BINARY STAR SYNTHETIC PHOTOMETRY AND DISTANCE DETERMINATION USING BINSYN

Synthetic photometry has become a widely used procedure. Sample references are Bessell & Brett (1988), Cohen et al. (1992), Bessell et al. (1998, hereafter BCP), Cohen et al. (2003a), Cohen et al. (2003b), Bohlin & Gilliland (2004), Bessell (2005), Holberg & Bergeron (2006, hereafter HB), Holberg (2007), Holberg et al. (2008), Bohlin et al. (2011) and Bessell & Murphy (2012, hereafter B&M).

These applications have considered individual stars. Whether most stars are single or multiple is still a subject of discussion (Lada 2006), but a large fraction of all stars are binary, including systems with planetary companions. Does synthetic photometry apply to components of binary star systems? In principle, if the individual component spectra can be calculated, individual synthetic magnitudes follow and, for the system, individual absolute magnitudes and a system distance determination is possible. By extension, eclipse effects can be calculated and a light curve produced based on synthetic photometry. The available observed spectrum will represent combined contributions from the separate components. The BINSYN suite produces separate synthetic spectra for the components and a combined synthetic spectrum. The analysis procedure would fit the combined spectrum to the observed spectrum and would use the combined spectrum for synthetic photometry.

This paper demonstrates procedures for applying synthetic photometry to individual binary star components; a paper in preparation describes the production of synthetic photometry light curves and their solution. This demonstration proceeds by placing photometric standards in fictitious binary systems and extracting relevant data from the resulting systems.

The BINSYN program suite provides synthetic photometry procedures for binary stars. A recent paper (Linnell et al. 2012) announced the public availability of the BINSYN program suite⁴; users should consult that paper and the Users Guide, available from the wiki, for a complete description of the program details.

The plan of this paper is as follows. Section 2 discusses the calculations involved in synthetic photometry. Section 3 discusses photometric distance determination. Section 3.1 considers the V photometric zero point while Section 3.2 describes our routine, Integsp, for calculating synthetic magnitudes. Section 4 describes spectrophotometric standardization for components of binary star systems, specifically fictitious binary systems that include a spectrophotometric standard. Section 4.1 discusses Vega as a spectrophotometric standard. In Section 4.2, we place Vega in a fictitious binary system and compare standardization results with the literature and Section 4.3 presents Vega spectrophotometry using routine Integsp. Section 4.4 describes Vega distance determination using synthetic photometry. Section 5 describes a corresponding study of Sirius. Section 6 presents a study of the white dwarf photometric standard GD153. Section 7 presents a study of the sun-like photometric standard HD209458. Section 8 discusses the relation of BINSYN and the Wilson WD and the Prs˘a & Zwitter PHOEBE procedures. Section 9 summarizes our results.

Bessell (2005), Bohlin et al. (2011), and B&M provide a detailed description of synthetic photometry.

Define a mean flux given by

$$\begin{equation} \langle f_{\lambda }\rangle = \left[\frac{\int _{0}^{\infty } F({\lambda })R({\lambda })d{\lambda }}{\int _0^{\infty }R({\lambda })d{\lambda }}\right], \end{equation} \tag{ 1 }$$

where F(λ) is the monochromatic measured flux in energy units and R(λ) is the filter bandpass function normalized to a peak response of 1.0 (Bessell (2005)).

B&M (their Equations (A4) and (A5)) define a synthetic V magnitude by

$$\begin{equation} V=-2.5 {\rm {log}}\langle f_{\lambda }\rangle -21.100. \end{equation} \tag{ 2 }$$

The equation is written for the V band but applies to other bands with the corresponding filter function. Bessell (2005) provides a very valuable discussion of different photometric systems and the filters used to define them.

Production of a simulated "catalog" magnitude for comparison with photometric catalogs requires a zero point correction for individual filters; the calculated catalog magnitude is

$$\begin{equation} V_{\rm {cat}}=V-V(\rm {zp}). \end{equation} \tag{ 3 }$$

BCP tabulate zero points (their Table A2) for the Cousins–Glass–Johnson system. Note that the labels of the last two rows in the BCP Table A2 are interchanged.

Oke & Gunn (1983) defined narrow band AB magnitudes by

$$\begin{equation} {\rm AB}_{\nu }=-2.5 {\rm {log}} F({\nu })-48.600, \end{equation} \tag{ 4 }$$

where F(ν) is the energy flux measured in bands of about 50 Å width.

