6 Magnitude effects

  As it is well known, the non-linear response of the detector, and/or an asymmetric point spread function (PSF), can introduce a magnitude-dependent shift in star positions recorded on two-dimensional detectors. Asymmetric PSF profiles can arise from several causes, the most usual being imperfect guiding. The non-linearity of photographic response makes the recorded PSF profile to be different for objects of different brightness. This effect, usually known as magnitude equation, induces magnitude-dependent trends that can be propagated to the final proper motions (Dinescu et al. 1996) and causes serious biases in the analysis of the results, if it is not properly taken into account.

We call inter-plate magnitude equation the magnitude effects involved in plate-to-plate transformations. Our specific treatment is fully discussed in Sect. 6.2. Another kind of magnitude-dependent effect is present in our MAMA data, due to the interaction between asymmetric PSFs with the multi-thresholding processing of the images. This effect, which we call in-plate magnitude equation, is discussed in the next section.

6.1 In-plate magnitude equation

  Let us consider an asymmetric PSF recorded on a photographic plate. When this PSF is sliced at different density thresholds, as was done in our MAMA measurements, the photocenters of the different cut profiles, in general, do not coincide. If a plate is affected by magnitude equation, the positional shift of the photocenters between any two thresholds could be different for stars of different brightness. Since simple visual inspection of several plates seemed to indicate the presence of guiding problems (Sect. 2.3), we decided to check the in-plate magnitude equation and to correct for it in all plates measured with MAMA.

One reference threshold was selected for each plate, trying to maintain the widest dynamical range. The difference of photocenter positions between the reference threshold and any other was plotted as a function of measured flux. The differences were averaged in bins of flux, and a cubic spline drawn through these means was assumed to represent the relative magnitude equation from each threshold to the reference threshold.

In general, in-plate magnitude equations were small. In most cases, the effects remained below $\pm1~\mu$m, reaching 2 $\mu$m at most in the bright end of some thresholds (T 109). In one case (T 6573), the equations reached $\pm2~\mu$m or $\pm3~\mu$m in some flux intervals. OCA plates (and, among them, that selected as master, OCA 3305) showed a virtually null in-plate magnitude equation.

The accepted positions of the stars for each plate were computed by transforming all the thresholds to the magnitude equation system of the reference one, and averaging the star position using the object area in each threshold as a weight. Of course, the correction of in-plate magnitude equation does not remove magnitude effects from the data: it simply reduces them to a common magnitude equation system (that of the reference threshold).

6.2 Inter-plate magnitude equation

  The classical method for dealing with magnitude effects in plate-to-plate transformations, consists in including magnitude-dependent terms in the plate models, as in Eqs. (4) and (5). Terms Q_x(x,y,m) and Q_y(x,y,m) are usually built as non-complete polynomials, with several terms depending only on magnitude (linear and quadratic), and adding some mixed coordinate-magnitude dependent terms if optical aberrations such as coma are important. In some cases, colour-dependent terms have been used in the geometrical transformations.

As indicated by Kozhurina-Platais et al. (1995), this is not the best way of dealing with magnitude effects. Magnitude equations are not well represented by polynomial functions. Furthermore, there exists a real, intrinsic dependence of proper motions on magnitude: bright stars are statistically closer to the Sun and, so, display larger proper motions. This effect could be mistaken by the fit and introduced into the magnitude-dependent terms, eliminating physical information from the data and biasing the results. Last but not least, magnitude equations usually are stronger for bright stars, among which we expect the most important contribution of cluster members (Paper I). These members have a specific kinematical behaviour which should allow to distinguish them from the field population, and which could be misinterpreted and deformed by the classical approach that incorporates magnitude-dependent terms into the plate models.

For the reasons just explained, and following Kozhurina-Platais et al. (1995) and Dinescu et al. (1996), we dealt with inter-plate magnitude equations outside the ICOA. We evaluated and removed magnitude effects from all plates following several steps.

The first step implied the correction of modern plates (and the master plate among them) through comparison with contemporary CCD astrometric measurements. CCD star positions from Paper I are free of magnitude effects, due to the maximum-finding algorithm used to locate stars (FIND algorithm in DAOPHOT, Stetson 1987) and to the linearity of the detector. Being the CCD frames only a few arcmin across, other optical defects, such as coma, are not affecting the CCD astrometry. The comparison of star positions measured on individual CCD fields and on contemporary photographic plates allows us to evaluate and to correct for the magnitude-dependent shift due to magnitude equation. This process is described and applied to modern epoch plates in Sect. 6.2.1.

After that, old epoch plates were compared with the corrected modern epoch ones. Proper motions interfere with magnitude equation in this treatment of old epoch plates. CCD photometry from Paper I and the calculation of preliminary proper motions permitted us to disentangle real kinematical effects from magnitude equation in the treatment of old epoch plates, as described in Sect. 6.2.3.

6.2.1 Modern epoch plates

  Since all modern epoch plates were obtained in a time interval of one year, any differences in star positions among them must be interpreted as caused not by proper motions, but by geometric effects or by magnitude equation. Assuming that the complete 2-D polynomials without photometric dependence for each source plate selected in Sect. 5.1 constitute a good representation of the geometric differences with the master plate, the residuals of the transformations should display only the magnitude equations.

The inspection of these residuals as a function of brightness displays the profile of the relative magnitude equations between each source plate and the master plate (Fig. 7 shows an example). These magnitude equations were represented by discrete points, calculated by averaging the residuals in flux intervals (incorporating a 3$\sigma$ clipping). The magnitude equation was represented by a cubic spline drawn through the points.

