# PRECISION ASTEROSEISMOLOGY OF THE WHITE DWARF GD 1212 USING A TWO-WHEEL CONTROLLED KEPLER SPACECRAFT

Abstract

We present a preliminary analysis of the cool pulsating white dwarf GD1212, enabled by more than 11.5 days of space-based photometry obtained during an engineering test of a two-reaction wheel controlled Kepler spacecraft. We detect at least 21 independent pulsation modes, ranging from $$369.8-1220.8$$s, and at least 17 nonlinear combination frequencies of those independent pulsations. Our longest uninterrupted light curve, 9.0 days in length, evidences coherent difference frequencies at periods inaccessible from the ground, up to 14.5hr, the longest-period signals ever detected in a pulsating white dwarf. These results mark some of the first science to come from a two-wheel controlled Kepler spacecraft, proving the capability for unprecedented discoveries afforded by extending Kepler observations to the ecliptic.

# Introduction

The endpoints of stellar evolution, white dwarf (WD) stars provide important boundary conditions on the fate of all stars with masses $$\leq 8$$ $${M}_{\odot}$$, as is the case for more than 97% of all stars in our Galaxy, including our Sun. When a WD cools to the appropriate effective temperature to foster a hydrogen partial ionization zone, roughly $$12{,}000$$K, global oscillations driven as non-radial $$g$$-modes become unstable and reach observable amplitudes.

These hydrogen-atmosphere pulsating WDs (so-called DAVs or ZZCeti stars) have spent hundreds of Myr passively cooling before reaching this evolutionary state. Global oscillations provide a unique window below the thin photosphere and deep into the interior of these relatively simple stars, enabled by matching the observed periods to theoretical models generated by adiabatic pulsation calculations.

Given the number of free parameters for full asteroseismic fits, the most reliable results require securing a large number of significant pulsation periods and uniquely identifying the oscillation modes. However, with typical $$g$$-mode periods ranging from $$100-1400$$ s, ground-based photometry suffers from frequent gaps in coverage, frustrating efforts to disentangle multiperiodic signals and alias patterns.

Multi-site campaigns coordinated across the globe via the Whole Earth Telescope (WET, Nather et al. 1990) have proved the richness of well-resolved WD pulsation spectra. For example, less than a week of nearly continuous observations of the helium-atmosphere (DBV) GD358 revealed more than 180 significant periodicities in the power spectrum, providing exquisite constraints on the helium-envelope mass, $$(2.0\pm1.0) \times 10^{-6}$$$${M}_{\star}$$, the overall mass, $$0.61\pm0.03$$$${M}_{\odot}$$, and the magnetic field strength, $$1300\pm300$$G (Winget et al., 1994). Similarly, roughly 11 days of nearly continuous photometry on the pre-WD PG1159$$-$$035 revealed 125 individual periodicities, accurately constraining the mass, rotation rate and magnetic field of this DOV (Winget et al., 1991).

DAVs have also been extensively studied by some half-dozen WET campaigns, with varying results. In part, this is a result of how pulsation modes excited in DAVs are characteristically influenced by the WD effective temperature: hotter DAVs tend to have fewer modes, lower amplitudes and shorter-period pulsations, while cooler DAVs driven by substantially deeper convection zones tend to have more modes at higher amplitude and longer periods (Mukadam et al., 2006). WET campaigns have borne this out. More than 5 days of nearly continuous monitoring of the hot DAV G226$$-$$29 revealed just one significant triplet pulsation mode (Kepler et al., 1995), whereas the cooler DAV G29$$-$$38 has more a dozen modes of relatively high amplitude (Kleinman et al., 1994).

In fact, G29$$-$$38 illustrates the challenges faced to performing asteroseismology of cooler DAVs: although the WD exhibits at least 19 independent oscillation frequencies, there is significant amplitude and phase modulation of these modes, which change dramatically from year-to-year (Winget et al., 1990; Kleinman et al., 1998). Another excellent example of this complex behavior is the cool DAV HLTau76 (citation not found: 2006A&A...446..237D), which shows 34 independent periodicities along with many oscillation frequencies at linear combinations of the mode frequencies. The complex mode amplitude and frequency variations are likely the result of longer-period pulsations having much shorter linear growth times, increasing the prevalence of amplitude and phase changes in cooler DAVs with longer periods (e.g., Goldreich et al. 1999).

