3 AC reduction onto the ICRS/Hipparcos system

  The Astrographic Catalogue reduction onto the modern reference system ICRS, as defined by the Hipparcos Catalogue, is of primary importance to any project aimed at the derivation of proper motions from the AC and modern-epoch observations. The TRC approach to this task (Kuimov et al. 1998) was basically the same as the one used for the AC reduction onto the FK5 system (Kuzmin et al. 1997a). This section outlines the approach, along with modifications necessitated by the change of reference system.

3.1 Plate reduction strategy

  In general, the reduction model used to derive equatorial coordinates of stars from their measurements on a photographic plate can be described by the following relations:

 

$$\begin{array} {lcl} x &=& f(\alpha, \delta, m, ci, c_1,~\l... ...y &=& g(\alpha, \delta, m, ci, c_1,~\ldots,c_n).\\ \end{array}$$ (1)

The meaning of individual symbols in Eqs. (1) is as follows: x and y - image coordinates of the star as measured on the plate; $\alpha, \delta$ - equatorial coordinates of the star referred to some chosen reference system; m, ci - its magnitude and colour; f and g - gnomonic projections supplemented by additional (usually polynomial) terms describing various plate-scale effects like tilt, coma etc.; $c_1,~\ldots,c_n$ - the unknown parameters of the latter (so-called plate constants).

The plate reduction proper is the derivation of the unknowns $c_1,~\ldots,c_n$through Eqs. (1). Obtained plate constants then allow to solve Eqs. (1) for the equatorial coordinates $\alpha$and $\delta$ of all stars measured on a plate or a set of plates.

Under the common approach to reduction, the so-called conventional plate adjustment, unknowns $c_1,\,\ldots,c_n$ are derived based on stars with known position at epoch of the plate exposure (i.e. reference stars). For the reduction to proceed, it is first necessary to investigate the significant plate-scale systematic errors or, in other words, to establish the functional form of f and g in Eqs. (1). Since neither normal astrographs nor measuring devices used throughout the AC project are anymore available for laboratory testing and calibration in their original state, the only method to investigate telescope-specific aberrations and measuring machine errors is to analyze the residuals after conventional plate adjustment. When the form of f and g - that is, the plate model - is established, a reduction of the AC data can proceed.

The establishment of the AC zone-specific plate models followed this approach. It started with a so-called Turner plate model, in which f and g are linear functions of the standard coordinates (rectangular coordinates on the plate derived from $\alpha$, $\delta$ through gnomonic projection with an assumed plate centre). After the least-squares solution of Eqs. (1) residuals from all plates of a zone were combined and analyzed for higher-order terms which significantly improve the model with respect to the chosen statistical criteria. The additional plate model terms defined in this way - the zone-specific corrections - may represent known physical phenomena (which is welcome), or be purely ad hoc (which cannot be entirely avoided).

To investigate significant plate-scale systematics in such a way, the number of reference stars per plate should be sufficient for a reliable determination of the parameters $c_1,\,\ldots,c_n$. However, the representation of the ICRS system by the Hipparcos Catalogue is too sparse for this purpose, providing an average 3 stars per square degree. Inapplicability of the Hipparcos for this purpose necessitated the employment of a denser intermediate reference catalogue and the development of a specific three-step plate adjustment procedure.

First, the plates constituting an AC zone were reduced on the "Astrographic Catalogue Reference Stars'' (ACRS, Corbin & Urban 1991) catalogue system by adjusting the 6 parameters of the Turner plate model. The residuals were then analyzed to yield additional terms. These then formed the advanced reduction model that accounts for the significant plate-scale systematic errors specific to the AC zone in question.

In step 2 the solution was repeated, but this time with the significant plate-scale terms found in step 1 added to the plate model. The ACRS again served as reference catalogue. The derived plate parameters were then analyzed for any dependence on observation/measurement conditions and other factors. Based on the results of this analysis, plates of the zone were, if necessary, discriminated into subsets. Subsequent averaging of the higher-order plate parameters yielded the final zone-specific (or subset-specific) systematic corrections.

