The Hipparcos–Gaia Catalog of Accelerations: Gaia EDR3 Edition

The DR2 version of the HGCA (B18) represents a cross-calibration of three proper motions: the Hipparcos proper motions, the Gaia DR2 proper motions, and the position difference between Hipparcos and Gaia divided by the difference in epoch between the two surveys. This long-term proper motion is the most precise measurement for nearly 98% of stars after calibrating the Gaia DR2 uncertainties. This permits a clean separation of Hipparcos and Gaia proper motions: both can separately be calibrated to the more precise long-term proper motions. The uncertainties and systematics in Hipparcos and Gaia DR2 are, as a result, relatively straightforward to disentangle.

B18 find, with ∼150σ significance, that a 60/40 weighted average of the ESA (1997) and van Leeuwen (2007) Hipparcos reductions outperforms either on its own. They also find that the Hipparcos proper motions display small-scale structure that can be removed. Doing so improves the agreement of the Hipparcos proper motions with the long-term proper motions in a cross-validation set at high significance. Finally, B18 find that an additional 0.2 mas yr⁻¹ has to be added in quadrature to the proper motion uncertainties to achieve a statistical agreement with the long-term proper motions. The calibrated Hipparcos proper motion uncertainties are comparable to the reported uncertainties in the 1997 catalog (ESA 1997) and larger than the uncertainties in van Leeuwen (2007), especially for bright stars.

For Gaia DR2, B18 recover the frame rotation seen by Lindegren et al. (2018) with modest additional small-scale variations. They determine a spatially variable error inflation (ratio of true to formal uncertainties) averaging 1.7. This error inflation varies spatially, primarily with ecliptic latitude and hence with the average number of observations according to Gaia's scanning pattern. Finally, B18 find that the stars in Gaia with the largest uncertainties (predominantly the very bright stars) have underestimated uncertainties even after error inflation.

Gaia EDR3 has implemented a number of improvements over DR2. The most important for the HGCA is a factor of nearly 4 improvement in proper motion uncertainties for bright stars (Fabricius et al. 2021; Lindegren et al. 2021b). A factor of (34 months/22 months)^3/2 ≈ 2 is attributable to the longer observational baseline; the rest is due to improved data processing and control of systematics (Fabricius et al. 2021). Gaia EDR3 has also removed most of the frame rotation that affected the bright star reference frame of DR2, and it has corrected the DOF bug (Lindegren et al. 2021b).

Figure 1 shows the improvements from DR2 (on the left) to EDR3 (on the right). In this and subsequent figures, we show proper motions in R.A. and decl. together. All of our quantitative analyses include the separate uncertainties and covariance of the two components of proper motion. From DR2 to EDR3, the formal proper motion uncertainties have fallen by a magnitude-dependent factor of 3 or 4, while the median residuals between the long-term proper motion and the Gaia proper motion (corrected for frame rotation in DR2's case) have fallen by a factor of 2 to 3. The long-term proper motions adopt the 60/40 mix of the two Hipparcos reductions found by B18. In DR2, the long-term proper motion uncertainties were dominated by Gaia errors for very bright magnitudes (G ≲ 5). They are now dominated by Hipparcos uncertainties at all magnitudes.

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In the DR2 edition of the HGCA, the median precision on the long-term proper motion was a factor of ∼4 better than the Gaia proper motions. Structure and systematics in the proper motion residuals were overwhelmingly due to the rotation of Gaia's bright star reference frame. The Gaia EDR3 and long-term proper motions are comparably precise. Their difference is now equally sensitive to systematic rotations in the Gaia reference frame and positional systematics in the Hipparcos reference frame. Despite the difficulty in interpreting structures and systematics in the proper motion residuals, we follow the same approach to cross-calibrating the catalogs.

Figure 2 shows the distribution of z-scores: the residuals between the Gaia EDR3 and the long-term proper motions, divided by their standard errors. In this and subsequent figures, we compute z-scores in R.A. and decl. separately and then plot them together. The uncertainties on both are Gaussian even when the two components of proper motion are covariant. The standard errors are the quadrature sums of the formal uncertainties in the EDR3 proper motions and the positional uncertainties in EDR3 and the merged Hipparcos catalog, scaled by the 24.75 yr baseline between the catalogs. The distributions deviate from a unit Gaussian. The width of the distribution increases at bright magnitudes where Gaia EDR3 proper motions dominate the error budget. This suggests that the EDR3 formal uncertainties underestimate the true uncertainties. This underestimate is expected (Lindegren et al. 2012) and also seen by the Gaia team (Fabricius et al. 2021; Lindegren et al. 2021b). The distributions of Figure 2 also have tails well in excess of those of a Gaussian; these are likely from stars that are astrometrically accelerating.

