TOWARD A NETWORK OF FAINT DA WHITE DWARFS AS HIGH-PRECISION SPECTROPHOTOMETRIC STANDARDS

Sub-percent global standardization of photometric calibration in astronomy remains elusive. Major ongoing and planned astronomical surveys (including LSST,⁷ PanSTARRS,⁸ SDSS,⁹ the Dark Energy Survey with DECam,¹⁰ JWST,¹¹ GALEX,¹² and WISE¹³ ) will have to attain band-to-band photometry and all-sky uniformity to better than 1% to realize their full scientific promise. As a concrete example, consider the use of SNe Ia to probe the history of cosmic expansion and determine the properties of dark energy. Currently, photometric calibration issues dominate the uncertainty budget on the equation of state of the dark energy, w (e.g., Sullivan et al. 2011; Suzuki et al. 2012; Betoule et al. 2014). Weak lensing tomography with LSST also demands sub-percent accuracies in color for reliable photometric redshift determination (e.g., Gorecki et al. 2014 require a systematic uncertainty of 0.005 mag or less for their photometric redshift simulations).

Most of these surveys (as well as all of the post-Hubble Space Telescope (HST) visible-band astronomy in the foreseeable future) are or will be pursued from the ground. For calibration at ground-based facilities, one source of uncertainty is the time-variable transmissivity of the atmosphere, in both chromatic (Rayleigh scattering, Ozone absorption, Mie scattering, molecular absorption, aerosol) as well as gray (cloud) terms. Attenuation from clouds, water-vapor, and aerosols can change rapidly and on differing angular scales and amplitudes, as well as over all timescales (e.g., Querel & Kerber 2014). A variety of methods are currently used to track and account for such effects, including monitoring with Light Detection and Ranging (LIDAR) techniques, using dual-band Global Positioning System (GPS) metrology (e.g., Li et al. 2014), and modeling the atmosphere (e.g., with MODTRAN).

There remain, however, key obstacles to obtaining fluxes/colors in physical units at the 1% level or better.

• 1.  
  Vega was one of six A0V stars used to establish the color zero point on the photometric system of Johnson & Morgan (1953) by defining the mean U − B and B − V colors of the six stars to be zero. This definition was further extended to Cousins ${R}_{C}-{I}_{C}$. Vega's spectral energy distribution (SED) was tied to tungsten-ribbon filament lamps and laboratory blackbody sources employed as fundamental standards (Oke & Schild 1970; Hayes & Latham 1975, and references therein). The laboratory sources (placed at distances of the order of a mile) and Vega were observed through a telescope and spectral scanner, but corrections are necessary to account for extinction through the atmosphere along very different paths. In particular, the wavelength dependence of the aerosol scattering could not be ascertained with sufficient confidence and the consequent uncertainty permeates the empirical determination of Vega's SED. Alternatively, stellar atmosphere models of Vega have also been used as its intrinsic SED, in lieu of the empirical comparison against a laboratory source (especially for extension into the UV and IR). Even if the models were perfect, there are other complications. Vega's SED is punctuated by several unusually shaped absorption lines. Vega has an excess of NIR emission longward of 1–2 μm, which is likely a result of its dust ring (Bohlin 2014) and possibly its rapid rotation (Peterson et al. 2006). It also has an excess of UV emission relative to a 9400 K model (a result of its rapid rotation; Bohlin et al. 2014). These may introduce systematic errors when such models are used. As a result, the absolute calibration of Vega has known deficiencies at the ∼1% level in the visible region and at the few percent level in the near-IR and near UV regions (Blackwell et al. 1983; Selby et al. 1983; Mountain et al. 1985).
• 2.  
  Calibrations transferred from any one primary standard to other objects around the sky using ground-based observations suffer from seasonal and site-specific systematic errors, especially in the gray component of extinction through the terrestrial atmosphere. Without a well-spaced network of calibrators, "self-calibration" (Schlafly et al. 2012) or "ubercal" (Padmanabhan et al. 2008) methods cannot guarantee systemic all-sky uniformity. They are especially vulnerable to any seasonal variations in the declination dependence of atmospheric extinction.
• 3.  
  Other currently used primary SED standards, such as BD+17°4708 and P330E, are too bright to be directly imaged by the large-aperture telescopes employed by dark energy studies. Secondary standards are typically no fainter than magnitude 15, which is not within the operating dynamic range of surveys such as PanSTARRS, DES, and LSST. Indirect links require bridging across ∼8 mag of dynamic range to calibrate the surveys. This allows various systematic errors to creep in.

The best way to circumvent the issues involved with terrestrial atmospheric extinction would be to fly a standard laboratory source, such as the National Institute of Standards and Technology (NIST) calibrated standard, above the atmosphere. The planning and execution of such an expensive experiment takes a long time, however, and much progress can be made toward establishing sub-percent accuracy spectrophotometry now using existing facilities and at a much lower cost.

DA white dwarfs (WDs) have atmospheres dominated by hydrogen, and thus are the simplest stellar atmospheres to model. The opacities are known from first principles and, in the temperature ranges (20,000–80,000 K) in which we are interested, the photospheres are purely radiative and photometrically stable. The SED from such an atmosphere can be defined by just two parameters: effective temperature (${T}_{\mathrm{eff}}$) and surface gravity ($\mathrm{log}g$ ). These parameters can be determined spectroscopically from a detailed analysis of the H i Balmer profiles without reference to any photometry. Thus, the SED, from the UV to the IR, can be calculated at arbitrary spectral resolution and used to calibrate any photometric passband or spectroscopic system. A flux normalization in any chosen passband and accurate derivation of reddening/extinction are the only other quantities required to fully characterize the received flux and to establish such objects as spectrophotometric standards. Bohlin (2014) used the HST Space Telescope Imaging Spectrograph (STIS) to observe three bright DA WDs spanning a range of temperatures. These objects are unaffected by reddening as interstellar absorption at Lyα implies a low H i column density corresponding to a reddening of $E(B-V)\lt 0.0005$. Bohlin (2014) found their relative flux distributions to be internally consistent with model predictions from spectroscopic ${T}_{\mathrm{eff}}$ and $\mathrm{log}g$ to better than 1% in the wavelength range 0.2–1.0 μm. This level of internal consistency is superior to direct comparison with Vega fluxes from HST/STIS (Bohlin & Gilliland 2004).

The HST photometric scale is thus defined relative to model SEDs for three nearby DA WDs (Bohlin 2007). We employ the HST "Vegamag" photometric scale throughout this work. Holberg & Bergeron (2006) used synthetic photometry of DA WDs to place UBVRI, 2MASS JHK, SDSS ugriz, and Stromgren ubvy magnitudes on the HST photometric scale to 1%. Holberg et al. (2008) confirmed this calibration by using a set of DA WDs with good trigonometric parallaxes that agreed at the 1% level with their photometric parallaxes from the Bergeron photometric grid.¹⁴

Modern large-telescope imaging surveys have an overlapping brightness range where the signal-to-noise ratio (S/N) is excellent for V ∼ 17–19 mag and where these objects are unsaturated in the survey data. Thus, we must extend the WD scale to fainter reference objects.

