A&A 451, 603-611 (2006)
DOI: 10.1051/0004-6361:20054600
P. L. Dufton1 - R. S. I. Ryans1 - S. Simón-Díaz2,3 - C. Trundle1,3 - D. J. Lennon2,3
1 - Department of Pure & Applied Physics, The Queen's University
of Belfast, BT7 1NN, Northern Ireland, UK
2 -
Isaac Newton Group of Telescopes, Apartado de Correos 368, 38700
Santa Cruz de La Palma, Canary Islands, Spain
3 -
Instituto de Astrofísica de Canarias, 38200
La Laguna, Tenerife, Spain
Received 28 November 2005 / Accepted 27 January 2006
Abstract
High-resolution spectra for 24 SMC and Galactic B-type supergiants
have been analysed to estimate the contributions of both
macroturbulence and rotation to the broadening of their metal lines.
Two different methodologies are considered, viz. goodness-of-fit
comparisons between observed and theoretical line profiles and
identifying zeros in the Fourier transforms of the observed profiles.
The advantages and limitations of the two methods are briefly discussed
with the latter techniques being adopted for estimating projected
rotational velocities (v sin i) but the former being used to estimate
macroturbulent velocities. The projected rotational velocity
estimates range from approximately 20 to 60 km s-1, apart from one SMC
supergiant, Sk 191, with a v sin i
90 km s-1.
Apart from Sk 191, the distribution of projected rotational
velocities as a function of spectral type
are similar in both our Galactic and SMC samples with larger
values being found at earlier spectral types. There is marginal evidence
for the projected rotational velocities in the SMC being higher than
those in the Galactic targets but any differences are only of the
order of 5-10 km s-1, whilst evolutionary models predict differences
in this effective temperature
range of typically 20 to 70 km s-1. The combined sample is consistent
with a linear variation of projected rotational velocity with effective
temperature, which would imply rotational velocities for supergiants
of 70 km s-1 at an effective temperature of 28 000 K (approximately B0 spectral type) decreasing to 32 km s-1 at 12 000 K (B8 spectral type).
For all targets, the macroturbulent broadening would appear to be
consistent with a Gaussian distribution (although other distributions
cannot be discounted) with an
half-width
varying from approximately 20 km s-1 at B8 to 60 km s-1 at B0 spectral types.
Key words: galaxies: Magellanic Clouds - stars: early-type - stars: supergiants
Evolutionary models, such as those by Heger & Langer (2000), Meynet & Maeder (2000) and Maeder & Meynet (2001), provide predictions of the evolution of both surface chemical compositions and rotational velocities of massive hot stars as a function of time. One common theme of these models is that an initially high rotational velocity (of the order of 300 km s-1) is required to produce significant mixing of nucleosynthetically processed material to the surface. Although rotational velocities decrease as their progenitor OB-type stars evolve away from the main sequence, B-type supergiants showing processed material at their surface are predicted to retain significant rotational velocities. However measurement of projected rotational velocities for a range of SMC early-type stars (O-type supergiants - Hillier et al. 2003; O-type dwarfs - Bouret et al. 2003; B-type giants and supergiants - Lennon et al. 2003; Trundle et al. 2004) appear to be systematically too low when compared with the predictions (see Fig. 7 of Trundle et al.). Additionally in most previous analyses, it has been assumed that the line broadening was predominantly due to rotation and if this were not the case the discrepancy between observation and theory would become larger.
Howarth et al. (1997) investigated the line widths in IUE spectra for a large sample of early-type evolved stars and noted that there were no narrow-lined objects. They interpreted this as evidence that there was another mechanism (hereafter designated as "macroturbulence''), which was contributing to the line broadening. More recently Ryans et al. (2002) have analysed high signal-to-noise (S/N) ratio (of the order of 500) spectra of Galactic supergiants in order to try and estimate the relative contributions of rotation and macroturbulence. They found that although the line widths implied projected rotational velocities in the range of 50 to 100 km s-1, their best estimates were typically in the range of 10 to 30 km s-1, with all upper limits being less than 60 km s-1. These results were lower than, for example, the calculations of Meynet & Maeder (2000) for solar metallicity objects that predicted rotational velocities in the range of 50 to 200 km s-1. Even allowing for projection effects, the observed values appeared to be systematically too low.
