3 Measurement of the rotational velocity

3.1 Method

As pointed out by Gray & Garrison (1987), there is no "standard'' technique for measuring projected rotational velocity. The first application of Fourier analysis in the determination of stellar rotational velocities was undertaken by Carroll (1933). Gray (1992) uses the whole profile of Fourier transform of spectral lines to derive the , instead of only the zeroes as suggested by Carroll. The measurement method we adopted is based on the position of the first zero of the Fourier transform (FT) of the line profiles (Carroll 1933). The shape of the first lobe of the FT allows us to better and more easily identify rotation as the main broadening agent of a line compared to the line profile in the wavelength domain. FT of the spectral line is computed using a Fast Fourier Transform algorithm. The value is derived from the position of the first zero of the FT of the observed line using a theoretical rotation profile for a line at 4350Å and equal to 1 (Ramella et al. 1989). The whole profile in the Fourier domain is then compared with a theoretical rotational profile for the corresponding velocity to check if the first lobes correspond (Fig. 3).

Figure 3: Profile of the Fourier transform of the Mg II 4481Å line (solid line) for the star HIP 95965 and theoretical rotational profile (dashed line) with $\ensuremath{v\sin i} =200$ .

If $\nu_0$ is the position of the first zero of the line profile (at $\lambda_0$) in the Fourier space, the projected rotational velocity is derived as follows:

$$\ensuremath{v\sin i} = {4350\over\lambda_0}\,{\nu_{\rm T}\over\nu_0},$$ (1)

where 4350Å and ${\nu_{\rm T}}$ respectively stand for the wavelength and the first zero of the theoretical profile.

It should be noted that we did not take into account the gravity darkening, effect that can play a role in rapidly rotating stars when velocity is close to break-up, as this is not relevant for most of our targets.

3.2 Continuum tracing

Determination of the projected rotational velocity requires normalized spectra.

As far as the continuum is concerned, it has been determined visually, passing through noise fluctuations. The MIDAS procedure for continuum determination of 1D-spectra has been used, fitting a spline over the points chosen in the graphs. Uncertainty related to this determination rises because the continuum observed on the spectrum is a pseudo-continuum. Actually, the true continuum is, in this spectral domain, not really reached for this type of stars. In order to quantify this effect, a grid of synthetic spectra of different effective temperatures (10000, 9200, 8500 and 7500K) and different rotational broadenings has been computed from Kurucz' model atmosphere (Kurucz 1993), and Table 1 lists the differences between the true continuum and the pseudo-continuum represented as the highest points in the spectra.

  

Table 1: Differences between the true continuum and the highest points in different spectral bands for the set of synthetic spectra.

$$\begin{tabular}{r@{\hspace*{0.5mm}}rcccc\vert c\vert ccccc} \... ...82 & 0.0191 & 0.0255 & 0.0064 & 0.0245 \cr \hline \end{tabular}$$

It illustrates the contribution of the wings of H$\gamma$, as the hydrogen lines reach their maximum strength in the early A-type stars, and in addition, the general strength of the metallic-line spectrum which grows with decreasing temperature. In the best cases, i.e. earliest type and low broadening, differences are about a few 0.1%. For cooler stars and higher rotators, they reach up to 3%. The points selected to anchor the pseudo-continuum are selected as much as possible in the borders of the spectra, where the influence of the wings of H$\gamma$ is weaker.

Continuum is then tilted to origin and the spectral windows corresponding to lines of interest are extracted from the spectrum in order to compute their FT.

3.3 Set of lines

3.3.1 A priori selection

The essential step in this analysis is the search for suitable spectral lines to measure the . The lines which are candidates for use in the determination of rotation (Table 2) have been identified in the Sirius atlas (Furenlid et al. 1992) and retained according to the following criteria:

• not blended in the Sirius spectrum;
• far enough from the hydrogen line H$\gamma$ to maintain relatively good access to the continuum.

These are indicated in Fig. 2.

  

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

$$\begin{tabular}{llll} \hline wavelength & element &\qquad wav... ... \cr & &\qquad 4491.405 & Fe {\sc ii} \cr \hline \end{tabular}$$

Wavelengths of both components are indicated for the magnesium doublet line.

