ABSOLUTE PROPER MOTIONS OUTSIDE THE PLANE (APOP)—A STEP TOWARD THE GSC2.4

Following the work by van Altena et al. (1990), Spagna et al. (1996), and, more recently, by Mahmud & Anderson (2008), Kallivayalil et al. (2013) and Pryor et al. (2014), we work under the hypothesis that the absolute proper motions of galaxies are always zero, i.e., they are not dependent on their plate position, magnitude, or color. To this we add a second hypothesis: objects (stars and galaxies) physically close on a photographic plate and with similar magnitudes/colors have similar systematic errors.

Based on these assumptions, and considering the available plate data, the adopted procedure is to choose a good quality plate as the reference plate and to use the objects classified as stars with good image quality to transform the relevant program plates to the reference plate system. Here good image quality means objects in the middle of the unsaturated magnitude range (i.e., 16.0 < R_F < 18.5) where the centering errors are minimized, i.e. ∼0.13 arcsec, see Section 2.

The major plate-based calibration steps are:

• 1.  
  Remove the PdE with a moving-mean filter⁷ using stellar objects with good image quality;
• 2.  
  Select galaxies from non-stars via their pseudo (common) motion relative to the reference stars; and
• 3.  
  Calibrate the MdE, CdE and the residual PdE of all objects with reference to the galaxies.

Finally, absolute proper motions are calculated from all of the detections at the different plate epochs.

The relevant equations are described below.

For each object, the difference of positions between the reference (plate 1) and one of the corresponding program plates (plate 2) is modeled as

$$\begin{eqnarray}{\rm{\Delta }}{x}_{{\rm{s}}} & = & {x}_{1}-{x}_{2}={\mu }_{x}{\rm{\Delta }}t+D({x}_{2},{y}_{2})\\ & & +E({m}_{2},{c}_{2},{x}_{2},{y}_{2})\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{x}_{{\rm{g}}}=D({x}_{2},{y}_{2})+E({m}_{2},{c}_{2},{x}_{2},{y}_{2}),\end{eqnarray} \tag{ 2 }$$

where Δx_s, Δx_g represent the individual positional difference of stars and galaxies, respectively, and μ_x is a star's absolute proper motion in the reference plate coordinate system; also, Δt is the epoch interval between the reference and program plates, D is the PdE, and E is the combined MdE and CdE. Similar equations hold for the y coordinate.

The absolute proper motion μ_x of each star can be separated into the average proper motions ${\bar{\mu }}_{x}$ for all reference stars on the plate and a remaining individual proper motion, relative to the reference stars, dμ_x , i.e.,

$$\begin{eqnarray}&&{\mu }_{x}={\bar{\mu }}_{x}+d{\mu }_{x}.\end{eqnarray} \tag{ 3 }$$

Removing the systematic error D, besides removing the average MdE and CdE of the reference stars, also removes their average proper motions ${\bar{\mu }}_{x}$, while the galaxies will attain a "pseudo" proper motion $-{\bar{\mu }}_{x}$.

After this step, the position differences ${\rm{\Delta }}x{^{\prime} }_{{\rm{s}}},{\rm{\Delta }}x{^{\prime} }_{{\rm{g}}}$ of stars and galaxies between the reference and program plates can be expressed as:

$$\begin{eqnarray}{\rm{\Delta }}x{^{\prime} }_{{\rm{s}}} & = & {x}_{1}-x{^{\prime} }_{2}={\rm{\Delta }}{x}_{{\rm{s}}}-\left(D+{\bar{\mu }}_{x}{\rm{\Delta }}t\right)\\ & = & d{\mu }_{x}{\rm{\Delta }}t+E\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}x{^{\prime} }_{{\rm{g}}}={\rm{\Delta }}{x}_{{\rm{g}}}-\left(D+{\bar{\mu }}_{x}{\rm{\Delta }}t\right)=-{\bar{\mu }}_{x}{\rm{\Delta }}t+E.\end{eqnarray} \tag{ 5 }$$

