A&A 393, 897-911 (2002)
DOI: 10.1051/0004-6361:20020943

Rotational velocities of A-type stars[*],[*]

II. Measurement of \ensuremath{v\sin i} in the northern hemisphere

F. Royer1,2 - S. Grenier2 - M.-O. Baylac2 - A. E. Gómez2 - J. Zorec3

1 - Observatoire de Genève, 51 chemin des Maillettes, 1290 Sauverny, Switzerland
2 - GEPI/CNRS FRE 2459, Observatoire de Paris, 5 place Janssen, 92195 Meudon Cedex, France
3 - CNRS, Institut d'Astrophysique de Paris, 98bis boulevard Arago, 75014 Paris, France

Received 25 February 2002 / Accepted 19 June 2002

This work is the second part of the set of measurements of \ensuremath{v\sin i} for A-type stars, begun by Royer et al. (2002). Spectra of 249 B8 to F2-type stars brighter than V=7 have been collected at Observatoire de Haute-Provence (OHP). Fourier transforms of several line profiles in the range 4200-4600 Å are used to derive \ensuremath{v\sin i} from the frequency of the first zero. Statistical analysis of the sample indicates that measurement error mainly depends on \ensuremath{v\sin i} and this relative error of the rotational velocity is found to be about 5% on average.
The systematic shift with respect to standard values from Slettebak et al. (1975), previously found in the first paper, is here confirmed. Comparisons with data from the literature agree with our findings: \ensuremath{v\sin i} values from Slettebak et al. are underestimated and the relation between both scales follows a linear law $\ensuremath{v\sin i} _{\rm new} = 1.03~\ensuremath{v\sin i} _{\rm old}+7.7$.
Finally, these data are combined with those from the previous paper (Royer et al. 2002), together with the catalogue of Abt & Morrell (1995). The resulting sample includes some 2150 stars with homogenized rotational velocities.

Key words: techniques: spectroscopic - stars: early-type - stars: rotation

1 Introduction

This paper is a continuation of the rotational velocity study of A-type stars, initiated in Royer et al. (2002, hereafter Paper I). The main goals and motivations are described in the previous paper. The sample of A-type stars described and analyzed in this work is the counterpart of the one in Paper I, in the northern hemisphere.

In short, it is intended to produce a homogeneous sample of measurements of projected rotational velocities ( \ensuremath{v\sin i}) for the spectral interval of A-type stars, and this without using any preset calibration. This article is structured in a way identical to the precedent, except for an additional section (Sect. 5) where data from this paper, the previous one and the catalogue of Abt & Morrell (1995) are gathered, and the total sample is discussed in statistical terms.

2 Observational data

Spectra were obtained in the northern hemisphere with the AURÉLIE spectrograph (Gillet et al. 1994) associated with the 1.52 m telescope at Observatoire de Haute-Provence (OHP), in order to acquire complementary data to HIPPARCOS observations (Grenier & Burnage 1995).

The initial programme gathers early-type stars for which \ensuremath{v\sin i} measurement is needed. More than 820 spectra have been collected for 249 early-type stars from January 1991 to May 1994. As shown in Fig. 1, B9 to A2-type stars represent the major part of the sample (70%). Most of the stars are on the main sequence and only about one fourth are classified as more evolved than the luminosity class III-IV.

These northern stars are brighter than the magnitude V=7. Nevertheless, three stars are fainter than this limit and do not belong to the HIPPARCOS Catalogue (ESA 1997). Derivation of their magnitude from TYCHO observations turned out to be: HD 23643 V=7.79, HD 73576 V=7.65 and HD 73763 V=7.80. These additional stars are special targets known to be $\delta $ Scuti stars.

\includegraphics[width=8.8cm,clip]{MS2413f1} \end{figure} Figure 1: Distribution of the spectral type for the 249 programme stars.
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AURÉLIE spectra were obtained in three different spectral ranges (Fig. 2):

Two thirds of the sample have observations in each of the three ranges. The $\Lambda _3$ range is particularly aimed at \ensuremath{v\sin i} measurement, and contains the largest number of lines selected for this purpose (twice as much as $\Lambda _1$ and $\Lambda _2$). Besides, it is the only one which covers the magnesium doublet at 4481 Å. This line remains often alone for measurement in fast rotators. It is thus significant to note that among the 249 stars of the sample, three were observed only in the $\Lambda _1$ range, eight only in $\Lambda _2$ and eleven in both $\Lambda _1$ and $\Lambda _2$ only. Overall 22 stars have no observation in $\Lambda _3$. The reason is that these stars have already known \ensuremath{v\sin i}. Moreover they are effective temperature standard stars or reference stars for chemical abundances. Some changes in the configuration of the instrument meant the central wavelengths of the $\Lambda _2$ and $\Lambda _3$ domains have been slightly modified during mission period. The entrance of the slit is a 600 $\mu$m hole, i.e. 3 $^{\prime\prime}$ on the sky, dedicated to the 1.52 m Coudé telescope. The dispersion of the collected spectra is 8.1 Å  ${\rm mm}^{-1}$ and the resolving power is about 16 000. The barrette detector is a double linear array TH 7832, made of 2048 photo-diodes. Reduction of the data has been processed using MIDAS[*] procedures. Flat field correction with a tungsten lamp and wavelength calibration with a Th-Ar lamp have been made with classical procedures. Nevertheless, a problem occurred when applying the flat-field correction with the tungsten calibration lamp. The division by the W lamp spectrum produced a spurious effect in the resulting spectrum at a given position in the pixels axis. This effect distorts the continuum, as it can be seen on the spectra of Vega (Fig. 2). This problem triggered the decision to change the instrumental configuration and central wavelengths of the spectral ranges.

