All-sky Faint DA White Dwarf Spectrophotometric Standards for Astrophysical Observatories: The Complete Sample

Hot DA white dwarfs (DAWDs) have fully radiative pure hydrogen atmospheres that are the least complicated to model. Pulsationally stable, they are fully characterized by their effective temperature T eff and surface gravity logg , which can be deduced from their optical spectra and used in model atmospheres to predict their spectral energy distributions (SEDs). Based on this, three bright DAWDs have defined the spectrophotometric flux scale of the CALSPEC system of the Hubble Space Telescope (HST). In this paper we add 32 new fainter (16.5 < V < 19.5) DAWDs spread over the whole sky and within the dynamic range of large telescopes. Using ground-based spectra and panchromatic photometry with HST/WFC3, a new hierarchical analysis process demonstrates consistency between model and observed fluxes above the terrestrial atmosphere to <0.004 mag rms from 2700 to 7750 Å and to 0.008 mag rms at 1.6 μm for the total set of 35 DAWDs. These DAWDs are thus established as spectrophotometric standards with unprecedented accuracy from the near-ultraviolet to the near-infrared, suitable for both ground- and space-based observatories. They are embedded in existing surveys like the Sloan Digital Sky Survey, Pan-STARRS, and Gaia, and will be naturally included in the Large Synoptic Survey Telescope survey by the Rubin Observatory. With additional data and analysis to extend the validity of their SEDs further into the infrared, these spectrophotometric standard stars could be used for JWST, as well as for the Roman and Euclid observatories.


INTRODUCTION
Most currently available spectrophotometric (and photometric) standards in the sky limit us to color accuracies of 1 to 2%.This accuracy is a limitation to the uncertainty budgets of key scientific investigations such as the determination of photo-redshifts.This in turn limits the uncertainties in determining the dark-energy equation of state, e.g.Betoule et al. (2013).The motivation for developing an all-sky network of DA white dwarfs (DAWDs) as more accurate spectrophotometric standards was described in considerable detail by Narayan et al. (2016, hereafter N16), and need not be repeated here.Salient features of those arguments are briefly re-cast in § 2 below.
This paper presents an all-sky set of 32 new spectrophotometric standard stars on an absolute scale.They are faint enough to be within the dynamic range of large telescopes (apertures 4m and higher), with two or more of them accessible from any site on the globe at any instant at airmass lower than 2. This study has utilized observational data from the Hubble Space Telescope (HST) through three proposals: GO-12967, GO-13711, and GO-15113 (PI: A. Saha), and spectroscopic observations from the ground utilizing data from Gemini Observatory, the MMT Observatory, and the SOAR telescope.
In prior publications we have presented results for a sample of 19 stars in the equatorial and northern regions of the sky with spectrophotometric accuracy in colors to sub-percent accuracy from the near ultraviolet (UV) through near infrared (IR).In this paper we add an additional 13 DAWDs in the Southern sky to extend the 19 Northern and Equatorial faint DAWD standards presented in Narayan et al. (2019, hereafter N19), thereby yielding an all-sky network of 32 faint spec-trophotometric standards.These 13 Southern standards (observed in HST Cycle 25) are analyzed using the same technique developed in N19 for observations from Cycles 20 and 22. Results from this analysis are then input to a new simultaneous analysis of the entire all-sky network of the 32 faint stars and three CALSPEC1 standards (Bohlin et al. 2014).For brevity, the exposition of the N19 analysis method is incorporated only by reference.Here, the emphasis is on the new Southern standards and the new simultaneous analysis method.
§2 recapitulates the motivation for establishing faint white dwarfs as spectrophotometric standards, as well as the concepts that underpin our approach to doing so.We outline how our analysis processes have evolved through our past publications and led to our complete "final" sample.Our calibration stands maximally independent of all other spectrophotometric systems and standards and is dependent only on how well the atmospheres of pure hydrogen white dwarfs can be modeled.§3 briefly describes the selection process for the new Southern DAWDs, referring extensively to N16 and N19, as well as Calamida et al. (2019, hereafter C19) and Calamida et al. (2022a, hereafter C22).§4 describes the observations and their reductions, all of which are closely similar to N19. §5.1 presents the N19 data reduction scheme as applied to the Southern candidates, while §5.2 and §5.3 present the simultaneous analysis of all Northern, equatorial, and Southern standards.§6 compares the results to CAL-SPEC and gives calculated magnitudes for DES, DECaLS, PaNSTARRS DR1, SDSS (DR 7), and Gaia (DR3).Finally, §7 presents the conclusions.