Broadband AB magnitudes are calculated with Equations (5) and (6),

$$\begin{equation} {\rm AB}_{\nu }=-2.5 {\rm {log}} \left[\frac{\int _{0}^{\infty } F({\nu })S({\nu })d{\nu }/{\nu }}{\int _0^{\infty }S({\nu })d{\nu }/{\nu }} \right] -48.600 \end{equation} \tag{ 5 }$$

(B&M, Equation (A29)), and

$$\begin{equation} {\rm AB}_{\lambda }=-2.5 {\rm {log}} \left[\frac{\int _{0}^{\infty } F({\lambda })S({\lambda }){\lambda }d{\lambda }}{\int _0^{\infty }S({\lambda }){\lambda }d{\lambda }} \right]-21.100 \end{equation} \tag{ 6 }$$

(B&M, Equations (A4), (A5), and (A11)), where F(λ) is the flux in energy units (see B&M) and S(λ) is the (normalized) filter bandpass function including detector quantum efficiency, telescope optics, and terrestrial atmosphere effects (if applicable). The functions S(λ) and S(ν) account for the difference between photon counting and energy integration (see B&M); F(λ) and F(ν) both are in energy flux units and are connected by the monochromatic equation

$$\begin{equation} F(\nu)=F(\lambda)(\lambda ^2/c). \end{equation} \tag{ 7 }$$

Equations (5) and (6) are described in detail in B&M; these authors note that the standard Hubble Space Telescope (HST) photometry packages, synphot and pysynphot, use AB_λ magnitudes. They are referenced as STmags in the HST literature.

Cohen et al. (1992) advocate the use of an isophotal wavelength for which the corresponding flux is the same value as the mean flux integrated across the band (B&M, Equation (A18)). Since the definitions are the same, we use the term mean flux:

$$\begin{equation} {F_{{\rm iso}}}{(\lambda)}=\langle F_{\lambda }\rangle \end{equation} \tag{ 8 }$$

$$\begin{equation} \langle F_{\lambda }\rangle = \left[\frac{\int _{0}^{\infty } F({\lambda })S({\lambda }){\lambda }d{\lambda }}{\int _0^{\infty }S({\lambda }){\lambda }d{\lambda }}\right]. \end{equation} \tag{ 9 }$$

HB use Equation (9) to calculate their flux-weighted magnitudes.

Cohen et al. (2003a, 2003b) tabulate mean flux values, 〈F_{λ, 0}〉, for a fictitious source whose calculated AB magnitude would be 0.0. Thus, for 〈F_λ〉 the calculated catalog magnitude is

$$\begin{equation} \rm {m_{{\rm cat},{\lambda }}}=-2.5{\times }log \left[\frac{\langle F_{\lambda }\rangle }{\langle F_{\lambda,0}\rangle } \right] + zp({\lambda }), \end{equation} \tag{ 10 }$$

where zp(λ) is a zero point. We call Equation (10) the Cohen procedure. Cohen et al. (2003a) determine zero points equal to 0.0 for (transformed) Landolt filters (see Cohen et al. (2003a, Section 3.3) and zero points for Two Micron All Sky Survey (2MASS) filters (Cohen et al. 2003b). HB separately determine a set of zero points (their ΔM) and 〈F_{λ, 0}〉. Equations (3) and (10) provide alternate routes to calculate synthetic "catalog" magnitudes. Cohen adds the zero point (Equation (10)), Bessell subtracts (Equation (3)).

Three related quantities are the effective wavelength, the pivot wavelength, and the mean photon wavelength. See B&M for details.

The synthetic photometry reported here involves three sets of photometric filters. The first is described by Bessell (1990); we label this set the Bessell90 filters. The second is described in B&M. The third is the filter set described by Cohen et al. (2003a, 2003b); we label this set the Cohen filters; it includes Landolt filters as modified by transmission through the Earth's atmosphere and the response efficiency of the detection electronics and, similarly, the 2MASS filters.

Let FM(λ) represent the system model flux at the distance, d, of the system in question. Applying the BCP (Equation (3)) procedure, if the model is an exact fit to the observed spectrum, then

$$\begin{eqnarray} \rm {m_{{\rm cat},{\lambda }}} &=& -2.5 {\rm {log}} \left[\frac{\int _{0}^{\infty } {\rm FM}({\lambda })S({\lambda }){\lambda }d{\lambda }}{\int _0^{\infty }S({\lambda }){\lambda }d{\lambda }} \right] \nonumber\\ &&-21.100-\rm {zp(m({\lambda })})+5.0 {\rm {log}} (d), \end{eqnarray} \tag{ 11 }$$

where m_{cat, λ} is the catalog magnitude in filter m and zp(m(λ)) is the corresponding zero point. Applying the Cohen procedure, let the synthetic mean physical flux at the system be 〈FM_λ〉. Then, if the synthetic spectrum scaled by the distance squared exactly fits the observed spectrum,

$$\begin{equation} \rm {m_{{\rm cat},{\lambda }}}=-2.5{\times }log \left[\frac{\langle FM_{\lambda }\rangle }{\langle F_{\lambda,0}\rangle }\right] + zp({\lambda })+5.0 {\rm {log}} (d). \end{equation} \tag{ 12 }$$

