3 Measurement of the rotational velocity

The method adopted for 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.

,		central wavelength (Å)
(K, )		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
9200,	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
8500,	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
7500,	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 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 determination is given in Table 2.

   

Table 2: List of the 23 spectral lines used (when possible) for the 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
	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

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 m_k is moment of kth order equal to

$$\forall k,\;m_k = {\displaystyle \sum_{i=1}^{L}\left[1-\maths... ...\over \displaystyle \sum_{i=1}^{L} 1-\mathscr{F}(\lambda_i) },$$ (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  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 and in the synthetic spectra.

		(K)
line	( )	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 $\ensuremath{v\sin i}\gtrsim 100$  . This non-blended behavior continues on the whole range of temperature, and the candidate lines remain reliable in most cases.

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 $\langle\ensuremath{v\sin i}\rangle>30$  .

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.

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 $\ensuremath{v\sin i} = 24$  . This limit coincides with what is observed in Fig. 3. For higher velocities ( $\langle\ensuremath{v\sin i}\rangle>30$  ), $\langle\ensuremath{v\sin i}\rangle$ becomes larger than $\ensuremath{v\sin i} _{\rm Mg {\sc ii}}$. A linear regression gives:

$$\ensuremath{v\sin i} _{{\rm Mg {\sc ii}}} = 0.88~\langle\ensuremath{v\sin i}\rangle +2.2.$$ (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 of about 10%.

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.

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 (solid line). Rotational broadening starts to make the number of lines decrease beyond about 70  . 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 from 30  to reach only one line (i.e. the Mg II doublet) at 100  , the number of lines with AURÉLIE is much sizeable: seven at 70  , still four at 100  and more than two even beyond 150  .

   
3.3 Precision

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

In Fig. 6, the differences between the individual 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 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:

$$\sigma_{\ensuremath{v\sin i} \vert\ensuremath{v\sin i} } = 0.... ...angle\ensuremath{v\sin i}\rangle + 0.14{\scriptstyle\pm 0.19}.$$ (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 .

Figure 6: Differences between individual 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.

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 . In the velocity range from 15 to 180  , 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 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 .

 

 

Table 4: Mean of differences between individual and average $\langle\ensuremath{v\sin i}\rangle$ over a spectrum, normalized by the formal error due to , 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	$\left\langle{\ensuremath{v\sin i} -\langle\ensuremath{v\sin i}\rangle\over \sigma_{\ensuremath{v\sin i} \vert\ensuremath{v\sin i} }}\right\rangle$	$\hat{\sigma}_{\ensuremath{v\sin i} \vert\Lambda}$
$\Lambda _1$	$-0.04 {\scriptstyle\pm 0.08}$	$1.00 {\scriptstyle\pm 0.09}$
$\Lambda _2$	$-0.10 {\scriptstyle\pm 0.12}$	$0.83 {\scriptstyle\pm 0.13}$
$\Lambda _3$	$-0.03 {\scriptstyle\pm 0.03}$	$0.92 {\scriptstyle\pm 0.04}$