THE ELM SURVEY. VII. ORBITAL PROPERTIES OF LOW-MASS WHITE DWARF BINARIES^*

We acquire spectra for the 15 ELM WD candidates using the Blue Channel spectrograph (Schmidt et al. 1989) on the 6.5 m MMT telescope. We configured the Blue Channel spectrograph to obtain 3650–4500 Å spectral coverage with 1.0 Å spectral resolution. We acquire additional spectra for 5 objects using the KOSMOS spectrograph (Martini et al. 2014) on the Kitt Peak National Observatory 4 m Mayall telescope on program numbers 2014B-0119 and 2015A-0082. We configured the KOSMOS spectrograph to obtain 3500–6200 Å spectral coverage with 2.0 Å spectral resolution. We also acquire spectra for objects with $g\lt 17$ mag using the FAST spectrograph (Fabricant et al. 1998) on the Fred Lawrence Whipple Observatory 1.5 m Tillinghast telescope. We configured the FAST spectrograph to obtain 3500–5500 Å spectral coverage with 1.7 Å spectral resolution.

All spectra were paired with comparison lamp exposures for accurate wavelength calibration. Flux calibration is performed with observations of blue spectrophotometric standards (Massey et al. 1988). We measure radial velocities using the RVSAO cross-correlation program (Kurtz & Mink 1998) with high signal-to-noise templates. Exposure times were chosen to yield a signal-to-noise ratio of 10–15 per resolution element, resulting in typical radial velocity errors of 10–15 km s⁻¹ for these WDs.

Our observing strategy is to acquire a single spectrum per target, identify candidate ELM WDs from the initial spectra, then re-observe candidate ELM WDs in subsequent observing runs. If a candidate showed velocity variability, we continue to observe the object until we constrain its orbital solution. If a candidate shows no velocity variability but is confirmed to have $5\lt \mathrm{log}g\lt 7$, we continue to observe it until we can rule out all $P\lt 1$ day period aliases at high confidence. As a result, each object has around two dozen irregularly spaced observations obtained over a baseline of a few years. The spectra published here were mostly acquired between 2013 January and 2015 October. Figure 2 presents the Balmer line profiles from the summed, rest-frame spectra for our 15 objects. The individual radial velocities are provided in a data table presented in the Appendix.

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We derive the stellar atmosphere parameters for each WD by fitting the summed, rest-frame spectra to a grid of pure hydrogen atmosphere models as described in Gianninas et al. (2011, 2014, 2015). In brief, the models cover 4000$\;{\rm{K}}\lt {T}_{{\rm{eff}}}\;\lt$ 35,000 K and $4.5\lt \mathrm{log}g\lt 9.5$ and include the Stark broadening profiles from Tremblay & Bergeron (2009). We also apply the Tremblay et al. (2013, 2015) three-dimensional stellar atmosphere model corrections to fix the so-called "$\mathrm{log}g$ problem." These corrections can lower the one-dimensional stellar atmosphere model parameters by up to 500 K in ${T}_{{\rm{eff}}}$ and 0.3 dex in $\mathrm{log}g$. Our statistical uncertainties are typically ±180 K in ${T}_{{\rm{eff}}}$ and ±0.07 dex in $\mathrm{log}g$.

The best-fit models are over-plotted on each spectrum in Figure 2. We mask the region around the Ca ii K line when doing the Balmer line fits. Twelve of our new WDs have $5\lt \mathrm{log}g\lt 7$ and are thus ELM WDs. Two objects, J0125+2017 and J1128+1743, have anomalously low surface gravities $\mathrm{log}g\simeq 4.7$ consistent with metal-poor, young, main-sequence stars, however, they both show short-period orbital motion and are probable ELM WDs (see Section 3.2). The last object, J1039+1645, is a slightly more massive $\mathrm{log}g\;=\;7.64$ WD that also shows orbital motion. Table 1 presents the stellar atmosphere parameters for all 15 objects.

Table 1.  WD Physical Parameters

Object	R.A.	decl.	${T}_{{\rm{eff}}}$	$\mathrm{log}g$	Mass	g0	Mg	dhelio
	(h:m:s)	(d:m:s)	(K)	(cm s−2)	(M⊙)	(mag)	(mag)	(kpc)
J0125+2017	1:25:16.758	20:17:44.63	11170 ± 200	4.709 ± 0.062	0.184 ± 0.010	17.153 ± 0.021	4.88 ± 0.18	2.855 ± 0.238
J0806−0905	8:06:53.308	−9:05:11.51	8510 ± 130	5.116 ± 0.104	0.166 ± 0.011	17.681 ± 0.006	6.88 ± 0.22	1.454 ± 0.145
J0940+6304	9:40:08.729	63:04:27.40	12910 ± 210	5.964 ± 0.050	0.180 ± 0.010	19.618 ± 0.026	7.69 ± 0.13	2.436 ± 0.150
J1017+1217	10:17:07.109	12:17:57.42	8330 ± 130	5.528 ± 0.062	0.142 ± 0.012	17.479 ± 0.020	8.13 ± 0.23	0.745 ± 0.083
J1039+1645	10:39:53.118	16:45:24.28	14310 ± 240	7.639 ± 0.046	0.458 ± 0.018	18.941 ± 0.024	10.71 ± 0.08	0.444 ± 0.018
J1128+1743	11:28:23.334	17:43:54.58	11260 ± 210	4.756 ± 0.063	0.183 ± 0.010	19.434 ± 0.026	4.99 ± 0.18	7.774 ± 0.656
J1130+0933	11:30:27.956	9:33:03.55	12020 ± 210	5.062 ± 0.057	0.179 ± 0.010	16.843 ± 0.029	5.62 ± 0.16	1.761 ± 0.133
J1241+0633	12:41:24.291	6:33:51.02	11280 ± 170	6.648 ± 0.047	0.199 ± 0.012	17.722 ± 0.021	9.64 ± 0.09	0.413 ± 0.018
J1355+1956	13:55:12.336	19:56:45.43	8050 ± 120	6.101 ± 0.064	0.156 ± 0.010	16.098 ± 0.024	9.76 ± 0.17	0.185 ± 0.014
J1518+1354	15:18:02.566	13:54:31.96	8080 ± 120	5.435 ± 0.071	0.147 ± 0.018	18.988 ± 0.019	8.00 ± 0.34	1.594 ± 0.246
J1555+2444	15:55:02.000	24:44:22.05	18170 ± 310	6.296 ± 0.051	0.190 ± 0.012	16.045 ± 0.015	7.82 ± 0.12	0.443 ± 0.025
J1631+0605	16:31:23.675	6:05:33.82	10150 ± 170	5.818 ± 0.069	0.162 ± 0.010	19.002 ± 0.019	8.08 ± 0.17	1.533 ± 0.118
J1735+2134	17:35:21.694	21:34:40.64	7940 ± 130	5.758 ± 0.081	0.142 ± 0.010	15.904 ± 0.011	9.06 ± 0.21	0.235 ± 0.023
J2139+2227	21:39:07.415	22:27:08.87	7990 ± 130	5.932 ± 0.121	0.149 ± 0.011	15.600 ± 0.011	9.42 ± 0.29	0.174 ± 0.023
J2309+2603	23:09:19.904	26:03:46.69	10950 ± 160	6.127 ± 0.057	0.176 ± 0.010	18.986 ± 0.016	8.54 ± 0.14	1.229 ± 0.080