3.1. V Photometric Zero Point

3.2. Integsp

A primary objective of this paper is to demonstrate that BINSYN spectrophotometry can be applied to an isolated component of a binary system. To this end, we compare BINSYN spectrophotometry of four standards, Vega, Sirius, GD153, and HD209458 with data from the literature. Sirius is a member of a binary system but it is simple to treat it as an isolated object. HD209458 has a (famous) planetary companion; we may treat it formally as a binary system although the influence of the planet is very small. We place each standard in a fictitious binary system with secondary component parameters chosen so that there is negligible intercomponent interaction. We stress that BINSYN calculates the properties of both components of a binary system separately as well as composite system properties; this isolation permits direct comparison of the BINSYN representation of the standards with their single star properties from the literature. By setting the component bolometric albedos = 0.0 we eliminate radiative component interaction. BINSYN provides an option to force a spherical model. However, because Vega is a rapidly rotating star we require a Roche model with rotational distortion, repeated for the other stars for consistency in comparisons. We therefore usually specify a fictitious orbital period large enough to make mutual component tidal distortion negligible.

4.1. Vega as a Spectrophotometric Standard

4.2. A Vega Model

The adopted parameters are in Table 1 and their continuation plus calculated parameters are in Table 2. The mass is from Yoon et al. (2008) and the inclination of the rotation axis, here called the orbital inclination, is from Hill et al. (2010). We note that Aufdenberg et al. (2006) determine a mass of 2.3 M_☉ and i = 47, and Gulliver et al. (1994) determine i = 51. The value of ψ, the orbital longitude, is arbitrary (because of the inclination no eclipse is possible; radiative interaction is 0.0 and tidal interaction is negligible). Similarly, the values of orbital period, mass ratio, Ω(3-I) (Roche potential), FV(3-I) (rotation parameter), and T_eff(3-I) are arbitrary as long as the component interaction is negligible (see below) and the binary system is physically acceptable. (I is an integer which designates as I = 1 the component eclipsed at orbital phase 0.0, in this case Vega, and 3-I is the companion.)

Table 1. Vega Fictitious Binary System Parameters

Parameter	Value	Parameter	Value
M(I) = Vega	2.09 M☉a	A(I)	0.0
Period(d)	20.0	BCF(I)	0.17
Mass ratio	0.2	Teff(I)(pole)	9700 K
Ω(I)	18.8	Teff(3-I)(pole)	4000 K
Ω(3-I)	10.0	i	57b
FV(I)	36.7	ψ	3.14159
FV(3-I)	1.0

Notes. ^aYoon et al. (2008). ^bHill et al. (2010).

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Table 2. Vega Parameters (continued)

Parameter	Value	Parameter	Value	Parameter	Value
R(pole)	2.2651 R☉	Teff(pole)	9700 K	log g(pole)	4.06
R(point)	2.8061 R☉	Teff(point)	8076 K	log g(point)	3.58
R(side)	2.8056 R☉	Teff(side)	8078 K	log g(side)	3.58
R(back)	2.8061 R☉	Teff(back)	8076 K	log g(back)	3.58

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The parameter A(I) is the bolometric albedo; the value 0.0 specifies no photospheric heating by the companion. (The output section dealing with the companion can show photospheric heating by the primary, and that heating can appear in the output section dealing with the entire system. The output section considered here isolates the primary.) Parameter BCF(I) specifies the gravity brightening exponent. (See Linnell 1984, Equation (31) for a definition and Kallrath & Milone 2009, Section 3.2.1 for a discussion.) The value of Ω(I), the photospheric Roche potential, combines with M(I) to set the polar radius and the polar log g. The value of FV(I) is the rotation rate in units of reciprocal orbital period (FV(I) = 1.0 represents synchronous rotation). BINSYN calculates a synthetic spectrum representing the component physical flux at the system. The flux at Earth follows by dividing by the square of the distance, neglecting any interstellar absorption. The Hipparcos parallax is 130.23 ± 0.36 mas; the corresponding distances is 7.678 (7.658, 7.700) pc; the Hipparcos distance squared, 5.614 × 10³⁸ cm², is a fixed scaling factor for superposing the synthetic spectrum on the observed spectrum.

According to Brown et al. (1974), the darkened disk angular diameter of Vega is 3.24 mas; Ciardi et al. (2001) cite a value of 3.28 mas and Aufdenberg et al. (2006) determine 3.345 mas. Given the Hipparcos distance and the Brown et al. (1974) angular diameter, the apparent radius of Vega is 2.68 R_☉. With the adopted component mass, the required radius sets an initial value of the Roche potential, Ω(I). The model analysis proceeds by calculating a model synthetic spectrum, using the scaling factor described above to place the scaled synthetic spectrum on the observed spectrum, and modifying the adjustable parameters empirically to produce the best-fitting model spectrum as judged visually. The adjustable parameters include T_eff(pole), Ω(I), BCF(I), and FV(I); the flux at 1000 Å is sensitive to T_eff(pole) and the flux at 30000 Å is sensitive to FV(I); the overall fit is sensitive to Ω(I). Experiment showed that the pole-to-equator temperature variation produced by variation of BCF(I) is important; allowing the gravity brightening value to change from the von Zeipel value for radiative atmospheres (0.25) produced an improved fit. The gravity brightening value finally selected, 0.17 (Table 1), is intermediate between the von Zeipel value and the value for convective atmospheres (0.08) and is consistent with the argument by Aufdenberg et al. (2006) that rotation-induced convection may be in play. Our Table 2 values of equatorial radii produce a Vega angular diameter of 3.40 mas at the Hipparcos distance, closely equal to the literature values.

