Testing systematics of Gaia DR2 parallaxes with empirical surface brightness -- color relations applied to eclipsing binaries

TL;DR

This study assesses Gaia DR2 parallax accuracy using eclipsing binaries and surface brightness-color relations, confirming a small negative zero-point shift consistent with Gaia team estimates.

Contribution

It provides an independent empirical validation of Gaia DR2 parallax zero-point shift using a large sample of eclipsing binaries and multiple SBC relations.

Findings

Gaia DR2 parallaxes have a zero-point shift of about -0.031 mas.

The shift is consistent with Gaia team estimates.

Different SBC relations yield slightly varying shifts.

Abstract

Using a sample of 81 galactic, detached eclipsing binary stars we investigated the global zero-point shift of their parallaxes with the Gaia Data Release 2 (DR2) parallaxes. The stars in the sample lay in a distance range of 0.04-2 kpc from the Sun. The photometric parallaxes {\pi}_Phot of the eclipsing binaries were determined by applying a number of empirical surface brightness - color (SBC) relations calibrated on optical-infrared colors. For each SBC relation we calculated the individual differences d{\pi}_i = ({\pi}_Gaia - {\pi}_Phot)_i and then we calculated unweighted and weighted means. As the sample covers the whole sky we interpret the weighted means as the global shifts of the Gaia DR2 parallaxes with respect to our eclipsing binary sample. Depending on the choice of the SBC relation the shifts vary from -0.094 mas to -0.025 mas. The weighted mean of the zero-point shift from all colors and calibrations used is d{\pi} = -0.054 +/- 0.024 mas. However, the SBC relations based on (B-K) and (V-K) colors, which are the least reddening dependent and have the lowest intrinsic dispersions, give a zero-point shift of d{\pi} = -0.031 +/- 0.011 mas in full agreement with results obtained by Lindegren et al. and Arenou et al. Our result confirms the global shift of Gaia DR2 parallaxes of d{\pi} = -0.029 mas reported by the Gaia team, but we do not confirm the larger zero-point shift reported by a number of follow-up papers.

Figures

Tables

Table 1. Table 1 Basic data on the selected detached eclipsing binaries

Name	GAIA DR2	RA(2000)	DEC(2000)	V	Spectral	Orbital	GAIA DR2 Parallax
	ID	h:m:s	deg:m:s	(mag)	Type	Period (d)	(mas)
MU Cas	429158427922077184	00:15:51.560	+60:25:53.64	10.808$\pm$0.007	B5V+B5V	9.652949	0.453$\pm$0.040
YZ Cas	539047365205648128	00:45:39.077	+74:59:17.06	5.653$\pm$0.015	A2m+F2V	4.4672235	10.473$\pm$0.094
V459 Cas	522461335380031104	01:11:29.913	+61:08:48.07	10.322$\pm$0.003	A1V+A1V	8.4582538	1.304$\pm$0.048
V505 Per	455772347387763840	02:21:12.964	+54:30:36.28	6.889$\pm$0.016	F5V+F5V	4.222020	15.970$\pm$0.039
DN Cas	513431183821175936	02:23:11.540	+60:49:50.18	9.878$\pm$0.010	O8V+B0V	2.31095111	0.485$\pm$0.032

Table 2. Table 2 Parameters of the Wilson-Devinney models

	Input parameters									Ref.	WD model parameters
Eclipsing	RV semiamplitude		Orientation of the orbit			Fractional radius		Effective temperature			Semimajor	Mass	${\Omega }_1$	${\Omega }_2$
binary	$K_1($km s-1)	$K_2($km s-1)	$e$	$\omega$ (rad)	$i$ (deg)	$r_1$	$r_2$	$T_1($K)	$T_2($K)		axis$\left(R_{\odot }\right)$	ratio
MU Cas	107.7(1.0)	105.8(9)	0.1930(3)	0.234(7)	87.02(7)	0.0917(9)	0.1048(10)	15900(500)	15525(500)	1	40.0234	1.0180	12.173	10.952
YZ Cas	73.05(19)	124.78(27)	0.0	–	88.33(7)	0.14456(56)	0.07622(33)	9520(120)	6880(240)	2	17.4753	0.5854	7.5141	8.8912
V459 Cas	81.70(60)	83.90(60)	0.0243(4)	4.19(1)	89.47(1)	0.07260(30)	0.07100(30)	9140(300)	9085(300)	3	27.6786	0.9738	14.776	14.757
V505 Per	89.01(8)	90.28(9)	0.0	–	87.95(4)	0.0860(9)	0.0846(9)	6512(21)	6460(30)	4	14.9715	0.9859	12.618	12.665
DN Cas	211(3)	292(5)	0.0	–	77.20(20)	0.3070(20)	0.2460(20)	32100(1000)	28500(1100)	5	23.5613	0.7226	4.0372	4.1129

