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%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. 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. Last but not least a summary of the current knowledge related to this study is given.
\section{Goals and current knowledge \label{sec:intro:goals}}
The goal of this study is to relate flares/superflares to the appearance of spots on the surfaces of stars of various spectral types. Flares are well studied for the sun (\cite{solar_flares_1}, \cite{solar_flares_2}, \cite{solar_flares_3}) as well as its impact on earths magnetic field (\cite{solar_flare_mag_field}). While the first stellar flares were discovered in middle of the last century (\cite{early_stellar_flares1}, \cite{early_stellar_flares2}), the topic gained a lot of traction with the launch of the likes of Kepler and the Transiting Exoplanet Survey Satellite (TESS). They allowed the survey of thousands of stars. With this, studies of flares and superflares on a large number of stars have been conducted (e.g. \cite{flare_study_1}, \cite{flare_study_2}, \cite{connection_starspots_flares_ms_kepler}, \cite{flare_occurance_periodicity}), but the origin of superflares (flares with an energy above $10^{33}$ erg) is still not clear. So far no superflare has been observed on our sun, but there have been studies focusing on the possible origin on superflares and their likelyhood to happen on our sun (\cite{superflares_on_sun}). They found that superflares on our sun would be rare events (every \textasciitilde800 years for superflares with $10^{34}$ erg).\\
This chapter gives an introduction to the goals of 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. Last but not least a summary of the current knowledge related to this study is given.\\
The goal of this study is to relate flares/superflares to the appearance of spots on the surfaces of stars of various spectral types. Flares are well studied for the sun (\cite{solar_flares_1}, \cite{solar_flares_2}, \cite{solar_flares_3}) with the first recorded event being the Carrington event from September 1859 (\cite{Carrington_event}). The same goes for the impact of stellar activity on earths magnetic field (\cite{solar_flare_mag_field}). A detailed description of spots and flares is found in section \ref{sec:intro:flares_and_spots}.
While the first stellar flares were discovered in middle of the last century (\cite{early_stellar_flares1}, \cite{early_stellar_flares2}), the topic gained a lot of traction with the launch of the likes of Kepler and the Transiting Exoplanet Survey Satellite (TESS). They allowed the survey of thousands of stars. With this, studies of flares and superflares on a large number of stars have been conducted (e.g. \cite{flare_study_1}, \cite{flare_study_2}, \cite{connection_starspots_flares_ms_kepler}, \cite{flare_occurance_periodicity}), but the origin of superflares (flares with an energy above $10^{33}$ erg) is still not clear. So far no superflare has been observed on our sun, but there have been studies focusing on the possible origin on superflares and their likelyhood to happen on our sun (\cite{superflares_on_sun}). They found that superflares on our sun would be rare events (every \textasciitilde800 years for superflares with $10^{34}$ erg).\\
A few proposed causes could be star-planet interaction (SPI) (\cite{au_mic_flaring_spi}, \cite{SPI_1}, \cite{SPI_2}), or just being scaled up version of normal flares which we see from our sun coming from large spots (\cite{superflares_1}, \cite{superflares_2}).\\
\section{State of the art \label{sec:intro:goals}}
\cite{doyle_2018} studied 34 M dwarfs from the K2 mission, using short cadence observational data. They confirmed that the stars in their dataset 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}).\\
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}).\\
There have also been studies on individual stars, for example \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}). Star-planet interaction 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}).\\
In coclusion, many flares and superflares have been found on stars, but the origin of the later is still not clear. There are a few ongoing possible origins like star-planet interaction (or interactions with other close companions) or them coming from larger, more complex starspots.
\cite{connection_starspots_flares_ms_kepler} investigated a sample of 119 stars from spectral types M to F. They used a subset of the Kepler long-cadence data. 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}).\\
Further analysis on periodic flare occurance was done by \cite{flare_occurance_periodicity}, who studied TESS lightcurves of 284 cool 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.\\
There have also been studies on individual stars, for example \cite{kepler_411_study} focused on Kepler-411, which is a K2V dwarf, 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 (a K dwarf with two planets), 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}).\\
Star-planet interaction was 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 state that a $>3\sigma$ detection requires more observations (\cite{au_mic_flaring_spi}).\\
In conclusion, many flares and superflares have been found on stars, but the origin of the latter is still not clear. Although it is widely accepted that superflares may be scales up normal flares, still the issue of star-planet interaction is discussed. Furthermore, the above mentioned studies could not relate flare/superflare occurrence distinctively to the spottedness of stars. With a large data set comprised of Kepler, K2 and TESS data we aim for a more distinct relation flare/superflare occurrence and spottedness of stars.