Note. ${T}_{{\rm{eff}}}$ and $\mathrm{log}g$ values are corrected for 3D effects following Tremblay et al. (2015). Mass and M_g values are estimated with Althaus et al. (2013) models.

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Figure 3 plots the distribution of ${T}_{{\rm{eff}}}$ and $\mathrm{log}g$ for the ELM Survey objects. Our ${(g-r)}_{0}$ target selection results in an approximate temperature selection of $8000{\rm{K}}\lt {T}_{{\rm{eff}}}\;\lt$ 22,000 K. The observed range of surface gravity reflects our choice to follow-up those objects with initial surface gravity estimates $5\lesssim \mathrm{log}g\lesssim 7$. The clump of new objects around 8000 K is from the non-variable objects.

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We validate our stellar atmosphere parameters by comparing the best spectroscopic model for each object against photometry from GALEX (Martin et al. 2005), SDSS (Alam et al. 2015), 2MASS (Skrutskie et al. 2006), UKIDSS (Lawrence et al. 2007), and WISE (Wright et al. 2010). We do not derive parameters from the spectral energy distributions because many of the fainter ELM WDs lack ultraviolet and infrared photometry. Instead, we compare available photometry to the best spectroscopic fit as a consistency check. We generally find excellent agreement between the models and the observed spectral energy distributions. J1241+0633, for example, has a reduced ${\chi }^{2}$ of 1.15 (see Figure 4). A notable exception is J1555+2444, the first WD + M dwarf in the ELM Survey, which has a reduced ${\chi }^{2}$ of 181. WD + M dwarf binaries are common (e.g., Rebassa-Mansergas et al. 2012), however our ugri color selection is designed to exclude such systems. In the case of J1555+2444, the M dwarf contributes significant flux in the near-infrared bands (see Figure 4). We discuss this system further below.

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We estimate WD mass and luminosity using the recent ELM WD evolutionary tracks of Althaus et al. (2013) and of Istrate et al. (2014). Both sets of tracks adopt neutron star companions for their evolutionary calculations, but an ELM WD's evolution should be unaffected by its companion once it detaches from the common envelope. The choice of progenitor metallicity, on the other hand, can have a significant impact on the hydrogen envelope mass and resulting cooling times (Althaus et al. 2015). Both sets of ELM WD tracks assume solar metallicity appropriate for disk objects; our ELM Survey sample contains both disk and halo objects.

For reference, we plot fiducial 0.17 M_⊙ and 0.24 M_⊙ ELM WD tracks in Figures 1 and 3. We draw Althaus et al. (2013) tracks in magenta and Istrate et al. (2014) tracks in cyan. In Figure 1, we plot estimated synthetic colors to compare with our color selection. In Figure 3, we plot the published tracks, and use thin lines to indicate the short-lived parts of the tracks and thick lines to indicate the long-lived parts of the tracks. Shell flashes, which generate loops and lead to faster evolution, are present in the 0.24 M_⊙ tracks and absent in the 0.17 M_⊙ tracks.

We interpolate the evolutionary tracks to determine the mass and luminosity for each of our objects, however the interpolation is complicated by the fact that the tracks map phase space in a discrete and irregular way. Our solution is to identify the two nearest tracks to an observed ${T}_{{\rm{eff}}}$ and $\mathrm{log}g$ value, and interpolate between those two tracks on the basis of $\mathrm{log}g$. We then Monte Carlo our ${T}_{{\rm{eff}}}$ and $\mathrm{log}g$ with their uncertainties to derive the mass and luminosity uncertainties.

The evolutionary tracks of Althaus et al. (2013) and Istrate et al. (2014) yield very similar mass and luminosity estimates for our objects. Consider the ELM Survey WDs in the mass range over which the two sets of tracks overlap: $0.16\lt M\lt 0.30$ M_⊙. For these 66 objects, the two sets of tracks differ on average by 0.007 M_⊙ (systematic) ±0.012 M_⊙ (statistical) in mass and 0.044 mag (systematic) ±0.068 mag (statistical) in absolute g-band magnitude M_g. Istrate et al. (2014) tracks yield slightly more massive and more luminous estimates, likely due to the lack of gravitational settling in the models resulting in mixed H/He atmospheres. In any case, Althaus et al. (2013) tracks span a wider range of mass and so are more convenient to use with the entire ELM Survey sample. We adopt Althaus et al. (2013) tracks for the following discussion, and add 0.01 M_⊙ in quadrature to the formal mass uncertainty to account for the uncertainty in the choice of model. Table 1 presents the parameters, including the de-reddened apparent g-band magnitudes and distances, for all 15 WDs.

Before determining orbital elements, we must first separate the radial velocity variable and non-variable objects. We use the F-test to quantify how the variance of the radial velocities around a constant mean velocity compares to the variance around a best-fit orbital solution (e.g., Bevington & Robinson 1992). The larger the F-test probability, the more likely the orbital solution is consistent with noise. Six of the ELM WDs published here are significantly non-variable, with F-test probabilities $\gt 0.01$ and typical 95% confidence upper-limits on semi-amplitude of $k\lt 30$ km s⁻¹. We discuss the non-variable ELM WDs further below.

Nine WDs have significant velocity variability, with F-test probabilities $\lt 0.002$. We calculate orbital elements for these nine WDs in the same way as in prior ELM Survey papers. In brief, we use the summed spectra as cross-correlation templates to maximize the velocity precision for each individual object, and then minimize ${\chi }^{2}$ for a circular orbit using the code of Kenyon & Garcia (1986). We search for orbital periods up to 5 days, which is the maximum time span of our individual observing runs and a period at which a 0.6 M_⊙ WD companion (assuming $i\;=\;60^\circ$) produces a detectable k = 75 km s⁻¹. Figure 5 shows periodograms and phased radial velocity plots for the nine WDs. We estimate the significance of the period aliases using ${\chi }^{2}$ values. For normally distributed errors, a ${\rm{\Delta }}{\chi }^{2}\;=\;13.3$ with respect to the minimum value corresponds to a 99% confidence interval for 4 degrees of freedom (Press et al. 1992). On this basis, J1130+0933 has significant period aliases around 2 days, but we consider these aliases uninteresting given that they are longer than the best 1.6 day period. For all other objects, the period aliases have ${\rm{\Delta }}{\chi }^{2}\;=\;$ 15–20 larger than the minimum and thus appear insignificant. Table 2 presents the orbital elements. The full ELM Survey sample of 88 objects is provided in a data table presented in the Appendix.

