Multiwavelength Ground and Space Observations of the Variable White Dwarf BOKS 53856: Nonuniform Metal Absorption in Dark Spots

BOKS 53856 was monitored during most of the Kepler mission (Borucki et al. 2010; Koch et al. 2010), with observations spanning 2009 June through 2013 May. Twelve long-cadence (Δt ≈ 30 minutes) data sets were obtained during Kepler observing quarters 2–5 and 10–17; BOKS 53856 was not observed by Kepler during quarters 6–9. Short-cadence (Δt ≈ 1 minute) data sets spanning the entire observing quarter were also obtained during Q2, Q10, and Q14. HH2011 used only the Q2 short-cadence and Q3 long-cadence data for their analysis.

We downloaded the Kepler observations for BOKS 53856 from the NASA Exoplanet Archive.⁷ Figure 1 shows the raw Kepler light curve data, as well as the division-normalized data sets used for the current analysis. The delineation of the observing quarters is apparent from the offsets in the raw long-cadence data, and some intraquarter gaps in the data collection are also visible. Short (≲1 day) interruptions for routine reasons (e.g., Earth point for data downlink, spacecraft rolls, reaction-wheel angular momentum dumps) are found in every quarter.

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Some quarters contain unique events that interrupted or otherwise affected data acquisition for longer periods (e.g., safe modes, loss of fine pointing). For example, the gap just before day 1300 resulted from the failure of one of Kepler's reaction wheels during Q14. The observation in Q16 was interrupted 5.2 days after the start of the quarter (just before day 1500) by a "wheel rest" safe mode to address an increase in friction in the three remaining reaction wheels. This lasted 11.2 days before Kepler returned to normal operations for the rest of the quarter. Approximately 100 days later, the Q17 observation was terminated after the end of the second month in the quarter due to the failure of a second reaction wheel, which effectively ended the Kepler mission (later reborn as the K2 mission; Howell et al. 2014).

As part of a joint Hubble/Spitzer program (Hubble program 13322; also see Section 2.3), we observed BOKS 53856 on 2014 September 29 UT using the Cosmic Origins Spectrograph (COS; Osterman et al. 2011)⁸ during a total of four Hubble orbits, thereby spanning the entire ≈6 hr period found by HH2011. COS was used with the G130M grating, centered at 1309 Å, in TIME-TAG mode. The total exposure times were 2222 s in orbit 1 and 2650 s each in orbits 2–4, for a grand total of 10,172 s (the orbit 1 exposure time is shorter due to initial acquisition images required at the start of the Hubble program). The exposure start time (UT) in each of orbits 1–4 was 06:35:47, 07:58:37, 09:34:08, and 11:09:40, respectively.

In order to help mitigate Lyα air glow (by spreading it around the detector), reduce high-frequency fixed pattern noise, and increase the signal-to-noise ratio (S/N) of the extracted spectra, we configured COS to use FP-POS mode. This splits the exposure obtained during each Hubble orbit into four equal-length subexposures, separated by short time gaps of a few minutes to allow for the dispersion direction offsets (Fischer et al. 2017, Section 5.8.2). We utilized the standard extracted spectrum data products obtained from the Mikulski Archive for Space Telescopes (MAST).⁹

Figure 2 shows our COS UV spectrum of BOKS 53856 for the entire exposure time. Other than the typically broad WD Lyα absorption line, the only apparent spectral features correspond to interstellar medium (ISM) absorption lines. For comparison, we also show a similar archival COS spectrum of the metal-polluted DAZ WD, WD 1929+011 (= GALEX J193156.8+011745; Debes et al. 2011; Gänsicke et al. 2012), which we retrieved from MAST. Numerous photospheric absorption lines of metals are present in the spectrum of WD 1929+011, primarily from Si, C, and Fe (for a detailed identification of the absorption features, see Gänsicke et al. 2012). Between 0.131 and 0.132 $\mu {\rm{m}}$, the spectrum of BOKS 53856 shows some weak features that appear to correspond to Si and Fe lines. However, as this region is at the extreme short-wavelength end of COS grating segment A, we cannot strongly advocate for the veracity of these features (in addition, other stronger Si features that would be expected—such as the 1265 Å doublet—are not present). None of the photospheric metal lines seen in WD 1929+011 are convincingly (or at all) detected in BOKS 53856.

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The Spitzer component of our joint observing program (Spitzer program 90105) was composed of an Infrared Array Camera (IRAC; Fazio et al. 2004) channel 1 (3.6 $\mu {\rm{m}}$) and channel 2 (4.5 $\mu {\rm{m}}$) observation of BOKS 53856. Single-frame exposure times of 30 s were used in both channels, with the standard medium-scale cycling dither pattern with 40 positions in channel 1 and 25 positions in channel 2 (for total on-source times of 1200 s and 750 s, respectively). The channel 1 observation was carried out on 2013 January 22 with start time 18:04:40 UT, while the channel 2 observation started on 2013 January 24 at 10:27:57 UT.

We utilized the corrected basic calibrated data (CBCD) products from the S19.2.0 pipeline, provided through the Spitzer Heritage Archive.¹⁰ We performed aperture photometry on the CBCD images using the Image Reduction and Analysis Facility (IRAF; v2.16.1)¹¹ with a three-pixel aperture to prevent contamination from nearby sources, following the processes described in the IRAC Instrument Handbook¹² and the Spitzer Data Analysis Cookbook.¹³ We applied corrections for aperture size, source color, pixel solid angle, pixel phase, and array location, as described in the relevant sections of the IRAC Instrument Handbook and in other documents available on the IRAC web page¹⁴ at the NASA/IPAC Infrared Science Archive (IRSA). The last two of these corrections were applied using the Interactive Data Language (IDL)¹⁵ routine irac_aphot_corr (2016 March 02 version).¹⁶ The photometry from the individual CBCD images was then averaged to produce the final values. The resulting mid-IR flux density measurements are listed in Table 1.