Equation (5) has a special role in our reduction procedure; we iterate exactly on this equation to reject non-stars that are not genuine galaxies. In practice, assuming that all real galaxies will show the common pseudo proper motion $-{\bar{\mu }}_{x}$ discussed before, we flag as outliers, i.e., probable blended objects, those that do not.⁸ In the end, the selected true galaxies constrain the average proper motions, and MdE and CdE terms. After this stage, the updated position differences ${\rm{\Delta }}x{^{\prime\prime} }_{{\rm{s}}},{\rm{\Delta }}x{^{\prime\prime} }_{{\rm{g}}}$ can be expressed as:

$$\begin{eqnarray}{\rm{\Delta }}x{^{\prime\prime} }_{{\rm{s}}} & = & {x}_{1}-x{^{\prime\prime} }_{2}={\rm{\Delta }}^{\prime} {x}_{{\rm{s}}}-{\rm{\Delta }}^{\prime} {x}_{{\rm{g}}}\\ & = & d{\mu }_{x}{\rm{\Delta }}t+E-(-{\bar{\mu }}_{x}{\rm{\Delta }}t+E)\\ & = & (d{\mu }_{x}+{\bar{\mu }}_{x}){\rm{\Delta }}t\\ & = & {\mu }_{x}{\rm{\Delta }}t\\ {\rm{\Delta }}x{^{\prime\prime} }_{{\rm{g}}} & = & {\rm{\Delta }}^{\prime} {x}_{{\rm{g}}}-{\rm{\Delta }}^{\prime} {x}_{{\rm{g}}}=0.\end{eqnarray} \tag{ 6 }$$

Once all of the plates have been reduced to one system, for every detected object a linear fit is applied to the corresponding transformed multi-epoch observations to obtain the absolute proper motions (and a reference position) as follows:

$$\begin{eqnarray}&&\left\{\begin{array}{l}{x}_{{\rm{t}}}={x}_{0}+{\mu }_{x}(t-{t}_{0})\\ {y}_{{\rm{t}}}={y}_{0}+{\mu }_{y}(t-{t}_{0})\end{array}\right.,\end{eqnarray} \tag{ 7 }$$

where x_t , y_t are the transformed plate (measured) coordinates at epoch t, and μ_x, μ_y, x₀, y₀ are the estimated absolute proper motions and positions in the same coordinate system at the chosen reference epoch t₀ = J2000.0 (see the paragraph on Step 7 in Section 3.3). The x₀, y₀ are the positions at the reference epoch from all plates in different colors and epochs; this decreases the influence of individual random errors and improves the final precision.

3.1.1. Neglecting Parallax

The complete version of Equation (1) usually includes the parallax term, πΔP_x, where π is the parallax and ΔP_x is the difference of the parallax factor at the two epochs of measurement. Therefore, it is important that we speak to the reasons for not including such a term in the reduction model and provide estimates of any possible proper motion bias this decision might cause.

There are three main reasons why we did not include a parallax term in our reductions: (i) the relatively low, plate-scale related, precision of the individual plate measurements (σ_x), (ii) the very coarse time sampling they realize, and (iii) repeated observations of the same fields occur around the same time of the year. Properties (ii) and (iii) are well illustrated in Figures (2) and (3): the upper panel in Figure 2 shows that most of the APOP entries have only three measurements with less than 10% receiving 10 or more; Figure 3, providing the distribution of the folded observing times (folded to the fraction of the year on each observation was taken), clearly reveals a very uneven sampling, unsuitable for properly tracing the parallax ellipse. Actually, the lack of data points in a large interval, ∼0.35 years, around 0.5 years suggests that parallax induced positional changes are naturally minimized in our data as parallax factor differences tend to be small.

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From Section 2, the precision of our best measurements is σ_x ∼ 0.15 arcsec; therefore, dividing this value by the square root of the number of measurements, Nobs, yields a rough estimate of sensitivity to actual parallax shifts. By setting, Nobs = 10, although this actually happens for a very small fraction of the cases (Figure 2), we estimate (${\sigma }_{\pi }\sim {\sigma }_{x}/\sqrt{\mathrm{Nobs}}\ \sim$ 0.050 arcsec), i.e., in the best case scenario, parallax would have a chance to show in our data (i.e., as nonlinear effects) but only for distances up to ∼20 pc. If we now consider again properties (ii), and especially (iii), from above, it becomes clear that parallax direct estimability with our data is unlikely. Under these circumstances forcing a parallactic term in Equation (1) is likely to correlate parallax with proper motion itself, therefore justifying our decision to drop the parallax factors from the adopted reduction model. Yet we still need to address here the possible effects on the APOP proper motions of the parallax term we left unaccounted.