RATIONALE, METHODOLOGY, AND EVOLUTION
If a DA white dwarf is hotter than ∼15000K, the atmosphere is completely radiative, making it the least complicated type of star to model.Such stars are characterizable by two parameters, T eff and log g (Holberg et al. 1985).The shapes and widths of the Balmer features, and the Balmer jump when available, determine these two parameters.An atmospheric model with these parameters then predicts the emergent spectral energy distribution (SED) for the DA white dwarf.Measurements of the incident SED made above the terrestrial atmosphere (by an instrument whose response stability can be independently monitored) can be used to verify if the model predicted SEDs agree with the measurements.
Investigations by Bohlin et al. (2014Bohlin et al. ( , 2020) ) and Bohlin et al. (2022), using HST and its instruments, provided empirical verification of this concept.Three DA white dwarfs with V ≈ 12 mag were established as SED standards covering UV, visible and near-IR wavelengths.These stars define the HST CALSPEC system.The relative flux vs. wavelength of these stars is based on the physical properties of the stars and rests on our understanding of the physics of their atmospheres, independently of the absolute flux level, which depends only on the absolute monochromatic flux of Vega at 5557.5 Å (vacuum) and the IR flux of Sirius (Bohlin et al. 2020).
To create standards in the high signal-to-noise ratio (S/N) range, but not saturated in observations from telescopes of up to 30m aperture, the standards must be fainter than V ≥16.5 mag, with a median brightness even fainter, say, V = 18 mag.This implies that they must be ≈ 10 times more distant than Bohlin's triad that define the CALSPEC system.Unfortunately, this also puts them at large enough distances that interstellar extinction can no longer be ignored.The comparison of model predicted SEDs with observations can account for this by allowing and solving for the extinction A V of each individual star when comparing observations with prediction (assuming that the total to selective extinction characterized by R V is the same for all the stars).Thus there are now three parameters that quantitatively characterize the SED received above the terrestrial atmosphere: T eff , log g, and A V , plus an overall achromatic normalization that scales the absolute brightness at all wavelengths.
In N16 these precepts were directly applied to four stars: the details are available there and are not repeated here.There were several areas in which improvement was desirable: • The parameters T eff and log g are determined in a separate step from the one for A V so that errors from the first propagate into the determination of A V which is undesirable.
• Given that the Balmer lines can be very broad and vary from object to object depending on its temperature and pressure, locating the continuum can be subjective, inducing errors in determining the atmospheric model parameters.
N19 introduced a hierarchical analysis that addresses the first issue, while utilizing a Gaussian process model for the flux calibration errors to mitigate the second.Figure 16 of N19 shows 1-σ residuals of 0.003 to 0.005 mag, except for the F160W band, which had mean of 0.009 mag and 1-σ of 0.013 mag.The reader is referred to the extensive discussion in N19 for details.
In N19, each of the stars is treated individually, with zeropoints in each passband determined by the CALSPEC calibration.It is instead possible to reduce the CALSPEC observations and observations of the 13 Southern and 19 Northern/Equatorial simultaneously, the equivalent of Bohlin's experiment, which is to see how self-consistent the predicted SEDs are for all the final selected candidates (including also the three CALSPEC standards), without reference to any preexisting calibration.This allows the relative band to band zero-point differences to be determined by the entire ensemble of stars, which potentially sets accuracies in colors more robustly than the CALSPEC calibration.The only unknown left is the monochromatic zero-point scalar that adjusts the flux level equally in all bands to match the canonical brightness of Vega at a reference wavelength of 5557.5 Å (vac) according to Megessier (1995), as adjusted to 3.44 × 10 −9 erg cm −2 s −1 Å −1 by Bohlin et al. (2014).Our next paper will adopt the revised value of 3.47 × 10 −9 erg cm −2 s −1 Å −1 (Bohlin et al. 2020).Details are in §5.2 and §5.3.