Table 3. Table 3 Corrections to original 2MASS magnitudes

ID	K (mag)	H (mag)	J (mag)
GG Lup	-0.345	-0.342	-0.334
KX Cnc	-0.426	-0.427	-0.429
WZ Oph	-0.535	-0.536	-0.540
YZ Cas	-0.162	-0.155	-0.137
BW Aqr	-0.524	-0.526	-0.532
SZ Cen	-0.454	-0.458	-0.467
V442 Cyg	-0.283	-0.283	-0.283
ASAS1800	-0.512	-0.505	-0.493
DW Car	-0.464	-0.465	-0.467

Table 4. Table 4 Physical and photometric parameters of individual components used to derive photometric parallaxes.

Eclipsing	E$\left(B-V\right)$	Radius ($R_{\odot }$)		Observed magnitudes					Light ratios
binary	mag	$R_1$	$R_2$	$B$ (mag)	$V$ (mag)	$J$ (mag)	$H$ (mag)	$K$ (mag)	$B$	$V$	$J$	$H$	$K$
MU Cas	0.478(10)	3.670(43)	4.194(48)	11.112(9)	10.808(7)	10.170(23)	10.092(23)	10.050(16)	1.253	1.257	1.273	1.276	1.277
YZ Cas	0.015(10)	2.526(11)	1.332(6)	5.715(26)	5.660(15)	5.490(20)	5.502(42)	5.475(22)	0.061	0.088	0.168	0.200	0.205
V459 Cas	0.246(24)	2.009(13)	1.965(13)	10.591(9)	10.322(3)	9.791(23)	9.715(25)	9.668(16)	0.936	0.941	0.949	0.950	0.951
V505 Per	0.003(5)	1.288(14)	1.267(14)	7.269(30)	6.846(20)	6.120(70)	5.793(39)	5.795(21)	0.924	0.935	0.952	0.958	0.959
DN Cas	0.917(50)	7.233(96)	5.796(82)	10.495(18)	9.878(10)	8.413(25)	8.217(19)	8.132(24)	0.492	0.500	0.521	0.527	0.529

Table 5. Table 5 The zero point shifts ϖ G ​ a ​ i ​ a − ϖ P ​ h ​ o ​ t subscript italic-ϖ 𝐺 𝑎 𝑖 𝑎 subscript italic-ϖ 𝑃 ℎ 𝑜 𝑡 \varpi_{Gaia}-\varpi_{Phot} determined with eclipsing binaries for the different SBCRs

SBC	Band	Color	Range	$\log g$	Number of	(${\varpi }_{Gaia}-{\varpi }_{Phot}$) (mas)
relation			of color	(dex)	systems	Unweighted		Weighted
Kervella et al. 2004	$B$	$B-K$	$-$0.25 - 2.36	$>3.5$	61	$-0.046$ $\pm$	0.021	$-0.025$ $\pm$	0.015
Kervella et al. 2004	$B$	$B-H$	$-$0.20 - 2.28	$>3.5$	60	$-0.050$ $\pm$	0.028	$-0.041$ $\pm$	0.018
Kervella et al. 2004	$B$	$B-J$	$-$0.17 - 2.08	$>3.5$	58	$-0.092$ $\pm$	0.037	$-0.067$ $\pm$	0.021
Kervella et al. 2004	$V$	$V-K$	$-$0.20 - 1.75	$>3.5$	62	$-0.051$ $\pm$	0.021	$-0.025$ $\pm$	0.015
Kervella et al. 2004	$V$	$V-H$	$-$0.15 - 1.66	$>3.5$	61	$-0.051$ $\pm$	0.029	$-0.040$ $\pm$	0.016
Kervella et al. 2004	$V$	$V-J$	$-$0.12 - 1.47	$>3.5$	60	$-0.136$ $\pm$	0.040	$-0.083$ $\pm$	0.020
di Benedetto 2005	$V$	$V-K$	$-$0.10 - 4.93	$>2.0$	62	$-0.124$ $\pm$	0.025	$-0.052$ $\pm$	0.020
Boyajian et al. 2014	$V$	$V-K$	$-$0.15 - 1.75	$>3.5$	62	$-0.105$ $\pm$	0.024	$-0.060$ $\pm$	0.019
Boyajian et al. 2014	$V$	$V-H$	$-$0.13 - 1.66	$>3.5$	61	$-0.137$ $\pm$	0.031	$-0.084$ $\pm$	0.021
Boyajian et al. 2014	$V$	$V-J$	$-$0.12 - 1.47	$>3.5$	60	$-0.188$ $\pm$	0.044	$-0.094$ $\pm$	0.019
Challouf et al. 2014	$V$	$V-K$	$-$0.60 - 4.93	$>2.0$	72	$-0.196$ $\pm$	0.030	$-0.092$ $\pm$	0.019
Bolometric	-	-	-	-	81	$-0.103$ $\pm$	0.026	$-0.067$ $\pm$	0.012