The Kepler mission has already uniquely contributed to long-term distinctions between the handful of hot and cool DAVs eventually found in the original pointing. The longest-studied by Kepler, the cool DAV ($$11{,}130$$K) KIC4552982 discovered from ground-based photometry (Hermes et al., 2011), shows considerable frequency modulation in the long-period modes present between $$770-1330$$s (Bell et al. 2014, in prep.). A much hotter DAV was also observed for six months, KIC11911480 ($$12{,}160$$K), which shows at least six independent pulsation modes from $$172.9-324.5$$s that are incredibly stable and evidence consistent splitting from a $$3.5\pm0.5$$day rotation rate (citation not found: 2014MNRAS.438.3086G).

After the failure of a second reaction wheel in 2013May, the Kepler spacecraft has been demonstrated a mission concept using two reaction wheel control, observing fields in the direction of the ecliptic. This mission conecpt aims to obtain uninterrupted observations of fields in the ecliptic for approximately 75days. As part of an initial test to monitor the two-wheel controlled pointing behavior on long timescales, short-cadence photometry was collected every minute on the cool DAV GD1212 during a preliminary engineering run in 2014January and February.

GD1212 ($$V=13.3$$ mag) was discovered to pulsate by Gianninas et al. (2006), with roughly 0.5% relative amplitude photometric variability dominant at 1160.7s. The most recent model atmosphere fits to spectroscopy of GD1212 find this WD has a $${T}_{\mathrm{eff}}$$ $$= 11{,}270\pm170$$ K and $$\log{g}$$ $$= 8.18\pm0.05$$, which corresponds to a mass of $$0.71\pm0.03$$ $${M}_{\odot}$$ (Gianninas et al., 2011). This puts GD1212 at a distance of roughly 17 pc, although GD1212 has the lowest proper motion of any WD within 25 pc of the Sun, $$33.6\pm1.0$$ mas yr$$^{-1}$$ (Subasavage et al., 2009).

In this paper we provide a preliminary analysis of the unique two-wheel controlled Kepler observations of GD1212. In Section 2 we outline the observations and reductions. We analyze the independent pulsation modes and nonlinear combination frequencies in Sections 3 and 4, respectively. We reserve Section 5 for a preliminary asteroseismic interpretation of the results, and conclude with a discussion of these results in the context of a two-wheel controlled Kepler mission observing into the ecliptic in Section 6.

# Observations and Reductions

\label{sec:observations}

We observed GD1212 (GJ4355, WD2336$$-$$079, catalog no. 60017836, WDJ233850.74$$-$$074119.9) for a total of 264.5hr using the Kepler spacecraft in two-wheel mode. With only two working reaction wheels, the spacecraft pointing cannot be stabilized in three axes. By pointing the bore-sight close to the plane of the spacecraft’s orbit (within about a degree of the ecliptic, Howell et al., 2014), the unconstrained spacecraft roll is placed in equilibrium with respect to the solar pressure, the dominant external force exerted on the spacecraft.

The unconstrained spacecraft roll is an unstable equilibrium, and the pointing needs to be corrected every $$3-6$$hr by firing the thrusters. No data is lost during a thruster firing (known as a reset), but the change in attitude causes a strong jump in the measured flux from the star. The spacecraft rolls by no more than 25 arcsec between resets, which corresponds to motion of a star at the extreme edge of the field of view by no more than 1 pixel. The spacecraft can point a single field for up to 80 days, before the angle of the sun with respect to the solar panels exceeds allowed limits. Putnam et al. (2014) provides further details of the capabilities and limitations two wheel operation.