Table 1.  Photometry

Band	λ ($\mu {\rm{m}}$)	Magnitudea		fν (mJy)	Symbolb	Note
COS01	0.1170	⋯		2.48(13)	bar	1
COS02	0.1190	⋯		2.20(12)	bar	1
COS03	0.1202	⋯		1.50(13)	bar	1
COS04	0.1208	⋯		0.96(16)	bar	1
COS05	0.1225	⋯		1.18(15)	bar	1
COS06	0.1232	⋯		1.73(13)	bar	1
COS07	0.1245	⋯		2.32(10)	bar	1
COS08	0.1270	⋯		2.47(11)	bar	1
COS09	0.1330	⋯		2.38(12)	bar	1
COS10	0.1355	⋯		2.38(11)	bar	1
COS11	0.1380	⋯		2.31(12)	bar	1
COS12	0.1405	⋯		2.26(12)	bar	1
COS13	0.1430	⋯		2.22(16)	bar	1
UVW2	0.1928	13.936(3)	I	1.767(5)	diamond	2
NUV	0.2271	16.141(6)	A	1.269(7)	diamond	3
U	0.36	15.934(23)	V	0.771(28)	circle	4
B	0.44	16.989(28)	V	0.661(26)	circle	4
gP1	0.481	16.930(9)	A	0.614(5)	circle	5
gKIS	0.4846	16.982(2)	Ac	0.585(1)	circle	6
V	0.55	17.150(22)	V	0.522(19)	circle	4
rP1	0.617	17.337(3)	A	0.422(1)	circle	5
rKIS	0.6240	17.381(4)	Ac	0.405(2)	circle	6
HaKIS	0.6568	17.537(9)	Ac	0.351(3)	circle	6
iP1	0.752	17.690(5)	A	0.305(1)	circle	5
iKIS	0.7743	17.717(6)	Ac	0.297(2)	circle	6
zP1	0.866	17.932(16)	A	0.244(4)	circle	5
yP1	0.962	18.114(13)	A	0.206(3)	circle	5
JMKO	1.235	17.726(39)	V	0.127(6)	circle	7
W1	3.35	>16.6	V	<0.07	triangle	8
IRAC1	3.55	⋯		0.0168(2)	square	9
IRAC2	4.493	⋯		0.0103(11)	square	9
W2	4.60	>15.6	V	<0.10	triangle	8
W3	11.56	>11.2	V	<1.05	triangle	8
W4	22.24	>8.0	V	<5.28	triangle	8

Notes. (1) Mean flux density (and standard deviation) in our COS spectrum (visit 2) in 5 Å bins centered on the listed wavelengths (see Section 3.3 and inset panel in Figure 10). (2) Swift Master Catalog, processing script version 3.17.06 and software version Hea_27Jul2015_V6.17_Swift_Rel4.5(Bld34)_27Jul2015_SDCpatch_2 (see Poole et al. 2008; Breeveld et al. 2011). (3) GALEX near-UV photometry from the Kepler field survey by Olmedo et al. (2015). (4) UBV Photometric Survey of the Kepler Field (Everett et al. 2012). (5) Pan-STARRS Data Release 1 (see https://panstarrs.stsci.edu/). (6) Kepler-INT Survey (KIS) Data Release 2 (Greiss et al. 2012a, 2012b). The listed value is the weighted mean of the three contemporaneous survey measurements. Note that the KIS U-band data are not used here due to the nonstandard filter and calibration issues described in Greiss et al. (2012a). (7) Kepler Input Catalog (KIC; Brown et al. 2011) value cited in HH2011; it is calibrated on the Mauna Kea Observatory (MKO) system (Tokunaga & Vacca 2005). (8) WISE All-Sky Release nondetection upper limits for S/N > 5 in unconfused regions with at least eight independent coverages, obtained from IRSA. (9) Spitzer aperture photometry (this work).

^aThe magnitude system is given as A = AB, V = Vega, I = instrumental. ^bShape of plotting symbol in Figure 10. ^cTransformed from Vega to AB using relations in Blanton & Roweis (2007) for the g filter and González-Solares et al. (2008) for the other filters.

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BOKS 53856 was serendipitously observed by the Ultra-Violet/Optical Telescope (UVOT; Roming et al. 2005) on the Swift satellite (Gehrels et al. 2004) on 2010 July 16 for a total of 4770 s through the UVW2 ultraviolet filter. We retrieved the UV photometry for BOKS 53856 from the Swift Master Catalog maintained at NASA's High Energy Astrophysics Science Archive Research Center (HEASARC).¹⁷

Although not in an area of the sky covered during the survey phase of the initial GALEX mission, BOKS 53856 was subsequently observed during the mission continuation phase (the GALEX Complete All-Sky UV Survey Extension, or CAUSE). Olmedo et al. (2015) used CAUSE observations to compile a deep UV photometric catalog of the Kepler field, including BOKS 53856. We retrieved these data from the Vizier web service (Ochsenbein et al. 2000).¹⁸

The UV photometry for BOKS 53856 is listed in Table 1.

In addition to utilizing a number of published ground-based spectroscopic and photometric measurements in this work (see Table 1), we have also obtained new optical spectroscopic observations. These are summarized in Table 2; the Palomar Observatory spectra from 2010 September 17 are the same data set referenced, and shown in limited fashion, in HH2011.¹⁹ The two-dimensional spectral images were processed into one-dimensional extracted spectra in the standard manner using IRAF. A representative composite spectrum is shown in Figure 3, displaying the typical blue continuum and broad Balmer absorption features of a DA WD.

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We calculated discrete Fourier transform power spectra (e.g., as described in Roberts et al. 1987, but without application of the clean algorithm; also see Bracewell 1965, Chapter 11) for the entire (Q2–Q17) Kepler long- and short-cadence data sets (see Figure 4), and for each individual quarter of long-cadence data (see Table 3). In all cases, the division-normalized data were used. We applied a single-pass sigma-clipping rejection to the data sets before calculating the power spectra (3σ or 5σ rejection for long or short cadence, respectively).

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We extracted the frequencies corresponding to peaks in the power spectra by obtaining the center of a five-term Gaussian fit (i.e., Gaussian function plus constant and linear offsets) using the IDL routine gaussfit. The corresponding peak-frequency statistical uncertainties were calculated by gaussfit via χ² considerations as described in Press et al. (1992, Chapter 15). Periods and their associated uncertainties were then calculated by propagating the inverse of the frequencies and their associated uncertainties. The frequency and period results are listed in Table 3.

