:K-Flarecount-10_Bins})/30 bins (\subref{fig:K-Flarecount-30_Bins}) of the phase, showing the number of flares per bin. The error bar shows the standard deviation for the histogram. The blue line indicates an idialized phase (sine curve), with the maximum at phase $0 \pi$/$2 \pi$ and the minimum at phase $1 \pi$.}
    \label{fig:K-Flarecount}
\end{figure}

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/K/K_minFlarePeak_1.05-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:K-Flarecount-10_Bins_1.05_peak}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/K/K_minFlarePeak_1.05-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:K-Flarecount-30_Bins_1.05_peak}
    \end{subfigure}
    \caption{Histograms showing the amount of flares with a normalized peak of greater than 1.05 per phase for all 37 K dwarfs used in this study. The x-axis represents the normalized phase of the folded lightcurves. There are 10 (\subref{fig:K-Flarecount-10_Bins_1.05_peak})/30 (\subref{fig:K-Flarecount-30_Bins_1.05_peak}) bins of the phase, showing the number of flares per bin. The error bar shows the standard deviation for the histogram. The blue line indicates an idialized phase (sine curve), with the maximum at phase $0 \pi$/$2 \pi$ and the minimum at phase $1 \pi$.}
    \label{fig:K-Flarecount-peaks_1.5_peak}
\end{figure}

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/K/K_maxFlarePeak_1.01-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:K-Flarecount-10_Bins_1.01_maxpeak}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/K/K_maxFlarePeak_1.01-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:K-Flarecount-30_Bins_1.01_maxpeak}
    \end{subfigure}
    \caption{Histograms showing the amount of flares with a normalized peak of less than 1.01 per phase for all 37 K dwarfs used in this study. The x-axis represents the normalized phase of the folded lightcurves. There are 10 (\subref{fig:K-Flarecount-10_Bins_1.01_maxpeak})/30 (\subref{fig:K-Flarecount-30_Bins_1.01_maxpeak}) bins of the phase, showing the number of flares per bin. The error bar shows the standard deviation for the histogram. The blue line indicates an idialized phase (sine curve), with the maximum at phase $0 \pi$/$2 \pi$ and the minimum at phase $1 \pi$.}
    \label{fig:K-Flarecount-peaks_1.01_maxpeak}
\end{figure}

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/K/K_maxFlarePeak_1.05-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:K-Flarecount-10_Bins_1.05_maxpeak}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/K/K_maxFlarePeak_1.05-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:K-Flarecount-30_Bins_1.05_maxpeak}
    \end{subfigure}
    \caption{Histograms showing the amount of flares with a normalized peak of less than 1.05 per phase for all 37 K dwarfs used in this study. The x-axis represents the normalized phase of the folded lightcurves. There are 10 (\subref{fig:K-Flarecount-10_Bins_1.05_maxpeak})/30 (\subref{fig:K-Flarecount-30_Bins_1.05_maxpeak}) bins of the phase, showing the number of flares per bin. The error bar shows the standard deviation for the histogram. The blue line indicates an idialized phase (sine curve), with the maximum at phase $0 \pi$/$2 \pi$ and the minimum at phase $1 \pi$.}
    \label{fig:K-Flarecount-peaks_1.05_maxpeak}
\end{figure}

\FloatBarrier
\section{G dwarfs \label{sec:results:g_dwarfs}}

This section shows the results for 71 G dwarfs in this study. Table \ref{apA:list_of_g_stars} contains a list of all G type stars used.\\
The first histogram over all data of G type dwarfs with 10 bins in figure \ref{fig:G-Flarecount-10_Bins} shows a significant increase of occuring flares well outside the errorbar range in the lightcurve minima at around phase $1 \pi$ compared to the maxima at phase $0 \pi$/$2 \pi$. The rise in flare occurances from maxima to minima (phase $0 \pi$ to $1 \pi$) seems to be gradual, while there is a steep fall off after the sixth bin at phase $1.1 \pi$.\\
Due to the lower number of detected flares on G type stars, the error bars in figure \ref{fig:G-Flarecount-30_Bins} are rather large. The increase of flares in the minima of the folded lightcurve from the previous figure is now splint into two peaks at phase $0.6 \pi$ and $1 \pi$. An additional peak bin appears at around phase $0.25 \pi$ in this figure. This peaks errorbar does not overlap with its surrounding bins errorbars. Right before this peak is a significant dip visible at phase $0.2 \pi$. On the other half of the phase after the peak at phase $1 \pi$, the near even distribution of figure \ref{fig:G-Flarecount-10_Bins} becomes more noisy, even though its still well within error.