\includegraphics[width=11.2cm,clip]{MS2413f2.eps} \end{figure} Figure 2:  Observed spectra of Vega are displayed for the different spectral ranges: top panel, range $\Lambda _1$; middle panel, range $\Lambda _2$; bottom panel, range $\Lambda _3$. Each domain covers nearly 200 Å. $\Lambda _1$) the first range extends from the red wing of H$\delta $ to the blue wing of H$\gamma $ which restricts the reliable normalization area. It contains seven of the selected lines. $\Lambda _2$) centered around H$\gamma $, this range only contains five selected lines. $\Lambda _3$) this range contains the largest number of selected lines, 16 in total, among which the doublet line Mg II 4481. The 23 selected lines (listed in Table 2) are indicated, and show up twice in the overlap areas. The instrumental feature coming from flat-fielding lamp is noticeable in the three spectra ($\sim $4280 Å in $\Lambda _1$, $\sim $4390 Å in $\Lambda _2$, $\sim $4560 Å in $\Lambda _3$).
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3 Measurement of the rotational velocity

The method adopted for \ensuremath{v\sin i} determination is the computation of the first zero of Fourier transform (FT) of line profiles (Carroll 1933; Ramella et al. 1989). For further description of the method applied to our sample, see Paper I. The different observed spectral range induces some changes, which are detailed below.

3.1 Continuum tracing

The normalization of the spectra was performed using MIDAS: the continuum has been determined visually, passing through noise fluctuations. The procedure is much like the normalization carried out in Paper I, except for a different spectral window. For the ranges $\Lambda _1$ and $\Lambda _2$, the influence of the Balmer lines is important, and their wings act as non negligible contributions to the difference between true and pseudo-continuum, over the major part of the spectral domain, as shown in Paper I. On the other hand, the $\Lambda _3$ range is farther from H$\gamma $. In order to quantify the alteration of continuum due to Balmer lines wings and blends of spectral lines, a grid of synthetic spectra of different effective temperatures (10 000, 9200, 8500 and 7500 K) and different rotational broadenings, computed from Kurucz' model atmosphere (Kurucz 1993), is used to calculate the differences between the true continuum and the pseudo-continuum. The pseudo-continuum is represented as the highest points in the spectra. The differences are listed in Table 1, for different spectral 20 Å wide sub-ranges. This table is a continuation of the similar one in Paper I, considering the spectral range 4200-4500 Å.


Table 1: Differences between the true continuum and the highest points in different spectral bands for the set of synthetic spectra in the $\Lambda _3$ domain. Wavelength indicates the center of the 20 Å wide range.

\ensuremath {T_{\rm eff}}, \ensuremath{v\sin i}
central wavelength (Å)
(K, \ensuremath{{\rm km}~{\rm s}^{-1}}) 4510 4530 4550 4570 4590

Data for wavelengths shorter than 4500 Å
are given in Table 1 of Paper I
10 000, 10 0.0005 0.0003 0.0002 0.0000 0.0000
10 000, 50 0.0008 0.0003 0.0003 0.0002 0.0003
10 000, 100 0.0011 0.0005 0.0016 0.0005 0.0013


10 0.0010 0.0006 0.0006 0.0006 0.0006
9200, 50 0.0017 0.0008 0.0010 0.0012 0.0012
9200, 100 0.0023 0.0012 0.0027 0.0012 0.0051


10 0.0017 0.0012 0.0010 0.0010 0.0010
8500, 50 0.0030 0.0020 0.0022 0.0025 0.0020
8500, 100 0.0042 0.0027 0.0062 0.0030 0.0093


10 0.0005 0.0005 0.0005 0.0005 0.0005
7500, 50 0.0032 0.0023 0.0036 0.0045 0.0032
7500, 100 0.0059 0.0050 0.0149 0.0059 0.0181

It is clear that the pseudo-continuum is much closer to the true continuum in $\Lambda _3$ than in both bluer ranges.

3.2 Set of lines

Put end to end, the spectra acquired with AURÉLIE cover a spectral range of almost 500 Å. It includes that observed with ECHELEC in Paper I. The choice of the lines for the determination of the \ensuremath{v\sin i} in Paper I is thus still valid here. Moreover, in addition to this selection, redder lines were adopted in order to benefit from the larger spectral coverage.

The complete list of the 23 lines that are candidate for \ensuremath{v\sin i} determination is given in Table 2.

Table 2: List of the 23 spectral lines used (when possible) for the \ensuremath{v\sin i} measurement, and the corresponding spectral range(s) to which they belong.
range wavelength element range
  4215.519 Sr II  
  4219.360 Fe I  
  4226.728 Ca I  
$\Lambda _1$ 4227.426 Fe I  
  4235.936 Fe I  
  4242.364 Cr II  
  4261.913 Cr II $\Lambda _2$
  4404.750 Fe I  
  4415.122 Fe I  
  4466.551 Fe I  
  4468.507 Ti II  
  4481 .126 .325 Mg II \dag  
  4488.331 Ti II  
$\Lambda _3$ 4489.183 Fe II  
  4491.405 Fe II  
  4501.273 Ti II  
  4508.288 Fe II  
  4515.339 Fe II  
  4520.224 Fe II  
  4522.634 Fe II  
  4563.761 Ti II  
  4571.968 Ti II  
  4576.340 Fe II  

\dag Wavelength of both components are indicated for the magnesium doublet line.

