5 Discussion of errors

5.1 Internal astrometric consistency

The large number of repeated observations of the individual stellar systems (Table 2) allows to estimate very well the internal astrometric and photometric consistencies.

The overall mean standard deviation (s.d.) in separation amounts to $\sigma(\rho) = 0.004\hbox{$^{\prime\prime}$ }$. This corresponds to about 0.01 $\times$the pixel width of the CCD. The small value is, of course, the consequence of the repeated measurements for the program stars and was confirmed by the preliminary results of the other observational runs of the network (Lampens et al. [1997]). The mean s.d. in position angle $\sigma(\theta) = 0.05\hbox{$^\circ$ }$.

Figure 1: Internal errors $\sigma (\rho )$ of the angular separation vs. mean separation $\rho$ and differential magnitude $\Delta V$of the binary components (larger symbols indicate larger $\Delta V$)

Figure 2: Differences between the astrometry in V and i( $\rho (\theta _V-\theta _i)=\rho \Delta \theta _{V,i}$ versus $\Delta \rho _{V,i}$)

Figure 3: Histogram of the mean positional differences rV,i

The dependence of the s.d. in separation $\sigma (\rho )$ from the separation $\rho$ and from the magnitude difference $\Delta V$ is illustrated in Fig. 1. The s.d. in separation between $4\hbox{$^{\prime\prime}$ }$ and $12\hbox{$^{\prime\prime}$ }$ is apparently not much dependent on the separation itself. Only for the systems with $\rho < 4\hbox{$^{\prime\prime}$ }$ and $\rho > 12\hbox{$^{\prime\prime}$ }$ an increase of the s.d. in separation is noticed. The small separations are close to the seeing limit and at larger separation the isoplanicity could be lost, resulting in a small degradation of the results for $\rho > 12\hbox{$^{\prime\prime}$ }$.

Figure 4: Internal errors $\sigma (\Delta V)$ of the differential magnitude vs. mean separation $\rho$ and differential magnitude $\Delta V$ (larger symbols indicate larger $\Delta V$)

The difference between astrometric parameters of our CCD observations in the passbands V and i is a check for the internal consistency. Taking the mean over all binary systems, the separation in the Vband is $0.002\hbox{$^{\prime\prime}$ }$ smaller than in i. The standard deviation of the differences in separation is $0.010\hbox{$^{\prime\prime}$ }$. The mean difference in position angle between V and i is $0.01\hbox{$^\circ$ }$ or $0.0012\hbox{$^{\prime\prime}$ }$ if expressed as $\rho\Delta\theta$ (s.d. is $0.011\hbox{$^{\prime\prime}$ }$). No indications of systematic differences were found, as can be seen in Fig. 2.

The quality of the measurements can also be read off the distances r_V,i as defined in Eq. (1). The value r_V,i is only zero if both, $\rho_V = \rho_i$ and $\theta_V~=~\theta_i$. The overall mean distance $\overline{r_{V,i}}$ is $0.011\hbox{$^{\prime\prime}$ }$. This is in agreement with the value expected according to the error propagation over r_V,i. The distribution of these distances is shown in Fig. 3.

Measurements of the same system on more than one night or in different seasons provide also a very good internal consistency check. The mean of the deviations from the mean value of the separation for each system tabulated in Table 2 is only $0.004\hbox{$^{\prime\prime}$ }$. This indicates again the very high level of accuracy attained in the differential astrometry.

Systems with high internal errors in separation and position angle (not included in the statistics) have almost always separations close to the seeing limit (e.g. HIC18414 and the photometric standard HD214509).

Figure 5: Differences between the Hipparcos and the Network astrometry ( $\rho (\theta _{\rm N}-\theta _{\rm H})=\rho \Delta \theta _{{\rm N,H}}$ versus $\Delta \rho _{{\rm N,H}}$)

5.2 Internal photometric consistency

The internal photometric errors (of the magnitude differences in Vand i) are listed in Table 1. Their overall mean values are small: 0.005mag in $\Delta V$ and 0.006mag in $\Delta i$. Again, the large number of individual observations (especially increasing for stars with increasing $\Delta V$) is the reason of the small errors. Figure 4 displays the internal errors as a function of angular separation $\rho$ and magnitude difference $\Delta V$. No real tendency in one or the other direction can be seen, except that there are some larger errors (exceeding 0.01mag) at small separations.

An independent check on the magnitudes can be done with the data of systems measured more then once (Tables 2 and 4). The mean value of the deviations from the mean of the differential values in V is 0.009 mag (median: 0.006 mag) and 0.016 mag in i(median: 0.006 mag). The standard photometry of the components is very compatible: mean values of the deviations on components A or B, in V, I or V-I are in the range 0.014 to 0.021 mag.

The cited mean values could even be a too high estimate of the errors, since variability for some of the components is not excluded and some systems were measured again only because there was some doubt on the quality of the first observations.

