Spectral energy distributions and colours of hot subluminous stars

Photometric surveys at optical, ultraviolet, and infrared wavelengths provide ever growing datasets as major surveys proceed. Colour-colour diagrams are useful tools to identify classes of stars and to provide large samples. Combining all photometric measurements of a star into a spectral energy distribution will allow quantitative analyses to be carried out. We demonstrate how to construct and exploit spectral energy distributions and colours for sublumious B (sdB) stars. The aim is to identify cool companions to hot subdwarfs and to determine atmospheric parameters of apparently single sdB stars as well as composite spectrum sdB binaries. We analyse two sdB stars with high-quality photometric data which serve as our benchmarks, the apparently single sdB HD 205805 and the sdB + K5 binary PG 0749+658, briefly present preliminary results for the sample of 142 sdB binaries with known orbits, and discuss future prospects from ongoing all-sky optical space- (Gaia) and ground-based (e.g. SkyMapper) as well as NIR surveys.


Introduction
Optical photometry and spectroscopy provide the observational basis for astronomy. Ongoing large photometric surveys provide a huge amount of photometric measurements in several optical passbands, which can be used to identify candidate hot subluminous stars, though spectroscopy is needed for proper spectral typing. However, optical photometry is much more than a mere target selection tool. Time-series photometry (light curves) are a crucial ingredient for asteroseismology of pulsating stars and to identify eclipses, reflection effects, and ellipsoidal variations, in compact binaries. Single-epoch observations, however, provide crucial information as well. Spectroscopic distances rely on at least one measured apparent magnitude. Ultraviolet and infrared surveys when combined with optical photometry allow us to construct broad spectral energy distributions (SED), which can be used to determine e.g. the effective temperature of a star, to identify an infrared excess hinting at the presence of a cool companion, and to quantify interstellar absorption from UV flux depression.
Here we shall not address light variation but restrict ourselves to colour-metric properties of hot subdwarf B (sdB) stars. Several investigations of hot subdwarf stars have made use of single-epoch photometry. With the advent of the International Ultraviolet Explorer (IUE) satel-lite crucial information to study hot stars arose and early attempts to analyse SEDs of sdB stars were carried out by Heber et al. (1984), Heber (1986), and Aznar Cuadrado & Jeffery (2001) by combining low resolution UV spectra from IUE with optical photometry. Colour-colour diagrams combining infrared and optical magnitudes are important tools to identify composite objects, such as sdB stars with F/G/K companions (see e.g. Stark & Wade 2003;Green et al. 2008).
Subdwarf B stars are core helium burning stars of half a solar mass and in the Hertzsprung Russell diagram they form the extreme horizontal branch. Because radial velocity surveys have shown that the fraction of close binaries amongst single-lined (SB1) sdB stars is as high as 50 %, common envelope evolution plays an important role in the formation of sdB stars. About 30% of the sdB stars show composite colours, that is they have companions of spectral types F, G, or K. In many cases the companions to SB1 systems have been found to be white dwarfs, but low mass main-sequence stars (spectral type M) and brown dwarfs have been found as well (see Heber 2009Heber , 2016, for reviews). Because it is often very difficult to clarify the nature of the companions from optical data alone, broad SEDs are the method of choice to constrain the companions' properties and constrain their nature.
We describe a method to construct broad SEDs by combining measurements in various photometric systems from the ultraviolet to the infrared in Sect. 2. Synthetic photometry for various photometric systems are calculated from grids of appropriate model atmospheres (Sect. 3). An objective method to derive various parameters from observed SEDs is presented in Sect. 4 and preliminary results are discussed in Sect. 5. We conclude with an outlook.

Constructing observational SEDs
The SEDs of the program stars were constructed from photometric measurements ranging from the ultraviolet to the infrared collected from literature. To eliminate the steep slope of the SED we plot the flux density times the wavelength to the power of three (F λ λ 3 ) as a function of wavelength throughout this paper.

Photometric data
The visual range is covered by SDSS (Alam et al. 2015) and APASS (Henden et al. 2016) data as well as magnitudes and colours in the Johnson-Cousins, Strömgren (see Fig. 1), and Geneva systems, which are collected using Vizier as well as the Subdwarf Database 1 by Østensen (2006). Ultraviolet fluxes are important to constrain the atmospheric parameters of a hot subdwarf and were extracted from observations by the International Ultraviolet Explorer (IUE), available in the MAST 2 archive.
Infrared photometry is of particular importance for binary systems that contain a cool companion because an IR excess is expected. Available infrared data were taken from ALLWISE (Wright et al. 2010;Cutri & et al. 2013), 2MASS (Skrutskie et al. 2006), and UKIDSS (Lawrence et al. 2007, see Fig. 1).
The photometric datasets are inhomogeneous, both with respect to bandwidth as well as to accuracy. Ultraviolet spectra from IUE cover the wavelength range from 1150Å to 3150Å at a spectral resolution of 6 Å. The optical spectral range is covered by several filters both narrow band (e.g. Strömgren) and wide-band (e.g. Sloan or Johnson), while the infrared is usually represented by five wide-band filters (J, H, K, W1, and W2).