TARGET SELECTION
N19 and C19 picked known probable DAWDs from SDSS observations (Kleinman et al. 2004;Eisenstein et al. 2006) and from the Villanova (McCook & Sion 1999) catalog 2 , which is now superseded by the Montreal (Dufour et al. 2017) WD database3 .All of these northern and equatorial candidates had published low-resolution spectra whose quality was sufficient to decide if they were DA, if they were hot enough to be fully radiative, and if there were no obvious issues such as magnetic line splitting or trace atmospheric elements.We observed all these Northern/Equatorial stars at Gemini (GMOS, 1. 5 or 1. 0 slit, 0.92 Å/mm, 3500-6360Å coverage) and/or at the MMT (Blue channel, 300 line grating, 1. 0 or 1. 25 slits, 1.95 Å/mm, 3400-8400Å coverage).These spectra, their reduction and analysis are presented in C19.
When searching for southern stars, there was no obvious equivalent list of faint WDs, so candidates were selected from the Supercosmos and VST surveys (Gentile Fusillo et al. 2017;Raddi et al. 2016Raddi et al. , 2017) ) by using photometry and proper motion selection criteria (absolute magnitude brighter than 9.0).For more details please see C22.Approximately 50 southern WD candidates were observed using the Goodman spectrograph at SOAR (1.07 slit, 1.99 A/pixel, 3850-7100A coverage) over several runs in 2016 and 2017.A list of candidates observed and a log of observations are in C22.To be conservative, the final list includes 15 DAWDs (two rejected in the next section) with T eff > 20000K observed with a total of three Cycle 25 HST orbits per star.The three primary CALSPEC spectrophotometric standards, GD 71, GD 153, and G191-B2B, have three visits each in HST Cycle 25 to mitigate possible WFC3 sensitivity changes.
Spectral reductions were completed as in N19.Using the machinery in N19, we derived T eff , log g, and A V , which, as will be discussed in §5.2, become input priors for the new analysis (along with their error distributions).
Final photometric reductions, as described in C19, were performed with Saha's ILAPH.This custom aperture photometry code offers interactive "growth curve" analysis for optimal sky subtraction.In addition to extracting the best possible count-rates from the images, it is critical to ensure that they are all on a fully self-consistent system of instrumental magnitudes.Since our HST/WFC3 data were acquired at different times over several years, and with different instrument configurations, special attention was paid to adjusting for possible systematic shifts.These are described in considerable detail in C19 and C22.The end result, presented in Table 2, puts the photometry on the existing AB magnitude system of CALSPEC.The fact that these are AB magnitudes is irrelevant per se for subsequent analysis, except that it was a convenient way to ensure that they are based on a self-consistent instrumental system.The eventual result is the derivation of new AB magnitudes that are not dependent on this particular starting point for the observed magnitudes.

Las Cumbres Time Series Photometry
The main purposes of the work presented in C22 were to show spectra of all the stars observed with SOAR and to test these stars for variability using the Las Cumbres Observatory (LCO) network of telescopes.While hot DAWDs are not expected to be intrinsically variable, they could be variable because of binary companions,"seeing variables" due to close faint red stars, or dust cloud remnants around the WD.N19 and C19 discuss the rejection of 4 candidate stars for spectroscopic and photometric reasons, while C22 rejects a total of six stars in the all-sky set leaving 32 faint stars, plus the three brighter CALSPEC standards, to form our network.The details of the resulting set of target stars are in Table 1, duplicated from C22, which contains further details of our target selection procedure.

Previous Analysis Procedure
The analysis presented in this paper incorporates the analysis in N19, which is based in turn on N16, as an integral part.In particular, we use the same Tlusty (Hubeny & Lanz 1995) v202 NLTE model atmosphere grid4 as N16.The grid has 31 uneven steps in T eff from 16,000-90,000 K, with a spacing of 2,000 K from 16,000-20,000 K and 2,500 K from 20,000-90,000 K.The grid has 6 even steps in log g from 7-9.5 dex, with 0.5 dex spacing.The grid covers a wavelength range of 1,350 Å -2.7 µm, in 1 Å steps from 1,350 Å ≤ λ ≤3,000 Å, 0.5 Å steps from 3,000 Å ≤ λ ≤ 7,000 Å, and 5 Å steps for λ > 7,000 Å. N16 used the shape of the observed spectrum, particularly the Balmer lines, to derive T eff and log g.Reddening was deduced, and the process iterated.In N19 we solved for the stellar parameters and the reddening simultaneously, while also using the entire spectrum.Uncertainties in flux calibration were taken into account.The out-put was a set of best values and distribution of errors for T eff , log g, and A V , assuming that the ratio of total to selective extinction is R V =3.1.No evidence was seen for variation in R V for these stars, though the data set is not well suited to detect it.N19 contains all the details.Synthetic magnitudes in this series of papers are AB magnitudes, defined by Fukugita et al. (1996): (1) where F(ν) is the energy flux per unit frequency and R(ν) is the system response function.
The calibration for both the Northern and equatorial DAWDs (Cycle 20 + 22) and for the Southern DAWDs (Cycle 25) is tied to the published flux values for the three primary CALSPEC standards.Note that our photometry is tied to the previous flux calibration from Bohlin et al. (2014), which defined CALSPEC until 2019.A flux calibration based on new models for the three CALSPEC primary standards was released in 2020 (Bohlin et al. 2020); subsequently, an updated time-dependent calibration for the WFC3 UVIS and IR detectors was also delivered in October 2020 (Calamida et al. 2022b).The WFC3 UVIS detector has indeed had an average sensitivity decline of ≈ 0.15%/year, differing depending on the filter (Calamida et al. 2022b).A sensitivity decline of the WFC3 IR detector has not been established yet; however, preliminary evidence indicate an av-erage decline of ≈ 0.1%/year (Bohlin & Deustua 2019; Bajaj  et al. 2022 5 ) Therefore, we verified our photometry for time sensitivity changes as discussed in detail in C19.Although we applied an offset to bring Cycle 20 photometry onto the Cycle 22 system, we did not measure a sensitivity change in the Cycle 22 photometry of the three CALSPEC primary DAWDs, spanning approximately 1.5 years.We then did not correct for time sensitivity changes the photometry of the Northern and equatorial DAWDs, nor the photometry for the Southern DAWDs, collected during Cycle 25.
The resulting HST photometry for the 32 established DAWDs and our measurements of the 3 CALSPEC standards is listed in Table 3.