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/G/G-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:G-Flarecount-10_Bins}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/G/G-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:G-Flarecount-30_Bins}
    \end{subfigure}
    \caption{Histogram showing the amount of flares per phase for all 71 G dwarfs used in this study. The x-axis represents the normalized phase of the folded lightcurves. There are 10 bins (\subref{fig:G-Flarecount-10_Bins})/30 bins (\subref{fig:G-Flarecount-30_Bins}) of the phase, showing the number of flares per bin. The error bar shows the standard deviation for the histogram. The blue line indicates an idialized phase (sine curve), with the maximum at phase $0 \pi$/$2 \pi$ and the minimum at phase $1 \pi$.}
    \label{fig:G-Flarecount}
\end{figure}

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/G/G_minFlarePeak_1.05-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:G-Flarecount-10_Bins_1.05_peak}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/G/G_minFlarePeak_1.05-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:G-Flarecount-30_Bins_1.05_peak}
    \end{subfigure}
    \caption{Histograms showing the amount of flares with a normalized peak of greater than 1.05 per phase for all 71 G dwarfs used in this study. The x-axis represents the normalized phase of the folded lightcurves. There are 10 (\subref{fig:G-Flarecount-10_Bins_1.05_peak})/30 (\subref{fig:G-Flarecount-30_Bins_1.05_peak}) bins of the phase, showing the number of flares per bin. The error bar shows the standard deviation for the histogram. The blue line indicates an idialized phase (sine curve), with the maximum at phase $0 \pi$/$2 \pi$ and the minimum at phase $1 \pi$.}
    \label{fig:G-Flarecount-peaks_1.5_peak}
\end{figure}

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/G/G_maxFlarePeak_1.05-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:G-Flarecount-10_Bins_1.05_maxpeak}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/G/G_maxFlarePeak_1.05-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:G-Flarecount-30_Bins_1.05_maxpeak}
    \end{subfigure}
    \caption{Histograms showing the amount of flares with a normalized peak of less than 1.05 per phase for all 37 K dwarfs used in this study. The x-axis represents the normalized phase of the folded lightcurves. There are 10 (\subref{fig:G-Flarecount-10_Bins_1.05_maxpeak})/30 (\subref{fig:G-Flarecount-30_Bins_1.05_maxpeak}) bins of the phase, showing the number of flares per bin. The error bar shows the standard deviation for the histogram. The blue line indicates an idialized phase (sine curve), with the maximum at phase $0 \pi$/$2 \pi$ and the minimum at phase $1 \pi$.}
    \label{fig:G-Flarecount-peaks_1.05_maxpeak}
\end{figure}

\FloatBarrier
\section{F dwarfs \label{sec:results:f_dwarfs}}

This section shows the results for all 13 F dwarfs in this study. The list of F type stars can be found in table \ref{apA:list_of_f_stars}.\\
Due to the low number of F type stars in this study, and the difficulty to detect flares on them, the detected number of flares in figures \ref{fig:F-Flarecount-10_Bins} and \ref{fig:F-Flarecount-30_Bins} is very low which causes the errorbars of the histogram to grow very large. Nontheless all detected flares were around the minimum of the lightcurves.

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/F/F-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:F-Flarecount-10_Bins}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/F/F-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:F-Flarecount-30_Bins}
    \end{subfigure}
    \caption{Histogram showing the amount of flares per phase for all 13 F dwarfs used in this study. The x-axis represents the normalized phase of the folded lightcurves. There are 10 bins (\subref{fig:F-Flarecount-10_Bins})/30 bins (\subref{fig:F-Flarecount-30_Bins}) of the phase, showing the number of flares per bin. The error bar shows the standard deviation for the histogram. The blue line indicates an idialized phase (sine curve), with the maximum at phase $0 \pi$/$2 \pi$ and the minimum at phase $1 \pi$.}
    \label{fig:F-Flarecount}
\end{figure}

\FloatBarrier
\section{Combined results \label{sec:results:combined}}

The results for all 265 stars in the study are shown in this chapter. This includes the stars from tables \ref{apA:list_of_m_stars} to \ref{apA:list_of_f_stars}. The histograms in figure \ref{fig:MKGF-Flarecount-10_Bins} and \ref{fig:MKGF-Flarecount-30_Bins} are stacked histograms. The flare amount of the individual stars are per bin are stacked on top of each other resulting in the final value.\\
The dip at phase $0.5 \pi$ which was present in the histogram for the M type stars (see figure \ref{fig:M-Flarecount-10_Bins} in section \ref{sec:results:m_dwarfs}) propagates and casues the dip to be also visible in figure \ref{fig:MKGF-Flarecount-10_Bins}. The peak at phase $1 \pi$ is also a result of propagation, but from the K and G type star data. Due to the large number of overall flares, the errorbars are small and the errorbars of the peak in the center at phase $1 \pi$ does not overlap with the other errorbars.\\
Looking at the same data with 30 bins over the phase in figure \ref{fig:MKGF-Flarecount-30_Bins}, the propagation of the variation in the data of the M type stars is clearly visible with the dip around phase $0.5 \pi$ and $1.4 \pi$. Additionally the peaks from the data of K (figure \ref{fig:K-Flarecount-30_Bins}) and G (figure \ref{fig:G-Flarecount-30_Bins}) type stars cause a wider peak at around phase $1 \pi$. Additionally there are also smaller, especially less wide peaks at around phases $0.4 \pi$, $0.7 \pi$ and $1.5 \pi$.