We collected data on GD1212 from 2014Jan17 to 2014Feb13 in short-cadence mode (SC), where each exposure is 58.8s. After the first 2.6 days, the observations were interrupted for 15.1days by a safe-mode event and subsequent engineering fault analysis. The final 9.0-day light curve can be found in Figure \ref{fig:GD1212lc}. We have removed all points falling more than 4$$\sigma$$ from the light curve mean, resulting in 2.6- and 9.0-day observations with a duty cycle of more than 98.2% and 98.9%, respectively.

In contrast to the primary mission, where only small masks were placed around each star, the data on GD1212 were collected using a $$50 \times 50$$ pixel “super aperture.” We expect the aperture size used in two-wheel mode will decrease as confidence in the spacecraft pointing ability increases. The observed pixels were processed through the CAL module of the Kepler pipeline (citation not found: 2010SPIE.7740E..64Q) to produce target pixel files (TPFs); light curve files were not produced for this engineering data. TPF data for all stars observed during this engineering run are available at the MAST Kepler archive1.

Our final 9.0-day dataset is nearly continuous and has a formal frequency resolution of 1.29 $$\mu$$Hz. The median noise level in the Fourier transform (FT) near 500 and 1500 $$\mu$$Hz for this 9.0-day run is roughly 0.0036% (36 ppm). For the entire 26.7-day (42.8% duty cycle) data set on the $$K_p=13.3$$ mag GD1212, the median noise level is roughly 0.0028% (28 ppm).

A portion of the K2 short-cadence photometry collected on the pulsating white dwarf GD1212 in 2014February. This 9.0-day light curve (shown without any smoothing) establishes the capabilities of the extended Kepler mission on this $$K_p=13.3$$ mag star. An additional 2.6days of data, not shown, were collected on GD1212 in 2014January and included in our analysis. \label{fig:GD1212lc}

# Independent Pulsation Modes

\label{sec:analysis}

As expected given the relatively cool spectroscopically determined temperature (e.g., Mukadam et al. 2006), the pulsation periods excited in GD1212 are relatively long, ranging from $$828.2-1220.8$$ s. A cool effective temperature is also borne out from model atmosphere fits to the photometry of this WD, which find $${T}_{\mathrm{eff}}$$ $$= 10{,}940\pm320$$ K and $$\log{g}$$ $$= 8.25\pm0.03$$ (Giammichele et al., 2012).

Here, however, we will adopt for our discussion the more precise parameters derived from spectroscopy: $${T}_{\mathrm{eff}}$$ $$= 11{,}270\pm170$$ K and $$\log{g}$$ $$= 8.18\pm0.05$$ (Gianninas et al., 2011). The spectroscopy of GD1212 is notable in that much of it was collected in order to identify the then-dominant mode at 1160.7s, 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 (Desgranges, 2008).

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 $$\mu$$Hz (14.5hr) down to 2703.9 $$\mu$$Hz (369.8s), which is shown in full in the top panel of Figure \ref{fig:GD1212ft}. The only periodicity detected with marginal significance at higher frequencies occurs at 4531.8 $$\mu$$hz (0.012% amplitude), which is an instrumental artifact sampling the long-cadence exposures of 29.4min that often appears in short-cadence Kepler data, so we do not include it in our analysis.

The highest-amplitude variability in GD1212 occurs in the region between $$810-1210$$$$\mu$$hz ($$826-1234$$s). This region shows at least 19 independent pulsation modes, which we illustrate 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 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 $$810-1210$$$$\mu$$hz 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$$\mu$$hz 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$$\mu$$hz is decreasing in amplitude, and appears further to bifurcate into two modes.

A common description of amplitude and frequency modulation in pulsating stars is the presence of additional nearby frequencies that are unresolved over the course of observations. The pulsational variability in DAVs is the result of non-radial stellar oscillations, 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., (citation not found: 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 $$2.5-19.0$$ $$\mu$$hz (see Fontaine et al. 2008; Kawaler 2004 and references therein).