Analysis Overview
N19 argued for better solutions to the spectroscopic and photometric parameters by doing a complete hierarchical Bayesian model (e.g., Loredo & Hendry (2019)), solving for all stars (both the stars presented in N19 and the new stars presented here) simultaneously.The analysis presented here takes a significant step in this direction.We go into detail below, but first lay out the general idea.
The new analysis draws on the lessons learned from N16 and N19 and attempts to make incremental improvements.Our goals are to • Perform a full Bayesian analysis incorporating the spectroscopy and photometry for all DAWDs simultaneously.
• Preserve the alternative analysis of N16 which removed the dependence of the results on the MAST zeropoints for the CALSPEC primary standards.
• Account for the count rate nonlinearity (CRNL) of the F160W data through free model parameters.
An analysis based on N19 for all the stars observed in Cycle 20, 22, and 25 provides input priors and error distributions for the spectroscopic parameters and HST photometry.
The input HST photometry can be in any magnitude system that is stable over time, including instrumental, as was used in N16.In the process of matching the observed photometry to the synthetic photometry from the DAWD models, all color dependent offsets in the input magnitude system relative to an AB system are corrected, leaving only an overall absolute flux calibration to determine.In the results presented here, the incoming magnitudes have been initially placed on the CALSPEC system using the procedure in N19.The absolute flux calibration is not altered from its CAL-SPEC value.The stellar parameters, T eff , log g, A V and distance modulus, are allowed to change independently for each star, and the per-band zeropoints are allowed to vary while keeping the overall flux normalization fixed.
The input photometry includes our observations of the CALSPEC standards as well as our observations of the program stars.
The errors in this new analysis method should be smaller than the errors in the N19 method, simply because band-toband differences effectively take out small errors in CAL-SPEC.This expectation is borne out, as shown in Figure 3.

Analysis Details
The first goal is limited by available computer power.Determining the posterior distribution for a model which accounts for the spectral and photometric data from all DAWDs simultaneously is judged to be impractical currently.Recognizing that the determination of T eff and log g relies almost exclusively on the spectroscopy and is nearly independent of the HST photometry, while the determination of Av and the distance modulus of each DAWD are nearly independent of the spectroscopy, we settled on a practical compromise with the following outline: 1.The analysis of N19 is performed as before for each DAWD separately.This yields for each DAWD, s, the posterior distribution for the apparent magnitudes in band λ, m λ s , and the SED parameters T eff , log g, and A V 2. Using the posteriors from the previous step as input priors, the photometry of all DAWDs are incorporated simultaneously in a Bayesian model, the posteriors of which yield a second determination of the T eff , log g, and A V posteriors, together with those for the perband zeropoint shifts δ λ and the F160W CRNL slope, α CRNL .
If necessary for convergence, the two steps of this calculation could be iterated, incorporating the zeropoint shifts and CRNL slope from step 2 as priors into step 1.We have determined that this iteration is not necessary, a conclusion which supports the assumption of very weak coupling between the modeling of the spectroscopy and the photometry.
The calculation proceeds as follows 1.For each DAWD, a 2D normal distribution is fit to the output T eff , log g chain from the N19 analysis (preceding section).These are used as priors.
2. Noninformative priors are used for A V , with the exception of those for the primary CALSPEC DAWDs.
The A V values of the three primary standards are constrained with an upper limit of 0.003, consistent with CALSPEC upper limits (Bohlin et al. 2020).This is a crucial element of the calculation and is the only way that the CALSPEC DAWDs play a special role.
3. The likelihood function is constructed, utilizing the same synthetic spectral model grid employed in N19.
4. A set of MCMC chains is run using emcee (Foreman-Mackey et al. 2013).
5. The posterior distributions are constructed for the output chains for the per-object T eff , log g, A V , the overall model per-band zeropoint shifts, ∆ λ , and the F160W CRNL slope, α CRNL .6.As a consistency check, the T eff , log g, and A V posteriors are compared with those from the separate N19 DAWD analysis performed above.Major differences would be cause for further investigation, but in practice have not been found.