5.3 Comparison with astrometric catalogue data

The discussion of the final comparison of the observations with the HIPPARCOS data will be presented elsewhere (Oblak et al., in preparation). Here we again point out that the calibration of our data was performed with the separations and position angles measured by HIPPARCOS (see Sect. 3). Therefore, the mean of the differences between the HIPPARCOS and the Network astrometry in the $(\rho,\theta)$ plane (see Fig. 5) is zero. The standard deviations of these differences are $0.014\hbox{$^{\prime\prime}$ }$ in separation and $0.18\hbox{$^\circ$ }$ in position angle (or $0.016\hbox{$^{\prime\prime}$ }$ if expressed as $\rho\Delta\theta$), respectively. For the distances $r_{{\rm N,H}}$ between the HIPPARCOS and the Network results, as defined in Eq. (2), we get a mean value of $0.019\hbox{$^{\prime\prime}$ }$, if neglecting the "outliers''. The forgoing errors and distances display again the high quality of the Network results.

In addition we report briefly on the comparison with the HIC annex (Turon et al. [1992], Vol. 6), which contains data mainly from the "Catalogue des Composantes d'Etoiles Doubles et Multiples'' (CCDM, Dommanget & Nijs [1995]). The following statements throw, of course, more light onto the reliability of the data before the "CCD era'' than on the recent observations. In any case one must admit that the astrometric and photometric characteristics may change due to binary motion, proper motion and light variability.

The mean difference between the angular separations CCD minus HIC is $0.1\hbox{$^{\prime\prime}$ }$ (leaving out here and in the following text a few very discordant values, probably caused by misidentifications). The median value is only $0.03\hbox{$^{\prime\prime}$ }$, but the standard deviation on the differences is $0.8\hbox{$^{\prime\prime}$ }$. No systematic effects could be be noticed.

The scatter of the HIC separations around the CCD values is increasing for separations in excess of 8 $^{\prime\prime}$. Likewise the scatter increases for small ( $\Delta V < 1.0$ mag) and large ( $\Delta V > 3.0$ mag) magnitude differences between the double star components.

For several stellar systems the position angle differs by about $180\hbox{$^\circ$ }$. This simply means that the new measurements and the reduction interchanged the components A and B: the former component B is now the brighter one. In the mean, the HIC position angles are the same as the CCD ones (mean difference: $0.1\hbox{$^\circ$ }$, median $0.0\hbox{$^\circ$ }$), but the scatter is very large: about $6\hbox{$^\circ$ }$.

Figure 6: Histogram of the differences in joint magnitudes between CCD photometry and photoelectric photometry (V)

5.4 Comparison with published magnitude data and photoelectric measurements

As sources for the magnitudes of double stars and double star components we used the annex of the HIC (Vol. 6), where component magnitudes are given to a tenth of a magnitude as in the CCDM, and the HIC itself. In the HIC mostly joint photometry is given, but here we can select the most reliable measurements by means of the error bars.

Figure 7: Differences in joint magnitudes between CCD photometry and photoelectric photometry vs. mean separation $\rho$

No systematic differences were found between our CCD measurements and the estimates of the magnitudes of the main components A as listed in the HIC annex. The mean value of the differences is -0.05 mag (median is -0.03 mag), but the scatter is large (0.4 mag) and values range from -1.7 to 1.1 mag. The estimates of the magnitude differences between components A and B ($\Delta V$) differ largely from the CCD measurements: from -1.0 to +1.1 mag with a s.d. of 0.3 mag, if we do not take probable misidentifications into account. The mean of 0.1 mag could be significant but we have no explanation for this.

For a few binary stars of our sample individual photometric data on the A components are now available in the literature, e.g. in the Lausanne General Catalogue of Photometric Data (Mermilliod et al. [1997]) or in the Besançon Data Base of Double and Multiple Stars (Kundera & Oblak [1998]). Some were already given in the HIC, but it is not always clear whether an entry is joint photometry or the A component magnitude. The indication of the sources of the V magnitude in the HIC is not complete: amongst the codes we find D for joint and P for photoelectric. As a consequence we could not always decide whether a photoelectric measurement was joint or only the A component.

We found 14 stars with errors smaller than 0.05 mag where the published V magnitudes refer without doubt to the A components. The source of the magnitudes is mostly photoelectric. The mean of the deviations with our CCD data is only 0.002 mag with a 0.04 mag scatter. Since the mean error on the published magnitudes is about 0.03 mag, this is in agreement with a similar error on our component photometry.

More comparison data are available for the joint photometry, i.e. for magnitudes taken of both components in common with a sufficiently large diaphragm. We found 30 double stars with Geneva (V), 42 with Johnson (V) and 55 with Strömgren (y) joint photometry in our sample. Several double stars have measurements in the three photometric systems with differences of about 0.01 mag.

Figure 8: Differences in joint magnitudes between CCD photometry and photoelectric photometry vs. difference $\Delta V$ in magnitude between components

No systematic difference between the CCD and photoelectric observations could be detected. The mean difference (in the sense CCD minus photoelectric measurements) is 0.010 mag for the Geneva data, -0.009 mag for the Johnson data and -0.007 mag for the Strömgren data. All are close to zero. The standard deviations of the differences are very small: 0.022, 0.030 and 0.022 mag respectively. A histogram with all data combined is given in Fig. 6. If we consider the errors on the published Geneva and Strömgren photometry as negligible and, as a consequence, estimate the error on the Johnson photometry as 0.02 mag, a good estimate for our CCD joint photometry error is 0.022 mag. This proves the high quality of our photometric data.

Although there is no overall dependence on separation, a few cases with angular separations above 6 $^{\prime\prime}$ and large magnitude differences between the components, have fainter photoelectric magnitudes (Figs. 7 and 8). This could be caused by loss of light when the two components were measured photoelectrically in a (too) small diaphragm.