6 Discussion of the errors

   
6.1 Internal errors on the astrometry

Because each frame was repeated at least three times, we provide statistics on the internal consistency of the data (in the form of standard deviations). A first example is shown in Fig. 1 where the internal errors on angular separation have been plotted as a function of angular separation for two different ranges of $\Delta V$. There is a very clear dependence on both angular separation and difference of magnitude in the sense that the mean error is larger for decreasing separation and increasing differential magnitude while it increases slightly again at the largest separations. The mean error for all 253 data is $0.0073 \hbox{$^{\prime\prime}$ }$ with a standard deviation of $0.0122 \hbox{$^{\prime\prime}$ }$ This is somewhat higher than expected (Lampens et al. 1997): the reason is the significant amount of systems with small separation and large magnitude difference. After having selected 217 data with angular separation $\geq$ $3 \hbox{$^{\prime\prime}$ }$ or $\Delta V \leq 1.5$ mag, the mean error becomes $0.0057 \hbox{$^{\prime\prime}$ }$ with a standard deviation of $0.0113 \hbox{$^{\prime\prime}$ }$. Similarly the mean error in position angle decreases from $0.167 \hbox{$^\circ$ }$ to $0.076 \hbox{$^\circ$ }$ while the standard deviation decreases from $0.561 \hbox{$^\circ$ }$ to $0.167 \hbox{$^\circ$ }$. Internal errors are much larger in the case of systems with angular separation < $3 \hbox{$^{\prime\prime}$ }$ and $\Delta V > 1.5$ mag: the mean error in separation for the 36 remaining systems is $0.0171 \hbox{$^{\prime\prime}$ }$ and the mean error in position angle is then $0.719 \hbox{$^\circ$ }$. We may safely state that mean astrometric errors vary from pixel width $\times 0.01$ in the best cases to pixel width $\times 0.05$ in the worst ones.

Internal errors on the astrometry may also be discussed from a consistency check between the observation sequences made in several filters. We verified the assumption that relative positions are independent from the filter choice by computing the "distance'' d, i.e. the difference in relative positions for the filters V and i, as

$$d = \sqrt{\rho_{V}^2 + \rho_{i}^2 - 2 \rho_{V} \rho_{i} \cos(\theta_{V} - \theta_{i})}.$$ (2)

Figure 2 shows the distribution of these values d where 225 data are represented with a mean difference of $0.022 \hbox{$^{\prime\prime}$ }$ and a standard deviation of $0.029 \hbox{$^{\prime\prime}$ }$. After elimination of 15 systems with a separation smaller than 3 $^{\prime \prime }$  and a differential i (and V in most cases) magnitude larger than 2 mag, the mean value decreases to $0.018 \hbox{$^{\prime\prime}$ }$ with a standard deviation of 0.018 $^{\prime \prime }$. This is larger than the mean internal error but probably reflects the true error of our measurements. This peak value is also larger by about $0.007 \hbox{$^{\prime\prime}$ }$ than the one determined in Paper II, again principally due to the significant number of double stars of separation smaller than 2 $^{\prime \prime }$  in our sample (we count 39 such systems, 15 of which have $\Delta V > 1.5$ mag, compared to 4 in Paper II). The largest inconsistencies are found for the more difficult configurations of smaller separation and larger magnitude difference: this is reflected by a mean difference of $0.084 \hbox{$^{\prime\prime}$ }$ and a standard deviation of $0.061 \hbox{$^{\prime\prime}$ }$ for the 15 previously mentioned data points. We also verified whether any systematics occurs between angular separations measured in both filters. A simple statistical test shows that the distribution for 207 data points is different from a normal distribution with zero mean at the 0.05 significance level ( $\overline{(\rho_{V}-\rho_{i})}$ of $-0.0025\hbox{$^{\prime\prime}$ }$; $\sigma$ of $0.018 \hbox{$^{\prime\prime}$ }$), in the sense that separations appear generally somewhat larger in the filter i: this effect is best seen for separations larger than about 5 $^{\prime \prime }$ (Fig. 3). We tried to find some explanation for this. For example, we have verified that the effect is:

• independent from the observer;
• not correlated with seeing or differential refraction;
• not due to a bad focusing in one of the filters in a systematic way;
• not correlated with the colour difference of the two components.

Therefore we did not introduce any correction. This will be checked further in the results of other campaigns.

  
6.2 External errors on the astrometry

Some programme and astrometric standard stars were observed twice or more (see Table 3). As mentioned before, these regularly observed double stars are important to check the consistency between observations obtained at different epochs. There is indeed an excellent agreement between the measurements of the same star as can be deduced from the small standard deviations both in angular separation and in position angle. For example, we can check the wide astrometric pairs with observations obtained at different epochs: BD +13 $\hbox{$^\circ$ }$3203 (+1303203a, n=5), BD -22 $\hbox{$^\circ$ }$1505 (32144a, n=4), BD -56 $\hbox{$^\circ$ }$256 (5843a, n=7) in Table 3. Their standard deviations fluctuate between 0.003 to 0.005 $^{\prime \prime }$ in angular separation and between 0.01 to 0.02$^\circ$ in positional angle.

