4 Iterative central overlap algorithm

  There are two different widespread approaches to the calculation of proper motions from photographic positions: the plate pairs method and the central overlap method. In the plate pairs technique, all plates are transformed into a common reference frame and, after that, they are paired following some reasonable criteria. Usually, pairs are constructed with plates spanning a wide time baseline and, if possible, having similar accuracies, limiting magnitudes and filters. For each star, the proper motion is calculated once in each plate pair, and the different determinations obtained are averaged. The plate pairs method performs best when the plate material is homogeneous (Schilbach et al. 1995; Tian et al. 1994; Zhao & He 1990).

The central overlap technique was first proposed by Eichhorn & Jefferys (1971). It intends to determine the plate-to-plate transformation parameters, the stars motions and the errors simultaneously, using all the data at the same time. This method has rigorous mathematical foundations (Eichhorn 1988), but its computational requirements are so huge that, in practice, it cannot be implemented in its strict formulation. The usual approach to the method is generally known as iterative central-overlap algorithm (ICOA), and implies the separation of the determination of plate and star parameters in consecutive steps that are iterated until convergence is achieved. This procedure is known to be equivalent, in practice, to the one-step block-adjustment approach (Tucholke 1992), and has been extensively used during the last decades (for instance, Cudworth et al. 1993; Van Altena et al. 1988; Tucholke et al. 1994).

Given the diversity of the plate material used in this study, we decided to use an ICOA for our astrometric analysis. Our implementation, similar to that by Jones & Walker (1988), is summarized in the flow chart of Fig. 3. One plate is selected as reference ("master'' plate). All the other plates ("source'' plates) are tied into the master reference system. Proper motions are computed from different positions at different epochs, and they are fed-back to improve the transformations of source plates to the master plate reference system. The whole process is iterated until the computed proper motions converge. The best modern epoch plate, OCA 3305, was selected as master.

To begin the process, an initial input list with a number of stars cross-identified with the master plate is needed for each source plate. These initial cross-identification files were generated by matching the equatorial coordinates contained in the preliminary astrometric catalogue of each plate (Sect. 2.8).

  

Figure 3: Flow chart of the iterative central overlap algorithm (ICOA) used for the determination of proper motions

4.1 The transformation-crossing loop

  The transformation-crossing loop was applied in turn, one by one, to all plate files. This essential piece of the algorithm consists of two steps: the computation of transformation equations from source plates to the master plate, and the application of these equations to generate cross-identification lists as complete as possible.

The code calculates the transformation coefficients by a least squares fit, with 3$\sigma$ clipping: the fit is performed iteratively, discarding at each step those stars whose residual is larger than 3$\sigma$, until no more stars are eliminated.

The program admits a general plate model of the following form:

 

x' = P_n,x(x,y) + Q_x(x,y,m); (4)

 

y' = P_n,y(x,y) + Q_y(x,y,m). (5)

In these equations, (x',y') are the coordinates on the master reference system; (x,y) are the coordinates on the source plate system; m is the apparent magnitude, and P_n,x(x,y), P_n,y(x,y) are complete two-dimensional polynomials of n-th degree. For instance,

P_2,x(x,y) = a₀ + a₁ x + a₂ y + a₃ x² + a₄ xy + a₅ y².

Q_x(x,y,m) and Q_y(x,y,m) are polynomial functions containing some magnitude-dependent terms.

So, the general model has a purely geometrical part, beside a mixed geometrical-photometric dependence. The specific plate models used in the calculations will be discussed in Sects. 5 and 6.

The cross-identification routine applies the just computed transformation equations to all stars in the source plate and performs a complete cross-identification with the master plate. The cross-identification takes into account a double position-brightness criterion:

1.

For a given master plate star, all the stars in the source plate within a specified search radius are candidates to be cross-identified;

2.

Among the candidates, those stars whose magnitude difference with the master plate star is larger than a specified magnitude tolerance are eliminated, and

3.

From the remaining candidates, the spatially closest is selected.

The search radius and magnitude tolerance are parameters set up for each plate. The search radius is a function of the epoch difference with the master plate. For contemporary plates, a search radius of 5'' was selected. The search radius grows linearly with time in such a way that for an epoch difference of 90 years, it becomes 9''.

