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

This chapter includes the main results of this study. The data for the plots in this section were generated using the "New FC" Button from the main GUI described in section \ref{sec:gui:data_processing}. As every star has a different rotational/spot modulation period, the value range for the phases of the folded lightcurves do not match. To compare different stars (or spot modulation periods for the same star) with each other, the phase was normalized to $0$ to $2 \pi$.\\
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. Moreover plots for stars according to their spectral sub 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{link}{<link to github>}.

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

This section shows the results for 49 M dwarfs for which flares could be detected. 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 flare count histograms, with 10 and 30 phase bins respectively, of the number of flares during the normalized phase.\\
Looking at figure \ref{fig:M-Flarecount-10_Bins} there is an even distribution within the 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 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 significantly below the surrounding bins. The same is true for the dips at phases $0.7 \pi$ and $1.3 \pi$. The dip at phase $1.4 \pi$ on the other is not significant. Due to the dips surrounding the center, it may look like there is an increased number of flares in the center.

\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 number of flares per phase of 49 M dwarfs for which flares could be detected. 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, meaning only taking flares into account which have a normalized flare peak of greater than a fixed threshold(an example of flare peak distributions is seen in figures \ref{fig:BD-08_995-flarepeaks_1.2}), is shown in figures \ref{fig:M-Flarecount-10_Bins_1.25_peak} to \ref{fig:M-Flarecount-30_Bins_1.5_peak}. There are 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 phase transition regions (phase regions between maximum and minimum).
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 those to the dip visible in figure \ref{fig:M-Flarecount-10_Bins}, it widened by 1 bin.
From phase \textasciitilde$1.7 \pi$ to $2 \pi$ there is a major peak.\\
The same data, just with 30 bins can be seen in figure \ref{fig:M-Flarecount-30_Bins_1.25_peak}. Peaks in this histogram appear at the same phases as in the histogram with 10 bins. Even though those bins are now partially seperated by bins with a low flare count at phases \textasciitilde$0.2 \pi$, $1 \pi$ and \textasciitilde$1.8 \pi$. Additionally, at around phase $0.6 \pi$, there is now a smaller peak visible, while there was none in the histogram with 10 bins.\\
Increasing the normalized minimal flare peak threshold further to 1.5 decreases the total number 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 it is visible that the largest flares seem to appear more often around phase $1 \pi$ and phase $0 \pi$/$2 \pi$.\\
Increasing the number of bins to 30 (see figure \ref{fig:M-Flarecount-30_Bins_1.5_peak}) causes very large error bars which do not allow any proper analysis.

\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 number of flares with a normalized peak of greater than 1.25 per phase of 27 M dwarfs for which such flares could be detected. 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 number of flares with a normalized peak of greater than 1.5 per phase for 20 M dwarfs for which such flares could be detected. 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}

Doing the opposite and only plotting the histograms for flares with normalized peaks of less than 1.01 can be seen in figure \ref{fig:M-Flarecount-peaks_1.01_maxpeak}.
Figure \ref{fig:M-Flarecount-10_Bins_1.01_maxpeak} shows a similar picture to the histogram with no filtering (see figure \ref{fig:M-Flarecount-10_Bins}). The difference here though is that there are less flares in the phase region surrounding the minima around phase $0.5 \pi$ to $1.5 \pi$. Increasing the resolution to 30 bins (figure \ref{fig:M-Flarecount-30_Bins_1.01_maxpeak}) shows overall a similar pattern, even though the scatter of bins, especially in the center during the phase minima are larger than twice the error.\\
Limiting the normalized flare peaks to <1.05 (figure \ref{fig:M-Flarecount-peaks_1.05_maxpeak}) shows again a similar pattern as seen in the histograms with all data. In figure \ref{fig:M-Flarecount-10_Bins_1.05_maxpeak} the major difference is that there is a peak around normalized phase $1.1 \pi$.
The difference becomes less obvious when increasing the bin size 30 (figure \ref{fig:M-Flarecount-30_Bins_1.05_maxpeak}). While it is similar to figure \ref{fig:M-Flarecount-30_Bins}, there appears a new dip around phase $0.25 \pi$ which appears right after the maximum. Also the shape of the peak consisting of multiple bins around phase $1.1 \pi$ changed slightly compared to the one in figure \ref{fig:M-Flarecount-30_Bins}.