In order to quantify effects of blends in the selected lines for later spectral types, we use the skewness of synthetic line profiles, as in Paper I. The same grid of synthetic spectra computed using Kurucz' model (Kurucz 1993), is used. Skewness is defined as $\gamma_1 = m_3
~ m_2^{-1.5}$, where mk is moment of kth order equal to

 \begin{displaymath}\forall k,\;m_k = {\displaystyle \sum_{i=1}^{L}\left[1-\maths...
...\over \displaystyle \sum_{i=1}^{L} 1-\mathscr{F}(\lambda_i) },
\end{displaymath} (1)

for an absorption line centered at wavelength $\lambda_{\rm c}$ and spreading from $\lambda_1$ to $\lambda_{\rm L}$, where $\mathscr{F}(\lambda_i)$ is the normalized flux corresponding to the wavelength $\lambda_i$. Ranges $[\lambda_1,\lambda_{\rm L}]$ are centered around theoretical wavelengths from Table 2 and the width of the window is taken to be 0.35, 0.90 and 1.80 Å for rotational broadening 10, 50 and 100  \ensuremath{{\rm km}~{\rm s}^{-1}} respectively (the width around the Mg II doublet is larger: 1.40, 2.0 and 2.3 Å).

Table 3: Variation of the skewness $\gamma _1$ of the lines with \ensuremath {T_{\rm eff}} and \ensuremath{v\sin i} in the synthetic spectra.
  \ensuremath{v\sin i} \ensuremath {T_{\rm eff}} (K)
line ( \ensuremath{{\rm km}~{\rm s}^{-1}}) 10 000 9200 8500 7500

Data for wavelengths shorter than 4500 Å
are given in Table 3 of Paper I
Ti II 4501 10     -0.05 -0.06 -0.07 -0.12
  50     -0.02 -0.03 -0.04 -0.04
  100     -0.03 -0.04 -0.05 -0.07
Fe II 4508 10     0.01 0.01 0.01 0.02
  50     -0.00 -0.00 -0.00 -0.00
  100     -0.01 -0.02 -0.03 -0.05
Fe II 4515 10     0.00 -0.00 -0.01 -0.06
  50     0.02 0.02 0.01 -0.04
  100     0.01 0.01 0.02 0.03
Fe II 4520 10     0.01 0.01 0.01 -0.01
  50     0.00 0.00 -0.00 -0.01
  100     -0.17 -0.19 -0.23 -0.30
Fe II 4523 10     -0.06 -0.06 -0.06 -0.05
  50     -0.01 -0.01 -0.01 0.01
  100     -0.12 -0.09 -0.01 0.08
Ti II 4564 10     0.04 0.04 0.05 0.06
  50     0.01 0.02 0.04 0.06
  100     0.03 0.04 0.08 0.16
Ti II 4572 10     -0.00 -0.00 -0.01 -0.02
  50     0.01 0.00 -0.01 -0.09
  100     0.01 0.01 -0.00 -0.04
Fe II 4576 10     0.01 0.01 0.02 0.05
  50     0.00 0.00 0.01 0.01
  100     0.01 0.02 0.04 0.07

Table 3 lists the skewness of the lines for each element of the synthetic spectra grid and is a continuation of Table 3 from Paper I for the lines with wavelength longer than 4500 Å. These additional lines are rather isolated and free from blends. Major part of the computed $\gamma _1$ for the hotter spectrum (10 000 K) is far lower than the threshold 0.15 chosen in Paper I to identify occurrence of blends. The only case where a line must be discarded is the blend occurring with Fe II 4520 and Fe II 4523 for $\ensuremath{v\sin i}\gtrsim 100$  \ensuremath{{\rm km}~{\rm s}^{-1}}. This non-blended behavior continues on the whole range of temperature, and the candidate lines remain reliable in most cases.

\includegraphics[width=8.8cm,clip]{MS2413f3} \end{figure} Figure 3: $\ensuremath{v\sin i} _{\rm Mg {\sc ii}}$ derived from the 4481 Mg II line versus $\langle\ensuremath{v\sin i}\rangle$ derived from other metallic lines for early A-type stars. The solid line stands for the one-to-one relation. The dashed line is the least-squares linear fit for $\langle\ensuremath{v\sin i}\rangle>30$  \ensuremath{{\rm km}~{\rm s}^{-1}}.
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\includegraphics[width=8.8cm,clip]{MS2413f4} \end{figure} Figure 4: Simulation of the doublet width behavior: FWHM of the sum of two Gaussian lines (separated with 0.2 Å) as a function of the FWHM of the components.
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The comparison between the rotational velocity derived from the weak lines and the one derived from the magnesium doublet was already approached in Paper I. It is here of an increased importance since the Mg II line is not present in all spectra (i.e. $\Lambda _1$and $\Lambda _2$ spectral ranges). Figure 3 shows this comparison between $\langle\ensuremath{v\sin i}\rangle$ and $\ensuremath{v\sin i} _{\rm Mg {\sc ii}}$ using AURÉLIE data. The deviation from the one-to-one relation (solid line) in the low velocity part of the diagram is due to the intrinsic width of the doublet. This deviation is simulated by representing the Mg II doublet as the sum of two identical Gaussians separated by 0.2 Å. The full-width at half maximum (FWHM) of the simulated doublet line is plotted in Fig. 4 versus the FWHM of its single-lined components. The relation clearly deviates from the one-to-one relation for single line FWHM lower than 0.6 Å. Using the rule of thumb from Slettebak et al. (1975, hereafter SCBWP): $FWHM{\scriptstyle [{\rm\AA}]} \approx
0.025~\ensuremath{v\sin i} {\scriptstyle [\ensuremath{{\rm km}~{\rm s}^{-1}} ]}$, this value corresponds to $\ensuremath{v\sin i} = 24$  \ensuremath{{\rm km}~{\rm s}^{-1}}. This limit coincides with what is observed in Fig. 3. For higher velocities ( $\langle\ensuremath{v\sin i}\rangle>30$  \ensuremath{{\rm km}~{\rm s}^{-1}}), $\langle\ensuremath{v\sin i}\rangle$ becomes larger than $\ensuremath{v\sin i} _{\rm Mg {\sc ii}}$. A linear regression gives:

 \begin{displaymath}\ensuremath{v\sin i} _{{\rm Mg {\sc ii}}} = 0.88~\langle\ensuremath{v\sin i}\rangle +2.2.
\end{displaymath} (2)

The effect is similar to the one found in Paper I, suggesting that blends in lines weaker than Mg II produce an overestimation of the derived \ensuremath{v\sin i} of about 10%.

