5 Merging the samples

Homogeneity and size are two crucial characteristics of a sample, in a statistical sense. In order to gather a sample obeying these two criteria, derived in this paper and in Paper I can be merged with those of Abt & Morrell (1995). The different steps consist of first joining the new data, taking care of their overlap; then considering the intersection with Abt & Morrell, carefully scaling their data to the new ones; and finally gathering the complete homogenized sample.

5.1 Union of data sets from Paper I and this work (I $\cup$ II)

Despite little differences in the observed data and the way were derived for the two samples, they are consistent. The gathering contains 760 stars. Rotational velocity of common stars listed in Table 7 are computed as the mean of both values, weighted by the inverse of their variance. This weighting is carried on when both variances are available (i.e. $\sigma_{{\rm N}}^2$ and $\sigma_{{\rm S}}^2$), except for low rotators and HD 198001, for which $\ensuremath{v\sin i} _{{\rm S}}$ is taken as the retained value.

5.2 Intersection with Abt & Morrell and scaling

In order to adjust by the most proper way the scale from Abt & Morrell's data to the one defined by this work and the Paper I, only non biased should be used. The common subsample has to be cleaned from spurious determinations that are induced by the presence of spectroscopic binaries, the limitation due to the resolution, uncertain velocities of high rotators with no measurement of the Mg II doublet, etc. The intersection gathers 308 stars, and Fig. 11 displays the comparison.

We have chosen to adjust the scaling from Abt & Morrell's data (AM) to ours (I $\cup$ II) using an iterative linear regression with sigma clipping. The least-squares linear fit is computed on the data, and the relative difference

$$\Delta = \left(\ensuremath{v\sin i} _{{\rm I}\cup {\rm II}}-... ...rm AM}}+B)\right)/\ensuremath{v\sin i} _{{\rm I}\cup {\rm II}},$$

where A and B are the coefficients of the regression line, is computed for each point. The standard deviation $\sigma_\Delta$ of all these differences is used to reject aberrant points, using the criterion:

$$\vert\Delta\vert > 1.1~\sigma_\Delta.$$

Then, the least-squares linear fit is computed on retained points and the sigma-clipping is repeated until no new points are rejected. One can see in previous section that points lying one sigma beyond their expected value are already significantly discrepant, this reinforces the choice of the threshold $1.1~\sigma_\Delta$.

The 23 points rejected during the sigma-clipping iterations are indicated in Fig. 11 by open symbols. They are listed and detailed in Appendix C. Some of them are known as spectroscopic binaries. Moreover, using HIPPARCOS data, nine of the rejected stars are indicated as "duplicity induced variable'', micro-variable or double star. Half a dozen stars are low stars observed with AURÉLIE, and the resolution limitation can be the source of the discrepancy

Figure 11:  Comparison of data for the 308 common stars between Abt & Morrell (1995) and the union of data from this work and from Paper I. Filled circles stand for stars from the "cleaned'' intersection, that are used in the fit of Eq. (8), whereas open symbols represent stars discarded from the scaling fit (see text). The different open symbols indicate the possible reason why the corresponding stars are discarded: open square: known spectral binary system; open triangle: variability flag (H52) or binary flag (H59) in HIPPARCOS; open diamond: very low from AURÉLIE data; open circle: no reason. The solid line stands for the one-to-one relation. The dashed line is the fit carried on filled circles. All the discarded objects (open symbols) are listed and detailed in Appendix C.

The "cleaned'' intersection, gathering 285 stars, is represented in Fig. 11 by filled circles. The solid line is the one-to-one relation and the dashed line represents the relation given by the iterative linear fit:

$$\ensuremath{v\sin i} _{{\rm I}\cup {\rm II}} = 1.05~\ensuremath{v\sin i} _{\rm AM}+7.5.$$ (8)

Rotational velocities from Abt & Morrell are scaled to the derived by Fourier transform (union of data sets from Paper I and this work), according to Eq. (8), in order to merge homogeneous data.

5.3 Final merging

Table 9 lists the 2151 stars in the total merged sample. It contains the following data: Col. 1 gives the HD number, Col. 2 gives the HIP number, Col. 3 displays the spectral type as given in the HIPPARCOS catalogue (ESA 1997), Col. 4 gives the derived value of (uncertain , due to uncertain determination in either one of the source lists, are indicated by a colon).