In the third step the final reduction onto the Hipparcos system was performed using the Turner model and the Hipparcos Catalogue as reference catalogue. The zone-specific systematic corrections derived in step 2 were applied beforehand. If, for example, step 1 suggested that a sizeable cubic distortion is present for a specific astrograph, the distortion coefficient was estimated for all plates in step 2. The estimates from individual plates were then averaged to yield a unique value, assumed to be characteristic for the astrograph. At step 3, that cubic distortion correction was then applied to all stars, as function of the actually measured plate coordinates.

The three-step approach using zone-specific corrections may be considered as an abridged plate overlap solution (Eichhorn 1960) in the sense that it assumes the instrument-specific parameters (e.g. the distortion coefficient) the same for all plates of the set, thus utilizing an important piece of information being lost under the individual plates adjustment approach. On the other hand, zone-specific plate models established under the present study provide the necessary basis for the future global block adjustment of the AC data.

3.2 Intermediate reference catalogue

  The Hipparcos Catalogue cannot be used directly for the AC reduction due to its low star density: Hipparcos typically provides about 12 stars per $2.1\hbox{$^\circ$}\times 2.1\hbox{$^\circ$}$ AC plate. This number, though low, would be tolerable for the reduction with a linear model, provided the plate-scale systematic errors of the Astrographic Catalogue were known. Since the investigation of the plate-scale systematic errors present in the AC data is, in fact, one of the main subjects of the present study, an intermediate reference catalogue, tied to the Hipparcos system and providing at least 30-40 reference stars per AC plate (an equivalent of 10 stars per square degree), is necessary for the purpose.

At present, only two reference catalogues of the required density are available: "Astrographic Catalogue Reference Stars'' (ACRS, Corbin & Urban 1991), and "Positions and Proper Motions'' (PPM, Röser & Bastian 1991; Bastian et al. 1993), both utilizing practically all meridian and photographic observations of stars brighter than about B=9.5 mag available by 1990. PPM is distinguished as the first major application of the Astrographic Catalogue for the derivation of proper motions. It should be noted that the AC reduction performed for PPM was limited to stars brighter than about B=9.5 mag. Though it amounts only to about 10% of the Astrographic Catalogue stars, PPM is nonetheless correlated with AC and, consequently, cannot be applied for the investigation of systematic errors in AC. This leaves only ACRS as a suitable intermediate reference catalogue.

Prior to the application of ACRS to the investigation of systematic errors present in the AC data, it is necessary to remove the systematic errors of ACRS itself. Such errors were found in a comparison with Hipparcos during the TRC pilot project study (Kuzmin et al. 1997b). The errors amount to 250 mas in position at mean epoch and 5 mas/yr in proper motion (cf. Fig. 2). If unaccounted for, they would result in systematic errors of up to 500 mas when propagated to a typical AC epoch (the mean epoch difference between ACRS and AC is about 50 years).

  

Figure 2: Systematic differences ACRS-Hipparcos in position at mean epoch of the ACRS catalogue (left) and in proper motion (right). Mean differences averaged over cells of 25 square degrees are shown as two-dimensional vectors

Systematic patterns similar to the ones found in the ACRS-Hipparcos differences were previously reported from comparisons of FK5 and PPM with Hipparcos (Lindegren et al. 1995). This is not surprising: first, ACRS is on the FK5 system and, second, ACRS and PPM are based on nearly the same photographic material (apart from the AC data). This fact, along with the clear zonal structure of the systematic differences, makes it possible to attribute the latter mainly to systematic errors of the positional catalogues ACRS is based on.

Systematic differences of positions or proper motions given in any two catalogues may depend on position, magnitude and colour of the stars. The dependence on the latter two parameters, known as magnitude and colour equation respectively, has previously been difficult to investigate due to the absence of homogeneous set of magnitudes and colours for the large number of stars. Fortunately, the Tycho Catalogue photometry now provides the necessary basis for such an analysis.

