Measuring He i Stark Line Shapes in the Laboratory to Examine Differences in Photometric and Spectroscopic DB White Dwarf Masses

Accurate helium White Dwarf (DB) masses are critical for understanding the star’s evolution. DB masses derived from the spectroscopic and photometric methods are inconsistent. Photometric masses agree better with currently accepted DB evolutionary theories and are mostly consistent across a large range of surface temperatures. Spectroscopic masses rely on untested He i Stark line-shape and Van der Waals broadening predictions, show unexpected surface temperature trends, and are thus viewed as less reliable. To test this conclusion, we present in this paper detailed He i Stark line-shape measurements at conditions relevant to DB atmospheres (T electron ≈12,000–17,000 K, n electron ≈ 1017 cm−3). We use X-rays from Sandia National Laboratories’ Z-machine to create a uniform ≈120 mm long hydrogen–helium mixture plasma. Van der Waals broadening is negligible at our experimental conditions, allowing us to measure He i Stark profiles only. Hβ, which has been well-studied in our platform and elsewhere, serves as the n e diagnostic. We find that He i Stark broadening models used in DB analyses are accurate within errors at tested conditions. It therefore seems unlikely that line-shape models are solely responsible for the observed spectroscopic mass trends. Our results should motivate the WD community to further scrutinize the validity of other spectroscopic and photometric input parameters, like atmospheric structure assumptions and convection corrections. These parameters can significantly change the derived DB mass. Identifying potential weaknesses in any input parameters could further our understanding of DBs, help elucidate their evolutionary origins, and strengthen confidence in both spectroscopic and photometric masses.


Introduction
White Dwarfs (WDs) are the evolutionary endpoint for ∼98% of all stars. Approximately 20% of all known WDs have helium-dominated atmospheres. Among them, only those showing neutral helium lines are called DBs (Sion et al. 1983;Koester & Kepler 2015). Accurate DB masses are important to multiple areas of astrophysics, including stellar evolution (e.g., Nather et al. 1981;Fontaine & Wesemael 1987;Werner & Herwig 2006), and stellar mass loss processes (e.g., Choi et al. 2016;Cummings et al. 2018). Most DB masses are measured using either the spectroscopic or photometric methods.
The accuracy of the spectroscopic method, which relies on fitting observed spectra with calculations, has been questioned due to uncertain input physics (e.g., Bergeron 1993;Bergeron et al. 2011;Tremblay et al. 2013;Koester & Kepler 2015;Genest-Beaulieu & Bergeron 2019b;Cukanovaite et al. 2021). Atomic line shapes relating the width of an observed line to a stellar surface gravity (log g)/mass are difficult to calculate and are the critical ingredient in the spectroscopic method. Details of all line broadening processes found in DB atmospheres must be included in spectroscopic models. The DB community uses the semi-analytical Beauchamp et al. (1997) line shapes (henceforth called B97) to account for the Stark broadening (caused by charged plasma particles), while the Van der Waals contribution (caused by plasma neutrals) is modeled using the Unsold (1955) and Deridder & van Renspergen (1976) predictions (henceforth called U55 and D76). Tremblay et al. (2020) recently used the computer-simulation approach (e.g., Gigosos et al. 2014) to reinvestigate He I Stark profiles, and the resulting line shapes agreed very well with the B97 calculations. To our knowledge, no laboratory He I Stark line-shape experiments have ever been performed at conditions applicable to DB atmospheres. However, He I full width at half maximum (FWHM) data have been published at electron densities (n e ) higher and lower than those found in DBs (e.g., Chiang et al. 1977;Perez et al. 1991;Büscher et al. 1995;Milosavljević and Djeniže 2002). No experimental data for the Van der Waals broadening models exist at DB atmosphere conditions. The photometric method, considered simpler and more reliable than the spectroscopic technique, has recently been applied to many DBs due to the release of Gaia parallax measurements (Genest-Beaulieu & Bergeron 2019a, 2019b. This approach combines the parallax data with a fit to the measured stellar flux to arrive at a mass (Tremblay & Cukanovaite 2019). Only DB atmospheric flux calculations, a mass-radius relationship, and a dereddening algorithm are needed as an input. Each of these are thought to be easier to calculate and validate than atomic line shapes. However, as pointed out by Genest-Beaulieu & Bergeron (2019b), photometric temperatures can be highly sensitive in the 22,000 K  T eff  26,000 K range, where the main opacity source switches from He I bound-free to He II free-free. Furthermore, because the photometric method is heavily dependent on measured parallaxes, large uncertainties in those values lead to large mass uncertainties (Genest-Beaulieu & Bergeron 2019b).
Genest-Beaulieu & Bergeron (2019b) published a detailed investigation of photometric and spectroscopic DB masses. Their results are shown in Figure 1, where the top panel shows photometric DB masses as a function of surface temperature (T eff ) and the bottom panel depicts the spectroscopic values. We plot a black dotted line at M/M e = 0.6 in each panel to guide the reader, along with a moving average fit of the masses (green). The photometric masses exhibit almost no T eff dependency and match the expected mass. Spectroscopic masses, however, show clear T eff trends. At T eff > 16,000 K, Stark broadening is thought to dominate spectra because the atmospheric plasma contains more charged particles due to a higher ionization fraction compared to cooler temperatures. Weaknesses in the B97 calculations are blamed for the observed mass trends. At T eff < 16,000 K, Van der Waals broadening is believed to control line broadening, because more neutrals are present in the plasma. The increased scatter in that temperature range is ascribed to deficiencies in the U55 and D76 predictions (e.g., Bergeron et al. 2011;Koester & Kepler 2015;Genest-Beaulieu & Bergeron 2019a, 2019b. The combination of the simpler input physics for photometric method, lack of T eff -mass trends, and the agreement with the expected DB mean mass seem to validate the assumption that photometric masses are more reliable than spectroscopic ones.
Developing a deeper understanding of the spectroscopic mass problems is essential to our comprehension of DBs. If it is shown that, contrary to present-day assumptions, spectroscopic mass are reliable, our current understanding of stellar evolution would be incorrect. DBs cool as they age, and the spectroscopic mass trend shown in Figure 1 would indicate that different DB generations have varying masses. Such a finding would counter fundamental tenets of modern stellar evolution and would also invalidate the currently accepted DB evolutionary model. A natural starting point for such an investigation is benchmark laboratory experiments aimed at testing line-shape calculations used in DB atmospheres. If laboratory data do not agree with calculations, the culprit for the spectroscopic mass trends has been identified. However, if line-shape models agree with experiments, other spectroscopic method input parameters, such as atmospheric structure assumptions, may be flawed.
In this paper, we perform validation tests for the B97 Stark model using two strong He I lines found in DB atmospheres: the 3p → 2s He I line at 5875 Å (henceforth referred to as He5875) and the 3d → 2p He I line at 5015 Å (henceforth referred to as He5015). The U55 and D76 models predict <0.07 Å Van der Waals broadening at our highest experimental neutral densities (n neutral ≈ 1 × 10 18 cm −3 ). The neutral density in our experiments varies by a factor of ≈ 2 and no dependence of line width on n neutral was found, confirming that Van der Waals broadening is insignificant. In n e and electron temperature (T e ) space, our experiments are performed at DB atmosphere conditions (T e ≈ 12,000-17,000 K, n e ≈ 10 17 cm −3 ). For additional insight into He line shapes and their effect on DB masses, we also validate our own, computer-simulation-based line shapes, which represent an improvement and update of those presented in Gigosos et al. (2014). Plasma T e , n e , and n neutral are diagnosed sufficiently well to qualify for benchmark experiment status.
We find that our experimental line shapes agree with B97 and our own calculations within experimental uncertainties. The DB mass uncertainty associated with our experimental He I n e uncertainties is estimated to be smaller than the spectroscopic-photometric mass differences in the temperature regimes accessed by our validation tests. Our results therefore indicate that perceived weaknesses in Stark predictions may not be responsible for the DB spectroscopic mass discrepancies. Identifying which other spectroscopic input parameters may be responsible for the observed mass trends is difficult. Cukanovaite et al. (2021) showed that small changes in DB atmospheric modeling assumptions can have significant effects on the inferred spectroscopic mass. Other spectroscopic mass inputs such as the assumed atmospheric structure or data analysis methods could also introduce undetected errors. Closer investigation of the latter would have an impact on all WD astrophysics, because these data analysis methods are used for deriving non-DB spectroscopic masses as well (e.g., Dufour et al. 2008;Tremblay et al. 2013). Finally, the photometric method and its sensitivities to the input physics should also be scrutinized further.
This paper is organized as follows: In Section 2, we give an overview of the WDPE platform on Sandia National Laboratories' Z-machine. Section 3 describes our data extraction and analysis techniques. The overlap of the plasma conditions reached in our experiment and those found in DBs is discussed in Section 4. Experimental results are presented in Section 5, followed by conclusions in Section 6.