 

 

Table 9: (extract) Results of the merging of samples. Only the 15 first stars are listed below. The whole table is available electronically at the CDS. $\in$ stands for the membership and flags which sample stars belong to: 1, sample from Paper I; 2, sample from this work; 4, sample from Abt & Morrell (1995). This flag is set bitwise, so multiple membership is set by adding values together.

HD	HIP	Spect. type		$\in$
			( )
3	424	A1Vn	228	4
203	560	F2IV	170	4
256	602	A2IV/V	241	5
315	635	B8IIIsp...	81	4
319	636	A1V	59	5
431	760	A7IV	97	4
560	813	B9V	249	1
565	798	A6V	149	1
905	1086	F0IV	36	6
952	1123	A1V	75	4
1048	1193	A1p	28	4
1064	1191	B9V	128	1
1083	1215	A1Vn	233	4
1185	1302	A2V	128	4
1280	1366	A2V	102	4

The are attributed as the mean of available values weighted by the inverse of their variance. Trace of the membership to the different subsamples is kept and listed in Col. 5. The composition in terms of proportions of each subsample is represented as a pie chart in Fig. 12. The catalogue of Abt & Morrell contributes to the four fifths of the sample, and the remaining fifth is composed of new measurements derived by Fourier transforms.

Figure 12: Pie chart of the subsample membership of the stars in the total sample. Multiple membership is represented by superimposed patterns.

The total sample is displayed in Fig. 13a, as a density plot in equatorial coordinates. This distribution on the sky partly reflects the distribution in the solar neighborhood, and the density is slightly higher along the galactic plane (indicated by a dashed line). Note that the cell in equatorial coordinates with the highest density (around $\alpha=5$ h, $\delta = 23~\hbox{$^\circ$ }$) in Fig. 13a corresponds to the position of the Hyades open cluster. The lower density in the southern hemisphere is discussed hereafter in terms of completeness of the sample.

5.4 Completeness

Except for a handful of stars, all belong to the HIPPARCOS catalogue. The latter is complete up to a limiting magnitude $V_{\rm lim}$ which depends on the galactic latitude b (ESA 1997):

$$V_{\rm lim} = 7.9 + 1.1~\sin\vert b\vert.$$ (9)

This limit $V_{\rm lim}$ is faint enough for counts of A-type stars among the HIPPARCOS catalogue to allow the estimate of the completeness of the sample. This sample is north-south asymmetric because of the way it is gathered. Abt & Morrell observed A-type stars from Kitt Peak, and the range of declinations is limited from $\delta = -30~\hbox{$^\circ$ }$ to $\delta = +70~\hbox{$^\circ$ }$, and these limits can be seen in Fig. 13a. Whereas the northern part of the sample benefits from the large number of stars in the catalogue from Abt & Morrell, the southern part mainly comes from Paper I. Thus the completeness is derived for each equatorial hemisphere. Figures 13b and c display the histograms in V magnitude of the sample compared to the HIPPARCOS data, for $\delta > 0~\hbox{$^\circ$ }$ and $\delta < 0~\hbox{$^\circ$ }$ respectively. For both sources, only the spectral interval from B9 to F0-type stars is taken into account. Moreover data are censored, taking V=8.0 as the faintest magnitude.

Figure 13: a) Density of the sample on the sky. Counts over 15$^\circ$$\times$15$^\circ$ bins in equatorial coordinates are indicated by the grey scale. The dashed line stands for the galactic equator. b) and c) represent the counts in magnitude bins of the sample compared to the A-type stars in the HIPPARCOS catalogue for the northern and southern hemisphere respectively.

The completeness of the northern part is 80% at V=6.5 mag. This reflects the completeness of the Bright Star Catalogue (Hoffleit & Jaschek 1982) from which stars from Abt & Morrell are issued. In the southern part, it can be seen that the distribution of magnitudes goes fainter, but the completeness is far lower and reaches 50% at V=6.5 mag. These numbers apply to the whole spectral range from B9 to F0-type stars, and they differ when considering smaller spectral bins. For the A1-type bin for instance, the completeness reaches almost 90% and 70% for the northern and southern hemispheres respectively, at V=6.5 mag.