The DAV G226$$-$$29 is an excellent example of a closely spaced multiplet affecting pulsation amplitudes in a WD (Kepler et al., 1983). In this WD, the 109s pulsation is actually a superposition of three signals at 109.08648s, 109.27929s, and 109.47242s, which are separated by roughly 16.1$$\mu$$Hz and require more than 17.2hr of observations to resolve the beat period of the closely spaced signals. Shorter observations show the signal sinusoidally varying at the beat period of 17.2hr.

However, rotational splittings are unlikely to explain the incoherent changes in the running FT of GD1212, nor the disappearance of $$f_5$$ near 847.2 $$\mu$$Hz, 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, as has been witnessed during long campaigns on other cool DAVs (e.g., Kleinman et al. 1998).

The frequency, amplitude, and possibly phase variability observed in GD1212 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 in the star (e.g., (citation not found: 2014MNRAS.438.3086G)). However, none of the peaks in either the 26.7-day FT or the 9.0-day FT of GD1212 can be smoothly pre-whitened in the standard way because fitting a pure sine wave does not completely remove the pulse shape.

One can always decompose the broad peaks in an FT with a linear combination of sine waves, but not all of these sinewaves represent physical modes. For example, using just the final 9.0 days of data, the highest-amplitude signal near 840.0$$\mu$$Hz requires five sine waves to reproduce the broad peak: 840.211 $$\mu$$Hz (0.248% amplitude), 839.183 $$\mu$$Hz (0.154%), 842.016 $$\mu$$Hz (0.110%), 838.000 $$\mu$$Hz (0.082%), and 836.827 $$\mu$$Hz (0.038%). This series of pre-whitened peaks in all likelihood corresponds to only one independent pulsation mode.

The pre-whitening method does provide a quantitative test for significance. We can iteratively fit and pre-whiten the 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 $$\pm200$$$$\mu$$Hz 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 48 significant signals.

We do not, however, use this technique to define adopted periods and the associated period uncertainties of the independent pulsation modes, which form the input for asteroseismic modeling. Performing a linear least-squares fit of the periods determined from the iterative pre-whitening technique would greatly underestimate the period uncertainties.

Instead, we have fit a Lorentzian profile to a $$\pm5$$$$\mu$$Hz range of the regions of power defined significant using our 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 371.1s and 369.8s 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}).

When establishing the coherence of variability over long timescales, it is often instructive to 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 most coherent regions of the FT.

Figure \ref{fig:GD1212ft} shows such an average FT for four equal-length (2.6-day) subsets of our data on GD1212. 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 FT of our 9.0-day light curve.

It is 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 average half-width-half-maximum of the profiles fit to the highest seven peaks of the averaged FT is 3.14$$\mu$$Hz, compared to 1.12$$\mu$$Hz for the 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 all $$\ell=2$$ modes.

An inferred rotation rate of $$1.9-17.2$$ days provides one of the first constraints on 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 (Fontaine et al., 2008; Kawaler, 2004).

As expected given the relatively cool spectroscopically determined temperature for GD1212, the pulsation periods excited in this WD are relatively long, ranging from $$369.8-1190.5$$ s. A cool effective temperature is also borne out from model atmosphere fits to the photometry of GD1212, which find $${T}_{\mathrm{eff}}$$ $$= 10{,}940\pm320$$ K and $$\log{g}$$ $$= 8.25\pm0.03$$ (Giammichele et al., 2012). Here we will adopt the more reliable parameters derived from spectroscopy: $${T}_{\mathrm{eff}}$$ $$= 11{,}270\pm170$$ K and $$\log{g}$$ $$= 8.18\pm0.05$$ (Gianninas et al., 2011).

The spectroscopy of GD1212 is notable in that much of it was collected in order to identify the then-dominant mode at 1160.7s, 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 low amplitude of the variability (Desgranges, 2008).

A Fourier transform (FT) of our entire dataset is shown in Figure\ref{fig:GD1212ft}, which orients us to the dominant frequencies of variability.

# Nonlinear Combination Frequencies

\label{sec:nonlinear}

In addition to the 19 significant independent pulsation modes we identify in Section \ref{sec:analysis}, we see a number of nonlinear combination frequencies in our data that arise at summed and difference frequencies of the independent modes. For example, the peak at 2107.3$$\mu$$Hz is, within the uncertainties, a linear sum of the two independent pulsations $$f_4=940.3$$$$\mu$$Hz and $$f_7=1166.2$$$$\mu$$Hz.