The WDPE at SNLʼs Z-machine
Benchmark experimental line-shape data must be collected under conditions that fulfill several key requirements. First, reliable and accurate plasma diagnostics are needed. The He I line-shape data presented in this paper require T e , n e , and n neutral diagnostics. Second, line-shape data can only be extracted from accurate absorption measurements that are free from any self-emission effects. In this section, we describe the  WDPE platform and demonstrate that it allows us to collect all data needed for a benchmark experiment.
A thorough description of the WDPE platform and our experimental setup is given in Falcon et al. (2013Falcon et al. ( , 2015 and Schaeuble et al. (2019). Recent updates to allow for accurate measurements of He I line shapes are described in Schaeuble et al. (2021). Only a short overview of the WDPE platform is given here.
We perform the WDPE He line-shape validation experiments at the currently most energetic pulsed X-ray source on earth: SNLʼs Z-machine. This pulsed power driver converts a ≈26 MA current into an X-ray drive using a dynamic hohlraum (McDaniel et al. 2002;Bailey et al. 2006;Rose et al. 2010;Jones et al. 2014;Rochau et al. 2014;Sinars et al. 2020). We can use the Z-machine X-rays to reproduce a DB atmosphere in the laboratory by filling the 120 mm long WDPE gas cell (see Figure 2) with a H/He gas mixture. Our gas cell is placed ≈324 mm away from the Z-pinch source and designed such that X-rays overfill the cell. This ensures uniform heating of the macroscopic plasma. Z-pinch X-rays (green arrows in Figure 2) traverse the H/He gas mixture cell and deposit their energy into the gold back wall and backlighting surface. The photons reradiated (dark red arrows in Figure 2) from these two hardware pieces then heat the H/He gas mixture and turn it into a plasma.
Due to continuous heating from the gold back wall, the H/ He plasma steadily increases in T e and n e throughout our experiment (Falcon et al. 2015;Schaeuble et al. 2019). Use of time-resolved spectrometers allows us to collect and test line shapes at multiple plasma parameter values in a single experiment. Gas pressure and composition can be used to control n e , n ion , and n neutral . In DB astronomy, n neutral = n H I + n He I . We adopt the same definition here. At our plasma conditions, hydrogen is the main electron donor while helium largely stays neutral. Plasma n H I therefore decreases as a function of time resulting in each experiment covering a range of plasma conditions (12,000 K T e 17,000 K, 5 × 10 16 cm −3 n e 5 × 10 17 cm −3 , 5 × 10 17 cm −3 n neutral 1 × 10 18 cm −3 ).
Representative T e , n e , and n neutral trends observed during an experiment are shown in Figure 3. T e is shown in the top panel, n e is depicted in the center, and n neutral trends are plotted on the bottom. We list the gas mixtures, pressures, average n e , n neutral , and diagnostics setups (discussed below) for all shots considered in this paper in Table 1. The gas fill pressures listed in Table 1 were collected using an in situ piezoresistive pressure sensor attached to the gas cell during the experiment Schaeuble et al. 2021). Extraction methods for each experimental parameter plotted in Figure 3 and listed in Table 1 are discussed in Section 3.1.
When using absorption spectra to measure line shapes, it is crucial to ensure that self-emission contamination in the collected data is either negligible or accounted for. WDPE absorption spectra consist of the backlighter spectrum attenuated by the plasma and plasma self-emission processes. If the backlighter is weak, plasma self-emission processes cannot be overcome and the absorption spectrum is thus contaminated and unsuitable for line-shape studies. We collect absorption and emission spectra on each shot, and these enable us to determine the self-emission contamination for the He5015 and He5875 lines. We find that no significant self-emission is present for He5015, while He5875 shows large self-emission effects. We therefore also collect high-resolution He5875 absorption and emission data on CCD spectrometers. These two datasets are calibrated and then subtracted. The He5015 data is processed without performing a self-emission correction. To collect the needed experimental data, the photons emitted from the H/He plasma are transported to two CCDbased and one film-based streaked (i.e., time-resolved) spectrometer system via fiber optics. The absorption line-ofsight (LOS; red in Figure 2) runs along the gold back wall terminating on the backlighter, which is also heated by Z-pinch X-rays. Since the backlighter is hotter than the plasma (backlighter T ≈ 25,000 K, plasma T ≈ 12,000 K), an absorption spectrum results. The emission LOS (blue in Figure 2) also runs along the gold back wall and exits the gas cell. Only emission data are captured on this LOS.
The CCD systems record higher-resolution data (λ/ Δλ ≈ 1000 at 5000 Å) compared to the film system (λ/ Δλ ≈ 500 at 5000 Å). All He I line-shape data presented in this paper are collected on the CCD systems for accurate line-shape testing (Schaeuble et al. 2021). The Hβ line for all our experiments is captured on film, except for shot z3462, during which it was recorded on the CCD. Since the Hβ line is broad and theoretically well-understood, λ/Δλ ≈ 500 is sufficient to accurately infer n e from this line. Sample experimental spectra are shown in Figure 4. The top panel shows the spectrum containing Hβ, He5015, He5875, and the He I line at 4713 Å. Analysis of the 4713 Å He I line is beyond the scope of this paper. The relative intensities of the absorption lines shown in    Figure 4 shows the emission and absorption data for He5875. More details about the spectrometer systems and other attributes of the data are discussed in Schaeuble et al. (2021). In Table 1, we list the diagnostic setups for each shot considered in this paper.