The lines selected in the Sirius spectrum are valid for early A-type stars. When moving to stars cooler than about A3-type stars, the effects of the increasing incidence of blends and the presence of stronger metallic lines must be taken into account. The effects are: (1) an increasing departure of the true continuum flux (to which the spectrum must be normalized) from the curve that joins the highest points in the observed spectrum, as mentioned in the previous subsection, and (2) an increased incidence of blending that reduces the number of lines suitable for measurements. The former effect will be estimated in Sect. 3.4. The latter can be derived from the symmetry of the spectral lines. Considering a line, continuum tilted to zero, as a distribution, moments of kth order can be defined as:

$$\forall k,\;m_k = {\displaystyle \sum_{i=1}^{L}\left[1-\math... ...\over \displaystyle \sum_{i=1}^{L} 1-\mathscr{F}(\lambda_i) },$$ (2)

for an absorption line centered at wavelength $\lambda_{\rm c}$ and spreading from $\lambda_1$ to $\lambda_L$, where $\mathscr{F}(\lambda_i)$ is the normalized flux corresponding to the wavelength $\lambda_i$. Ranges $[\lambda_1,\lambda_L]$ 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). Skewness is then defined as

$$\gamma_1 = {m_3 \over (m_2)^{3/2}}\cdot$$ (3)

Variations of skewness of a synthetic line profile with temperature and/or rotational broadening should be caused only by the presence of other spectral lines that distort the original profile. Table 3 gives skewness of the selected lines for the different synthetic spectra.

 

 

Table 3: Variation of the skewness $\gamma _1$ (Eq. (3)) of the lines with and in the synthetic spectra.

The most noticeable finding in this table is that $\vert\gamma_1\vert$ usually increases with decreasing and increasing . This is a typical effect of blends. Nevertheless, high rotational broadening can lower the skewness of a blended line by making the blend smoother.

Skewness $\gamma _1$ for the synthetic spectrum close to Sirius' parameters ( $\ensuremath{T_{\rm eff}} =10\,000$K, $\ensuremath{v\sin i} =10$ ) is contained between -0.09 and +0.10. The threshold, beyond which blends are regarded as affecting the profile significantly, is taken as equal to 0.15. If $\vert\gamma_1\vert>0.15$ the line is not taken into account in the derivation of the for a star with corresponding spectral type and rotational broadening. This threshold is a compromise between the unacceptable distortion of the line and the number of retained lines, and it ensures that the differences between centroid and theoretical wavelength of the lines have a standard deviation of about 0.02 Å.

As can be expected, moving from B8 to F2-type stars increases the blending of lines. Among the lines listed in Table 2, the strongest ones in Sirius spectrum (Sr II 4216, Fe I 4219, Cr II 4242, Fe I 4405 and Mg II 4481) correspond to those which remain less contaminated by the presence of other lines. Only Fe I 4405 retains a symmetric profile not being heavily blended at the resolution of our spectra and thus measurable all across the grid of the synthetic spectra.

The Mg II doublet at 4481Å is usually chosen to measure the : it is not very sensitive to stellar effective temperature and gravity and its relative strength in late B through mid-A-type star spectra makes it almost the only measurable line in this

Figure 4: $\ensuremath{v\sin i} _{\rm Mg {\sc ii}}$ 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 $\langle\ensuremath{v\sin i}\rangle>30$  .

spectral domain for high rotational broadening. However the separation of 0.2Å in the doublet leads to an overestimate of the derived from the Mg II line for low rotational velocities. Figure 4 displays deviation between the $\ensuremath{v\sin i} _{\rm Mg {\sc ii}}$ measured on the Mg II doublet and the mean $\langle\ensuremath{v\sin i}\rangle$ derived from weaker metallic lines, discarding automatically the Mg II line. For low velocities, typically $\langle\ensuremath{v\sin i}\rangle\lesssim 25$ , the width of the doublet is not representative of the rotational broadening but of the intrinsic separation between doublet components. That is why $\ensuremath{v\sin i} _{\rm Mg {\sc ii}}$ is stagnant at a plateau around 19 , and gives no indication of the true rotational broadening. In order to take this effect into account, the Mg II doublet is not used for determination below 25 . For higher velocities, the derived from weak lines are on the average overestimated because they are prone to blending, whereas Mg II is much more blend-free. On average, for $\langle\ensuremath{v\sin i}\rangle>30$  , the relation between $\langle\ensuremath{v\sin i}\rangle$ and $\ensuremath{v\sin i} _{\rm Mg {\sc ii}}$ deviates from the one-to-one relation as shown by the dashed line in Fig. 4, and a least-squares linear fit gives the equation