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Table 2.  Binary Orbital Parameters

Object	${N}_{{\rm{obs}}}$	P	k	γ	M2	τ
		(days)	(km s−1)	(km s−1)	(M⊙)	(Gyr)
J0125+2017	16	0.88758 ± 0.00004	65.4 ± 2.1	79.4 ± 2.2	>0.14	<938.3
J0806−0905	16	...	$\lt 28.5$	109.4 ± 3.7	...	...
J0940+6304	20	0.48438 ± 0.00001	210.4 ± 3.2	32.6 ± 2.3	>0.73	<51.27
J1017+1217	14	...	$\lt 30.2$	24.8 ± 3.6	...	...
J1039+1645	26	0.82470 ± 0.02214	83.4 ± 4.0	−32.9 ± 2.8	>0.31	<186.2
J1128+1743	21	2.16489 ± 0.03889	41.2 ± 2.0	−24.5 ± 1.4	>0.11	<12350
J1130+0933	16	1.55935 ± 0.00014	69.0 ± 3.9	−145.6 ± 1.8	>0.19	<3241
J1241+0633	40	0.95912 ± 0.00028	138.2 ± 4.8	90.0 ± 3.3	>0.51	<377.6
J1355+1956	33	...	$\lt 40.9$	−41.8 ± 3.9	...	...
J1518+1354	16	0.57655 ± 0.00734	112.7 ± 4.6	−43.4 ± 3.0	>0.23	<237.6
J1555+2444	20	...	$\lt 23.0$	1.0 ± 3.2	...	...
J1631+0605	9	0.24776 ± 0.00411	215.4 ± 3.4	−9.6 ± 3.7	>0.47	<13.15
J1735+2134	20	...	$\lt 31.6$	−16.3 ± 3.4	...	...
J2139+2227	37	...	$\lt 22.0$	8.5 ± 2.7	...	...
J2309+2603	15	0.07653 ± 0.00001	405.8 ± 3.5	−14.8 ± 2.6	>0.79	<0.359

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Over the magnitude range $15\lt {g}_{0}\lt 20$, we have obtained spectra for 99% of the stars in the HVS Survey target selection region (Brown et al. 2012b) and 80% of the stars in the ELM Survey target selection region (Figure 1). In absolute numbers, 120 of 589 ELM Survey targets remain unobserved. Almost all of the 120 unobserved targets are located between R.A. = 21 hr and R.A. = 3 hr due to telescope time allocation and weather. Based on current detection rates, we estimate that 26% of the remaining targets are likely DA WDs, of which one third (or about 10) are likely ELM WD binaries.

Our multi-epoch follow-up spectroscopy is less complete. We currently have 26 ELM WD candidates that require confirming observations, and another 17 velocity variable ELM WDs that have poorly constrained orbital parameters. Summing these numbers together with the unobserved targets suggests there may be another $\simeq 53$ ELM WD binaries in our target selection regions. Our published sample currently contains 76 binaries. We therefore estimate that the published ELM Survey sample is approximately 60% complete for ELM WD binaries.

SDSS J230919.904+260346.69 (hereafter J2309+2603) is a 0.176 M_⊙ ELM WD with a 1.8367 ± 0.0002 hr orbital period and a 412.4 ± 2.7 km s⁻¹ semi-amplitude. Given Kepler's 3rd Law, the minimum mass companion to this ELM WD is a 0.82 M_⊙ object at an orbital separation of 0.76 R_⊙. If we adopt the mean inclination angle for a random stellar sample, $i\;=\;60^\circ$, the companion is a 1.14 M_⊙ object at an orbital separation of 0.83 R_⊙. There is no evidence for mass-transfer in this system. Thus the ELM WD's most likely companion is another WD.

There remains a 20% likelihood, assuming a random distribution of inclinations, that the companion is a $1.4\lt {M}_{2}\lt 3.0$ M_⊙ neutron star. Given that this putative neutron star would have accreted material during the common envelope evolution of the ELM WD progenitor, J2309+2603 is possibly a millisecond pulsar binary system. However no gamma-ray source exists at this position in the Fermi Large Area Telescope source catalog (Acero et al. 2015). A targeted Chandra search of other high-probability candidates in the ELM Survey has also found no neutron star companions (Kilic et al. 2014b). We conclude that J2309+2603 is most likely a double WD binary with a total binary mass near the Chandrasekhar mass.

J2309+2603 is losing energy and angular momentum to gravitational wave radiation. The timescale for the binary to shrink and begin mass transfer via Roche-lobe overflow is given by the gravitational wave merger time

$$\begin{eqnarray}&&\tau \;=\;47925\displaystyle \frac{{({M}_{1}+{M}_{2})}^{1/3}}{{M}_{1}{M}_{2}}{P}^{8/3}\;{\rm{Myr}},\end{eqnarray} \tag{ 1 }$$

where the masses are in M_⊙ and the period P is in days (Kraft et al. 1962). The merger time is 350 Myr for the minimum mass companion, and shorter if the companion is more massive.

The other eight double degenerate binaries are longer orbital period, lower semi-amplitude systems (see Figure 5). The radial velocity constraints vary by object, and there is little to say about the best-measured system. J0940+6304 has complete phase coverage and excellent constraints on period and semi-amplitude.

J1241+0633 and J1518+1354 have excellent phase coverage and orbital period constraints, but increased semi-amplitude uncertainties due to a relative lack of observations near velocity minimum. J1631+0605 is a $P\;=\;5.946\pm 0.099\;{\rm{hr}}$ binary with a well-constrained period and semi-amplitude—observations were obtained at quadrature on back-to-back nights—however incomplete phase coverage allows for the possibility of a modestly ($e\leqslant 0.3$) eccentric orbit. Given that none of the other 75 binaries in the ELM Survey have significant eccentricity, nor do we expect much eccentricity from the common envelope origin of the ELM WD, we adopt the circular orbit solution. J1039+1645, J1128+1743, and J1130+0933, by comparison, have excellent phase coverage but exhibit period aliases clumped around their best-fit periods. Given that the observations rule out significantly different orbital periods for each object, we consider the 3% uncertainty in period acceptable. Systems with $P\gt 16\;{\rm{hr}}$ orbital periods have $\gt 100\;{\rm{Gyr}}$ merger times. The median likelihood for a neutron star companion in each of the eight longer-period binaries is 4%.