To this end, as recalled in Section 2, pairing plate measurements together (object matching) was done via the GSC2.3 object IDs (no actual re-matching of plate entries); therefore, APOP inherited the same 4 arcsec max matching radius utilized for the construction of the GSC2.3 (Lasker et al. 2008). We anticipate here the characteristics of the proper motions we decided to export to the APOP catalog: each proper motion entry was generated after (1) fitting at least three plate measurements, and (2) rejecting fits with total epoch separation of less than 10 years (see lower panel of Figure 2).

At 20 pc a full parallax shift would show in our results, because of the 10-year condition (No.2) above, as a proper motion bias of ∼0.005 arcsec yr⁻¹. At this distance even disk population stars exhibit proper motions of the order of 0.1 arcsec yr⁻¹, for a bias-to-proper-motion ratio of only 5%. With local thick disk and halo stars moving at much higher spatial velocities (up to 10–20 times larger), this bias-to-proper-motion ratio drops to below 1% and 0.3%, for thick disk and halo stars, respectively. As both parallax and proper motion decrease with distance, this ratio could be seen as a value characterizing the quality of the APOP proper motions of the different Galactic populations independent of distance.

In conclusion, the considerations above provide evidence that the proper motion reduction model presented in the previous section is quite robust against parallax effects.

Finally, we still need to address here the effect of the 4 arcsec matching radius recalled before. Actually, for high proper motion samples (roughly higher than 0.4 arcsec yr⁻¹ if below −30° decl., and 0.1 arcsec yr⁻¹ if above; see the lower panel of Figure 2) involving Galaxy populations like WD and RD stars, this spatial filter can cause the loss of those measurements with the largest epoch differences and therefore may not be the most useful for the best proper motion accuracy. Potential users in need of studying the properties of these important, but relatively small, high proper motion samples will have to decide if the APOP provided values are adequate for their applications.

The overall procedure just discussed only removes the systematic errors between reference and program plates; it can not remove the systematics inherent to the differences between plate-based coordinates and the standard coordinates.

This last registration is obtained using an external reference catalog and an account of the procedure follows.

We use a gnomonic projection to transform the estimated measured coordinates to an equatorial system. Since the Schmidt design provides an equidistant projection, we correct the measured coordinates following Dick (1991):

$$\begin{eqnarray}&&\left\{\begin{array}{l}{x}_{G}={x}_{0}+\displaystyle \frac{1}{3}{x}_{0}({x}_{0}^{2}+{y}_{0}^{2})\\ {y}_{G}={y}_{0}+\displaystyle \frac{1}{3}{y}_{0}({x}_{0}^{2}+{y}_{0}^{2})\end{array}\right.,\end{eqnarray} \tag{ 8 }$$

where x_G, y_G are the measured coordinates in a gnomonic projection. By differentiating Equation (8) as a function of time we find the corresponding relations for proper motions:

$$\begin{eqnarray}&&\left\{\begin{array}{l}{\mu }_{{x}_{G}}={\mu }_{x}(1+{x}_{0}^{2}+\displaystyle \frac{1}{3}{y}_{0}^{2})+\displaystyle \frac{2}{3}{x}_{0}{y}_{0}{\mu }_{y}\\ {\mu }_{{y}_{G}}={\mu }_{y}(1+{y}_{0}^{2}+\displaystyle \frac{1}{3}{x}_{0}^{2})+\displaystyle \frac{2}{3}{x}_{0}{y}_{0}{\mu }_{x}\end{array}\right.,\end{eqnarray} \tag{ 9 }$$

where ${\mu }_{{x}_{G}}$, ${\mu }_{{y}_{G}}$ are the proper motions in a gnomonic projection.

The transformation between the measured coordinates x_G, y_G and their celestial standard coordinate ξ, η can be expressed as

$$\begin{eqnarray}&&\left\{\begin{array}{l}\xi ={{ax}}_{G}+{{by}}_{G}+c+\varepsilon ({x}_{G},{y}_{G})\\ \eta =a^{\prime} {x}_{G}+b^{\prime} {y}_{G}+c^{\prime} +\varepsilon ^{\prime} ({x}_{G},{y}_{G})\end{array}\right.,\end{eqnarray} \tag{ 10 }$$

where $a,b,c,a^{\prime} ,b^{\prime} ,c^{\prime}$ are the coefficients of the linear terms of the plate model, i.e., axis direction, scale and origin difference between the measured and standard coordinate systems; $\varepsilon ({x}_{G},{y}_{G})$ and $\varepsilon ^{\prime} ({x}_{G},{y}_{G})$ represent the higher order terms.