$$\ensuremath{v\sin i} _{{\rm Mg {\sc ii}}} = 0.9\,\langle\ensuremath{v\sin i}\rangle + 0.6,$$ (4)

which suggests that blends can lead to a 10% overestimation of the .

3.3.2 A posteriori selection

Among the list of candidate lines chosen according to the spectral type and rotational broadening of the star, some can be discarded on the basis of the spectrum quality itself. The main reason for discarding a line, first supposed to be reliable for determination, lies in its profile in Fourier space. One retains the results given by lines whose profile correspond to a rotational profile.

In logarithmic frequency space, such as in Figs. 3 and 5, the rotational profile has a unique shape, and the effect of simply acts as a translation in frequency. Matching between the theoretical profile, shifted at the ad hoc velocity, and the observed profile, is used as confirmation of the value of the first zero as a .

Figure 5: Example of line profiles in the Fourier space for HD 75063 (A1III type star) whose $\ensuremath{v\sin i} =30\,\ensuremath{{\rm km}\,{\rm s}^{-1}}$. The theoretical rotational profile (grey solid line), computed for the average of the star, matches perfectly the FT of the Fe I 4405 line (black solid line), whereas Fourier profiles of Sr II 4216 and Fe I 4415 differ from a rotational shape. Among these three observed lines, only Fe I 4405 is retained for determination.

This comparison, carried out visually, allows us to discard non suitable Fourier profiles as shown in Fig. 5.

A discarded Fourier profile is sometimes associated with a distorted profile in wavelength space, but this is not always the case. For low rotational broadening, i.e. $\ensuremath{v\sin i}\lesssim 10$  , the Fourier profile deviates from the theoretical rotational profile. This is due to the fact that rotation does not completely dominate the line profile and the underlying instrumental profile is no longer negligible. It may also occur that an SB2 system, where lines of both components are merged, appears as a single star, but the blend due to multiplicity makes the line profile diverge from a rotational profile.

To conclude, the number of measurable lines among the 15 listed in Table 2 also varies from one spectrum to another according to the rotational broadening and the signal-to-noise ratio and ranges from 1 to 15 lines.

Figure 6: Average number of measured lines (running average over 30 points) is plotted as a function of the mean $\langle\ensuremath{v\sin i}\rangle$.

As shown in Fig. 6, the average number of measured lines decreases almost linearly with increasing , because of blends, and reaches one (the Mg II 4481 doublet line) at $\ensuremath{v\sin i}\approx 100$  . Below about 25  , the number of measured lines decreases with for two reasons: first, Mg II line is not used due to its intrinsic width; and more lines are discarded because of their non-rotational Fourier profile, instrumental profile being less negligible.

   
3.4 Systematic effect due to continuum

The measured continuum differs from the true one, and the latter is generally underestimated due to the wings of H$\gamma$ and the blends of weak metallic lines. One expects a systematic effect of the pseudo-continuum on the determination as the depth of a line appears lower, and so its FWHM. We use the grid of synthetic spectra to derive rotational broadening from "true normalized'' spectra (directly given by the models) and "pseudo normalized'' spectra (normalized in the same way as the observed spectra). The difference of the two measurements is

$$\delta\ensuremath{v\sin i} = \ensuremath{v\sin i} - \ensuremath{v\sin i} ',$$ (5)

where is the rotational broadening derived using a pseudo-continuum and $\ensuremath{v\sin i} '$ using the true continuum.

