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\section{Independent Pulsation Modes}
\label{sec:analysis}
As expected given the relatively cool spectroscopically determined temperature
for GD\,1212 (e.g., \citealt{2006ApJ...640..956M}), the pulsation periods excited in
this cool WD GD\,1212 are relatively long, ranging from
$369.8-1220.8$ $828.2-1220.8$ s. A cool effective temperature is also borne out from model atmosphere fits to the photometry of
GD\,1212, this WD, which find \teff\ $= 10{,}940\pm320$ K and \logg\ $= 8.25\pm0.03$ \citep{2012ApJS..199...29G}.
Here, however, we will adopt for our discussion the more precise parameters derived from spectroscopy: \teff\ $= 11{,}270\pm170$ K and \logg\ $= 8.18\pm0.05$ \citep{2011ApJ...743..138G}. The spectroscopy of GD\,1212 is notable in that much of it was collected in order to identify the then-dominant mode at 1160.7\,s, by measuring the pulsation amplitude as a function of wavelength with time-resolved spectroscopy from the Bok 2.3 m Telescope at Kitt Peak National Observatory. However, the identification was inconclusive given the
variable, low amplitude of the pulsations \citep{Desgranges08}.
% ++++++++++++++++++++++++++++ GD1212 FTs ++++++++++++++++++++++++++++ %
\begin{figure*}
\centering{\includegraphics[width=0.98\textwidth]{f2.eps}}
\caption{Fourier transforms (FTs) of our {\em K2} observations of GD\,1212. The top panel shows the FT of the entire data set out to 2800 \muhz; there are no significant periodicities at longer frequencies. The middle and bottom panels show the independent pulsation modes in more detail. The blue FT represents only the final 9.0-days of data, while the black FT was calculated from the entire 26.7-day data set. Additionally, we have split the data into four 2.6-day subsets, and take an average FT of these subsets, shown in grey for the bottom two panels. We mark the independent pulsation modes adopted, detailed in Table~\ref{tab:freq}, with red points and associated uncertainties at the top of each panel. Additionally, the magenta lines in the top panel mark nonlinear combination frequencies also detected in our data, shown in more detail in Figure~\ref{fig:GD1212combos}. \label{fig:GD1212ft}}
\end{figure*}
% ==================================================================== %
A Fourier transform (FT) of our entire dataset is shown in Figure~\ref{fig:GD1212ft}, which orients us to the dominant frequencies of variability. An FT of our entire dataset finds significant variability ranging from 19.2 \muhz\ (14.5\,hr) down to 2703.9 \muhz\ (369.8\,s), which is shown in full in the top panel of Figure~\ref{fig:GD1212ft}. The only periodicity detected with marginal significance at longer frequencies occurs at 4531.8 \muhz\ (0.012\% amplitude), which is an instrumental artifact sampling the long-cadence exposures of 29.4\,min that often appears in short-cadence {\em Kepler} data, so we do not include it in our analysis.
The highest-amplitude variability in GD\,1212 occurs in the region between $810-1210$\,\muhz\ ($826-1234$\,s). This region bears the evidencing at least 19 independent pulsation modes, which we show in more detail in the bottom panels of Figure~\ref{fig:GD1212ft}. However, the amplitudes of variability in this region are not stable over our 26.7 days of observations, so we additionally display the FT of just our final 9.0-day
run. light curve. We also find a number of nonlinear combination frequencies of these highest-amplitude pulsations, which we discuss further in Section~\ref{sec:nonlinear}.
The amplitude and frequency variability of the pulsations in the region between $810-1210$\,\muhz\ are best shown by a running Fourier transform of our 9.0-day light curve using a 4.0-day sliding window, shown in Figure~\ref{fig:GD1212runningFT}. For example, the highest-amplitude peak in the FT near 840.0\,\muhz\ is essentially monotonically increasing in amplitude over these last 9.0 days, and is broadly unstable in frequency. Conversely, the running FT shows that the second-highest peak in the FT near 910.5\,\muhz\ is decreasing in amplitude, and appears further to bifurcate into two modes.