  
6.3 Internal errors on the photometry

Figure 3: Differences in angular separation between filters V and i. Unfilled diamonds represent the 15 systems with $\rho < 3$ $^{\prime \prime }$ and $\Delta i > 2$ mag.

Figure 4: Internal errors on mean differential V magnitude vs. angular separation for two classes of $\Delta m$.

We discuss here both internal errors: those on the differential photometry only and those of the absolute photometry (obtained through calibration of standard stars). In the former case, the internal photometric errors depend on the repeatability of the differences of the component magnitudes. The internal consistency of these differences can be assessed by inspection of the standard deviations listed in Table 3. Figure 4 represents the distribution of the internal errors on the differential V magnitude plotted as a function of angular separation for two different ranges of $\Delta V$. The mean internal error is well below 0.01 mag but there is evident degradation at larger differential magnitudes. In the worst case of a combination of a small separation ($\rho$ < 3 $^{\prime \prime }$) and a large difference of magnitude ( $\Delta {V} > 2$ mag), this error tends to increase to a few tenths of a magnitude. A similar figure applies to the differential I magnitudes. The multiply observed astrometric standard stars BD +13 $\hbox{$^\circ$ }$3203 (n=5), BD -22 $\hbox{$^\circ$ }$1505 (n=4) and BD -56 $\hbox{$^\circ$ }$256 (n=7) from Table 3 show the same tendency: we find millimag consistency on the differential magnitudes in the case of BD +13 $\hbox{$^\circ$ }$3203 ( $\Delta V < 1$), some hundreths of a magnitude for $1 < \Delta V < 4$ and more than 0.1 mag in the case of BD -56 $\hbox{$^\circ$ }$256 ( $\Delta i > 5$) shown hereafter to be variable. In the latter case - under favourable photometric conditions - several standard stars have been observed to which classical colour equations have been applied. We have taken into consideration a transformation error (generally 0.02-0.03 mag) depending on the quality of each night as well as an error (usually insignificant) on the joint magnitude of the system to compute the individual errors on the component magnitudes which are also listed in Table 4 (Paper I, Sect. 4.3). The mean errors on the magnitudes and the indices (V-I) of the components A and B deduced from Table 4 respectively give the following values: $\sigma_{V_{\rm A}} = 0.006$, $\sigma_{V_{\rm B}} = 0.009$, $\sigma_{(V-I)_{\rm A}} = 0.009$ and $\sigma_{(V-I)_{\rm B}} = 0.024$ mag. For the sample of binaries with $\rho$ < 3 $^{\prime \prime }$ and $\Delta m \geq 2$, these values rise to $\sigma_{V_{\rm A}} = 0.020$, $\sigma_{V_{\rm B}} = 0.027$, $\sigma_{(V-I)_{\rm A}} = 0.027$ and $\sigma_{(V-I)_{\rm B}} = 0.041$ mag.

  
6.4 External errors on the photometry

 

 

Table 5: CCD I photometry for 4 Hipparcos and 1 astrometric standard double stars.

Identifier	Jul. Dat.	nI	$I_{\rm A}$	$\sigma I_{\rm A}$	$I_{\rm B}$	$\sigma I_{\rm B}$	$\Delta I$
	2440000+		(mag)	(mag)	(mag)	(mag)	(mag)
005843a	9348.4085	3	5.904	.017	11.173	.061	5.269
079902	8851.5582	16	7.816	.020	10.801	.020	2.985
088603	8850.6143	13	8.658	.020	10.720	.020	2.062
109156	8850.7526	4	9.102	.020	12.706	.020	3.604
117316	8850.8339	11	9.647	.020	12.601	.020	2.954

 

 

Table 6: Comparison of CCD VI photometric observations: our data (LO) versus Cuypers & Seggewiss (1999) (CS).