Equations

$$K_J-K_{2M}$$

$$(J-K{)}_J$$

$$(H-K{)}_J$$

$${\varpi }_i(\mathrm{mas})=2.956\cdot 10^{10}{R}_i^{-1}{T}_i^{-2}10^{-0.2(B{C}_i+V_i)}$$

$$\varphi (\mathrm{mas})=10^{0.2\cdot (S-m_0)}$$

$${\varpi }_{Phot}(\mathrm{mas})=107.52\cdot \varphi (\mathrm{mas})/R({\mathcal{R}}_{\odot }),$$

Full text

1. Introduction

The Gaia mission (Gaia Collaboration, 2016) is a milestone in understanding the Milky Way structure and its chemo-dynamical evolution, but it is also extremely important for the recalibration of standard candles and standard rulers used to construct the extragalactic distance ladder. A number of studies devoted to the recalibration of Cepheids and RR Lyr stars with Gaia DR2 parallaxes has already appeared (e.g. Groenewegen, 2018; Muraveva et al., 2018).

The possible global systematics of the Gaia DR2 parallaxes were evaluated with different methods by the Gaia consortium. Lindegren et al. (2018) used quasars and internal validation solutions to estimate the systematics and reported that the parallaxes are, on average, by about 0.03 mas too small, and that some local significant correlations of parallaxes occur. Arenou et al. (2018) made a comparison of Gaia DR2 parallaxes with a number of external catalogs to evaluate the global shift. Individual shifts vary from $-0.118\pm 0.003$ mas (HIPPARCOS) to $+0.09\pm 0.07$ mas (Phoenix dwarf), where the minus sign signifies that Gaia parallaxes are too small. However, most catalogs point to a global systematic offset of about $-0.03$ mas, e.g. comparisons with the Large and Small Magellanic Cloud stars catalogs, which have the smallest formal uncertainties, yield shifts of $-0.0380\pm 0.0004$ mas and $-0.0268\pm 0.0006$ mas, respectively.

The global shift was also evaluated by a number of external works. A comparison with distance moduli of 50 Galactic cepheids derived from HST photometry resulted in an estimate of global zero-point offset of Gaia DR2 parallaxes of $-0.046\pm 0.013$ mas (Riess et al., 2018). Using a sample of 3475 red giant branch stars in the Kepler field from the APOKASC-2 catalog and asteroseismic relations, Zinn et al. (2018) found a systematic shift of $-0.053\pm 2$(stat)$\pm 1$(syst) mas. The derived shift is specific to the Kepler field but it would also be a measure of the global zero-point shift. Another approach was presented by Stassun & Torres (2018) who used the eclipsing binary method. They calculated observed reddening-free bolometric fluxes $F_{bol}$ by fitting the spectral energy distribution (SED) and compared them to the bolometric luminosities $L_{bol}$ to obtain distances. Using a subsample of 89 suitable eclipsing binaries from their catalog (Stassun & Torres, 2016) they derived a global shift of $-0.082\pm 0.033$ mas. The method used by Stassun & Torres (2018) is, from a theoretical point of view, the most robust one because it utilizes a very wide range of photometry from UV to midinfrared. However, the $L_{bol}$ are calculated from the effective temperature of each component and this radiative parameter is almost always the least constrained fundamental parameter of an eclipsing binary. There are two main reasons for this: 1) the wide range of methods used to determine effective temperatures that exhibit significantly different zero-points of the resulting temperature scales, and 2) the notorious difficulty in establishing an absolute temperature scale with a precision better than 1% (e.g. Casagrande et al., 2014). Thus from a practical point of view the method used by Stassun & Torres (2018) is not the optimal one because it has some systematics that are still difficult to properly evaluate. The need of an eclipsing binary approach which would minimize systematic effects, and would be based on direct and precise empirical relations, was the prime motivation for this work. As in our previous work (Graczyk et al., 2017, hereafter G17) we focus on an application of surface brightness - color relations.

