Gaia EDR3 Parallax Zero-point Offset Based on W Ursae Majoris-type Eclipsing Binaries

Chen et al. (2018a) determined optical-to-mid-IR PLRs based on 183 EWs with Tycho-Gaia parallaxes. We rederived the W1 PLR using Gaia EDR3 parallaxes to improve the PLR zero-point. Although only the maximum EW magnitudes, i.e., those outside eclipses, are directly related to the periods, a tight relationship also exists between mean magnitudes and periods; the dispersion between maximum and mean magnitudes is just σ = 0.05 mag (Chen et al. 2018a). Our most important reason for deriving the mean-magnitude PLR is that it is more appropriate and convenient for large samples. Maximum magnitudes cannot be determined easily, especially not for EWs collected from different catalogs.

We selected nearby (<500 pc), bright EWs with accurate parallaxes (σ_π /π < 0.01, where π and σ_π are the parallax and its uncertainty, respectively) and W1 magnitudes (σ_W1 < 0.05 mag). We only consider nearby stars to reduce the systematic error, since the bias (zero-point offset versus parallax) is smaller for nearby stars. The systematic error associated with the zero-point offset in our PLR fits is proportionally reduced when applied to additional EWs (see Section 4.1 for further details). Extinction values were estimated using the three-dimensional (3D) dust reddening map of Green et al. (2019) and A_W1/A_V = 0.039 (Wang & Chen 2019). We determined absolute magnitudes via ${M}_{W1}={m}_{W1}\,-5\mathrm{log}(1000/\pi )+5\,-{A}_{W1}$, where the unit of π is mas and m_W1 is the mean magnitude. The W1 PLR was determined from a linear fit to the 1138 objects contained within the 3σ envelope (see Figure 1, top), ${M}_{W1}\,=(-6.27\pm 0.15)\mathrm{log}P\ {\rm{(days)}}-0.24\,\pm 0.07,\sigma =0.16$ mag. The green and blue lines in Figure 1 indicate the linear fit and the 1σ range, respectively.

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The average extinction for our 1138 EWs is A_W1 = 0.013 mag (A_V = 0.322 mag). This is reliable and appropriate for a sample at an average distance of 368 pc. Considering a 10% uncertainty in the extinction, reflecting uncertainties due to our choice of extinction law, the prevailing systematic bias is around 0.0013 mag.

The G_BP and G_RP PLRs were determined similarly (see Figure 1, middle and bottom): ${M}_{{BP}}=(-10.71\pm 0.40)\mathrm{log}P\,-0.38\pm 0.19,\sigma =0.43$ mag and ${M}_{{RP}}=(-8.98\,\pm 0.29)\mathrm{log}P-0.44\pm 0.13,\sigma =0.31$ mag. They were used for extinction estimates in the part of the southern sky not covered by the 3D extinction map (see Section 3.2).

The W1 PLR thus determined can be used to estimate the absolute magnitudes of all EWs. If the extinction is known, we can obtain an object's distance. This distance is affected by the PLR zero-point rather than the Gaia parallax zero-point and can therefore be used to determine the systematic offset in the Gaia parallaxes. For sources covered by the 3D extinction map, we iteratively obtained the best-fitting extinction, using A_W1(μ₀) = m_W1 − M_W1 − μ₀. For sources not covered by the 3D extinction map, we estimated the extinction using the G_BP and G_RP PLRs, i.e., ${A}_{W1}=\tfrac{{A}_{W1}}{{A}_{V}}{A}_{V}=\tfrac{{A}_{W1}}{{A}_{V}}\tfrac{({m}_{\mathrm{BP}}-{M}_{\mathrm{BP}})-({m}_{\mathrm{RP}}-{M}_{\mathrm{RP}})}{\tfrac{{A}_{\mathrm{BP}}}{{A}_{V}}-\tfrac{{A}_{\mathrm{RP}}}{{A}_{V}}}$. Here, μ₀ is the distance modulus, A_W1(μ₀) the extinction at μ₀, m_λ , and M_λ are the apparent and absolute magnitudes in the corresponding band, λ. We adopted the Wang & Chen (2019) extinction law.

Figure 2 shows a comparison of the parallaxes derived from the PLR with those from Gaia EDR3. Here, π_EW, π_EDR3, and ${\pi }_{\mathrm{EDR}3}^{\mathrm{corr}}$ represent parallaxes obtained from PLR distances, Gaia EDR3, and Gaia EDR3 after zero-point correction based on Lindegren et al. (2020b), respectively. Δπ_EDR3 = π_EDR3 − π_EW and ${\rm{\Delta }}{\pi }_{\mathrm{EDR}3}^{\mathrm{corr}}={\pi }_{\mathrm{EDR}3}^{\mathrm{corr}}-{\pi }_{\mathrm{EW}}$ represent parallax differences.