\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 number of flares with a normalized peak of less than 1.01 per phase of 17 M dwarfs for which such flares could be detected. 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 number of flares with a normalized peak of less than 1.05 per phase of 32 M dwarfs for which such flares could be detected. 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 21 K dwarfs for which flares could be detected. 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 number of flares per phase with 10 bins of 21 K type dwarfs used in the present study. One can see a slight increase in the number of flares between the maximum and minimum of the lightcurve at phase $0.5 \pi$, and a slight decrease between minimum and maximum at phase $1.5 \pi$. There is also a peak around $1 \pi$ with \textasciitilde95 flares compared to the 60-80 flares per bin in the rest of the histogram. One of the major contributions is V* V471 Tau. The individual results for this star are presented 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 number of flares between phase $0 \pi$ and $0.5 \pi$ shows a slightly increasing trend in the number of flares. We see 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} (10 bins). The major dips at phase $0.7 \pi$, $1.3 \pi$, $1.6 \pi$ and $1.9 \pi$ are significant to 1 $\sigma$, while the smaller ones between phase $0 \pi$ and $1 \pi$ are within the errors.

\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 number of flares per phase of 37 K dwarfs for which flares could be detected. 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}

The histograms with normalized flare peaks limited to >1.05 are shown in figure \ref{fig:K-Flarecount-peaks_1.5_peak}. In figure \ref{fig:K-Flarecount-10_Bins_1.05_peak} the histogram with 10 bins is shown, while figure \ref{fig:K-Flarecount-30_Bins_1.05_peak} shows the histogram with 30 bins.
There are multiple significant peaks visible in figure \ref{fig:K-Flarecount-10_Bins_1.05_peak}. The two highest being at phases $0.7 \pi$ as well as $1.1 \pi$. From phase $1.1 \pi$ onwards we see a decrease in the number of flares. These two peaks are seperated by a bin at phase $0.9 \pi$ which is significantly lower (roughly twice the error). Moreover, we also see a peak at phase $0 \pi$ which is slightly smaller than the other two at $0.7$ and $1.1 \pi$.\\
A similar picture forms when increasing the resolution to 30 bins (figure \ref{fig:K-Flarecount-30_Bins_1.05_peak}). The main differences to figure \ref{fig:K-Flarecount-10_Bins_1.05_peak} is though that the heights of the afformentioned peaks is now the same, and that the errorbars are ofcourse larger than for the 10 bin representation. Additionally a new peak in the last bin, during the phase maximum, appeared having the same height as the other three peaks.

\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 number of flares with a normalized peak of greater than 1.05 per phase of 13 K dwarfs for which such flares could be detected. 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}

Similar to figures \ref{fig:M-Flarecount-peaks_1.01_maxpeak} and \ref{fig:M-Flarecount-peaks_1.05_maxpeak}, we also show the histograms of flares with a normalized peak threshold of 1.01 and 1.05. In figure \ref{fig:K-Flarecount-10_Bins_1.01_maxpeak}, where we show the histogram of flares with a peak <1.01, there appears a siginificantly large peak around phase $1.2 \pi$ which is at the phase minimum. Another, slightly smaller peak appears at the phase maximum at $1.9 \pi$. Both of these peaks are significant when comparing them to their surrounding bins. Checking for more specific phases for the peaks, by increasing the resolution to 30 bins, shows siginificant peaks at phases $1.6 \pi$ and $1.9 \pi$. These two bins stand out with respect to their neighbouring bins.\\
Increasing the maximum flare peak threshold to 1.05 in figure \ref{fig:K-Flarecount-peaks_1.05_maxpeak} reveals a similar pattern as shown in figure \ref{fig:M-Flarecount-peaks_1.05_maxpeak} with the data for M dwarfs. The histogram is similar to the ones using all available flare data for K dwarfs (figure \ref{fig:K-Flarecount}). The differences of figure \ref{fig:K-Flarecount-10_Bins_1.05_maxpeak}, which shows the histogram with 10 bins with flare peaks limited to <1.05, and \ref{fig:K-Flarecount-10_Bins}, which shows the histogram with 10 bins for all detected flares of K dwarfs, is that that the two bins at around phase $0.3 \pi$ to $0.5 \pi$ are now more pronounced, even though still not outside of the error of their surrounding bins. Similarly the last bin at around phase $1.9 \pi$ is more pronounced.\\
Increasing the bins to 30 (figure \ref{fig:K-Flarecount-30_Bins_1.05_maxpeak}) shows a very similar figure compared to figure \ref{fig:K-Flarecount-30_Bins}.