If we differentiate Equation (10) with respect to time, we get the proper motions in the standard coordinate system.

$$\begin{eqnarray}&&\left\{\begin{array}{l}{\mu }_{\xi }=a{\mu }_{{x}_{G}}+b{\mu }_{{y}_{G}}\\ {\mu }_{\eta }=a^{\prime} {\mu }_{{x}_{G}}+b^{\prime} {\mu }_{{y}_{G}}\end{array}\right.,\end{eqnarray} \tag{ 11 }$$

where we have neglected higher order terms, which we estimate to be less than 0.03% of the μ_x, μ_y.

For the objects on the plate, there is a precise geometrical relationship between the standard coordinates and equatorial coordinates i.e.,

$$\begin{eqnarray}&&\left\{\begin{array}{l}\mathrm{tan}(\alpha -{\alpha }_{0})=\displaystyle \frac{\xi }{\mathrm{cos}{\delta }_{0}-\eta \mathrm{sin}{\delta }_{0}}\\ \mathrm{tan}(\delta )=\displaystyle \frac{\eta \mathrm{cos}{\delta }_{0}+\mathrm{sin}{\delta }_{0}}{\mathrm{cos}{\delta }_{0}-\eta \mathrm{sin}{\delta }_{0}}\mathrm{cos}(\alpha -{\alpha }_{0})\end{array}\right.,\end{eqnarray} \tag{ 12 }$$

where α₀, δ₀ are the equatorial coordinates of the tangent point.

Finally, the rigorous relationship between proper motions in the equatorial and standard systems is obtained by differentiating Equation (12) as a function of time, i.e.,

$$\begin{eqnarray}&&\left\{\begin{array}{l}{\mu }_{\alpha }={\mathrm{cos}}^{2}(\alpha -{\alpha }_{0})\\ \left(\begin{array}{l}\displaystyle \frac{1}{\mathrm{cos}{\delta }_{0}{\rm{}}\;{\rm{-}}\;{\rm{}}\eta \mathrm{sin}{\delta }_{0}}{\mu }_{\xi }\\ +\;\displaystyle \frac{\xi \mathrm{sin}{\delta }_{0}}{{\left(\mathrm{cos}{\delta }_{0}{\rm{}}\;{\rm{-}}\;{\rm{}}\eta \mathrm{sin}{\delta }_{0}\right)}^{2}}{\mu }_{\eta }\\ \end{array}\right)\\ {\mu }_{\delta }={\mathrm{cos}}^{2}(\delta )\\ \left(\begin{array}{l}\displaystyle \frac{\mathrm{cos}(\alpha -{\alpha }_{0})\mathrm{cos}{\delta }_{0}}{\mathrm{cos}{\delta }_{0}{\rm{-}}\eta \mathrm{sin}{\delta }_{0}}{\mu }_{\eta }\\ +\;\displaystyle \frac{\mathrm{cos}(\alpha -{\alpha }_{0})\left(\eta \mathrm{cos}{\delta }_{0}+\mathrm{sin}{\delta }_{0}\right)\mathrm{sin}{\delta }_{0}}{{\left(\mathrm{cos}{\delta }_{0}{\rm{}}\;{\rm{-}}\;{\rm{}}\eta \mathrm{sin}{\delta }_{0}\right)}^{2}}{\mu }_{\eta }\\ -\;\displaystyle \frac{\left(\eta \mathrm{cos}{\delta }_{0}+\mathrm{sin}{\delta }_{0}\right)\mathrm{sin}(\alpha -{\alpha }_{0})}{\mathrm{cos}{\delta }_{0}{\rm{}}\;{\rm{-}}\;{\rm{}}\eta \mathrm{sin}{\delta }_{0}}{\mu }_{\alpha }\\ \end{array}\right)\\ \end{array}\right..\end{eqnarray} \tag{ 13 }$$

We developed a FORTRAN pipeline based on the above principles to determine absolute proper motions from the COMPASS database. The flowchart (Figure 4) summarizes the specific steps and techniques used to remove the systematic errors and to acquire the absolute proper motions. Key steps in the flow chart are detailed below.