Combination frequencies are a common feature in the pulsation spectra of variable WDs, which are created by a nonlinear distortion of an underlying linear oscillation signal. Work by Brickhill (1992) attributed these nonlinear distortions to the changing thickness of the convection zone of a DAV, a result of local surface temperature variations from the underlying global stellar oscillations. The convection zone depth is extremely sensitive to temperature ($$\propto T^{-90}$$, Montgomery 2005), distorting the observed emergent flux. For the hottest DAVs, the nonlinear response of the flux to a temperature perturbation (i.e., the “$$T^4$$” nonlinearity) may also play a role in creating these signals (Brassard et al., 1995).

Observations of the rate of period change of combination frequencies show that they match identically the rates of their parent modes, establishing that these signals are not independently excited pulsations but rather nonlinear distortions (Hermes et al., 2013). Thus, these signals provide no additional asteroseismic constraints on the interior structure of the star. However, the amplitude ratios of these combination frequencies to their parent modes can yield insight into the depth of the convection zone, which is responsible for driving the underlying independent pulsations (e.g., Montgomery et al. 2010; Provencal et al. 2012).

We detail the observed nonlinear combination frequencies in GD1212 and identify their likely parent modes in Table \ref{tab:freq}. Their amplitudes and frequencies were determined in an identical manner to the independent pulsation modes, as discussed in Section \ref{sec:analysis}. However, we have used a less stringent test for significance, and include as significant the signals that exceed 4$$\sigma$$.

We mark these adopted nonlinear combination frequencies as magenta points in our FT of all the data, shown in the top panel of Figure \ref{fig:GD1212ft}. We show a more detailed view of the regions of these combination frequencies in Figure \ref{fig:GD1212combos}.

Good evidence that these are in fact combination frequencies comes qualitatively from the region in the FT near 2006$$\mu$$Hz (bottom panel of Figure \ref{fig:GD1212combos}) as compared to the region near 847$$\mu$$Hz (middle panel, Figure \ref{fig:GD1212ft}). There appear two combination frequencies, 2006.6$$\mu$$Hz and 2011.5$$\mu$$Hz, which are combination of $$f_1+f_7$$ and $$f_5+f_7$$, respectively. The power for the $$f_5+f_7$$ combination frequency is nearly nonexistent in our final 9.0 days of data, exactly as the power for $$f_5$$ near 847.2$$\mu$$Hz is diminished in our final 9.0 days of data.

Unfortunately, the underlying frequency variability of the independent parent modes extends to these combination frequencies, broadening their power in an FT. Thus, these nonlinearities do not provide much assistance in refining the periods of the parent modes. Still, the eight significant low-frequency difference frequencies, ranging from $$19.2-170.0$$$$\mu$$Hz ($$1.6-14.5$$ hr), are the longest-period signals ever detected in a pulsating WD, accessible only because of the exceptionally long and uninterrupted observations provided by Kepler.

We preliminarily identify the variability at $$f_{21}=371.1$$s and $$f_{19}=369.8$$s as independent pulsation modes, but they arise at substantially shorter periods, far isolated from the other independent pulsations, which are all longer than 828.2s. The signals $$f_{21}$$ and $$f_{19}$$ may instead be more complicated nonlinear combination frequencies. For example, $$f_{21}$$ may be the sum $$2f_1+f_2$$. However, no obvious frequency combination can produce the higher-amplitude $$f_{19}$$.

It is also possible that the signals near 370s may be new instrumental artifacts in K2. Spurious signals occur in the FTs of the two-wheel concept engineering test data at harmonics of the 3-hr reset, where pointing is reset by thruster firing. This frequency range is near 30 times the frequency of the pointing reset.