Using the notation $\mu_{\alpha*}=\mu_{\alpha}\cos{\delta}$, we define systematic differences between ACRS and Hipparcos as

 

$$\begin{array} {lcl} \Delta\alpha_*&=&\langle(\alpha_{\mathrm... ...ta\mathrm{ACRS}}-\mu_{\delta\mathrm{HIP}}\rangle,\\ \end{array}$$ (2)

where the angle brackets denote averages over suitable sets of stars (e.g. cells of 25 square degrees, as in Fig. 2).

The analysis of the ACRS-Hipparcos differences had shown that the main contribution has the form of a declination-dependent magnitude equation and, to lesser extent, a colour equation, to the first order given by:

 

$$\Delta p = b_0 + b_1(B_{\mathrm{T}}-9.15) + b_2[(B_{\mathrm{T}}-V_{\mathrm{T}})- 0.768].$$ (3)

Here $\Delta p$ denotes one of the systematic differences defined in Eqs. (2), $B_{\mathrm{T}}$ and $(B_{\mathrm{T}}-V_{\mathrm{T}})$ are Tycho blue magnitude and colour, 9.15 mag is the mean of $B_{\mathrm{T}}$ and 0.768 mag the mean of $(B_{\mathrm{T}}-V_{\mathrm{T}})$ of the stars common to ACRS and Hipparcos.

Figure 2 clearly identifies the zonal structure of the systematic differences between ACRS and Hipparcos positions and proper motions. Consequently, the coefficients b₀, b₁, and b₂ in (3) should depend on spherical coordinates, mainly on declination. To account rigorously for the latter, a fairly complex analytical or numerical model would be necessary. Moreover, the number of stars common to ACRS and Hipparcos is insufficient for the statistically reliable determination of the coefficients of an analytical expansion of the differences (e.g. by spherical harmonics). A reasonable approach to correct ACRS systematic errors would thus be to re-reduce the positional catalogues underlying ACRS. Since this time-consuming effort did not fit into the time schedule of the TRC project, a simple numerical method was used:

• A linear model of the type (3) was fitted to the ACRS-Hipparcos differences in position at mean ACRS epoch and in proper motion, both in $5\hbox{$^\circ$}\times 5\hbox{$^\circ$}$ cells, based on the residuals of 85794 astrometric standards of the Hipparcos catalogue identified in ACRS.
• Corrections to the position and proper motion of an individual ACRS star were derived by linear two-dimensional interpolation of the corrections for the three closest cell centres, using Tycho magnitude and colour of the actual star.

The residuals after the systematic correction (Fig. 3) confirm that the major part of the ACRS systematic errors was removed. The residual systematic errors are below 100 mas in position at mean epoch and 1 mas/yr in proper motion. This improved version of the ACRS was used as an intermediate reference catalogue to derive the zone-specific corrections to the Astrographic Catalogue data.

  

Figure 3: ACRS-Hipparcos differences following the systematic corrections (compare with Fig. 2)

3.3 Identification of reference stars on AC plates

  Prior to the plate reduction, it is necessary to identify the images of the reference catalogue stars on the AC plates. The problem is complicated by the difference between the coordinate systems which the measured rectangular coordinates and the standard coordinates are referred to. The two systems may differ by a displacement and a rotation, which for an AC plate may well amount to a few minutes of arc. Thus, to identify the reference stars among the measured images, at least an approximate transformation between the two coordinate systems is necessary. This could in principle be given by the provisional plate constants published along with the AC measurements. However, this approach would lead to a lot of additional efforts to transform the published constants for more than 22000 plates into machine-readable form. Furthermore, this would be unreliable due to typographical and keypunching errors.