WDPE Data Extraction and Analysis Procedures
The data collected using the hardware and instruments described in Section 2 must be processed to extract all parameters needed for a benchmark line-shape experiment: plasma T e , n e , and n neutral values, as well as line-shape profiles. Each of these parameters must be extracted with uncertainties of 20%. The data extraction approach for these parameters is discussed in Sections 3.1 and 3.2. Benchmark theoretical lineshape validation tests also require an appropriate line-shapefitting methodology and accurate instrumental broadening measurements, which are described in Section 3.3. The analysis associated with the He5875 data must be approached with special care, due to the unique data collection setup for that line. These effects are discussed in Section 3.4.

T e , n e , and n neutral Extraction Methods
We use Hβ to extract plasma n e and T e values needed for validating Stark broadening predictions. This transition is particularly suitable to serve as an n e diagnostic, because it is well-studied in the WDPE platform.  and Falcon et al. (2015) showed that four well-tested Hβ line-shape theories predict the same n e values in the WDPE if plasma n e < 3 × 10 17 cm −3 . Schaeuble et al. (2019) demonstrated that potential plasma T e and n e gradients do not affect n e values derived from Hβ. In the same publication, it was also shown that detailed line fits are not necessary to extract an accurate plasma n e from Hβ. Full width at half area (FWHA) measurements of Hβ line profiles that have not been emission-corrected result in the same plasma n e as line-shape fits to fully processed experimental data. We therefore adopt the FWHA n e extraction approach here. As in Schaeuble et al. (2019), we use Xenomorph (Gomez et al. 2016;Cho et al. 2022) to translate our experimental FWHA values to plasma n e . However, the choice of line-shape theory is ultimately unimportant for the present study, because Falcon et al (2015) showed that all major theories predict the same n e at densities reached in the experiments presented here. The combination of the work presented in Falcon et al. (2015) and Schaeuble et al. (2019) gives us great confidence in the accuracy of the Hβ n e values.
The Hβ n e values are converted to a local thermodynamic equilibrium (LTE) temperature under the assumption that all electrons are being donated by H and that every neutral H atom is in the ground state (see Falcon et al. 2015). The resulting LTE temperatures are slight overestimates since our plasma is initially "overionized," leading to a larger than expected n = 2 population (e.g., Kawasaki et al. 2002). However, line shapes are much more sensitive to n e and n neutral , making T e measurements of limited importance for the purposes of this paper.
The neutral densities (n He I , n H I ) required for estimating the Van der Waals broadening contribution to the total measured line width are obtained through in situ gas pressure measurements (Schaeuble et al. 2021). The measured pressures in combination with the known gas composition allow for the derivation of n H 2 and n He in the gas cell just prior to the experiment. The WDPE gas cell is repeatedly filled and purged before an experiment, to minimize the risk of gas contamination. The pressure data extraction procedure is described in . The estimated pressure and particle density uncertainties are ≈1%. We convert the measured n H 2 and n He to n neutral by assuming that all He stays neutral and that H donates all plasma e − . This assumption is based on both experimental data and simulation results. We find no evidence of any He II lines in our experimental spectra. Further, all WDPE H/He mixture experiments are fielded at the same partial H pressure as the pure H experiments presented in Falcon et al. (2015) and Schaeuble et al. (2019). H/He experimental n e values match those derived from pure H experiments, providing further evidence that He is not significantly ionized in the data presented in this paper. Finally, simulations of our plasma indicate that 10% of He is ionized at our plasma conditions. Subtracting n e at every time step in the experiment from the measured n H therefore results in n neutral . n e is constantly increasing as a function of time, n H I is always decreasing. Adding the constant n He I to the time-dependent n H I then results in a n neutral value at each time step in the experiment (see Figure 3). The U55 and D76 Van der Waals models predict that neutral hydrogen atoms provide more broadening than neutral helium. We add the predicted contributions in quadrature to arrive at the total width. Given our assumptions, the n neutral values shown in Figure 3 might be lower estimates.