After determining and correcting magnitude equation for each modern epoch source plate, the residuals of the transformation to the master plate were checked as a function of colour and the quantities xm and ym, where m is a brightness estimator. No systematic trends were found.

  

Figure 7: The residuals of the transformation from source plate A 550 to the master plate OCA 3305 through a complete 4th degree polynomial without magnitude terms, as a function of flux. The graph permits the determination and removal of the relative magnitude equation among both plates

At this point, all modern plates are affected by the same magnitude equation: that of the master plate. In order to remove this last effect, CCD astrometric data were used. The master plate positions can be directly compared with the CCD positions, but, having a complete set of four good quality plates in the same magnitude equation system, we preferred to average them before comparing with the CCD, in order to reduce the random positional errors.

6.2.2 Master plate magnitude equation and CCD pseudo-plate

  The comparison of the average modern plate with the CCD data was done independently for each CCD field. We took the star positions given by the FIND algorithm of DAOPHOT program (Stetson 1987). Most often, there were two different exposures of each field (long and short, as described in Paper I). In these cases, we adopted the star positions determined in the long exposure, except for saturated stars, whose positions were taken from the short exposures and transformed into the long exposure frames by means of a simple shift of coordinates. The shift was determined using all the stars in common among both frames.

  

Figure 8: Magnitude equation of the master plate, as deduced from comparison of the average modern photographic plate with CCD data

Each CCD field was transformed onto the average modern plate using a complete 2-D polynomial without magnitude-dependent terms. The mean number of stars cross-identified between one CCD field and the average plate was 88. Since the CCD fields slightly overlap, the comparison of the different transformed coordinates obtained for stars in the overlapped regions allowed us to perform a realistic test of the absolute accuracy of the resulting star positions. A total of 735 stars were contained in more than one CCD field. The transformed positions found for 14 of them differed by more than 2 $\mu$m (0.13'') in x, in y or in both coordinates. For the other 721 stars, the differences of their positions displayed a standard deviation of 0.41 $\mu$m (0.027'') in x and in y.

The positions of stars matched in the overlapping areas were averaged, and the whole set of CCD astrometric information was assembled into a CCD pseudo-plate and introduced in the ICOA as one more plate.

Our CCD astrometry is free of magnitude effects. For this reason, the residuals of the transformations from the CCD frames onto the averaged modern photographic plate, when plotted as a function of magnitude, display the magnitude equation of the master plate (Fig. 8). This equation was determined using the same procedure described in Sect. 6.2.1. In this way, our modern epoch plates and master reference system are completely free of magnitude effects.

6.2.3 Magnitude equation of old epoch plates

  Magnitude equation of old epoch plates is determined through comparison with the average modern epoch plate corrected for magnitude effects. For plates with a significant epoch difference with the master plate, proper motions not only introduce a higher dispersion in the plate-to-plate transformations, but also interfere with the determination of magnitude equations, since there exists an intrinsic dependence of proper motion with magnitude. This drawback can be overturned by determining magnitude equations not using the whole set of cross-identified stars, but a subsample of stars known to have an homogeneous kinematical behaviour.

In order to select a kinematically homogeneous sample, we did our best to pick up a set of stars belonging to one of the star clusters present in the surveyed area. We selected as preliminary members those stars from Paper I close to the main sequence seen in the CCD colour-magnitude diagram. The resulting list was cleaned using additional kinematical data from raw proper motions.

The ICOA described in Sect. 4 was applied to derive these raw proper motions. The plate-to-plate transformations incorporated magnitude-dependent terms in quite a classical style. When solving Eqs. (6) and (7), we assigned a fixed weight to each plate, as just the focal length of each instrument (or, equivalently, the inverse of plate scales).

The stars previously selected as preliminary members from a photometric point of view, were analyzed regarding their raw proper motions. After suppressing a few high proper motion stars, the raw proper motions of the remaining objects were very similar in x, but two distinct groups in y, separated by about 5 mas yr^-1, appeared. The spatial distribution of these two groups agreed rather well with what could be expected if the two clusters NGC 1750 and NGC 1758 were real. The stars with raw proper motion similar to that displayed by most of the stars in the area of NGC 1750, were selected to form the kinematically homogeneous sample of preliminary cluster members.

To begin the treatment of old epoch plates, they were grouped into sets of similar epoch and same telescope. Almost in all cases, these groups were formed by just one plate pair, except for AC plates, that were treated as only one set. A geometric transformation was computed to convert the plates in each group into one of them. The residuals of the transformations allowed us to determine and remove the relative magnitude equations inside each group. By averaging the plate sets, we reduced the uncertainty of star positions due to random measurement errors. Each averaged group of old plates was transformed into the average modern plate previously corrected for magnitude effects. The residuals of our kinematically homogeneous sample were analyzed in order to draw the magnitude equation of the old plate group. Figure 9 displays, as an example, the magnitude equation found for the average Palomar plate.

  

Figure 9: Magnitude equation of the average POSS 1461 plate. The residuals of the transformation from POSS plate to the modern average plate are measured in the master plate system. Dots represent all the stars in common, while open circles correspond to the kinematically homogeneous sample used for defining the magnitude equation (solid line)

After correcting for magnitude effects, the residuals of the transformation from the old to the modern plates were also checked as a function of colour, xm and ym. Only the residuals of Astrographic Catalog plates showed a very slight trend with coordinate$\times$magnitude, which could be attributed to the presence of coma. The trend was small, and we decided not to correct for it.