Identifier	Jul. Dat.	nV	$V_{\rm A}$	$\sigma_{V_{\rm A}}$	$V_{\rm B}$	$\sigma_{V_{\rm B}}$	nI	$(V-I)_{\rm A}$	$\sigma_{(V-I)_{\rm A}}$	$(V-I)_{\rm B}$	$\sigma_{(V-I)_{\rm B}}$	$\Delta V$	$\Delta$(V-I)	Code
011219	9668.7130	10	8.864	0.020	10.129	0.020	0	-	-	-	-	1.265	-	LO
011219	8549.7389	4	8.842	0.016	10.132	0.016	4	0.969	0.017	0.630	0.021	1.290	-0.340	CS
022463	8941.7542	3	8.958	0.015	9.763	0.016	2	0.51	0.02	0.64	0.02	.805	0.13	LO
022463	8550.7998	6	8.960	0.014	9.770	0.014	3	0.539	0.017	0.661	0.017	0.810	0.122	CS
042581a	9345.6621	3	7.738	0.020	9.910	0.025	0	-	-	-	-	2.172	-	LO
042581a	8671.6577	6	7.712	0.016	9.933	0.024	3	0.734	0.018	0.734	0.027	2.221	0.000	CS
114378a	8850.7841	3	6.543	0.020	10.275	0.021	1	0.60	0.03	1.56	0.03	3.731	0.97	LO
114378a	8548.6123	16	6.572	0.011	10.276	0.020	13	0.594	0.019	1.564	0.017	3.704	0.970	CS

External photometric errors can be evaluated from observations of the same target acquired during different nights or different campaigns. For this we consider the multiple observations of some astrometric standard stars (BD +13 $\hbox{$^\circ$ }$3203 (+1303203a, n=5), BD -22 $\hbox{$^\circ$ }$1505 (32144a, n=2), BD -45 $\hbox{$^\circ$ }$13443 (97593a, n=2)) as well as some programme stars (HIP 2438, HIP 20020) listed in Table 4. In addition, we also consider the objects in common with Paper II (see Table 6). The case of BD -56 $\hbox{$^\circ$ }$256 (5843a, n=5) is not included since we note significant variations between individual measurements as well as a difference of 0.1 mag between our measurements and those reported in Paper II. This confirms the variability detected by Hipparcos for this red giant of spectral type M2/3III. Taking into consideration the nine objects previously cited, we find $\sigma_{V_{\rm A}} = 0.025 \pm 0.029$ and $\sigma_{V_{\rm B}} = 0.032 \pm 0.037$ mag. If we remove only one star (HIP 20020) we obtain $\sigma_{V_{\rm A}} = 0.016 \pm 0.012$ and $\sigma_{V_{\rm B}} = 0.017 \pm 0.021$ mag. We thus are confident that the errors quoted in Table 4 are very realistic upper limits.

  
6.5 Comparison with ground-based photometry

Figure 5: Difference between V component magnitudes in the sense CCD minus photoelectric versus angular separation.

Figure 6: Difference between V component magnitudes in the sense CCD minus photoelectric versus differential magnitude.

Figure 7: Difference between total $V_{\rm A+B}$ magnitudes in the sense (CCD minus photoelectric) versus angular separation for two classes of $\Delta m$ (filled symbols are used for $\Delta V > 1.5$  mag).

Figure 8: Histogram of differences between total $V_{\rm A+B}$ magnitudes in the sense (CCD minus photoelectric) for $\Delta V < 1.5$  mag only.

We searched existing data bases for component photoelectric photometry. We have systematically searched in the UBV, Strömgren and Geneva photometries since these systems have the highest probability of success. Apart from the component photometry of the Double and Multiple Systems Annex (vol. 10 of the Hipparcos Catalogue, ESA 1997), we found individual information for 11 primary and 5 secondary components of our sample only making use mostly of the Lausanne Photometric data base (Mermilliod et al. 1996) and for some systems of the Besançon Double and Multiple Star data base (Kundera et al. 1999). In Figs. 5 and 6 we illustrate the comparison as a function of angular separation and magnitude difference: HIP 3397 ( $\rho \simeq$ 14 $^{\prime \prime }$ and colours of a giant) has a small deviation for the primary but a large difference (>0.1 mag) for the secondary component; the primary component of HIP 9258 ( $\rho \simeq$ 9 $^{\prime \prime }$) turned out to be a variable star (a discordant previous measurement had mistakenly been attributed to the B companion) so the deviation of 0.05 mag is not surprising. Except for HIP 113386 ( $\rho \simeq$ 9 $^{\prime \prime }$) where both Strömgren and Geneva values are off by as much as 0.08 mag (no reason found); the same effect is also seen on the difference between the joint V Johnson and CCD magnitude), the agreement is excellent for the primaries, with a mean deviation of +0.006 mag and a scatter of 0.022 mag only. In general the agreement is really good for the primary star but worse on the secondary component. The published mean error is of the same order as the scatter found in the differences between the CCD and the photoelectrically measured component magnitudes ($\simeq$0.02 mag). Even though the data are few, there is a possible trend when one considers both figures together: the discrepancies are more frequent at large separations and large differences of magnitude.