He5015 and He5875 Line-shape Extraction Methods
He5015 and He5875 WDPE line shapes are extracted for line-shape fitting using two separate methods. Since the He5015 is a weak line (τ ≈ 0.3) that shows little to no selfemission, this feature can be extracted by simply defining a continuum across the absorption line, converting it to transmission (), and then to optical depth (τ = −ln()). The optical depth of the He5875 line is much larger (τ ≈ 5) and exhibits significant self-emission (see Figure 4). More careful treatment is therefore required before extracting the line shape.
The data collected on the He5875 absorption LOS can be described as follows: where I backlighter is the backlighter intensity and I self−emission is the self-emission intensity along the LOS. Near the He5875 line center,  approaches zero, making the self-emission intensity a significant contributor to the recorded absorption signal. By subtracting the self-emission from the absorption data, we recover the true absorption that can be used to extract line-shape data. All He5875 data presented in this paper were calibrated using the process described in Schaeuble et al. (2021). The effect of the self-emission correction on the experimental He5875 profile can be seen in Figure 5, where we compare the calibrated absorption (red dashed) and emission (blue dashed) to the self-emission-corrected absorption (solid green). The green line is used for all He5875 analyses presented in this paper. Past studies have identified the incorrect/incomplete accounting for such self-emission/ absorption effects as the most likely reason for faulty He5875 experimental measurements (e.g., Heading et al. 1992;Büscher et al. 1995). Cool boundary layer plasmas cause the observed emission-absorption line-shape discrepancy in Figure 5. These plasmas absorb from the center of each respective line shape. For emission, this results in a noticeable dip in the line-profile center, while the effect on absorption is much less noticeable.

Instrumental Broadening and Line-fitting Methods
Instrumental broadening measurements, the final ingredient needed to correctly fit experimental line shapes with calculations, are determined using a process outlined in Schaeuble et al. (2021). As described in that paper, we use laser lines to determine that the instrumental broadening of the CCD and film systems used in the WDPE has a Gaussian shape with FWHM of ≈6 Å. Recorded WDPE line shapes are broadened by both neutrals and charged particles in the plasma and the instruments used to record spectra. Theoretical line shapes only account for plasma processes. Instrumental effects must therefore be accounted for when fitting line shapes to experimental data.
Deconvolving the instrumental profile from the recorded spectra is a possibility, but numerically unstable and difficult. A much more straightforward approach is to convolve the theoretical line shapes with the instrumental profile during fitting, which is what we adopt for this paper. The convolution of the theoretical line shapes with the instrumental profile is performed in transmission (Schaeuble et al. 2019). The lowest WDPE FWHM measurements are comparable to the measured instrumental broadening (FWHM ≈ 6 Å; Schaeuble et al. 2021). Experimental line shapes are therefore significantly broadened by our data collection setup, and such effects need to be properly accounted for. See Schaeuble et al. (2021) for more details. The instrumental broadening convolved theoretical profiles are fit to experimental data using a Levenberg-Marquardt algorithm (Levenberg 1944;Marquardt 1963).

He5875 n e Data Extraction Methodology
Special care is required when using Hβ to extract n e values associated with He5875 line-shape data, because two separate LOS are used to collect each data set. As explained in Section 2, we capture both emission and absorption data for He5875. Absorption data captured on the CCD and film spectrometer systems are collected by arranging a fiber bundle containing two fibers side-by-side on the backlighter (see Figure 6). A top-to-bottom arrangement is not possible due to the geometric limitations of the backlighter. The adopted sideby-side arrangement results in one absorption LOS being closer to the gold back wall compared to the other. Since the gold wall is the main heating source in the WDPE gas cell, we expect the LOS closer to the gold wall to have a higher n e compared to the one farther away. In the case presented in Figure 6, we expect the hi-res fiber to probe a higher n e environment compared to the lo-res fiber.
Since the Hβ line is used to supply the n e values needed to validate the He5875 line-profile calculations, we must ensure that the difference in probed plasma n e is accounted for. Figure 7 shows the results of an experiment where Hβ was collected using the side-by-side arrangement shown in Figure 6. The blue curve in Figure 7 shows the Hβ n e values for the LOS closer from the gold wall (hi-res fiber in Figure 6), while the red curve shows the Hβ n e values for the LOS farther away (lo-res fiber in Figure 6). The data shown in that figure were collected on shot z3462. As expected, the LOS closer to the gold wall exhibits a higher n e compared to that farther away. The n e percent difference between the two LOS (dashed purple with purple right hand y-axis as reference) is very close to 30% for the entire duration of the experiment. We therefore apply this correction factor to the n e values of all our He5875 experiments. While we only have a single data set in which we tracked the front-back alignment n e differences, comparing the z3462 experimental results to other shots and applying the derived density differences to z3195 and z3402 produces consistent results across all shots, lending further credibility to this approach. For experiment z3195, the Hβ data were collected on the farther LOS and we therefore multiply the n e values by 1.3 for line-shape validation. Experiment z3402 had the reverse setup, leading us to multiply its Hβ n e values by 0.7.