The three WDs used to define the CALSPEC calibration are bright and near enough that line-of-sight extinction to them can be ignored. However, the WDs in our desired brightness range are expected to have mild but easily characterizable reddening that any experiment we design must be able to quantify. There may be some faint DA WDs at 17 to 18 mag in the Galactic polar caps, but for all-sky coverage, we cannot get around having to deal with extinction. Theoretically, spectroscopy to derive ${T}_{\mathrm{eff}}$ and $\mathrm{log}g$ of suitably chosen DA WDs between 17 and 19 mag would be enough to define their SEDs. However, these brightness levels place the WDs at sufficiently large distances where interstellar extinction cannot be ignored.

Alternatively, we could choose a purely empirical spectrophotometric extension of the CALSPEC calibration by observing a more diverse class of objects at the desired brightness with HST/STIS (so that the measurements do not suffer from the effects of the terrestrial atmosphere). Such a sample could include redder objects and the question of reddening would be inconsequential as we would measure the net energy distribution. However, to build a network of a couple of dozen of such stars, so that some subset of them are available at all times from all sites though a variety of airmasses, would require an amount of observing time with STIS that would be prohibitively large. It would also preclude the possibility of extrapolating the calibration to wavelengths outside the range of the observations. The STIS exposure time calculator (ETC) reports integration times of several thousand seconds to reach a S/N of 100 for a 40,000 K blackbody with V = 18 mag in just one of the three configurations that would be minimally required to cover the wavelength range from 3000–18000 Å.

The approach we take utilizes the principle behind the CALSPEC basis for calibration, but builds up the system independently. By doing so, we also test the proposition that the DA WD models are adequate, since we intercompare the results from many more objects for internal self-consistency. Our results need to stand independent of any prior calibrations to be sufficiently robust, which we think is worth the additional complications that our approach invokes.

In Table 1, we list nine putative standard stars that are part of our Cycle 20 HST Program (GO 12967, P.I.: Saha), including the best-available ground-based V-band magnitudes. Our aim in this program is to derive the effective temperature, surface gravity, and interstellar reddening for these stars, which then fully parameterize (via the stellar atmosphere model) the SED incident on top of the Earth's atmosphere, which is normalized with respect to the HST flux scale.

The essential goal of this program is to establish a set of spectrophotometrically self-consistent calibrated standard stars, tied as well as possible to an absolute physical scale. Along the way, we are empirically testing the proposition that DA WD atmospheric models result in realistic SEDs by examining whether the observed flux ratios from stars with different atmospheric parameters are consistent with model predictions after derivation and application of corrections to account for interstellar reddening. In this paper, we present initial findings that define the level of consistency it is possible to achieve using these procedures.

We aim to place a select set of DA WDs between magnitudes 17 and 19 on a common physical scale for the flux incident at the top of the terrestrial atmosphere. This will remove uncertainties in the currently used SED standards, which are plagued by uncertain corrections for extinction within the terrestrial atmosphere. The expectation is that sub-percent internal consistencies can be achieved, which will be an improvement by several factors over any currently employed photometric or spectrophotometric system. Thus, a set of better-calibrated standard SED sources, referred to as a physical (absolute) system, a subset of which is always visible from any ground-based observatory, will serve as universal standard calibration sources for spectrophotometry and photometry.

In Section 2, we describe the data attributes and reductions for the HST imaging as well as the ground-based spectroscopy. In Section 3, the photometry from the HST imaging is described, as is with the procedure used to place the measured magnitudes on the HST photometric scale defined by the three bright DA white dwarfs. The modeling of the emergent and received flux from each of the white dwarfs is described in Section 4, as is the derivation of reddening/extinction to reconcile the model and observed fluxes. In Section 5, we present an alternate approach to the reconciliation of model fluxes and observed counts that bypasses some of the assumptions made in Section 4 and show that the two complementary approaches produce the same results to within the accuracy limitations of the data at hand. This cross-validates some key assumptions and results of both procedures. In Section 6, we present the predicted magnitudes of four stars in the respective passbands of three current imaging surveys. We conclude (in Section 7) with a discussion of caveats and lessons learned that have informed the observing strategy for our HST Cycle 22 program.

This program has three operational components. The first consists of multi-band photometry from HST/WFC3 images of each star that accurately tie the fluxes to the HST photometric scale. The second is a grid of appropriate model atmospheres used to generate detailed, absolutely calibrated flux distributions and model spectra (in particular, the line profiles of Balmer lines) for each star. The third is high S/N Balmer line spectra, nominally obtained with the Gemini Multi-Object Spectrograph (GMOS) instrument on Gemini South. The observed Balmer line profiles are analyzed in reference to the model profiles to accurately determine ${T}_{\mathrm{eff}}$ and $\mathrm{log}g$ for each star. The model SEDs for stars with atmospheric parameters determined in this manner are then compared to the observed HST/WFC3 photometry in multiple bands with respect to a to-be-determined contribution from interstellar reddening along the line of sight to each star. The best match between model and measurement determines the reddening/extinction parameter(s).

2.1. WFC3 Observations

Wide Field Camera 3 (WFC3) observations have been obtained for all nine of the target stars listed in Table 1, These observations yield absolute photometric fluxes in the five bands ($F336W$, $F475W$, $F625W$, $F775W$, and $F160W$) that are used to normalize the reddening-corrected model spectrum that defines the stellar flux distributions. In Figure 1, we show the five WFC3 images for one of our targets, SDSS-J102430.93, along with an SDSS finder chart image of this star.

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All targets were placed near the center of the WFC3 UVIS2 detector. Each object is observed two or three times in each band with small dithers to facilitate cosmic-ray rejection. The bands were chosen to ensure that the temperature, surface gravity, and reddening could be independently determined without significant covariance. Stromgren u ($F336W$) is chosen because it sits entirely shortward of the Balmer jump, and thus allows for a way to measure it. For our targets, the Balmer jump is sensitive to temperature. This dependency, however, is different from that of the Paschen continuum slope. The $F336W$ observations therefore help to mitigate degeneracies between reddening and temperature—a critical prerequisite for our study. $F475W$, $F625W$, and $F775W$ emulate the Sloan gri bands, respectively, and were chosen so that, in addition to measuring the Paschen continuum, they also provide a direct measure in the bands being used in large surveys today (e.g., SDSS, PanSTARRS, and DES) and those planned for the future (LSST). $F160W$ is the HST/WFC3 equivalent for the H band, and it anchors measurements of our objects as far into the infrared as HST will allow.

The exposure times were chosen to allow for an S/N better than 200 for each of the targeted white dwarfs while staying shorter than half of the time to saturation. The arrangements of the various exposures times within an orbit were crafted to minimize "dead time" between exposures and optimize program efficiency. A post-flash flux was added to ensure the background level reached 12 electrons to mitigate charge transfer efficiency (CTE) losses. Additional parallel observations that include stars within a few arcminutes from each of our target white dwarfs were obtained with the Advanced Camera for Surveys (ACS). Photometry for them, as well as an evaluation of their usefulness as supplementary standard stars, will be presented in a future paper.

2.2. GMOS Observations

We used GMOS (Hook et al. 2004) at Gemini South to obtain moderate-resolution ($R\sim 1000$), high S/N spectroscopy of our WDs. The analysis of the detailed shape of the H i Balmer profiles (Hβ to Hζ) provides primary spectroscopic estimates of ${T}_{\mathrm{eff}}$ and $\mathrm{log}g$ which are independent of photometry or SED slopes.

