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 and are the proper motions in either RA or Dec in catalogues x and y, respectively, and and their respective errors. We now introduce which gives the probability of observing the proper motion when the real proper motion is and the real proper-motion error is . We define to be a Gaussian with a mean and a standard deviation . Similarly we define to be a Gaussian with a mean and a standard deviation . Under the assumption that and are independent realizations of the same proper motion we find that the mean of is
which should equal unity when the quoted errors and in the catalogues reflect the real errors and , respectively. However, when the proper-motion errors in catalogue y are over- or underestimated by a factor q, i.e., (assuming that ), is not equal to unity and q can be expressed as,
The normalized proper-motion difference distribution 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 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 were rejected before the fit.
For each pair of catalogues we calculate for all stars in common between the two catalogues (), and for all stars in common between the two catalogues which are also contained in the HIP Catalogue (). To construct 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 , we also divide the sample into four magnitude intervals: (1) all magnitudes, (2) bright stars ( mag), (3) intermediate stars ( mag), and (4) faint stars (). is one of the broad-band filters of the Tycho experiment, and is similar to the Johnson B filter.
3.2.1. Underestimation of ACT proper-motion errors
The third panel of Fig. 2 shows that the standard deviation of is larger than unity for all zones but one. Using mas yr-1 (HIP proper-motion error) and 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 30%. The standard deviation of is closest to unity around the equator and increases towards the equatorial poles. The large standard deviation of is partly due to the shape of the normalized proper-motion difference, which is clearly non-Gaussian (see Fig. 3). The wings of the 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 shows no magnitude dependencies indicating that this effect is inherent to the construction of the ACT. The shape of the distribution can, of course, also be due to overestimated proper-motion errors of the stars around the mode of the distribution.
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 distribution are mostly consistent with unity, except for declinations between and . Nine of the 20 zones of the distribution have standard deviations larger than unity while for the distribution this is the case for 19 of the 20 zones.
3.2.2. Faint Tycho stars
The and 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 and 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 distribution show a large scatter around zero (see Fig. 1). For 13 of the 20 zones the mean of 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 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 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 mean with Astrographic Catalogue zone.
3.2.4. TRC and ACT correlated
The standard deviations for and 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. and 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 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 and 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 standard deviations. While most of the zones have 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 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.
© European Southern Observatory (ESO) 2000
Online publication: July 27, 2000