We note that J0125+2017 and J1128+1743 are abnormally low surface gravity objects, with $\mathrm{log}g\simeq 4.7$, ${T}_{{\rm{eff}}}\;\simeq$ 11,200 K, and masses around 0.19 M_⊙ if they are ELM WDs. Their low $k\simeq 50$ km s⁻¹ semi-amplitudes imply that the unseen companions are comparable 0.2 M_⊙ mass objects at 2.5 R_⊙ orbital separations (assuming $i\;=\;60^\circ$). Alternatively, it is possible these objects are very metal-poor, young, main-sequence stars. Padova tracks show that a 1.6 M_⊙, Z = 0.0001 star has similar ${T}_{{\rm{eff}}}$ and $\mathrm{log}g$ at $\lt 100\;{\rm{Myr}}$ ages (Bressan et al. 2012). If J0125+2017 and J1128+1743 are very metal-poor, main-sequence stars, their orbital motion would be due to 0.4 M_⊙ mass companions at 4.5 R_⊙ orbital separations. Very metal-poor, main-sequence stars should not be forming near the Sun, but blue stragglers are possible (e.g., Brown et al. 2008). However, observed blue stragglers have typical orbital periods of ∼1000 days (Geller & Mathieu 2012; Geller et al. 2015). Because J0125+2017 and J1128+1743 have orbital periods of 21 hr and 52 hr, respectively, we believe they are most likely ELM WD binaries.

J1555+2444 is a possible ELM WD + M dwarf system. The M dwarf is classified as an 0.43 M_⊙ M2 star (Rebassa-Mansergas et al. 2010; Schreiber et al. 2010). Assuming that the M2 star dominates the z-band flux and has absolute magnitude ${M}_{z}\;=\;+8.0$ (Bressan et al. 2012), its distance is about 0.51 kpc. This is consistent with the 0.44 kpc and 0.49 kpc distance estimates to the hot ${T}_{{\rm{eff}}}$ = 18,170 K ELM WD using Althaus et al. (2013) and Istrate et al. (2014) tracks, respectively. Thus J1555+2444 could be a legitimate WD + M dwarf system.

The ELM WD shows no orbital motion, however. We constrain its orbital semi-amplitude to have a 95% confidence upper limit of $k\lt 23$ km s⁻¹ on the basis of 6 epochs of radial velocities obtained over 2 years. This result is in tension with the work of Nebot Gómez-Morán et al. (2011), who find that the period distribution of WD + main-sequence binaries peaks around $P\;=\;10.3\;{\rm{hr}}$. The orbital period of J1555+2444 would have to be 3 months, assuming $i\;=\;60^\circ$, to be consistent with the observed semi-amplitude limit. Alternatively, the orbital inclination would have to be $i\lt 8^\circ$, assuming $P\;=\;10.3\;{\rm{hr}}$, to be consistent with the observed semi-amplitude limit, an inclination that has a 1% likelihood in a random distribution.

An ELM WD + M dwarf binary is also unlikely from a stellar evolutionary standpoint. The standard ELM WD formation scenario requires double common-envelope evolution in which the companion star evolves first. Other possibilities are that this is a triple system in which the M dwarf is an outer third, or else a chance super-position of an M dwarf and ELM WD. J1555+2444 thus shares some similarities to J0935+4411, the 20 minutes orbital period double WD binary that appears to have an M dwarf along the line of sight (Kilic et al. 2014a). Additional observational constraints are needed to understand the nature of this object.

Given that ELM WDs are the likely end product of compact binary evolution, the discovery of J1555+2444 and the five other non-variable objects is surprising. An obvious question is whether our observations have missed potential companions. We explore the sensitivity of our radial velocities to different possible orbital periods and semi-amplitudes following the procedure described by Brown et al. (2013): we Monte-Carlo the observed velocities with their errors, and count the number of 10,000 orbital fits that have F-test probabilities $\lt 0.01$ for orbital periods between 0.1 and 2 days.

Figure 6 presents the averaged result for the six non-variable ELM WD candidates. The lines in Figure 6 represent the probability of detecting k = 50, 75, and 100 km s⁻¹ orbits as a function of P given our set of observations. The objects have 6 or 7 epochs of time-series spectroscopy and a total of a couple dozen observations each. Figure 6 shows that while our ground-based observations are naturally less sensitive around 12 and 24 hr orbital periods, we are sensitive to almost any orbit with $k\gt 75$ km s⁻¹. On average, our observations should detect 95% of binary systems with k = 75 km s⁻¹ and 99% of binary systems with k = 100 km s⁻¹ out to P = 2 day orbital periods. Yet our observations show that the non-variable systems have a typical 95% confidence upper limit of $k\lt 30$ km s⁻¹.

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While we expect to find some binaries in face-on orientations, it is hard to explain all of the non-variable ELM WD candidates in this way. Restricting ourselves to the 78 best ELM WD candidates in our Survey, those objects with $\mathrm{log}g\lt 7.15$, we find 67 binaries and 11 non-variable objects. The median semi-amplitude of the ELM WD binaries is k = 220 km s⁻¹, a semi-amplitude that would appear $\lt 75$ km s⁻¹ for $i\lt 20^\circ$ inclinations. Assuming a random distribution of inclinations, only 5 of the 78 objects with $\mathrm{log}g\lt 7.15$ should have $i\lt 20^\circ$. Yet 11 of the 78 objects are non-variable. If we adopt the uncertainty as $\sqrt{N}$, the excess of non-variable objects appears significant at the 2.5-σ level. This excess suggests that the non-variable ELM WD candidates in our sample may not all be face-on binary systems.

To avoid possible bias in relying on observed semi-amplitudes, we can instead assume a companion mass distribution M₂ and ask what fraction of binaries should fall below our detection threshold. If we adopt the lognormal orbital period distribution (allowing $0\lt P\lt \infty$) and the normal M₂ distribution derived below (Section 4) and assume a random distribution of inclinations, 6.5 of 78 binaries should have $k\lt 75$ km s⁻¹. This number is in 2-σ disagreement with the number of observed non-variable ELM WD candidates. While not formally significant, it is suggestive that Maxted et al. (2000) find a similar excess of non-variable objects in their sample of $\lt 0.5$ M_⊙ low-mass WDs. Thus our non-variable objects are possibly binaries with long orbital periods, or they are possibly something else altogether.