The observations were obtained in queue-scheduled mode on GMOS with the B600 grating and a long slit having a width of 15. Night sky lines had a FWHM of ∼12 Å. In general, 6 × 1500 s exposures were made of each target. The inter-chip gaps between the three GMOS CCDs were filled by taking three observations at one grating tilt and three at another. The entire combined spectrum contiguously covers the wavelength region between 3500 and 6360 Å with a dispersion of 0.92 Å/pix. We resampled the spectra onto a single linear scale at 1 Å per bin. We used standard IRAF¹⁵ routines to process the CCD data and optimally extract (Horne 1986) the spectra. The overall S/N per resolution element is very high, averaging from 80 to 170 for our targets. We used our own IDL routines (Matheson et al. 2008) to flux calibrate the spectra using Feige 110 (Stone 1977). Strictly speaking, the flux calibration is not needed (and in any case rests on a less accurate basis than the ones we seek to establish), and the analysis presented in Section 4.1 is independent of the flux calibration of the spectra. Nevertheless, getting the continuum slopes as correct as possible helps when analyzing the shapes of the wide Balmer lines.

At this time, there are five targets (of the nine in Table 1) for which we have complete spectra and photometry. Their spectra are shown in Figure 2. One of these, SDSS-J203722.16, exhibits narrow emission lines in the cores of the Balmer absorption features. The emission lines shift with time. This could indicate the presence of a low-luminosity companion, such as an M star, although there are other possibilities. We exclude this object from all further analysis. The remaining four spectra are very characteristic of DA white dwarfs and appear to be free of contamination from companions.

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Because the calibration standard star was not always observed contemporaneously with the targets, we validated the calibration by comparisons with the expected fluxes obtained by preliminary reddened model fluxes. In Figure 3, we show such a comparison. As discussed in Section 4, the method used to analyze individual Balmer profiles is relatively insensitive to any residual calibration uncertainties.

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The overall response of the grating and detector yielded a count-rate spectrum that peaks very near Hβ and falls off significantly to the blue. At Hζ, the signal is approximately 30% of peak, leading to a significant S/N gradient across the Balmer lines. Fortunately, the overall S/N is very high, averaging from 80 to 170 per bin for our targets. For some targets, we have obtained supplemental spectroscopic observations using IMACS (Dressler et al. 2011) at Magellan and the Blue Channel (Schmidt et al. 1989) at the MMT. They have similar S/N and resolution, and cover the same spectral range as the GMOS configuration.

As this program requires extremely accurate and precise determinations of the white dwarf flux, we have elected to use two independent methods for photometry of our WFC3 observations. The drizzled (drz) images from the Mikulski Archive for Space Telescopes (MAST) are processed using SExtractor (Bertin & Arnouts 1996). The images are first masked to remove pixels in the flt image file which are only derived from a single exposure (determined from the ctx extension) and sources above the saturation threshold. A 64-pixel background mesh is used with an 8-pixel filter size to smooth the image and locally estimate the background. This procedure removes virtually all spurious objects (cosmic rays, drizzling artifacts, etc.). We use a range of apertures from 4 to 25 pixels to measure the flux of each source. As WFC3 has a physical image scale of 004/pix on UVIS and 013/pix on the IR channel, this corresponds to 016–10 on UVIS and 052–325 on the IR channel. We construct curves of growth for each image from objects with stellar point-spread functions (PSFs). We use a simple filtering routine to remove any objects that are more than 3σ away from the mean curve of growth in more than three apertures. This removes any objects with close companions and any non-stellar sources. We use the mean curve of growth of the image to interpolate between the measured aperture flux in each pixel to determine the precise flux with a physical aperture size of radius 04. The weight map output from multidrizzle is used to estimate the uncertainties.

Measurements were also made using an interactive program written by one of us (A.S.). A curve of growth (COG) as a function of aperture size is constructed and the background level is set to be the value for which the COG is flat in the radius range of 9–12 pixels (036–048) in the UVIS images and 6–8 pixels (078–104) in the IR channel data. The "background" thus measured is actually within the low-level wings of the stellar PSF, but allows the use of a tight aperture so that the S/N is not compromised and chances of encountering any unremoved cosmic rays or other pathologies is minimized. Using published encircled energy curves,¹⁶ measurements made this way can be corrected systematically for "infinite" radius apertures (and referred to the true background), which include all of the incident light from a given object. Look-up tables for these corrections were made for each passband and applied as needed. These interactively made measurements, although time-consuming, are judged to be a good investment toward quality assessment because they quickly reveal the common kinds of image pathologies that defy assumptions inherent in our data models. These interactive measurements were made on both flt and drz images, where the flt images were corrected for variations in pixel area. Concordant measurements on both sets of images indicate whether or not the multidrizzle process compromises data quality (e.g., by being overzealous about CR rejection and cropping the tips of bright stars). Agreement across the two measuring methods establishes that methodology-dependent systematic errors are not being injected.

As the WFC3 detector sensitivity is a function of time, the zeropoints are time dependent. Rather than rely on the fiducial zeropoints supplied by MAST, we measured the zeropoints directly using contemporaneous observations of the CALSPEC primary standard, GD153, obtained through HST GO program 13575 (P.I.: S. Deustua). These are applied as an additional adjustment to the fiducial MAST zero point. Subrastered images of GD153 are available with the target at the same position on the detector as our DA white dwarfs, however, not in all of our passbands ($F775W$ is missing). We are obtaining contemporaneous observations of all three primary standards in Cycle 22 to determine instrumental zeropoints and to tie our measurements directly to this photometric system.

In this work, we have measured GD153's instrumental magnitudes using the same process for our targets as described above. We applied the fiducial MAST zeropoints to our instrumental magnitude, and compared these magnitudes against the synthetic photometry of the CALSPEC model of GD153 (gd153_fos_003)¹⁷ through the WFC3 passbands. The synthetic photometry is described in Section 4 by Equation (5). While this procedure is not as optimal as tying our measurements directly to the three HST primary standards, it is the best methodology possible given the extant data.

We found a difference between the synthetic and measured magnitudes that exhibits a strong linear trend with passband frequency. The measurements in each passband, and the linear trend, is shown in Figure 4. There is no process through which our measurement technique, which treats all of the images individually, can introduce such a linear trend with frequency. The largest discrepancy was found in the ultraviolet and the smallest differences in the optical. No significant difference was found in the WFC3/IR channel, which has a completely different detector and light path through the camera. The observed trend therefore indicated a possible difference between the reported and true efficiency of the UVIS detectors.

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This finding was independently replicated by staff scientists at STScI. As the two WFC3/UVIS CCDs were manufactured on different wafers and foundry production runs, their physical properties, such as quantum efficiency and thickness, differ. Studies at STScI have found that UVIS2 is $\sim 30\%$ more sensitive in the UV than UVIS1. In the range 3500–7000 Å, the two CCDs have comparable sensitivity, while at wavelengths longer than 7000 Å, UVIS1 is a few percent more sensitive. Furthermore, the two detectors have different detection layer thicknesses. For UVIS1, the median thickness is 16.04 ± 0.23 μm and, for UVIS2, it is 15.42 ± 0.58 μm.¹⁸

New flat-fields that treat the two UVIS detectors individually have been created and will be applied to future MAST products. For this work, we correct the fiducial MAST zeropoints for each of our passbands with the linear relation we found. These corrected zeropoints are applied to the measured flux to produce the observed magnitudes for the targets used in this work. The photometry and measurement uncertainties for these targets are listed in column 3 of Table 2. The measured magnitudes are therefore described by

$$\begin{eqnarray}&&{m}_{T,i}^{O}=-2.5\;{\mathrm{log}}_{10}({\phi }_{T,i}^{O})+{\mathrm{ZP}}_{T}^{\mathrm{MAST}}+{\rm{\Delta }}{\mathrm{ZP}}_{T}^{\mathrm{GD}153},\end{eqnarray} \tag{ 1 }$$

where m is the natural magnitude, ϕ the measured flux of the object, i, ZP${}_{T}^{\mathrm{MAST}}$ is the instrumental zero point published in MAST, and ΔZP${}_{T}^{\mathrm{GD}153}$ is the correction to the fiducial MAST zero point derived, as described above, using observations and synthetically predicted photometry of GD153, observed through passband T. "O" is used to denote observed quantities.