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/MKGF-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:MKGF-Flarecount-10_Bins}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/MKGF-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:MKGF-Flarecount-30_Bins}
    \end{subfigure}
    \caption{Histograms showing the amount of flares per phase for all 265 dwarfs used in this study. The x-axis represents the normalized phase of the folded lightcurves. There are 10 bins (\subref{fig:MKGF-Flarecount-10_Bins})/30 bins (\subref{fig:MKGF-Flarecount-30_Bins}) of the phase, showing the number of flares per bin. The colors show the individual amount for each spectral type with the amount being stacked ontop of each other. The error bar shows the standard deviation for the histogram. The blue line indicates an idialized phase (sine curve), with the maximum at phase $0 \pi$/$2 \pi$ and the minimum at phase $1 \pi$.}
    \label{fig:MKGF-Flarecount}
\end{figure}

\FloatBarrier
\section{Individual stars \label{sec:results:individual}}

This section contains a selection of results for individual stars. The results for this section were selected because either they match what this study was looking for or because the opposite is the case or were in some other way interesting.

\subsection{BD-08   995}

BD-08 995, also known by TIC 43472154, is a G type star with a surface temperature of 5231.2 K, which is \textasciitilde87 pc away. It is a very active solar like star, producing over 200 superflares per year (\cite{tess_1st_year_superflares}). It has a rotational period of 2.8 days (\cite{tess_1st_year_superflares}). There are two TESS lightcurves available, sectors 5 and 32.\\
Looking at figure \ref{fig:BD-08_995-Flarecount-10_Bins}, which shows the flare distribution across the normalized phase of the folded lightcurves with 10 bins, it shows a clear peak of flares appearance in the lightcurve minima at phase $1 \pi$. The errorbars of this wide peak only overlap with the first bin of the plot at phase $0 \pi$. 
This bin belong to the phase maxima which also shows slight increase in flare activity at phase $0/2 \pi$ compared to the transition regions at around phase $0.5 \pi$ (maxima to minima) and $1.5 \pi$ (minima to maxima). Considering errors for this, the errorbars of the bins at the phase maxima overlap with those of the transition regions.\\
Figure \ref{fig:BD-08_995-Flarecount-30_Bins} shows the same data just with 30 bins instead of 10. Ignoring the errorbars, it shows a similar picture as the previous figure. But due to the low number of total detected flares, and a the relatively large bin count, the errorbars become large compared to the individual bins. Due to this, the errorbars of most bins overlap with each other, with the exceptions of the the first bin at phase $0 \pi$ and the bin at phase \textasciitilde$1.1 \pi$, whichs erorbars only overlap with other higher bins like the ones at phase \textasciitilde$0.9 \pi$ and \textasciitilde$1.4 \pi$.


\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/BD-08_995/BD-08\space\space\space995-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:BD-08_995-Flarecount-10_Bins}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/BD-08_995/BD-08\space\space\space995-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:BD-08_995-Flarecount-30_Bins}
    \end{subfigure}
    \caption{Histograms of BD-08 995 across the phase showing the number of flares in each bin. The error bar shows the standard deviation for the histogram. The blue lines show the various fits for the folded lightcurves used to generate the data, with the phase minimum at $1 \pi$ and phase maximum at $0 \pi$/$2 \pi$.}
    \label{fig:BD-08_995-Flarecount}
\end{figure}

The distribution of normalized flare peaks for BD-08 995 in figure \ref{fig:BD-08_995-flarepeaks_1.2} shows that the highest flare peak appeared at around phase $0.1 \pi$ with a peak of \textasciitilde1.11. Other high flare peaks were detected at phases \textasciitilde$1 \pi$ and \textasciitilde$1.9 \pi$. Furthermore the flares around phase $1 \pi$ seem to have higher average peak.

\begin{figure}[pt!]
    \includegraphics[width=.95\textwidth]{plots/sine/BD-08_995/BD-08\space\space\space995-Flarepeaks_maxY-1.2.png}
    \caption{Distribution of flare peaks in relation to the normalized phase at which they occured for BD-08 995. Y-Axis shows the flare peak and is limited to the value of the highest peak detected. The x-axis shows the normalized phase.}
    \label{fig:BD-08_995-flarepeaks_1.2}
\end{figure}

Looking at the folded TESS lightcurves of sector 5 (\ref{fig:BD-08_995-TESS5_foldedLC}) and 32 (\ref{fig:BD-08_995-TESS32_foldedLC}) one can see the earlier presented distribution of flares and their peaks. In TESS sector 5 there appear comparatively many flares at around phase $-0.2$ (not normalized). The highest detected flare peak which was mentioned prior was detected in the TESS sector 32 lightcurve at phase $-1.2$.