Likelihood function
The likelihood function for each DAWD is a small modification of that employed in N19.
where m s,λ is the observed magnitude for a DAWD star s in HST/WFC3 passband λ ∈ {F275W, F336W, F475W, F625W, F775W, F160W}, with photometric measurement error described by an estimated standard deviation σ sλ , M s,λ (T eff , log g, A V , R V ) is the synthetic magnitude of the reddened SED through passband λ, and N(m, σ) is the normal distribution.µ s is a per-star achromatic normalization parameter which is added to the synthetic reddened magnitudes in all passbands to account for the distance and radius of the DAWD s. ∆ λ was introduced in N16 and is the star independent offset to the observed magnitudes in passband λ to convert them to AB magnitudes (the magnitude system for the synthetic magnitudes).If the ∆ λ were left unconstrained, there would be a degeneracy between their mean value and the mean value of the µ s over the full set of stars.To break the degeneracy, we require The observed magnitudes for this analysis are already on the CALSPEC AB system, and the ∆ λ are expected to be quite small, accounting only for small errors in the measured HST passbands and/or aperture corrections, an expectation that is realized, as shown in Table 4.However, this constraint is not the only possible choice for breaking the degeneracy.One could, for example, instead require that the synthetic SED of a selected calibration star at a given wavelength match a value determined outside the system (e.g., CALSPEC).
The likelihood function for the entire model is then the product over all DAWDs of the likelihood for each individual DAWD.For the results reported here, R V = 3.1.
There is one further refinement beyond the previous analysis.The HST detector for the F160W band is known to have a dependence of the counts from a source integrated over the exposure time on the rate of those counts, commonly referred to as the "count rate nonlinearity" (CRNL) (Bohlin & Deustua 2019;Riess et al. 2019).To account for this effect, a synthetic magnitude in F160W M s,F160W , is observed as We include α CRNL in the free parameters of the model.We have found no significant effects from varying β CRNL , and it is arbitrarily fixed at 15.The λ = F160W term in the product in Equation 2 then becomes: It is convenient to express equation 2 as where the observed magnitude, corrected for the zero point shift, and in the case of F160W, the CRNL, is The values of m corr s,λ are given in Table 3 alongside those for m s,λ .Note that it is these values which the SEDs integrated over the WFC3 passbands are expected to match.
The free model parameters, then, include T eff , log g, A V , and µ s for each star, the five element array ∆ λ , and α CRNL , a total of 146.

RESULTS
We used the emcee implementation of MCMC to sample the posterior probability density function (pdf) of the model parameters employing 400 walkers, each producing a chain of 20000 steps after a 100 step burn-in.Corner plots of the resulting pdf show good convergence in all parameters.Two examples are shown in Figures 10 and 11.Derived parameters for each DAWD are presented in Table 5.

Comparison with CALSPEC
Despite the input photometry being on the 2014 CALSPEC system, there are small differences between our model SEDs for the CALSPEC primary DAWDs and the 2014 CALSPEC model SEDs of Bohlin et al. (2014).The absolute flux scale reported herein and our previous (N19) paper is based on the absolute flux calibration of the WFC3 filters described in C19.These spectral energy distributions (SEDs), i.e., absolute flux in physical units, are based on models of the three primary DAWDs, G191B2B, GD153, and GD71 (Bohlin et al. 2014).However, those models were improved with new NLTE grids computed by Ivan Hubeny and Thomas Rauch (Bohlin et al. 2020), which resulted in changes to the basis of the HST flux scale by up to 3% at some wavelengths.A future paper will report the SEDs of our DAWD standards, as adjusted to the more recent Bohlin et al. (2020) flux scale.See the Conclusion Section for more details.

Comparison with Gentile Fusillo 2021
A new catalog of white dwarfs based on Gaia EDR3 was recently published by Gentile Fusillo et al (Gentile Fusillo et al. (2021, hereafter GF21)).This catalog contains values for the stellar parameters T eff and log g based on WD model atmospheres in conjunction with Gaia photometry, and A V from a three dimensional extinction model.It is useful to compare the GF21 values for our WDFS stars with our results.These are shown in Figure4.The comparison for A V is particularly useful, showing good agreement between values determined by two completely independent methods.The T eff and log g comparisons are likewise based on independent methods, but the a priori confidence in the GF21 values must be lowered by the lack of spectroscopic input.

Count Rate Nonlinearity
The value determined for α CRNL is −1.74 ± 0.32 mmag per mag.This is significantly less than the published value of −3.12 ± 0.32 mmag/mag in Bohlin & Deustua (2019), or the combined result of −3.0 ± 0.24 mmag/mag in Riess et al. (2019).However, the CRNL is consistent with the value of −2.36 ± 0.48 mmag/mag for the subset of our stars analyzed in Riess et al. (2019).As shown in Figure 11, the posterior distribution for α CRNL is tightly constrained.