Our spectra use the line lists originally supplied with program SYNSPEC.⁷ See Linnell et al. (2012) for details on our use of the programs TLUSTY and SYNSPEC. According to that web site, the adopted solar abundances are from Grevesse & Sauval (1998). The line lists divide into three ranges: 900 Å to 3000 Å, 3000 Å to 10,000 Å, and 10,000 Å to 100,000 Å. Because the James Web Space Telescope will permit surveys to 300,000 Å, R. C. Bohlin (2013, private communication) recommended that where possible, our spectra extend to that limit. Although the SYNSPEC line list does not extend beyond 100,000 Å, the continuum can be extended to the longer limit. The structure of the SYNSPEC line lists differs from Kurucz (http://kurucz.harvard.edu/linelists.html) and Castelli (http://wwwuser.oat.ts.astro.it/castelli/linelists), and the latter line lists cannot be used with SYNSPEC.

It is known that Vega has lower than solar abundances. We initially adopted abundances from Fitzpatrick (2010; there designated FIT A) and the He abundance from Yoon et al. (2008). That list of abundances led to appreciable residuals (extra flux) in the 900 Å to 2500 Å region. We used the output lists of calculated line strengths for the individual synthetic spectra to identify atomic species with strong lines in the sensitive region and used that information empirically to modify adopted abundances to reduce the discrepancies. Other parameters that might affect the SED, such as rotation, microturbulence, etc., have been ignored. Table 3 lists final abundances used by SYNSPEC in calculating the Vega synthetic spectrum.

Table 3. fort.56 Vega Control File for SYNSPEC

Item	Atomic Number	Abundance	Species
1	2	0.03	He
2	6	8.35	C
3	7	7.91	N
4	8	8.48	O
5	12	7.10	Mg
6	13	6.10	Al
7	14	7.10	Si
8	15	5.00	P
9	16	7.00	S
10	20	6.00	Ca
11	22	4.21	Ti
12	23	3.40	V
13	24	5.50	Cr
14	25	5.00	Mn
15	26	6.00	Fe
16	27	4.70	Co
17	28	5.75	Ni
18	30	4.81	Zn

Notes. The abundance is X/H = log (N_X/N_H) + 12.0, where N_X/N_H = the number abundance of element X relative to that of hydrogen.

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Our model represents Vega by a grid of 91 latitude profiles and 151 longitude profiles. We interpolate from a two-dimensional grid of model atmosphere synthetic spectra to the T_eff and log g of each photospheric segment and sum the results. To cover the range of T_eff values dictated by gravity brightening, the geometric model required synthetic spectra for T_eff values of 8000 K, 8250 K, 8500 K, 8750 K, 9000 K, 9250 K, 9500 K, 9750 K, and 10,000 K and log g values of 3.5, 4.0, and 4.5. We used models from the Castelli (wwwuser.oat.ts.astro.it/castelli/) grids of models, specifically, the ap00k2C125odfnew Atlas9 grid, and applied the SYNSPEC program to calculate the 27 synthetic spectra at a resolution of 0.1 Å in the ultraviolet (UV) and optical and 1.0 Å in the infrared (IR).

Figure 1 shows a sample screenshot of the control program for calculating a 1.0 Å resolution IR synthetic spectrum. The sample program is abbreviated from the one actually used to enable presentation of its entirety on a single page. The first three fields on the top line specify that 20,000 wavelengths will be considered, starting at 10,000 Å, and at a spacing of 1.0 Å. Line 4 specifies 5 temperatures and 3 log g values for the synthetic spectra supplied by SYNSPEC. Lines 5–12 list the T_eff and log g values. Lines 13–15 specify corresponding data for the companion. Lines 16–30 list the synthetic spectra from SYNSPEC for Vega and line 31 lists the single synthetic spectrum for the companion. For a given SYNSPEC spectrum (lines 16-30), the final three fields provide information on how to use the spectrum. In the case of Vega spectra, the first field, with the value 10, indicates that the Vega synthetic spectrum will be calculated using the 10 intensity spectra from the SYNSPEC file, and the second field indicates that the SYNSPEC intensity spectra were calculated at a cos(γ) spacing of 0.1. The third field is not used in this instance. See Linnell et al. (2012) for details.

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The synthetic Vega spectrum result of this calculation is in erg cm⁻² s⁻¹ hz⁻¹; a subsequent program converts the spectrum to $\rm {erg}\ {\rm cm^{-2}\ s^{-1}\ {\rm \mathring{\rm{A}}}^{-1}}$. We call this the high resolution IR spectrum. We then used the IDL program rsmooth.pro, kindly supplied by Dr. Ralph Bohlin, to calculate the final IR spectrum at a resolution of 500 for comparison with the CALSPEC Vega spectrum.