Ultraviolet fluxes from IUE spectra
The IUE satellite provided UV spectra for two wavelength ranges; the short (SW, 1150 and long (1910-3150Å) wavelength range. Each spectrograph offered both high and low resolution modes, with spectral resolutions of 0.2 and 6 Å respectively, as well as two entrance apertures each, a small circular aperture with a 3 arcsec diameter and a large rectangular aperture of 10 by 20 arcsecs. We discarded spectra at high-resolution as well as those taken through a small aperture, because the flux calibration is less accurate than that for the largeaperture, low-resolution spectra. Because we have to combine them with broad and intermediate band optical and infrared photometry we defined a suitable set of filters to derive UV-magnitudes from IUE spectra (see Fig. 2). Three box filters, which cover the spectral ranges 1300-1800 Å, 2000-2500 Å, and 2500-3000 Å, are defined to extract magnitudes from the IUE spectra. The box filters were designed in order to avoid the boundaries of the SW and LW wavelength ranges because of the increasing noise level and the region around the Lyman-alpha line because of the contribution by interstellar gas absorption. The mid-UV filter was designed to include the UV absorption bump at ≈ 2200Å of interstellar absorption (see Fig. 3), which is important to determine the interstellar reddening parameter E(B-V). Box filters to convert IUE low resolution spectra to UV magnitudes. One box is placed in the wavelength range covered by the short wavelength camera (SWP, upper panel) and two in the regime covered by the long wavelength cameras (LWR/LWP, lower panel).

Synthetic SEDs and colours
The magnitude mag x of an arbitrary photometric passband x is defined as where r x (λ) is the response function of the filter (see Fig. 1 for examples) and f (λ) the flux at the photoncounting detector. The flux of a reference star (usually Vega) f ref is needed to set the zero point of the filter to a predefined magnitude mag ref x . Note that we assume photon-counting detectors, which explains the additional factor λ in the arguments of the integrals (see, e.g. Bessell et al. 1998, for details).
The stellar flux at Earth f (λ) can be calculated from the model flux at the stellar surface F (λ) and the angular diameter of the star Θ (= 2R /d ), which is two times the stellar radius R divided by the distance, from which we obtain f (λ) = Θ 2 F (λ)/4.
To account for interstellar extinction, the synthetic flux is multiplied with a reddening factor 10 −0.4A(λ) . The extinction in magnitude at wavelength λ, A(λ), as a function of the colour excess E(B − V ) and the extinction pa- Fig. 3). The final expression to calculate a synthetic magnitude, therefore, reads as (2)

Grids of synthetic SEDs and colours
We aim at modelling the observed SEDs and colours of single sdB stars or SB1 binaries, as well as composite spectrum systems consisting of a hot subdwarf and a latetype main-sequence star.

SEDs of hot subdwarf stars
Subluminous B stars are known to show peculiar chemical abundance patterns (Heber 2009(Heber , 2016 characterized in general by depletions of light metals (C to Ca) and enrichment of heavy metals by very large factors with respect to solar composition. However, star-to-star scatter is large. Naslim et al. (2013) suggested an average abundance pattern, which we adopted for the model calculations. For the elements not listed in Naslim et al. (2013) the reference abundance is solar (Asplund et al. 2009). In order to synthesize the SED of a hot subdwarf star a grid of model atmospheres was calculated using the AT-LAS12 code (Kurucz 1996) with effective temperatures ranging from 15000 K to 55000 K and surface gravities from 4.6 to 6.2. The helium abundance was fixed at a low value of one hundredth solar and the logarithmic "metallicities" z are scale factors with respect to the abundance pattern of Naslim et al. (2013). The synthetic spectra cover the wavelength range form 300 Å to 100 000 Å (far UV to mid infrared). The logarithmic metallicity z is allowed to vary between -1 and +1 (a tenth or ten times the typical composition of a subdwarf B star). Please note that iron and nickel are the dominant absorbers and have the greatest influence on the metallicity z because they have many absorption lines in the FUV and their absolute abundance is high. As demonstrated in Fig. 4, the metallicity essentially affects the UV spectral range, most significantly the short wavelength UV box filter. Hence it might be possible to derive the metallicity of the sdB if such UV measurements were available.
Recently, several improvements have been implemented in the ATLAS12 code (Irrgang, in prep.), the most important of which is the treatment of high series members of the hydrogen and ionized helium line series, following Hubeny et al. (1994). This is of particular importance to model the Balmer jump.