\includegraphics[width=8.8cm,clip]{MS2413f5} \end{figure} Figure 5: The average number of measured lines (running average over 30 points) is plotted as a function of the mean $\langle\ensuremath{v\sin i}\rangle$. Solid lines stands for the spectra collected with AURÉLIE ($\Lambda _3$ range) whereas dotted line represents ECHELEC spectra from Paper I.
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The number of measurable lines among the 23 listed in Table 2 varies from one spectrum to another according to the wavelength window, the rotational broadening and the signal-to-noise ratio. The number of measured lines ranges from 1 to 17 lines. The $\Lambda _3$ range offers a large number of candidate lines. Figure 5 shows the variation of this number with \ensuremath{v\sin i} (solid line). Rotational broadening starts to make the number of lines decrease beyond about 70  \ensuremath{{\rm km}~{\rm s}^{-1}}. Nevertheless additional lines in the spectral domain redder than 4500 Å makes the number of lines larger than in the domain collected with ECHELEC (Paper I; dotted line). Whereas with ECHELEC the number of lines decreases with \ensuremath{v\sin i} from 30  \ensuremath{{\rm km}~{\rm s}^{-1}} to reach only one line (i.e. the Mg II doublet) at 100  \ensuremath{{\rm km}~{\rm s}^{-1}}, the number of lines with AURÉLIE is much sizeable: seven at 70  \ensuremath{{\rm km}~{\rm s}^{-1}}, still four at 100  \ensuremath{{\rm km}~{\rm s}^{-1}} and more than two even beyond 150  \ensuremath{{\rm km}~{\rm s}^{-1}}.

3.3 Precision

3.3.1 Effect of $v\,\sin\, i$

In Fig. 6, the differences between the individual \ensuremath{v\sin i} values from each measured line in each spectrum and the associated mean value for the spectrum are plotted as a function of $\langle\ensuremath{v\sin i}\rangle$. In the same way the error associated with the \ensuremath{v\sin i} has been estimated in Paper I, a robust estimate of the standard deviation is computed for each bin of 70 points. The resulting points (open grey circles in Fig. 6) are adjusted with a linear least squares fit (dot-dashed line). It gives:

 \begin{displaymath}\sigma_{\ensuremath{v\sin i} \vert\ensuremath{v\sin i} } = 0....
...angle\ensuremath{v\sin i}\rangle + 0.14{\scriptstyle\pm 0.19}.
\end{displaymath} (3)

This fit is carried out using GaussFit (Jefferys et al. 1998a,b), a general program for the solution of least squares and robust estimation problems. The resulting constant of the linear fit has an error bar of the same order than the value itself, and then the formal error is estimated to be 5% of the \ensuremath{v\sin i}.

\includegraphics[width=8.8cm,clip]{MS2413f6} \end{figure} Figure 6: Differences between individual \ensuremath{v\sin i} and mean over a spectrum $\langle\ensuremath{v\sin i}\rangle$. Variation of the standard deviation associated with the measure with the $\langle\ensuremath{v\sin i}\rangle$ is shown by the open circles. A linear least-square fit on these points (dot-dashed line) gives a slope of 0.05.
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The slope is lower with AURÉLIE data than with ECHELEC spectra (Paper I): $4.8{\scriptstyle\pm 1.0}$ % against $5.9{\scriptstyle\pm 0.3}$ %. This trend can be explained by the average number of lines for the computation of the mean \ensuremath{v\sin i}. In the velocity range from 15 to 180  \ensuremath{{\rm km}~{\rm s}^{-1}}, the number of measured lines (Fig. 5) is on average 2.4 times larger with AURÉLIE than with ECHELEC, which could lower the measured dispersion by a factor of $\sqrt{2.4}\approx 1.5$.

3.3.2 Effect of spectral range

As shown in Fig. 1, the distribution of spectral types is mainly concentrated towards late-B and early-A stars, so that a variation of the precision as a function of the spectral type would not be very significant. On the other hand, as the observed spectral domain is not always the same, this could introduce an effect due to the different sets of selected lines, their quantity and their quality in terms of \ensuremath{v\sin i} determination. For each of the three spectral domains, the residuals, normalized by $\sigma_{\ensuremath{v\sin i} \vert\ensuremath{v\sin i} }$ (Eq. (3)), are centered around 0 with a dispersion of about 1 taking into account their error bars, as shown in Table 4. This suggests that no effect due to the measurement in one given spectral range is produced on the derived \ensuremath{v\sin i}.


Table 4: Mean of differences between individual \ensuremath{v\sin i} and average $\langle\ensuremath{v\sin i}\rangle$ over a spectrum, normalized by the formal error due to \ensuremath{v\sin i}, are indicated for each spectral range as well as the standard deviations $\hat{\sigma}_{\ensuremath{v\sin i} \vert\Lambda}$ of these means.