We used the ACP6.DAT control file, Figure 1, modified, to calculate high resolution optical and UV spectra at a resolution of 0.1 Å followed by calculation of a resolution = 500 spectrum for the optical region and 200 for the UV region. The spectrum of the secondary component and the system spectrum (sum of the two components) were discarded; if the study concerned an actual binary system, then those spectra would be involved in further processing. The use of intensity spectra automatically handles the issue of limb darkening.

Figure 2 shows a fit to the CALSPEC spectrum (black) with the BINSYN resolution = 200 synthetic spectrum (red). In producing this fit, we have scaled the synthetic Vega spectrum by 5.614 × 10³⁸, the square of the Hipparcos distance. We stress that there is no empirical adjustment of the scaling factor.

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Figure 3 shows the fit details of the high Balmer series and the continuum shortward of the Balmer jump. There are some discrepancies between the synthetic and observed spectra in the regions devoid of strong lines. These may very likely be attributed to NLTE effects in metals, such as C, Mg, Si, and perhaps Fe, which could be alleviated by fine-tuning abundances, using NLTE line blanketed models. For a detailed discussion see García-Gil et al. (2005). We did not embark on such a study since our aim here is not to perform detailed fitting of individual UV spectral features but instead to show overall agreement of predicted and observed spectra for synthetic photometry. Figure 4 shows details of the 3000 Å–10,000 Å fit. Figure 5 shows details of the 10000 Å–30000 Å fit and Figure 6 shows the fit in the Brackett series region. Figure 7 shows the ratio of the synthetic spectra to the CALSPEC spectrum, extending to 100,000 Å. The continuation to 400,000 Å maintains a ratio of 1.0 within 3%. The separate wavelength segments were concatenated into a single model spectrum. For a 3000 Å to 100,000 Å model fit, the standard deviation is 1.16 × 10⁻¹¹ erg s⁻¹ cm⁻² Å⁻¹ (compare with the ordinal scale; Figure 4); the reduced χ² is 1.0018 for 96,925 degrees of freedom. The presence of a debris disk (Gray 2007) contaminates the observed spectrum beginning at about 200,000 Å (Cohen et al. 1992). A copy of the digital model spectrum is available in the BINSYN Web site.⁸

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4.3. Vega Spectrophotometry with Integsp

4.4. Vega Distance Determination via Synthetic Photometry

Cohen et al. (1992) make the case for Sirius as a fundamental spectrophotometric standard.

No CALSPEC spectrum of Sirius has been available until very recently. Dr. Ralph Bohlin has kindly supplied a new STIS spectrum (sirius_stis_001.dat) prior to its public release. According to the (now available) sirius_stis_001.fits file heading, the file spectrum consists of three segments: an IUE spectrum over the range 1150 Å to 1675 Å, merged HST spectra over the range 1675 Å to 10,000 Å, and a special Kurucz synthetic spectrum over the range 10,000 Å to 2,996,860 Å. We finally adopted a polar T_eff of 9950 K, different from the isothermal T_eff = 9850 K value by Kurucz.⁹ The mass is 2.12 M_☉ (Landstreet 2011). Other model parameters are in Tables 10 and 11. The orbital inclination is from Gatewood & Gatewood (1978). The adopted orbital period is arbitrary and bears no relation to the actual 50.09 yr orbit with Sirius B. The adopted orbital period is used with FV(I), the rotation parameter described below, to set the projected equatorial rotation velocity to its observed value of $\rm {v \times sin(i)}=16.5\ \rm {km\ s^{-1}}$ (Landstreet 2011). The value of Ω(I), the Roche potential of Sirius, determines the radius, listed in Table 11. The mass and radius determine the photospheric gravity. Simbad lists the parallax as 379.21 ± 1.58 mas; the corresponding distance is 2.637 (2.626, 2.648) pc. The Table 11 radius produces an angular diameter of 6.10 mas: Brown et al. (1974) list a limb-darkened observed value of 5.89 mas while Cohen et al. (1992) determine a value 6.02 mas for their model. Given the model radius, the observed v dictates the value of FV(I), the ratio of the component rotation rate to the synchronous rotation rate.

Table 10. Sirius Fictitious Binary System Parameters

Parameter	Value	Parameter	Value
M(I) = Sirius	2.12 M☉a	A(I)	0.0
Period(d)	00.0	BCF(I)	0.25
Mass ratio	0.4717	Teff(I)(pole)	9950 K
Ω(I)	77.89	Teff(3-I)(pole)	4000 K
Ω(3-I)	9000.0	i	4347b
FV(I)	78.84	ψ	1.5707963
FV(3-I)	1.0

Notes. ^aLandstreet (2011). ^bGatewood & Gatewood (1978).

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We used SYNSPEC to calculate synthetic spectra for T_eff = 9750 K and 10,000 K and logg = 4.0, 4.5 for each T_eff. The model atmospheres were from the Castelli data base (Section 4.3) for solar composition. A Sirius control file corresponding to Figure 1 permitted interpolation to the specific local model values of T_eff and log g. Note that rotational distortion produces a variation in photospheric T_eff of 100 K between the pole and equator.