There are a few ongoing possible origins like star-planet interaction (or interactions with other close companions) or them coming from larger, more complex starspots.
\section{Spectral Types \label{sec:intro:spectral_types}}
@@ -83,15 +87,25 @@ This section gives a short overview on the space mission from which the data was
\subsection{Kepler and K2}
The Kepler space telesope was named after Johannes Kepler and was launched in 2009 (\cite{nasa_kepler_in_depth}) and its primary mission lasted till May 2013 (\cite{kepler_missions_and_data_mast}). Its main scientific goal was the discovery of exoplanets, especially Earth-sized ones, in the habitable zone around their host star (\cite{kepler_missions_caltech}, \cite{Kepler_first_results}). This was done by observing transits of planets around their host star (\cite{kepler_missions_caltech}, \cite{kepler_missions_and_data_mast}, \cite{Kepler_first_results}). \\
It provides two different types of data: short cadence (1 minute exposure) and long cadence (30 minute exposure) data (\cite{Kepler_first_results}, \cite{kepler_missions_and_data_mast}). While short cadence data provides a higher temporal resolution, it was used for fewer targets (\cite{Kepler_first_results}). It had a fixed exposure time of 6.02 seconds for its photometer with a field of view of 115 $deg^{2}$ near the cygnus constellation (\cite{Kepler_first_results}, \cite{kepler_missions_and_data_mast}). The data was then for selected targets accumulated to either short or long cadence, and then downloaded (\cite{Kepler_first_results}). The space telescope could accomondate data of around 170,000 targets with long cadence and 512 targets with short cadence (\cite{Kepler_first_results}). The resolution of each of the 42 CCDs was 4 arcseconds per pixel (\cite{doyle_2019}, \cite{kepler_mission_stellar_and_instrument_noise}).\\
The K2 mission is the continuation mission of Kepler after the second of four of its reaction wheels broke in 2013 (\cite{nasa_kepler_in_depth}, \cite{k2_missions_and_data_mast}). Due to this, a new mission was proposed, keeping the now limited capabilities in mind (\cite{nasa_kepler_in_depth}). The mission was active from February 2014 till September 2018 when its fuel ran out (\cite{k2_missions_and_data_mast}, \cite{kepler_end_nytimes}, \cite{kepler_end_nasa_press}). It discovered a total of over 2600 planets as of 2018 (\cite{kepler_end_nytimes}, \cite{kepler_end_nasa_press}).
The Kepler space telesope was named after Johannes Kepler and was launched in 2009\footnote{\label{fn:nasa_kepler_in_depth}\href{https://science.nasa.gov/mission/kepler/in-depth/}{https://science.nasa.gov/mission/kepler/in-depth/}} and its primary mission lasted till May 2013\footnote{\label{fn:kepler_missions_and_data_mast}\href{https://archive.stsci.edu/missions-and-data/kepler}{https://archive.stsci.edu/missions-and-data/kepler}}. Its main scientific goal was the discovery of exoplanets, especially Earth-sized ones, in the habitable zone around their host star\footnote{\label{fn:kepler_missions_caltech}\href{https://exoplanetarchive.ipac.caltech.edu/docs/KeplerMission.html}{https://exoplanetarchive.ipac.caltech.edu/docs/KeplerMission.html}} (\cite{Kepler_first_results}). This was done by observing transits of planets around their host star\cref{fn:kepler_missions_caltech}\cref{fn:kepler_missions_and_data_mast} (\cite{Kepler_first_results}).\\
It provides two different types of data: short cadence (1 minute exposure) and long cadence (30 minute exposure) data\cref{fn:kepler_missions_and_data_mast} (\cite{Kepler_first_results}). While short cadence data provides a higher temporal resolution, it was used for fewer targets (\cite{Kepler_first_results}). It had a fixed exposure time of 6.02 seconds for its photometer with a field of view of 115 $deg^{2}$ near the cygnus constellation\cref{fn:kepler_missions_and_data_mast} (\cite{Kepler_first_results}) and is most sensitive in the visual to near infrared as seen in figure \ref{fig:kepler_bands}. The data was then for selected targets accumulated to either short or long cadence, and then downloaded (\cite{Kepler_first_results}). The space telescope could accomondate data of around 170,000 targets with long cadence and 512 targets with short cadence (\cite{Kepler_first_results}). The resolution of each of the 42 CCDs was 4 arcseconds per pixel (\cite{doyle_2019}, \cite{kepler_mission_stellar_and_instrument_noise}).\\