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Figure 2 compares favorably to Figure 1 of B18, the equivalent plot for Gaia DR2. The systematics from frame rotation are mostly gone and the formal uncertainties appear to be closer to the true uncertainties. A cross-calibration of Hipparcos and Gaia remains necessary, but the catalogs as published are in much closer agreement than they were for Gaia DR2.

We use the Gaia EDR3 parallax measurements and uncertainties to refine the other Hipparcos astrometric parameters in the same way as B18. We refer to Section 4 of that paper for details. For Gaia DR2, B18 found an improvement of ∼1% in the agreement between Hipparcos and Gaia proper motions after incorporating Gaia parallax measurements to refine the other Hipparcos astrometric parameters. We do not apply error inflation to the Gaia EDR3 parallaxes for this purpose. El-Badry et al. (2021) found that the EDR3 parallax uncertainties were underestimated by ≲30% for stars with well-behaved fits.

We find that the EDR3 parallaxes offer the same ∼1% improvement to the Hipparcos proper motions as the DR2 parallaxes did. The DR2 parallaxes were precise enough relative to the Hipparcos measurements that EDR3 provides a negligible gain. The parallaxes also improve the Hipparcos positions: the mean absolute deviation between the long-term proper motions (whose uncertainties are dominated by Hipparcos ) and the Gaia EDR3 proper motions fall by about 0.4% in R.A. and 0.2% in decl. when incorporating Gaia parallaxes into the Hipparcos solution.

Finally, we assess whether correcting for the parallax bias according to the prescriptions derived by Lindegren et al. (2021a) can further improve Hipparcos astrometry. These bias corrections are functions of source magnitude, color, and ecliptic latitude. We find that applying these corrections to EDR3 parallaxes offers a tiny improvement in the agreement of Hipparcos proper motions with long-term proper motions. However, it slightly degrades the agreement between long-term proper motions and Gaia EDR3 proper motions. As a result, we do not apply any bias corrections to the EDR3 parallaxes. The effects of applying a bias correction to the EDR3 parallaxes are at least two orders of magnitude smaller than the improvement from using the EDR3 parallaxes directly.

Our analysis says little about the accuracy of the Lindegren et al. (2021a) bias corrections at bright magnitudes. The biases are much smaller than the Hipparcos parallax uncertainties, especially around G ∼ 12 where the corrections change abruptly with magnitude (Lindegren et al. 2021a). Further, we can only measure the agreement between Gaia parallaxes and true parallaxes in the Hipparcos frame, which could be subject to their own biases and systematics.

Our next step is to propagate positions to the central epochs of the Hipparcos observations (the epochs that minimize the positional uncertainties), exactly as was done by B18. The central epochs generally differ in R.A. and decl., and they also differ from the catalog epoch of 1991.25. This step results in a smaller uncertainty on position and removes covariance between position and proper motion. It also provides a more accurate approximation to the true epoch at which the proper motion is effectively measured. We apply this exact same approach to Gaia EDR3 using its covariance matrix.

The perturbation to the reference epoch in R.A. is given by

$$\begin{eqnarray}&&\delta {t}_{\alpha * }=-\displaystyle \frac{\mathrm{Cov}(\alpha * ,{\mu }_{\alpha * })}{{\sigma }^{2}[{\mu }_{\alpha * }]},\end{eqnarray} \tag{ 1 }$$

where Cov(α*, μ_α*) is the covariance between position and proper motion in R.A. ($\alpha * \,\equiv \,\alpha \cos \delta$) and σ²[μ_α*] is the variance in proper motion. We compute Equation (1) and its counterpart for decl. separately and propagate the positions to ${t}_{\mathrm{catalog}}+\delta {t}_{\alpha * }$ and ${t}_{\mathrm{catalog}}+\delta {t}_{\delta }$. We then update the covariance matrices for the astrometric parameters as described in B18.