The strongest peak in each of the full long- and short-cadence power spectra corresponds to essentially the same fundamental frequency (3.910386 d⁻¹ and 3.910385 d⁻¹, respectively); we adopt the long-cadence period of 0.2557292(9) days as the refined rotation period of the WD (under the assumption that the photometric variability is caused by the changing view of surface features on the WD as it rotates). This result is in agreement with the power spectrum analysis by HH2011, which gave a frequency of 3.91037(3) d⁻¹ and corresponding period of 0.255730(2) days. The frequencies that we measured from the individual quarter long-cadence data sets all agree to within better than 2.5σ with the adopted frequency from the full data set. The other strong peaks in both of the full data set power spectra are integer harmonics of the fundamental frequency.

Figure 5 shows the Kepler light curve data phased on the adopted WD rotation period. The shape of the phased light curve is essentially identical between the long- and short-cadence data sets, as well as in comparison to the phased light curve constructed by HH2011 from only the Q2 short-cadence data. Based on our analysis of the full Kepler data set, we define the following updated ephemeris for WD rotation in BOKS 53856:

$$\begin{eqnarray}&&\mathrm{BJD}=\mathrm{BJD}\,2,\,455,\,002.95252(70)+0.2557292(9){\rm{E}},\end{eqnarray} \tag{ 1 }$$

where phase 0.0 corresponds to minimum light.

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For illustrative purposes, we show in Figure 6 a comparison between a sine curve with the adopted period and the Kepler long-cadence data. Although a single sine curve is not a completely representative match to the shape of the folded, binned BOKS 53856 light curve from Figure 5 (see the bottom panel in Figure 6), this simple comparison shows that the sine curve maintains coherence with the unfolded Kepler data from near the start of the data set (Figure 6, upper panel) to near its end (Figure 6, middle panel) more than 5000 cycles later.

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Figure 7 shows the individual quarters of long-cadence Kepler data folded on the ephemeris of Equation 1 and averaged into 100 phase bins, with the bin-averaged, phased light curve of the entire long-cadence data set (Figure 5, bottom panel) subtracted. These residual light curves are generally flat with scatter of less than about ±0.1%. This indicates that the variability on the WD rotation period in BOKS 53856 is quite stable over intervals of months to years.

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Some periodic structure with amplitude in excess of the scatter is apparent in the residual light curves in only a few cases (most notably Q14 and Q15), which could possibly indicate small net changes in variability. However, there is no obvious pattern to these changes from quarter to quarter; for example, the residual light curve from Q14 indicates that the bright peak of the light curve was fainter than average, while in Q15 it was brighter than average.

It is also possible that these apparent systematic changes in the WD variability are actually photometric artifacts caused by small changes in the Kepler pointing or attitude control linked to the reaction wheel failure in Q14 and a subsequent safe-mode event due to loss of pointing in Q15.²⁰ The residual light curves for the final two quarters show increased scatter, likely linked to the ongoing reaction-wheel problems that Kepler experienced during Q16 and Q17 (culminating in the final wheel failure two months into Q17).

In Figure 8, we show the bin-averaged long-cadence Kepler (optical) light curve of BOKS 53856 (from Figure 5), along with a UV continuum (excluding air glow and ISM lines) light curve constructed by summing the TIME-TAG data from our COS spectrum in 10 s bins (as described in Sandhaus et al. 2016, their Section 2.1). In order to apply our improved rotation ephemeris, the JD_UTC times of the Hubble data were converted to BJD_TDB using the conversion routines described in Eastman et al. (2010).

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Figure 8 also shows light curves constructed using light-curve inversion (LI) with a nonlinear inversion algorithm (Wild 1989, 1991; Harmon & Crews 2000; Roettenbacher et al. 2011, 2013), in which the stellar surface is subdivided into a set of uniformly radiating "patches" defined in latitude and longitude. A set of specific intensities for the patches is then deduced that mimics as closely as possible the true surface specific intensity distribution, constrained by comparison between observed and calculated light curves. We used 3434 approximately equal area patches across the WD surface. Each patch was varied independently to best reproduce the observed light curve given photospheric and spot temperatures, limb-darkening coefficients, inclination, and assumed noise level.

We assumed a photospheric temperature of 31,000 K (see Section 3.3) and allowed for the presence of cooler (dark) spot(s) with minimum temperature 25,000 K. That is, the brightest LI patch was assigned a temperature of 31,000 K, and the darkest patch was assigned a temperature of 25,000 K. We estimated limb-darkening values from the coefficients in Gianninas et al. (2013). The results shown here use a value of i = 60° for the (unknown) inclination of the WD. Tests with lower inclinations showed very poor agreement between the observed and constructed light curves for i < 45°, and reasonable results for a range of i = 45°–90^◦. Maps of the surface brightness distributions on the WD are shown in Figure 9. We discuss this further in Section 4.4.

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In both surface maps, most of the surface (hence, most of the LI patches) stayed at the defined temperature of 31,000 K (white in the maps). We can quantify the spot coverage by calculating the area of patches that are fractionally darker than the baseline provided by the brightest patches. For the Kepler light curve, the dark spot coverage (all LI patches whose intensity is less than 95% of that of the brightest patch) is 7.1% of the surface. For the Hubble light curve, the spot coverage (all LI patches whose intensity is less than 93% of that of the brightest patch) is 8.6% of the surface. We utilized a slightly lower threshold in the latter case because about one-third of the WD surface satisfies the 95% threshold in the Hubble map.

It should be noted that the 25,000 K minimum temperature for the spot(s) was somewhat arbitrarily chosen as a representative value, since there is a degeneracy between spot size and temperature. Changing (reducing) the minimum temperature of a spot primarily changes (increases) its size, not its location (longitude). However, the latitudes (and to some extent shapes) of the spots should also be considered representative, since these are poorly constrained in monocolor light curves. The distribution of dark spots in both the optical and UV maps is similar, but not identical, and reflects the differences in the light curves (see Figure 8, top panel). An understanding of why the light curves differ in appearance depends on the currently uncertain physical origin of the spots (also see the discussion in Section 4.3).

We used the photometric data from Table 1, along with our COS spectrum (from orbit 2; i.e., the brightest part of the UV light curve that most avoids being affected by the dark spots), to construct a spectral energy distribution (SED) for BOKS 53856 spanning the UV to the IR. The SED is shown in Figure 10, with a best-fitting WD model SED superimposed. The details of the model-fitting process and resultant system parameters are described below.