We assumed an initial composition profile (Landstreet 2011, Table 1, LII) and listed it in a fort.56 composition modification file (for displacement from the solar abundance adopted by the input Castelli models). There were appreciable excess ultraviolet flux residuals in the early trial fits. Modification of the composition profile, based on the tabulated line strengths from a given trial fit, led to the final adoptions listed in Table 12.

We calculated the SYNSPEC spectra at a spacing of 0.2 Å, then smoothed the output model spectrum to a resolution of 500 (200 in the UV). The initial model adopted a Ω(I) for which the model radius corresponded to the Hanbury Brown angular diameter. We subsequently modified Ω(I), in analogy to the Vega procedure, to achieve a best fit to the STIS spectrum.

Figure 8 illustrates our ultraviolet region synthetic spectrum and Figure 9 shows the Balmer jump region. As with Figure 3, the discrepancies can be attributed to NLTE effects. Figure 10 shows the fit in the optical region and Figure 11 in the near infrared. This plot covers the wavelength range of concern for the 2MASS simulation. Figure 12 shows the fit to the Brackett series. Figure 13 shows the ratio of the model spectrum to the STIS spectrum over the range 2000 Å to 30,000 Å, the wavelength region of our interest for synthetic photometry. The extension to 400,000 Å maintains a flux ratio within 1% of 1.0. A copy of the digital model spectrum is available on the BINSYN Web site.¹⁰

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Table 13 compares synthetic photometry with BCP data (their Table A1 with magnitudes extracted from their colors). We have used the BCP observed V = −1.43 in calculating individual magnitudes. Our synthetic photometry used the Bessell90 filters, the Bessell procedure (Equation (3)), and the STIS spectrum. Our synthetic magnitudes are in the row listed as calc. mag. The corresponding observed data are in the row listed as BCP.

Table 14 compares synthetic photometry, using the STIS spectrum, with photometric data listed in the Simbad Sirius entry. The calculations use Equation (9). The row labeled 〈F_{λ, 0}〉 is the mean flux for a zero magnitude star as determined by HB. The row labeled calc. mag. is the magnitude offset tabulated by HB, Column 2. The J, H, K_s entries are 2MASS photometry with respective 1σ values of 0.109, 0.184, and 0.214. The synthetic photometry closely fits the observational data. We assumed zero interstellar extinction or reddening.

For the interval 3000 Å to 100,000 Å, the fit of the synthetic spectrum to the STIS Sirius spectrum has a reduced χ² of 1.0022 and a standard deviation of residuals of $5.53\times 10^{-11}\ {\rm {erg}\ {\rm cm^{-2}\ {\rm s}^{-1}\ {\rm \mathring{\rm{A}}}^{-1}}}$ (compare with the ordinate scale of Figure 10). Table 15 lists distance determinations for Sirius using Equation (12); the procedure follows that of Table 9. Note that the distance determinations are accurate to 1%, including the J, H, K_s photometry, in spite of the brightness of Sirius.

GD153 is one of a set of white dwarf spectrophotometric standards discussed by Bohlin et al. (1995). Additional discussion can be found in Bohlin (1996, 2000, 2007). The present analysis used the CALSPEC spectrum.¹¹ The spectrum consists of two segments: a merged STIS spectrum covering the range 1140 Å to 10,040 Å, and a merged NICMOS spectrum covering the range 10,040 Å to 24,976 Å.

The adopted system parameters are in Table 16. The WD mass is from Finley et al. (1997, hereafter F97). The period is fictitious but, together with the component masses, it sets the component separation and in turn controls the value of the Roche potential, Ω(I), required to produce a photospheric log g, 7.662, equal to the tabulation of F97. We used TLUSTY and SYNSPEC (Section 4.2) to calculate a series of pure H NLTE models and corresponding synthetic spectra. Initial tests adopted the value of T_eff(I), 38,686 K, from F97. However, it was not possible to fit a 1100 Å–3000 Å synthetic spectrum segment to the observed spectrum using the same scaling factor as a 3000 Å–10,000 Å spectrum segment. We finally achieved consistency with a 40,000 K synthetic spectrum and the original log g. Table 17 completes the listing of system parameters.

Table 18 tabulates synthetic photometry magnitudes using the CALSPEC spectrum and Cohen filters. The observed UBVRI and J, H, K_s magnitudes are from Simbad. Table 19 substitutes the synthetic spectrum for the CALSPEC spectrum; the calculated magnitudes track Table 18 accurately. Table 20 follows the procedure used for the previous standards to calculate photometric distances. Scaling from the synthetic spectrum at the star to the synthetic spectrum at Earth assumed zero interstellar absorption. The table omits the observed r and i photometry because of their large residuals from the synthetic photometry. The consistency of the calculated distances is reasonably good. Note that the calculated distances depend directly on the WD radius, which in turn is set by the adopted component mass and photospheric log g. Over the range 1140 Å to 24,976 Å, the fit of the synthetic spectrum (at Earth) to the observed spectrum has a reduced χ² of 1.0003 for 23,836 degrees of freedom.