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The parallax differences trace a roughly symmetric, normal distribution with a negative shift, where the Gaia parallaxes are systematically smaller. The offsets are Δπ_EDR3 = −28.6 ± 0.6 μas and −25.4 ± 4.0 μas, respectively, for the five- and six-parameter solutions (Figure 2, blue histograms), where the errors are the standard deviations. These offsets are slightly larger than that derived from the quasar sample (−17 μas; Lindegren et al. 2020a). The ${\rm{\Delta }}{\pi }_{\mathrm{EDR}3}^{\mathrm{corr}}$ distributions are shown as red histograms in Figure 2; the mean values are 4.2 ± 0.5 μas and 4.6 ± 3.7 μas for the five- and six-parameter solutions, respectively. This suggests that the parallax zero-point correction provided by the Gaia team adopting the quasar reference frame significantly reduces the bias in the Gaia EDR3 parallaxes, but it may slightly overcorrect the bias for Galactic objects.

Equipped with over 100,000 EWs, we can now assess which parameters contribute to the systematic offset. Figure 3 shows the binned parallax difference distributions, Δπ_EDR3 (blue dots) and ${\rm{\Delta }}{\pi }_{\mathrm{EDR}3}^{\mathrm{corr}}$ (red dots), as a function of G magnitude, effective wavenumber, ν_eff, ecliptic latitude, $\sin \beta$, and Galactic latitude, $\sin b$, for the five-parameter solutions. Following Lindegren et al. (2020a), we use ν_eff as a proxy for the color information. It results from processing of the BP and RP spectra and can be converted directly to G_BP − G_RP. The parallax offsets estimated from distant quasars (Lindegren et al. 2020a) are shown as gray points for comparison. Representing one of the largest external comparison catalogs (e.g., Fabricius et al. 2020, their Table 1), the number of EWs is only smaller than the quasar sample. Meanwhile, EWs have good coverage in the Galactic plane, spanning a wide range of colors and magnitudes. The detailed dependence of the parallax differences derived from EWs is fully complementary to that derived from quasar analysis; it is more suitable for Galactic stars.

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In Figure 3(a), Δπ_EDR3 exhibits an increasing, roughly linear trend as a function of magnitude, similar to that shown by the quasars at fainter magnitudes. ${\rm{\Delta }}{\pi }_{\mathrm{EDR}3}^{\mathrm{corr}}\simeq 0$ is stable for 14.5 < G < 17 mag. Since 55% of our EWs are found in this range, we conclude that the offset analysis based on EWs and quasars is consistent here. In Figure 3(b), the Δπ_EDR3 trends based on EWs and quasars are consistent for ν_eff < 1.5 μm⁻¹. For effective wavenumbers of 1.5–1.64 μm⁻¹, Δπ_EDR3 based on EWs is systematically lower by 10–20 μas. After correction, ${\rm{\Delta }}{\pi }_{\mathrm{EDR}3}^{\mathrm{corr}}\simeq 0$, but the pattern persists. This effective wavenumber range corresponds to the range of F- and G-type main-sequence stars. The different offsets may be associated with different types of stars and this requires additional data to verify.

We also checked for trends as a function of spatial position. Any trend in ${\rm{\Delta }}{\pi }_{\mathrm{EDR}3}^{\mathrm{corr}}$ with ecliptic latitude is weak (Figure 3(c)). However, a clear trend is seen as a function of Galactic latitude (Figure 3(d)). Both Δπ_EDR3 and ${\rm{\Delta }}{\pi }_{\mathrm{EDR}3}^{\mathrm{corr}}$ exhibit sharp drops of 10 μas in the Galactic plane (∣b∣ ≲ 10°). Elsewhere, Δπ_EDR3 and ${\rm{\Delta }}{\pi }_{\mathrm{EDR}3}^{\mathrm{corr}}$ exhibit stable distributions around the mean. This trend is not a result of either extinction or metallicity variations. It is also present for low-extinction sources. Based on 320 EWs with Large-Area Multi-Object Spectroscopic Telescope (LAMOST) metallicity measurements, we obtain ΔM_W1 = M_W1,obs −M_W1,PLR = 0.222[Fe/H] + 0.014 mag. The slope agrees with the near-IR metallicity effect found by Chen et al. (2016). If metallicity effects are taken into account, the trend becomes steeper rather than flatter. Since only quasars with ∣b∣ > 20° are used to model the correction for the five-parameter solutions, the correction is reliable for disk sources if the model used for the disk sources is similar to that of the sources at ∣b∣ > 20°. However, Lindegren et al. (2020a; their Figure 13) showed that the rise and fall of the parallaxes becomes obvious when they approach the Galactic plane. The likely reason for the trend is, instead, that the correction based on the five-parameter solutions is insufficient for some regions in the Galactic disk.

