% !TEX root = ../thesis-example.tex
%Example chapter on Asteroseismology
\chapter{Introduction \label{sec:intro}}
%\cleanchapterquote{Shoot for the moon. Even if you miss, you'll land among the stars.}{Les Brown} %optional, if you want to place something here.
This chapter gives an introduction to the goals of this study. It also gives a summary of the current knowledge related to this study. Afterwards there will be summaries on topics related to this study, giving explanations on the various spectral types of stars, flares and spots, as well as giving an introduction to the space missions whichs data was used.

\section{Goals and current knowledge \label{sec:intro:cur_knowledge}}

The goal of this study is to relate the appearance of flares/superflares to that of star spots on a large sample of stars of various spectral types. This is done by folding the lightcurves and the normalizing the phases. Due to the appearance of spots on the stars surface, the star is slightly dimmed, causing a local minimum in the lightcurve. This dimming can be observed with the Kepler and TESS (Transiting Exoplanet Survey Satellite) space missions. 
\cite{kepler_411_study} focused on Kepler-411 by investigating the relation between superflares and star spots on that star. They found a positive correlation between the energy of flares and the area of star spots (\cite{kepler_411_study}) on Kepler-411. They then compared their results for Kepler-411, which produced multiple superflares, with Kepler-210, which did not produce superflares while having the same number of spots (\cite{kepler_411_210_comparison}). They found the spots on Kepler-210 to be larger, warmer and therefor being magnetically weaker/less complex compared to Kepler-411 and concluded that the area of starspots is not the only relevant parameter for superflare occurance (\cite{kepler_411_210_comparison}).\\
\cite{doyle_2018} studied 34 M dwarfs from the K2 mission, using short cadence observational data. They confirmed that stars with a rotational period of less than 10 days showed more flares, which was already shown previously (\cite{faster_rot_stars_more_flares1}, \cite{faster_rot_stars_more_flares2}). Furthermore they found no star with a preference for when flares occured during the rotational phase (\cite{doyle_2018}). A similar study using TESS 2 minute cadence data has been conducted by \cite{doyle_2019}. In this study they used data of 167 M dwarfs and found a total of 1834 flares. Similar to the study on K2 data, they found no preference for roational phase (\cite{doyle_2019}).\\
Another possible cause for a periodic increase in flares is star-planet interaction (SPI), which is studied by \cite{au_mic_flaring_spi} for the star AU Mic. While they found a signal in their used TESS lightcurves correlating with the orbital period of AU Mic b, they require more observation time to get a $>3\sigma$ detection (\cite{au_mic_flaring_spi}).\\
Further analysis on periodic flare occurance was done by \cite{flare_occurance_periodicity}, who studied lightcurves of 284 M dwarfs. They found three targets (TIC 80427281, TIC 95328477, TIC 220432563) with a confirmed flare periodicity, which correlates to their rotational period or half of it.\\
\cite{connection_starspots_flares_ms_kepler} investigated a sample of 119 stars from spectral types M to F. They found that flares which increase the stellar flux by 1\% to 5\% appear more often while larger starspots are visible, while flares which increase the flux by more that 5\% do not seem to have this dependency (\cite{connection_starspots_flares_ms_kepler}).


\section{Spectral Types \label{sec:intro:spectral_types}}

% Different Spectral Types

\section{Flares and Starspots \label{sec:intro:flares_and_spots}}

% Stellar Activity, 

\section{Space missions \label{sec:intro:space_missions}}

% TESS, Kepler/K2

\citet{Thompson2012}.

\begin{figure}[t!]
  \includegraphics[width=\linewidth]{gfx/rghbHRD.pdf}
  \caption{Position of the 18 red giant heartbeat stars from Tables 1 and 2 in
the HR diagram, where the colour shows the mass of the red giant, derived from seismology. The size of the dots represents the orbital period,
ranging between 20 and 438 d. The contour surfaces reflect the density
distribution of 1000 pulsating red giants. The darkest areas mark the
position of the densely populated red clump. Numbers in red indicate
the star count per bin, for which the contour surfaces have been drawn.
Lines of equal radii in the HR diagram have been drawn for selected
stellar radii between 5 and 30 R. \citep[Figure taken from][]{Beck2014}}
  \label{fig:singelPanelPlot}
\end{figure}

\textit{Kepler} \ref{sec:conclusion}

\begin{eqnarray}
c^2 &=& a^2 + b^2, \label{eq:pyth1}\\
c &=& \sqrt{a^2 + b^2} \label{eq:pyth2},
\end{eqnarray}