Notably, 8 of the 11 non-variable ELM WD candidates are found clumped together around ${T}_{{\rm{eff}}}\;\simeq \;$ 8000 K at the cool edge of our Survey. Kepler et al. (2016) recently identified thousands of similar "sdA" stars with ${T}_{{\rm{eff}}}\;\sim$ 8000 K and $\mathrm{log}g\sim 6$ in the SDSS spectroscopic catalog. The non-variable ELM candidates are possibly linked to these sdA stars, however the sdA stars cannot be ELM WDs on the basis of their numbers. According to the evolutionary tracks, an ELM WD spends about as much time with $7500\lt {T}_{{\rm{eff}}}\;\lt$ 8500 K as it spends at $8500\;{\rm{K}}\lt {T}_{{\rm{eff}}}\;\lt$ 22,000 K. Shell flashes can cause the time spent in these temperature ranges to differ by a factor of 2, but not by a factor of 10. The models also show that 8000 K ELM WDs are about 3 mag fainter in M_g than 12,000 K ELM WDs. Taken together, a magnitude-limited survey should observe fewer ELM WDs at 8000 K than at higher temperatures. Yet the sdA stars of Kepler et al. (2016) are over 10 times more abundant than our higher-${T}_{{\rm{eff}}}$ ELM WD binaries in the same footprint of sky. We conclude that the sdA stars are probably unrelated to ELM WDs.

Our 8 non-variable objects with ${T}_{{\rm{eff}}}\;\simeq \;$ 8000 K are possibly related to the sdA stars. If true, this would explain why we find so many non-variable objects around 8000 K. The three hotter non-variable objects would then be consistent at the 1σ level with the number of ELM WDs we would expect to find in face-on binaries in our color-selected survey. Given the ambiguity about the nature of the non-variable objects, we focus the remainder of this paper on the ELM WD binaries.

We begin by defining a clean sample of ELM WD binaries, and then use that sample to derive the orbital period and secondary mass distributions of the sample. We construct a clean sample of ELM WD binaries by first taking the 88 published ELM Survey objects and excluding the 12 non-variable objects. We further restrict the sample to those binaries with $k\gt 75$ km s⁻¹, a semi-amplitude threshold above which our catalog should be 95% complete (Figure 6). The ELM Survey color selection provides a built-in temperature selection, but the $\mathrm{log}g$ selection is more ambiguous. Tremblay et al. (2015) 3D corrections push some of the WD surface gravities below $\mathrm{log}g\;=\;5$. We have also obtained observations for every WD near $\mathrm{log}g\simeq 7$ during the course of our survey. Since we would like to maximize our number statistics, we choose to restrict our sample to $4.85\lt \mathrm{log}g\lt 7.15$, a range over which the ELM Survey observations should be reasonably complete. Taken together, these cuts leave us with a clean sample of 62 ELM WD binaries. We present the clean sample in Table 3 sorted by orbital period.

