Minor typo corrections

appendices/lyttle21.tex

@@ -220,7 +220,7 @@ \section{The Sun as a Star}\label{sec:sun-res}
 
 \begin{table}
     \centering
-    \caption[Solar results from the NP model.]{Solar results from the NP model. The second column shows the median marginalised posterior samples for each parameter with their respective upper and lower 68 per cent credible intervals.}
+    \caption[Solar results from the NP model.]{Solar results from the NP model. We show the median marginalised posterior samples for each parameter with their respective upper and lower 68 per cent credible intervals.}
     \label{tab:sun-out}
     \input{tables/sun_outputs.tex}
 \end{table}

chapters/conclusion.tex

@@ -34,7 +34,7 @@ \section*{Improving the Hierarchical Model}
 
 We also expect the HBM to scale to red giant solar-like oscillators for which observations are abundant. We trained the emulator on a grid of stellar models from the zero-age main sequence to the base of the red giant branch for masses from \SIrange{0.8}{1.2}{\solarmass}. The upper mass limit was motivated by the diminishing outer convective envelope (responsible for driving solar-like oscillators) in these stars on the main sequence. However, extending the emulator to model red giant solar-like oscillators would require expanding the grid up to \(\sim\SI{2.0}{\solarmass}\). We would need to compute twice as many more evolutionary tracks and evolve existing models further. Stars with \(M \gtrsim \SI{1.1}{\solarmass}\) have a convective core on the main sequence which introduces an additional model uncertainty from mixing at its boundary. Parametrising this process would further multiply the number of input tracks, increasing dimensionality and grid computation time. Therefore, we should research ways of selectively computing stellar models. For example, we could augment the grid \citep[e.g.][]{Li.Davies.ea2022} by upsampling where the neural network error is large.
 
-There are a few additional systematic uncertainties we could also include in the HBM. For example, in Chapter \ref{chap:hmd} we did not consider the inaccuracies of near-surface physics which effect modelled mode frequencies. So-called `surface correction' methods exist \citep[e.g.][]{Ball.Gizon2014,Kjeldsen.Bedding.ea2008} but vary across the HR diagram when compared with 3D hydrodynamical simulations \citep{Sonoi.Samadi.ea2015}. \citet{Compton.Bedding.ea2018} found a range of surface corrections can shift modelled frequencies at \(\numax\) by up to \(\sim 0.5\) per cent. This would amount to a systematic effect on \(\dnu\) which we would expect to correlate with other stellar parameters. Therefore, a future iteration of the HBM should account for the surface term systematic.
+There are a few additional systematic uncertainties we could also include in the HBM. For example, in Chapter \ref{chap:hmd} we did not consider the inaccuracies of near-surface physics which affect modelled mode frequencies. So-called `surface correction' methods exist \citep[e.g.][]{Ball.Gizon2014,Kjeldsen.Bedding.ea2008} but vary across the HR diagram when compared with 3D hydrodynamical simulations \citep{Sonoi.Samadi.ea2015}. \citet{Compton.Bedding.ea2018} found a range of surface corrections can shift modelled frequencies at \(\numax\) by up to \(\sim 0.5\) per cent. This would amount to a systematic effect on \(\dnu\) which we would expect to correlate with other stellar parameters. Therefore, a future iteration of the HBM should account for the surface term systematic.
 
 \section*{Current and Future Data}
 
@@ -62,7 +62,7 @@ \section*{Current and Future Data}
 
 Towards the end of the 2020s, the \emph{PLATO} mission will observe tens of thousands of dwarf and subgiant solar-like oscillators \citep{Rauer.Catala.ea2014}. \emph{PLATO} aims to discover hundreds of exoplanets orbiting solar-type stars across a wider proportion of the sky than observed by \emph{Kepler}. Among its targets are around \num{20000} bright (\(V\) < 11) oscillating F-K dwarf stars to be observed over a baseline of around 2 years \citep{Goupil2017}. Using our HBM method on a sample this size could see a reduction in uncertainty (\(\sigma\)) on helium abundance from 0.01 to 0.0005. While this is the maximum expected uncertainty reduction (as discussed in Chapter \ref{chap:hbm}), it shows that we can start to consider more complex population distributions in helium and other stellar parameters.
 