With the astrometric reference epoch defined as above, we compute the long-term proper motions as, e.g.,

$$\begin{eqnarray}&&{\mu }_{{HG},\alpha * }=\left(\displaystyle \frac{{\alpha }_{{\rm{Gaia}}}-{\alpha }_{{\rm{Hip}}}}{{t}_{{\rm{Gaia}},\alpha }-{t}_{{\rm{Hip}},\alpha }}\right)\cos \left[\displaystyle \frac{{\delta }_{{\rm{Gaia}}}+{\delta }_{{\rm{Hip}}}}{2}\right].\end{eqnarray} \tag{ 2 }$$

Once calibrated relative to the Hipparcos and Gaia EDR3 reference frames, this serves as one of the three proper motions supplied by the catalog.

B18 adopted a 60/40 mixture of the two Hipparcos reductions for both position and proper motion, with an additional uncertainty of 0.2 mas yr⁻¹ added in quadrature with the Hipparcos proper motions. The relative weights of the ESA (1997) and van Leeuwen (2007) reductions and the additional proper motion uncertainty were driven almost entirely by the Gaia DR2 position measurements and the Hipparcos proper motions. While the Gaia positions have improved with EDR3, they were already so precise with DR2 that the additional precision has no effect on the main results of B18.

We perform one more check on the Hipparcos catalog mixture. The Gaia EDR3 proper motions are now comparable in precision to the long-term proper motions. We therefore compute the Hipparcos weights that give the best agreement between the Gaia EDR3 proper motions and the long-term proper motions. This is sensitive to the Hipparcos positions but independent of the Hipparcos proper motions. We again find that a 60/40 mix is optimal, in agreement with the optimal mix found by B18 for the Hipparcos proper motions. We adopt this 60/40 linear combination of the two Hipparcos catalogs, and an error inflation of 0.2 mas yr⁻¹ added in quadrature, from B18.

A star moving uniformly through space will apparently accelerate when its motion is projected onto spherical coordinates. We take each star's instantaneous position, proper motion, parallax, and radial velocity in Gaia EDR3 and convert it into a three-dimensional position and space velocity. For stars without radial velocities in EDR3, we use radial velocities taken from the Extended Hipparcos compilation (XHIP; Anderson & Francis 2012). About 80% of Hipparcos stars have radial velocities from one source or the other. We assume zero radial velocity when computing three-dimensional velocities for the remaining 20% of stars.

We propagate three-dimensional positions backwards using each star's three-dimensional velocity. We propagate by the time difference between Hipparcos and Gaia EDR3; this time varies from star to star and between R.A. and decl. (Section 4.2). We then measure the difference between the correctly propagated position δ x and the linearly propagated position δ x_lin = μ δ t divided by this time. We apply this nonlinearity correction to the difference between the long-term proper motion and the Gaia proper motion. We apply the same correction between the Hipparcos proper motion and the long-term proper motion; this is equivalent to applying twice the correction to the difference between the Hipparcos and the Gaia proper motions.

We apply our computed nonlinearity corrections before calibrating the reference frames between Hipparcos, Gaia EDR3, and the long-term proper motions. The corrections are small for most stars but reach 16 mas yr⁻¹ (in decl.) for Barnard's Star. For all but 15 stars, the nonlinearity corrections in both R.A. and decl. are less than 1 mas yr⁻¹.

The Hipparcos data themselves have not changed since Gaia DR2 (apart from tiny changes in the Gaia parallax used to refine Hipparcos astrometry). As a result, our cross-calibration between the Hipparcos proper motions and the long-term proper motions is nearly identical to the results of B18. It does differ very slightly because we use a different set of stars to fit the reference frame and for cross-validation: B18 used stars with Hipparcos IDs ending in zero, while we use stars with a Gaia EDR3 random index ending in zero.

Figure 3 shows the local corrections to the Hipparcos proper motions to bring them into agreement with the long-term proper motions. These plots are very similar to Figure 6 of B18. Similarly to the DR2 edition of the HGCA, these corrections to the Hipparcos proper motion frame increase the likelihood of the 8828 stars used for cross-validation by about e³⁴⁰ over a constant frame rotation and by about e⁵⁵⁰ over no frame rotation at all. For the 87,386 stars consistent at 10σ with constant proper motion, the correction of Figure 3 increases the likelihood by a factor of about e⁵⁰⁰⁰ over a constant frame rotation. This is somewhat more than the tenth power of e³⁴⁰, suggesting a modest degree of overfitting just as for the DR2 edition of the HGCA.