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We used the Theoretical Stellar Spectra Access (TheoSSA) service²¹ (Reindl & Rauch 2015; Rauch et al. 2018) to obtain a grid of synthetic H-only WD spectra that were calculated with the Tübingen NLTE Model-Atmosphere Package (TMAP; Rauch & Deetjen 2003; Werner et al. 2003; Rauch et al. 2013). Our spectral grid spans the temperature range T_WD = 20,000–50,000 K in 1000 K steps and gravity range of log g = 7.0–8.0 in steps of 0.01 dex.

To perform the fitting, we utilized the amoeba algorithm (Press et al. 1992, Chapter 10) implemented in a custom-written IDL routine. Amoeba minimizes a multidimensional function using the downhill simplex method (Nelder & Mead 1965). In this case, the function that was minimized was the χ² of the observed SED and model spectrum, where the latter is, itself, a multidimensional function of WD effective temperature (T_WD), log g, WD radius (R_WD), distance (d), and reddening (parameterized by the color excess, E_{B − V}). Finer steps in temperature and log g were obtained by linear interpolation between adjacent synthetic spectra.

Because of the degeneracy between distance and radius, these parameters were combined during the fitting process into the single normalization factor N = (R_WD/d)². The best-fit value of the normalization factor was then disentangled back into d and R_WD by finding the intersection of three functions: the WD mass–radius(–temperature) relation, R_WD = f(M_WD, T_WD), the standard expression for surface gravity, R_WD = f(M_WD, log g), and the normalization factor, R_WD = f(N, d). For the WD mass–radius relation, we used the relation for C-O core WDs with thick H layers discussed by Parsons et al. (2017, see their Figure 9; also see Benvenuto & Althaus 1999 and Fontaine et al. 2001). Reddening was applied using the IDL Astronomy Library²² (Landsman 1993) subroutine fm_unred, which reddens (or dereddens) a flux vector using the Fitzpatrick (1999) parameterization with the R-dependent Galactic extinction curve of Fitzpatrick & Massa (1986, 1988, 1990).

We started the model-fitting process by making initial estimates of the best parameter values. We assumed d = 320 pc, T_WD = 32,500 K, and log g = 8.0 from HH2011.²³ We estimated a radius of R_WD = 0.015 ${R}_{\odot }$, corresponding to M_WD = 0.6 ${M}_{\odot }$ for a 30,000 K WD (Parsons et al. 2017). The initial estimate for reddening was determined manually by calculating several spectra at various values of color excess while keeping the other initial conditions fixed, then visually comparing them with the observations. This yielded an initial estimate of E_B–V = 0.05. These initial estimates were then used to construct a multidimensional grid of 3150 sets of plausible starting conditions composed of all possible combinations of the parameters T_WD = 28,000–34,000 K in steps of 1000 K, log g = 7.6–8.4 in steps of 0.2 dex, reddening of 0.03–0.07 mag in steps of 0.02 mag, and normalization factor constructed from distance and WD radius values of d = 200–400 pc in steps of 50 pc and R_WD = 0.010–0.020 ${R}_{\odot }$ in steps of 0.002 ${R}_{\odot }$, respectively. The amoeba routine was allowed to crawl over the entire parameter space range of this grid (with boundary conditions in T_WD and log g that did not allow it to exceed the parameter space defined by the available grid of synthetic spectra).

The Hubble spectrum is composed of more than 10,000 individual data points. To prevent it from overwhelmingly dominating the χ² calculation and, therefore, driving the minimization over only a narrow range of wavelength, we restricted the spectral data to 13 composite points. These were calculated as the average flux densities in 5 Å wide bins centered around the following wavelengths: 0.117, 0.119, 0.1202, 0.1208, 0.1225, 0.1232, 0.1245, 0.127, 0.133, 0.1355, 0.138, 0.1405, and 0.143 $\mu {\rm{m}}$. These values were chosen to uniformly sample the entire wavelength range of the Hubble spectrum, with dense coverage in the wings of the Lyα line, which strongly constrains the temperature of the model (see inset panel in Figure 10). Conversely, we gave the three data points longward of 1 $\mu {\rm{m}}$ a total weighting equal to that of the subset of points at shorter wavelengths in the SED, to prevent the fit from undervaluing the IR photometry (which has sparser wavelength sampling).

After obtaining the grid of 3150 best-fit solutions, we compiled histograms for each parameter by counting the number of best-fit values in each of 3000 bins spanning the entire available range for that parameter. We then fit a Gaussian function to each histogram using the IDL routine gaussfit. The centers and standard deviations of the Gaussian fits are adopted as the overall best values for the fit parameters and their associated uncertainties. The final Gaussian fit was restricted to use only the histogram bins for data values within ±10σ of the central (peak) value determined from an initial Gaussian fit to the entire histogram. (In practice, the majority of the solutions were distributed in a few hundred adjacent bins.)

We then reran the amoeba fit using the parameter values determined in this fashion as starting values, and we confirmed that no better (i.e., lower χ²) solution was found. The best normalization factor (along with the log g and T_WD values) was disentangled into distance and WD radius as described above, which also gave a value for the WD mass. The R_WD uncertainty includes an additional 2% to account for the uncertainty in the mass–radius relation (Figure 10 in Parsons et al. 2017), which propagates into the uncertainty for the distance.

The best-fit parameters and uncertainties determined in this fashion are T_WD = 31,056(19) K, log g = 7.913(5), R_WD = 0.0142(4) ${R}_{\odot }$, M_WD = 0.603(2) ${M}_{\odot }$, d = 277(10) pc, and E_B−V = 0.0533(1). The expected reddening based on Pan-STARRS1 and 2MASS stellar photometry (Green et al. 2015, 2018)²⁴ at the position of BOKS 53856 for a distance of 280 pc is E_B − V = 0.03 ± 0.02 (from the current Bayestar17 version of the reddening map), which is consistent with our fit results.