Table 2.  DA White Dwarf Observed Magnitudes, Fit Parameters, and Synthetic Magnitudes

Object	Passband	Observed	AV	${T}_{\mathrm{eff}}$	$\mathrm{log}g$	Synthetic	Extinction	Extinguished	Residual
Name		Magnitudea				Magnitude	(${R}_{V}=3.1$)	Magnitude
		(mag)	(mag)	(K)		(mag)	(mag)	(mag)	(mag)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
	$F336W$	17.344 (0.003)	⋯	⋯	⋯	17.176 (0.001)	0.171	17.347 (0.008)	−0.003
	$F475W$	19.189 (0.003)	⋯	⋯	⋯	19.058 (3E-4)	0.125	19.183 (0.006)	+0.006
SDSS-J010322.19	$F625W$	19.424 (0.003)	0.101	59530	7.53	19.340 (4E-5)	0.088	19.428 (0.004)	−0.004
	$F775W$	19.588 (0.004)	(0.005)	(260)	(0.02)	19.522 (7E-5)	0.068	19.590 (0.003)	−0.002
	$F160W$	20.105 (0.005)	⋯	⋯	⋯	20.081 (7E-5)	0.021	20.102 (0.001)	+0.004
	$F336W$	17.315 (0.004)	⋯	⋯	⋯	16.856 (7E-4)	0.453	17.309 (0.008)	+0.006
	$F475W$	19.004 (0.003)	⋯	⋯	⋯	18.679 (1E-4)	0.331	19.010 (0.006)	−0.006
SDSS-J102430.93	$F625W$	19.170 (0.003)	0.269	40620	7.75	18.937 (1E-4)	0.234	19.171 (0.004)	−0.002
	$F775W$	19.291 (0.003)	(0.005)	(124)	(0.01)	19.111 (2E-4)	0.180	19.291 (0.003)	−0.001
	$F160W$	19.718 (0.005)	⋯	⋯	⋯	19.658 (2E-4)	0.055	19.712 (0.001)	+0.005
	$F336W$	17.301 (0.003)	⋯	⋯	⋯	17.213 (0.002)	0.089	17.302 (0.013)	−0.000
	$F475W$	18.776 (0.002)	⋯	⋯	⋯	18.708 (4E-4)	0.064	18.773 (0.009)	+0.003
SDSS-J120650.4	$F625W$	18.912 (0.002)	0.052	23650	7.89	18.871 (1E-4)	0.046	18.916 (0.006)	−0.004
	$F775W$	19.033 (0.003)	(0.008)	(46)	(0.01)	19.000 (2E-4)	0.035	19.035 (0.004)	−0.002
	$F160W$	19.443 (0.004)	⋯	⋯	⋯	19.428 (2E-4)	0.011	19.439 (0.001)	+0.004
	$F336W$	17.113 (0.003)	⋯	⋯	⋯	16.778 (0.001)	0.337	17.115 (0.008)	−0.002
	$F475W$	18.925 (0.002)	⋯	⋯	⋯	18.676 (2E-4)	0.247	18.923 (0.006)	+0.002
SDSS-J163800.36	$F625W$	19.136 (0.003)	0.200	64610	7.43	18.961 (5E-5)	0.174	19.135 (0.004)	+0.001
	$F775W$	19.279 (0.003)	(0.005)	(425)	(0.02)	19.144 (9E-5)	0.134	19.278 (0.002)	+0.001
	$F160W$	19.743 (0.004)	⋯	⋯	⋯	19.704 (9E-5)	0.041	19.745 (0.001)	−0.002

Note.

^aAll of the magnitudes are reported in the HST VEGAmag system. Uncertainties are quoted in parentheses.

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4.1. Line Profile Fitting

We have analyzed the Balmer profiles of each star in a manner similar to that described by Bergeron et al. (1992) and Liebert et al. (2005). We use Tlusty non-local-thermodynamic-equilibrium model atmospheres (Hubeny & Lanz 1995, version 203)¹⁹ to match the models used by Bohlin et al. (2014). Each Balmer profile is individually extracted with a window centered on the profile and extending well into the wings. These extracted profiles are "flattened" by constructing a linear interpolation between fixed points in the far wings of the profiles. This flattening places the burden of determining ${T}_{\mathrm{eff}}$ and $\mathrm{log}g$ solely on the quasi-symmetrical line profiles and decouples the analysis from temperature-dependent slopes in the underlying SED. The model profiles are flattened in precisely the same fashion. Minimum ${\chi }^{2}$ fits are sought between models and observations, and contours enclosing 68% and 95% of the total likelihood around the maximum likelihood estimate are constructed. The profiles, fit, and likelihood surface for SDSS-J102430.93 are shown in Figure 5.

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The likelihood surfaces show very low covariance between ${T}_{\mathrm{eff}}$ and $\mathrm{log}g$. This is unsurprising as the two parameters change the shape of the Balmer profiles in distinct ways. Uncertainties are directly estimated from the one-dimensional (1D) marginalized likelihoods for both parameters. As the spectra have to be dereddened before the line profiles are fit, we also examine the covariance between ${T}_{\mathrm{eff}}$ and $\mathrm{log}g$ and the fiducial value of $E(B-V)$ used to deredden the spectrum, derived from SDSS DR12 photometry. We find a weak linear covariance of both ${T}_{\mathrm{eff}}$ and $\mathrm{log}g$ over a range of ±0.05 in $E(B-V)$. This range in $E(B-V)$ is much larger than the uncertainty in the value determined from our WFC3 photometry. We find that the maximum difference in the best-fit values for ${T}_{\mathrm{eff}}$ and $\mathrm{log}g$ over the entire range of $E(B-V)$ are less than 0.3 of the uncertainty in ${T}_{\mathrm{eff}}$ and $\mathrm{log}g$. The changes in ${\chi }^{2}$ over the same value are not statistically significant and reflect the different centering of the grid of ${T}_{\mathrm{eff}}$ and $\mathrm{log}g$ over which the minimum χ² fit is determined. This demonstrates that the temperature and surface gravity determined from spectra are completely insensitive to the fiducial interstellar $E(B-V)$ used to deredden the spectrum.

Our methodology is insensitive to the choice of model. While model atmospheres are generated as a function of ${T}_{\mathrm{eff}}$ and $\mathrm{log}g$, these are nuisance parameters that serve as labels for a specific line shape. Two different models may have different values of ${T}_{\mathrm{eff}}$ and $\mathrm{log}g$ for the same line shape. Our procedure is only affected by relative flux differences at constant line shape. We estimate the systematic error arising from the choice of a specific set of model atmospheres by fitting the Rauch et al. (2013, hereafter RWBK) models for the HST primary standards derived in Bohlin (2014). We fit the RWBK models using the same procedure as for the observed white dwarf spectra. The resulting model atmospheres are normalized at 5556 Å. The relative flux ratios are shown in Figure 6.