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We follow the same approach for Gaia. We use the same Gaussian mixture model as for DR2, but decrease the width of the broader, outlier Gaussian from 1 mas yr⁻¹ to 0.5 mas yr⁻¹ given Gaia's much improved precision. B18 used a coarser tiling of the sky than for Hipparcos proper motions, with more stars in each tile, to fit for the local frame rotation. Here, we fit for frame rotations between Gaia EDR3 and the long-term proper motions in 2578 tiles, with an average of 45 stars per tile, compared to 4586 tiles (with 25 stars per tile) for Hipparcos. Accounting for accelerating and cross-validation stars, we fit about 30 and 17 stars per tile for Gaia EDR3 and Hipparcos, respectively. The likelihood improvement for the cross-validation set of EDR3 proper motions is about e²⁹⁰ over a constant frame rotation and e³⁵⁰ over no frame rotation; the corresponding improvements for the entire sample are about e⁴⁹⁰⁰ and e⁵⁴⁰⁰. This again indicates only a modest degree of overfitting. Increasing the number of tiles (decreasing the number of stars per tile) offers a negligible improvement in the cross-validation sample at the cost of substantial overfitting; decreasing the number of tiles by 40% incurs a penalty of ∼e²⁰ in the likelihood of the cross-validation stars. The top panels of Figure 4 show the resulting corrections to the proper motions as a function of position in equatorial coordinates.

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Fabricius et al. (2021) find that the Gaia EDR3 reference frame does show some dependence on magnitude and color. We test this possibility by using a coarser division of the sky and dividing our sample (again excluding the cross-validation set) by color and by magnitude. We use 920 tiles for this step, with ∼100 stars per tile. We split along the median color, ν_eff ≈ 1.57 μm⁻¹, and at G = 7.1 (slightly below the median G ≈ 8). We choose G = 7.1 as it turns out to provide the best agreement with the data, and it is close to the median G magnitude. It is also approximately the magnitude that separates the use of Gaia's lowest and its second-lowest gatings (gates being used to limit exposures and avoid saturation).

We divide the sample sequentially. We first split by color, separately fitting the redder half and the bluer half of stars. We then split by magnitude and again separately fit the brighter and the fainter stars. Our Gaussian process regression will then operate on the difference between the correction for red and blue stars, or between bright and faint stars.

We next choose the functional forms that we assume for the dependence of an individual star's correction on ν_eff and G magnitude. We assume that the correction will be proportional to ν_eff relative to the catalog median normalized by its median absolute deviation, capping this value at ±3 based on our results in the cross-validation data set. For magnitude, we instead choose a smoothed version of a Heaviside step function. This is inspired by the shift in gating and based on improved results over a linear relationship. We adopt the functional form

$$\begin{eqnarray}&&f(G)=1-\displaystyle \frac{2}{1+\exp \left(5(G-7.1)\right)}.\end{eqnarray} \tag{ 5 }$$

The factor of 5 in the exponent allows for a smooth transition over about a magnitude. It performs slightly better in the cross-validation set than a step function and avoids a discontinuity.

Variations in the reference frame as a function of color and magnitude are shown in Figure 4 and are highly significant. A color correction to the Gaia proper motions improves the likelihood of the cross-validation sample by a factor of e¹⁶⁸, equivalent to about 18σ. The improvement to the likelihood from the magnitude correction is lower at e⁸² but still significant at about 13σ. Including both corrections improves the likelihood by e²⁵⁷. This is nearly equal to the product of the two improvements and indicates that the corrections may be regarded as independent. Intriguingly, the correction maps for color and magnitude look almost identical. Combining the two correction maps before smoothing with a Gaussian process regression slightly improves the likelihood of the cross-validation sample; it also reduces the potential for overfitting. We therefore compute and use a single set of maps to correct for frame rotation as a function of both color and magnitude. These maps are the ones that we use to compute likelihood ratios and that we show in Figure 4. The actual correction applied to an individual star is the value in the map multiplied by a factor

$$\begin{eqnarray}&&f(G,{\nu }_{{\rm{eff}}})=1-\displaystyle \frac{2}{1+\exp \left(5(G-7.1)\right)}+\displaystyle \frac{{\nu }_{{\rm{eff}}}-{M}({\nu }_{{\rm{eff}}})}{{M}(| {\nu }_{{\rm{eff}}}-{M}({\nu }_{{\rm{eff}}})| )},\end{eqnarray} \tag{ 6 }$$