\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 number of flares with a normalized peak of less than 1.01 per phase of 20 K dwarfs for which such flares could be detected. 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 number of flares with a normalized peak of less than 1.05 per phase of 20 K dwarfs for which such flares could be detected. 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 21 G dwarfs for which flares could be detected. Table \ref{apA:list_of_g_stars} contains a list of all G type stars used.\\
The first histogram of all data of G type dwarfs with 10 bins in figure \ref{fig:G-Flarecount-10_Bins} shows a significant increase of flares occuring around phase $1 \pi$ compared to the ones at phase $0 \pi$/$2 \pi$. The rise in flare number from phase $0 \pi$ to $1.3 \pi$ is 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 maximum of the histogram shown in figure \ref{fig:G-Flarecount-10_Bins} is now split into two peaks at phase $0.6 \pi$ and $1 \pi$. An additional peak appears at phase $0.3 \pi$ in figure \ref{fig:G-Flarecount-30_Bins}. This peak is significant with respect to its neighbouring bins. Right before this peak a significant dip is visible at phase $0.2 \pi$. From phase $1$ to $2 \pi$ we do not see any significant pattern.\\
Limiting the flare peaks to <1.05 (figure \ref{fig:G-Flarecount-peaks_1.05_maxpeak}) does not significantly affect the histograms.

\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 number of flares per phase of 21 G dwarfs for which flares could be detected. 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}

On the other side, looking at flares with peaks >1.05 reveals completely different histograms. Figure \ref{fig:G-Flarecount-10_Bins_1.05_peak} shows the histogram with 10 bins, while figure \ref{fig:G-Flarecount-30_Bins_1.05_peak} shows the one with 30 bins. The histogram with 10 bins shows 3 significant peaks. The first at around phase $0.5 \pi$ with a width of 3 bins, the second at around $1.1 \pi$ and the last at around $1.7 \pi$. All of these peaks lie significantly above their surrounding bins.\\
Increasing the bins to 30, which is seen in figure \ref{fig:G-Flarecount-30_Bins_1.05_peak}, does not give a clear picture, due to the overall lower total number of flares due to the threshold, and the increased bin count. Even though there are still singular bins at $0.4 \pi$, $1.1 \pi$ and $1.7 \pi$ which are larger than their surrounding, but well within the error range.

\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 number of flares with a normalized peak of less than 1.05 per phase of 16 G dwarfs for which such flares could be detected. 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}

\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 number of flares with a normalized peak of greater than 1.05 per phase of 13 G dwarfs for which such flares could be detected. 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}

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

This section shows the results for four F dwarfs for which flares could be detected. The list of F type stars can be found in table \ref{apA:list_of_f_stars}.\\
The detected number of flares in the already low number of F type stars in this study in figures \ref{fig:F-Flarecount-10_Bins} and \ref{fig:F-Flarecount-30_Bins} is very low which causes large errorbars of the histogram. Nontheless all detected flares were around the minimum of the phasefolded lightcurves. However, due to the above mentioned large errorbars, the results of F-type main-sequence stars is not significant.