To solve the problem in a consistent way, a pattern-matching algorithm was developed (Kuimov 1992). It does not require a transformation between the rectangular coordinate systems of the source (reference stars) and target (measurements) to be known. The algorithm analyzes the triangles formed by stars of the source and target lists in search for pairs of similar triangles (similarity is tested by matching the respective angles, so the approach does not even require source and target coordinate system scales to be the same). The pattern-matching algorithm was successful on all the 22652 Astrographic Catalogue plates, including all sorts of special cases.

3.4 Reduction quality estimators

  As mentioned above, the establishment of an optimal reduction model for the plates of an Astrographic Catalogue zone was based on the analysis of the residuals after adjusting a linear plate model to AC data, by looking for significant additional plate-scale terms. This approach, close to step-wise regression, requires criteria according to which the optimal model is chosen. Since the aim of the AC reduction is to minimize the plate-scale systematic errors, it would be natural to consider that model as optimal which minimizes the so-called reduction error - a combination of the variance of the plate parameters and the residuals resulting from the incompleteness of the plate model. To estimate the reduction error, it is common practice to use criteria based on the (properly normalized) rms scatter of the residuals. For example, under the step-wise regression approach the optimal model is usually chosen on the basis of a Fisher test applied to the rms scatters of the successive models.

The residual scatter of a conventional plate adjustment represents four main factors: the variance of the adopted plate parameters, systematic plate-scale effects unmodelled by the adopted plate model, random measurement errors, and the random errors of the reference star positions at the epoch of observation. Due to the early epoch of the Astrographic Catalogue, the reference catalogue's positional accuracy at the epoch of the plates is comparable to the measurement errors (e.g., about 200 mas for Hipparcos stars). Consequently, these two components dominate the rms scatter, which under such circumstances becomes a fairly insensitive estimate of the reduction quality. Another disadvantage of the residual scatter as a reduction quality estimate is that it is limited to the magnitude interval of the reference stars. Thus its application as an estimate for other, usually fainter, stars is a mere extrapolation.

Fortunately, the two-fold overlap of the Astrographic Catalogue plates provides a natural estimator of the reduction quality - the variance of positions of the same star derived from overlapping plates. To avoid a bias introduced by proper motions, the estimate must be based only on plates with insignificant epoch difference (less than, say, 5 years). Consider a subset of AC plates containing $N_\mathrm{S}$ stars with at least two measurements. Let $N_\mathrm{O}$ denote the total number of measurements, S_i the dispersion of the individual positions of star i, $1\leq i \leq N_\mathrm{S}$, and S the sum of these dispersions over all stars. Then the mean error of an individual position s and its standard deviation $\sigma(s)$ can be estimated as follows (Hudson 1964)

 

$$s^2= \frac{S}{N_\mathrm{O}-N_\mathrm{S}},\ \sigma^2(s) = \frac{s^2}{2(N_\mathrm{O}-N_\mathrm{S})}\cdot$$ (4)

The quantities $s_{\alpha*}$ and $s_{\delta}$, given by Eqs. (4) for the variance in right ascension and declination respectively, were used as a reduction quality estimates. The error of an individual star position includes a random part (measurement error) and a systematic part (reduction error). Since the former is not expected to vary significantly if measurements of the same star on overlapping plates are considered, smaller $s_{\alpha*}$ and $s_{\delta}$ values should correspond to smaller systematic errors. Moreover, due to the complete overlap, the estimates are based on all AC stars observed at least twice, and thus they solve the accuracy estimation problem for faint stars. Finally, the estimates are calculated from measurements of the same star, not from the residuals between measurements and reference star positions, and thus are not directly influenced by the errors of the reference catalogue.

These reasons make the variance of positions from overlapping plates suitable for the assessment of the reduction quality, both for the establishment of the optimal reduction model and for the estimation of the accuracy of an individual star positions after the final reduction step (cf. Sect. 3.7).