Applicability of WDPE Plasma Conditions to DB Atmospheres
Performing meaningful Stark line-shape validation experiments for the problems associated with spectroscopic DB masses requires laboratory data collected at the T e and n e parameters found in He WD atmospheres where Stark broadening is thought to dominate. DB atmospheres span many orders of magnitude in n e , and several factors in T e . Identifying relevant plasma conditions at which to perform meaningful line-shape validation experiments is therefore challenging.
For modeling purposes, the astrophysical community generally considers atmospheres to be plane parallel and uses the Rosseland mean optical depth (τ R ) as a depth scale. τ R is defined as ∑ i κ R,i × l i where κ R is the Rosseland mean opacity and l is the length of layer i. Deeper atmospheric layers have higher τ R , and also higher T e , n e , and n neutral values. The layer in which τ ν = 2/3 (not to be confused with τ R ) is the so-called "line-forming" region in an atmosphere. Line cores will form at significantly lower τ R values compared to continuum regions. This is caused by the additional opacity/optical depth provided by a line, which pushes the τ ν = 2/3 point into lower τ R regions (i.e., higher in the atmosphere). We define relevant DB atmosphere plasma conditions for this paper to be those found in the atmospheric layers where τ ν = 2/3, or τ R (τ ν = 2/3). Layers with lower τ ν may also contribute to the emergent stellar spectrum, but they have lower T e and n e values compared to the line-forming region. The relevant plasma conditions we list below are thus the highest T e and n e values expected to contribute to an emergent DB spectrum. We use the Montreal WD atmosphere models (Dufour et al. 2007;Bergeron et al. 2011) at standard DB conditions (T eff = 17,500 K, log g = 8 cm s −2 , log(H/He) = −6; see Genest-Beaulieu & Bergeron 2019b) to predict plasma conditions in He WD line-forming regions. The log(H/He) parameter in a DB atmosphere model specifies the number abundance ratio of hydrogen to helium. A value of −6 indicates the presence of 1 hydrogen atom per 1 million helium atoms.
The τ R (τ ν = 2/3) structure and the corresponding T e and n e values for the He5015 and He5875 lines are plotted in Figure 8. The top panel in that figure identifies the atmospheric layer in which τ ν = 2/3 (i.e., the line-forming region). We also identify the directions of the stellar surface and core in the top panel. Panels (B) and (C) show the n e and T e values found in the lineforming region. The WDPE n e and T e ranges are highlighted in gray in each panel. The atmospheric data shown in Figure 8 are only weakly dependent on log g and log(H/He).
The overlap between WDPE n e and T e ranges and the atmospheric conditions over which He5015 and He5875 are formed demonstrates the applicability of our data to investigating DB spectroscopic mass trends observed at 16,000 K T eff 20,000 K. As evident from panel (B) of Figure 8, the WDPE achieves a slightly higher n e compared to the lineforming regions found in DBs. However, the line-forming regions only constitute the visible part of an observed DB spectrum. Higher and lower atmospheric layers play an important role in the radiation transport and overall structure of a stellar atmospheres. Further, Stark broadening is more difficult to model at higher n e , because the perturbation approximation fails, making validation at those n e values more interesting (e.g., Falcon et al. 2015). WDPE T e values agree well with DB atmospheric temperatures at which the He5015 and He5875 lines are formed.  . Comparison of Hβ n e values as a function of time extracted from the same experiment (z3462), but using different LOS. Data depicted with the blue curve were collected closer to the gold wall and therefore exhibit higher Hβ n e values, just as expected. Hβ n e data collected on the farther LOS are plotted in red and show lower measurements. The difference between the two LOS is almost consistently ≈30%.

Experimental Results
WDPE absorption and emission spectra represent a convolution of three different line-broadening mechanisms: instrumental, Stark, and Van der Waals. Section 3 describes how we incorporate instrumental broadening into our data analysis. The Stark and Van der Walls broadening models are examined simultaneously by comparing Hβ n e to the n e values inferred from line-profile fits to He5015 and He5875 using the B97 and our models (see Section 3). If these line-profile fits result in higher n e values compared to Hβ, neutrals most likely contribute to the total observed line width. In this case, we deconvolve the Stark contribution predicted by models from the experimental line profile and ascribe the remaining broadening to Van der Waals processes. If the line-profile fits using the B97 or our models result in n e values that are lower than or match the Hβ measurements, Van der Waals broadening is not important at WDPE plasma conditions. The U55 and D76 predictions indicate that, at n neutral ≈ 1 × 10 18 cm −3 (WDPE maximum), the Van der Waals broadening contribution to the experimental line shape should be at least two orders of magnitude lower than Stark broadening contribution.
Since the main goal of our analysis is to investigate the accuracy of Stark He I line shapes used in the WD community to derive DB masses, we use the comparison between Hβ n e and the line-shape-derived n e as our validation metric, as opposed to the more traditional reduced χ 2 6 value (e.g., Falcon et al. 2015). This approach has multiple benefits. First, comparing Hβ n e to He n e allows us to more directly assess the validity of spectroscopically determined DB masses. The c red 2 metric would only show whether a theoretical line shape is a good match to experimental data. A line fit could have a c red 2 ≈ 1, but still result in an incorrect n e value. Second, using Hβ n e and He I n e differences allows more insightful study into potential systematic errors in the spectroscopic method. The numerical c red 2 metric is of limited use in that respect. We therefore mainly analyze n e trends in this section, but also calculate c red 2 to elucidate differences between our Stark He line-shape calculations and B97.
We find that Hβ n e and He n e values agree well, indicating that observed He5015 and He5875 line widths can be explained using only Stark broadening. As predicted, Van der Waals processes are not needed to explain the data. We can therefore perform a full validation of the B97 and our Stark line-profile models. Careful examination of the He5015 data indicates that current Stark broadening theories may overestimate the FWHM of this line for a given n e . Agreement between WDPE FWHM data and those presented in Chiang et al. (1977) further supports this conclusion.

Stark Broadening Results
We use WDPE He5015 and He5875 line-shape data to validate two Stark broadening line-shape theories: the semianalytical B97 calculations (standard in WD community) and our computer-simulation-based Stark model. For this paper, the Stark energy level structure for our He I Stark calculations takes into account the state mixing effect due to the perturbing plasma electric microfield between all nonperturbed 1snl states with n = 2, 3, and 4. This leads to more accurate results for both line shifts and shapes compared to the former work presented in Gigosos et al. (2014)-where state mixing was only considered within the n = 3 manifold-and it also allows us to simultaneously obtain the calculations for all the involved line transitions at given plasma conditions from the same computer-simulation run. Furthermore, the new calculations include the so-called interference term between all transitions consistent with the improved energy level diagram. The experimental line shapes and the associated uncertainties presented in this section have been extracted according to the description given in Section 3 and Schaeuble et al. (2021).
Our validation procedure consists of two steps: first, we compare the n e values resulting from the line-profile fits to the Hβ n e values. Next, we determine how well theoretical line shapes match experimental data using c red 2 . The first validation step is particularly important for this paper because it ensures that line-shape calculations can accurately predict n e values, a vital ingredient in determining DB masses (see Section 1).