\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 number of flares per phase of four F dwarfs for which flares could be detected. 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 combined for all 95 stars in the study for which flares could be detected is presented in this section. 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 number of the individual stars per bin are stacked on top of each other resulting in a total flare number histogram. Each color (red for M, orange for K, yellow for G and green for F type stars) represents the number of flares counted in the bin by the respective spectral type. For easier comparison, the histograms shown in the previous sections for M, K and G stars are shown again in figures \ref{fig:MKGF-Flarecount-10_Bins_M} to \ref{fig:MKGF-Flarecount-30_Bins_G}.\\
The dip at phase $0.5 \pi$ which was present in the histogram for the M type stars (see figure \ref{fig:MKGF-Flarecount-10_Bins_M}) dominates the total flarenumber histogram. From figures \ref{fig:MKGF-Flarecount-10_Bins_K}/\ref{fig:MKGF-Flarecount-30_Bins_K} and \ref{fig:MKGF-Flarecount-10_Bins_G}/\ref{fig:MKGF-Flarecount-30_Bins_G} one can see that no local minimum at phase $0.5 \pi$ exists. 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 peak in the center at phase $1 \pi$ is significant.\\
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:MKGF-Flarecount-30_Bins_K}) and G (figure \ref{fig:MKGF-Flarecount-30_Bins_G}) 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]{.46\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/MKGF-Flarecount-10_Bins.png}
        \caption{M, K, G and F type stars, 10 bins}
        \label{fig:MKGF-Flarecount-10_Bins}
    \end{subfigure}
    \begin{subfigure}[b]{.46\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/MKGF-Flarecount-30_Bins.png}
        \caption{M, K, G and F type stars, 30 bins}
        \label{fig:MKGF-Flarecount-30_Bins}
    \end{subfigure}
    \begin{subfigure}[b]{.46\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/M/M-Flarecount-10_Bins.png}
        \caption{M type stars, 10 bins}
        \label{fig:MKGF-Flarecount-10_Bins_M}
    \end{subfigure}
    \begin{subfigure}[b]{.46\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/M/M-Flarecount-30_Bins.png}
        \caption{M type stars, 30 bins}
        \label{fig:MKGF-Flarecount-30_Bins_M}
    \end{subfigure}
    \begin{subfigure}[b]{.46\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/K/K-Flarecount-10_Bins.png}
        \caption{K type stars, 10 bins}
        \label{fig:MKGF-Flarecount-10_Bins_K}
    \end{subfigure}
    \begin{subfigure}[b]{.46\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/K/K-Flarecount-30_Bins.png}
        \caption{K type stars, 30 bins}
        \label{fig:MKGF-Flarecount-30_Bins_K}
    \end{subfigure}
    \begin{subfigure}[b]{.46\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/G/G-Flarecount-10_Bins.png}
        \caption{G type stars, 10 bins}
        \label{fig:MKGF-Flarecount-10_Bins_G}
    \end{subfigure}
    \begin{subfigure}[b]{.46\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/G/G-Flarecount-30_Bins.png}
        \caption{G type stars, 30 bins}
        \label{fig:MKGF-Flarecount-30_Bins_G}
    \end{subfigure}
    \caption{Histograms showing the number of flares per phase for all 95 dwarfs for which flares could be detected in panels \subref{fig:MKGF-Flarecount-10_Bins} and \subref{fig:MKGF-Flarecount-30_Bins}. Panels \subref{fig:MKGF-Flarecount-10_Bins_M} to \subref{fig:MKGF-Flarecount-30_Bins_G} show the histograms for spectral types M, K and G respectively as direct comparison (same as histograms shown in figures \ref{fig:M-Flarecount}, \ref{fig:K-Flarecount} and \ref{fig:G-Flarecount}).}
    \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 represent the expected case of more flares on the more spotted hemisphere of the star but also the other way round, i.e. less flares on the more spotted hemisphere.

\subsection{BD-08   995}

BD-08 995, also known as TIC 43472154, is a G type main-sequence star with a surface temperature of 5316 K (\cite{revised_tess_input_catalogue}), and a distance of \textasciitilde87 pc (\cite{simbad}). 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 in 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, one can see a clear peak at phase $1 \pi$. The bins forming the peak are significantly enhanced with respect to the neighbouring bins.
The bins of the phase maxima also show a slight increase in flare activity at phase $0/2 \pi$ compared to the phase transition regions around phase $0.5 \pi$ (maximum to minimum) and $1.5 \pi$ (minimum to maximum). The peaks at the phase maxima are not as significant as the peaks in phase minimum.\\
Figure \ref{fig:BD-08_995-Flarecount-30_Bins} shows the same data just with 30 bins instead of 10, but less significant. It shows a similar picture as the previous figure. But due to the low number of total detected flares, and a 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$1.1 \pi$, which errorbars only overlap with other larger 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 around phase $0.1 \pi$ with a peak of \textasciitilde1.11. Other high flare peaks were detected at phases \textasciitilde$1 \pi$ and \textasciitilde$1.9 \pi$.

\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. The y-Axis shows the flare peak and 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 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 show the calculated sine fits. 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 largest 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 significant with respect to their neighboring bins and errors, while the third peak is only partly significant.
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 pronounced. But compared to figure \ref{fig:TYC_1360-957-1-Flarecount-10_Bins}, the low number of flares and the large number of bins result in a rather large error. Therefore these two peaks are not significant as those are within the error of the histogram.