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In Steps 1–3 we extract all matches on the plates overlapping with the reference plate. Only objects that appear in all plates are used to calibrate the fit to the reference plate system.

In Step 4, a moving-mean-like method is applied to remove the mean shift (PdE and the mean proper motions of reference stars) in x and y between the program and reference plate. Only stellar objects are used in the calculation of the moving mean but the result is applied to all detections. In order to attain the most accurate coordinate transformation, we select reference stars with good image quality (i.e., with low centering errors, see Section 3.1). In particular, we use detections classified as star in the range $\left({m}_{\mathrm{lim}}-4.5\right)\leqslant m\leqslant \left({m}_{\mathrm{lim}}-2.0\right)$, where ${m}_{\mathrm{lim}}$ is the limiting magnitude of the plate, ∼20.5 in R_F. In general we find around 500 stars degree⁻² well distributed on the plate. There are several methods for removing the PdE, such as a global plate solution using high-order polynomials, the astrometric MASK (Taff et al. 1990; Lattanzi & Bucciarelli 1991), the Infinitely Overlapping Circles (IOC, Taff et al. 1992), the sub-plate method (Taff 1989) etc. All these methods have pros and cons: the Schmidt plate distortions would require very complicated global solutions, the MASK method is very sensitive to the number of reference objects and the grid size, the IOC has problems at the plate boundaries and the sub-plate method can lead to non-uniformities.

We developed a variant of the sub-plate method, the moving sub-plate method, that is sensitive to local signals while ensuring an increased level of uniformity. The moving sub-plate method starts by transforming the measured coordinates x₂, y₂ on the program plates to the corresponding coordinates x₁, y₁ on the reference plate using cubic polynomials (Equation (14)). This will remove most of the large-scale errors (e.g., spherical deformation).

$$\begin{eqnarray}&&\left\{\begin{array}{l}{x}_{1}=f({x}_{2},{y}_{2})\\ {y}_{1}=g({x}_{2},{y}_{2})\end{array}\right.,\end{eqnarray} \tag{ 14 }$$

where the functions f and g are complete cubic polynomials.

The residuals Δx, Δy, from the global (whole plate) cubic polynomial fit are then used in the following linear relations:

$$\begin{eqnarray}&&\left\{\begin{array}{l}{\rm{\Delta }}x={{ax}}_{2}+{{by}}_{2}+c\\ {\rm{\Delta }}y=a^{\prime} {x}_{2}+b^{\prime} {y}_{2}+c^{\prime} \end{array}\right.,\end{eqnarray} \tag{ 15 }$$

where a, b, c, a', b', c' allow for zero point, rotation, and scale difference at a local level.

To fit Equation (15) we select nearby reference stars within a radius of 20 arcmin from the program object. If more then 15 stars were found, they were ranked with distance from the program star and the first 15 were used for the fit; with less then 3 reference stars no correction was computed. Therefore depending on actual star density, the effective radius of a moving sub-plate can be much smaller then 20 arcmin. Applying this correction to all program objects removes most of the small scale errors (See Figure 5). After Step 4, the mean displacement due to proper motion of the reference objects is locally zero, while that of galaxies is not.

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In Step 5 we use Equation (16) on all the non-stars to fit a linear (coordinate only) model to estimate the mean proper motion components for any given plate pair, i.e., the $-{\bar{\mu }}_{x}{\rm{\Delta }}t$ in Equation (5), and its analog for the y direction.

$$\begin{eqnarray}&&\left\{\begin{array}{l}{\rm{\Delta }}x={x}_{1}-{x}_{2}={{ax}}_{2}+{{by}}_{2}+c\\ {\rm{\Delta }}y={y}_{1}-{y}_{2}=a^{\prime} {x}_{2}+b^{\prime} {y}_{2}+c^{\prime} \end{array}\right.,\end{eqnarray} \tag{ 16 }$$

where x₂, y₂ are intended as the measured coordinates of non-stars on the program plate corrected for the systematics calculated in Step 4. As per Equation (5), we iterate rejecting any objects with residuals greater than 2.6 standard deviations⁹ per coordinate.

The objects that survive this selection are likely to be, or act like (i.e., non-stellar objects with zero proper motions), genuine galaxies. Figure 6 shows the final mean (pseudo) proper motion of the galaxies after completing Step 5 for the same plate pair as that of Figure 5.