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/BD-08_995/BD-08\space\space\space995_TESS-5-foldedLC-marked_fit_flares.png}
        \caption{TESS Sector 5}
        \label{fig:BD-08_995-TESS5_foldedLC}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/BD-08_995/BD-08\space\space\space995_TESS-32-foldedLC-marked_fit_flares.png}
        \caption{TESS Sector 32}
        \label{fig:BD-08_995-TESS32_foldedLC}
    \end{subfigure}
    \caption{Folded lightcurves for BD-08 995. The blue lines shows the sine fits calculated. The red crosses indicate the detected flare peaks.}
    \label{fig:BD-08_995-TESS_foldedLCs}
\end{figure}

\FloatBarrier
\subsection{TYC 1360-957-1}

Not much is published about TYC 1360-957-1. It has been categorized as spectral type K on SIMBAD (\cite{simbad}). It has the TESS Input Catalogue number 247117382, and has been observed by TESS in sectors 44, 45 and 46.\\
Looking at the flare distribution histogram with 10 bins for this star in figure \ref{fig:TYC_1360-957-1-Flarecount-10_Bins} one can see that the highest peaks in order are at phase \textasciitilde$0.7 \pi$, \textasciitilde$0.4 \pi$ and \textasciitilde$1.2 \pi$. The last one being two bins wide. The first two are, including error, above their surrounding bins, while for the third peak, the second peaks errorbar overlaps with those of the smaller bins.
Looking at the histogram with 30 bins of the same dataset (figure \ref{fig:TYC_1360-957-1-Flarecount-30_Bins}), the first 2 peaks mentioned for the histogram with 10 bins are still visible. But compared to the privious figure, the low number of flares result in a rather large error. This causes the errorbars of the two mentioned bin peaks to overlap with the surrounding bins which could be estimated to be the average.

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/TYC_1360-957-1/TYC 1360-957-1-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:TYC_1360-957-1-Flarecount-10_Bins}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/TYC_1360-957-1/TYC 1360-957-1-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:TYC_1360-957-1-Flarecount-30_Bins}
    \end{subfigure}
    \caption{Histograms of TYC 1360-957-1 across the phase showing the number of flares in each bin. The error bar shows the standard deviation for the histogram. The blue lines show the various fits for the folded lightcurves used to generate the data, with the phase minimum at $1 \pi$ and phase maximum at $0 \pi$/$2 \pi$.}
    \label{fig:TYC_1360-957-1-Flarecount}
\end{figure}

Figure \ref{fig:TYC_1360-957-1-flarepeaks_1.31} presents the overall distribution on the normalized phase detected for TYC 1360-957-1 in this study. The three highest flare peaks detected were at a normalized phase of \textasciitilde$0.8 \pi$, which is very close to the phase minimum at $1 \pi$. Three more higher flare peaks stand out at phase $1.2 \pi$. While they are not as high as the previous three, they have a higher peak than the rest of the detected flares.

\begin{figure}[pt!]
    \includegraphics[width=.95\textwidth]{plots/sine/TYC_1360-957-1/TYC 1360-957-1-Flarepeaks_maxY-1.3130771478482715.png}
    \caption{Distribution of flare peaks in relation to the normalized phase at which they occured for TYC 1360-957-1. Y-Axis shows the flare peak and is limited to the value of the highest peak detected. The x-axis shows the normalized phase.}
    \label{fig:TYC_1360-957-1-flarepeaks_1.31}
\end{figure}

The used folded lightcurves used can be seen in figure \ref{fig:TYC_1360-957-1-TESS_foldedLCs}. This figure is split into the three different TESS sectors. In \subref{fig:TYC_1360-957-1-TESS44_foldedLC} sector 44 is visible. The lowest point of the folded lightcurve is slightly shifted from the center, which represents the lowest point of the sine fit. One can also see that two large flares happened at this point.
No such large flares can be seen in \subref{fig:TYC_1360-957-1-TESS45_foldedLC}, unlike in \subref{fig:TYC_1360-957-1-TESS46_foldedLC} which shows a large flare again.