Figure 7: Systematic shift $\delta\ensuremath{v\sin i}$ as a function of the true rotational broadening $\ensuremath{v\sin i} '$ for: a) Fe I 4405 and b) Mg II 4481. The different symbols stand for the effective temperature of the synthetic spectra: fill circle: 10000 K; square: 9200 K; diamond: 8500 K; triangle: 7500 K.

The systematic effect of the normalization induces an underestimation of the as shown in Fig. 7. This shift depends on the spectral line and its relative depth compared to the difference between true and pseudo-continuum. For Fe I 4405 (Fig. 7a), the shift is about $0.15\,(\ensuremath{v\sin i} '-30)$, which leads to large differences. The effect is quite uniform in temperature as reflected by the symmetric shape of the line all along the spectral sequence (Table 3). Nevertheless, at $\ensuremath{v\sin i} '=150$  , the scattering can be explained by the strength of H$\gamma$ whose wing is not negligible compared to the flattened profile of the line. For Mg II 4481 (Fig. 7b), the main effect is not due to the Balmer line but to blends of metallic lines. The shift remains small for early A-type stars (filled circles and open squares): $\lesssim$5%, but increases with decreasing effective temperature of the spectra, up to 10%. This is due to the fact that Mg II doublet is highly affected by blends for temperatures cooler than about 9000 K.

This estimation of the effect of the continuum is only carried out on synthetic spectra because the way our observed spectra have been normalized offers no way to recover the true continuum. The resulting shift is given here for information only.

   
3.5 Precision

Two types of uncertainties are present: those internal to the method and those related to the line profile.

The internal error comes from the uncertainty in the real position of the first zero due to the sampling in the Fourier space. The Fourier transforms are computed over 1024 points equally spaced with the step $\Delta\nu$. This step is inversely proportional to the step in wavelength space $\Delta\lambda$, and the spectra are sampled with $\Delta\lambda=0.05$ Å. The uncertainty of due to the sampling is

$$\begin{array}{ccl} \Delta\ensuremath{v\sin i} & \propto & (\... ...approx & 4\times10^{-4}\,(\ensuremath{v\sin i} )^2. \end{array}$$ (6)

This dependence with to the square makes the sampling step very small for low and it reaches about 1 for $\ensuremath{v\sin i} =50$ .

The best way to estimate the precision of our measurements is to study the dispersion of the individual . For each star, is an average of the individual values derived from selected lines.

Effect of $\mathsfsl v$sin $\mathsfsl i$

The error associated with the is expected to depend on , because Doppler broadening makes the spectral lines shallow; that is, it reduces the contrast line/continuum and increases the occurrence of blends. Both effects tend to disrupt the selection of the lines as well as the access to the continuum. Moreover, the stronger the rotational broadening is, the fewer measurable lines there are. In Fig. 8, the differences between the individual values from each measured line in each spectrum with the associated mean value for the spectrum are plotted as a function of $\langle\ensuremath{v\sin i}\rangle$.

Figure 8: Differences between individual and mean over a spectrum $\langle\ensuremath{v\sin i}\rangle$. Variation of the standard deviation associated with the measure as a function of $\langle\ensuremath{v\sin i}\rangle$ is shown by the open circles. A linear least squares fit on these points (dot-dashed line) gives a slope of 0.06.

A robust estimate of the standard deviation is computed for each bin of 50 points; resulting points (open grey circles in Fig. 8) are adjusted with a linear least squares fit (dot-dashed line) giving:

$$\sigma_{\ensuremath{v\sin i} \vert\ensuremath{v\sin i} }=0.0... ...iptstyle\pm\, 0.003}\,\langle\ensuremath{v\sin i}\rangle\cdot$$ (7)

The fit is carried out using GaussFit (Jefferys et al. 1998,b), a general program for the solution of least squares and robust estimation problems. The formal error is then estimated as 6% of the value.

3.5.2 Effect of spectral type

Figure 9: 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 type by the open squares. The standard deviations of these means for each spectral class are plotted as filled circles, with their associated error bar.