\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 bars show 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 of the normalized phase detected for TYC 1360-957-1 in this study. The three largest flare peaks detected were at a normalized phase of \textasciitilde$0.8 \pi$, which is close to the phase minimum at $1 \pi$. This could indicate that they originate from the same spot or spot group. Three more larger flare peaks stand out at phase $1.2 \pi$. While they are not as large 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-1.5.png}
    \caption{Distribution of flare peaks in relation to the normalized phase at which they occured for TYC 1360-957-1. The y-Axis shows the flare peak and the x-axis shows the normalized phase.}
    \label{fig:TYC_1360-957-1-flarepeaks_1.31}
\end{figure}

The used folded lightcurves can be seen in figure \ref{fig:TYC_1360-957-1-TESS_foldedLCs}. This figure is split into the three different TESS sectors. In the upper left panel of \ref{fig:TYC_1360-957-1-TESS_foldedLCs} the folded lightcurve of sector 44 is shown. The minimum of the folded lightcurve is slightly shifted from the center. This is represented by the lowest point of the sine fit. One can also see that two large flares occured at the minimum of the phasefolded lightcurve.
No such large flares can be seen in sector 45 (upper right panel of figure \ref{fig:TYC_1360-957-1-TESS_foldedLCs}). Again in sector 46 (lower panel of figure \ref{fig:TYC_1360-957-1-TESS_foldedLCs}) we see a large flare at the minimum of the phasefolded lightcurve.

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\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]{.49\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]{.49\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. 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 an active G-type main-sequence 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}), a 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 seem 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 flare number is sufficiently large to be outside of the error of the average distribution. Another peak, even though less significant as its errorbar overlaps with the surrounding bins, appears at the bin at phase $1.9 \pi$. A similar, but a bit weaker 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 an increase in flare number around phase $0.6$ to $1 \pi$, as well as at $1.9$ to $0.1 \pi$ and $0.4 \pi$. Only the first mentioned peak managed to be large enough though, to be significantly above the error of the average distribution.

\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}

The flare peak of the normalized phase distribution can be seen in figure \ref{fig:TYC_4595-107-1-flarepeaks_1.27}. The strongest flare was detected at normalized phase \textasciitilde$0.8 \pi$. Also, pronounced flare peaks have been detected at phases $0.5 \pi$ and $2 \pi$ respectively. Apart from that we do not see any other energetic flares. The majority of flares reveal flare peaks in the range of 1.01 to 1.05.\\
All of the folded lightcurves of TYC 4595-107-1 can be found in appendix \ref{apB:TYC_4595-107-1}, figures \ref{apB:fig:TYC_4595-107-1-TESS_foldedLC1} and \ref{apB:fig:TYC_4595-107-1-TESS_foldedLC2}.

\begin{figure}[pt!]
    \includegraphics[width=.95\textwidth]{plots/sine/TYC_4595-107-1/TYC 4595-107-1-Flarepeaks_maxY-1.5.png}
    \caption{Distribution of flare peaks in relation to the normalized phase at which they occured for TYC 1360-957-1. The y-Axis shows the flare peak and the x-axis shows the normalized phase.}
    \label{fig:TYC_4595-107-1-flarepeaks_1.27}
\end{figure}

\FloatBarrier
\subsection{V* V471 Tau}
\label{results:v471_tau}

V471 Tau is a post-common envelope binary system consiting of a K2 and a white dwarf (\cite{v471tau_revised}). The K2 dwarf has a dominant spot facing the white dwarf all the time, as the system shows a bound rotation (\cite{V471tau_magnetic_activity}).
Figures \ref{fig:V471Tau-Flarecount-10_Bins} and \ref{fig:V471Tau-Flarecount-30_Bins} show the histograms of five TESS folded lightcurves for V471 Tau with 10 and 30 bins respectively. The TESS lightcurves used are from sectors 42, 43, 44, 70 and 71. The lightcurve of K2 target table ID 80 was rejected by the algorithm described in section \ref{sec:data:data_reduction}. Due to the lightcurve spanning 90 days, and the variability due to 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 remaining bins with 13/15 flares respectively compared to 1 to 8 in the other bins.\\
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 figure \ref{fig:V471Tau-Flarecount-10_Bins}, the errorbars are comparatively larger. There are still peaks at \textasciitilde$0.5 \pi$ and $1 \pi$.\\
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 (dip in the lightcurves) occurs phase 0 (in these figures) which translates to phase $1 \pi$ in the normalized phase. 