where M(ν_eff) denotes the median value of ν_eff; the denominator is the median absolute deviation. The color and magnitude-dependent corrections for a given star are the product of Equation (6) and the maps Ψ(α, δ) shown in the lower panels of Figure 4. The total cross-calibration term between the Gaia EDR3 and long-term proper motions is then

$$\begin{eqnarray}&&{\rm{\Delta }}\mu =\mathrm{Base}+f(G,{\nu }_{\mathrm{eff}})\times {\rm{\Psi }}.\end{eqnarray} \tag{ 7 }$$

For the full set of stars (about 10 times the cross-validation sample size), the likelihood improvements are nearly equal to the improvements in the cross-validation set raised to the tenth power. For color, magnitude, and color+magnitude corrections, they are approximately e¹⁵³⁰, e⁸⁵⁰, and e²³⁹⁰, respectively. These indicate an almost complete lack of overfitting and reflect the smoothness of the correction maps shown in Figure 4.

Most of the frame rotation that was present for the bright stars in Gaia DR2 was removed during the astrometric processing (Lindegren et al. 2021b) and is not present in the maps shown in Figure 4. Those rotations were easily the most apparent parts of their DR2 counterparts (Figure 6 of B18). The maps still retain considerable structure, though the typical magnitude of this structure is a factor of ∼20 smaller than for the Hipparcos proper motions. Both Hipparcos (when dividing the positions by 25 yr) and Gaia EDR3 have similar uncertainties, and both contribute to the structure seen. The magnitude and color-dependent effects are likely attributable to the Gaia proper motions; Fabricius et al. (2021) reported color- and magnitude-dependent frame rotations of up to 0.1 mas yr⁻¹. The structure seen in the top panels of Figure 4 probably has a component attributable to the Hipparcos positions and one attributable to the Gaia EDR3 proper motions.

To calibrate the uncertainties on the long-term proper motions and the Gaia EDR3 proper motions, we seek a sample of stars that are not accelerating within the precision of the measurements. For this purpose we turn to long-term radial velocity monitoring with precision spectrographs. We use the radial velocity time series as published by HIRES (Butler et al. 2017) and recalibrated by Tal-Or et al. (2019), the uniformly calibrated HARPS database (Trifonov et al. 2020), and the older but long-baseline Lick sample (Fischer et al. 2014).

We take all Hipparcos targets in each of our three radial velocity data sets and fit a linear trend, adding a jitter to enforce a reduced χ² of unity. This subsumes any real companions, like planets, into a jitter term and/or a trend. This is not a problem because we are searching for stars with low jitter and no trend. Once we have determined the jitter, a simple linear fit returns the standard error of the trend, the radial velocity acceleration. For stars that were observed by more than one spectrograph, we combine the measurements by each spectrograph using inverse-variance weighted means.

We next convert our radial velocity accelerations and their uncertainties to angular-equivalent units by multiplying by the Gaia EDR3 parallaxes. We then convert these values into proper motion units by multiplying by 12.375 yr. This is half the time baseline between the Hipparcos and Gaia EDR3 catalog epochs, i.e., the effective time between the long-term proper motion and the EDR3 proper motion. The resulting precision is directly comparable to the precision of the difference between the long-term proper motion and the Gaia EDR3 proper motion.

We seek to define a sample of stars showing no astrometric acceleration. Each star in our sample of candidate proper motion standard stars must meet five criteria:

• 1.  
  It must not have a common parallax companion in Gaia EDR3 within 10,000 au in projection;
• 2.  
  Its radial velocity baseline must be at least 5 yr on a single spectrograph;
• 3.  
  Its radial velocity jitter must be <10 m s⁻¹ for HIRES and HARPS, or <15 m s⁻¹ for Lick, to identify well-measured stars without planetary companions;
• 4.  
  Its radial velocity trend must be consistent with zero at 1σ; and
• 5.  
  Its radial velocity trend precision must be better than the precision of the proper motion difference.

For the last step, we take the Gaia EDR3 formal uncertainties and add these in quadrature with the formal uncertainties of the Hipparcos positions divided by 24.75 yr. We combine the published variances of the two Hipparcos reductions using a 60/40 mixture but apply no further error inflation. We then average the proper motion precisions in R.A. and decl. to get a single value for each star.