3.5 The standard reduction model

Analysis of the residuals after the first step of the reduction procedure suggested the following plate model to be optimal with respect to the minimum systematic errors criterion:  

$$\begin{array} {lcl} f&=&\frac{c_1\xi_1 + c_2\eta_1 + c_3}{... ...} + c_{12}x_1 + c_{13}y_1 + c_{16}by + c_{17}b.\\ \end{array}$$ (5)

The meaning of individual symbols in Eqs. (5) is as follows: $c_1,~\ldots,c_{17}$ - unknown plate parameters; x and y - measured coordinates of a star image in units of mm; b - measure of the star's brightness; $\xi$ and $\eta$ - standard coordinates; $\xi_1=\xi+c_9\xi(\xi^2+\eta^2)$ and $\eta_1=\eta+c_9\eta(\xi^2+\eta^2)$ - distorted standard coordinates. The terms with $x_1=(x+70)\ \mathrm{mod}\ p$and $y_1=(y+70)\ \mathrm{mod}\ p$ are included to account for the systematic errors of the measuring device (see Sect. 2.1.1); p stands for the réseau interval (typically 5 mm, with the only exception of Vatican zone where due to the use of a very special measuring technique p should be set to 10 mm).

The standard coordinates $\xi$ and $\eta$ in Eqs. (5) were computed from the apparent positions of the reference stars. This approach ensures that differential refraction and annual aberration are taken into account prior to plate reduction, so that the plate parameters $c_7,~\ldots,c_{17}$ of Eqs. (5) characterize exclusively the astrograph and measuring device errors.

3.6 Reduction of specific AC zones

To illustrate the reduction approach outlined above, the next few sections cover the reduction of three AC zones: Melbourne, Hyderabad, and Algiers. Of these, the Melbourne zone is described in most detail; for the other two the emphasis is put on the zone-specific features.

3.6.1 The Melbourne zone

  The Melbourne Observatory treated the zone $-90\hbox{$^\circ$}$ to $-65\hbox{$^\circ$}$(declination limits of AC zones are referred to the equinox B1900.0). A description of the observation and measurement procedures can be found in the introduction to the published data (Baldwin 1926). A total of 1149 plates were taken between 1892 and 1940, the mean plate epoch being J1897.9.

The instrument used for taking the plates was a normal astrograph made by "Grubb''. A total of 392584 images were measured with the aid of six different measuring devices, some of micrometer screw and some of eyepiece scale type. The measured coordinates are given to 1 $\mu$m (0$.\!\!^{\prime\prime}$06); the distribution of the digits in the last decimal place is close to uniform. The brightness estimates are given in the form of measured image diameters for bright stars and eye estimates for faint ones.

The residuals after the first step of the reduction procedure (using ACRS as the reference catalogue and a linear plate model) are shown in Figs. 4a and b. Two type of plots illustrate the plate-scale systematic errors and the errors of the secondary coordinate system, respectively.

Figure 5 shows the residuals in right ascension and declination as functions of measured image diameter, with the magnitude equation in right ascension clearly indicated. It is nonlinear and reaches 0$.\!\!^{\prime\prime}$3 for faint stars. A magnitude equation of this type cannot be attributed to ordinary aberrations of the optical system, nor to the atmospheric dispersion (plates were taken close to the meridian, so the latter may cause a magnitude equation in declination only). The most probable cause is a defect of the objective lens, e.g. a decentering, or imperfect guiding.

  

Figure 4: Systematic differences between AC Melbourne and the reference catalogue. Residuals from 1149 plates were averaged in $5 \times 5$ mm cells over the plate (plots a and c), and in $0.5 \times 0.5$ mm cells over the measuring microscope field b) and d). a) and b): residuals after the first step of the reduction procedure, with the AC data reduced onto the ACRS system by linear plate model; c) and d): residuals after the systematic correction and the final reduction onto the Hipparcos system

  

Figure 5: AC Melbourne residuals in right ascension ($\epsilon_{\alpha*}$) and declination ($\epsilon_\delta$), in units of arcseconds, as functions of the brightness measure (measured image diameter). Left: AC Melbourne-ACRS residuals before magnitude equation correction; right: AC Melbourne-Hipparcos after the correction. The curves represent 6th order polynomial fits to the residuals