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.95\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/TYC_1360-957-1/TYC 1360-957-1_TESS-44-foldedLC-marked_fit_flares.png}
        \caption{TESS Sector 44}
        \label{fig:TYC_1360-957-1-TESS44_foldedLC}
    \end{subfigure}
    \begin{subfigure}[b]{.95\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/TYC_1360-957-1/TYC 1360-957-1_TESS-45-foldedLC-marked_fit_flares.png}
        \caption{TESS Sector 45}
        \label{fig:TYC_1360-957-1-TESS45_foldedLC}
    \end{subfigure}
    \begin{subfigure}[b]{.95\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/TYC_1360-957-1/TYC 1360-957-1_TESS-46-foldedLC-marked_fit_flares.png}
        \caption{TESS Sector 46}
        \label{fig:TYC_1360-957-1-TESS46_foldedLC}
    \end{subfigure}
    \caption{Folded lightcurves for TYC 1360-957-1. The blue lines shows the sine fits calculated. The red crosses indicate the detected flare peaks.}
    \label{fig:TYC_1360-957-1-TESS_foldedLCs}
\end{figure}

\FloatBarrier
\subsection{TYC 4595-107-1}

TYC 4595-107-1, also known as TIC 394030788, is a G type star with an effective temperature of 5231 K, a radius of 0.9 $R_\odot$ (\cite{tess_2nd_year_superflares}, \cite{superflare_rate_variation_g_type}), mass of 0.89 $M_\odot$ (\cite{superflare_rate_variation_g_type}) and a rotational period of 3.3 days (\cite{tess_2nd_year_superflares}, \cite{superflare_rate_variation_g_type}) and was observed in 20 TESS sectors. The folded lightcurves for TYC 4595-107-1 can be found in appendix \ref{apB:TYC_4595-107-1}.\\
The first histogram, figure \ref{fig:TYC_4595-107-1-Flarecount-10_Bins}, with 10 bins shows the accumulated data for all 20 observed lightcurves. Due to the high number of available lightcurves, and TYC 4595-107-1 being a very active star (\cite{tess_2nd_year_superflares}, \cite{superflare_rate_variation_g_type}), the total flare count is very high. There seems to be a base level of flares per bin of around 30 over the phase, with a large and two bin wide peak at around phase $0.6 to 0.8 \pi$. This increase in flares is high enough to be outside of the error of the base amount of flares. Another peak, even though less certain as its errorbar overlaps with the surrounding bins, appears at the bin at phase $1.9 \pi$. A similar increase in flares can be seen at the bin at $0.2 \pi$.\\
The histogram with 30 bins (figure \ref{fig:TYC_4595-107-1-Flarecount-30_Bins}) shows a similar result. It shows a large increase in flares around phase $0.6 to 1 \pi$, as well at at $1.9 to 0.1 \pi$ and $0.4 \pi$. Only the first mentioned peak managed to be large enough though, to not have overlapping errors with the surrounding lower count bins.

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/TYC_4595-107-1/TYC 4595-107-1-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:TYC_4595-107-1-Flarecount-10_Bins}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/TYC_4595-107-1/TYC 4595-107-1-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:TYC_4595-107-1-Flarecount-30_Bins}
    \end{subfigure}
    \caption{Histograms of TYC 4595-107-1 across the phase showing the number of flares in each bin. The error bar shows the standard deviation for the histogram. The blue lines show the various fits for the folded lightcurves used to generate the data, with the phase minimum at $1 \pi$ and phase maximum at $0 \pi$/$2 \pi$.}
    \label{fig:TYC_4595-107-1-Flarecount}
\end{figure}

\FloatBarrier
\subsection{V* V471 Tau}

V471 Tau is a post-common envelope binary system consiting of a K2 type dwarf and a white dwarf (\cite{v471tau_revised}). The K2 dwarf has a always present dominant spot, which faces the white dwarf (\cite{V471tau_magnetic_activity}).
Figures \ref{fig:V471Tau-Flarecount-10_Bins} and \ref{fig:V471Tau-Flarecount-30_Bins} show the histograms of 5 TESS folded lightcurves for V471 Tau with 10 and 30 bins respectively. The TESS lightcurves used are of sectors 42, 43, 44, 70 and 71. The lightcurve of K2 target table ID 80 was rejected by the algorithm. Due to the lightcurve spanning 90 days, and the variability in spots, the folding and fitting algorithm could not produce a reliable output.
The blue lines show the used fits for the individual folded lightcurves.
In the first figure, there appear two different peaks. The first being at phase $0.5 \pi$, and the second one at phase $1.1 \pi$. These bins are significantly higher than the surrounding bins with 13/15 flares respectively compared to 1 to 8 on the other bins.\\
As seen in figures \ref{fig:V471Tau-TESS42_foldedLC}, \ref{fig:V471Tau-TESS43_foldedLC} and \ref{fig:V471Tau-TESS44_foldedLC} the transit of the white dwarf occurs at around phase 0 (in these figures) which translates to phase $1 \pi$ in the normalized phase.\\
Increasing the bins to 30 (figure \ref{fig:V471Tau-Flarecount-30_Bins}) does not change the result much. Due to the individual flare count being lower per bin compared to the previous figure, the errorbars increase in size comparatively. There are still peaks at around $0.5 \pi$ and $1 \pi$.