2. The Sample

We extended the sample of 35 eclipsing binaries compiled by G17 by searching for detached systems in the literature suitable for a precise distance determination. In order to make the extension we used catalogs of eclipsing binary systems compiled by a number of authors: Bilir et al. (2008); Torres et al. (2010); Eker et al. (2014); Southworth (2015); Stassun & Torres (2016).

We used the same selection criteria as in G17, however we relaxed the condition on a volume-limited sample ($d<300$ pc) by accepting also more distant systems. Because the $\beta$ Aur system is too bright to have a Gaia parallax, and AI Phe is a confirmed triple system (Konacki priv. communication) we drop these systems from our sample. We included also a system from our unpublished work: AL Dor. Following G17 we retained systems with a fractional precision of recent Gaia DR2 parallaxes (Gaia Collaboration, 2016, 2018; Lindegren et al., 2018) better then 10%. A trigonometric parallax $\varpi$ with a fractional error $f=\sigma_{\varpi}/\varpi$ smaller than 0.1 is a good and weakly biased estimator of the true distance (e.g. Bailer-Jones, 2015; Bailer-Jones et al., 2018). In our sample only two early and distant systems have $f$ marginally larger than 0.1 i.e.: EM Car and DW Car. Most of the sample systems have fractional errors of their parallaxes smaller than 0.06 for which any bias can be completely neglected. Finally our sample contains 81 systems (51 on the northern hemisphere, 30 on the southern one). Their basic parameters are presented in Table 1 and the distribution of the parallax fractional errors is presented in Fig. 1.

Arenou et al. (2018) proposed three quality controls to estimate reliability of astrometric solutions of Gaia DR2; they are defined by their equations (1–3). Those quality controls help to filter out most of spurious astrometric solutions, but they filter out also some well-defined solutions. In our sample 9 stars do not fulfill the first criterium (possible double stars, suspected binary motion or calibration problems) and 7 other stars do not fulfill the third criterium (i.e. isibility_periods_used }$\; <8$). Howeer, the third criterium is useful mostly to very crowded fields in the Bulge and as no star in our sample is there, we kept these 7 systems during analysis. The 9 stars that do not fulfill the first criterium are denoted as squares in Fig. 1. Only 3 of them have the relative parallax uncertainty larger by a factor of two than expected. We investigated our sample further by looking for the proper motion anomaly (PMa; Kervella et al., 2019) which could be a sign of a photocenter movement or a third body in a system. The PMa results from a comparison of the proper motions vectors at the Hipparcos and Gaia epoch with the mean proper motion computed from a coordinate shift between Hipparcos and Gaia epochs. We found a significant (SNR=5) detection of the PMa for 8 stars, but interestingly none of them are common with the subsample of 9 suspected stars. The stars with the PMa are denoted as diamonds in Fig. 1. All of them have well defined 5-parametric astrometric solutions, and the PMa, if confirmed with later Gaia data releases, is likely due to long-period changes caused by an unrecognized triple companion.

2.1. Photometry

2.1.1 Optical

We used Tycho-2 $B_{T}$ and $V_{T}$ photometry (Høg et al., 2000) downloaded from Vizier (Ochsenbein et al., 2000)111http://vizier.u-strasbg.fr: I/259/tyc2. The Tycho photometry was subsequently transformed onto the Johnson system using the method outlined by Bessell (2000). Whenever possible we used Johnson $B,V$ photometry from the compilation of Mermilliod (1997) and also absolute optical photometry from literature sources.

2.1.2 Near infrared

We used NIR $JHK_{S}$ photometry of the Two Micron All Sky Survey (2MASS) (Skrutskie et al., 2006) from Vizier222http://vizier.u-strasbg.fr: II/281/2mass6x. For the purpose of using the SBC relations based on the ($V\!-\!K$) color expressed on the Johnson photometric system we transformed the 2MASS magnitudes using the equations given in Bessell & Brett (1988) and Carpenter (2001)333http://www.astro.caltech.edu/$\sim$jmc/2mass/v3/

transformations/. The transformation equations are as follows:

[TABLE]