The second step of the reduction procedure yielded 1146 collections of plate parameters $c_1,~\ldots,c_{17}$. Three plates were excluded from the derivation of the zone-specific corrections due to an insufficient number of reference stars. The plate parameters $c_7,~\ldots,c_{17}$ were subsequently averaged to yield coefficients of the zone-specific corrections; the parameters $c_{10},~\ldots,c_{13}$, describing the secondary coordinate system errors, were averaged for each measuring device.

In the third step, the final reduction on the Hipparcos system was performed using the Turner model and the Hipparcos Catalogue as reference. Corrections for the zone-specific systematic errors derived in step 2 were applied beforehand. The residuals after the final reduction are shown in Figs. 4-5.

3.6.2 The Algiers zone

 

  

Figure 6: The plate parameter c15, characterizing the magnitude equation in right ascension, given for the AC Algiers plates as function of the year of observation

The Algiers Observatory treated the zone $-2\hbox{$^\circ$}$ to $+4\hbox{$^\circ$}$ of declination. Description of the observation and measurement procedures can be found in the introduction to the published data (Trepied 1903). A total of 1260 plates were taken between 1891 and 1912, the mean plate epoch being J1904.2.

The plates were taken with the normal astrograph manufactured by "Henry Brothers''. Coordinates of the 330449 images were measured with the aid of micrometer screw devices, the usual practice for the French AC zones. The measured coordinates are given to 0.1 $\mu$m (0$.\!\!^{\prime\prime}$006). Magnitudes derived from the measured image diameters are given for all stars.

The Algiers observations were found to be among the best with respect to both random and systematic errors. Both plate-scale and measuring machine systematic errors are small and could be corrected very well. The overall magnitude equation in right ascension is close to zero, except for plates taken between 1900 and 1905 (Fig. 6). However, a subdivision of the plates into two sets according to the exposure date did not improve the overall reduction quality as estimated by the position variance criterion. Therefore it was rejected. It should be noted that the same feature was detected in the AC Bordeaux and Oxford data. In the latter case, the change in the magnitude equation may be associated with the adoption of a new emulsion. Unfortunately, the French observatories did not document the emulsion type being used.

3.6.3 The Hyderabad zone

 

  

Figure 7: The cubic distortion coefficient of the Hyderabad Observatory telescope (dots, referred to the left-hand axis) and monthly mean air temperature in Hyderabad (boxes, referred to the right-hand axis), as functions of season. The solid line is described in the text

The Nizamiah Observatory, located at Hyderabad, India, actually observed two zones: $-23\hbox{$^\circ$}$ to $-17\hbox{$^\circ$}$ (referred to as Hyderabad South), and $+36\hbox{$^\circ$}$ to $+39\hbox{$^\circ$}$ (Hyderabad North). The reduction of the southern zone is described in the following, though the results apply to the northern one as well.

A description of the observation and measurement procedures can be found in the introduction to the published data (Pocock 1918). A total of 1260 plates for the Hyderabad South zone were taken between 1914 and 1929, the mean plate epoch being J1919.0. The coordinates of 521867 images were measured with an eyepiece scale device.

The Nizamiah Observatory was the only one of the participating institutions to use a non-standard objective, probably the so-called "Cook Triplet'', with the focal length of 3.37 m. The cubic distortion coefficient c₉ of the Hyderabad telescope was found to depend on the air temperature at exposure time (cf. Fig. 7). The dependence was approximated by a one-year sinusoid, as shown in Fig. 7, to account for the effect. No such feature has been found in any other AC zone.

