A GRAVITATIONAL REDSHIFT DETERMINATION OF THE MEAN MASS OF WHITE DWARFS: DBA AND DB STARS

The helium-atmosphere WDs in our sample are derived from the SPY objects analyzed by Voss et al. (2007). Out of their 38 DBAs with atmospheric parameter determinations, we exclude one magnetic WD, one DO, and 12 cool WDs with T_eff ≲ 16,500 K. For these 12, Voss et al. (2007) fix the surface gravity at log g = 8 in order to obtain spectral fits; this forfeits our ability to properly compare these stars with our gravitational redshift results. Due to data quality, we are unable to satisfactorily measure apparent velocities (v_app) of Hα for an additional eight, which leaves us with 16 DBAs. Keep in mind that for these objects, the atmospheric abundance of hydrogen is small and Hα is often barely visible.

Out of the 31 measured DBs from Voss et al. (2007), we exclude nine cool WDs and two known to have cool companions (Zuckerman & Becklin 1992; Farihi et al. 2005; Hoard et al. 2007). We end up with 20 normal DBs and 36 helium-atmosphere WDs overall.

Four of our DBAs show Ca ii lines in their spectra, giving them the additional classification as DBAZ (Koester et al. 2005b; Voss et al. 2007), and two DBs are classified as DBZs (Voss et al. 2007). Since the SPY Collaboration has not reported any of our WDs to have stellar companions, we presume them to not be in close binary systems, though two of our DBAs (WD 0948+013 and WD 2154−437) and two of our DBs (WD 0615−591, WD 0845−188) are in common proper motion or wide binary systems (Luyten 1949; Wegner 1973; Caballero 2009). The lack of detectable Zeeman splitting implies that our targets also do not harbor significant (≳100 kG) magnetic fields (e.g., Koester et al. 2009).

The WDs in our sample have not been classified in the literature as potential members of the thick disk or halo stellar populations. We therefore make the assumption that all of these WDs belong to the thin disk—a necessary assumption for our analysis (see Section 3.3). Nearly all (>90%) of the SPY WDs that have been studied kinematically are classified as thin disk objects (Pauli et al. 2006; Richter et al. 2007).

In the weak-field limit, the general relativistic effect of gravitational redshift can be observed as a velocity shift in absorption lines and is expressed as

$$\begin{equation} v_g=\frac{c\Delta \lambda }{\lambda }=\frac{GM}{Rc}, \end{equation} \tag{ 1 }$$

where G is the gravitational constant and c is the speed of light. In our case, M is the WD mass and R is the WD radius.

The apparent velocity of an absorption line is the sum of this gravitational redshift and the stellar radial velocity: v_app = v_g + v_r. These two components cannot be explicitly separated for individual WDs without an independent v_r measurement or mass determination. We break this degeneracy for a group of WDs by assuming that the sample is comoving and local. After we correct each v_app to the local standard of rest (LSR), only random stellar motions dominate the dynamics of our sample. These average out, and the mean apparent velocity becomes the mean gravitational redshift: 〈v_app〉 = 〈v_g〉.

Collisional (Stark) broadening effects cause asymmetry in the wings of absorption lines for the hydrogen Balmer series, making it difficult to measure a velocity centroid (Shipman & Mehan 1976; Grabowski et al. 1987). However, these effects are not significant in the sharp, non-LTE Hα and Hβ Balmer line cores. We use both of these line cores in Falcon et al. (2010). In the SPY spectra of DBAs, though, we do not observe the non-LTE line cores, but the Hα line centers are still distinct. For Hα the pressure shift should remain very small (<1 km s⁻¹) within a few Å of the line center (Grabowski et al. 1987).

Reid (1996) measures the gravitational redshift of the DBA WD 0437+138 using the line center since he observes no sharp line core. While cautioning that pressure effects may be significant, he determines a high mass of 0.748 ± 0.037 M_☉. Bergeron et al. (2011) determine a spectroscopic mass of 0.74 ± 0.06 M_☉ for this WD. For at least this one case, then, gravitational redshifts of the line center give a result consistent with the spectroscopic method.

van der Waals broadening due to neutral helium may also significantly affect line shapes below T_eff = 16,500 K (Koester et al. 2005a; Voss et al. 2007). We avoid this systematic by not including targets in that range of T_eff.