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In Step 6 once the galaxies positions are corrected using the fitted coefficients of Equation (16), a two-dimensional map of the galaxies residuals shows that some PdE are still present. These are smoothed out using again the moving sub-plate method but this time using the selected galaxies as reference objects. The residual corrections found at this stage are applied to all of the objects. The subset of these values relative to the galaxies is then spatially binned and plotted against magnitude and color to seek for any remaining MdE and/or CdE signals. In case, these are treated by simple one-dimensional linear interpolation in between the magnitude or color bins. This procedure removes most of the MdE and CdE between program and reference plates.

At this step we apply the mean motion to the stellar objects and remove it from the galaxies. From Figures 7 to 8 we note that both MdE and CdE were significantly reduced.

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In Step 7 for all of the plate objects we fit Equation (7) to determine absolute proper motions and positions at the given epoch J2000.0. We then correct the measured coordinates and absolute proper motions derived in an equidistant projection to a gnomonic projection by using Equations (8) and (9).

In Step 8 we use the Fourth US Naval Observatory CCD Astrograph Catalog (UCAC4, Zacharias et al. 2013) and Equations (10)–(13) to transfer absolute proper motions and positions to the celestial reference frame. Moreover, we corrected the frame bias between APOP and the International Celestial Reference System (ICRS) by using 1288 objects (most of them are faint Quasi-stellar objects; QSOs) common to the radio ICRF2 catalog (Ma et al. 2009). To establish the orientation to the standard frame, we use the following formula

$$\begin{eqnarray}{{\boldsymbol{r}}}_{\mathrm{ICRF}2} & = & {R}_{Y}({\xi }_{y}){R}_{X}({\xi }_{x}){R}_{Z}({\xi }_{z})\cdot {{\boldsymbol{r}}}_{\mathrm{APOP}}\\ & = & \left(\begin{array}{ccc}1 & {\xi }_{z} & -{\xi }_{y}\\ -{\xi }_{z} & 1 & {\xi }_{x}\\ {\xi }_{y} & -{\xi }_{x} & 1\end{array}\right)\cdot {{\boldsymbol{r}}}_{\mathrm{APOP}}\end{eqnarray} \tag{ 17 }$$

where ${{\boldsymbol{r}}}_{\mathrm{ICRF}2}$ and ${{\boldsymbol{r}}}_{\mathrm{APOP}}$ are the direction vectors to the common objects in both ICRF2 and APOP, respectively; ξ_x, ξ_y, ξ_z are the three small rotation angles needed to register the APOP coordinate frame about the x, y and z axes of the ICRF2. A least-squares fit to Equation (17) provides estimates of those three angles: ξ_z = 27.73 ± 0.12 mas, ξ_x = −7.30 ± 0.15 mas, and ξ_y = 2.21 ± 0.06 mas; these are used to bring APOP onto ICRF2. The ξ_x and ξ_y values are similar to those found in Zacharias & Zacharias (2014), while the ξ_z rotation difference is considerably large (ξ_z = 7.61 ± 1.21 in Zacharias' work). The actual cause is not clear at the moment but it could reside in the different samples used for the registration. After performing the transformation, the coordinate axes realized by our astrometric data are believed to be aligned, at least formally, with the extragalactic radio frame to within ±0.2 mas at the reference epoch J2000.0.

For each object, we calculated positions and proper motions by fitting Equation (7) utilizing all of the measurements from the different epochs and colors. This provides the formal errors of the calibrated parameters (μ_α* = μ_α × cos(δ), μ_δ, α, δ), and an internal check of the APOP quality.

In Figure 9 we plot the mean formal errors and find them to be consistent in both R.A. and decl. even though the two coordinates were treated independently. For stellar objects, the internal accuracy of the proper motions is better than ±4 mas yr⁻¹ for objects brighter than R_F = 18.5, increasing to 9 mas yr⁻¹ at R_F = 20.0 and to 14 mas yr⁻¹ for objects with 20.0 < R_F < 20.8. The internal accuracy of the stellar positions is better than ±100 mas for objects brighter than R_F = 18.5, increasing to 260 mas for objects with magnitude R_F ∼ 20.8. The offset between the star and non-star objects is consistent with the ∼1.5 times larger measurement errors of non-stellar objects.