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/V471Tau/V_star_ V471 Tau-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:V471Tau-Flarecount-10_Bins}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/V471Tau/V_star_ V471 Tau-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:V471Tau-Flarecount-30_Bins}
    \end{subfigure}
    \caption{Histograms of V471 Tau across the phase showing the number of flares in each bin. The error bar shows the standard deviation for the histogram. The blue lines show the various fits for the folded lightcurves used to generate the data, with the phase minimum at $1 \pi$ and phase maximum at $0 \pi$/$2 \pi$.}
    \label{fig:V471Tau-Flarecount}
\end{figure}

Looking at the folded lightcurves for this star (see figures \ref{fig:V471Tau-TESS42_foldedLC}, \ref{fig:V471Tau-TESS43_foldedLC} and \ref{fig:V471Tau-TESS44_foldedLC}), the transit of the white dwarf (sudden dip in the folded lightcurve) is clearly visible, and always happens around phase 0.

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.95\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/V471Tau/V_star_ V471 Tau_TESS-42-foldedLC-marked_fit_flares.png}
        \caption{TESS Sector 42}
        \label{fig:V471Tau-TESS42_foldedLC}
    \end{subfigure}
    \begin{subfigure}[b]{.95\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/V471Tau/V_star_ V471 Tau_TESS-43-foldedLC-marked_fit_flares.png}
        \caption{TESS Sector 43}
        \label{fig:V471Tau-TESS43_foldedLC}
    \end{subfigure}
    \begin{subfigure}[b]{.95\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/V471Tau/V_star_ V471 Tau_TESS-44-foldedLC-marked_fit_flares.png}
        \caption{TESS Sector 44}
        \label{fig:V471Tau-TESS44_foldedLC}
    \end{subfigure}
    \caption{Folded lightcurves for V* V471 Tau. The blue lines shows the sine fits calculated. The red crosses indicate the detected flare peaks.}
    \label{fig:V471Tau-TESS_foldedLCs}
\end{figure}


Figure \ref{fig:V471Tau-flarepeaks_1.053} shows the normalized phase and peak of each flare in the data during which part of the phase it happened. It shows that the largest flares happened at around phase $0.5 \pi$, $1 \pi$ and $1.5 \pi$. The phase in which the white dwarf transit happens is at around phase $1 \pi$.

\begin{figure}[pt!]
    \includegraphics[width=.95\textwidth]{plots/sine/V471Tau/V_star_ V471 Tau-Flarepeaks_maxY-1.0539907609848342.png}
    \caption{Distribution of flare peaks in relation to the normalized phase at which they occured. Y-Axis shows the flare peak and is limited to the value of the highest peak detected. The x-axis shows the normalized phase.}
    \label{fig:V471Tau-flarepeaks_1.053}
\end{figure}

\FloatBarrier
\subsection{V* HK Aqr}

HK Aqr is a M dwarf with a mass of 0.57 $M_\odot$, a radius of 0.53 $R_\odot$ and is around 22.3 pc away from our solar system. Its effective temperature is aroun 3800 K (\cite{conch_shell_m_dwarfs}). It was observed in four TESS sectors.
HK Aqr is mentioned here, as it is a star, for which the optimize fold algorithm partially broke. This happened for its lightcurves for the sectors 29 and 42. The algorithm found for both periodograms a second signal for a possible period. This resulted in the folded lightcurves seen in figures \ref{fig:HKAqr-TESS29_foldedLC} and \ref{fig:HKAqr-TESS42_foldedLC}. Figures \ref{fig:HKAqr-Flarecount-10_Bins} and \ref{fig:HKAqr-Flarecount-30_Bins} were created with this dataset.\\
In figure \ref{fig:HKAqr-Flarecount-10_Bins} one can see an increase in flares at around phase $1.3 \pi$. This peak is high enough to be outside the error range of the lower flare count bins between phase $0 to 1 \pi$. Figure \ref{fig:HKAqr-Flarecount-30_Bins} paints a similar picture. Additionally it shows a peak at phase $0 \pi$ though, which is during the phase maximum, as well as one at around phase $1.6 \pi$ which is in the transition from minima to maxima.\\
Figures \ref{fig:HKAqr-Flarecount-10_Bins_Period} and \ref{fig:HKAqr-Flarecount-30_Bins_Period} show the historgrams for the data of the two folded TESS lightcurves of sectors 29 and 42 which were flagged as period folded with 10 and 30 bins respectively. The folded lightcurves can be found in figure \ref{fig:HKAqr-TESS_foldedLCs_Period}.
In figure \ref{fig:HKAqr-Flarecount-10_Bins_Period} the overall flare distribution seems to be relatively high from phase $0.2 \pi$ to $1.4 \pi$, with a peak at phase $1.2 \pi$. The amount of detected flares during the maximum of the lightcurves at phase $1.8$ to $0.2 \pi$ is low compared to that. Even including the errorbars it cannot reach the previous bins. An additional dip of the same flare count can be seen at phase $1.5 \pi$.
The flare count in figure \ref{fig:HKAqr-Flarecount-30_Bins_Period} is very low. Due to this all errorbars overlap. Nontheless there are two bins which stand out at phases \textasciitilde$0.4 \pi$ and \textasciitilde$1.25 \pi$ which would be in the transition from maxima to minima, and shortly after the minima respectively.