3. Method

3.1. Collection of fundamental parameters

For each system we collected orbital and photometric parameters from the literature including the most recent publications. We were searching for basic parameters describing dynamical and geometrical parameters of each system: the radial velocity semi-amplitudes $K_{1,2}$, the orbital period $P$, the orbital inclination $i$, the eccentricity $e$, the longitude of periastron $\omega$, the photometric relative radii $r_{1,2}$. These parameters were supported by radiative parameters: the effective temperatures $T_{1,2}$ of both components. The radiative parameters are usually known with the least precision and accuracy. Whenever several papers independently presented the analysis of a given eclipsing binary we adopted the weighted mean for the parameters. However, if there was a significant improvement on the precision of parameters reported in one of papers, then we used only values from that paper. It turned out that we could not always directly trace all the above parameters from literature sources. In some cases when modeling of an eclipsing binary was done with numerical codes based on the Roche formalism (the Wilson-Devinney code, the ELC code, etc.) we had to calculate relative radii and radial velocity semiamplitudes from the absolute dimensions presented in the relevant papers. The collected parameters are summarized in Table 2.

3.2. The Wilson-Devinney model of the systems

For the purpose of obtaining homogenous parameters for the eclipsing binary sample we decided to create a model of each system following G17. The models were built using the Wilson-Devinney (WD) code version 2007 (Wilson & Devinney, 1971; Wilson, 1979, 1990; van Hamme & Wilson, 2007) while parameters of the models were based on solutions published in the literature (see Sec. 3.1). All models were checked for internal consistency of the parameters and it turned out that in many cases they had to be fine-tuned. In particular, the temperature ratio and the absolute temperature scale, being important for a precise prediction of infrared light ratios, were inspected carefully.

The procedure was as follows. Dynamical and geometrical parameters were transformed into the semi-major axis of the system $a$, the mass ratio $q$ and into dimensionless Roche potentials $\Omega_{1,2}$ using equations given in Torres et al. (2010) and Wilson (1979). Both $\Omega$ and $q$ are parameters directly fitted or used within the WD program. We usually fixed the temperature of the primary star $T_{1}$ and then, using published light ratios in different photometric bands, we adjusted the temperature of the companion $T_{2}$. In very few cases, however, we also rederived $T_{1}$ using two temperature - color calibrations (Flower, 1996; Worthey & Lee, 2011). However the temperature shifts are small and within errors given in literarure. None of the eclipsing binaries in our sample has infrared $J,H,K$ light curves suitable for deriving direct light ratios in those bands. Thus, in order to calculate intrinsic infrared colors of the components of each system we employed eclipsing binary models based on optical light curves and we extrapolated them into the infrared. The extrapolation was done using the atmosphere approximation within the WD code which uses precomputed intensities based on Kurucz’s ATLAS9 models (Kurucz, 1993).

The rotation parameter $F_{1,2}$ was kept to 1 (synchronous rotation), unless there was a direct spectroscopic determination of $F$ significantly different from unity. The albedo $A$ and the gravity brightening $g$ were set in a standard way for a convective atmosphere cooler then 7200 K and radiative ones for a hotter surface temperature. This was done only for the sake of consistency because the two parameters have negligible effect on the light ratios. The input and derived parameters used to create the appropriate WD models are listed in Table 2.

3.3. Correction of 2MASS magnitudes taken during eclipses

Nine systems from our sample have 2MASS observations taken during the eclipses. To account for the light lost during the minima we used our WD models to calculate the appropriate corrections which are given in Table 3.

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3.4. Reddening

Reddenings to each object were taken from the literature. We also derived independently values of $E(B\!-\!V)$ using the extinction maps of Schlegel et al. (1998) following the prescription given in Suchomska et al. (2015). Usually as a final value of the extinction we used an average, unless determinations were discrepant or we have at our disposal only one reddening estimate. The adopted reddenings are reported in Tab. 4. To check them we compared them with independent extinction estimates from 3D extinction maps from Stilism (Lallement et al., 2014; Capitanio et al., 2017) – see Fig. 2. On average Stilism extinctions are slightly smaller, but an overall agreement is good, and although the spread is quite large both sets of reddenings are consistent within errors.

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3.5. Intrinsic magnitudes

In Table 4 we summarize all parameters used to derive the intrinsic photometric indexes of the component stars. In order to calculate them the photometry was dereddened using the mean Galactic interstellar extinction curve from Fitzpatrick & Masana (2007) assuming $R_{V}=3.1$. Then the light ratios in the Johnson $BVJHK$ bands were derived using the WD models following G17, and employed to calculate the individual magnitudes and colors. Both the extinction correction and extrapolation of light ratios into the infrared add to uncertainty on derived intrinsic magnitudes. The extinction correction error is given in Table 4 but for the extrapolation uncertainty we did not add additional error. While for systems with similar temperature of components extrapolation leads to negligible additional error in some systems with the large temperature ratio of components this uncertainty may be significant. However when calculating a parallax (or a distance) to a particular eclipsing binary as an average from two components this uncertainty largely cancels out.