We measure v_app for each DBA in our sample by fitting a Gaussian profile to the Hα line center using GAUSSFIT, a nonlinear least-squares fitting routine in IDL. If multiple epochs of observation exist, we combine the measurements as a mean weighted according to the uncertainties returned by the fitting routine. Eight out of our 16 DBAs have two epochs of observation. Table 1 lists these measurements for the DBA sample including those for each epoch of observation and the final adopted values.

By inspecting the v_app measurements between epochs, we note that they differ by more than a typical measurement uncertainty; the mean difference between epochs of the same target is 9.20 km s⁻¹ while the mean measurement uncertainty for all observations is 3.05 km s⁻¹. Since we presume none of our targets to be in close binary systems (Section 2.1), reflex orbital motion cannot be the culprit, and therefore it is evident that we are underestimating our individual measurement uncertainties. To more accurately represent our adopted δv_app values for each target, we multiply all observation δv_app values by the factor we find above: the ratio of the mean difference in apparent velocity between epochs and the mean apparent velocity measurement uncertainty. For the DBA sample, this factor is 3.02. The final adopted values for δv_app listed in Column 3 in Table 1 include this adjustment, and the observation values in Column 8 are the values before the adjustment.

For the DBAs and the DBs, we also measure v_app using the He i 5876 Å line and list the results in Tables 2 and 3, respectively. Here, too, we apply a multiplicative factor to all observation δv_app values to better represent our measurement uncertainties. These factors are 2.12 and 2.49 for the two samples. The methodology for line center fitting is the same as used for the Hα Balmer line.

We measure a mean gravitational redshift by assuming that our WDs are a comoving, local sample. With this assumption, only random stellar motions dominate the dynamics of our targets; this falls out when we average over the sample.

For this assumption to be valid, at least as an approximation, our WDs must belong to the same kinematic population; in our work, this is the thin disk. We achieve a comoving group by correcting each measured v_app to the kinematical LSR described by standard solar motion (Kerr & Lynden-Bell 1986). Column 4 in Tables 1–3 lists the LSR corrections applied to each target. In Falcon et al. (2010), we show in detail that this is a suitable choice of reference frame for a sample consisting of thin disk WDs.

If the targets in our sample belong predominantly to the thick disk, for example, our chosen reference frame will fail to produce a group of objects that is at rest with respect to the LSR, introducing a directional bias into our measured 〈v_app〉. This is because the thick disk population lags behind the thin disk as it rotates about the Galactic center (∼40 km s⁻¹; Gilmore et al. 1989).

In contrast to the work on DAs in Falcon et al. (2010), we lack the sample size to empirically demonstrate whether or not our WDs move with the LSR. Therefore, as we state in Section 2.1, we must assume our targets are thin disk objects.

Our WDs must also reside at distances that are small compared to the size of the Galaxy so that systematics introduced by the Galactic kinematic structure are not significant. Nine of our DBAs and 12 of our DBs have published distance determinations in the range of ∼25 to ∼260 pc (Routly 1972; Wegner 1973; Koester et al. 1981; Farihi et al. 2005; Castanheira et al. 2006; Limoges & Bergeron 2010; Bergeron et al. 2011). We assume the others are at distances comparable to the rest of the SPY sample which Pauli et al. (2006) determine spectroscopically to be mostly (≳90%) within 200 pc with a mean of ∼100 pc. Over these distances, the velocity dispersion with varying scale height above the disk remains modest (Kuijken & Gilmore 1989), and differential Galactic rotation is negligible (Fich et al. 1989).

To translate the mean apparent velocity 〈v_app〉 to a mean mass, we invoke two constraints: (1) we need the mass–radius relation from an evolutionary model; and (2) since the WD radius does slightly contract during its cooling sequence, we need an estimate of the position along this track for the average WD in our sample (i.e., a mean T_eff).

Our evolutionary models use M_He/M_⋆ = 10⁻² for the helium surface-layer mass, and, since we are interested in WDs with helium-dominated atmospheres, we use M_H/M_⋆ = 0 for the hydrogen surface-layer mass. See Montgomery et al. (1999) for a more complete description of our models. Our dependency on evolutionary models is small. We are interested in the mass–radius relation from these models, and this is relatively straightforward since WDs are mainly supported by electron degeneracy pressure, making the WD radius a weak function of temperature. We estimate that varying the C/O ratio in the core affects the radius by less than 0.7%.