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We note that objects close to the magnitude limit (i.e., 20 < R_F < 21) have larger errors than predicted; APOP parameters and errors should be used with caution for objects fainter than R_F = 20.

An internal check of the accuracy of APOP proper motions is provided by multiple estimates for the same stellar objects appearing in overlap regions between adjacent reference plates. In Figure 10 we display the mean offset between the proper motions calibrated on two adjacent reference plates. The differences are plotted based on the position in the central reference XP216 (l = 1514, b = 628), which has a typical 15 × 65 overlap with four adjacent plates XP215 (left), XP217 (right), XP170 (top) and XP266 (bottom). There are no large scale offsets in the overlap regions, but some shifts of ∼1.8 mas yr⁻¹ appear at small scales, particularly at the bottom left of the central plate. These are very likely caused by the uncertainty in the correction of the absolute zero point by the galaxies.

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QSOs have star-like images and since they are extragalactic, they do not exhibit any time-dependent displacement. Thus, we can use the mean and dispersion of their measured motions to evaluate the zero point and overall precision of stellar proper motions. Here we use QSOs as an independent and direct determination of the APOP catalog quality.

The Gaia Initial QSO Catalog (GIQC; Andrei et al. 2009) is chosen as the source list for known QSOs. The objects are broadly distributed within the SDSS region, though their density is not uniform (see Figure 11). Figures 12 –14 show the mean proper motions of the GIQC QSOs and indicate that there is a very good agreement between the external and theoretical error estimates of proper motions for the magnitude range R_F < 20.5. In particular, from this sample of QSOs we find a proper motion zero point error of 0.6 mas yr⁻¹. As a verification of the internal estimates, Figure 15 shows the formal errors of positions (α, δ) of QSOs as a function of magnitude, and indicate that they are consistent with stellar objects.

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In addition to the GIQC, we also compared APOP with other external catalogs. The most natural comparison would be with the Positions and Proper Motion XL catalog (herafter PPMXL, Roeser et al. 2010) the largest most recent catalog with absolute proper motions. This was constructed by combining USNO-B1.0 and the Two Micron All Sky Survey (Skrutskie et al. 2006) catalogs. Figures 16 and 17 illustrate that there is a systematic offset between these two catalogs in absolute proper motions.

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We also selected a subset of the GIQC QSOs that were in both the APOP and the PPMXL for a comparison. Figure 18 shows that the offset between the two catalogs is mostly due to a zero point problem in the PPMXL. Also, for most of the magnitude range the APOP has smaller proper motion dispersion.

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Since the field of view of the Schmidt plate is very large (6.5 × 6.5 sq. deg) the mean proper motion of stellar objects could vary across the plate. For example, clusters generally cover a significant part of a photographic plate. Using the stars in Step 4 to remove the PdE we risk also "removing" the proper motions for this group of objects. Theoretically, we believe our procedure returns the mean proper motions to those groups with Steps 5 and 6. The proper motions of the Praesepe cluster in Figure 19 indicate that at least for this cluster the procedure has worked.

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The catalog is available at Strasbourg's astronomical Data Center (http://cds.u-strasbg.fr, catalog I/331); the description of the catalog data is detailed in Table 1. In particular, APOP derived positions and proper motions are at epoch J2000.0 and on the ICRS-defined equator.

Given the evidence provided, we believe that APOP's proper motions are reliable for operational applications as well as astrophysical research. However, the potential users should remember the following.

• 1.  
  The accuracy is best for objects with δ ≥ −30° because of the 45-year epoch difference between first and second generation survey plates; at lower declinations the quality can be about four times worse than that due to a time base of only 12 years.
• 2.  
  Our data around the Large Magellanic Cloud region (center at α_o = 809, δ_o = −697 and 108 × 92 in size) are not entirely reliable, as the region is too crowded, which caused object detection to generate many spurious objects.
• 3.  
  The accuracy may not be sufficiently good for objects with magnitude R_F < 15.0 due to a lack of bright galaxies on some plates and decreasing measuring precision due to heavier saturation effects. At this bright end, users should at least compare APOP's values to those listed in bright absolute catalogs like, e.g., UCAC4 (Zacharias et al. 2013).
• 4.  
  As explained in Section 3.1.1, when used in accurate work, especially with high proper motion stars, APOP values should be taken with caution because of possible residual parallax effects.

The items above provide users with the proper information to decide if the APOP entries are adequate for their applications.