3.6. Photometric parallaxes

The eclipsing binary method gives the photometric distance as well as the photometric parallax to a particular target, because both quantities are inferred from observables. Because of this we decided to work in a regime of parallaxes instead of distances to avoid an additional bias, which may arise in the conversion of parallaxes into distances. However, we underline here that working with a regime of parallaxes does not solve the whole problem because, for fractional parallax uncertainties larger then 15%–20% the trigonometric parallax becomes a poor prior on the distance.

3.6.1 Parallaxes from the bolometric flux scaling

In order to directly compare our results with those of Stassun & Torres (2018) we employed the bolometric flux scaling method utilizing the $V$-band bolometric corrections $BC$ to derive photometric parallaxes. The photometric parallax $\varpi$ to the $i$-th component of the system was calculated using equation:

[TABLE]

where index $i=\{1,2\}$, $R$ is the radius of a component expressed in solar radii, $T$ is its effective temperature, $BC$ is a bolometric correction interpolated from the Flower (1996) tables for a given temperature and $V$ is the intrinsic magnitude of a component. The parallax to a particular system was calculated as the unweighted average parallax of the two components.

3.6.2 Parallaxes from the SBC relations

The eclipsing binary method of distance determination is very flexible and has many different approaches. We decided to use as our main approach the method based on the empirical surface brightness - color relations. In order to derive the photometric parallax to an eclipsing binary we need to calculate the individual angular diameters of the components. An angular diameter is calculated with the formula:

[TABLE]

where $S$ is the surface brightness in a given band and $m_{0}$ is the dereddened magnitude of a star in that band. In most cases $S$ is calibrated as a function of an intrinsic color of a star, usually in the form of a polynomial. The photometric parallax $\varpi_{Phot}$ follows from the equation:

[TABLE]

where $R$ is the radius of the star and the conversion factor is equal to ${\rm 1AU}/2\mathcal{R}_{\odot}$ assuming the solar nominal radius $\mathcal{R}_{\odot}=695700$ km (Habbereiter et al., 2008; Mamajek et al., 2015) and a length of the astronomical unit ${\rm 1AU}=149597871$ km (Pitjeva & Standich, 2009).

We determined parallaxes using several SBC relations from the literature calibrated on infrared colors (Kervella et al., 2004; di Benedetto, 2005; Boyajian et al., 2014; Challouf et al., 2014). The most useful calibrations of $S$ are based on $(B\!-\!K)$ and $(V\!-\!K)$ colors, because the reddening vector is almost parallel to the SBC relations for intermediate- and late-type stars. Moreover, these relations have very low intrinsic dispersion (e.g. Kervella et al., 2004; Graczyk et al., 2017). However, we also used calibrations based on other infrared colors to have a more detailed view of possible systematics. The parallax to a given eclipsing binary was calculated as the unweighted average of the individual parallaxes of its components.

3.7. Calculation of the shift with respect to Gaia parallaxes

We calculated individual parallax differences $d\varpi_{i}=(\varpi_{Gaia}-\varpi_{Phot})_{i}$ for every $i$-th eclipsing binary and for each SBC relation used. For a given SBC relation we then determined the zero-point shift between the photometric and Gaia parallaxes as unweighted and weighted means of the differences. The individual differences for all used SBC relations are presented in Fig. 3. The errors of the differences were calculated taking into account systematic uncertainties of the SBC relations: 2% for calibrations by Kervella et al. (2004) and di Benedetto (2005), 3% for calibrations by Boyajian et al. (2014) and 3.5% for calibration by Challouf et al. (2014) – their Eq. 13. Weighted means are dominated by more distant systems for which the error on $d\varpi$ is much smaller than for nearby eclipsing binaries. This is a bit counterintuitive and it needs an explanation. Error on $d\varpi$ comes from two uncertainties: that of the Gaia parallax and that of the eclipsing binary photometric parallax. While the former for stars with $G<14$ mag (all our sample) is almost independent of the distance and equal to 0.04 mas on average (Lindegren et al., 2018), the latter decreases with the distance because it is, in the majority of cases, dominated by a systematic uncertainty of the SBCR. Thus for a systematic error of 2% on the photometric parallax the uncertainty is about 0.2 mas for a system lying 100 pc from the Sun and about 0.02 mas for a system lying 1 kpc away. In Fig. 4 we present histograms of the differences for all the SBCR used, and also for parallaxes derived with the bolometric flux scaling method. The Gaussians correspond to a standard uncertainty distribution on weighted (continuous line) and unweighted means (dashed line).