To better investigate the distribution of the parallax differences as a function of spatial position, maps of Δπ_EDR3 and ${\rm{\Delta }}{\pi }_{\mathrm{EDR}3}^{\mathrm{corr}}$ for the five-parameter solutions, in both ecliptic and Galactic coordinates, are shown in Figure 4. The parallax correction varies more significantly with Galactic than ecliptic latitude. The maps are more intuitive to evaluate the corrected Gaia EDR3 parallaxes. After correction, the parallax offset is less than 10 μas across 40% of the sky, and only 15% of the sky has a parallax offset greater than 30 μas. This shows that the correction for Gaia EDR3 parallaxes is effective in reducing deviations in the Gaia parallaxes.

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Here, we present an estimate of the systematic errors in our PLR-based EW distances. This is important for assessment as to how accurate our derived parallax offset is. The systematic uncertainties include four components: (i) the PLR zero-point offset, (ii) the internal PLR spread, (iii) the unresolved third components, and (iv) the errors in our extinction estimates.

The W1 PLR in Section 2 was obtained based on the parallaxes (calibrated using the five-parameter solution) of 1138 objects located within 500 pc of the Sun. This sample has an average parallax of 3.06 mas. Taking into account the systematic uncertainty of 4.2 ± 0.5 μas (Figure 2), the systematic error propagating to the PLR contributes 0.15%. The systematic error associated with the internal PLR spread is $0.16/\sqrt{1138}=0.0047$ mag, where the 1σ dispersion of the W1 PLR is 0.16 mag. Based on a study of 75 nearby EWs (D'Angelo et al. 2006), the presence of unresolved third components would affect the parallaxes of our sample objects by 0.3%.

For the adopted extinction, the systematic error is contributed by the extinction difference resulting from application of different methods of extinction determination and the choice of extinction law. Based on 98,466 EWs, the average extinction difference between the 3D extinction map used and the extinction calculated from the G_BP and G_RP PLRs is ΔA_W1 = 0.0377–0.0394 = −0.0017 mag. The average extinction for all EWs is A_W1 = 0.035 mag (A_V = 0.895 mag), which is reliable for an average distance of 2.41 kpc. Considering a 10% uncertainty in the extinction law, the systematic bias caused by extinction differences is about $\sqrt{{0.0035}^{2}+{0.0017}^{2}}=0.0039$ mag.

We do not consider systematic errors in the PLR caused by metallicity effects, for two reasons. First, our EWs are distributed uniformly around the Sun, at an average distance of 2.4 kpc. EW ages are between 1 and 10 Gyr, and they do not tend to be distributed preferentially in either the metal-poor halo or the metal-rich Galactic disk. Therefore, the assumption of an average solar abundance for the EW PLR as a whole is appropriate. Second, if metallicity effects are significant, Figure 3(d) shows that the parallax offset decreases for decreasing ∣b∣ (we only consider ∣b∣ > 10°).

Combining the individual error estimates, the systematic uncertainty affecting our results is σ = 415 μas × ${\left[{(0.0015)}^{2}+{(0.0047\times \mathrm{ln}{(10)/5)}^{2}+(0.003)}^{2}+{\left(0.0039\times \mathrm{ln}(10)/5\right)}^{2}\right]}^{1/2}=1.8$ μas.

Recently, much work has been done on Gaia EDR3 parallaxes based on other tracers. Stassun & Torres (2021) obtained offsets of −37 ± 20 μas and −15 ± 18 μas, respectively, before and after correction, based on 76 EBS. Huang et al. (2021) found a mean parallax offset of −26 μas based ∼70,000 red clump stars observed with LAMOST, which was reduced to around 4 μas after correction. Zinn (2021) and Riess et al. (2021) also found an overestimated zero-point correction of 15 ± 3 μas and 14 ± 6 μas based on, respectively, 2000 first-ascent red-giant-branch stars with asteroseismic parallaxes in the Kepler field and 75 classical Cepheids. Ren et al. (2021) found offsets of −42.1 ± 1.9(stat.) ± 12.9 (syst.) μas and −10.9 ± 2.9(stat.) ± 12.9 (syst.) μas, respectively, before and after correction, based on 2334 EWs in the northern Galactic plane.

Overall, for Galactic stars the Gaia EDR3 parallax offsets are [−20, −30] μas and 4–10 μas before and after correction, respectively. For specific regions—the Galactic disk, the bulge, and high-latitude regions—there is an additional deviation of about 10 μas. Compared with previous results, our new results have smaller errors and higher completeness because of the much larger sample size afforded by our EW sample and their more complete coverage in magnitude, color, and spatial distribution.