Synthetic Magnitudes for Common Survey Passbands
As in N19, we have calculated the synthetic magnitudes for our standards in a number of common survey passbands.The filter passbands are obtained from the Spanish Virtual Observatory (SVO) Filter Profile Service (Rodrigo & Solano 2020;Rodrigo et al. 2012).To calculate the synthetic magnitudes, we utilize the full MCMC chains from our analysis run, and for each point on the chain calculate the associated synthetic magnitudes.This gives a probability density function for each magnitude, which we characterized by its median and standard deviation.The standard deviations are typically less than one milli-mag (0.001 mag), significantly less in most cases than the survey reported observational uncertainties, and certainly less than the (unknown) systematic errors.The standard deviations therefore do not reflect the real uncertainties in our synthetic magnitudes, particularly in passbands not closely aligned to the HST passbands, and we do not include these values in the tables below.
For each survey system, we include below a table of our synthetic magnitudes, and where available the magnitudes and uncertainties reported by the survey.A plot for each band of each photometric system shows the magnitude differences (in the sense synthetic -observed) as a function of magnitude.We note some caveats.The observed magnitudes for each star are derived from differing photometric systems, especially the broad Gaia G filter.That these independent photometric systems demonstrate good agreement between their observed and our synthetic magnitudes is further evidence for the robustness of our system.
We provide magnitudes for our stars in the DES, DECaLS, PaNSTARRS1, SDSS, and Gaia systems.
DECaLS observed magnitudes are from DECaLS DR9 (Schlegel et al. 2021).Uncertainties are estimated as 2.5/ lg 10/snr.Results are in Table 7, and in Figure 6.
PaNSTARRS1 observed PSF magnitudes are from the mean table of the DR2 (Flewelling et al. 2020).Results are in Table 8, and in Figure 9.
The observed SDSS data for the CALSPEC DAWD standards are from Holberg & Bergeron (2006), modified to correct a typographical error in the i-band flux of GD153 (Holberg, private communication).The observed magnitudes for the fainter DAWDs come from SDSS DR7 (Abazajian et al. ( 2009)).
Gaia observed magnitudes are from DR3 (Gaia Collaboration et al. 2022).For comparison to DR3 magnitudes, our synthetic magnitudes are transformed from AB to Vega using passband data from Rodrigo & Solano (2020).Results are in Table 10, and Figure 7.
Our synthetic magnitudes are anchored on observations made above the atmosphere.With the exception of Gaia, all other surveys in this comparison are from ground-based surveys, which have required accounting for the constantly changing extinction from the terrestrial atmosphere.We hope that the network of spectrophotometric standard stars presented in this paper will be a useful tool for resolving any discrepancies between different surveys.Beyond this, we have not attempted to ascribe specific causes for the discrepancies between our synthetic magnitudes for these surveys and those published by the surveys themselves.To do so would require significant expertise from the survey teams, and could form a focus for future work.Similarly, the impact of any downstream scientific results from any re-calibration of existing surveys is best left to the discretion of topical experts, and the specific problems they have at hand. 7. CONCLUSIONS

Major results of this paper
This and prior papers (N19, C19, and C22) work towards an all-sky network of faint spectrophotometric standard stars (called WDFS) composed of hot DAWDs whose fluxes are tied to the three primary CALSPEC standards.Our initial goal of 1% absolute and 0.5% relative flux calibration in the visual band is realized and our network of 35 White Dwarf Flux Standards (32 WDFS and 3 CALSPEC) is available for general use.In addition, each stellar reddened SED provides predicted magnitudes in several common survey systems (6.4).This set of stars covers the entire sky, such that, at any time, two or more standards are above 2 airmasses, at any ground-based observatory.Because the HST/WFC3 photometry that defines our system is above the atmosphere, ground-based, atmospheric extinction problems do not exist.Our standards are suitable for many of the existing and future large telescope surveys.
The conversion of our derived SEDs to magnitudes in ground based surveys must necessarily include the filter functions of the surveys, which are often available (Rodrigo & Solano 2020;Rodrigo et al. 2012).Our published SEDs can be convolved with any filter function for any telescope.If these functions are not known, a later paper in this series, using parallel ACS images, can define color terms for conversion of native ground-based magnitudes to magnitudes on the space-based system.
Item 4 of the next section discusses extrapolations of our SEDs shortward of the HST F275W passband and longward of the infrared F160W passband.