4. Results

Tab. 5 presents a summary of the zero-point $\varpi_{Gaia}-\varpi_{Phot}$ shifts for the different SBC relations, and in Fig. 5 we show exemplary relations between trigonometric Gaia DR2 and photometric, eclipsing binary parallaxes. The photometric parallaxes derived from bolometric fluxes correspond well with Gaia DR2 parallaxes, although the offset between both is clear. The amount of the offset of $-0.067\pm 0.012$ mas is marginally consistent with the zero-point shift reported by Stassun & Torres (2018). Our sample agrees in about 80% with a sample used by Stassun & Torres (2018), we also adopted the same solar constants; however, we used a different method to derive $L_{bol}$ which may account for the difference with their result. We point out that the accuracy of the derived offset is limited by the accuracy of the zero-points of the different methods used to determine the effective temperatures and the choice of solar constants. However, the problem of homogenization of the temperature determination for all the sample and the proper calibration of the temperature zero-point scale is beyond the scope of this paper. We therefore prefer to establish the Gaia zero-point shift solely based on the empirical SBC relations, which allows for a homogenous treatment of the sample (and its subsamples).

As pointed out already by Stassun & Torres (2018) there is almost perfect agreement between the photometric and Gaia DR2 parallaxes. However, some of the SBC relations result in a larger offset and larger differences (e.g. Challouf et al., 2014) – see Fig. 3 and Fig. 4. The possible reasons for this are discussed in the next section.

The SBC relations, which result in the smallest differences and best agreement with Gaia DR2 parallaxes are those based on $(B\!-\!K)$ and $(V\!-\!K)$ colors and calibrated by Kervella et al. (2004). They are also the relations that result in the smallest internal dispersions of the differences. Calibrations based on these two colors are the least reddening dependent because the reddening line on the surface brightness $S$ - color diagram is almost parallel to the relations themselves. Moreover, the calibrations by Kervella et al. (2004) were done for main-sequence dwarfs and subgiants, which constitute the overwhelming majority of the eclipsing binary component stars in our sample. The resulting zero-point shift based on the $(B-K)$ and $(V-K)$ relations is $-0.025\pm 0.011$ mas. Relations by Kervella et al. (2004) calibrated on the other optical-infrared colors are more reddening dependent and also show significantly larger scatter of the differences.

Another SBC relation that we used extensively for the determination of the distances to the Magellanic Clouds (Pietrzyński et al., 2013; Graczyk et al., 2014, e.g.) was calibrated by di Benedetto (2005). This relation is calibrated on a mixture of giant and dwarf stars. The resulting zero-point shift is $d\varpi=-0.052\pm 0.020$ mas and is within $1.3\sigma$ consistent with the $d\varpi$ derived from the Kervella et al. SBC relations. Combining these two determinations we obtain a value of $d\varpi=-0.031\pm 0.011$.

5. Discussion

Using the most reliable SBC relations available in the literature, we derived a zero-point shift for Gaia DR2 parallaxes of $d\varpi=-0.031\pm 0.011$ mas. However, there are significant differences of the zero-point shifts resulting from applying 1) different SBC calibrations, and 2) different colors.

The reasons for the existence of the SBC calibration dependent zero-point shifts are somewhat unclear. For sure a choice of different interferometric stellar angular diameter sets will result in different calibrations. Most notably a calibration derived from a mixture of all kinds of stars (e.g. Challouf et al., 2014) may lead to a systematic difference with a calibration based on a mixture of dwarf and subgiant stars only (e.g. Kervella et al., 2004). However, more worrisome is a systematic difference of the zero-point shifts derived from the same colors but from two different calibrations (Kervella et al., 2004; Boyajian et al., 2014) that are based on angular diameters of dwarf and subgiants stars and calibrated on the Johnson system. We searched relevant papers to find the differences. Kervella et al. (2004) use a small but well characterized sample of stars, while Boyajian et al. (2014) used a much larger sample but containing multiple and variable stars, and also utilizing saturated 2MASS photometry for some bright stars. Because of this we tend to prefer the former calibrations over the latter. The systematic differences between the calibrations (see Tab. 5) vary from $-0.011$ mas to $-0.044$ mas with a weighted average of $-0.031$ mas.

