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Astron. Astrophys. 360, 391-398 (2000)

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3. Comparison

3.1. Method

We compare the proper motions in the different catalogues by examining the normalized proper-motion difference in Right Ascension (RA) and Declination (Dec) for each pair of catalogues. We define the normalized proper motion difference as


where [FORMULA] and [FORMULA] are the proper motions in either RA or Dec in catalogues x and y, respectively, and [FORMULA] and [FORMULA] their respective errors. We now introduce [FORMULA] which gives the probability of observing the proper motion [FORMULA] when the real proper motion is [FORMULA] and the real proper-motion error is [FORMULA]. We define [FORMULA] to be a Gaussian with a mean [FORMULA] and a standard deviation [FORMULA]. Similarly we define [FORMULA] to be a Gaussian with a mean [FORMULA] and a standard deviation [FORMULA]. Under the assumption that [FORMULA] and [FORMULA] are independent realizations of the same proper motion we find that the mean of [FORMULA] is


because [FORMULA] (the real proper motion of the same star). For the standard deviation of [FORMULA] we find


which should equal unity when the quoted errors [FORMULA] and [FORMULA] in the catalogues reflect the real errors [FORMULA] and [FORMULA], respectively. However, when the proper-motion errors in catalogue y are over- or underestimated by a factor q, i.e., [FORMULA] (assuming that [FORMULA]), [FORMULA] is not equal to unity and q can be expressed as,


The normalized proper-motion difference distribution [FORMULA] should in principle be a normal distribution with zero mean and unit variance. Any deviations from this distribution are indicative of systematic errors in the catalogues or systematic differences between catalogues. To obtain the mean and standard deviation of the observed [FORMULA] distribution we determine the best fitting Gaussian using a maximum likelihood scheme. Stars deviating more than five times the width (from the 16th to the 84th percentile) of the [FORMULA] were rejected before the fit.

For each pair of catalogues we calculate [FORMULA] for all stars in common between the two catalogues ([FORMULA]), and for all stars in common between the two catalogues which are also contained in the HIP Catalogue ([FORMULA]). To construct [FORMULA] we use the formal errors of the individual stars as quoted in the Catalogues. The comparison is done for each Astrographic Catalogue Declination zone to detect any zonal dependence of the results. Furthermore, to investigate if any magnitude effect is present in [FORMULA], we also divide the sample into four magnitude intervals: (1) all magnitudes, (2) bright stars ([FORMULA] mag), (3) intermediate stars ([FORMULA] mag), and (4) faint stars ([FORMULA]). [FORMULA] is one of the broad-band filters of the Tycho experiment, and is similar to the Johnson B filter.

3.2. Results

Fig. 1 and Fig. 2 show the mean and standard deviation of [FORMULA] resulting from the Gaussian fits. We first discuss the zone-independent results and then report on some peculiar zones.

[FIGURE] Fig. 1. Mean of the normalized proper-motion difference distribution, [FORMULA], resulting from the Gaussian fit (see Sect. 3.1). The zones, denoted by observatory and declination, run from bottom (south pole) to top (north pole). The Potsdam Observatory stopped its work soon after the First World War, and its zone was later reobserved at Oxford, Hyderabad, and Uccle. The catalogues compared are indicated at the top of each panel. The open and filled symbols denote [FORMULA] in RA and Dec, respectively. Bright stars are denoted as triangles ([FORMULA] mag), intermediate stars as squares ([FORMULA] mag),faint stars as pentagons ([FORMULA] mag), and the complete sample as circles. The symbols with arrows indicate values of [FORMULA] which lie outside the plotted range. The dotted line indicates the expected mean of zero.

[FIGURE] Fig. 2. Standard deviation of the normalized proper-motion difference distribution, [FORMULA]. Labels and symbols identical to Fig. 1. The dotted line indicates the expected standard deviation of one. The dramatic trend of the [FORMULA] and [FORMULA] standard deviations versus magnitude indicates that the faint TYC stars have underestimated proper-motion errors.

3.2.1. Underestimation of ACT proper-motion errors

The third panel of Fig. 2 shows that the standard deviation of [FORMULA] is larger than unity for all zones but one. Using [FORMULA] mas yr-1 (HIP proper-motion error) and [FORMULA] mas yr-1 (ACT proper-motion error) as the typical errors in Eq. (4), and assuming the HIP data are correct, the average standard deviation of 1.4 indicates an underestimate of the ACT proper-motion errors, or a fraction of them, by [FORMULA]30%. The standard deviation of [FORMULA] is closest to unity around the equator and increases towards the equatorial poles. The large standard deviation of [FORMULA] is partly due to the shape of the normalized proper-motion difference, which is clearly non-Gaussian (see Fig. 3). The wings of the [FORMULA] distributions are much broader than for a Gaussian distribution. Therefore, we expect that only a fraction of the ACT proper motions or proper-motion errors are incorrect. However, we can not identify which stars have unreliable data. The standard deviation of [FORMULA] shows no magnitude dependencies indicating that this effect is inherent to the construction of the ACT. The shape of the [FORMULA] distribution can, of course, also be due to overestimated proper-motion errors of the stars around the mode of the distribution.