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/HKAqr/V_star_ HK Aqr-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:HKAqr-Flarecount-10_Bins}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/HKAqr/V_star_ HK Aqr-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:HKAqr-Flarecount-30_Bins}
    \end{subfigure}
    \caption{Histograms of V* HK Aqr across the phase showing the number of flares in each bin. The error bar shows the standard deviation for the histogram. The blue lines show the various fits for the folded lightcurves used to generate the data, with the phase minimum at $1 \pi$ and phase maximum at $0 \pi$/$2 \pi$.}
    \label{fig:HKAqr-Flarecount}
\end{figure}

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.95\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/HKAqr/V_star_ HK Aqr_TESS-2-foldedLC-marked_fit_flares.png}
        \caption{TESS Sector 2}
        \label{fig:HKAqr-TESS2_foldedLC}
    \end{subfigure}
    \begin{subfigure}[b]{.95\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/HKAqr/V_star_ HK Aqr_TESS-29-foldedLC-marked_fit_flares.png}
        \caption{TESS Sector 29}
        \label{fig:HKAqr-TESS29_foldedLC}
    \end{subfigure}
    \caption{Folded lightcurves for V* HK Aqr. The blue lines shows the sine fits calculated. The red crosses indicate the detected flare peaks.}
    \label{fig:HKAqr-TESS_foldedLCs1}
\end{figure}
\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.95\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/HKAqr/V_star_ HK Aqr_TESS-42-foldedLC-marked_fit_flares.png}
        \caption{TESS Sector 42}
        \label{fig:HKAqr-TESS42_foldedLC}
    \end{subfigure}
    \begin{subfigure}[b]{.95\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/HKAqr/V_star_ HK Aqr_TESS-69-foldedLC-marked_fit_flares.png}
        \caption{TESS Sector 69}
        \label{fig:HKAqr-TESS69_foldedLC}
    \end{subfigure}
    \caption{Folded lightcurves for V* HK Aqr. The blue lines shows the sine fits calculated. The red crosses indicate the detected flare peaks.}
    \label{fig:HKAqr-TESS_foldedLCs2}
\end{figure}

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/HKAqr/V_star_ HK Aqr-Flarecount-10_Bins_Period.png}
        \caption{10 bins}
        \label{fig:HKAqr-Flarecount-10_Bins_Period}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/HKAqr/V_star_ HK Aqr-Flarecount-30_Bins_Period.png}
        \caption{30 bins}
        \label{fig:HKAqr-Flarecount-30_Bins_Period}
    \end{subfigure}
    \caption{Histograms of V* HK Aqr across the phase showing the number of flares in each bin. The error bar shows the standard deviation for the histogram. The blue lines show the various fits for the folded lightcurves used to generate the data, with the phase minimum at $1 \pi$ and phase maximum at $0 \pi$/$2 \pi$.}
    \label{fig:HKAqr-Flarecount_Period}
\end{figure}

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.95\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/HKAqr/V_star_ HK Aqr_TESS-29-periodFoldedLC-marked_fit_flares.png}
        \caption{TESS Sector 29}
        \label{fig:HKAqr-TESS29_foldedLC_Period}
    \end{subfigure}
    \begin{subfigure}[b]{.95\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/HKAqr/V_star_ HK Aqr_TESS-42-periodFoldedLC-marked_fit_flares.png}
        \caption{TESS Sector 42}
        \label{fig:HKAqr-TESS42_foldedLC_Period}
    \end{subfigure}
    \caption{Folded lightcurves for V* HK Aqr. The blue lines shows the sine fits calculated. The red crosses indicate the detected flare peaks.}
    \label{fig:HKAqr-TESS_foldedLCs_Period}
\end{figure}