Our procedure results in a set of 374 astrometric standard stars. Figure 5 shows their z-scores, the ratio of the best-fit radial velocity trend normalized to three different uncertainties. First, we normalize to the standard error of the radial velocity trend (after using jitter to enforce a reduced χ² of unity). Second, we normalize to the formal uncertainty in the proper motion difference after converting units and averaging the precisions in R.A. and decl.. Finally, we normalize to the calibrated uncertainties. These latter two distributions, shown with progressively thicker lines in Figure 5, are much narrower than unit Gaussians: the limits on line-of-sight accelerations are significantly smaller than Gaia EDR3's precision for tangential accelerations. Given our crude approach to modeling radial velocity jitter, we do not expect the distribution of z-scores to be accurately Gaussian, and we make no inference on the true distribution of radial velocity accelerations using this data set.

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Some of our radial velocity standard stars could have low levels of real astrometric acceleration, e.g., from widely separated planets or brown dwarfs, but the purity of this sample will be far higher than for the catalog at large. In the next section we will derive an error inflation by applying a slightly revised Gaussian mixture model to the radial velocity standards.

We use the Gaussian mixture model given in Equation (3) to fit for two error inflation terms: an additive term (in quadrature) with the Hipparcos positional uncertainties, and a multiplicative term for the Gaia EDR3 uncertainties:

$$\begin{eqnarray}&&\begin{array}{l}{\sigma }^{2}={a}^{2}{\sigma }^{2}[{\mu }_{G}]\\ \quad +\ \displaystyle \frac{0.6{\sigma }^{2}[{x}_{H2}]+0.4{\sigma }^{2}[{x}_{H1}]+{a}^{2}{\sigma }^{2}[{x}_{G}]+{b}^{2}}{{\left({t}_{G}-{t}_{H}\right)}^{2}},\end{array}\end{eqnarray} \tag{ 8 }$$

using x to denote position in either R.A. or decl. and the subscript to denote the catalog (H1 = ESA 1997, H2 = van Leeuwen 2007, G = Gaia EDR3). The additive term could be identified with Hipparcos positions and/or with Gaia EDR3 proper motions, while the multiplicative term is clearly associated with Gaia EDR3. The covariance terms between R.A. and decl. retain the factor a but omit the term b. A possible multiplicative error inflation on Hipparcos was ruled out by B18, and we see no evidence of one in the residuals that we derive below.

We make two modifications to our fitting approach for Equation (3). First, we take our outlier distribution to be a Gaussian with a covariance matrix 4 times that given by Equation (8) (equivalent to 2 times the standard deviation) rather than one with a static width. This better accommodates stars with widely varying precision. Second, we fit for the probability g that a star is not an outlier, i.e., that it is a good astrometric reference. We also tried this alternative outlier distribution in the analysis of Section 5 but found that it performed slightly worse under those circumstances.

Our approach returns a best-fit a = 1.37, b = 0, and g = 1: all of the stars may be fit acceptably well by Equation (8) and do not require an outlier distribution. Figure 6 shows the z-scores of the astrometry of the radial velocity standards before (left) and after (right) applying this calibration to the uncertainties: the final distribution is accurately Gaussian with unit variance. Our result for a is smaller than the error inflation found by B18 for Gaia DR2 and confirms EDR3's better control of systematic uncertainties (Fabricius et al. 2021; Lindegren et al. 2021b). The value of b = 0 mas may be compared to the value of 0.2 mas yr⁻¹ found by B18 for proper motion uncertainties in Hipparcos. It suggests that no error inflation is needed on the Hipparcos positions and may, in part, be due to the use of Hipparcos positions to establish Gaia EDR3's bright star reference frame (Lindegren et al. 2021b).

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We perform several consistency checks on our results. Modest variations in the sample size and inclusion criteria have a minor effect on our inferred error inflation factor a. We describe these variations, and the effects on our results, below.