-While \emph{PLATO} will offer unprecedented numbers of main sequence solar-like oscillators, we already have large samples of more evolved asteroseismic stars to include in a future HBM. Combined, \emph{Kepler}, \emph{K2}, and \emph{TESS} have yielded \(\sim 150,000\) red giant solar-like oscillators to date \citep{Hon.Huber.ea2021,Yu.Huber.ea2018}. Providing that we can extend our stellar model emulator to more these stars, expanding our dataset will allow us to test more complex population-distributions. For example, since \emph{TESS} is an all-sky survey, we could include kinematics and galactic positions in the helium enrichment law. Additionally, observations of open clusters and binary star systems motivate population distributions over age, distance and chemical abundances.
+While \emph{PLATO} will offer unprecedented numbers of main sequence solar-like oscillators, we already have large samples of more evolved asteroseismic stars to include in a future HBM. Combined, \emph{Kepler}, \emph{K2}, and \emph{TESS} have yielded \(\sim 150,000\) red giant solar-like oscillators to date \citep{Hon.Huber.ea2021,Yu.Huber.ea2018}. Providing that we can extend our stellar model emulator to more of these stars, expanding our dataset will allow us to test more complex population-distributions. For example, since \emph{TESS} is an all-sky survey, we could include kinematics and galactic positions in the helium enrichment law. Additionally, observations of open clusters and binary star systems motivate population distributions over age, distance and chemical abundances.
 
 % The number of dwarf and subgiant solar-like oscillators expected from \emph{PLATO} will be comparable to the number of red giant oscillators already found with \emph{Kepler} and \emph{TESS} \needcite. In the meantime we could test extending our method to red giant stars to make use of the abundance of data. This comes with additional challenges. Oscillating red giants include masses \(\gtrsim \SI{1.2}{\solarmass}\) which would have had a convective core during their hydrogen-burning phase of evolution. In this case, we would have to consider overshooting at the convective core boundary. This is an approximation of the physics to simulate mixing at the boundary bringing fresh hydrogen fuel into the core and extending the main sequence lifetime.
 

chapters/glitch.tex

@@ -415,7 +415,7 @@ \subsubsection{The Effect of a Change in \(\gamma\) on p Mode Frequencies}
 
 \subsubsection{A Functional Form of the Helium Glitch Signature}
 
-We have shown that a change in \(\gamma\) can induce a periodicity in \(\omega\). The functional from of this periodicity depends on \(\delta\gamma/\gamma\). As shown in Figures \ref{fig:gamma-zones} and \ref{fig:gamma-temp-density}, the dominant change in \(\gamma\) due to a change in helium abundance is from the second ionisation of helium. There have been different attempts to approximate \(\delta\gamma/\gamma\,|_\heII\) in the literature, for example using a Dirac delta function or a triangular function \citep{Monteiro.Christensen-Dalsgaard.ea1994,Monteiro.Thompson2005}. In recent years, work modelling the glitch has used the formulation from \citet{Houdek.Gough2007} where the change in \(\gamma\) due to He\,\textsc{ii} ionisation is modelled with a Gaussian shape,
+We have shown that a change in \(\gamma\) can induce a periodicity in \(\omega\). The functional form of this periodicity depends on \(\delta\gamma/\gamma\). As shown in Figures \ref{fig:gamma-zones} and \ref{fig:gamma-temp-density}, the dominant change in \(\gamma\) due to a change in helium abundance is from the second ionisation of helium. There have been different attempts to approximate \(\delta\gamma/\gamma\,|_\heII\) in the literature, for example using a Dirac delta function or a triangular function \citep{Monteiro.Christensen-Dalsgaard.ea1994,Monteiro.Thompson2005}. In recent years, work modelling the glitch has used the formulation from \citet{Houdek.Gough2007} where the change in \(\gamma\) due to He\,\textsc{ii} ionisation is modelled with a Gaussian shape,
 %
 \begin{equation}
     \left.\frac{\delta\gamma}{\gamma}\right|_\heII \simeq - \frac{\Gamma_\heII}{\Delta_\heII \sqrt{2\pi}} \, \ee^{- \frac12{(\tau - \tau_\heII)^2}/{\Delta_\heII^2} }, \label{eq:he-gamma}