The K2 mission is the continuation mission of Kepler after the second of four of its reaction wheels broke in 2013\cref{fn:nasa_kepler_in_depth} (\cite{k2_missions_and_data_mast}). Due to this, a follow-up mission was proposed, keeping the now limited capabilities in mind\cref{fn:nasa_kepler_in_depth}. The mission was active from February 2014 till September 2018 when its fuel ran out\footnote{\label{fn:k2_missions_and_data_mast}\href{https://archive.stsci.edu/missions-and-data/k2}{https://archive.stsci.edu/missions-and-data/k2}}\footnote{\label{fn:kepler_end_nytimes}\href{https://web.archive.org/web/20181030211627/https://www.nytimes.com/2018/10/30/science/nasa-kepler-exoplanet.html}{https://web.archive.org/web/20181030211627/https://www.nytimes.com/2018/10/30/science/nasa-kepler-exoplanet.html}}\footnote{\label{fn:kepler_end_nasa_press}\href{https://www.jpl.nasa.gov/news/nasa-retires-kepler-space-telescope/}{https://www.jpl.nasa.gov/news/nasa-retires-kepler-space-telescope/}}. It discovered a total of over 2600 planets as of 2018\cref{fn:kepler_end_nytimes}\cref{fn:kepler_end_nasa_press}.
\begin{figure}[pt!]
\centering
\includegraphics[width=.6\linewidth]{gfx/kepler/kepler_bandpass_jason1.jpg}
\caption{Imagine showing the Johnson B, V, R and I response curves and comparing it to the Kepler, MOST and CoRoT missions. Image from \cite{kepler_sensitivity}.}
\label{fig:kepler_bands}
\end{figure}
\subsection{TESS}
The Transiting Exoplanet Survey Satellite (TESS) was launched in 2018, with its primary mission lasting two years (\cite{tess_nasa}). Since 2020 TESS is in its extended missions, the first being from 2020 to 2022 (\cite{tess_extended_mission_1}), the second from 2022 to 2024 (\cite{tess_extended_mission_2}). TESS has four identical cameras, with each providing a 24 times 24 degree field of view (\cite{tess_tess}, \cite{TESS2014}). TESS's CCDs have a pixel scale of 21 arcseconds (\cite{TESS2014}, \cite{doyle_2019}). The data is provided in 2 minute cadence for individual stars and 30 minute cadence for full frame images (\cite{tess_tess}).\\
TESS observes its targets in so called sectors. Each sector is observed for 27 days (\cite{tess_tess}).
%\section{Current knowledge \label{sec:intro:current_knowledge}}
The Transiting Exoplanet Survey Satellite (TESS) was launched in 2018, with its primary mission lasting two years\footnote{\href{https://exoplanets.nasa.gov/tess/}{https://exoplanets.nasa.gov/tess/}}. Since 2020 TESS is in its extended missions, the first being from 2020 to 2022\footnote{\label{fn:tess_extended_mission_1}\href{https://archive.stsci.edu/contents/newsletters/august-2020/tess-extended-mission}{https://archive.stsci.edu/contents/newsletters/august-2020/tess-extended-mission}}, the second from 2022 to 2024\cref{fn:tess_tess}. TESS has four identical cameras, with each providing a 24 times 24 degree field of view\footnote{\label{fn:tess_tess}\href{https://tess.mit.edu/science/}{https://tess.mit.edu/science/}} (\cite{TESS2014}). TESS's CCDs have a pixel scale of 21 arcseconds (\cite{TESS2014}, \cite{doyle_2019}) and is most sensitive in the red and infrared (see figure \ref{fig:tess_bands}). The data is provided in 2 minute cadence for individual stars and 30 minute cadence for full frame images (\cite{tess_tess}).\\
TESS observes its targets in so called sectors. Each sector is observed for 27 days\cref{fn:tess_tess}.
\begin{figure}[pt!]
\centering
\includegraphics[width=.6\linewidth]{gfx/tess/The-TESS-spectral-response-function-black-line-defined-as-the-product-of-the-long-pass.png}
\caption{Imagine showing the Johnson V, R\textsubscript{C}, I\textsubscript{C} and SDSS z response curves and comparing it to TESS. Image from \cite{kepler_sensitivity}.}
\label{fig:tess_bands}
\end{figure}