The magnitude difference tolerance was set to $3\sqrt{\sigma^2_{\rm master} + \sigma^2}$, where $\sigma_{\rm master}$ and $\sigma$ are the dispersions of the fit performed on the master and source plates for cross-identification purposes (Sect. 3.3), respectively.

The resulting list of cross-identifications was re-introduced into the first element of the loop, in order to compute an improved plate-to-plate transformation. The loop was iterated until the number of cross-identifications in the output list stabilized. This resulted in one file for each source plate, containing the positions of all matched objects for the epoch of the source plate in the reference system of the master plate.

The programs used in the transformation-crossing loop were adapted from C-codes kindly provided by C. Alard, and originally designed for the DUO (Disk Unseen Objects) project (Alard & Guibert 1997).

4.2 Calculation of proper motions

  The set of output plate files in the master plate reference system were compiled. The resulting table was scanned in order to detect non cross-identified bright stars, for which it was impossible to compute a magnitude estimator for cross-identification (Sect. 3.3). Since the raw magnitudes used for cross-identification purposes were in some cases computed by fitting the photographic flux to a filter not similar to the band determined by the plate filter/emulsion, stars with very red or very blue colours could result unmatched. For this reason, the recovery of bright stars extended not only to saturated stars, but also to other unmatched bright stars (down to $V\approx13$ mag).

After the automated recovery of unmatched bright stars, proper motions were computed in the master plate reference system by means of least squares fits of position as a function of epoch:  

$$x_{i,p}-x_{i,{\rm master}} = \xi_{i} + \mu_{i,x} (t_p-t_{\rm master})$$ (6)

 

$$y_{i,p}-y_{i,{\rm master}} =\eta_{i} + \mu_{i,y} (t_p-t_{\rm master})$$ (7)

In these equations, (x_i,p, y_i,p) are the coordinates for star i, measured on plate p and transformed to the master plate reference system. $(x_{i,{\rm master}}, y_{i,{\rm master}})$ are the coordinates for the same star, measured on the master plate. t_p and $t_{\rm master}$ are the epochs of plate p and the master plate, respectively. The calculation was performed only for stars having positions spanning a minimum epoch interval of 30 yr. The fitted parameters are $(\xi_i, \mu_{i,x}; \eta_i, \mu_{i,y})$. $\mu_{i,x}$ and $\mu_{i,y}$ are the proper motions. $\xi_i$ and $\eta_i$ should be understood (Jones & Walker 1988) as corrections to the master plate measured positions.

The index p run over the whole plate set, including the master plate. The fits were performed including a weighting system that took into account the plate scale (Sect. 7.1). The proper motions fits included a 3$\sigma$ clipping. For a given star, after performing a proper motion calculation in both coordinates, the largest residual in x and y was checked. If one or both of them were larger than 3$\sigma$, the plate was removed from both fits for this star and the fit parameters re-computed. The checking and rejection was repeated until all the residuals were smaller than 3$\sigma$. The resulting parameters and standard errors were recorded.

4.3 Second and further iterations

High proper motion stars can distort the reference frame, if their motion is not taken into account in the plate-to-plate transformations. At the end of the first iteration, a search for high proper motion stars was performed through the output list. The stars with modulus of proper motion larger than 6 mas yr^-1 were labelled. All proper motions were set again to zero, and the whole process was repeated. But now, in the second iteration, high proper motion stars were not used for computing the plate-to-plate transformations. However, they were included in the last cross-identification performed in the transformation-crossing loop, and they continued the rest of this and further iterations as any other star.

From the third iteration on, positions of all stars in the master plate were converted to the epoch of each source plate, by applying the parameters ($\xi_i, \eta_i$) and ($\mu_{i,x},\mu_{i,y}$), before entering into the transformation-crossing loop. At this stage, the reference frame is no longer linked only to the physical positions measured on the master plate, but depends on the whole set of astrometric data.

At the end of the third and following iterations, the mean change of the proper motions from the previous to the last iterations was computed, using the formal proper motion errors as a weight. When the average change was well below the mean error, the iterations stopped.