Possible Improvements and Enhancements
The following items discuss some limitations of this sample and some possible future improvements.and Sirius in the IR (Bohlin 2014;Bohlin et al. 2020), which have their own uncertainties.Ongoing and proposed ground-based and space-based efforts seek to establish stellar calibrations with respect to NIST (National Institute of Standards and Technology) laboratory radiometry with 0.5% absolute and 0.3% relative uncertainties in the visual.When available, these improvements can be applied to our existing WDFS by making global corrections at the few tenths of a percent level.
3. Future expansion of our standard star network is possible.The size of our network was ultimately dictated by the observational effort required to locate and validate suitable candidates, as well as monitoring each star for photometric stability, by obtaining the spectroscopic time on large telescopes, and by obtaining the WFC3 photometry in six bands.Future efforts can rely on cur- show that our optically estimated values of A V predict the observed UV fluxes for most stars to a precision better than 3%.Outliers might be explained by adjusting A V and R V , the ratio of absolute extinction A V to selective extinction E(B-V), within our uncertainties.There are no observational tests longward of the F160W passband, but there is exquisite agreement between models and observations at shorter wavelengths.
5. The placement of our WDFS SEDs on the CALSPEC absolute flux scale has several inaccuracies at the percent level that will be addressed in our next paper.First, our model SEDs are in air above 2000 Å.The air to vacuum correction that is applied to our final SEDs is adequate, except for a small unphysical discontinuity at 2000 Å, but these models extend to only 1350 Å in the FUV and 2.7µ in the IR.CALSPEC now uti- lizes the NLTE grids computed by I. Hubeny and T. Rauch (Bohlin et al. 2020) that cover 900 Å to 30µ with native vacuum wavelengths and show emission lines at HI line centers where these features are actually observed.Furthermore, these newer NLTE grids include many more IR lines, including some important features like Paschen α.Measurements of radial velocities would improve our model SEDs slightly.
Perhaps the most important WDFS future improvement will be to place the absolute fluxes on the updated scale of Bohlin et al. (2020).Figure 12 illustrates quantitative comparisons with these old and new flux scales (GD153 is corrected for the published radial velocity of 8.3 km s −1 Napiwotzki et al. 2020).The blue curve compares our GD153 model with the 2014 gd153_mod_010.fits and is generally within 1% of agreement, except in the line profiles and at the short-est wavelengths.Because our flux scale is based on the 2014 models and the WFC3 calibration of Calamida et al. (2019) that is used in in our previous papers, the blue trace represents the small offset between our flux scale and the 2014 CALSPEC flux system.The difference between the red and blue is the amount of change in 2020 to the gd153_mod_011.fitsmodel of Bohlin et al. (2020).

DATA AVAILABILITY
We have created a Zenodo url8 that contains the SEDs derived in this paper, the WFC3 passbands we employed to create magnitudes in commonly used systems, and derived parameters for each star (Table 5 in the text), along with all of the other tables.These data and the corrected, "c",  We thank an anonymous referee for comments which allowed us to improve the paper.

Figure 1 .
Figure 1.Aitoff projection of our network of spectrophotometric standard DAWDs illustrating the full-sky nature of our system (HST Cycles 20 and 22 in green, Cycle 25 in blue, and the three bright CALSPEC DAWDs in red).

Figure 2 .
Figure2.Calculated SEDs for all DAWDs in our network with synthetic HST magnitudes overlaid (colored points).Each spectrum is arbitrarily shifted in AB mag for clarity.

Figure 3 .
Figure 3. HST photometric residuals (in the sense of corrected observed magnitude − model magnitude) for our network of DA white dwarf stars.X axis coordinates within each band are uniformly spaced, ordered by the F275W magnitude with the brightest at the left.The mean and RMS are weighted by the photometric uncertainties.

Figure 4 .
Figure 4. Comparison between our values for log g Teff and AV and those from Gentile Fusillo et al. (2021), denoted in the plots as GF21.The dotted lines show the identity relation.Note that a point in the log g plot at approximate value 9.0 is off scale.

Figure 9 .
Figure9.Pan-STARRS1 observed minus synthetic magnitudes in griz.X-axis coordinates within each band are uniformly spaced, ordered by the g magnitude with the brightest at the left.The mean and RMS are weighted by the photometric uncertainties.The black dashed line indicates zero difference.The mean value for each filter is represented by a dashed line through those filter's points in the same color.Note that some points are off scale for this figure.See Table8.

Figure 10 .Figure 11 .Figure 12 .
Figure 10.Corner plot showing the posterior distribution of the model parameters for WDFS1514+00.δ275 etc are the residuals as in Table3.Note the scales.

Table 1 .
Gaia DR3 astrometry and photometry for the candidate spectrophotometric standard DA white dwarfs Coordinates are from Gaia DR3 at epoch J2016.0.* This star was excluded from the final network of spectrophotometric standard DAWDs.See text of C22 for more details.

Table 2 .
HST observed photometry.Units are AB mag. a a Values in parentheses are 1σ errors in mmag.Values in cF columns are corrected values (see equation 7).Note that it is these corrected values which the SEDs integrated over the WFC3 passbands are expected to match.

Table 3 .
HST Synthetic Photometry and Residuals when Compared with Measured Values.Units are AB mag. a a Residual value is observed photometry minus synthetic photometry.

Table 4 .
Values for ∆ λ and Their Uncertainties

Table 5 .
Derived Object Parameters a ObjectTeff(K) Teffσ (K) log g (dex) log g σ (dex) AV (mag) AV σ (mag)a Parameter values are the medians of the posterior distributions.

Table 6 .
Observed and Synthetic Photometry in DECam Bands.Units are AB mag.