3.7 Overview of the reduction results

 

  

Table 2: Reduction quality estimates and rms scatter of the residuals (in units of mas) after the first step of the reduction procedure (ACRS as reference catalogue, Turner model, no systematic corrections) and after the third step (Hipparcos as reference catalogue, Turner model, zone-specific systematic corrections applied). The last column denotes whether the final plate model had magnitude-dependent terms added

The plate model given by Eqs. (5) and, for some AC zones, the shortcut without magnitude-dependent terms, yielded the minimum values of the reduction quality estimates $s_{\alpha*}$ and $s_{\delta}$ among all reasonable models tried. Consequently, the AC reduction onto the Hipparcos system relied on this plate model, considered as optimal with respect to the reduction quality estimates. Table 2 lists, for all zones of the Astrographic Catalogue, the reduction quality estimates along with rms scatter of the residuals (in units of mas), both after the first step of the reduction procedure (reduction onto the ACRS system by a linear model without systematic corrections) and after the final reduction onto the Hipparcos system, with zone-specific systematic corrections applied to the AC data.

The differentiation between plate models with and without magnitude-dependent terms is due to the fact that the magnitude equation is by far the most complicated type of systematic error found in the AC data. A noteworthy example of the non-linearity and complexity of the magnitude equation is the one detected in the Melbourne zone (cf. Fig. 5). Also a colour equation, though less pronounced, was found in the AC data.

Our approach to the treatment of the magnitude equation was a compromise between the statistical uncertainty of its determination by the conventional plate adjustment technique and the necessity to eliminate the effect, at least in the most prominent cases. The decision as to which type of plate model given by Eqs. (5) - with or without magnitude-dependent terms - should be adopted for each AC zone was made on the basis of two criteria: the size of the magnitude equation from the AC-Hipparcos residuals, and the comparison of the reduction quality estimates of the two plate models. It should be noted that the two criteria estimate the magnitude equation in distinct magnitude intervals: the comparison with Hipparcos provides information on bright stars, while the reduction quality estimates, being dominated by faint stars, estimate the magnitude equation at the faint end. If the introduction of the magnitude-dependent terms increased the reduction quality, or if a sizeable magnitude equation was found from the comparison with Hipparcos, the magnitude-dependent terms were adopted. The AC zones reduced by plate model (5) with the magnitude-dependent terms are checked in the last column of Table 2.

As can be seen (cf. Table 2), the values for the reduction quality estimates after the first step of the reduction procedure are always lower than those after the third step. This is due to the higher density of the ACRS Catalogue, which leads to smaller variances of the plate parameters and, thus, to smaller values of the reduction quality estimates. On the other hand, the better systematic quality of the reduction onto the ACRS system (as indicated by the smaller values for the reduction quality estimates) is limited to the scale of an AC plate, while on larger scales the systematic errors of the ACRS still persist. Using Hipparcos at the final step of the reduction ensures better systematic quality at the global scale, which is most important for many applications. However, it is achieved at the price of larger systematic errors on the scale of about one degree.

The reduction quality estimates and the rms scatter of the residuals represent two different attempts to assess the accuracy of the derived star positions. In the ideal case of an error-free reference catalogue and equal measurement errors of all stars, the two should coincide. In the real world, the reduction quality estimates may exceed the rms scatter or vise versa, depending on the actual balance between the reference catalogue errors, the plate model completeness and the presence of magnitude equation. The dominant role in both estimates is played by the random errors of the AC observations. However, the rms scatter of the residuals additionally reflects the reference catalogue errors. Accordingly, Table 2 shows that rms scatter with respect to Hipparcos is 1.5 times less than that with respect to ACRS. This is partly due to the superior quality of the Hipparcos proper motions, and partly to the higher correlation of the reduced AC positions of Hipparcos stars with the Hipparcos data. The latter is caused by the significantly smaller number of reference stars per plate.

Reduction quality estimates are based on all stars with two or more observations, and thus these estimates of the accuracy of the reduced AC measurements refer to faint stars. A test computation of $s_{\alpha*}$ and $s_{\delta}$using only Tycho stars (which on average are about one magnitude brighter) resulted in values which are about 30 per cent smaller than those given in Table 2.