Spectral range
$\left\langle{\ensuremath{v\sin i} -\langle\ensuremath{v\sin
i}\rangle\over \sigma_{\ensuremath{v\sin i} \vert\ensuremath{v\sin i} }}\right\rangle$ $\hat{\sigma}_{\ensuremath{v\sin i} \vert\Lambda}$

$\Lambda _1$
$-0.04 {\scriptstyle\pm 0.08}$ $1.00 {\scriptstyle\pm 0.09}$
$\Lambda _2$ $-0.10 {\scriptstyle\pm 0.12}$ $0.83 {\scriptstyle\pm 0.13}$
$\Lambda _3$ $-0.03 {\scriptstyle\pm 0.03}$ $0.92 {\scriptstyle\pm 0.04}$

4 Rotational velocities data

4.1 Results


Table 5: (extract) Results of the \ensuremath{v\sin i} measurements. Only the 15 first stars are listed below. The whole table is available electronically at the CDS. Description of the columns is detailed in the text.
HD HIP Spect. type \ensuremath{v\sin i} $\sigma$ # Remark
      ( \ensuremath{{\rm km}~{\rm s}^{-1}}    
905 1086 F0IV 35 1 6  
2421 2225 A2Vs 14 1 9  
2628 2355 A7III 21 2 9  
2924 2565 A2IV 31 2 16  
3038 2707 B9III 184 - 1  
4161 3572 A2IV 29 2 9  
4222 3544 A2Vs 38 2 17  
4321 3611 A2III 25: 4 14 SS
5066 4129 A2V 121 - 1  
5550 4572 A0III 16 3 5  
6960 5566 B9.5V 33 4 7  
10293 7963 B8III 62 - 1  
10982 8387 B9.5V 33 3 3  
11529 9009 B8III 36 4 8  
11636 8903 A5V... 73 2 11  

In total, projected rotational velocities were derived for 249 B8 to F2-type stars, 86 of which have no rotational velocities in Abt & Morrell (1995).

The results of the \ensuremath{v\sin i} determinations are presented in Table 5 which 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), Cols. 4, 5, 6 give respectively the derived value of \ensuremath{v\sin i}, the associated standard deviation and the corresponding number of measured lines (uncertain \ensuremath{v\sin i} are indicated by a colon), Col. 7 presents possible remarks about the spectra: SB2 ("SB'') and shell ("SH'') natures are indicated for stars showing such feature in these observed spectra, as well as the reason why \ensuremath{v\sin i} is uncertain - "NO'' for no selected lines, ``SS'' for variation from spectrum to spectrum and "LL'' for variation from line to line (see Appendix A).

4.1.1 SB2 systems

Nine stars are seen as double-lined spectroscopic binary in the data sample. Depending on the \ensuremath{v\sin i} of each component, their difference in Doppler shift and their flux ratio, determination of \ensuremath{v\sin i} is impossible in some cases.

...\includegraphics{MS2413f7e}\hspace*{3mm}\includegraphics{MS2413f7f}}\end{figure} Figure 7: Part of the spectra are displayed for the six SB2 stars that have been observed only once: a) HD 35189, b) HD 40183, c) HD 42035, d) HD 181470, e) HD 203439, f) HD 203858. Three of them are well separated b), d), f), allowing measurement of \ensuremath{v\sin i} for both components. The three others a), c), e) have low differential Doppler shift ($\le $60  \ensuremath{{\rm km}~{\rm s}^{-1}}) which makes all the lines blended. No \ensuremath{v\sin i} has been determined for these objects.
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...includegraphics{MS2413f8e}\hspace*{3mm}\includegraphics{MS2413f8f}} \end{figure} Figure 8: The three following SB2 stars have been observed twice, in $\Lambda _1$ (upper panels) and $\Lambda _3$ (lower panels): a) HD 79763 at HJD 2449025, b) HD 98353 at HJD 2448274, c) HD 119537 at HJD 2449025, d) HD 79763 at HJD 2449365, e) HD 98353 at HJD 2449413, f) HD 119537 at HJD 2449415. SB2 nature of these objects is not detected in $\Lambda _1$ spectral range, and the derived \ensuremath{v\sin i} is a "combined'' broadening. The triple system HD 98353 is observed close to conjunction, and lines remain blended. For HD 79763 d) and HD 119537 f), the difference in radial velocity is large enough to measure separately the rotational velocities.
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Table 6 displays the results for the stars in our sample which exhibit an SB2 nature. Spectral lines are identified by comparing the SB2 spectrum with a single star spectrum. Projected rotational velocities are given for each component when measurable, as well as the difference in radial velocity $\Delta V_{\rm r}$computed from a few lines in the spectrum.


Table 6: Results for stars seen as SB2. Rotational velocities are given for each component when measurable. $\Delta V_{\rm r}$ stands for the difference in radial velocity between the two components. Dash indicates a non possible measurement (either for \ensuremath{v\sin i} or $\Delta V_{\rm r}$).
HD HIP Spect. type \ensuremath{v\sin i} $\Delta V_{\rm r}$ Fig.
      ( \ensuremath{{\rm km}~{\rm s}^{-1}}) ( \ensuremath{{\rm km}~{\rm s}^{-1}})  
      A B    
35189 25216 A2IV - 37 7a
40183 28360 A2V 37 37 127 7b
42035 29138 B9V see text 12: 7c
79763 45590 A1V 29 - 8a
      34: 21: 67 8d
98353 55266 A2V 44   8b
      34 64: 8e
119537 67004 A1V 20: - 8c
      17 18 98 8f
181470 94932 A0III 15 20 229 7d
203439 105432 A1V - 56 7e
203858 105660 A2V 14 15 106 7f

4.2 Comparison with existing data

4.2.1 South versus North

Fourteen stars are common to both the southern sample from Paper I and the northern one studied here. Matching of both determinations allows us to ensure the homogeneity of the data or indicate variations intrinsic to the stars otherwise. Results for these objects are listed in Table 7.