A common cause for amplitude and frequency modulation in pulsating stars is the presence of additional nearby frequencies that are unresolved over the course % ++++++++++++++++++++++++++++ GD1212 FTs ++++++++++++++++++++++++++++ %
\begin{figure}
\centering{\includegraphics[width=0.485\textwidth]{f3.eps}}
\caption{A running Fourier transform of
observations. The observable oscillations in WDs are always non-radial, so rotation acts to break the
spherical symmetry of a pulsation and generates multiplets about an independent mode, spaced by an amount proportional to the rotation rate (e.g., \citealt{1981A&A...102..375D}). WDs with identified pulsations all vary in $\ell=1,2$ $g$-modes, and thus DAVs which have measured rotational splittings range
from $2.5-19.0$ \muhz\ (see \citealt{2008PASP..120.1043F} and references therein).
The DAV G226$-$29 is an excellent example of
a closely spaced multiplet affecting independent pulsation
amplitudes modes detected in
a WD \citep{1983ApJ...271..744K}. Here the 109\,s pulsation is actually a superposition our 9.0-day, high-duty-cycle {\em K2} light curve of
three signals at 109.08648\,s, 109.27929\,s, and 109.47242\,s, which are separated by roughly 16.1\,\muhz\ GD\,1212. This figure uses a four-day sliding window, and
require more than 17.2\,hr of observations darker greyscale corresponds to
resolve the beat period of the closely spaced signals; shorter observations show higher amplitudes. This demonstrates the
signal sinusoidally varying at the beat period lack of
17.2\,hr.
However, rotational splittings are unlikely to explain frequency stability for the
incoherent changes pulsations in
the running FT of GD\,1212,
nor the disappearance of the mode $f_5$ near 847.2 \muhz, which was strong during the first 2.6 days and was virtually unseen in our final 9.0 days of data. Instead, we are likely observing genuine frequency variability caused by changes in driving conditions, as has been observed with long observing campaigns on cool DAVs (e.g., \citealt{1998ApJ...495..424K}). even over less than a week. \label{fig:GD1212runningFT}}
\end{figure}
% ==================================================================== %
The frequency, amplitude, and possibly phase variability observed in GD\,1212 A common cause
difficulty in defining the true pulsation periods. For the hotter DAVs, which show exceptionally stable oscillations, pre-whitening the light curve by the highest peaks in an FT removes most power in that region, and allows for
a relatively simple extraction of the periods present amplitude and frequency modulation in
pulsating stars is the
star (e.g., \citealt{2014MNRAS.438.3086G}). However, none presence of
additional nearby frequencies that are unresolved over the
peaks course of observations. The observable variability in
either the 26.7-day FT or DAVs is the
9.0-day FT result of
GD\,1212 can be smoothly pre-whitened in non-radial stellar oscillations, so rotation acts to break the
standard way because fitting spherical symmetry of a
pure sine wave does not completely remove pulsation and generates multiplets about an independent mode, spaced by an amount proportional to the
pulse shape. rotation rate (e.g., \citealt{1981A&A...102..375D}). WDs with identified pulsations all vary in $\ell=1,2$ $g$-modes, and thus DAVs which have measured rotational splittings range from $2.5-19.0$ \muhz\ (see \citealt{2008PASP..120.1043F} and references therein).
One can always decompose the broad peaks in The DAV G226$-$29 is an
FT with excellent example of a closely spaced multiplet affecting pulsation amplitudes in a WD \citep{1983ApJ...271..744K}. In this WD, the 109\,s pulsation is actually a
linear combination superposition of
sine waves, but not all three signals at 109.08648\,s, 109.27929\,s, and 109.47242\,s, which are separated by roughly 16.1\,\muhz\ and require more than 17.2\,hr of
these sinewaves represent physical modes. For example, using just observations to resolve the
final 9.0 days beat period of
data, the
highest-amplitude closely spaced signals. Shorter observations show the signal
near 840.0\,\muhz\ requires five sine waves to reproduce sinusoidally varying at the
broad peak: 840.211 \muhz\ (0.248\% amplitude), 839.183 \muhz\ (0.154\%), 842.016 \muhz\ (0.110\%), 838.000 \muhz\ (0.082\%), and 836.827 \muhz\ (0.038\%). This series beat period of
pre-whitened peaks in all likelihood corresponds to only one independent pulsation mode. 17.2\,hr.
The pre-whitening method does provide a quantitative test for significance. We can iteratively fit and pre-whiten However, rotational splittings are unlikely to explain the
highest peaks incoherent changes in the
running FT
until there are none above some threshold; we estimate this locally for each frequency by calculating of GD\,1212, nor the
median value, $\sigma$, disappearance of
$f_5$ near 847.2 \muhz, which was strong during the
FT in a $\pm200$\,\muhz\ sliding window, first 2.6 days and
mark as significant the signals that exceed 5$\sigma$. Since the 15.1-day gap was virtually unseen in our
whole dataset strongly degrades the window function, we have adopted this approach for our final 9.0 days of
data and our initial 2.6-day light curve. This provides an input list of 48 significant signals. data. Instead, we are likely observing genuine frequency variability caused by changes in driving, as has been witnessed during long campaigns on other cool DAVs (e.g., \citealt{1998ApJ...495..424K}).