Figure 14 presents the fit of the model synthetic spectrum, at Earth, to the observed spectrum over the range 1100 Å to 3000 Å. The scaling factor to convert from the synthetic spectrum at the star to the spectrum at Earth is 1.7932 × 10⁻⁴¹. Figure 15 shows the fit to Lyα. This spectrum segment had no resolving power smoothing; the synthetic spectrum had a calculated spacing of 0.05 Å. Figure 16 shows the fit over the range 3000 Å to 10,000 Å. This spectral segment was smoothed for a resolution of 500. The fit to Hδ appears in Figure 17. Figure 18 shows the fit over the range 10,000 Å to 30,000 Å; the smoothing resolution was 350. The fit to Pα is in Figure 19. Figure 20 presents the ratio of the synthetic spectrum to the observed spectrum over the range 1100 Å to 25,000 Å. A copy of the synthetic spectrum is available from the Web site.¹²

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HD209458 is one of the solar analog stars selected as an HST spectrophotometric standard (Bohlin 2010). Our analysis uses the CALSPEC spectrum.¹³ The CALSPEC spectrum consists of an STIS spectrum covering the range 2905 Å to 10,150 Å; a NICMOS spectrum covering the range 10,150 Å to 24,920 Å, and a synthetic spectrum described in Bohlin (2010) covering the range 24,920 Å to 40,0000 Å.

The simulation assumed zero interstellar extinction or reddening. As an interesting example of the BINSYN potential to simulate both components of stars with planets, we have modeled both components but have excluded radiative interaction. The adopted parameters are in Tables 21 and 22. Note that component B (the planet) shows a small distortion (Table 22, elongated R(point) and shortened R(pole)). This arises from the requirement that the photosphere be represented by a single Roche potential, Ω(3-I), (Table 21); the elongated R(point) represents a tidal distortion and the R(pole) that is smaller than the other three radii represents rotational distortion arising from the specified synchronous rotation, FV(3-I), Table 21. A projected view of the system at orbital phase 0.15 (Table 21) is in Figure 21. Synthetic light curves give a reasonable representation of secondary minimum using the primary component synthetic spectrum described below and a T_eff = 2500 K MARCS¹⁴ spectrum for the planet, but the primary component grid is too coarse compared with the size of the planet, leading to a representation of the primary minimum that is unsuitable for light curve analysis.

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Table 21. HD209458 Fictitious Binary System Parameters

Parameter	Value	Parameter	Value
M(I) = HD209458	1.101 M☉a	A(I)	0.0
Period(d)	3.5247b	BCF(I)	0.25
Mass ratio	0.000555c	Teff(I)(pole)	6100 Kd
Ω(I)	8.91	Teff(3-I)(pole)	2500 Ke
Ω(3-I)	1.5417	i	86929f
FV(I)	1.0	ψ	0.15
FV(3-I)	1.0

Notes. ^aBBH ^bBBH ^cBBH ^dKurucz, see text. ^eBBH, see text. ^fBBH

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The widely adopted T_eff = 6117 K log g = 4.48 primary component calibration originates with Santos et al. (2004). These parameters and other adopted parameters, found in Table 21, are tabulated in Burrows et al. (2008), hereafter BBH. Their values of component radii are from light curve analyses referenced in their paper.

We have adjusted Ω(I) and Ω(3-I), assuming synchronous rotation, to produce the adopted geometric parameters as specified. Calculation of a synthetic spectrum for HD209458 requires a model atmosphere synthetic spectrum that extends far enough into the infrared to permit simulation using the 2MASS filters. There is a special HD209458 synthetic spectrum by Kurucz.¹⁵ This model has a T_eff of 6100 K, log g = 4.38; in using this single synthetic spectrum in the HD209458 equivalent of Figure 1, we force an isothermal model atmosphere representation of the primary, which is component A in Table 22. The slight variation of calculated log g in Table 22 is purely formal.

With the BBH radius of the primary and the adopted T_eff, there is a marginal inconsistency with the Hipparcos distance. The Hipparcos parallax is 20.15 ± 0.80 mas, corresponding to a distance of 49.628(47.733, 51.680) pc. With our adopted parameters, an accurate fit of the Kurucz model primary component spectrum to the observed spectrum requires a distance of 45.832 pc, a 2σ discrepancy. The calculated angular diameter of the primary is 0.227 mas, using the Hipparcos distance.

Claret (2009) has discussed the disagreement between model atmosphere limb darkening coefficients and the observed values from light curve analysis and shows that in the range shortward of 5000 Å, a 5250 K, log g = 4.5 solar composition synthetic spectrum provides a better representation; however, the lower temperature model produces a poorer fit at longer wavelengths.