# Discussion and Conclusions

\label{sec:discussion}

We report a detailed analysis of pulsations excited in the cool WD GD1212 using more than 11.5 days of data collected during a two-wheel concept engineering test of the repurposed Kepler spacecraft. Our results suggest the presence of at least 19 independent pulsation modes excited to observable amplitudes, with up to 2.5% peak-to-peak variability.

Observed frequency and amplitude modulation complicates a precise determination of the pulsation periods in GD1212, likely the result of these relatively long-period oscillations ($$828.2-1220.8$$s) having short linear growth times. We have defined a conservative estimate of the excited pulsations by fitting Lorentzian functions to the broad bands of power in a Fourier transform of our entire data set, yielding an average uncertainty of 3.0s ($$<0.5$$%) for each adopted period.

A complete asteroseismic solution of the data in hand would be premature until we have more secure identification of the spherical degree ($$\ell$$ values) of the independent pulsation modes. We can make a first attempt at these identifications given that such long-period modes likely have high radial order ($$k>17$$), suggesting that the modes are near the asymptotic limit and patterns in consecutive period spacing should yield insight into the spherical degree of the modes.

There appear to be two regions of semi-evenly spaced independent pulsations, one at longer periods (involving $$f_1$$-$$f_7$$-$$f_2$$-$$f_5$$-$$f_9$$-$$f_3$$) that have a weighted mean period spacing of $$41.5\pm 2.5$$s. Additionally, four shorter-period modes ($$f_{15}$$-$$f_8$$-$$f_{10}$$-$$f_6$$) have a roughly even period spacing of $$14.4\pm1.5$$s.

The asymptotic period spacing for $$\ell=1$$ modes of a cool ($$11{,}147$$K), 0.837$${M}_{\odot}$$WD with a $$10^{-5.408}$$$${M}_{\star}$$ hydrogen layer mass is roughly 40.3s, according to the models of (citation not found: 2012MNRAS.420.1462R). However, this model does not accurately predict the other period spacings, as they calculate $$\ell=2$$ modes to have a 23.3s mean period spacing.

The asymptotic period spacing for higher spherical degrees decreases as $$\sqrt{\ell(\ell+1)}$$, so it is possible that the shorter-period sequence is composed of a series of $$\ell=3$$ modes. However, $$\ell=3$$ modes have never been previously inferred in pulsating WDs. Limb-darkening effects greatly reduce the observed amplitudes of $$\ell=1,2$$ modes compared to $$\ell=3$$ modes at shorter (near UV) wavelengths (e.g., (citation not found: 2000ApJ...539..379K)), so multicolor observations of the pulsation amplitudes of GD1212 through different bandpasses could conclusively test this hypothesis.

It is also possible that the two seemingly evenly spaced regions are not consecutive radial order modes of the same spherical degree at all, but rather a coincidence of mixed $$\ell=1$$ and $$\ell=2$$ modes. More quantitative answers will require a full asteroseismic analysis, matching all 19 independent pulsation modes to the periods expected from theoretical models.

Our final 9.0-day light curve displayed in Figure \ref{fig:GD1212lc} is to date the highest-quality continuous look at a pulsating WD, unmolested by differential atmospheric effects, clouds, or sunrises. Much of the long-term variability about the mean of the light curve is real, caused by nonlinear combination frequencies of the excited pulsation modes at periods extending up to 14.5hr. Such long-period signals have never been detected from ground-based campaigns, accessible only due to unique space-based photometry afforded by missions such as Kepler.

This relatively short two-wheel concept engineering test proves the impact a two-wheel controlled Kepler spacecraft will have on new discoveries in the exploration of stellar interiors and asteroseismology by extending the mission and field of view of the Kepler spacecraft to the ecliptic.

We acknowledge members of Working Group 11 of the Kepler Asteroseismology Consortium for their eager discussion and analysis of these two-wheel concept engineering test data, which led to this rapid result. J.J.H. acknowledges funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 320964 (WDTracer). M.H.M. and D.E.W. gratefully acknowledge the support of the NSF under grants AST-0909107 and AST-1312678 and the Norman Hackerman Advanced Research Program under grant 003658-0252-2009.

Facilities: Kepler, K2

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