We do not, however, use this technique to define adopted periods The frequency, amplitude, and
possibly phase variability observed in GD\,1212 cause difficulty in defining the
associated period uncertainties of the independent true pulsation
modes, periods. For the hotter DAVs, which
form show exceptionally stable oscillations, pre-whitening the light curve by the
input highest peaks in an FT removes most power in that region and allows for
asteroseismic modeling. Performing a
linear least-squares fit relatively simple extraction of the periods
determined from present in the
iterative pre-whitening technique would greatly underestimate star (e.g., \citealt{2014MNRAS.438.3086G}). However, none of the peaks in either the 26.7-day FT or the 9.0-day FT of GD\,1212 can be smoothly pre-whitened in the standard way because fitting a pure sine wave does not completely remove the
period uncertainties. pulse shape.
Instead, we have fit One can always decompose the broad peaks in an FT with a
Lorentzian profile to $\pm5$\,\muhz\ linear combination of
the regions sine waves, but not all of
power defined significant these sinewaves represent physical modes. For example, using
our pre-whitening method, and use just the
central peak, half-width-half-maximum, and intensity final 9.0 days of
this Lorentzian to define data, the
adopted periods, period uncertainties, and amplitudes of highest-amplitude signal near 840.0\,\muhz\ requires five sine waves to reproduce the broad
peaks in the FT. These values provide the red points peak: 840.211 \muhz\ (0.248\% amplitude), 839.183 \muhz\ (0.154\%), 842.016 \muhz\ (0.110\%), 838.000 \muhz\ (0.082\%), and
uncertainties shown in Figure~\ref{fig:GD1212ft} that conservatively represent the periods 836.827 \muhz\ (0.038\%). This series of
excited variability, and are detailed pre-whitened peaks in
Table~\ref{tab:freq}. all likelihood corresponds to only one independent pulsation mode.
% ++++++++++++++++++++++++++++ GD1212 Fs ++++++++++++++++++++++++++++++ %
\begin{deluxetable}{lccc}
...
$f_{13}$ & 849.13 $\pm$ 0.76 & 1177.68 $\pm$ 1.05 & 0.023 \\
$f_{10}$ & 842.96 $\pm$ 1.02 & 1186.30 $\pm$ 1.43 & 0.046 \\
$f_{6}$ & 828.19 $\pm$ 1.79 & 1207.45 $\pm$ 2.61 & 0.053 \\
$f_{21}$ &
{\em 371.05 $\pm$
0.36 0.36} &
{\em 2695.04 $\pm$
2.63 2.63} &
0.007 {\em 0.007} \\
$f_{19}$ &
{\em 369.83 $\pm$
0.17 0.17} &
{\em 2703.91 $\pm$
1.24 1.24} &
0.015 {\em 0.015} \\
\multicolumn{4}{c}{\bf Nonlinear Combination Frequencies} \\
$f_{10}-f_8$ & $50{,}453.61 \pm 10{,}580.57$ & 19.82 $\pm$ 4.16 & 0.024 \\
$f_7-f_1$ & 31{,}540.61 $\pm$ 3584.19 & 31.70 $\pm$ 3.60 & 0.027 \\
...
\end{deluxetable}
% ==================================================================== %
When establishing The pre-whitening method does provide a quantitative test for significance. We can iteratively fit and pre-whiten the
coherence of variability over long timescales, it is often instructive to calculate FTs from multiple different subgroups highest peaks in the FT until there are none above some threshold; we estimate this locally for each frequency by calculating the median value, $\sigma$, of the
FT in a $\pm200$\,\muhz\ sliding window, and mark as significant the signals that exceed 5$\sigma$. Since the 15.1-day gap in our whole dataset strongly degrades the window function, we have adopted this approach for our final 9.0 days of data
and our initial 2.6-day light curve. This provides an input list of
identical resolution 48 significant signals.
We do not, however, use this technique to define adopted periods and
then average the
resultant FTs. This effectively throws away associated period uncertainties of the independent pulsation modes, which form the
phase information input for asteroseismic modeling. Performing a linear least-squares fit of the
different datasets, and identifies periods determined from the
regions iterative pre-whitening technique would greatly underestimate the period uncertainties.