[FIGURE] Fig. 3. This figure shows a typical example of the normalized proper-motion difference distributions, [FORMULA] (the Melbourne zone, [FORMULA]). The labels in the panels indicate the catalogues compared and the number of stars used in the comparison. The black and grey lines indicate [FORMULA] for RA and Dec, respectively. The smooth lines denote the Gaussian fits to [FORMULA].

Since the binaries have been treated differently in the ACT than in TRC construction we investigate wether the large standard deviations might be caused by binary contamination. We remove all HIP and TYC entries with the slightest indication of duplicity 1 from the sample and redo the comparison. We find only marginal differences. The large standard deviations are thus not due to contamination by binaries.

The standard deviations of the [FORMULA] distribution are mostly consistent with unity, except for declinations between [FORMULA] and [FORMULA]. Nine of the 20 zones of the [FORMULA] distribution have standard deviations larger than unity while for the [FORMULA] distribution this is the case for 19 of the 20 zones.

3.2.2. Faint Tycho stars

The [FORMULA] and [FORMULA] standard deviations show a very dramatic trend with magnitude (see Fig. 2). This trend is similar for all zones. Fig. 2 shows that the samples of faint stars, and therefore also the complete samples, have standard deviations on the order of 1.6 whereas the bright and intermediate samples have standard deviations close to unity. The typical errors in the Tycho Catalogue are an order of magnitude larger than those in the TRC and ACT, and therefore dominate the normalized proper-motion difference distribution. This means that the faint stars in the TYC Catalogue have underestimated proper-motion errors up to 40% (Eq. (4)). We do not see these large standard deviations in the faint sample of the [FORMULA] and [FORMULA] distributions. The standard errors of the proper motions given in the Tycho catalogue are known to be underestimates for the faint stars (see ESA 1997: Vol. 1 p. xv and p. 142 and Vol. 4 Sect. 18.5).

3.2.3. Systematic differences between TRC and ACT

The mean values of the [FORMULA] distribution show a large scatter around zero (see Fig. 1). For 13 of the 20 zones the mean of [FORMULA] in either RA or Dec differs by more than 0.1 from zero (see also Fig. 3 for an example). Assuming typical errors of 3 mas yr-1 for both the TRC and ACT, this amounts to more than 0.4 mas yr-1 systematic difference in proper motion between the two catalogues. Some zones show mean values of [FORMULA] as large as 0.3 (i.e., 1.2 mas yr-1 difference). Furthermore, for almost half of the zones the difference between the means of [FORMULA] in RA and Dec differ by more than 0.1 from each other. So not only are the means inconsistent with zero, they are also inconsistent for the two components of the proper motion. The majority of these differences are consistent with the systematic errors quoted in the TRC (less than 1.0 mas yr-1) (see Hog et al. 1998), and we expect that those in the ACT to be of a similar magnitude. There appears to be no systematic trend of the [FORMULA] mean with Astrographic Catalogue zone.

3.2.4. TRC and ACT correlated

The standard deviations for [FORMULA] and [FORMULA] are systematically smaller than unity, with only a few exceptions. This is indicative of a correlation between the proper motions in the catalogues. This comes as no surprise as both catalogues originate from the same material, the Astrographic Catalogue and the Tycho Catalogue, and have been constructed in a similar manner. [FORMULA] and [FORMULA] are independent of magnitude except for the Cape zone (see Sect. 3.2.5).

3.2.5. Peculiar zones

Only two of the 20 zones show some peculiarities. These are the Cape and Vatican zones. The Cape zone is peculiar in that the standard deviations of its [FORMULA] distributions show a magnitude dependence which is not present in any of the other zones. The trend with magnitude is similar to that found for the [FORMULA] and [FORMULA] distributions. The normalized proper-motion difference distributions for other catalogue pairs in the Cape zone do not show any peculiarities.

The Vatican zone is special in the sense that it has large [FORMULA] standard deviations. While most of the zones have [FORMULA] standard deviations close to unity, the standard deviation of the Vatican zone is as large as 1.5. The Vatican zone also has one of the largest [FORMULA] standard deviations. The other catalogue comparisons for the Vatican zone do not show any deviating characteristics. We do not know what caused these peculiarities in these two zones. The median epochs are 1903 for the Cape zone and 1909 or the Vatican zone. The peculiarities can thus not be due to a small epoch difference for these two zones.

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© European Southern Observatory (ESO) 2000

Online publication: July 27, 2000