The parallax of BOKS 53856 from Data Release 2 of Gaia (Gaia Collaboration et al. 2016, 2018) is π = 3.80(7) mas,²⁵ corresponding to a distance of d_π = 263(5) pc. The distance determined from our model fit, d, and d_π agree within 1σ. This level of agreement is achieved even without considering the likely ∼8%–12% underestimate of the parallax uncertainty for sources with G ≳ 16 mag (G is measured in the Gaia white light band; see Gaia Collaboration et al. 2018; G = 17.1 for BOKS 53856). Because even the upper limit estimate of the parallax uncertainty is <20%, we have obtained d_π and its uncertainty from a simple inversion of the parallax (Bailer-Jones 2015) rather than applying the Bayesian approach described by Luri et al. (2018). Assuming the nominal Gaia parallax distance for BOKS 53856 with the normalization factor determined from our model fit results in a 5% decrease in the corresponding WD radius (R_WD = 0.0135 ${R}_{\odot }$).

As already noted in Section 2.2, it is apparent from the UV spectrum of BOKS 53856 shown in Figure 2 that there is no strong (or at all apparent) signature of accreted photospheric metals. In Figure 11, we examine the time-resolved spectra of BOKS 53856 obtained from each of our four Hubble orbits. These correspond to sections of different brightness in the light curve of BOKS 53856 (see Figure 8). Although there are small differences between the spectra, these are at low significance: they possibly suggest the presence of weakly rotational phase-dependent spectral features, but no apparent metal lines (other than the previously identified ISM lines).

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Part of our motivation for proposing UV observations with Hubble was to follow up on a feature found in our ground-based optical spectra of BOKS 53856. Specifically, this is the very weak presence in some spectra of absorption from the Ca ii K line. Figure 12 shows representative samples of spectra from our Palomar Observatory data of 2010 September 18 that exhibit the presence (and absence) of this weak Ca absorption. The fact that it returns in multiple spectra at the same wavelength and with similar profile and strength, coupled with the ease with which its absence is apparent in comparison, suggests that this is a real feature, despite its admittedly low S/N.

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In addition, the appearance and disappearance of the weak Ca ii absorption repeats cyclically on the rotation period of the WD, over multiple cycles spanning several years. In Figure 13, we show an equivalent width (EW) curve of this line from our ground-based optical spectra. The EWs were measured manually using the "e" routine in the IRAF task splot, on spectra that had been normalized by dividing by a highly smoothed version of each spectrum. The Ca ii H line at 3968.5 Å could not be measured because it is entangled with the H i λ 3970.1 line, but it is possibly also present.

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The cyclic behavior of the EWs is also notable for having the largest values at rotational phases in the range ≈0.65–0.1. This range corresponds to the longitudes of the primary dark-spot regions on the WD surface inferred from the optical Kepler light curves (see Figure 9). This implies that the Ca ii absorption is both photospheric and preferentially confined to the dark-spot regions on the WD surface. Owing to its weakness relative to the quality of the currently available data, additional verification and characterization of this feature are desirable (see Section 4.6).

Our Spitzer observations effectively rule out the presence of an IR excess in BOKS 53856 out to λ ≈ 5 $\mu {\rm{m}}$ (see Figure 10). Dust grains are inefficient radiators at long wavelengths, which shifts the Wien peak of their thermal emission to shorter wavelengths for a given temperature relative to blackbody emission. Wien's displacement law for "graybody" dust grains (e.g., see Whittet 1992, Chapter 6, and Kruegel 2003, Chapters 5 and 8) is

$$\begin{eqnarray}&&{\lambda }_{\max }=\displaystyle \frac{b}{T}\displaystyle \frac{5}{(\beta +5)},\end{eqnarray} \tag{ 2 }$$

where λ_max is the wavelength of maximum emission (in $\mu {\rm{m}}$), b is Wien's constant (≈2898 $\mu {\rm{m}}$ K), and β is the exponent of the wavelength dependence of the emissivity (Q ∝ λ^−β). The value of β is typically in the range from 0 (blackbodies) to 1 (amorphous or layered lattice grains) or 2 (metallic or crystalline grains, typical of ISM dust grains); for example, see Draine & Lee (1984) and Tielens & Allamandola (1987).

If a dust disk is present, then it would have to be extremely paltry or contain very cool dust, with a disk inner-edge temperature ≲400–600 K (assuming that a significant Wien peak of thermal emission for dust at this temperature or higher would have been detected by our Spitzer observations). This would correspond to an inner-edge disk radius of ≳190–110 R_WD, respectively (see Section 3 in Hoard et al. 2013, and references cited therein). This is comfortably outside the tidal disruption radius for a typical WD (≈70–90 R_WD, assuming densities of 10⁶ g cm⁻³ for the WD and 3–5 g cm⁻³ for the parent body of the dust), implying that such a cool disk is unlikely to form from the tidal disruption of an asteroid or comet, as is usually assumed (Jura 2003). Of course, this does not exclude the possibility that cool dust with a different origin could be present at larger distances from the WD.

HH2011 suggested that BOKS 53856 might have a magnetic field of B ≈ 350 kG, based on perceived Zeeman splitting in the core of the Hα line. Figure 14 shows the Hα region of our optical spectra. The spectrum shown in the left panel (from 2010 September 17) at WD rotational phase ≈0.3 (second from the bottom) is the same spectrum presented by HH2011 in their Figure 7 to illustrate the presumed Zeeman splitting. In our time series spectra, the Hα profile shape appears to be more consistent with a variable-strength, and somewhat asymmetric, central emission peak in the Hα absorption line. This is especially apparent in the spectra from 2013 April 22, in which the emission peak is quite pronounced and reaches close to the continuum level. That said, we cannot entirely rule out the possibility of a very weak magnetic field in BOKS 53856, especially if it has a complex, nonuniform field geometry.

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BOKS 53856 is a moderately hot (31,000 K) DA WD located at a distance of 277 pc. Its light is slightly reddened (consistent with this distance), as indicated by a small color excess revealed by our modeling of its UV–IR SED. The most prominent features in its UV spectrum—other than Lyα—are low-excitation ISM lines.

BOKS 53856 displays a low-amplitude, periodic photometric variability that persisted, unchanged, for the duration of the Kepler mission. The origin of this variability is consistent with the presence of dark spots of varying intensity at several locations on the WD surface, viewed as the WD rotates. The most prominent spot coincides with the minimum brightness in the Kepler (optical) light curve.