Besides atmospheric composition, the difference between the WD sample in Falcon et al. (2010) and the ones in this paper is sample size; the DA sample boasts 449 targets while our DBA and DB samples contain 16 and 20, respectively. For large numbers such as for the DAs, the (arithmetic) mean is the preferred estimator of central location of the distribution. For small numbers, however, this is not necessarily so.

In lieu of the mean, we use the biweight central location estimator and the biweight scale estimator as recommended by Beers et al. (1990) for a kinematic sample of our size. For a discussion on the biweight and its statistical properties of resistance, robustness, and efficiency as they relate to sample size and type of distribution, see Beers et al. (1990).

The biweight central location estimator is defined as

$$\begin{equation} C_{\rm BI}=M+\frac{\sum _{|u_i|<1}(x_{i}-M)\big(1-u_i^2\big)^2}{\sum _{|u_i|<1}\big(1-u_i^2\big)^2}, \end{equation} \tag{ 2 }$$

where x_i are the data, M is the sample median, and u_i are given by

$$\begin{equation} u_i=\frac{(x_i-M)}{c\,{\rm MAD}}. \end{equation} \tag{ 3 }$$

We set the "tuning constant" c = 7.0 (a slight increase from the recommended c = 6.0 listed in Beers et al. 1990) so that no data rejection occurs. Note that C_BI approaches the arithmetic mean in the limit where c → ∞. MAD (median absolute deviation from the sample median) is defined as

$$\begin{equation} {\rm MAD}={\rm median}(|x_i-M|). \end{equation} \tag{ 4 }$$

We calculate C_BI iteratively, taking M as a first guess, substituting the calculated C_BI as the improved guess, and continuing to convergence.

The biweight scale estimator is defined as

$$\begin{equation} S_{\rm BI}=n^{1/2}\frac{\big[\sum _{|u_i|<1}(x_i-M)^2\big(1-u_i^2\big)^4\big]^{1/2}}{\big|\sum _{|u_i|<1}\big(1-u_i^2\big)\big(1-5u_i^2\big)\big|}. \end{equation} \tag{ 5 }$$

Here we set c = 9.0 (within the expression for u_i) as recommended by Beers et al. (1990).

To determine the uncertainty of the biweight, we use a Monte Carlo approach. We adopt the standard deviation of biweight values from 30,000 simulated v_app distributions. Each simulated distribution randomly samples from a convolution of two Gaussians. One represents the measured v_app distribution by using the biweight scale estimator S_BI (analogous to standard deviation) of the sample as the width of the Gaussian. The other accounts for the individual apparent velocity measurement uncertainties by drawing from a Gaussian with a width equal to each δv_app from the sample.

From here on we use the subscript BI to denote the biweight central location estimator, C_BI, and biweight scale estimator, S_BI, as used in lieu of the arithmetic mean and standard deviation, respectively.

As described in Section 1, helium line centroids are expected to shift due to pressure effects, and the magnitude and direction of the shift is thought to depend on the specific line. For that reason, we focus on measurements of a single helium line, He i 5876 Å. Theoretical calculations give a negative (blue) Stark shift of the He i 5876 Å line for electron densities corresponding to WD photospheres and in the temperature range relevant to our investigation (e.g., Griem et al. 1962; Dimitrijević & Sahal-Brèchot 1990); the experiment by Heading et al. (1992) confirms the sign of the shift.

We search for this expected blueshift using our sample of DBAs, whose spectra show both the He i 5876 Å line and the Hα Balmer line center that should not be affected by pressure shifts. Figure 1 shows a comparison of apparent velocities measured from these lines, and indeed the intersection (filled, red square) of the average (biweight) values lies below the unity line.

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Using the calculated shifts from Dimitrijević & Sahal-Brèchot (1990), we plot a dashed line in Figure 1 indicating the deviation from unity for plasma conditions corresponding to temperature T = 16,500 K and electron density n_e = 10¹⁷ cm⁻³. This should represent an overestimate of the expected shift based on theory. T_eff = 16,500 K is the lower limit for the WDs in our sample, and not only does the magnitude of the shift increase with decreasing temperature, but it tracks exponentially so that the effect is more exaggerated at lower T. Furthermore, although n_e = 10¹⁷ cm⁻³ is typical for WD photospheres, we are concerned with absorption line centers. These are formed higher in the atmosphere where the density is lower and the shift should be less. As expected, the intersection of our measured biweight values lies between the unity line and the blueshift overestimate.