3d → 2p 5015 Å He I Results
We have four WDPE data sets that allow testing the B97 and our He5015 line-shape calculations: z3403, z3409, z3462_B, and z3462_F (see Table 1). The z3462 data sets were captured using the side-by-side backlighter alignment discussed in association with Figure 6. We denote the z3462 LOS closer to the gold back wall with "_B" (for back) and the one further away with "_F" (for front). All four data sets used for our He5015 investigation were collected using the CCD experimental setup described in Section 2. Sample line-profile fits to z3462_F He5015 data are shown in Figure 9 for three time steps: 0 ns (top), 20 ns (middle), and 40 ns (bottom). Each time step represents a 20 ns integration, with the time given in each panel referencing the initial integration time (i.e., the 0 ns step integrates from 0 to 20 ns). The individual panels in Figure 9 show the experimental absorption data in optical depth units (black dots); the B97 (solid blue) and our (dashed red) line-profile fits (see Section 3.3). The N e and c red 2 values for each time step are also given in the panels. Here, c red 2 ≈1, indicating a good fit with appropriately estimated uncertainties for each time step. No significant differences between the B97 and our own line profiles in either n e or c red 2 are apparent from the data presented in Figure 9. Fits to all other shots result in similar c red 2 values and similarly good agreement between c red 2 for B97 and our line profiles. We therefore conclude that both the B97 and our line-profile calculations reproduce the experimental data well and produce n e values within 5%.
Hβ n e values agree well with He5015 n e values for the B97 and our line-shape calculations. Figure 10 shows this comparison for the B97 profiles, while Figure 11 depicts the Hβ-He5015 n e plot for our calculations. The bottom panel of each figure shows the c red 2 for each fit shown in the top panel. B97 line-shape fits to He5015 data result in n e values that are ≈15 % lower than those derived from Hβ. For our lineshape calculations, the He5015 n e values are, on average, 17% lower than those inferred from Hβ. Both the B97 and our theories therefore seem to overestimate line widths for a given plasma n e .
To compare our experimental results to other experiments, we show a He5015 FWHM comparison as a function of n e in  Hβ n e compared to n e derived from fitting B97 line-shape models to WDPE experimental He5015 line data (see Figure 9). The Hβ n e = He5015 n e line is shown as a black dashed line. z3462_F and z3462_B refer to front and back data sets for this experiment. See Figure 7 and associated discussion for more details. Bottom: c red 2 values for all data points shown in the top panel. All c red 2 values 1, indicating good model-data fits. Figure 11. Top: Hβ n e compared to n e derived from fitting our line-shape models to WDPE experimental He5015 line data (see Figure 9). The Hβ n e = He5015 n e line is shown as a black dashed line. z3462_F and z3462_B refer to front and back data sets for this experiment. See Figure 7 and associated discussion for more details. Bottom: c red 2 values for all data points shown in the top panel. All c red 2 values 1, indicating good model-data fits. Figure 12. Other experiments (Chiang et al. 1977;Perez et al. 1991) only measured the FWHM, making this the only method of comparison. Data presented in Figure 12 demonstrate that, while our uncertainties are higher compared to other experiments, we still show the same line width trends. Furthermore, we also show the theoretical FWHM-n e predictions of the B97 (solid blue) and our line shapes (red dashed). The agreement between other experiments, our experiments, and the theoretical predictions are good at lower densities (n e 1 × 10 17 cm −3 ), but data collected at higher densities seem to indicate that both B97 and our line-shape theories may overpredict the FWHM at a given n e . This mirrors trends observed in Figures 10 and 11. He5015 n e uncertainties shown in Figures 10 and 11 are derived by considering instrumental broadening, spectral line normalization, and CCD photon-counting uncertainties. Instrumental broadening uncertainties (see Schaeuble et al. 2021 for detailed derivation) are by far the largest contributor (>90%) to the final He5015 n e error. This is not surprising, given that the instrumental broadening is comparable to the He5015 line width in the WDPE experiments. Small changes in the measured instrumental broadening will have measurable effects on the derived n e value. Spectral line-normalization errors are the nextbiggest contributor (≈10%) to the He5015 n e uncertainty. The normalization contribution was derived by defining a continuum across the He5015 line in the WDPE data three times: (i) the ideal case, which we believe to be the true line-normalization continuum; (ii) the "upper" case, where the continuum was placed through the highest continuum data points available; and (iii) the "lower" case, during which the continuum was defined though the lowest continuum points. The He5015 line was renormalized with each continuum and fit with the B97 and our own line profiles. The resulting n e values changed little as a result. The continuum placement test is particularly important for the He5015 line because it is located close to the red wing of the Hβ feature (see Figure 4). Defining an accurate continuum can therefore be challenging, especially at higher densities. Tests revealed that, despite the proximity of He5015 to Hβ, continuum placement does not introduce significant uncertainties into our analysis. If the continuum placement played an important role in our analysis, we expect the effects of this to be more severe at higher densities, given that the Hβ and He5015 lines are broader at those conditions, making accurate continuum placement more challenging due to line overlap. However, we see no such trends in Figures 9, 10, or 11. Taking the He5015 and Hβ n e uncertainties into account and combining that with the information presented in Figures 10 and 11, we conclude that both the B97 and our He5015 line profiles are accurate within WDPE errors at the n e conditions tested in our experiments. 5.1.2. 3p → 2s 5875 Å He I results WDPE experiments z3195 and z3402 (see Table 1) are used to validate B97 and our He5875 line shapes. The experimental data for this transition is collected using three separate LOS: the lowresolution LOS captures the Hβ n e diagnostic on film, and the two higher-resolution LOS capture the emission and absorption data on CCDs. Due to the high (≈8) optical depths and significant selfemission of this line at WDPE experimental conditions, the absorption data must be corrected for self-emission. A detailed description of this process and the associated errors are given in Schaeuble et al. (2021). Because Hβ and He5875 data are collected on separate LOS with two different distances from the gold back wall (see Figure 6 and Section 2), we have to adjust the Hβ n e values accordingly (see Section 3.4). We remind the reader that these Hβ n e adjustments are not performed to increase agreement between Hβ n e and He5875 n e values, but rather to account for the different plasma regions probed by the LOS.
Sample fits of B97 (solid blue) and our He5875 line profiles (dashed red) to z3195 He5875 data are shown in Figure 13. All these fits are performed to self-emission-corrected absorption data of shot z3195 (black dots). N e and the c red 2 parameters for each line-shape theory are also given in Figure 13. Each lineprofile model appears to fit the data well. The c red 2 < 1 values Figure 12. Comparison of FWHM-n e data from previous experiments (Chiang et al. 1977;Perez et al. 1991) to WDPE measurements and predictions obtained from B97 (solid blue) and our own line-shape predictions (red dashed). indicate that we overestimated the errors for the line profiles. Uncertainties in the plotted profiles include contributions from photon-counting noise (≈2%) and uncertainty in the absolute scaling of the emission and absorption data sets (≈15%). As evident from Figure 5, the self-emission correction is most significant in the line core and has a negligible effect on the continuum. To estimate the total line-profile uncertainty, we thus adopt a value of 15% in the core and assign optical-depthweighted uncertainties to the rest of the line profile, terminating with a 2% photon-counting error at the continuum. Comparisons between Hβ n e and He I n e derived using B97 and our He5875 line-profile calculations are shown in Figures 14 and 15, respectively. The n e equivalency line in those figures is plotted in dashed black. The bottom panel contains the c red 2 values for each of the data points shown in the top panel. In contrast to the He5015 data shown in Figures 10  and 11, He5875 n e values are higher than those derived from Hβ. On average, He5875 n e values are 23% higher than Hβ n e values for the B97 line shapes, and 28% higher for our own line shapes. These higher densities are largely driven by z3195. If only z3402 values are considered, the Hβ-He5875 n e difference shrinks to 12% for B97 line shapes and 15% for our calculations.
Taken at face value, the c red 2 shown in the bottom panel of Figures 14 and 15 suggest that the He5875 line-profile calculations do not reproduce the experimental data as accurately as the He5015 shown in Figures 10 and 11. This is particularly true for fits to z3402 that lead to c red 2 of 2.29 and 3.58 for B97 and 1.79 and 3.12 for our profiles. Many factors influence the c red 2 value, including the quality of fit between theory and experiment and the experimental uncertainties. As demonstrated in Figure 13, theoretical calculations fit well to experimental data. The measured c red 2 values therefore indicate that experimental uncertainties are overestimated for z3195 and underestimated for z3402. These two data sets have different optical depths and different levels of self-emission, and are therefore subject to different systematic uncertainties. It is most likely these differences as well as inaccurate experimental uncertainties that are responsible for the c red 2 discrepancies between z3195 and z3402. However, because the main purpose of this paper is to validate the predicted n e of B97 and our own line shapes, we do not consider these c red 2 discrepancies to be significant.
A comparison of our He5875 FWHM measurements to previous experiments as well as predictions by B97 and our own line shapes is shown in Figure 16. Just like with He5015, other He5875 experiments (Büscher et al. 1995;Milosavljević & Djeniže 2002) only measured FWHM values. Data presented in Figure 16 demonstrate that WDPE data again show the same trends as previous data and theoretical predictions. The agreement between other experiments, our experiments, and the theoretical predictions is excellent at lower densities (n e 2.5 × 10 17 cm −3 ), but data collected at higher densities seems to indicate that both B97 and our line-shape theories may underpredict the FWHM at a given n e . Similar trends are also observed in Figures 14 and 15.
The much larger (≈30%) He5875 n e errors and elevated n e values derived for z3195 compared to z3402 in Figures 14 and  15 can be traced back to significant differences in optical depths between the two experiments. The highest optical depth measured for z3195 is ≈10, while that for z3402 is only ≈5. This optical depth difference is most likely caused by the z3195 He5875 data being collected closer to the gold wall compared to z3402. A closer proximity to the gold wall results in more  available photons to drive the 2s (He5875 lower level) populations and thus increase the optical depth of the line. As part of the fitting process, we convolve the line profile with the instrumental broadening. A line profile with a higher peak (as is the case for z3195 compared to z3402) will be more susceptible to changes in the measured instrumental broadening. At later times (higher n e values), the measured line optical depth decreases, causing the measured instrumental broadening to have a smaller influence. The line is also wider at higher n e values, leading to yet higher insensitivities regarding the measured instrumental broadening. We would therefore expect the He5875 n e errors to decrease at higher n e . This trend is confirmed by the z3195 data presented in Figures 14 and 15. We note here that the data presented in Figure 13 seem to indicate that the He5875 optical depth seems to slightly increase/stay constant with experiment time. However, the data shown in that figure are the recorded experimental profiles. To extract the true optical depth of the line, we must take instrumental broadening effects into account. Increases in experimental n e result in broader He5875 lines, which means that instrumental effects become less important at higher densities. As a result, fitted optical depths decrease as a function of time. The larger optical depth for z3195 also put the line core close to the CCD detection limit. While there is no direct evidence that this line is saturated in our data, it would explain why the FWHM and n e values measured for this line are higher than those for an experiment conducted at nominally the same conditions-z3402.
N e uncertainties for each shot shown in Figures 14 and 15 were derived in a similar manner to the He5015 data presented in Figures 10 and 11, except that now the influence of the selfemission processes is also included. Just like with the He5015 data, instrumental broadening uncertainties dominate the errors shown in Figures 14 and 15. Self-emission correction uncertainties, including the relative scaling and shift of the emission and absorption spectra during the correction process, have a negligible effect (<10%) on the He5875 n e value. The He5875 line is isolated in our spectrum (see Figure 4), which allows for easier continuum definition compared to the He5015 data. Continuum normalization errors also have a negligible effect on the final line shape and n e . CCD photon-counting uncertainties are also negligible for the He5875 line.