Table 7: Comparison of the computed \ensuremath{v\sin i} for the stars in common in the northern and southern samples (N $\equiv $ this work, S $\equiv $ Paper I). CFF is a flag indicating the shape of the cross-correlation function carried out by Grenier et al. (1999) using the ECHELEC spectra (0: symmetric and Gaussian peak, 4: probable double, 5: suspected double, 6: probable multiple system).
HD Sp. type CCF $\ensuremath{v\sin i} _{{\rm N}}$ $\sigma_{{\rm N}}$ $\ensuremath{v\sin i} _{{\rm S}}$ $\sigma_{{\rm S}}$
27962 A2IV 0 16 2 11 1
30321 A2V 4 132 4 124 -
33111 A3IIIvar 6 196 - 193 4
37788 F0IV 0 29 1 33 4
40446 A1Vs - 27 5 27 5
65900 A1V 0 35 3 36 2
71155 A0V 4 161 12 137 2
72660 A1V 0 14 1 9 1
83373 A1V 0 28 - 30 2
97633 A2V 0 24 3 23 1
98664 B9.5Vs - 57 1 61 5
109860 A1V 5 74 1 76 6
193432 B9IV 0 24 2 25 2
198001 A1V 0 130 - 102 -

Instrumental characteristics differ from ECHELEC to AURÉLIE data. First of all, the resolution is higher in the ECHELEC spectra, which induces a narrower instrumental profile and allows the determination of \ensuremath{v\sin i} down to a lower limit. Taking the calibration relation from SCBWP as a rule of thumb ( $FWHM{\scriptstyle [{\rm\AA}]} \approx
0.025~\ensuremath{v\sin i} {\scriptstyle [\ensuremath{{\rm km}~{\rm s}^{-1}} ]}$), the low limit of \ensuremath{v\sin i} is:

 \begin{displaymath}\ensuremath{v\sin i} _{\rm lim} = {1\over 0.025}~ FWHM_{\rm inst} = {1\over 0.025}~ {\lambda\over R},
\end{displaymath} (4)

where R is the power of resolution, and $\lambda$ the considered wavelength. For ECHELEC spectra ( $R\approx 28~000$) this limit is 6.4  \ensuremath{{\rm km}~{\rm s}^{-1}} at 4500 Å, whereas for AURÉLIE data ( $R\approx 16~000$), it reaches 11.3  \ensuremath{{\rm km}~{\rm s}^{-1}}. These limits correspond to the "rotational velocity'' associated with the FWHMof the instrumental profile. There is no doubt that in Fourier space, the position of the first zero of a line profile dominated by the instrumental profile is rather misleading and the effective lowest measurable \ensuremath{v\sin i} may be larger. This effect explains the discrepancy found for slow rotators, i.e. HD 27962 and HD 72660 in Table 7. The \ensuremath{v\sin i} determination using AURÉLIE spectra is 5  \ensuremath{{\rm km}~{\rm s}^{-1}} larger than using ECHELEC spectra. The two stars are slow rotators for which \ensuremath{v\sin i} has already been derived using better resolution. HD 27962 is found to have $\ensuremath{v\sin i} =12$ and 11  \ensuremath{{\rm km}~{\rm s}^{-1}} by Varenne & Monier (1999) and Hui-Bon-Hoa & Alecian (1998) respectively. HD 72660 has a much smaller \ensuremath{v\sin i}, lower than the limit due to the resolution of our spectra: 6.5  \ensuremath{{\rm km}~{\rm s}^{-1}} in Nielsen & Wahlgren (2000) and 6  \ensuremath{{\rm km}~{\rm s}^{-1}} in Varenne (1999).

Second of all, one other difference lies in the observed spectral domain. HD 198001 has no observation in the $\Lambda _3$ domain using AURÉLIE, so that $\ensuremath{v\sin i} _{{\rm N}}$ in Table 7 is not derived on the basis of the Mg II line. The overestimation of $\ensuremath{v\sin i} _{{\rm N}}$ reflects the use of weak metallic lines instead the strong Mg II line for determining rotational velocity.

Using the same ECHELEC data, Grenier et al. (1999) flagged the stars according to the shape of their cross-correlation function with synthetic templates. This gives a hint about binary status of the stars. Three stars in Table 7 are flagged as "probable binary or multiple systems'' (CCF: 4 and 6).

When discarding low rotators, probable binaries and data of HD 198001 that induce biases in the comparison, the relation between the eight remaining points is fitted using GaussFit by:

 \begin{displaymath}\ensuremath{v\sin i} _{\rm S} = 1.05{\scriptstyle\pm 0.04}~\ensuremath{v\sin i} _{\rm N}-0.2{\scriptstyle\pm 1.5}.
\end{displaymath} (5)

Although common data are very scarce, they seem to be consistent. It suggests that both data sets can be merged as long as great care is taken for cases detailed above, i.e. extremely low rotators, high rotators with no \ensuremath{v\sin i} from Mg II line, spectroscopic binaries.

4.2.2 Standard stars

\includegraphics[width=8.8cm,clip]{MS2413f9}\end{figure} Figure 9: Comparison of \ensuremath{v\sin i} data for the 163 common stars between this work and Abt & Morrell (1995). The solid line stands for the one-to-one relation.
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\includegraphics[width=8.8cm,clip]{MS2413f10}\end{figure} Figure 10: Comparison between \ensuremath{v\sin i} data from this work and from Slettebak et al. (1975). The solid line stands for the one-to-one relation. The 21 standard stars are plotted with error bar on both axes (see text). HD number of the stars that deviate most from the one-to-one relation are indicated and these stars are listed in Table 8 and detailed in Appendix B.
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A significant part of the sample is included in the catalogue of Abt & Morrell (1995). The intersection includes 163 stars. The comparison of the \ensuremath{v\sin i} (Fig. 9) shows that our determination is higher on average than the velocities derived by Abt & Morrell (AM). The linear relation given by GaussFit is:

 \begin{displaymath}\ensuremath{v\sin i} _{\rm this\;work} = 1.18{\scriptstyle\pm 0.04}~\ensuremath{v\sin i} _{\rm AM}+3.8{\scriptstyle\pm 0.8}.
\end{displaymath} (6)

