sec_observations.tex

As discussed in Section \ref{sec:parameterizations}, we are trying to distinguish between different models of thermohaline mixing based not on the amount of mixing they predict, but on their trends as a function of the reduced density ratio $r$. As discussed in Section \ref{sec:mesa_results}, this requires stars of a wide range of masses and metallicities. It is therefore quite convenient that 
modern spectroscopic surveys have recently begun collecting measurements of mixing diagnostics for large samples of stars whose masses are also well constrained. 
%, making it possible to compare observed mixing trends to the predictions of the 
%variety of 
%theoretical models discussed above in Section \ref{sec:parameterizations} by using one dimensional stellar evolution models to estimate the relevant fluid parameters of these stars given their masses and metallicities (Section \ref{sec:mesa_results}). 
%
We choose for this work to use the carbon-to-nitrogen ratios, [C/N], measured from the Apache Point Galactic Evolution Experiment \citep[APOGEE, ][]{Majewski2017}. APOGEE is a Sloan Digital Sky Survey III and IV \citep{Blanton2017} project using the 2.5 meter Sloan Telescope \citep{Gunn2006} and the APOGEE spectrograph \citep{Wilson2019} to obtain medium resolution (R $\sim$ 22,500) spectra of large numbers of stars across the galaxy \citep{Zasowski2017, Beaton2021,Santana2021}. These spectra are homogeneously reduced and analyzed using the ASPCAP pipeline \citep{Nidever2015, Zamora2015, GarciaPerez2016} and the resulting stellar parameters are then calibrated using asteroseismic, cluster, and field data \citep{Holtzman2015,Holtzman2018, Jonsson2020}. We choose to use the APOGEE data because this calibration work has already been done and an asteroseismic overlap sample is already available to provide stars with precise and accurate masses, though %\sout{we acknowledge that} 
similar work could likely be done with, for example, the lithium abundances measured by the GALAH survey \citep{buder2019} or the \ctwelvecthirteen data estimated from the APOGEE data using the Brussels Automatic Code for Characterizing High accUracy Spectra \citep[BACCHUS,][]{Masseron2016_BACCHUS} pipeline (C. Hayes, submitted). 

%FIGURE omgcomp---------------------------------------------------
\begin{figure}[!tb]
\begin{center}
\includegraphics[width=9cm,clip=true, trim=0.5in 0in 0in 0in]{Mfeh5c_mixingCRGBdr14v7.pdf}%[width=9cm, clip=true, trim=1in 1in 1in 1in]{./Figs/omgcomp.eps}
\caption{
The difference in average [C/N] (indicated by box color, with negative values in orange and positive values in purple) between stars significantly below the RGB bump and those significantly above the bump is shown as a function of stellar mass and metallicity. The gradient is consistent with previous work, with lower mass, lower metallicity stars having more extra mixing (purple), but there is clearly an unphysical `unmixing' trend (orange) that needs to be removed (see text). We highlight only bins with a sufficient number of stars both below and above the bump. }
\label{fig:obssquare}
\end{center}
\end{figure}
%FIGURE omgcomp---------------------------------------------------




We also note that the evolution of [C/N] is in some ways simpler for these low-mass stars than other mixing diagnostics. Unlike lithium, its abundance at the surface does not change significantly during the main sequence due to the effects of rotational and other mixing processes \citep{Iben1967}. Its initial ratio seems to be somewhat metallicity dependent \citep{Shetrone2019}, with higher values at lower metallicity. As stars reach the first dredge up, there is a strong, rapid, mass--\textbf{ and metallicity--}dependent change in the surface [C/N] ratio \citep{Salaris2015, MasseronGilmore2015, Martig2016, Ness2016, Spoo2022}. The [C/N] ratio at the surface then remains constant until stars reach the red giant branch bump, after which there seems to be another rapid drop in the [C/N] ratio, particularly in stars of low metallicity \citep[e.g.][]{Gratton2000,Shetrone2019}; it is this drop that has been associated with thermohaline mixing.  For stars of particularly low metallicities, there are some suggestions of Upper RGB extra mixing \citep{Shetrone2019}, but this is not well motivated theoretically and is distinct from the processes we are discussing here.