More comparison data are available for the combined photometry of these systems: 122 double stars have either UBV or Strömgren or Geneva combined photometry. We have 62 common pairs with Johnson, 70 with Strömgren and 64 with Geneva photometry. Total CCD magnitudes have been recomputed from standard component magnitudes. A histogram of the differences is shown in Fig. 10 where the gray zone refers to the differences with the Strömgren photoelectric photometry. After removal of the "outlier'' cases at the 3-$\sigma$ level for which the differences are larger than 0.1 mag in absolute value (including 10 different objects), the mean deviations and scatters are:

• for 58 CCD minus Johnson V magnitudes: (+0.0097, 0.033);
• for 63 CCD minus Strömgren y magnitudes: (+0.0090, 0.036);
• for 61 CCD minus Geneva V magnitudes: (+0.0118, 0.031).

Consistent features are a mean difference of +0.01 mag, slightly more pronounced for the Geneva system, and a scatter of 0.03 mag that perfectly matches the abovecited error distributions. In contrast to Paper II, we detect a small systematic difference of $\simeq$0.01 mag in the sense that our CCD magnitudes tend to be somewhat fainter than the photoelectric ones, even though it must be said that our errors are a little bit larger (due to different photometric conditions). Following the same reasoning as in the former paper, if we adopt a mean error of 0.02 mag for the photoelectric magnitudes, we find a typical error of 0.025 mag for our combined CCD magnitudes. This shows that our CCD component magnitudes are generally reliable to within much better than 0.03 mag and successfully competing with precise photoelectric photometry. Figures 7 and 9 again illustrate the differences in the sense CCD - photoelectric, this time as a function of separation and differential magnitude. The distribution of the differences in these figures does not look as expected from our previous considerations: several large deviations can be noted. However, different effects are in play here and we will discuss each one in turn. Let us first consider the largest differences found a) at values >0.1 mag and b) at values <-0.1 mag. In Figs. 7 and 9, from the nine values > 0.1 mag, two are flagged by larger errors than usual in the CCD data (HIP 10722 (red giant), 31042). One is clearly wrong (HIP 110654; $\rho=12.6\hbox{$^{\prime\prime}$ }$). The remaining data show no problem of data reduction and concern three systems with separations between 4 and 6 $^{\prime \prime }$  (HIP 13199, 85685, 116737). We do not discard that the reason of this discrepancy could lie in our CCD data. From the 12 cases with values <-0.1 mag (only nine plotted between -0.1 and -0.3 mag in Fig. 9), one has a flag indicated by larger errors than usual in the CCD data (HIP 13815; $\rho=1.7\hbox{$^{\prime\prime}$ }$), one has a separation very close to the seeing limit (HIP 33499 with red colours has $\rho=1.7\hbox{$^{\prime\prime}$ }$). From the remaining cases, three have separations at or larger than 10 $^{\prime \prime }$. Although no immediate explanation can be found for the other cases (HIP 20020, 23480, 26401, 31833, 34898 and 93521) with separations between 3 and 7 $^{\prime \prime }$, we are confident that the reason does not lie in the CCD data but in the way how photoelectric photometry was performed on these systems: joint photoelectric magnitudes are fainter because part of the light of both components is lost when measured in a diaphragm too small for the given pair separation. This effect is found to be most pronounced when the companion has $\Delta m$ around or larger than 1 mag and for separations which are significant compared to the used diaphragm size. This may be the cause for the important negative deviations noted for double stars with separations above $\simeq$5 $^{\prime \prime }$. Next we discuss the gross of the data with differences situated between -0.1 and 0.1 mag in both figures. We have seen that the general agreement is of order 0.02-0.03 mag. This is reflected by the scatter in the data with $\Delta V < 1.5$ mag which appears to show a bi-modal distribution centered on two values: $\simeq$0.00 mag (Strömgren system, to a lesser extent Geneva system) and $\simeq$0.03 mag (Johnson and Geneva systems, to a lesser extent Strömgren system) in the histogram presented in Fig. 8 (whereas this effect was seen only in the differences with the Geneva photometry (Paper II)). If we now consider the region with $\Delta V > 1$ mag in Fig. 9, we see a tendency of larger negative deviations in the range $1.5 < \Delta V < 3$ mag associated with angular separations above 5 $^{\prime \prime }$. The same effect plays a role in the few cases having $1.5 < \Delta V < 3$ mag and $\rho > 6$ $^{\prime \prime }$ in Paper II: it can be explained by the loss on the system's total light when the two components are simultaneously measured in a diaphragm whose size is about the size of the system's angular separation (Oblak et al. 1997). We thus repeat that 0.03 mag is a conservative upper limit of the mean error of the CCD V magnitudes.

Figure 9: Same data as in Fig. 7: difference between total $V_{\rm A+B}$ magnitudes in the sense CCD minus photoelectric versus differential magnitude for two classes of $\rho$ (filled symbols are used for $\rho > 6$ $^{\prime \prime }$).

Figure 10: Histogram of differences between total $V_{\rm A+B}$ magnitudes.