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The Tlusty models at the fiducial temperature and surface gravity of the RWBK model are a close match to the Bohlin (2014) models. Fitting the line profiles of the RWBK models improves the agreement between the two models, and the synthetic photometry difference is less than 0.003 mag in all of the passbands. The residual difference in the line shape arises from a difference in the shapes of the line profiles between the RWBK and Tlusty models, which is likely a result of the different prescriptions used for Stark broadening. Sharp discontinuities in the relative flux ratio arise from differences in the continuum between the two models. This is most evident in the models of G191B2B. Both models are interpolated from model atmosphere grids, piece-wise over different wavelength ranges, and these different sections are combined together to create a model atmosphere spanning the full wavelength range at the desired values of ${T}_{\mathrm{eff}}$ and $\mathrm{log}g$. These discontinuities reflect differences in the grids and the different interpolation schemes adopted by the two models, rather than any difference in the underlying physics.

4.2. Synthetic Photometry and Interstellar Reddening

In most prior work on photometric standards, interstellar reddening has not been a significant issue. The stars presented here, though, are from one to three magnitudes fainter and significantly more distant, and so interstellar reddening needs to be considered, estimated, and applied. Our working model is to assume that the DA white dwarf models correctly predict the emergent flux from the star, and so the difference between observed magnitudes and model predictions must show the color signature from the Galactic reddening law. Since we have photometry in multiple bands, failure of either the Galactic reddening law or the SED model for emergent flux will show up as discrepancies. If all of the passbands are consistent with these assumptions, then the SED model plus the reddening/extinction value fully describes the received SED from the white dwarf. Accordingly, we proceed as described below.

4.2.1. Synthetic Photometry

With the best-fit temperature and surface gravity, we generate a full model SED for the white dwarf from 1300–25000 Å. We use the pysynphot package²⁰ and convolve the model SED with the response function of the observed HST passbands. We model the passband transmissions as the product of three components: the optical train (Opt), the filter (PB), and the Quantum Efficiency of the WFC3 detectors (OE):

$$\begin{eqnarray}&&T(\lambda )={T}_{\mathrm{Opt}}(\lambda )\times {T}_{\mathrm{PB}}(\lambda )\times \mathrm{QE}(\lambda ).\end{eqnarray} \tag{ 2 }$$

The published passband transmissions are based on pre-flight laboratory measurements and represent the fiducial system throughput. In principle, it is possible to determine the count-rate yielded by an object with an $F(\lambda )$ spectrum throughput by simply computing the integral of the flux through the passband. However, all three components of the transmission are time variable, and this is reflected in periodic updates to the published MAST zeropoints. We therefore derive synthetic zeropoints for each passband, T, using the definition of the HST "Vegamag" system that defines Vega as having a magnitude of zero at all wavelengths. The synthetic flux and magnitude, ϕ and m, respectively, are given by

$$\begin{eqnarray}&&{\phi }_{T,i}^{S}=\displaystyle \frac{{\int }_{0}^{\infty }\lambda {F}_{i}^{S}(\lambda )T(\lambda )\;d\lambda }{{\int }_{0}^{\infty }\lambda T(\lambda )\;d\lambda }\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{\phi }_{T,\mathrm{Vega}}^{S}=\displaystyle \frac{{\int }_{0}^{\infty }\lambda {F}_{\mathrm{Vega}}^{S}(\lambda )T(\lambda )\;d\lambda }{{\int }_{0}^{\infty }\lambda T(\lambda )\;d\lambda }\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}{m}_{T,i}^{S} & = & -2.5\;{\mathrm{log}}_{10}({\phi }_{T,i}^{S})+{\mathrm{ZP}}_{T}^{\mathrm{Vega}}\\ & = & -2.5\;{\mathrm{log}}_{10}({\phi }_{T,i}^{S})+2.5\;{\mathrm{log}}_{10}({\phi }_{T,\mathrm{Vega}}^{S})\\ & = & -2.5\;{\mathrm{log}}_{10}\left(\displaystyle \frac{{\displaystyle \int }_{0}^{\infty }\lambda {F}_{i}^{S}(\lambda )T(\lambda )\;d\lambda }{{\displaystyle \int }_{0}^{\infty }\lambda {F}_{\mathrm{Vega}}^{S}(\lambda )T(\lambda )\;d\lambda }\right),\end{eqnarray} \tag{ 5 }$$

where "S" is used to indicate synthetic quantities. The resulting synthetic magnitudes are tied to Vega. The SED, ${F}^{S}(\lambda )$, is not adjusted by the distance and radius of the white dwarf, and thus is unreddened. The reddening and normalization for the distance and radius are treated in Section 4.2.2. We use the CALSPEC composite stellar spectrum of Vega (alpha_lyr_stis_005)²¹ as our model for Vega's SED, ${F}_{\mathrm{Vega}}^{S}$. The CALSPEC model for Vega and GD153 used in Section 3 are normalized consistently internally, and thus synthetic magnitudes produced relative to the Vega model are directly comparable to the observed magnitudes that have been tied to GD153. In Cycle 22, we will obtain observations of all three primary standards in all of our bands and tie our observations to the average of all three, obviating the need to put our observations on the "Vegamag" system.

4.2.2. Interstellar Reddening

We compute the difference between the observed and synthetic magnitudes for each star, and fit these with (i) a constant to account for the normalization, which is the same at all wavelengths (and reflects the radius of and distance to the white dwarf), and (ii) a slope corresponding to a scaling of the Fitzpatrick (1999) reddening law, with the canonical ${R}_{V}=3.1$, appropriate for the Milky Way. The extinction values in each passband are given by the weighted extinction for each band as calculated via

$$\begin{eqnarray}{m}_{T,i}^{O}-{m}_{T,i}^{S} & = & {A}_{T}(\langle \lambda \rangle )\;\ast \;{A}_{V}^{i}+{c}_{i}\\ \langle \lambda \rangle & = & \displaystyle \frac{{\int }_{0}^{\infty }{\lambda }^{2}T(\lambda )\;d\lambda }{{\int }_{0}^{\infty }\lambda T(\lambda )\;d\lambda },\end{eqnarray} \tag{ 6 }$$

where $A(\langle \lambda \rangle )$ is the extinction computed at the effective wavelength of the passband T, $\langle \lambda \rangle$, defined for units of $E(B-V)$, again using the extinction of Fitzpatrick (1999) for ${R}_{V}=3.1$, and c_i is a wavelength-independent constant to account for a passband-independent overall brightness normalization for each white dwarf. The normalization constant, c_i, must be added to the synthetic magnitudes, ${m}_{T,i}^{S}$ to compute correctly the normalized magnitudes.

The results of applying the above procedure to the four DA white dwarfs for which the spectroscopic and photometric data are complete (discarding the discrepant object with the emission components in the Balmer lines) are shown in Table 2 and are displayed graphically in Figure 7. Included in the Table are the three fitted parameters, A_V, ${T}_{\mathrm{eff}}$, and $\mathrm{log}g$, intrinsic normalized magnitude before reddening, the extinction in each band, and the net predicted magnitude per band. Both synthetic quantities include the normalization constant, c_i. Comparison of column 9 with column 3 (predicted versus observed magnitudes with WFC3/HST) corresponds to the fit residuals per band given in column 10. Note that the extreme residuals are $+/-0.006$ with rms $=\;0.003$ mag.