\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 occurs happens around phase 0. Additionally lightcurves folded by the rotational period for the sectors which have a spot modulation period different from the rotational period can be seen in figure \ref{apB:fig:V471Tau-TESS_foldedLC} from appendix \ref{apB:V471_tau}.

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\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]{.49\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]{.49\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 show the calculated sine fits. The red crosses indicate the detected flare peaks.}
    \label{fig:V471Tau-TESS_foldedLCs}
\end{figure}


In figure \ref{fig:V471Tau-flarepeaks_1.1} the flare peak of the normalized phase distribution can be seen. It reveals that the largest flares occured around phase $0.5 \pi$, $1 \pi$ and $1.5 \pi$. The phase in which the white dwarf transit occurs is around phase $1 \pi$.

\begin{figure}[pt!]
    \includegraphics[width=.95\textwidth]{plots/sine/V471Tau/V_star_ V471 Tau-Flarepeaks_maxY-1.1.png}
    \caption{Distribution of flare peaks in relation to the normalized phase at which they occured. The y-Axis shows the flare peak and the x-axis shows the normalized phase.}
    \label{fig:V471Tau-flarepeaks_1.1}
\end{figure}

\FloatBarrier
\subsection{V* HK Aqr}
\label{results:hk_aqr}

HK Aqr is an M dwarf with a mass of 0.57 $M_\odot$, a radius of 0.53 $R_\odot$ and a distance of 22.3 pc. Its effective temperature is 3800 K (\cite{conch_shell_m_dwarfs}). It was observed in four TESS sectors.
For the analysis of HK Aqr the optimize fold algorithm partially broke. This happened for its lightcurves for TESS 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 flare number at phase $1.3 \pi$. This peak is sufficiently large 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} shows a similar picture. Additionally it shows a peak at phase $0 \pi$ though, which means during the phase maximum, as well as one around phase $1.6 \pi$ which is in the transition from minimum to maximum.\\
Figures \ref{fig:HKAqr-Flarecount-10_Bins_Period} and \ref{fig:HKAqr-Flarecount-30_Bins_Period} show the histograms 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 number of detected flares during the maximum of the lightcurves at phase $1.8$ to $0.2 \pi$ is low compared to that. 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} together with larger number of bins results in large errors. 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 maximum to minimum, and shortly after the minimum respectively. In figure \ref{fig:HKAqr-TESS_foldedLCs1} we see that the three strongest (according to their flare peak) flares all occur at the same phase. Also here, and similar to the strong flares seen in figure \ref{fig:TYC_1360-957-1-flarepeaks_1.31}, those may originate from the same spot or spot group.

\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]{.49\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]{.49\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]{.49\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]{.49\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 show the calculated sine fits. 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]{.49\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]{.49\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 show the calculated sine fits. The red crosses indicate the detected flare peaks.}
    \label{fig:HKAqr-TESS_foldedLCs_Period}
\end{figure}

\FloatBarrier
\subsection{KOI-256}
\label{results:koi_256}

KOI-256 is, similarly to V471 Tau, a binary system consisting of an 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 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 wrongly detected. This figure shows an increase in flare count around phase $1 \pi$, which peaks at phases $0.7 \pi$ and a bit weaker 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 more distinct peaks at phase $0.7 \pi$ as well as at $1.2 \pi$. The mentioned peaks of both figures are significantly 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. Therefore only the period folded lightcurves from Kepler target table id 37, 38, 53 and TESS sector 80 were used. The folded lightcurves can be found in appendix \ref{apB:koi-256}, figures \ref{apB:fig:KOI-256-Kepler_foldedLC} and \ref{apB:fig:KOI-256-TESS_foldedLC}.
In figure \ref{fig:KOI-256-Flarecount-10_Bins_Period}, which shows the histogram with 10 bins for the flare distribution across the normalized phase, there is a clear increase of flare number seen around the minimum at phase $1 \pi$. The distribution is nearly comparable to a normal distribution, except for the last two bins.\\
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 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 shows gradual increases/descreases before and after, with the exception of the dip in the bin right before 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 (folded lightcurves in appendix \ref{apB:koi-256}, figures \ref{apB:fig:KOI-256-Kepler_periodfoldedLC} and \ref{apB:fig:KOI-256-TESS_periodfoldedLC}). Comparing flare peaks which exist in both data samples, one can see that the highest peak of \ref{fig:KOI-256-flarepeaks_1.3_period} around phase $0.8 \pi$ and a normalized peak of \textasciitilde1.3 was moved to phase $0.4 \pi$ in figure \ref{fig:KOI-256-flarepeaks_1.7_spot} due to the additional detected periodicity.