Residual around this formal error can be expected to depend on the effective temperature of the star. Figure 9 displays the variations of the residuals as a function of the spectral type. Although contents of each bin of spectral type are not constant all across the sample (the error bar is roughly proportional to the logarithm of the inverse of the number of points), there does not seem to be any trend, which suggests that our choice of lines according to the spectral type eliminates any systematic effect due to the stellar temperature from the measurement of the .

3.5.3 Effect of noise level

Although noise is processed as a high frequency signal by Fourier technique and not supposed to act much upon determination from the first lobe of the FT, signal-to-noise ratio (SNR) may affect the measurement. SNR affects the choice of the lines' limits in the spectrum as well as the computation of the lines' central wavelength.

Figure 10: Differences between individual and mean over a spectrum $\langle\ensuremath{v\sin i}\rangle$, normalized by the formal error due to . Variation of the standard deviation associated to the measure with the noise level (SNR-1) is shown by the open circles. A linear least squares fit on these points (dot-dashed line) gives a slope of $\sim$100.

The differences $\ensuremath{v\sin i} - \langle\ensuremath{v\sin i}\rangle$, normalized by the formal error $0.059\,\langle\ensuremath{v\sin i}\rangle$, are plotted versus the noise level (SNR^-1) in Fig. 10 in order to estimate the effect of SNR. Noise is derived for each spectrum using a piecewise-linear high-pass filter in Fourier space with a transition band chosen between 0.3 and 0.4 times the Nyquist frequency; standard deviation of this high frequency signal is computed as the noise level and then divided by the signal level. The trend in Fig. 10 is computed as for Fig. 8, using a robust estimation and GaussFit. The linear adjustment gives:

$$\hat{\sigma}_{\ensuremath{v\sin i} \vert{SNR}} = 93\,{\scriptstyle\pm \,16}\,{SNR}^{-1} + 0.5\,{\scriptstyle\pm \,0.1}.$$ (8)

The distribution of mean signal-to-noise ratios for our observations peaks at SNR = 190 with a standard deviation of 78. This means that for most of the observations, SNR does not contribute much to the formal error on ( $\hat{\sigma}_{\ensuremath{v\sin i} \vert{SNR}} \approx 1$). Finally, the formal error associated with the can be quantified as:

$$\begin{array}{ccl} \sigma_{\ensuremath{v\sin i} } & = & \sig... ...splaystyle {\ensuremath{v\sin i}\over{SNR}}}\cdot \end{array}$$ (9)

3.5.4 Error distribution

Distribution of observational errors, in the case of rotational velocities, is of particular interest during a deconvolution process in order to get rid of statistical errors in a significant sample.

To have an idea of the shape of the error law associated with the , it is necessary to have a great number of spectra for the same star. Sirius has been observed on several occasions during the runs and its spectrum has been collected 48 times. Sirius spectra typically exhibit high signal-to-noise ratio ( $SNR\gtrsim 250$). The 48 values derived from each set of lines, displayed in Fig. 11, give us an insight into the errors distribution. The mean is $16.22\,{\scriptstyle \pm \, 0.04}$ and its associated standard deviation $0.27\,{\scriptstyle \pm \, 0.03}$ ; data are approximatively distributed following a Gaussian around the mean .

Figure 11: The determinations for the 48 spectra of Sirius are distributed around the mean $16.22\,{\scriptstyle \pm \, 0.04}$ with a dispersion of $0.27\,{\scriptstyle \pm \, 0.03}$ . The optimal normal distribution $\mathscr{N}(16.23,0.21)$ that fits the histogram with a 96% significance level is over-plotted. The optimal log-normal distribution merges together with the Gaussian.

A Kolmogorov-Smirnov test shows us that, with a 96% significance level, this distribution is not different from a Gaussian centered at 16.23 with a standard deviation equal to 0.21. In the case of Sirius (low ) the error distribution corresponds to a normal distribution. We may expect a log-normal distribution as the natural error law, considering that the error on is multiplicative. But for low , and low dispersion, log-normal and normal distributions do not significantly differ from each other.

Moreover, for higher broadening, the impact of the sampling effect of the FT (Eq. (6)) is foreseen, resulting in a distribution with a box-shaped profile. This effect becomes noticeable for $\ensuremath{v\sin i}\gtrsim 100$  .