Table 3.  Clean Sample of ELM WD Binaries

Object	P	k	M1	${M}_{1}/{M}_{2}$	${M}_{{\rm{tot}}}$	τ
	(days)	(km s−1)	(M⊙)		(M⊙)	(Gyr)
0651+2844	0.00886 ± 0.00001	616.9 ± 5.0	0.247 ± 0.015	${0.50}_{-0.01}^{+0.01}$	${0.74}_{-0.01}^{+0.01}$	${0.001}_{-0.0001}^{+0.0001}$
0935+4411	0.01375 ± 0.00051	198.5 ± 3.2	0.313 ± 0.019	${0.42}_{-0.10}^{+0.20}$	${1.07}_{-0.24}^{+0.25}$	${0.002}_{-0.0004}^{+0.0008}$
0106−1000	0.02715 ± 0.00002	395.2 ± 3.6	0.189 ± 0.011	${0.33}_{-0.09}^{+0.05}$	${0.75}_{-0.07}^{+0.21}$	${0.027}_{-0.006}^{+0.003}$
1630+4233	0.02766 ± 0.00004	295.9 ± 4.9	0.298 ± 0.019	${0.39}_{-0.10}^{+0.17}$	${1.06}_{-0.23}^{+0.24}$	${0.015}_{-0.003}^{+0.005}$
1053+5200	0.04256 ± 0.00002	264.0 ± 2.0	0.204 ± 0.012	${0.27}_{-0.06}^{+0.12}$	${0.97}_{-0.23}^{+0.23}$	${0.068}_{-0.012}^{+0.021}$
0056−0611	0.04338 ± 0.00002	376.9 ± 2.4	0.180 ± 0.010	${0.22}_{-0.03}^{+0.03}$	${1.00}_{-0.10}^{+0.13}$	${0.076}_{-0.008}^{+0.008}$
1056+6536	0.04351 ± 0.00103	267.5 ± 7.4	0.334 ± 0.016	${0.44}_{-0.10}^{+0.18}$	${1.10}_{-0.22}^{+0.23}$	${0.045}_{-0.008}^{+0.014}$
0923+3028	0.04495 ± 0.00049	296.0 ± 3.0	0.275 ± 0.015	${0.36}_{-0.09}^{+0.14}$	${1.04}_{-0.22}^{+0.24}$	${0.059}_{-0.011}^{+0.017}$
1436+5010	0.04580 ± 0.00010	347.4 ± 8.9	0.234 ± 0.013	${0.30}_{-0.07}^{+0.10}$	${1.02}_{-0.20}^{+0.23}$	${0.071}_{-0.012}^{+0.017}$
0825+1152	0.05819 ± 0.00001	319.4 ± 2.7	0.279 ± 0.021	${0.35}_{-0.08}^{+0.11}$	${1.07}_{-0.19}^{+0.23}$	${0.113}_{-0.019}^{+0.026}$
1741+6526	0.06111 ± 0.00001	508.0 ± 4.0	0.170 ± 0.010	${0.14}_{-0.01}^{+0.00}$	${1.34}_{-0.04}^{+0.07}$	${0.154}_{-0.006}^{+0.004}$
0755+4906	0.06302 ± 0.00213	438.0 ± 5.0	0.184 ± 0.010	${0.19}_{-0.03}^{+0.02}$	${1.15}_{-0.10}^{+0.17}$	${0.178}_{-0.019}^{+0.015}$
2338−2052	0.07644 ± 0.00712	133.4 ± 7.5	0.258 ± 0.016	${0.34}_{-0.09}^{+0.16}$	${1.01}_{-0.24}^{+0.25}$	${0.261}_{-0.049}^{+0.089}$
2309+2603	0.07653 ± 0.00001	405.8 ± 3.5	0.176 ± 0.010	${0.19}_{-0.03}^{+0.02}$	${1.11}_{-0.10}^{+0.17}$	${0.317}_{-0.036}^{+0.029}$
0849+0445	0.07870 ± 0.00010	366.9 ± 4.7	0.179 ± 0.010	${0.21}_{-0.04}^{+0.04}$	${1.04}_{-0.14}^{+0.19}$	${0.359}_{-0.048}^{+0.049}$
0751−0141	0.08001 ± 0.00279	432.6 ± 2.3	0.194 ± 0.010	${0.20}_{-0.00}^{+0.00}$	${1.17}_{-0.01}^{+0.01}$	${0.317}_{-0.004}^{+0.003}$
2119−0018	0.08677 ± 0.00004	383.0 ± 4.0	0.159 ± 0.010	${0.19}_{-0.03}^{+0.01}$	${0.99}_{-0.05}^{+0.14}$	${0.530}_{-0.053}^{+0.024}$
1234−0228	0.09143 ± 0.00400	94.0 ± 2.3	0.227 ± 0.014	${0.30}_{-0.07}^{+0.14}$	${0.97}_{-0.24}^{+0.24}$	${0.478}_{-0.091}^{+0.160}$
1054−2121	0.10439 ± 0.00655	261.1 ± 7.1	0.178 ± 0.011	${0.23}_{-0.05}^{+0.09}$	${0.95}_{-0.21}^{+0.23}$	${0.829}_{-0.141}^{+0.217}$
0745+1949	0.11240 ± 0.00833	108.7 ± 2.9	0.164 ± 0.010	${1.11}_{-0.77}^{+0.27}$	${0.31}_{-0.03}^{+0.33}$	${3.95}_{-2.41}^{+0.81}$
1108+1512	0.12310 ± 0.00867	256.2 ± 3.7	0.179 ± 0.010	${0.23}_{-0.05}^{+0.08}$	${0.96}_{-0.21}^{+0.22}$	${1.27}_{-0.21}^{+0.32}$
0112+1835	0.14698 ± 0.00003	295.3 ± 2.0	0.160 ± 0.010	${0.22}_{-0.04}^{+0.02}$	${0.90}_{-0.05}^{+0.15}$	${2.35}_{-0.29}^{+0.13}$
1233+1602	0.15090 ± 0.00009	336.0 ± 4.0	0.169 ± 0.010	${0.17}_{-0.02}^{+0.02}$	${1.15}_{-0.09}^{+0.16}$	${1.96}_{-0.20}^{+0.15}$
1130+3855	0.15652 ± 0.00001	284.0 ± 4.9	0.288 ± 0.018	${0.32}_{-0.06}^{+0.05}$	${1.18}_{-0.12}^{+0.19}$	${1.40}_{-0.19}^{+0.18}$
1112+1117	0.17248 ± 0.00001	116.2 ± 2.8	0.176 ± 0.010	${0.23}_{-0.06}^{+0.11}$	${0.93}_{-0.24}^{+0.24}$	${3.25}_{-0.59}^{+1.06}$
1005+3550	0.17652 ± 0.00011	143.0 ± 2.3	0.168 ± 0.010	${0.22}_{-0.05}^{+0.10}$	${0.92}_{-0.23}^{+0.24}$	${3.62}_{-0.66}^{+1.15}$
0818+3536	0.18315 ± 0.02110	170.0 ± 5.0	0.165 ± 0.010	${0.22}_{-0.05}^{+0.10}$	${0.92}_{-0.23}^{+0.24}$	${4.05}_{-0.73}^{+1.26}$
1443+1509	0.19053 ± 0.02402	306.7 ± 3.0	0.201 ± 0.013	${0.20}_{-0.03}^{+0.02}$	${1.20}_{-0.09}^{+0.16}$	${3.06}_{-0.31}^{+0.27}$
1840+6423	0.19130 ± 0.00005	272.0 ± 2.0	0.182 ± 0.011	${0.21}_{-0.04}^{+0.04}$	${1.04}_{-0.14}^{+0.20}$	${3.76}_{-0.52}^{+0.52}$
2103−0027	0.20308 ± 0.00023	281.0 ± 3.2	0.161 ± 0.010	${0.18}_{-0.03}^{+0.03}$	${1.05}_{-0.13}^{+0.19}$	${4.85}_{-0.61}^{+0.57}$