% ++++++++++++++++++++++++++++ GD1212 FTs ++++++++++++++++++++++++++++ %
\begin{figure*}
\centering{\includegraphics[width=0.98\textwidth]{f4.eps}}
\caption{Fourier transforms (FTs) of the
nonlinear combination frequencies detected in our {\em K2} observations of GD\,1212. As in Figure~\ref{fig:GD1212ft}, the blue FT
most coherent. Figure~\ref{fig:GD1212ft} shows such an represents only the final 9.0-days of data, while the black FT was calculated from the entire 11.6 days of data spread over 26.7 days. Additionally, the average FT
for of four
different, equal-length (2.6-day) 2.6-day subsets
is shown in grey. We mark the adopted combination frequencies, detailed in Table~\ref{tab:freq}, with magenta points and associated uncertainties at the top of each panel. \label{fig:GD1212combos}}
\end{figure*}
% ==================================================================== %
Instead, we have fit a Lorentzian profile to a $\pm5$\,\muhz\ range of the regions of
power defined significant using our
data on GD\,1212. pre-whitening method, and use the central peak, half-width-half-maximum, and intensity of this Lorentzian to define the adopted periods, period uncertainties, and amplitudes of the broad peaks in the FT. These values provide the red points and uncertainties shown in Figure~\ref{fig:GD1212ft} that conservatively represent the periods of excited variability, and are detailed in Table~\ref{tab:freq}.
This method strongly underestimates the adopted pulsation amplitudes, but theoretical pulsation calculations do not account for mode amplitudes, and our results reasonably represent the two quantities requisite for a seismic analysis: the pulsation periods and associated uncertainties. We italicize the signals at 371.1\,s and 369.8\,s in Table~\ref{tab:freq} because they may not in fact be independent pulsation modes (see discussion at the end of Section\,\ref{sec:nonlinear}).
This averaged FT establishes When establishing the coherence of
the adopted regions of variability. But perhaps more importantly, variability over long timescales, it
shows that smaller 2.6-day subsets are not sufficient is often instructive to
resolve some closely spaced variability that can be resolved within longer subsets, such as calculate FTs from multiple different subgroups of the data of identical resolution and then average the resultant FTs. This effectively throws away the phase information of the different datasets, and identifies the
FT most coherent regions of
our 9.0-day light curve. the FT.
It is quite possible that this frequency broadening for the averaged FTs is the result of unresolved rotational multiplets embedded within the observed, natural frequency variability of the underlying pulsation modes. The Figure~\ref{fig:GD1212ft} shows such an average
half-width-half-maximum FT for four equal-length (2.6-day) subsets of
the profiles fit our data on GD\,1212. We find that smaller 2.6-day subsets are not sufficient to
resolve some closely spaced variability that can be resolved within longer subsets, such as the
averaged FT
is 3.14\,\muhz, compared to 1.12\,\muhz\ for the of our 9.0-day
FT. Rotational splittings falling between this range would arise from a WD rotation rate between $1.9-5.2$ days if these are all $\ell=1$ modes and between $6.1-17.2$ days if these are $\ell=2$ modes.
%Test with longer subsets light curve.
An inferred rotation rate It is possible that this frequency broadening for the averaged FTs is the result of
$1.9-17.2$ days provides one unresolved rotational multiplets embedded within the observed, natural frequency variability of the
first constraints on underlying pulsation modes. The average half-width-half-maximum of the
rotation rate profiles fit to the highest seven peaks of
the averaged FT is 3.14\,\muhz, compared to 1.12\,\muhz\ for the 9.0-day FT. Rotational splittings falling between this range would arise from a
cool DAV, albeit a very coarse constraint. Hotter, stable DAVs show detected WD rotation
rates from rotational splittings rate between
$0.4-2.3$\,days \citep{2008PASP..120.1043F}. $1.9-5.2$ days if these are all $\ell=1$ modes and between $6.1-17.2$ days if these are all $\ell=2$ modes.
%Period spacings? Include in section An inferred rotation rate of $1.9-17.2$ days provides one of the first constraints on
preliminary asteroseismic solution? the rotation rate of a cool DAV, albeit a very coarse estimate. Hotter, stable DAVs show detected rotation rates from rotational splittings between $0.4-2.3$\,days \citep{2008PASP..120.1043F}.