Our synthetic spectrum fit using the Kurucz model is in Figure 22, with the infrared continuation in Figure 23. Figure 24 shows the ratio of the synthetic spectrum (at the Earth) to the observed spectrum. A copy of the digital model spectrum is available from the BINSYN Web site.¹⁶

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The σ of the fit is 3.78 × 10⁻¹⁴ erg cm⁻² s⁻¹ Å⁻¹ for comparison with the ordinate scale of Figure 22. The reduced χ² of the fit is 1.0018 with 26,845 degrees of freedom over the range 3000 Å to 30,000 Å (note that the observed spectrum terminates at 25,000 Å).

Table 23 presents calculated magnitudes (Equation (10)) using the STIS spectrum and Cohen filters. Table 24 presents calculated magnitudes using the model synthetic spectrum as seen at the Earth and Cohen filters. Calculated distances are in Table 25. The presentation is as in Tables 9, 15, and 20. The synthetic photometry presented here determines distances with internal consistency error limits as least as tight as with Hipparcos parallaxes.

Wilson (2008), hereafter W2008, and Prs˘a & Zwitter (2005) have introduced a procedure simulating synthetic photometry in their respective programs WD and PHOEBE. The heart of their simulation depends on the tabulations by Van Hamme & Wilson (2003, hereafter VH). VH start with intensities for the 1993 Kurucz model atmospheres (Kurucz 1993) as contained on CD-ROMs 16 and 17. Intensities are available at 1221 wavelengths between 9 and 160,000 nm, 11 log g values, and 19 abundances at T_eff values between 3500 K and 50,000 K. VH integrate the product of the intensities, filter transmission functions, and tabulate bandpass intensities as a function of T_eff. VH then establish a series of T_eff subintervals and, within those subintervals, represent bandpass intensities by Legendre polynomials of degree up to 10. Where data are needed outside the available tabular boundaries, VH provide a smooth transition to black body intensities. Use of bandpass intensity for the radiative properties of eclipsing binary components in the Wilson–Devinney (WD) program leads to absolute flux representation at filter effective wavelengths and permits, with calibrated observed flux, distance determination: a process W2008 terms Direct Distance Estimation. In a procedure paralleling our simulations in this paper, W2008 places the Sun in a fictitious binary system at 10pc and determines its U, B, and V absolute magnitudes. Given the observational difficulty of this comparison, the results appear quite accurate.

The PHOEBE program builds on the WD program with enhancements. One innovation of PHOEBE is to use parameters from a light curve solution to interpolate in the Zwitter et al. (2004) grid of synthetic spectra and construct a synthetic spectrum of the system for comparison with an observed spectrum. The procedure currently is limited to phases outside of eclipse.

We believe the BINSYN approach has a number of advantages. It more closely simulates the actual observing process. As synthetic spectra improve (e.g., ATLAS12 versus ATLAS9) the new synthetic spectra can be used immediately (the WD procedure would require repeating the calculations for the VH results). Synthetic spectra with higher resolution than the VH tables can be tested for sensitivity of synthetic photometry magnitudes to spectral resolution. Similarly, new photometric systems with new photometric passbands can be simulated immediately. BINSYN uniquely permits inclusion of fits to observed spectra at arbitrary orbital phases as part of the solution process. Given the TLUSTY, ATLAS12, or other routine scalability to calculate synthetic spectra for extreme cases of T_eff or composition, BINSYN can process data for systems that fall outside the limits of the VH tables. We emphasize that BINSYN provides synthetic spectra for the individual components as well as the system at arbitrary phases, including eclipse.

A valid criticism of BINSYN is that the processing time is longer than competing procedures. We do not believe this is a serious objection in the case of the important systems that will be the subject of individual papers. The BINSYN simulation can run concurrently with other activities.

This paper extends synthetic photometry to components of binary star systems. The paper demonstrates accurate recovery of single star photometric properties for four photometric standards, Vega, Sirius, GD153, and HD209458, ranging over the HR diagram, when their model synthetic spectra are placed in fictitious binary systems and subjected to synthetic photometry processing. For Vega, a physically accurate model requires a range of synthetic spectra to represent the variation of T_eff and log g over the rotationally distorted photosphere. The same is true, to a smaller extent, for Sirius. Both GD153 and HD209458 can be represented with a single value of T_eff and log g. Techniques for photometric distance determination have been validated for all four photometric standards. Data files of the synthetic spectra are available from the on-line electronic version of Table 26 (Vega), Table 27 (Sirius), Table 28 (GD153), and Table 29 (HD209458).

Although this study has considered the photometric standards as single objects, we emphasize that BINSYN produces separate and combined synthetic spectra for both binary components as a function of orbital phase, including eclipse effects. With synthetic photometry, introduced by this paper, BINSYN provides a means to study both observed spectra and observed multicolor photometry of binary star systems.

Thanks are due Dr. Ralph Bohlin for extensive consultation during preparation of this paper, to Dr. Mike Bessell for sending calibrations of the UBVRI filter set and for extensive email correspondence, and to Dr. Jay Holberg for numerous emails. This research has used the Simbad data base.