It is estimated that 1%–3% of WDs currently have CDDs, while ≈20% show spectroscopic evidence for accreted metals with no detectable disk (e.g., Jura 2008; Farihi 2016). From their extensive COS survey of WDs, Koester et al. (2014) suggest that the latter fraction could actually be as high as ≈50%. These statistics are based on multiple studies of WDs cooler than about 25,000 K; however, at least a few hotter WDs have been identified as likely hosting CDDs (e.g., see Figure 5 in Hoard et al. 2013). Yet, as shown by Bonsor et al. (2017, their Figure 1), IR excesses denoting CDDs in WDs hotter than 25,000 K that are drawn from unbiased population samples are exceptionally rare (also see Xu & Jura 2012). Koester et al. (2014) note that no WDs with T_eff > 23,000 K having strong metal pollution are found, likely because the luminosity of these hotter WDs is sufficient to vaporize circumstellar dust grains.

Based on the conditions required to produce a region in a dynamically cold disk where small bodies are stable against sublimation and coagulation, von Hippel et al. (2007) have suggested that no WDs with temperatures above a critical value, T_WD > 32,000 K, can host CDDs. This result depends on a number of assumptions, such as dust grains that act like blackbodies and have a sublimation temperature of 2000 K, and CDDs that are optically thin. With our new temperature determination, BOKS 53856 does not quite exceed this limit; however, changing the assumptions about the dust grains can affect the value of the critical temperature. For example, at least one model for nonblackbody grains lowers the critical temperature (von Hippel et al. 2007), while higher sublimation temperatures (e.g., Rafikov & Garmilla 2012 and discussion in Hoard et al. 2013) would raise it. Additionally, if the CDD is not optically thin (in whole or in part), then a much higher critical temperature might be applicable.

Supported by both the lack of detectable photospheric metal absorption lines in its Hubble UV spectrum and the absence of an IR excess out to λ ≈ 5 $\mu {\rm{m}}$ in our Spitzer observations, we conclude that BOKS 53856 is apparently not encircled by—and accreting from—a prominent CDD produced by tidal disruption of an asteroid, comet, or other planetesimal (however, see below).

As described in Section 4.2, the fraction of WDs that lack an IR excess but still have detectable metal contamination exceeds, by a factor of ≳10, the fraction of WDs that have an IR excess indicating the presence of a CDD. The former objects are possibly previous hosts of CDDs, but are near the end of these transient events (lifetimes of CDDs around WDs are ∼10⁴–10⁶ years; Girven et al. 2012). Little or no dust is left in the disk (not enough to produce a measurable signal in the IR), or collisions and sublimation in the disk have rendered the remaining dust into optically thin metallic gas (i.e., the pollution mechanism proposed for WDs with T_eff > 20,000 K by Bonsor et al. 2017). The remaining material continues to trickle onto the WD, producing only a low level of observable metal contamination.

In the absence of ongoing accretion, processes such as winds, convective mixing, and thermal and ordinary diffusion can influence the presence of metals in the atmosphere of a DA WD, but gravitational settling and radiative levitation (for T_eff ≳ 20,000 K) predominate (e.g., see reviews in Fontaine & Wesemael 1987, 1991). The interplay between these processes determines the overall diffusion timescale (τ_D) that governs how long metals remain observable at the WD surface.

To first order, the strength of these processes is a function of WD effective temperature. At all temperatures, the overall diffusion timescale is more rapid than the evolutionary (cooling) timescale for a given DA WD. Koester & Wilken (2006, their Table 2 and Figure 2) show the expected diffusion timescale for Ca in a log g = 8 DA WD, which ranges from about a week at T_eff ≈ 24,000 K to τ_D ≈ 200,000 years at T_eff ≈ 5000 K (also see Koester 2009). After decreasing slightly as the WD cools from hotter temperatures, τ_D increases dramatically (by more than four orders of magnitude) as the WD cools in the interval T_eff ≈ 12,500 K (τ_D ≈ 5 days) to T_eff ≈ 10,000 K (τ_D ≈ 25 years), due to the increase in the depth of the convective zone in this temperature range. Extrapolating the Koester & Wilken (2006) calculations of Ca diffusion timescales to a 30,000 K, log g = 8 WD suggests a value of τ_D ≈ 10–15 days for BOKS 53856.

Meanwhile, the equilibrium abundance of Ca in the presence of radiative levitation for a 30,000 K DA WD is [Ca/H] ≲ −10 (Chayer et al. 1995), more than four orders of magnitude less than the cosmic (ISM) abundance. The sample of metal-enriched DA WDs observed by Koester & Wilken (2006) in the T_eff = 25,000–30,000 K range (≈30 WDs) had Ca abundances of [Ca/H] ≈ −6 to −4, respectively (which is typically less than a factor of 10 higher than their detection limit in that temperature range).

This situation—the presence of Ca in numerous T_eff ≈ 30,000 K DA WDs, including BOKS 53856—requires there be some ongoing mechanism for accretion or retention of Ca in order for it to be observable at all. Yet, BOKS 53856 shows neither of the tell-tale signatures of accretion from a CDD (i.e., IR excess and UV metal lines) at detectable levels. Hallakoun et al. (2018) show that the upper-limit abundances defined by nondetections of Si and C in spectra of the WD sample of Koester et al. (2014) are comparable to the actual detections in more sensitive spectra, suggesting that a large fraction—even up to 100%—of WDs might display photospheric metals if observed at high enough sensitivity. The lack of corroborating features in its UV spectrum could indicate that BOKS 53856 is at the extreme end of a CDD event, in which the disk material is all but depleted (and the remaining material has likely been sublimated into gas; e.g., see Gänsicke et al. 2006; Bonsor et al. 2017) and accretion onto the WD occurs at a very low rate. As suggested by Hallakoun et al. (2018), the correspondingly weak signatures of accreted metals in BOKS 53856—other than the Ca reported here—might be revealed by spectroscopic observations of suitably higher sensitivity.

Jura (2008) suggested that a likely path for metal pollution in WDs is via accretion of multiple small asteroids: the first disrupted asteroid creates a disk around the WD, while subsequent asteroids encounter the disk (on a timescale of every few hundred years) and are effectively vaporized. This produces a circumstellar environment containing a large amount of gas, little dust, and a correspondingly negligible IR excess. This scenario differs subtly, but importantly, from the one in which the CDD is created by the tidal disruption of a single, larger asteroid or even planet. The multiple-small-asteroids scenario was supported by Debes et al. (2012), who observed weak (EW ≲ 10 mÅ) circumstellar Ca ii H and K absorption lines in optical spectra of the cooler (T_eff = 9420 K) DA WD, WD 1124−293, spanning an interval of several years. A stronger (tens of mÅ) photospheric Ca ii absorption component is also present. Like BOKS 53856, WD 1124–293 does not have a detectable IR excess (out to 8 $\mu {\rm{m}}$; Farihi et al. 2008).