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If we use the extinction law of O'Donnell (1994) instead of Fitzpatrick (1999), the derived A_V values change by 1 millimag, thereby indicating that our experiment is insensitive to differences between these forms of the reddening law.

The above analysis is contingent on the state of calibration of WFC3 and the assertion that CALSPEC standards have SEDs which are known to high accuracy. The shape of the WFC3 passbands is implicitly assumed to be correct. The SED of GD153, a DA white dwarf itself, is used to adjust the overall normalization of the effective passbands, and the CALSPEC SED of Vega is used to establish the zeropoints of color. These assumptions create a formal dependency on the state of calibration of WFC3, and on the currently adopted SEDs of GD153 and Vega. The "constructionist" approach taken in the above section is thus vulnerable to the state of calibration of WFC3. Thus, we have had to correct the zeropoints as in Equation (1).

However, there is some indication that the WFC3 zeropoints are uncertain at the 0.01 mag level. For example, the current MAST zeropoints are prefaced by a discussion that states "The tables below summarize results determined from a new reduction of all SMOV4, Cycle 17, and Cycle 18 observations of GD153, G191B2B, GD71, and P330E. The independent calibrations from the four stars agree to within 1% in most filters, and the photometric zero point is set to the average of the measurements." All four of these stars nominally have zero reddening, and have equal weight in the determination of the WFC3 zeropoints.

Given this reality, and our goal of achieving precisions substantially better than 1%, we have explored an alternate approach to optimally determining the reddening and SEDs of our white dwarfs. In this approach, we determine an optimal set of stellar parameters and passband-dependent aperture corrections by fitting the synthetic magnitudes for all stars simultaneous with the observed instrumental magnitudes, including for each star an extinction value and a passband-independent zero point. The only assumptions here are that DA white dwarf models do predict SEDs which are accurate to a few millimag or better, that the standard Galactic reddening law is valid for our targets, that the published WFC3 passbands are approximately correct in shape, and that the bright DA white dwarf GD153 has zero reddening.

We develop the following formalism, using the same notation adopted in Section 4.2.

The instrumental magnitude is just $-2.5\;{\mathrm{log}}_{10}\;N$, where N is the count rate of detected electrons.

$$\begin{eqnarray}&&{N}_{T,i}=k{\displaystyle \int }_{0}^{\infty }\lambda {F}_{i}^{O}(\lambda )T(\lambda )\;d\lambda \end{eqnarray} \tag{ 7 }$$

where k is a band independent constant that depends only on the telescope aperture. We can write

$$\begin{eqnarray}&&{m}_{T,i}^{X}=-2.5\;{\mathrm{log}}_{10}\;k-2.5{\mathrm{log}}_{10}{\displaystyle \int }_{0}^{\infty }\lambda {F}_{i}(\lambda )T(\lambda )\;d\lambda .\end{eqnarray} \tag{ 8 }$$

We subsequently use $K=-2.5\;{\mathrm{log}}_{10}\;k$.

To compare a synthetic magnitude to a measured HST instrumental magnitude, the synthetic magnitude is defined in terms of the model flux, ${F}_{i}^{S}(\lambda )$, as

$$\begin{eqnarray}&&{m}_{T,i}^{S}=-2.5\;{\mathrm{log}}_{10}\;{\displaystyle \int }_{0}^{\infty }\lambda {F}_{i}^{S}(\lambda )T(\lambda )\;d\lambda .\end{eqnarray} \tag{ 9 }$$

We define another band independent constant, C_i, to account for the geometry (radius of the star and distance to the star) that modulates for each target star, the flux at the stellar surface to that incident at the top of the terrestrial atmosphere. The instrumental magnitude, m^X, can then be written in terms of synthetic (S) quantities as

$$\begin{eqnarray}&&{m}_{T,i}^{X}=K-2.5\;{\mathrm{log}}_{10}\;{C}_{i}^{X,S}-2.5\;{\mathrm{log}}_{10}\;{\displaystyle \int }_{0}^{\infty }\lambda {F}_{i}^{S}(\lambda )T(\lambda )\;d\lambda .\end{eqnarray} \tag{ 10 }$$

Denoting $K-2.5\;{\mathrm{log}}_{10}\;{C}_{i}^{X,S}$ by ${K}_{i}^{X,S}$, this is simply

$$\begin{eqnarray}&&{m}_{T,i}^{X}={K}_{i}^{X,S}+{m}_{T,i}^{S}.\end{eqnarray} \tag{ 11 }$$

If the model fluxes are correct, then the synthetic instrumental magnitudes will equal the observed instrumental magnitudes. However, the synthetic magnitudes implicitly count all of the photons detected from the object, regardless of where they end up in the PSF, while the observed magnitudes count detected photons only within a finite radius (04 in our case). The resulting difference is the aperture correction and is band dependent because the PSF is also band dependent.A nominal value of the aperture corrections is available from MAST, but we leave them as values to be determined from the data. Denoting the aperture correction as ${{\rm{\Delta }}}_{T}$, we then expect that

$$\begin{eqnarray}&&{m}_{T,i}^{X}={K}_{i}^{X,S}+{m}_{T,i}^{S}+{{\rm{\Delta }}}_{T}\end{eqnarray} \tag{ 12 }$$

or, expressed another way,

$$\begin{eqnarray}&&{\delta }_{T,i}\equiv {m}_{T,i}^{X}-{m}_{T,i}^{S}-{{\rm{\Delta }}}_{T}={K}_{i}^{X,S}.\end{eqnarray} \tag{ 13 }$$

Note that although we expect the dominant contribution to ${{\rm{\Delta }}}_{T}$ to come from the aperture correction, any bandpass dependent error, including putative errors in calibrating the instrument, will contribute. The dependence of ${\delta }_{T,i}$ on the passband, T, is a powerful diagnostic. Equation (13) tells us that the wavelength dependence should be fully accounted for by the aperture correction and the synthetic model. If there is a discrepancy, then it can arise from several causes.

• 1.  
  The aperture corrections ${{\rm{\Delta }}}_{T}$ are incorrect.
• 2.  
  There is extinction affecting the real object that is not represented in the model.
• 3.  
  The model fluxes have errors not accounted for by extinction.

Given the distance to our stars, it is likely that they will be affected by some amount of extinction (interstellar and possibly circumstellar); we can fit for the reddening, and analyze the remaining residuals for the other causes listed above. The extinguished synthetic magnitude is

$$\begin{eqnarray}&&{m}_{T,i}^{S}=-2.5\;{\mathrm{log}}_{10}\;{\displaystyle \int }_{0}^{\infty }\lambda {F}_{i}^{S}(\lambda ){\alpha }_{i}(\lambda )T(\lambda )\;d\lambda ,\end{eqnarray} \tag{ 14 }$$

where ${\alpha }_{i}(\lambda )$ is the transmission function of the extinction for object i.