Van der Waals Broadening Contributions
The data presented in Figures 10, 11, 14, and 15 strongly suggest that, despite a significant and varying neutral density in the WDPE plasma, Van der Waals broadening does not need to be invoked to explain our experimental results. At the maximum WDPE neutral densities (n neutral ≈ 1 × 10 18 cm −3 ), both the U55 and D76 Van der Waals broadening theories predict negligible broadening for H I and He I, making this result not entirely unsurprising.
Our conclusions regarding Van der Waals broadening are confirmed by the He5015 data. The experiments presented in Figures 10 and 11 do not show a need to invoke any Van der Waals broadening models to explain any n e disagreements between Hβ and He5015. In fact, current Stark broadening theories seem to slightly overestimate the line width for a given n e value. If Stark broadening was not sufficient to fully explain experimental data, we would expect an increased influence at lower Hβ n e values, given that n neutral is highest at that time. Figure 17 depicts the n e percent difference between Hβ and He5015, plotted as a function of experimental n neutral . If Van der Waals broadening were important, we would expect to see a positive correlation between the plotted parameters. No such correlation is observed. The upper panel shows the results for B97, while the lower panel shows our line-shape trends. We include the average n e of each plotted experiment in the legend.
He5875 FWHM and n e measurements also do not necessitate any Van der Waals broadening. Figure 18 shows Hβ-He5875 n e percent differences as a function of n neutral , and again no obvious trends are observed. Shots z3195 and z3402 (see Figures 14 and 15) contained ≈7 × 10 17 cm −3 neutral helium atoms. Potential reasons for why z3195 FWHM and n e measurements are higher than those for z3402 are discussed in Section 5.1.2. We do not attribute these differences to Van der Figure 16. Comparison of FWHM collected during other experiments (Büscher et al. 1995;Milosavljević and Djeniže 2002) to WDPE data and predictions obtained from B97 (solid blue) and our own line-shape predictions (red dashed). Figure 17. Top: Hβ-He5015 n e percent differences plotted as a function of n neutral for B97 line-shape data. No trends are visible, indicating that Van der Waals broadening is not needed to explain measured line shapes. The average n e for each shot is given in parentheses after the shot designation. Bottom: Same as top, but for our line shapes.
Waals broadening. Figure 18 shows trends similar to those in Figure 17, leading to the same conclusions.