The presence of hydrogen in a helium-dominated atmosphere may also affect the atmospheric pressure and therefore electron density because of its relatively high opacity; in our temperature range, the opacity of hydrogen can be an order of magnitude greater than that of helium. From Voss et al. (2007), the mean atmospheric hydrogen abundance for the WDs in our DBA sample is H/He = 10^−3.29 with the highest abundance of any WD being H/He = 10⁻². At these abundances, this opacity effect should be negligible, and the dependency of the distribution of pressure shifts on electron density should only be due to the intrinsic mass distribution.

The average apparent velocity of the DBAs in Figure 1 as measured from Hα is 〈v_app〉_BI = 40.8 ± 4.7 km s⁻¹; the average velocity as measured from the He i 5876 Å line is 〈v^He_app〉_BI = 34.0 ± 5.0 km s⁻¹. The measured difference of 6.9 ± 6.9 km s⁻¹ is blueshifted and exactly at the 1σ boundary of our measurement precision.

With assumptions this measured blueshift can be used as a correction to apply to the average apparent velocity of DB WD samples in order to estimate an average gravitational redshift. First, one must assume that DBAs are not fundamentally different from DBs in a way that would manifest as different mean masses. Voss et al. (2007) detect various amounts of hydrogen in most (55%) of the helium-dominated WDs in their sample and find similar spectroscopic mass distributions between the DBAs and DBs. Bergeron et al. (2011) also find no significant differences between the masses of the two groups. This provides evidence to the idea of Weidemann & Koester (1991) that the DBA subclass is not distinct from its parent class but rather the observationally detectable end of a continuous distribution of hydrogen abundances.

Also, though our value is indeed a measure of the average blueshift due to pressure effects, this average is not straightforward. The magnitude and sign of the shift depend on temperature and electron density (or pressure) (e.g., Kobilarov et al. 1989; Dimitrijević & Sahal-Brèchot 1990), and the atmospheres of our DBAs sample a distribution of these plasma conditions. For our measured value to be representative of the average blueshift for another sample of WDs, it is important that this sample have a similar T_eff distribution. Assessing the similarity between samples when applying this correction will deserve further scrutiny when a future analysis is performed which uses significantly larger sample sizes and hence higher precision.

We apply this correction so that we may, for the first time, estimate an average gravitational redshift for DB WDs in the field. The precision of this correction is given by the combined uncertainty of the method using the Hα Balmer line and of the one using the He line (i.e., δ〈v_app〉_BI and δ〈v^He_app〉_BI added in quadrature; ∼7 km s⁻¹) and can be improved by increasing the sample size of DBA WDs.

4.1.1. Analysis with the Hα Balmer Line

We observe hydrogen absorption in 16 WDs classified as helium-dominated by Voss et al. (2007). Figure 2 shows the distribution of v_app measured from the Hα Balmer line. Table 1 lists these measured values, which are described in Section 3.2.

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For this sample, 〈v_app〉_BI = 40.8 ± 4.7 km s⁻¹. We find that this is different from 〈v_app〉 of DAs (32.57 km s⁻¹) at the 91% confidence level.

Given the small sample size, it is possible that a systematic uncertainty is skewing our result. For example, by chance kinematic substructures may not average out with only 16 targets as they would for a larger sample. As a check, we perform the following test: let us assume that the SPY DBAs have the same intrinsic mean mass as the SPY DAs. If we randomly pick 16 apparent velocities from the measured v_app distribution from Falcon et al. (2010), which contains 449 objects, how often is the 〈v_app〉_BI of these 16 greater than 〈v_app〉_BI = 40.8 km s⁻¹, our result for DBAs? Performing the random selection 30,000 times, we find that this false detection occurs 9% of the time. The synthetic biweights reproduce the 〈v_app〉 of DAs with a standard deviation of 4.17 km s⁻¹. There is a rare but not insignificant possibility that our small sample size is fooling us.

Using spectroscopically determined T_eff from Voss et al. (2007), 〈T_eff〉_BI = 17,890 ± 250 K. We use the mass–radius relation from our evolutionary models and interpolate to find 〈M〉_BI = 0.71^+0.04_{− 0.05} M_☉. This is larger than the mean mass for normal DAs determined using the same gravitational redshift method (0.647^+0.013_{− 0.014} M_☉; Falcon et al. 2010).

Though suggestive of a real difference in mean mass between the samples, this is not a stringent result. It remains plausible that no intrinsic difference exists. We can say, however, that the difference between 〈v_app〉_BI of the DBAs and 〈v_app〉 of normal DAs is no larger than 9.2 km s⁻¹, at the 95% confidence level. With a note of caution, because of the subtleties in translating a mean apparent velocity to a mean mass, this corresponds to roughly 0.10 M_☉.