SEDs of cool stars
For cool stars a grid of PHOENIX models calculated by Husser et al. (2013) is used 3 . The synthetic SEDs cover the wavelength range from 500-55000 Å. The parameter range is confined to effective temperatures between 2300 K and 12000 K, surface gravities between 2 and 5 dex, and the helium content is set at the solar value.

Combining SEDs of sdB and cool stars
In order to combine the spectra of the two components the surface ratio S needs to be determined, which adds another parameter, from which the angular diameter of the companion Θ c can be derived.

Photometric analysis methodology
To facilitate objective and efficient photometric analyses, we have developed a grid-based fitting routine. It is based on χ 2 minimization tools provided by the Interactive Spectral Interpretation System (Houck & Denicola 2000) to find the global best-fit in the multi-parameter space.
The six parameters considered to model SEDs and colours of the sdB stars are -the angular diameter Θ -the effective temperature: T sdB eff -the surface gravity: log g sdB -Helium Abundance: log(n(He)/n(all)) -"Metallicity" z (scaled typical abundance pattern (Naslim et al. 2013)) -the interstellar reddening parameter E(B-V) Adopting the canonical mass for the sdB star, we also derive the stellar distance.

Composite spectra
In the case of binary stars we may observe a composite spectrum, which increases the parameter space by the parameters describing the companion as well as the surface ratio S of both stars. -Effective temperature: T c eff -Surface gravity: log g c -Metallicity: [Fe/H] However, usually the surface gravity and metallicity of the cool star are unconstrained. Therefore, they were kept fixed to log g=4.5 and 1/10 solar metallicity.

Determination of uncertainties
Uncertainties are derived from the χ 2 statistics. The parameter under consideration is increased/decreasedwhile all remaining parameters are fitted to account for possible correlations -until a certain increment ∆χ 2 from the minimum χ 2 is reached. The value chosen for ∆χ 2 determines the confidence level of the resulting interval. For instance, ∆χ 2 = 1 yields single-parameter 1σ uncertainties.
The photometric data were compiled from various sources and, thus, are quite inhomogeneous, in particular with respect to the stated uncertainties. The following strategy was employed to cope with this: (i) Data flagged in catalogs as uncertain and obvious outliers are omitted. (ii) Magnitudes and colours without given errors are assigned typical uncertainties of 0.05 and 0.025 mag, respectively. (iii) To account for systematic shortcomings (e.g. in the system response curves, synthetic SEDs, or calibration of the data), a generic error of 0.015 mag is added in quadrature to all observed values. (iv) Eventually, all uncertainties are rescaled by a common factor to ensure a reduced χ 2 of 1 at the best fit.

Results
We present preliminary results for two sdB stars, the apparently single HD 205805 and the composite spectrum sdB binary PG 0749+658. These stars were chosen because extensive, high-quality photometric observations in all relevant wavelength regimes are available.

HD 205805 -a benchmark single sdB star
HD 205805 is one of the brightest sdB stars and, therefore, ample photometric measurements are available for the optical regime including Strömgren indices, in particular the Hβ index, which employs a narrow band filter centered at the Hβ line to measure its strength. An ultraviolet spectrum has also been observed by IUE as well as infrared fluxes. HD 205805 is one of a handful of sdB stars that have such good photometric data coverage and, therefore provides our benchmark for SED and colour fitting.
High resolution optical spectra taken with the FEROS spectrograph at the ESO 2.2m telescope became also available through the ESO archive. We analysed five FEROS spectra using an updated version of the grid of synthetic hydrogen and helium spectra calculated from metal-line blanketed LTE models described by Heber et al. (2000) The resulting atmospheric parameters are T eff =25114±214 K, log g=4.96±0.09, and a helium to hydrogen ratio of log(n He /n H )=-1.93±0.03 by number.
The SED fit and the corresponding confidence map for the error estimation of T eff are shown in Figs. 6 and 8, respectively. The resulting parameters (T eff =25338 +463 −423 K, log g=5.21±0.21) are in perfect agreement with those derived from spectroscopy. The metal abundance parameter (z=0.09 +0.17 −0.28 ) points to a normal sdB composition for HD 205805 consistent with the metal abundances derived by Geier (2013). The resulting interstellar reddening towards HD 205805 is very low (E(B-V)=0.016 ± 0.005 mag).