First, we vary the width of the outlier distribution in Equation (8). Changing its variance by a factor from 2 to 16 (1.4 to 4 in standard deviation) does not affect the results. As the outlier distribution becomes narrower still, it begins to merge with the central distribution and slightly decreases our inferred error inflation to a ≈ 1.30 However, this brings a negligible benefit to the likelihood: about e^0.2 or ${\rm{\Delta }}\mathrm{ln}{ \mathcal L }=0.2$. As the outlier distribution merges with the central distribution, the best-fit g decreases; g becomes effectively an additional free parameter. The data themselves do not demand this extra flexibility. For any relative width of the outlier distribution, we always infer an error inflation factor for Gaia EDR3 proper motions of at least 1.30.

We also perform consistency checks using bootstrap resampling on our set of radial velocity standard stars and by varying the criteria for inclusion in this set. Bootstrap resampling gives a mean value of a = 1.35 with a standard deviation of 0.08 and a mean value of b = 0.08 with a standard deviation of 0.11. The values of a and b are covariant: if we hold b fixed at zero, the mean and standard deviation of a become 1.37 and 0.07, respectively. Loosening the criteria for inclusion in our radial velocity standard sample slightly increases our inferred error inflation and makes it more sensitive to the outlier distribution, likely due to the inclusion of astrometric accelerators. Increasing our stringency for radial velocity precision to require 0.7σ consistency with zero trend and a trend precision twice that of the astrometric precision yields just 142 reference stars, but this sample gives consistent best-fit values of a = 1.34, g = 1, and b = 0.

Varying the sample of standard stars and the width of the outlier distribution changes our inferred error inflation factor by only a few percent. We adopt a = 1.37 at b = 0 as our final values for the cross-calibration. Though we adopt b = 0, it is difficult to distinguish a small value of b from a small change in a given our limited sample size: part of our inferred error inflation might be properly ascribed to the Hipparcos positions. Some of the Hipparcos systematic uncertainties will also be shared by the Gaia EDR3 proper motions given the use of Hipparcos positions to establish the bright star reference frame (Lindegren et al. 2021b). Any systematics shared by the two catalogs will cancel out in a cross-calibration. If we adopt an additional uncertainty of 0.2 mas in Hipparcos (c.f. the additional proper motion uncertainty inferred by B18), this would bring a close to 1.3. Adopting a = 1.30 and b = 0.3 mas rather than a = 1.37 and b = 0 produces distributions of residuals that are indistinguishable by eye from those shown in Figure 6. Continued radial velocity monitoring will ultimately produce better and larger sets of standard stars and enable a better characterization of Gaia's true astrometric uncertainties.

The cross-calibration we provide assumes that the model sky paths used by Hipparcos and Gaia EDR3 are good approximations to the actual motions of the stars. This is appropriate for single stars and for stars with much fainter companions on long-period orbits. For these stars the acceleration between Hipparcos and Gaia will be large compared to the acceleration within either Hipparcos or Gaia . Our cross-calibrated proper motions will be unreliable for stars with companions of comparable brightness. In this case, Hipparcos, Gaia, or both will measure something closer to the photocenter motion of the system. Hipparcos provides seven- and nine-parameter fits to stars that showed astrometric acceleration within Hipparcos. The Gaia EDR3 proper motions for these systems will likely be unreliable until the treatment of nonsingle stars in DR3.

The cross-calibration should be taken with caution for extremely high proper motion stars. As an example, Barnard's Star has a χ² between the Gaia EDR3 and long-term proper motions of about 80, corresponding to more than 8σ. However, its discrepancy in proper motion in decl. (0.56 mas yr⁻¹) is less than 4% of the 16 mas yr⁻¹ correction for nonlinear motion. The proper motion residual thus represents the apparent acceleration over about 6 months (4% of 12.375 yr). Uncertainty in the true central epoch in which the proper motion is measured could account for much of this difference. When astrometric accelerations are measured at very high significance, uncertainties in the effective epoch at which proper motions are measured can be an important contribution to uncertainties in orbit fitting. This situation will be resolved when future Gaia data releases include epoch astrometry.

Finally, we caution the user against making inferences using the distribution of χ² residuals. The catalog is intended to identify astrometrically accelerating stars for follow-up and to fit orbits to these systems. The residuals will reflect a mixture of astrometric acceleration due to faint companions, photocenter motion for stars with bright companions, short-period orbits for which the Gaia sky paths are not yet good approximations, and possible residual systematics in the catalog. However, for stars with faint, long-period companions, the combination of Hipparcos and Gaia provides a powerful tool for identifying systems, constraining orbits, and measuring masses.