An alternative scenario for accretion of metals by some WDs was proposed by Koester & Wilken (2006). They suggested that some DA WDs showing very low levels of metal pollution (and no evidence for CDDs) might be accreting at a steady state from the ISM due to their space motion through locally enriched regions ("clouds"), that is, via Bondi accretion (Bondi 1952). In particular, they suggest this possibility to explain the presence of Ca in WDs. Barstow et al. (2014), however, reiterate the potential problems with ISM accretion (e.g., multispecies abundance discrepancies; also see Kilic & Redfield 2007). They suggest from their extensive analysis of far-UV spectra of metal-polluted WDs that the hot WDs (T_eff > 20,000 K) are accreting from circumstellar gas derived from tidally disrupted, and subsequently sublimated, exoplanetary material. As noted above, this seems to be a likely scenario for BOKS 53856 as well.

Turning to the question of magnetism, we find that, when considered en masse, the time series spectra of BOKS 53856 do not strongly support the presence of previously reported Zeeman splitting indicative of a magnetic field (HH2011). Rather, there appears to be a variable emission component in the Hα line core; however, its origin is somewhat mysterious. It could be related to emission from nonuniformly distributed, optically thin gas in a depleted CDD remnant (possibly hydrogen dissociated from water that was present in the source body of the circumstellar material; Farihi et al. 2011, 2013; Raddi et al. 2015; Fusillo et al. 2017). Alternatively, the emission component might result from non-LTE effects in the WD atmosphere (e.g., Lanz & Hubeny 1995), but these would have to be nonhomogeneous over the WD surface to explain the variability of the emission.

Although we cannot confirm the presence of a magnetic field in BOKS 53856 based on the Hα line profiles, we also cannot rule out the existence of a magnetic field of tens or even hundreds of kilogauss. This is especially true if the field geometry is complex. We are reluctant to draw a direct analogy between the dark spots inferred for BOKS 53856 and the sunspots/starspots that form on the Sun and other main-sequence stars due to magnetic activity. The extreme differences in photospheric and convective depth, gravitational gradient, and so on, between WDs and main-sequence stars suggest against making a casual assumption that spots on the former have the same origin as spots on the latter (also see the discussion by Maoz et al. 2015 of cold or hot spots as the origin of variability on WDs). That said, it is difficult to conceptualize a scenario for spot formation on a WD that does not involve magnetism. If this is the case, then possibly the small amount of accreted metals that contaminate the WD photosphere are magnetically confined to the spots (e.g., via enhanced magnetic levitation that slows their diffusion out of the photosphere). The decreased brightness in the spots might then be due to enhanced opacity from the metals.

Alternatively, the entire WD surface might be uniformly contaminated with Ca, but a localized temperature decrease in the spots (but why?) produces conditions that strengthen the Ca ii absorption features. Normally, the Ca ii H and K lines are strongest at T ≈ 4000–5000 K and become extremely weak above 10,000 K. A simple estimation using the Saha equation suggests that the Ca ii absorption strength increases by ∼10% when the stellar atmosphere temperature changes from 31,000 K to 25,000 K. Yet, this small effect might be enough to make the Ca ii lines go from being undetectable over most of the (hot) surface of the WD, to weakly detectable in the dark (cool) spots.

We basically have a "chicken or the egg" situation with regard to the origin of the dark spots, with two potential scenarios that could operate separately or in combination. In the first scenario, accreted metals are confined and preserved in particular regions on the WD surface (likely via magnetism), and the localized increase in opacity creates the dark spots. In the second, the spots are formed as regions of intrinsically lower temperature and brightness (again, likely via magnetism, as in sunspots), which coincidentally enhance the absorption strength of accreted metals in the WD photosphere.

Either scenario would explain why the Ca ii absorption line is strongest (or present at all) primarily at rotational phases when the dark spots are most directly viewed. In essence, this combines the suggestions made by HH2011, namely, that the photometric modulation could result from surface temperature variations caused by either frozen-in magnetic poles or spots or localized photospheric chemical abundance enhancements. We suggest that these two mechanisms likely work hand-in-hand.

Although additionally supported and refined by our investigation of BOKS 53856, the idea of magnetically confined metals on the surface of a WD producing photometric variability is not original to this work. As described above, HH2011 already mentioned this possibility for BOKS 53856, and there are a number of other similar WDs in the literature. The common presence of photometric variability (typically attributed to spots) in WDs with strong (from a few to hundreds of megagauss) magnetic fields has been demonstrated (Brinkworth et al. 2013; Valeev et al. 2017). BOKS 53856 and similar objects could extend this correlation to the weakest (kilogauss and below) WD magnetic fields.

Maoz et al. (2015) and Hallakoun et al. (2018) have explored a small sample of 14 WDs similar to BOKS 53856 that were observed by Kepler and found to have low-amplitude variability. In particular, Hallakoun et al. (2018) detected photospheric metals in Hubble UV spectra of four out of seven of these WDs, but were unable to reject their null hypothesis that the variability is unrelated to debris accretion. Instead, they conclude that some other source of surface inhomogeneity on the WD—possibly related to magnetism—must be at work. Table 4 contains a summary of the physical characteristics of notable examples of similar objects (i.e., weakly magnetic DA WDs displaying periodic photometric variability linked to the rotation of the WD); we discuss several of them in the remainder of this section.