Relating Experimental n e Uncertainties to Spectroscopic-Photometric DB Mass Differences
Drawing astrophysically meaningful conclusions using our experimental results requires understanding the relationship between our experimental He I n e uncertainties, DB atmospheric n e values, and the corresponding stellar mass. Because both lines under investigation in this paper form over large n e ranges in DB atmospheres (see panel (B) in Figure 8), it is difficult to develop a comprehensive understanding of this relationship without performing detailed fits to DB observations that take our experimental line-shape and n e uncertainties into account. Such a study is beyond the scope of this paper.
To derive an approximate atmospheric n e -stellar mass relationship, we calculate the average n e value over which the He5015 and He5875 lines are formed at a given log g and T eff using the same data as plotted in Figure 8 and relate the results to stellar mass. For He5875, we average over all n e within 125 Å of line center in Figure 8, while He5015 lineforming n e is calculated by averaging all n e values within 25 Å of line center. We vary log g from 7.0 to 9.0 in 0.1 step increments and set T eff = 17,500 K for the purposes of this test. The log g and T eff combinations are converted to a stellar mass using the Montreal White Dwarf Database 7 (Dufour et al. 2017). The results in Figure 19 show the resulting DB lineforming n e -mass relationship for both He5015 (red) and He5875 (blue). As outlined in the discussion accompanying Figure 8, the WDPE reaches higher n e as compared to DB atmospheric line-forming regions. This is again reflected in Figure 19. However, we are only interested in deriving the change in mass as a function of the change in n e for the purposes of this test (i.e., d(mass)/d(n e )). Since both of the presented mass-n e relationships are nearly linear, we use the slope of the linear fits to quantify the mass uncertainty associated with our experimental He I n e uncertainty. The slope is independent of overall n e , limiting the significance of the small mismatch between experimental and DB line-forming n e .
The derived DB atmospheric n e -stellar mass trends reveal a roughly one-to-one relationship between mass and atmospheric n e . Hence, our experimental He I n e uncertainties approximately correspond to DB mass uncertainties. The shots with the lowest He I n e uncertainties are z3402 for He5875 and z3462_F, both of which have data points with a ≈10% uncertainty, implying a 10% DB mass uncertainty. This is smaller than the ≈20% mass difference between spectroscopic and photometric masses at 16,000 K T eff 20,000 K. Other experiments presented in this paper have larger He I n e uncertainties, but all presented data sets are internally consistent and agree with other experimental data (see Figures 12 and 16). Therefore, repeating these experiments with a platform that results in lower uncertainties would be unlikely to change these conclusions. It is important to note, however, that the approach presented here is only an estimate of the atmospheric n e -stellar mass relationship. A fully integrated test, where many detailed DB fits that incorporate all our experimental uncertainties (n e , line shape, etc.) are performed, is needed to solidify these conclusions.

Conclusions
In this paper, we present experimental data to validate lineshape calculations for the He5015 and He5875 lines using two separate theories: the semi-analytical B97 and our computersimulation-based He I Stark line shapes. The U55 and D76 Van der Waals theories predict negligible neutral broadening at WDPE conditions, which we confirm with our experimental data. The B97 theory is the standard in the WD community, and spectroscopically derived DB masses are thought to be dependent on their accuracy.
Our results show that Stark He5015 and He5875 line-shape theories (B97 and our theory) are accurate at n e values found in DB atmospheres. Furthermore, our experimental uncertainties indicate that differences between spectroscopic and photometric masses most likely cannot be attributed solely to Stark Figure 18. Top: Hβ-He5875 n e percent differences plotted as a function of n neutral for B97 line-shape data. No trends are visible, indicating that Van der Waals broadening is not needed to explain measured line shapes. The average n e for each shot is given in parentheses after the shot designation. Bottom: Same as top, but for our line shapes. Figure 19. Relationship between line-formation averaged n e and stellar mass for the He5015 (red) and He5875 (blue). We also show the linear fits to these relationships as dashed lines. The slopes of these linear fits (d(mass)/d(n e )) allow us to estimate how n e uncertainties in our data affect derived spectroscopic DB masses.