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\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]{.49\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{Distributions of flare peaks for KOI-256 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. Panel \subref{fig:KOI-256-flarepeaks_1.7_spot} flare peak distribution on the normalized phase for the detected spot modulation. Panel \subref{fig:KOI-256-flarepeaks_1.3_period} shows the flare distribution for the detected rotational periods.}
    \label{fig:KOI-256-flarepeaks}
\end{figure}

\FloatBarrier
\subsection{2MASS J19230963+3739397}
\label{results:2MASS_J19230963p3739397}

2MASS J19230963+3739397, also known as KIC 2300039 or TIC 122672447, is an M dwarf around 213 pc away from our solar system (\cite{simbad}). It was observed in three Kepler target table IDs as well as four TESS sectors. The results for its flare distributions can be seen in figure \ref{fig:2MASS_J19230963p3739397-Flarecount}. It was selected as an example as it shows the opposite of what is expected. As seen in the histogram with 10 bins (figure \ref{fig:2MASS_J19230963p3739397-Flarecount-10_Bins}) it shows an increased flare occurence during the transition between phase minimum and maximum and during the maximum (phase $1.3 \pi$ to $2 \pi$). An additional significant peak is seen in the second bin. The flare count during the phase minimum at phase $1 \pi$ is lowest. Increasing the bin count to 30 (see figure \ref{fig:2MASS_J19230963p3739397-Flarecount-30_Bins}) creates a similarly shaped histogram. The errorbars are larger, and the distribution less significant. The folded lightcurves for the Kepler target table IDs 47, 48 and 49 can be seen in figure \ref{fig:2MASS_J19230963p3739397-Kepler_foldedLCs}. There were no flares detected in the TESS lightcurves. As seen in the Kepler lightcurves, flare peaks as high as 1.5 have been detected, all around the phase maxima. 

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/2MASS J19230963+3739397/2MASS J19230963+3739397-Flarecount-10_Bins.png}
        \caption{10 bins}
        \label{fig:2MASS_J19230963p3739397-Flarecount-10_Bins}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/2MASS J19230963+3739397/2MASS J19230963+3739397-Flarecount-30_Bins.png}
        \caption{30 bins}
        \label{fig:2MASS_J19230963p3739397-Flarecount-30_Bins}
    \end{subfigure}
    \caption{Histograms of 2MASS J19230963+3739397 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:2MASS_J19230963p3739397-Flarecount}
\end{figure}

\begin{figure}[pt!]
    \centering
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/2MASS J19230963+3739397/2MASS J19230963+3739397_Kepler-47-foldedLC-marked_fit_flares.png}
        \caption{Kepler target table ID 47}
        \label{fig:2MASS_J19230963p3739397-Kepler47_foldedLC}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/2MASS J19230963+3739397/2MASS J19230963+3739397_Kepler-48-foldedLC-marked_fit_flares.png}
        \caption{Kepler target table ID 48}
        \label{fig:2MASS_J19230963p3739397-Kepler48_foldedLC}
    \end{subfigure}
    \begin{subfigure}[b]{.49\textwidth}
        \centering
        \includegraphics[width=\linewidth]{plots/sine/2MASS J19230963+3739397/2MASS J19230963+3739397_Kepler-49-foldedLC-marked_fit_flares.png}
        \caption{Kepler target table ID 49}
        \label{fig:2MASS_J19230963p3739397-Kepler49_foldedLC}
    \end{subfigure}
    \caption{Folded lightcurves for 2MASS J19230963+3739397. The blue lines show the calculated sine fits. The red crosses indicate the detected flare peaks.}
    \label{fig:2MASS_J19230963p3739397-Kepler_foldedLCs}
\end{figure}