Abt & Morrell use the standard stars of SCBWP to calibrate the relation FWHM- \ensuremath{v\sin i}. There are 21 stars in common between our sample and these standard stars. Figure 10 displays the \ensuremath{v\sin i} derived in this paper versus the \ensuremath{v\sin i} from SCBWP for these 21 common stars. The solid line represents the one-to-one relation. A clear trend is observed: \ensuremath{v\sin i} from SCBWP are on average 20% lower. A linear least squares fit carried out with GaussFit on these values makes the systematic effect explicit:

 \begin{displaymath}\ensuremath{v\sin i} _{\rm this\;work} = 1.11{\scriptstyle\pm...
...7}~\ensuremath{v\sin i} _{\rm SCBWP}+7.1{\scriptstyle\pm 1.7}.
\end{displaymath} (7)

The relation is computed taking into account the error bars of both sources. The error bars on the values of SCBWP are assigned according to the accuracy given in their paper (10% for $\ensuremath{v\sin i} <200~\ensuremath{{\rm km}~{\rm s}^{-1}} $ and 15% for $\ensuremath{v\sin i}\geq 200~\ensuremath{{\rm km}~{\rm s}^{-1}} $). Our error bars are derived from the formal error found in Sect. 3.3 (Eq. (3)).

Table 8: Highlight of the discrepancy between \ensuremath{v\sin i} values from SCBWP and ours (standard deviation of our measurement is indicated; dash "-'' stands for only one measurement). Comparison with data from the literature for the twelve stars that exhibit the largest differences. \ensuremath{v\sin i} are classified in three subgroups according to the way they are derived: by-product of a spectrum synthesis, frequency analysis of the lines profiles or infered from a FWHM- \ensuremath{v\sin i} relation independent from SCBWP's one. Flags from HIPPARCOS catalogue are indicated: variability flag H52 (C: constant, D: duplicity-induced variability, M: possibly micro-variable, U: unsolved variable, -: no certain classification) and double annex flag H59 (O: orbital solution, G: acceleration terms, -: no entry in the Double and Multiple Systems Annex).

HD Sp. type \ensuremath{v\sin i} ( \ensuremath{{\rm km}~{\rm s}^{-1}}) HIPPARCOS
      SCBWP this work depth 0pt height 0.4pt width 3.0cm literature depth 0pt height 0.4pt width 3.0cm H52 H59
           spec. synth. freq. analysis FWHM    

$\gamma $ Gem
47105 A0IV <10 15 ${\scriptstyle\pm
1}$ 11.2(1) $10.2{\scriptstyle\pm 0.2}^{(2)}$, 19.0(3)   - X
30 Mon 71155 A0V 125 161 ${\scriptstyle\pm 12}$       C -
$\beta$ UMa 95418 A1V 35 47 ${\scriptstyle\pm 3}$ 44.8(1), 39(4) 44.3(3)   - -
$\theta$ Leo 97633 A2V 15 24 ${\scriptstyle\pm 3}$ 21(5), 22.1(1) 24(6), 27.2(3) 23(7) - -
$\gamma $ UMa 103287 A0V SB 155 178 ${\scriptstyle\pm 9}$   $154{\scriptstyle\pm 4}^{(8)}$   M -
$\alpha$ Dra 123299 A0III SB 15 25 ${\scriptstyle\pm 2}$   27(9)   M O
$\sigma$ Boo 128167 F3Vwvar 10 15 ${\scriptstyle\pm
1}$ $7.5{\scriptstyle\pm 1}^{(10)}$ 7.5(11) 7.8(12), 8.1(13) - -
$\alpha$ CrB 139006 A0V 110 139 ${\scriptstyle\pm 10}$   $127{\scriptstyle\pm 4}^{(8)}$   U O
$\tau$ Her 147394 B5IV 30 46 ${\scriptstyle\pm 3}$   32(6)   P -
$\alpha$ Lyr 172167 A0Vvar <10 25 ${\scriptstyle\pm 2}$ 22.4(1), 23.2(14) $23.4{\scriptstyle\pm 0.4}^{(16)}$, 24(6)   U -
           $21.8{\scriptstyle\pm 0.2}^{(15)}$ 29.9(3)      
$\gamma $ Lyr 176437 B9III 60 72 ${\scriptstyle\pm 2}$       M -
$\epsilon$ Aqr 198001 A1V 85 130- 95(17), 108.1(1)     - -

(1) Hill (1995). (6) Smith & Dworetsky (1993). (11) Gray (1984). (16) Gray (1980b).
(2) Scholz et al. (1997). (7) Fekel (1998). (12) Fekel (1997). (17) Dunkin et al. (1997).
(3) Ramella et al. (1989). (8) Gray (1980a). (13) Benz & Mayor (1984).  
(4) Holweger et al. (1999). (9) Lehmann & Scholz (1993). (14) Erspamer & North (2002).  
(5) Lemke (1989). (10) Soderblom (1982). (15) Gulliver et al. (1994).  

The standard stars for which a significant discrepancy occurs between our values and those derived by SCBWP - i.e. their error box does not intersect with the one-to-one relation - have their names indicated in Fig. 10. They are listed with data from the literature in Table 8 and further detailed in Appendix B.