We parameterize ${\alpha }_{i}(\lambda )$ in terms of the quantities R_V and ${E}_{(B-V,i)}$, and calculate it using the prescription of O'Donnell (1994). We can then determine the extinction parameters, along with the ${{\rm{\Delta }}}_{T}$ and ${K}_{i}^{X,S}$ by minimizing

$$\begin{eqnarray}&&{\chi }^{2}(\{{E}_{B-V,i}\},\{{K}_{i}^{X,S}\},\{{{\rm{\Delta }}}_{T}\})=\displaystyle \sum _{T,i}\;({m}_{T,i}^{X}-{m}_{T,i}^{S}\\ &&\quad -{{A}_{T}\times {R}_{V}\times {E}_{i}(B-V)-{{\rm{\Delta }}}_{T}-{K}_{i}^{X,S})}^{2}\end{eqnarray} \tag{ 15 }$$

with the constraint that ${E}_{B-V,i}\geqslant 0\;\forall i$.

With our current four stars and five WFC3 passbands, ${\chi }^{2}$ depends on 12 parameters, while we have 20 measurements, so that the minimization problem is potentially well constrained. Additional stars and/or passbands will further constrain the solution. We keep R_V = 3.1 for the present, although it could in principle be included in the parameters being optimized.

There are two difficulties with this approach, however. The first one is trivial. There is a partial degeneracy between ${{\rm{\Delta }}}_{T}$ and ${K}_{i}^{X,S}$, in that an arbitrary constant can be added to all of the ${{\rm{\Delta }}}_{T}$ and then subtracted from all of the ${K}_{i}^{X,S}$ without changing the value of ${\chi }^{2}$. This needlessly complicates the job of the optimizer and makes the results harder to interpret. We deal with this by enforcing

$$\begin{eqnarray}&&\displaystyle \sum _{T}\;{{\rm{\Delta }}}_{T}=0.\end{eqnarray} \tag{ 16 }$$

The second difficulty is more serious. Experience with minimizing ${\chi }^{2}$ as given above shows that the minimum is consistently achieved by choosing ${{\rm{\Delta }}}_{T}$ so that the extinction of the least-extinguished star comes out to be zero. This is readily achieved, since the right choice for ${{\rm{\Delta }}}_{T}$ can mimic an arbitrary extinction, which is then effectively subtracted from the extinctions of all of the stars. This behavior is unphysical and must be remedied by an additional data-based constraint. We have chosen to use the observations of GD153 discussed in Section 3 in conjunction with the conclusion of Bohlin et al. (2014) that its extinction is zero to high precision.

To make use of this constraint, we treat the GD153 data in the same way as the program stars, but its contribution to ${\chi }^{2}$ is treated differently in that, when calculating the model flux, we assume ${E}_{B-V,\mathrm{GD}153}=0$. The GD153 contribution to the overall ${\chi }^{2}$ is multiplied by a weighting factor that is large enough to ensure that any significant deviation from this assumption is suppressed. The summation over bands includes only $F336W$, $F475W$, and $F625W$, since we currently lack GD153 data for the other bands. The model flux for GD153 uses the G11 values for ${T}_{\mathrm{eff}}$ and $\mathrm{log}g$ from Bohlin (2014):

$$\begin{eqnarray}{\chi }_{\mathrm{GD}153}^{2} & = & {w}_{\mathrm{GD}153}\\ & & \displaystyle \sum _{i}{({m}_{T,\mathrm{GD}153}^{X}-{m}_{T,\mathrm{GD}153}^{S}-{{\rm{\Delta }}}_{T}-{K}_{\mathrm{GD}153}^{X,S})}^{2}.\end{eqnarray} \tag{ 17 }$$

The weighting factor ${w}_{\mathrm{GD}153}=1/{0.01}^{2}$ for the results quoted below ensures that the GD153 reddening is very close to zero.

5.1. Results

The ${\chi }^{2}$ minimization of Equation (15) was carried out using IDL routine CONSTRAINED_MIN. To perform error analysis of the results, minimization was carried out a large number of times using independent samples from Gaussian distributions of ${T}_{\mathrm{eff}},\mathrm{log}g$ and $\{{m}_{T,i}^{O}\}$. The covariance matrix for these quantities is assumed to be diagonal. The uncertainties in the distributions come from Table 2. Table 3 shows the resulting mean values of A_V for the four stars, their statistical uncertainties, and the mean standard deviation of the fit to the observed magnitudes for each star. Table 4 presents the mean values of the derived aperture corrections, ${{\rm{\Delta }}}_{T}$, for the five WFC3 bands that we employ, and the nominal values from MAST. It is noteworthy that the agreement between our derived aperture corrections, ${{\rm{\Delta }}}_{T}$, and the fiducial MAST values is excellent, with discrepancies of the order of 0.01 mag, as expected.

Table 3.  Fit Results for Primary DA White Dwarf Targets

Object	AV	${\sigma }_{{A}_{V}}$	$\sigma ({fit})$
		(mag)
SDSS-J010322.19-002047.7	0.095	0.005	0.007
SDSS-J102430.93-003207.0	0.264	0.005	0.004
SDSS-J120650.40+020142.4	0.051	0.008	0.004
SDSS-J163800.36+004717.7	0.197	0.005	0.006

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Table 4.  Derived and Fiducial Aperture Corrections for HST Passbands

Passband	Aperture Correction	Mean Subtracted
	${{\rm{\Delta }}}_{T}$	MAST Aperture Correction (mag)
F336W	0.011	0.033
F475W	−0.039	−0.023
F625W	−0.023	−0.024
F775W	−0.095	−0.020
F160W	0.060	0.064

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Comparison of the A_V values in Table 3 with those in Table 2 which are derived by the other procedure shows differences of a few millimag, with consequent differences in SED fluxes over the observed passband range of 10 or 20 mmag. We expect that with the Cycle 22 data, which includes observation in $F275W$ and contemporaneous constraining observation of all three HST white dwarf calibrators, the SEDs from both approaches will result in closer agreement for the derived final SEDs. We emphasize that all of the quoted uncertainties are purely statistical.

As our targets are within the dynamic range of present wide-field survey facilities, we provide expected synthetic magnitudes for SDSS, PanSTARRS, and DECam photometric systems in Table 5. The passband throughputs used to synthesize magnitudes include the attenuation from the terrestrial atmosphere at a representative airmass of 1.3 at the respective observatory sites. Where available, we have compared our synthetic photometry to published photometry from surveys from their respective facilities.

Table 5.  DA White Dwarf Synthetic and Observed Magnitudes on Common Photometric Systems