chapters/lyttle21.tex

@@ -450,7 +450,7 @@ \subsection{Mixing-Length Theory}\label{sec:mlt}
 
 There are a few prior studies which look at the spread in $\mlt$ for a population of stars, typically by fitting $\mlt$ as a function of $\metallicity$, $\teff$, and $\log g$ \citep[e.g.][]{Bonaca.Tanner.ea2012,Viani.Basu.ea2018}. For example, results from \citet{Viani.Basu.ea2018} for stellar models including diffusion, predict $\mlt$ in the range \numrange{1.5}{2.3} across our sample. This dispersion would be more compatible with the larger spread obtained by our PPS model. However, in future work we should further investigate how $\mlt$ varies with stellar parameters, as our assumption of a normal distribution may not be accurate.
 
-We found a greater difference in $\mlt$ between the models with and without the Sun when we max-pooled $\mlt$. The MP models yielded a global $\mlt$ in-line with $\mu_\alpha$ from the PP model. However, when we added the Sun, the model yielded $\mlt \approx 2.1$ which is in common with the solar results (see Appendix \ref{sec:sun-res}). This had a similar affect as assuming a solar calibrated value, because the model favoured fitting to data with the best observational precision. The change in $\mlt$ between the MP and MPS models resulted in a mean difference of $\sim 20$ per cent between the individual stellar ages. This is an example of how adopting a solar calibrated value can bias stellar ages. We argue that carefully including the Sun as a part of the population with an intrinsic spread is a better way to calibrate the stellar models than assuming as solar $\mlt$ across the sample.
+We found a greater difference in $\mlt$ between the models with and without the Sun when we max-pooled $\mlt$. The MP models yielded a global $\mlt$ in-line with $\mu_\alpha$ from the PP model. However, when we added the Sun, the model yielded $\mlt \approx 2.1$ which is in common with the solar results (see Appendix \ref{sec:sun-res}). This had a similar affect as assuming a solar calibrated value, because the model favoured fitting to data with the best observational precision. The change in $\mlt$ between the MP and MPS models resulted in a mean difference of $\sim 20$ per cent between the individual stellar ages. This is an example of how adopting a solar calibrated value can bias stellar ages. We argue that carefully including the Sun as a part of the population with an intrinsic spread is a better way to calibrate the stellar models than assuming a solar $\mlt$ across the sample.
 
 In all observables except for $L$, the Sun is near the centre of our distribution of stars. However, we found no relationship between $L$ and $\mlt$ in both our NP and PP models. A possible explanation for the difference in $\mlt$ with and without the Sun could be some systematic offset in our observational data for the sample. Here, we point to our choice of spectroscopic $\teff$ which typically underestimates $\teff$ compared to photometric scales, as noted in \citetalias{Serenelli.Johnson.ea2017}. We ran a solar model with an additional parameter, $\Delta \teff = T_{\rm eff, obs} - \teff$ which represents a bias in the observed effective temperature. The estimated covariance between $\Delta \teff$ and $\mlt$ was \SI{0.452}{\kelvin}\footnote{\mlt~is dimensionless, hence the units of covariance are \si{\kelvin}.} (with a correlation of \num{0.517}). Therefore, underestimating $\teff$ by about \SI{100}{\kelvin} could underestimate $\mlt$ by about \num{0.1}. If we extend this result to the other stars in the sample, the lower $\mlt$ obtained without including the Sun as a star could be caused by underestimating $\teff$ relative to the Sun. Alternatively, the $\mlt$ of the Sun could have been higher than the rest of the sample to compensate for neglecting additional sources of mixing required to reproduce the higher precision solar observables. A deeper quantification of systematic uncertainties is left to future work.