1238+1946	0.22275 ± 0.00009	258.6 ± 2.5	0.210 ± 0.011	${0.24}_{-0.04}^{+0.04}$	${1.08}_{-0.13}^{+0.19}$	${4.91}_{-0.66}^{+0.62}$
1249+2626	0.22906 ± 0.00112	191.6 ± 3.9	0.160 ± 0.010	${0.21}_{-0.05}^{+0.08}$	${0.92}_{-0.22}^{+0.23}$	${7.51}_{-1.31}^{+2.09}$
0345+1748	0.23503 ± 0.00013	273.4 ± 0.5	0.218 ± 0.012	${0.27}_{-0.01}^{+0.01}$	${1.02}_{-0.02}^{+0.02}$	${5.80}_{-0.21}^{+0.23}$
0822+2753	0.24400 ± 0.00020	271.1 ± 9.0	0.191 ± 0.012	${0.21}_{-0.03}^{+0.03}$	${1.12}_{-0.11}^{+0.17}$	${6.52}_{-0.77}^{+0.63}$
1631+0605	0.24776 ± 0.00411	215.4 ± 3.4	0.162 ± 0.010	${0.21}_{-0.05}^{+0.07}$	${0.95}_{-0.19}^{+0.22}$	${8.94}_{-1.46}^{+2.05}$
1526+0543	0.25039 ± 0.00001	231.9 ± 2.3	0.161 ± 0.010	${0.20}_{-0.04}^{+0.05}$	${0.97}_{-0.17}^{+0.22}$	${9.04}_{-1.41}^{+1.72}$
2132+0754	0.25056 ± 0.00002	297.3 ± 3.0	0.187 ± 0.010	${0.17}_{-0.02}^{+0.01}$	${1.26}_{-0.07}^{+0.13}$	${6.43}_{-0.52}^{+0.33}$
1141+3850	0.25958 ± 0.00005	265.8 ± 3.5	0.177 ± 0.010	${0.19}_{-0.03}^{+0.03}$	${1.10}_{-0.11}^{+0.18}$	${8.29}_{-0.97}^{+0.82}$
1630+2712	0.27646 ± 0.00002	218.0 ± 5.0	0.170 ± 0.010	${0.21}_{-0.05}^{+0.06}$	${0.97}_{-0.17}^{+0.22}$	${11.3}_{-1.8}^{+2.2}$
1449+1717	0.29075 ± 0.00001	228.5 ± 3.2	0.171 ± 0.010	${0.21}_{-0.04}^{+0.05}$	${1.00}_{-0.15}^{+0.20}$	${12.5}_{-1.8}^{+2.0}$
0917+4638	0.31642 ± 0.00002	148.8 ± 2.0	0.173 ± 0.010	${0.23}_{-0.05}^{+0.10}$	${0.94}_{-0.23}^{+0.23}$	${16.5}_{-2.8}^{+5.0}$
0152+0749	0.32288 ± 0.00014	217.0 ± 2.0	0.169 ± 0.010	${0.20}_{-0.04}^{+0.05}$	${1.00}_{-0.16}^{+0.21}$	${16.8}_{-2.5}^{+3.0}$
1422+4352	0.37930 ± 0.01123	176.0 ± 6.0	0.181 ± 0.010	${0.23}_{-0.05}^{+0.08}$	${0.96}_{-0.21}^{+0.22}$	${25.2}_{-4.3}^{+6.4}$
1617+1310	0.41124 ± 0.00086	210.1 ± 2.8	0.172 ± 0.010	${0.20}_{-0.04}^{+0.04}$	${1.02}_{-0.14}^{+0.20}$	${30.8}_{-4.3}^{+4.4}$
1538+0252	0.41915 ± 0.00295	227.6 ± 4.9	0.168 ± 0.010	${0.18}_{-0.03}^{+0.03}$	${1.09}_{-0.11}^{+0.18}$	${31.4}_{-3.7}^{+3.1}$
1439+1002	0.43741 ± 0.00169	174.0 ± 2.0	0.181 ± 0.010	${0.23}_{-0.05}^{+0.08}$	${0.97}_{-0.19}^{+0.22}$	${36.8}_{-6.0}^{+8.6}$
0837+6648	0.46329 ± 0.00005	150.3 ± 3.0	0.181 ± 0.010	${0.24}_{-0.06}^{+0.09}$	${0.95}_{-0.22}^{+0.23}$	${43.7}_{-7.7}^{+12.0}$
0940+6304	0.48438 ± 0.00001	210.4 ± 3.2	0.180 ± 0.010	${0.20}_{-0.03}^{+0.03}$	${1.08}_{-0.12}^{+0.18}$	${43.9}_{-5.4}^{+4.8}$
0840+1527	0.52155 ± 0.00474	84.8 ± 3.1	0.192 ± 0.010	${0.25}_{-0.06}^{+0.12}$	${0.95}_{-0.24}^{+0.24}$	${57.5}_{-10.7}^{+18.2}$
0802−0955	0.54687 ± 0.00455	176.5 ± 4.5	0.198 ± 0.012	${0.24}_{-0.05}^{+0.06}$	${1.02}_{-0.17}^{+0.22}$	${59.3}_{-9.1}^{+11.0}$
1518+1354	0.57655 ± 0.00734	112.7 ± 4.6	0.147 ± 0.018	${0.19}_{-0.05}^{+0.09}$	${0.90}_{-0.24}^{+0.26}$	${97.4}_{-19.4}^{+31.0}$
2151+1614	0.59152 ± 0.00008	163.3 ± 3.1	0.181 ± 0.010	${0.23}_{-0.05}^{+0.07}$	${0.97}_{-0.18}^{+0.22}$	${81.7}_{-13.3}^{+17.3}$
1512+2615	0.59999 ± 0.02348	115.0 ± 4.0	0.250 ± 0.014	${0.33}_{-0.08}^{+0.14}$	${1.01}_{-0.22}^{+0.24}$	${65.1}_{-12.1}^{+19.7}$
1518+0658	0.60935 ± 0.00004	172.0 ± 2.0	0.224 ± 0.013	${0.27}_{-0.05}^{+0.06}$	${1.06}_{-0.15}^{+0.21}$	${70.0}_{-10.4}^{+11.4}$
0756+6704	0.61781 ± 0.00002	204.2 ± 1.6	0.182 ± 0.011	${0.19}_{-0.03}^{+0.02}$	${1.14}_{-0.10}^{+0.17}$	${79.8}_{-8.7}^{+7.1}$
1151+5858	0.66902 ± 0.00070	175.7 ± 5.9	0.186 ± 0.011	${0.22}_{-0.04}^{+0.04}$	${1.03}_{-0.14}^{+0.19}$	${105}_{-15}^{+15}$
0730+1703	0.69770 ± 0.05427	122.8 ± 4.3	0.182 ± 0.010	${0.24}_{-0.06}^{+0.10}$	${0.94}_{-0.22}^{+0.23}$	${130}_{-23}^{+38}$
0308+5140	0.80590 ± 0.00109	78.9 ± 2.7	0.149 ± 0.015	${0.20}_{-0.05}^{+0.09}$	${0.90}_{-0.24}^{+0.25}$	${233}_{-45}^{+75}$
0811+0225	0.82194 ± 0.00049	220.7 ± 2.5	0.179 ± 0.010	${0.14}_{-0.01}^{+0.01}$	${1.45}_{-0.05}^{+0.10}$	${141}_{-7}^{+4}$
1241+0633	0.95912 ± 0.00028	138.2 ± 4.8	0.199 ± 0.012	${0.25}_{-0.05}^{+0.07}$	${1.00}_{-0.18}^{+0.22}$	${270}_{-44}^{+58}$
2236+2232	1.01016 ± 0.00005	119.9 ± 2.0	0.186 ± 0.010	${0.24}_{-0.06}^{+0.09}$	${0.96}_{-0.21}^{+0.23}$	${338}_{-58}^{+91}$
0815+2309	1.07357 ± 0.00018	131.7 ± 2.6	0.200 ± 0.021	${0.25}_{-0.06}^{+0.08}$	${1.00}_{-0.19}^{+0.24}$	${367}_{-66}^{+85}$