Survey	Target	Sourcea	u	g	r	i	z	y
			(mag)b
DECam	SDSS-J010322.19	Syn	18.744	19.096	19.629	19.999	20.293	20.474
	SDSS-J102430.93	Syn	18.661	18.914	19.365	19.694	19.956	20.127
	SDSS-J120650.40	Syn	18.630	18.683	19.115	19.444	19.721	19.879
	SDSS-J163800.36	Syn	18.499	18.831	19.332	19.683	19.961	20.137
PS1c	SDSS-J010322.19	Syn	⋯	19.109	19.573	19.938	20.212	20.484
		Obs	⋯	19.104 (0.007)	19.432 (0.016)	19.944 (0.021)	20.132 (0.032)	20.120 (0.124)
		Res	⋯	−0.005	−0.141	+0.006	−0.080	−0.364
	SDSS-J102430.93d	Syn	⋯	18.925	19.315	19.642	19.886	20.136
	SDSS-J120650.40	Syn	⋯	18.689	19.065	19.386	19.643	19.888
		Obs	⋯	18.688 (0.007)	19.030 (0.010)	19.375 (0.014)	19.645 (0.026)	19.742 (0.052)
		Res	⋯	−0.001	−0.035	−0.011	+0.002	−0.146
	SDSS-J163800.36	Syn	⋯	18.843	19.279	19.628	19.886	20.147
		Obs	⋯	18.879 (0.008)	19.281 (0.010)	19.610 (0.011)	19.880 (0.021)	19.963 (0.101)
		Res	⋯	+0.036	+0.002	−0.018	−0.006	−0.184
SDSS	SDSS-J010322.19	Syn	18.657	19.045	19.553	19.918	20.255	⋯
		Obs	18.66 (0.02)	19.03 (0.01)	19.54 (0.01)	19.94 (0.02)	20.37 (0.12)	⋯
		Res	0.00	−0.02	−0.01	+0.02	+0.11	⋯
	SDSS-J102430.93	Syn	18.597	18.874	19.296	19.623	19.922	⋯
		Obs	18.57 (0.02)	18.84 (0.01)	19.28 (0.01)	19.58 (0.02)	19.77 (0.08)	⋯
		Res	−0.03	−0.03	−0.02	−0.04	−0.15	⋯
	SDSS-J120650.40	Syn	18.586	18.648	19.046	19.367	19.681	⋯
		Obs	18.55 (0.02)	18.63 (0.01)	19.03 (0.01)	19.34 (0.02)	19.70 (0.10)	⋯
		Res	−0.04	−0.02	−0.02	−0.03	+0.02	⋯
	SDSS-J163800.36	Syn	18.418	18.783	19.259	19.608	19.925	⋯
		Obs	18.44 (0.01)	18.81 (0.01)	19.28 (0.01)	19.58 (0.02)	19.79 (0.08)	⋯
		Res	+0.02	+0.03	+0.02	−0.03	−0.14	⋯

Notes.

^aSynthetic magnitudes are labeled "Syn." Observed magnitudes are provided where available and are labeled "Obs." Uncertrainties are provided in parentheses. Residuals between Observed and synthetic magnitudes are labeled by "Res." ^bAll magnitudes are on the AB system. The transmission/response function for each survey facility is calculated assuming an airmass of 1.3. ^cThe PS1 observed magnitudes are from the calibration of Tonry et al. (2012) and Schlafly et al. (2012). They have been adjusted by the offsets in Scolnic et al. (2015). These offsets are $({\rm{\Delta }}g,{\rm{\Delta }}r,{\rm{\Delta }}i,{\rm{\Delta }}z,{\rm{\Delta }}y)=(0.020,0.033,0.024,0.028,0.011)$ mag. ^dThere are no entries for SDSS-J102430.93 in the catalogs of Tonry et al. (2012) and Schlafly et al. (2012).

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In all cases, we quote magnitudes on the AB system, as defined for photon proportional detection systems in Fukugita et al. (1996, see their Equation (7)). We have converted from our HST-based photometric system to AB by determining the AB magnitude of Vega.

The mean absolute value of the residuals between the observed and synthetic photometry are within 0.025 mag for the ugri bands within SDSS DR12, completely in line with the 2%–3% systematic error associated with the absolute calibration of the SDSS photometry onto the AB system.

The observed PanSTARRS photometry is from Tonry et al. (2012) and Schlafly et al. (2012), and is adjusted by the offsets reported in Scolnic et al. (2015). There are no measurements reported for SDSS-J102430.93. The residuals to the PanSTARRS photometry show a negative bias in the y band, however, this bias is within 2–3σ of the large reported uncertainties of the photometry. The PanSTARRS 3pi survey images have significantly more cosmetic issues than the SDSS photometry, and careful analysis of these images with imporoved masking of diffraction spikes, ghosts, and cosmic rays will likely improve the agreement between the observed and synthetic photometry.

We have presented the results from four faint DA white dwarfs that demonstrate the proposition that DA white dwarf fluxes as predicted from models are consistent with observations. This proposition has been the basis for HST calibrated fluxes for some time now. In this paper, we have demonstrated that for fainter objects where modest amounts of reddening must be taken into account, the simple extension that includes solving for and applying individual, self-determined reddening corrections over a wide range of wavelengths using a standard Milky Way reddening law works quite well, producing residuals of a few millimag. Nevertheless, there are several caveats to consider, some of which will be addressed in our continuing work as more data on more objects become available.

What our results really show is that the models accurately predict SED ratios between DA white dwarfs, i.e., the models are validated as differential predictors. It does not, however, test whether the models predict the absolute SEDs correctly. This is likely the dominant source of systematic uncertainty. We can get some purchase on it by comparing different sets of DA white dwarf models. A comprehensive discussion can be found in Bohlin et al. (2014) where current models are found to differ by more than 1% (and as much as 5%) shortward of $0.27\;\mu {\rm{m}}$ and longward of $5\;\mu {\rm{m}}$. It is also possible that our understanding of the relevant atomic and atmospheric processes is yet incomplete. Nevertheless, the validation of models as a good differential predictor of SEDs is already a tool that will enable us to calibrate photometric systems to a few millimag accuracy. The data we present here (and in future papers to come) will continue to anchor results, as model improvements and validation take place.

There are also additional caveats, but for these we have specific mitigation plans.

• 1.  
  There are improvements underway to our processing steps. In particular, there is covariance between effective temperature and reddening. The data analysis flow is being redesigned to fit both of these parameters (along with $\mathrm{log}g$) in the same process.
• 2.  
  For the nine targets observed in the HST GO 12967 (Cycle 20) program, there are additional photometric data being gathered in the Cycle 22 HST GO-13711 program. The additional images will not only secure the photometry, but an additional passband (F275W in the near UV) will allow us to examine if the standard extinction law for reddening is indeed adequate.
• 3.  
  We are obtaining contemporaneous observations of all three HST primary standards, GD153, GD71, and G191B2B in our Cycle 22 program. This will allow us to tie our measurements directly to the CALSPEC photometric scale without the additional step of tying to the "Vegamag" scale, which in turn is tied to CALSPEC.
• 4.  
  There has been little or no long-term monitoring of these objects for constancy. We seek to pursue this from the ground.
• 5.  
  It is known that a small fraction of DA white dwarfs have gaseous and planetary debris around them and some display pollution from metals (e.g., Koester et al. 2014). These can produce anomalous SEDs. Metal pollution is detectable only in the far UV. A program to observe our target DA white dwarfs with COS on HST would be necessary to constrain this potential systematic.
• 6.  
  Magnetic fields have sometimes been observed in DA stars in our temperature range. The effect of such fields can be two-fold. First, undetected Zeeman splitting in the Balmer lines can bias temperature and gravity determinations. Second, a rotating magnetic white dwarf can exhibit observable flux variations (Holberg & Howell 2011). Flux monitoring and spectroscopy of the Hα line can flag the presence of such effects.

As a result of the above cautionary points, and with specific plans underway for their mitigation, we urge that the results for the four DA white dwarfs presented in Tables 2 and 5 be treated as provisional. Nevertheless, we will be very surprised if they change by more than a percent and we publish them so that they can be tested by observers.

Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX13AC07G and by other grants and contracts.

Based on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência, Tecnologia e Inovação (Brazil) and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina) (Programs GS-2013A-Q-8 and GS-2013B-Q-22).

The authors thank the anonymous referee for incisive questions and helpful suggestions.

The authors wish to thank Dr. Annalisa Calamida for useful discussion and her critical reading of the paper.

Facilities: HST (WFC3, ACS), Gemini:South (GMOS), MMT (Blue Channel), Magellan:Baade (IMACS) - .