5 Merging the samples

Homogeneity and size are two crucial characteristics of a sample, in a statistical sense. In order to gather a \ensuremath{v\sin i} sample obeying these two criteria, \ensuremath{v\sin i} 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 \ensuremath{v\sin i} 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 \ensuremath{v\sin i} 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

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

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:

\begin{displaymath}\vert\Delta\vert > 1.1~\sigma_\Delta.\end{displaymath}

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 \ensuremath{v\sin i} stars observed with AURÉLIE, and the resolution limitation can be the source of the discrepancy

\includegraphics[width=10cm,clip]{MS2413f11.eps}\end{figure} Figure 11:  Comparison of \ensuremath{v\sin i} 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 \ensuremath{v\sin i} 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.
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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:

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

Rotational velocities from Abt & Morrell are scaled to the \ensuremath{v\sin i} 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 \ensuremath{v\sin i} (uncertain \ensuremath{v\sin i}, due to uncertain determination in either one of the source lists, are indicated by a colon).


Table 9: (extract) Results of the merging of \ensuremath{v\sin i} 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 \ensuremath{v\sin i} $\in $
      ( \ensuremath{{\rm km}~{\rm s}^{-1}})  
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 \ensuremath{v\sin i} 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.

\includegraphics[width=8.8cm,clip]{MS2413f12} \end{figure} Figure 12: Pie chart of the subsample membership of the stars in the total \ensuremath{v\sin i} sample. Multiple membership is represented by superimposed patterns.
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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):

 \begin{displaymath}V_{\rm lim} = 7.9 + 1.1~\sin\vert b\vert.
\end{displaymath} (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 \ensuremath{v\sin i} 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 \ensuremath{v\sin i} 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.

\par\resizebox{9cm}{!}{\includegraphics{MS2413f13a}\hspace*{3mm}\includegraphics{MS2413f13bc}} \end{figure} Figure 13: a) Density of the \ensuremath{v\sin i} 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 \ensuremath{v\sin i} sample compared to the A-type stars in the HIPPARCOS catalogue for the northern and southern hemisphere respectively.
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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.

6 Summary and conclusions

The determination of projected rotational velocities is sullied with several effects which affect the measurement. The blend of spectral lines tends to produce an overestimated value of \ensuremath{v\sin i}, whereas the lowering of the measured continuum level due to high rotation tends to lower the derived \ensuremath{v\sin i}. The solution lies in a good choice of candidate lines to measure the rotational velocity. The use of the additional spectral range 4500-4600 Å, compared to the observed domain in Paper I, allows for the choice of reliable lines that can be measured even in case of high rotational broadening and reliable anchors of the continuum, for the considered range of spectral types. The \ensuremath{v\sin i} is derived from the first zero of Fourier transform of line profiles chosen among 23 candidate lines according to the spectral type and the rotational blending. It gives resulting \ensuremath{v\sin i} for 249 stars, with a precision of about 5%.

The systematic shift with \ensuremath{v\sin i} standard stars from SCBWP, already detected in Paper I, is confirmed in this work. SCBWP's values are underestimated, smaller by a factor of 0.8 on average, according to common stars in the northern sample. When joining both intersections of northern and southern samples with standard stars from SCBWP, the relation between the two scales is about $\ensuremath{v\sin i} =1.03~{\ensuremath{v\sin i} }_{\rm SCBWP}+7.7$, using these 52 stars in common. This is approximately our findings concerning the catalogue made by Abt & Morrell (1995). They derive their \ensuremath{v\sin i} from the calibration built by SCBWP, and reproduce the systematic shift.

In the aim of gathering a large and homogeneous sample of projected rotational velocities for A-type stars, the new data, from the present paper and from Paper I, are merged with the catalogue of Abt & Morrell. First, the \ensuremath{v\sin i} from the latter catalogue are statistically corrected from the above mentioned systematic shift. The final sample contains \ensuremath{v\sin i} for 2151 B8- to F2-type stars.

The continuation of this work will consist in determining and analyzing the distributions of rotational velocities (equatorial and angular) for different sub-groups of spectral type, starting from the \ensuremath{v\sin i}.

We insist on warmly thanking Dr. M. Ramella for having provided the programme of determination of the rotational velocities. We are also very grateful to Dr. R. Faraggiana for her precious advice about the analysis of the spectra. We would like to acknowledge Dr. F. Sabatié for his careful reading of the manuscript.

Appendix A: Notes on stars with uncertain rotational velocity

A.1 Stars with no selected line

In a few cases, the selected lines are all discarded either from their Fourier profile or from their skewness (Table 3). For these stars, an uncertain value of \ensuremath{v\sin i} is derived from the lines that should have been discarded. They are indicated by a colon and flagged as "NO'' in Table 5. These objects are listed below. It is worth noticing that none of them have spectra collected in $\Lambda _3$ spectral range:

A.2 Stars with high external error

A few stars of the sample exhibit an external error higher than the estimation carried on in Sect. 3.3. It can be the signature of a multiple system. The following stars have variable \ensuremath{v\sin i} from spectrum to spectrum and are labeled as "SS'' in Table 5:

Appendix B: Notes on \ensuremath{v\sin i} standard stars with discrepant rotational velocity

The common stars, among SCBWP's data and this sample, which exhibit the largest differences in \ensuremath{v\sin i} between both studies, are listed in Table 8. They are detailed below.

Appendix C: Notes on common stars with Abt & Morrell (1995) rejected by sigma-clipping

When merging the sample from Abt & Morrell (1995) with the new measurements using Fourier transforms, common data are compared in order to compute the scaling law between both samples. Aberrant points are discarded using a sigma-clipping algorithm. These stars are listed and discussed, and their \ensuremath{v\sin i} are indicated in \ensuremath{{\rm km}~{\rm s}^{-1}} ( $\ensuremath{v\sin i} _{{\rm I}\cup
{\rm II}} / \ensuremath{v\sin i} _{{\rm AM}}$):

$^\star$: these stars show very coherent values between both samples, but these low \ensuremath{v\sin i} are considered "discrepant'' with respect to the scaling law (Eq. (8)).



Copyright ESO 2002