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Next, we consider the orbital period distribution of the clean sample, plotted in Figure 7. The median orbital period of the clean sample is 0.226 days, or 5.4 hr.

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We use the non-parametric Anderson–Darling test to evaluate the goodness of fit between a model distribution and the observations (Scholz & Stephens 1987). The p-value of the Anderson–Darling test is the probability that the model and observations share a common distribution. A p-value of zero rejects the null hypothesis that the model and observations share a common distribution.

We fit different functional forms to the period distribution and find that the observations are best matched by a lognormal distribution,

$$\begin{eqnarray}&&{f}_{P}(P;{\mu }_{P},{\sigma }_{P}^{2})\;=\;\displaystyle \frac{1}{P\sqrt{2\pi }{\sigma }_{P}}{\rm{exp}}\left(-\displaystyle \frac{{(\mathrm{ln}P-{\mu }_{P})}^{2}}{2{\sigma }_{P}^{2}}\right),\end{eqnarray} \tag{ 2 }$$

where P is the period, ${\mu }_{P}$ is the lognormal mean, and ${\sigma }_{P}$ is the standard deviation. We restrict the fit to $P\lt 1.1$ day to match the absence of orbital periods greater than $\simeq 1$ day in the clean sample. The green lines in Figure 7 show the best fit for lognormal mean ${\mu }_{P}\;=\;-1.28$ and standard deviation ${\sigma }_{P}\;=\;1.34$. The Anderson–Darling p-value is 0.999, indicating a strong consistency between the model and the observations.

We now derive the (unseen) secondary mass distribution by considering the distribution of observables. Formally, the observed orbital period P, semi-amplitude k, and derived ELM WD mass M₁ are related to the secondary mass M₂ and orbital inclination i by the binary mass function,

$$\begin{eqnarray}&&\displaystyle \frac{{{Pk}}^{3}}{2\pi G}\;=\;\displaystyle \frac{{({M}_{2}\mathrm{sin}i)}^{3}}{{({M}_{1}+{M}_{2})}^{2}}.\end{eqnarray} \tag{ 3 }$$

Given our surface gravity constraints, the ELM WD mass distribution M₁ is narrow and centered around ∼0.2 M_⊙. Thus most of the information about M₂ is encoded in the semi-amplitude distribution. Inclinations of individual binaries are unknown, however the target selection is known: the ELM WDs were targeted by color. Thus we will assume that the inclination distribution is random in $\mathrm{sin}i$ above our detection threshold.

Our computational approach is to Monte Carlo different trial distributions of M₂. We draw inclination from a random $\mathrm{sin}i$ distribution and P from the observed lognormal distribution. We clip the simulations to match the observational $k\gt 75$ km s⁻¹ sample limit, a threshold that translates into an $i\gtrsim 20^\circ$ limit in inclination. We then use the Anderson–Darling test to quantify how likely the simulated and observed distributions of semi-amplitude share a common distribution.

Figure 8 presents three trial M₂ distributions and their corresponding simulated k distributions compared to the observations. At the top is the flat companion mass distribution observed in sdB binaries (Kupfer et al. 2015). The flat M₂ distribution produces a k distribution that clearly differs from the observations (p = 0.0012). In the middle is the WD mass distribution observed by SDSS (Kepler et al. 2007). The SDSS WD mass distribution produces a k distribution that agrees somewhat better with the observations (p = 0.028). Finally, we present what we find to be the best match: a normal distribution with mean ${\mu }_{M2}\;=\;0.76$ M_⊙ and standard deviation ${\sigma }_{M2}\;=\;0.25$ M_⊙. This normal distribution produces a k distribution that agrees with the clean sample of ELM WD binaries with p = 0.240.

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Andrews et al. (2014) derive essentially the same normal distribution using an earlier version of the ELM Survey data and a different Bayesian modeling approach. Boffin (2015) also finds a similar result using an earlier version of the ELM Survey data and an inversion technique. This agreement provides us with confidence in the normal distribution result. Clearly, to fit the observed distribution of k requires a large fraction of relatively massive WD companions.

With the M₂ distribution in hand, we can go back and constrain the mass ratio, total mass, and merger time of the individual ELM WD binaries. For some of the systems we have additional constraints on inclination: two systems are eclipsing (Brown et al. 2011b; Kilic et al. 2014b), eight have ellipsoidal variations (Kilic et al. 2011c; Hermes et al. 2012a, 2014), and many more have X-ray and/or radio observations that rule out massive neutron star companions (Kilic et al. 2011a, 2012, 2014b). Combining this information with our measurements of P, k, and M₁, we calculate the distributions of ${M}_{1}/{M}_{2}$, ${M}_{{\rm{tot}}}$, and τ for each system. The results are presented in Table 3. The columns list the median value of each distribution; the uncertainty comes from the 0.1587 and 0.8413 percentile values, equivalent to 1σ uncertainties if the quantities are normally distributed.

We find that the average total mass of the ELM WD binaries is 1.01 ± 0.15 M_⊙. Statistically speaking, 95% of the ELM WD binaries have a total mass below the Chandrasekhar mass and thus are not type Ia supernova progenitors. This is unsurprising given the mass of the ELM WDs. However, their short orbital periods and relatively massive binary companions yield a median gravitational wave merger time of 5.8 Gyr for the sample.

The outcome of the ELM WD binary merger process depends in part on the stability of mass transfer. The ELM WD has the largest diameter of two objects in a double-degenerate binary and thus will become the donor star, the star that will fill its Roche lobe and will transfer matter to its companion when the binary evolves into contact.

A clear implication of the M₂ distribution is that the binary components have fairly extreme mass ratios, on average M₁:${M}_{2}\;=\;1$:${3.9}_{-0.8}^{+1.2}$. For this mass ratio, theorists predict that helium mass transfer rates from the donor WD should be below the rate of stable helium burning on the accretor WD (Marsh et al. 2004; Kaplan et al. 2012; Kremer et al. 2015). Thus mass transfer should proceed stably over billion year timescales, and the ELM WD binaries will become AM CVn systems (Warner 1995; Solheim 2010). However, this statement ignores an ELM WD's hydrogen envelope. If the initial hydrogen mass transfer rate is super-Eddington, a common envelope may form and cause the two WDs to quickly merge into a single massive WD (Webbink 1984; Han & Webbink 1999; Dan et al. 2011). Material ejected during shell flashes may also form a common envelope and result in a merger (Shen 2015).

Interestingly, a small peak around 1 M_⊙ is seen in the mass distribution of all WDs within 20 pc (Giammichele et al. 2012). The same peak is also seen in the mass distribution of WDs in the SDSS WD catalog (e.g., Kepler et al. 2007, 2015). Rebassa-Mansergas et al. (2015) argue that the $\simeq 1$ M_⊙ peak in the WD mass distribution is more substantial than suggested in these earlier surveys. Perhaps part of the 1 M_⊙ peak in the WD mass distribution is related to the ELM WD binaries. The merger of a He+CO WD should eliminate the hydrogen atmosphere and look like an extreme helium star or R CrB star (Paczyński 1971). Whether the ELM WD binaries evolve into stable mass-transfer AM CVn systems or single massive WDs remains unclear, but it is fair to say that the binary mass ratio favors the AM CVn outcome.