Regarding 2) there is a clear trend that relations calibrated on shorter-wavelength colors (and therefore being more reddening dependent) give systematically larger zero-point shifts and larger scatter of the differences, with the largest shifts coming from SBC relation calibrations based on the $(V\!-\!J)$ color. We tried to remove the shifts by a rescaling of all individual reddening estimates according to $E(B\!-\!V)_{\rm new}=C\cdot E(B\!-\!V)$, where $C>1$ is a scaling factor. Unless $C$ is larger then about 4 there is no possibility to obtain an agreement between zero-point shifts derived from different colors. But such a large value of the scaling factor would lead to an unrealistically large extinction for most of the targets in our sample, and would be strongly at odds with independent reddening determinations by Stassun & Torres (2016). Another, but less likely, possibility is that the transformation equations from Sec. 2.1.2 contain some systematic error leading to color dependent zero-point shifts. And at last the SBC relations calibrated on colors containing $J$ and $H$ may be problematic by themselves because of a proper calibration of the ground-based photometry in these filters is difficult (strong and variable atmospheric extinction) and because of stronger sensibility to the interstellar reddening.

We investigated how the global shift depends on a distance. We binned up individual $d\varpi_{i}$ in several bins – see Fig. 3; however, no clear systematic trend can be noticed. We also investigated a possible Gaia color $G_{BP}-G_{RP}$ and $G$ magnitude dependency. We plotted $d\varpi$ differences for a one specific SBC relation – Fig. 6. As previously we do not detect any systematic trend. We also investigated the correlation of $d\varpi$ with sky position. We calculated separately shifts for equatorial and ecliptic hemispheres. The differences between hemispheres are in both cases within 1$\sigma$ uncertainty and thus not statistically significant.

The comparison with previous determinations of $d\varpi$ shows excellent agreement with the Gaia Team results (Lindegren et al., 2018; Arenou et al., 2018). In fact we confirm here the zero-point shift of $d\varpi=-0.029$ mas reported by the Gaia Team. However, comparisons with other independent determinations show some discrepancies. Although the $d\varpi$ value reported by Riess et al. (2018) agrees with our value at the $1\sigma$ level, the zero-point shifts reported by Stassun & Torres (2018) and Zinn et al. (2018) are somewhat discrepant. The problem with the shift reported by Zinn et al. (2018) is its unrealistically small systematic uncertainty, which ignores systematics of asteroseismic relations themselves and the zero-point uncertainty of the temperature scale used by the authors. Because of this a comparison of the shift by Zinn et al. (2018) with other reported shifts is vague.

Of special interest is the knowledge of how binarity will impact Gaia parallaxes, and the determination of the global zero-point shift. In order to quantify this, we calculated the photocenter movement for our $\sim$80 eclipsing binaries and compared them with the uncertainties on Gaia DR2 parallaxes – see Fig. 7. For the overwhelming majority of the systems the photocenter movement is only a very small fraction of the parallax itself. Only in two systems could Gaia DR2 probably recognize the photocenter movement (EPIC 211409263 and V380 Cyg). Thus we can consider the Gaia DR2 parallaxes of the eclipsing binaries in our sample as practically unaffected by binarity.

We dedicate this paper to our colleague Zbigniew (”Zibi”) Kołaczkowski who passed away much too early. We thank the anonymous referee for valuable comments and remarks on the manuscript. The research leading to these results has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 695099). We are grateful for financial support from Polish National Science Center grant MAESTRO UMO-2017/26/A/ST9/00446. Support from the BASAL Centro de Astrofísica y Tecnologías Afines (CATA) grant AFB-170002, the Millenium Institute of Astrophysics (MAS) of the Iniciativa Cientifica Milenio del Ministerio de Economia, Fomento y Turismo de Chile, project IC120009 and the IdP II 2015 0002 64 grant of the Polish Ministry of Science and Higher Education is also acknowledged. We are also thankful to the staff in La Silla Observatory (ESO) and Las Campanas Observatory (Carnegie) for their excellent support. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular, the institutions participating in the Gaia Multilateral Agreement. This research has made extensive use of the excellent astronomical SIMBAD database and of the VizieR catalog access tool, operated at CDS, Strasbourg, France, and made also use of NASA’s Astrophysics Data System Bibliographic Services (ADS). This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.