\FloatBarrier
\subsection{KOI-256}

KOI-256 is, similarly to V471 Tau, a binary system consisting of a M dwarf and a white dwarf (\cite{eclipsing_binaries_koi_256}, \cite{koi_256_effects_of_magnetic}). It has a mass of 0.51 M$_\odot$, a radius of 0.540 R$_\odot$ and an effective temperature of 3450 K (\cite{eclipsing_binaries_koi_256}). Similarly to HK Aqr, a second periodicity was detected, which was the used to fold for spot modulation. This happened for Kepler target table ids 37, 38, 53 and TESS sector 80.\\
Figure \ref{fig:KOI-256-Flarecount-10_Bins} shows the histogram with 10 bins for KOI-256 with all detected spot modulations, including the mentioned ones that were wrongfully detected. This figure shows a increase in flare count around phase $1 \pi$, which peaks at phases $0.7 \pi$ and around $1.4 \pi$. Additionally there appears a peak at phase $1.7 \pi$. 
Increasing the bins to 30 (figure \ref{fig:KOI-256-Flarecount-30_Bins}) shows clearer peaks at phase $0.7 \pi$ as well as $1.2 \pi$. Including errorbars, the mentioned peaks of both figures are above their surrounding bins.


\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/KOI-256/KOI-256-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:KOI-256-Flarecount-10_Bins}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/KOI-256/KOI-256-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:KOI-256-Flarecount-30_Bins}
    \end{subfigure}
    \caption{Histograms of KOI-256 across the phase showing the number of flares in each bin. The error bar shows the standard deviation for the histogram. The blue lines show the various fits for the folded lightcurves used to generate the data, with the phase minimum at $1 \pi$ and phase maximum at $0 \pi$/$2 \pi$.}
    \label{fig:KOI-256-Flarecount}
\end{figure}

Figures \ref{fig:KOI-256-Flarecount-10_Bins_Period} and \ref{fig:KOI-256-Flarecount-30_Bins_Period} show the histograms based on the, by the optimized fold algorithm determined, period folded lightcurves. Therefor only the period folded lightcurves from Kepler target table id 37, 38, 53 and TESS sector 80 were used.
In figure \ref{fig:KOI-256-Flarecount-10_Bins_Period}, which shows a histogram with 10 bins for the flare distribution across the normalized phase, there is a clear increase of flares seen around the minima at phase $1 \pi$. The distribution nearly reminds one of a normal distribution, with the exception that the last two bins around phase $1.8 \pi$ to $2 \pi$ do not fall off as much as the first bin at phase $0 \pi$ does compared to the bins in the center.\\
Looking at figure \ref{fig:KOI-256-Flarecount-30_Bins_Period} which shows the same data, just with 30 bins, the peak at the phase minimum at $1 \pi$ is still present. Additionally there are three more peaks visible at around phases $0.3 \pi$, $1.4 \pi$ and $2 \pi$. These three peaks are only 1 bin wide and stand out far from their surrounding by around 1.5 times the errorbar. The peak in the center on the otherhand has gradual increases/descreases before and after, with the exception of the dip in the bin right before at around phase $0.9 \pi$.

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/KOI-256/KOI-256-Flarecount-10_Bins_Period.png}
        \caption{10 bins}
        \label{fig:KOI-256-Flarecount-10_Bins_Period}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/KOI-256/KOI-256-Flarecount-30_Bins_Period.png}
        \caption{30 bins}
        \label{fig:KOI-256-Flarecount-30_Bins_Period}
    \end{subfigure}
    \caption{Histograms of KOI-256 across the phase showing the number of flares in each bin. The error bar shows the standard deviation for the histogram. The blue lines show the various fits for the folded lightcurves used to generate the data, with the phase minimum at $1 \pi$ and phase maximum at $0 \pi$/$2 \pi$.}
    \label{fig:KOI-256-Flarecount_Period}
\end{figure}

Comparing the results of the flare peak distributions across the normalized phases of the two datasets in figure \ref{fig:KOI-256-flarepeaks} shows that the highest normalized flare peaks of up to 1.7 were detected in the lightcurves with proper period detection. Comparing flare peaks which exist in both data sample, one can see that the highest peak of \subref{fig:KOI-256-flarepeaks_1.3_period} at around phase $0.8 \pi$ and a normalized peak of \textasciitilde1.3 was moved to around phase $0.4 \pi$ in figure \subref{fig:KOI-256-flarepeaks_1.7_spot} due to the additional found periodicity.

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.95\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/KOI-256/KOI-256-Flarepeaks_maxY-1.7092845644155863.png}
        \caption{Spot modulation folded}
        \label{fig:KOI-256-flarepeaks_1.7_spot}
    \end{subfigure}
    \begin{subfigure}[b]{.95\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/KOI-256/KOI-256-Flarepeaks_maxY-1.3100095148043271_Period.png}
        \caption{Period folded}
        \label{fig:KOI-256-flarepeaks_1.3_period}
    \end{subfigure}
    \caption{Distribution of flare peaks in relation to the normalized phase at which they occured. Y-Axis shows the flare peak and is limited to the value of the highest peak detected. The x-axis shows the normalized phase.}
    \label{fig:KOI-256-flarepeaks}
\end{figure}