Table 7 .
Observed and Synthetic Photometry in grz DECaLS Survey Bands.Units are AB mag.
7of about two thirds of our WDFS stars in the UV down to 1150 Å.Preliminary results from this program 7 GO 16764, PI: G. Narayan

Table 8 .
Observed and Synthetic Photometry in Pan-STARRS1 griz Bands.Units are AB mag.

Table 9 .
Observed and Synthetic Photometry in SDSS ugriz Bands.Units are AB mag.

Table 10 .
Observed and Synthetic Photometry in Gaia Bands.Units are Vega mag. the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation Grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation.This project used public archival data from the Dark Energy Survey (DES).Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology FacilitiesCouncil of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e Tecnológico and the Ministério da Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft, and the Collaborating Institutions in the Dark Energy Survey.The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciències de l'Espai (IEEC/CSIC), the Institut de Física d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universität München and the associated Excellence Cluster Universe, the University of Michigan, the National Optical Astronomy Observatory, the University of Nottingham, The Ohio State University, the OzDES Membership Consortium, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Labo-ratory, Stanford University, the University of Sussex, and Texas A&M University.Based in part on observations at Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.NOIRLab; the Bok telescope, Steward Observatory, University of Arizona; and the Mayall telescope, Kitt Peak National Observatory, NOIRLab.Pipeline processing and analyses of the data were supported by NOIRLab and the Lawrence Berkeley National Laboratory (LBNL).The Legacy Surveys project is honored to be permitted to conduct astronomical research on Iolkam Du'ag (Kitt Peak), a mountain with particular significance to the Tohono O'odham Nation.NOIRLab is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.LBNL is managed by the Regents of the University of California under contract to the U.S. Department of Energy.This project used data obtained with the Dark Energy Camera (DECam), which was constructed by the Dark Energy Survey (DES) collaboration.Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cientifico e Tecnologico and the Ministerio da Ciencia, Tecnologia e Inovacao, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey.The Collaborating Institutions are Argonne National Labo-ratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energeticas, Medioambientales y Tecnologicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciencies de l'Espai (IEEC/CSIC), the Institut de Fisica d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig Maximilians Universitat Munchen and the associated Excellence Cluster Universe, the University of Michigan, NSF's NOIRLab, the University of Nottingham, the Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A&M University.BASS is a key project of the Telescope Access Program (TAP), which has been funded by the National Astronomical Observatories of China, the Chinese Academy of Sciences (the Strategic Priority Research Program "The Emergence of Cosmological Structures" Grant # XDB09000000), and the Special Fund for Astronomy from the Ministry of Finance.The BASS is also supported by the External Cooperation Program of Chinese Academy of Sciences (Grant # 114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant # 12120101003, # 11433005).The Legacy Survey team makes use of data products from the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE), which is a project of the Jet Propulsion Laboratory/California Institute of Technology.NEOWISE is funded by the National Aeronautics and Space Administration.The Legacy Surveys imaging of the DESI footprint is supported by the Director, Office of Science, Office of High Energy Physics of the U.S. Department of Energy under Contract No. DE-AC02-05CH1123, by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract; and by the U.S. National Science Foundation, Division of Astronomical Sciences under Contract No. AST-0950945 to NOAO.Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions.SDSS-IV acknowledges support and resources from the Center for High Performance Computing at the University of Utah.The SDSS website is www.sdss4.org.SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Center for Astrophysics | Harvard & Smithsonian, the Chilean Participation Group, the French Participation Group, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University. by Figure5.DES observed minus synthetic magnitudes in grizy.X-axis coordinates within each band are uniformly spaced, ordered by the g magnitude with the brightest at the left.The mean and RMS are weighted by the photometric uncertainties.The black dashed line indicates zero difference.The mean value for each filter is represented by a dashed line through those filter's points in the same color.Note that some points are off scale for this figure.See Table6.Figure 6.DECaLS Survey observed minus synthetic magnitudes in grz.X-axis coordinates within each band are uniformly spaced, ordered by the g magnitude with the brightest at the left.The mean and RMS are weighted by the photometric uncertainties.The black dashed line indicates zero difference.The mean value for each filter is represented by a dashed line through those filter's points in the same color.See Table 7.Figure 7. Gaia DES observed minus synthetic magnitudes in G, RP, and BP.X-axis coordinates within each band are uniformly spaced, ordered by the G magnitude with the brightest at the left.The mean and RMS are weighted by the photometric uncertainties.The black dashed line indicates zero difference.The mean value for each filter is represented by a dashed line through those filter's points in the same color.See Table10.Figure8.SDSS observed minus synthetic magnitudes in ugrizy.X-axis coordinates within each band are uniformly spaced, ordered by the g magnitude with the brightest at the left.The mean and RMS are weighted by the photometric uncertainties.The black dashed line indicates zero difference.The mean value for each filter is represented by a dashed line through those filter's points in the same color.Note that some points are off scale for this figure.See Table9.