Our 〈M〉_BI for DBAs is also larger than 0.62 ± 0.02 M_☉, the biweight of the spectroscopic mass determinations for these targets from Voss et al. (2007). The value for the mean is the same as the biweight. We estimate the uncertainty using the same method as for our work, except that, since Voss et al. (2007) do not list individual uncertainties, we assign each mass an uncertainty equal to the standard deviation of the mass distribution. The biweight (and mean) spectroscopic mass of the 22 DBAs with T_eff ⩾ 16,500 K from Bergeron et al. (2011) agrees well at 0.67 ± 0.02 M_☉. For this value, we use the mass uncertainties from Bergeron et al. (2011).

4.1.2. Analysis with the He i 5876 Å Line

We now use the He i 5876 Å line to perform the gravitational redshift method on DBs. We measure v^He_app for 20 such WDs and find 〈v^He_app〉_BI = 36.0 ± 5.6 km s⁻¹. Table 3 lists individual measurements. After applying the correction for the blueshift due to pressure effects measured in Section 3.6, 〈v^He_app〉_BI = 42.9 ± 8.9 km s⁻¹. This uncertainty comes from adding the measured uncertainty and that of the correction in quadrature. Using 〈T_eff〉_BI = 20, 570 ± 660 K (Voss et al. 2007) with the corrected 〈v^He_app〉_BI, we determine 〈M〉_BI = 0.74^+0.08_{− 0.09} M_☉.

This is different from 〈v_app〉 of DAs at the 75% confidence level. The difference between 〈v^He_app〉_BI (with the blueshift correction) of the DBs and 〈v_app〉 of DAs is ⩽11.5 km s⁻¹, at the 95% confidence level. This corresponds to roughly 0.12 M_☉.

From Voss et al. (2007), we find 〈M*〉_BI = 0.58 ± 0.02 M_☉ for these targets (M* denotes spectroscopic mass); the mean is 0.59 ± 0.02 M_☉. The biweight spectroscopic mass of the 35 DBs with T_eff ⩾ 16,500 K from Bergeron et al. (2011) is 0.63 ± 0.01 M_☉; the mean is 0.64 ± 0.01 M_☉. The mean of the 82 DBs with T_eff > 16,000 K from Sloan Digital Sky Survey (SDSS) and whose spectra have S/N ⩾ 20 is 0.646 ± 0.006 M_☉ (Kepler et al. 2010).

As mentioned in Section 3.6, let us assume there are no fundamental differences between DBAs and DBs that would manifest as different mean masses. We can then apply the correction for the blueshift due to pressure effects, and we can improve both the precision and accuracy of our mean value determinations by combining the two samples. We do not discard the possibility, however, that the two groups are indeed fundamentally different.

Using the apparent velocity of the He i 5876 Å line for the combined sample of DBAs and DBs, we find 〈v^He_app〉_BI = 36.5 ± 3.9 km s⁻¹. Figure 3 shows the distribution of v^He_app. Applying the blueshift correction, 〈v^He_app〉_BI = 43.4 ± 7.9 km s⁻¹. Using 〈T_eff〉_BI = 19, 260  ±  390 K, we determine 〈M〉_BI = 0.74^+0.07_{− 0.08} M_☉.

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This is different from the mean v_app of DAs at the 82% confidence level. The difference between 〈v^He_app〉_BI of this combined sample of DBAs + DBs and 〈v_app〉 of DAs is ⩽7.7 km s⁻¹, at the 95% confidence level. This corresponds to roughly 0.08 M_☉.

〈M*〉_BI = 0.60 ± 0.02 M_☉ for these targets using the spectroscopic masses from Voss et al. (2007). This is significantly lower than our determination. The biweight (and mean) spectroscopic mass of the 57 DBAs and DBs with T_eff ⩾ 16,500 K from Bergeron et al. (2011) is 0.65 ± 0.01 M_☉.

We list our derived 〈v_app〉_BI in Table 4, denoting which samples use the He i 5876 Å line in lieu of the Hα Balmer line. Since the 〈v_app〉_BI values are entirely model independent, we list them apart from the 〈M〉_BI we determine. These are in Table 5. For comparison, we list the corresponding results for DAs from Falcon et al. (2010).