To estimate the amount of extra mixing in these stars near the bump---which thermohaline models suggest should correlate with the mixing coefficient $\Dth$ described above---\citet{Shetrone2019} estimated the drop in [C/N] just above the red giant branch bump. Their work used $\alpha$-element enhanced, and therefore old and low-mass ($\sim$0.9 \msun), first ascent red giant branch stars and binned them in bins of 0.2 dex in metallicity. The location of the red giant branch bump was identified empirically as an overdensity of stars at a particular surface gravity in each bin. They then identified the $\log g$ regime around the red giant branch bump and fit a hyperbolic tangent function to measure the location and size of the drops in the [C/N] ratio. For simplicity, we have reproduced their results in Table \ref{tab:obsdata}. 

\input{table_data}

We add to their analysis a sample of higher metallicity stars with asteroseismic masses from the APOGEE-Kepler overlap sample \citep[APOKASC,][]{Pinsonneault2014, Pinsonneault2018}. We do this because, according to our analysis in Section \ref{sec:mesa_results}, higher mass, higher metallicity stars probe larger values of the reduced density ratio, $r$. %Specifically,
We first bin the stars in mass (0.2 \msun) and  metallicity (0.2 dex). For consistency with \citet{Pinsonneault2018} and \citet{Shetrone2019}, we use the Data Release 14 \citet{DR14} carbon and nitrogen abundances. We note however that while the abundance scale seems to shift between releases, the rank ordering does not change very much \citep{Spoo2022}, which means that the conclusions of this analysis are not strongly affected by the choice of Data Release or seismic parameters.

Unlike in the \citet{Shetrone2019} analysis (e.g. their Figure 2), there is not a sufficient number of stars near the bump in each bin to detect and measure the extra mixing directly in the asteroseismic sample. Instead, we define a `pre-mixing' bin of stars between \logg\ of 3.4 and 2.8 dex whose oscillations have identified them as first ascent red giants \citep{Elsworth2019}, as well as a `post-mixing' bin of RGB stars with surface gravities between 2.3 and 1.0 dex. We then compute the average [C/N] of stars in each of the pre-mixing and post-mixing bins. If both bins had at least three stars, then the difference between the pre-mixing and post-mixing average [C/N] is plotted in Figure \ref{fig:obssquare}. Because of the calibration and choices in the analysis pipeline
%We then identify the range of surface gravities that represent stars around the red giant branch bump, adopting 1.5 $<$ \logg $<$ 2.9 dex for consistency with \citet{Shetrone2019}. Finally, we fit a hyperbolic tangent function to the [C/N] ratio as a function of surface gravity, reporting the midpoint of the transition, and the total change in [C/N] in Table \ref{tab:obs}.
%For this analysis, we use the most recent Data Release 17 \citep{DR17} measurements of carbon and nitrogen, combined with the most recent astroseismic results (M. Pinsonneault et al. 2022, in prep). Because of the challenges of calibration, as well as the changes to the analysis pipelines between data releases 
\citep[see e.g.][]{Holtzman2018,Jonsson2020, vsmith_apogee_dr16_2021}, the scale of the abundances, particularly for carbon and nitrogen, is somewhat uncertain.
%often change between data releases. While the rank ordering of stars of a particular surface gravity tends to be robust \citep{Martig2016,Ness2016}, 
There sometimes exist small trends with surface gravity and temperature that are not fully removed in the calibration process. This is notable in our measurement results here; in the highest mass, highest metallicity bins, we formally measure `unmixing' near the red giant branch bump, i.e. an increase rather than a decrease in the average [C/N] near the red giant branch bump, which is inconsistent both with theoretical expectations and with measurements from other sources. Following discussions with the APOGEE team (C. Hayes, private communication), we have decided to correct for these effects by correcting the bin with the most `unmixing' to have 0 mixing, and subtracting that change from all of the other bins under the assumption that the systematic measurement errors are consistent for stars of similar temperatures and gravities. \textbf{Such calibrations are common in the literature \citep{Holtzman2015, Buder2021}, and because} we are most interested here in the trend in mixing amounts as a function of the stellar parameters, we do not expect this shift to alter the results of this analysis, but we emphasize that care should be taken by future users of this data.