The composite spectrum sdB binary PG 0749+658
The sdB star PG 0749+658 was classified as sdB-O by Green et al. (1986). A spectral analysis of the optical spectrum resulted in an effective temperature T eff =24600K, surface gravity log g=5.54 (Saffer et al. 1994). Its composite nature was realised by Allard et al. (1994) from BVRI photometry and the faint companion was classified as spectral type K5.5, but no significant radial velocity variations were found (Maxted et al. 2001;Aznar Cuadrado & Jeffery 2002). Aznar Cuadrado & Jeffery (2001) determined the effective temperatures for both components from the SED to be T eff =25050±675 K and T c eff =5600±300 K, while Aznar Cuadrado & Jeffery (2002) analysed the composite spectrum of PG 0749+658 and derived similar tem-   Fig. 6. The contour lines refer to single parameter uncertainties of 68%, 90% and 99%, respectively. Single parameter 1 σ uncertainties are ∆T eff =400 K, ∆log g = 0.2 dex. Table 1. Results of the analyses of the SED (see Fig. 7) of PG 0749+658. The resulting interstellar redening parameter is zero to within error limits (E(B-V)<0.01mag. peratures T eff =25400±500 K and T c eff =5000±500 K. The corresponding gravities were found to be 5.7±0.11 and 4.58±0.24 for the sdB and the late-type companion, respectively. Heber et al. (2002) derived a considerably lower effective temperature of T eff =25050 K for the sdB component from the optical SED and attempted to resolve the binary spatially using the Wide Field and Planetary Camera 2 on-board the Hubble Space Telescope, but found it to be unresolved to a limiting angular separation <0.2", which at a distance of 580 pc translates into a separation < 116 AU. Ohl et al. (2000) determined metal abundances from FUV spectra obtained with FUSE and showed that the sdB is somewhat metal poor in comparison to the typical sdB abundance pattern.
The fit of the observed SED of PG 0749+658 is shown in Fig. 7 and the resulting parameters are listed in Table  1. The resulting temperatures of both stars are lower than those derived from spectroscopy. The resulting gravity of the sdB is consistent with the spectroscopic one derived by Saffer et al. (1994) to within error limits, but lower than that of Aznar Cuadrado & Jeffery (2002).

Outlook
HD 205805 and PG 0749+658 are amongst the best cases, both in terms of available data quality and wavelength coverage. For most of the other known sdB stars, available datasets are less complete. Hence we can not expect to achieve similar accuracy for the parameters derived from SED fitting, in particular the surface gravity log g sdB will likely be unconstrained as well as the metal abundance parameter z when no IUE data are available. Because of their large systematic uncertainties, FUV and NUV fluxes from the GALEX mission are not sufficient to replace UV magnitudes from IUE (Kawka et al. 2015).
Using mock datasets we shall investigate the quality requirements for observed photometric datasets to derive atmospheric parameters to be conclusive. Kupfer et al. (2015) and Kawka et al. (2015) compiled a list of close binary sdB stars with known orbits and studied their properties (see Fig. 9). We restrict ourselves to the single-lined spectroscopic binaries. Because the companions are unseen, they could be white dwarfs, low mass main sequences stars, or substellar objects. From light variations (reflection effect or ellipsoidal variations) and the mass function, the nature of the companions could be inferred only for about half of the sample (Kupfer et al. 2015).

The sample of sdB binaries with known orbits
Hence, we embarked on an analysis of their SEDs in order to better constrain the nature of the companions. We compiled available photometric data from the data archives and constructed the SEDs as described in Sect. 2. The sample contains 142 stars. Twenty-six are reflection effect systems, hence the companions are normal stars, but were excluded from the study because of their light variability. Kupfer et al. (2015) suggested that 52 stars host a white dwarf companion. We could model the SEDs of 50 of them by a single synthetic SED, confirming the white dwarf nature of the companion. However, two binaries showed infrared excess and where modelled with a composite SED. The companions are most likely main-sequence stars. For the stars for which the nature of the companion was unclear, we were able to reproduce their observed SED with a single synthetic sdB one in fifty cases, but ten binaries require a composite SED, indicating that the companion is likely a late-type mainsequence star. In the case of the single-SED binaries addi- Fig. 9. Companion mass histogram of the sample of sdB binaries with known orbits (Kupfer et al. 2015). Systems with white dwarf companions are depicted in light grey, those with M-dwarf companions in grey, and systems for which the nature of the companion is unclear are marked in black. Adapted from Kupfer et al. (2015).
tional modelling is required to possibly clarify the nature of their companions. Details will be reported in a subsequent publication.

Gaia, SkyMapper, and other photometric surveys
Because subdwarf O and B stars are hot the Balmer jump is an important diagnostic tool, which requires measurements of optical UV (e.g. u, u' or U) or NUV magnitudes. Several ongoing surveys will provide such photometric data, in particular SkyMapper, which measures the Strömgren u-band, and the Gaia space mission, which will measure spectrophotometry in 30 bands with fine sampling of the Balmer jump. All-sky NIR surveys will be important to study composite spectrum sdB binaries and constrain the properties of both components. This will put us into an excellent position to constrain the properties of the known (> 5000) hot subdwarfs (Geier et al. 2017), out of which we expect 50% to be close binaries as well as to enlarge the sample enormously from new discoveries, in particular from Gaia.