Table 4.  Comparison of Similar Objects

Object	TWD	MWD	log g	Prot	Aa	$\langle | B| \rangle$ b	$\langle {B}_{{\rm{z}}}\rangle$ c	References
	(K)	(${M}_{\odot }$)		(d)	(%)	(kG)	(kG)
BOKS 53856	31,000	0.60	7.91	0.2557	5, 10 (UV)	< < 350	⋯	(1), (2)
WD 0009+501 (G 217-037)	6540	0.73	8.23	0.334:	2–3	150–300	−120–50	(3), (4), (5), (6), (7)
WD 0257+080 (G 76–48)	6620	0.39	7.78	0.869:	4.5	≤200	⋯	(6), (8), (9)
SDSS J152934.98+292801.9	11,580	1.02	8.65	0.02639	6	<70	⋯	(10)
WD 1953−011 (G 92-40)	7765	0.79	8.20	1.4418	2	≈70	⋯	(6), (9), (11), (12)
WD 2047+372 (G 210-36)	14,710	0.81	8.31	0.2432	<2:	60	−12–15	(5), (13)
WD 2111+498 (GD 394)	38,500	0.55	7.90	1.150	25 (EUV)	<12	<20	(3), (14), (15)
WD 2359−434 (GJ 915)	8650	0.78	8.29	0.1125	1	50–100	2–4	(5), (13), (16), (17)

Notes. (1) HH2011, (2) this work, (3) Schmidt & Smith (1995), (4) Valyavin et al. (2005), (5) Giammichele et al. (2012), (6) Brinkworth et al. (2013), (7) Valeev et al. (2015), (8) Lawrie et al. (2013), (9) Bédard et al. (2017), (10) Kilic et al. (2015), (11) Maxted et al. (2000), (12) Brinkworth et al. (2005), (13) Landstreet et al. (2017), (14) Barstow et al. (1996), (15) Dupuis et al. (2000), (16) Aznar Cuadrado et al. (2004), (17) Gary et al. (2013).

^aApproximate variability amplitude (minimum to maximum) in the optical (except where indicated). ^bMean surface magnetic field averaged over the visible hemisphere, typically measured from spectral line splitting due to the Zeeman effect; also denoted as B_s (Landstreet 1992). ^cMean line-of-sight magnetic field, typically measured from the circular polarization signature of the Zeeman effect; also denoted as B_ℓ, the mean longitudinal field, when weighted by cos θ, where θ is the angle between the line of sight and a normal to the WD surface, or as B_e, the effective field, when weighted by limb darkening (Landstreet 1992).

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4.4.1. SDSS J152934.98+292801.9 (and WD 1145+017)

4.4.2. GD 394

Like BOKS 53856, GD 394 is a relatively hot WD, but Bruhweiler & Kondo (1983), Dupuis et al. (2000), and others noted that GD 394 displays strong lines from accreted metals (primarily Si) in its UV and optical spectra. Dupuis et al. (2000) suggest that "episodic accretion is the likely source of heavy elements, with the accreted material concentrated on a spot by an hypothetical magnetic field." Yet, Mullally et al. (2007) found no evidence for an IR excess in the SED of GD 394, using Spitzer observations out to 8 $\mu {\rm{m}}$, and Dickinson et al. (2012) did not detect evidence for the presence of nonphotospheric (i.e., circumstellar) metals in a sensitive investigation of line velocities in archival UV spectra. Burleigh et al. (2010, 2011) even ruled out the presence of a circumstellar gas disk around GD 394 for some ionic species (Ca ii, Fe ii, Si ii).

In their analysis of GD 394, Dupuis et al. (2000) note that the model of accreted metals magnetically confined in a dark spot on the WD surface predicts antiphased variability in the UV and optical light curves due to flux redistribution. There is some indication of this effect in the Hubble (UV) and Kepler (optical) light curves of BOKS 53856 (see Figure 8). Shallow dips in the optical light curve at rotational phases 0.3–0.4 and 0.6–0.7 correspond to local peaks in the UV light curve, and the optical light curve maximum at phase 0.5 corresponds to an inferred dip in the UV light curve (between Hubble orbits 2 and 3).

To better illustrate this, Figure 15 shows the ratio of the Hubble and Kepler light curves (after scaling the latter to match the amplitude of the former). Regions in which the ratio has a slope near zero indicate that the two light curves are varying in phase, while regions with nonzero slope (regardless of sign) are antiphased. For a more quantitative assessment, we calculated the linear Pearson correlation coefficient of the two light curves in each of the four segments (which are delineated by the four Hubble orbits). For segments 1–4, the correlation coefficients are 0.96, −0.56, −0.90, and 0.98, respectively. This confirms that the two light curves are strongly correlated (varying in phase) in segments 1 and 4, while they are anticorrelated (varying in antiphase) in segments 2 and 3.

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4.4.3. WD 2359−434

It is possible that we confront a new class of WDs that has been revealed through the confluence of the availability of high-precision time series photometry and the contemporaneous growth of our understanding of accretion of circumstellar matter onto WDs enabled by recent UV and IR observational capabilities. Observationally, they present as WDs displaying low-amplitude (a few percent), periodic (on the WD rotation period), but not strictly sinusoidal, photometric variability. They lack an IR signature of a CDD, yet show spectroscopic evidence for accreted metals at levels ranging from strong to almost undetectable.

A member of this class has an inferred (but possibly not directly detected) kilogauss-scale magnetic field with complex geometry producing a nonuniform surface temperature distribution (i.e., dark spots), possibly via opacity effects caused by localized containment of accreted metals. These objects support the suggestion made by Wickramasinghe & Ferrario (2000) that WD magnetic field complexity and strength might not be correlated (i.e., even very low-strength magnetic fields can have a highly complex geometry conducive to the presence of multiple long-lasting spots on the WD surface).

Approximately one-half of a sample of 14 WDs (not including BOKS 53856) that were observed during the original Kepler mission show low-amplitude variability (Maoz et al. 2015). By making reasonable assumptions about the number of nearby, weakly magnetic, variable WDs and achievable ground-based photometric precision, Kilic et al. (2015, and references cited therein) suggest that the Large Synoptic Survey Telescope might discover another ∼10⁵ members of this class.

We end with a wish list of tasks that would help to further untangle the still poorly understood nature of BOKS 53856 and similar objects. First, additional monitoring of the Ca ii absorption could irrefutably confirm both its existence and correlation with the rotation period variability of the WD. Second, more sensitive IR and/or UV observations could set more stringent limits on the (non)detection of the tell-tale signs of CDDs and accretion of metals. Finally, polarimetric observations can be aimed at settling the question of whether or not BOKS 53856 has a magnetic field and, if so, its strength and field morphology. As with the lingering unanswered questions of many observational projects, the logistically challenging scenario of more time with larger telescopes is called for. In this case, the absence of Kepler to continue monitoring the small-amplitude photometric variability of BOKS 53856 presents an impediment to the analysis of future observations.