Recently there have been two non-LTE analyses of high quality spectroscopy of SMC B-type supergiants. Trundle et al. (2004) analysed the spectra of 7 supergiants (and a giant) using the unified code FASTWIND (Santolaya-Rey et al. 1997) to determine their photospheric and wind parameters. Dufton et al. (2005) used the static code TLUSTY (Hubeny 1988; Hubeny & Lanz 1995) to analyse 9 supergiants. Two of these targets were in common with those of Trundle et al. and a comparison of the two analyses showed excellent agreement. Here we analyse the spectroscopic data from these two SMC investigations using the techniques discussed in Ryans et al. (2002) and Simón-Díaz & Herrero (2006) to estimate projected rotational and macroturbulent velocities. These techniques utilise different methodologies and hence allow us to investigate their strengths and limitations. We conclude that the Fourier transform technique of Simón-Díaz & Herrero provides the better estimates for the projected rotational velocity and have hence used it to reanalyse the Galactic stellar data presented by Ryans et al. (2002).
Our principle aim is to obtain reliable estimates of the stellar projected rotational velocities while allowing for the contribution of other broadening mechanisms. In particular, we will investigate whether the low estimates found by Ryans et al. (2002) are confirmed and replicated in our SMC samples. One advantage of studying targets in the low metallicity environment of the SMC is that the presence and the amount of contamination of the stellar surface by nucleosynthetically processed material is far easier to characterise. This in turn makes it easier to search for any correlation between rotation and the amount of such contamination.
Our observational material consists of high resolution spectroscopy for 13 SMC supergiants obtained at the Anglo-Australian Telescope (AAT) and the ESO Very Large Telescope (VLT) and for 11 Galactic supergiants obtained at the McDonald Observatory and the William Herschel Telescope (WHT).
Trundle et al. (2004) obtained spectroscopy of 7 SMC targets
at the VLT with the UVES instrument in November 2001. Of particular relevance
here are the observations obtained in the blue arm of the spectrograph with
a wavelength coverage of 3900 to 5000 Å. Trundle et al. rebinned their
data to a pixel size of 0.2 Å (yielding signal-to-noise
ratios ranging from 120 to 190). Here we have used the unbinned
spectra which have a spectral resolution, R
40 000.
Two targets, AzV 104 (B0.5Ia) and AzV 216 (B1 III) have not been
included in the current investigation - the former as the quality
of the observational data was insufficient to accurately estimate
the projected rotational velocity and the latter as it is not a supergiant. The VLT data are summarized in Table 2 and further details of the observations and the reduction techniques can be found in Trundle et al. (2004).
Spectroscopy for 9 SMC B-type supergiants (two in common with the
VLT observations) was obtained at the Anglo-Australian Telescope in
September 1996, with the UCLES instrument. There was complete wavelength
coverage from 3900 to 4900 Å with a pixel size of 0.08 Å,
a spectral resolution, R
20 000 and
with signal-to-noise ratios ranging from 90 to 130. Note that although the
signal-to-noise ratios of the AAT data are lower than that of the binned VLT data, the differences in pixel size lead to the data being of similar quality (see, for example, Fig. 3 of Dufton et al. 2005). The stars observed are again summarized in Table 2
and further details of the reduction techniques can be found in Dufton
et al. (2005).
Ryans et al. (2002) has presented spectroscopy for the
Galactic supergiants with signal-to-noise ratios ranging from 250 to 700
and spectral resolutions of R
160 000 (McDonald) and
R
40 000 (WHT). HD 36371 (B4 Iab) has been excluded from
this investigation due to the difficulty of estimating the projected
rotational velocity with the other targets being listed in Table 2. Full details of the observations and their reduction can be found in Ryans et al.
We have considered two methodologies to investigate the broadening of the absorption lines in our supergiant sample and in particular to distinguish between rotational broadening and that due to other mechanisms that we characterise as macroturbulence. These utilise the goodness-of-fit between observed and theoretical profiles and the shape of the Fourier transforms of observed absorption lines and are discussed in detail by Ryans et al. (2002) and Simón-Díaz & Herrero (2006) respectively. Below we summarize the two methodologies.
The methodology of Ryans et al. is, in principle, straightforward.
Equivalent widths of relatively strong, unblended lines in each
star are measured, and then theoretical profiles of the same strength
are generated using a non-LTE code (e.g. TLUSTY - Hubeny
1988) with appropriate atmospheric parameters.
It is assumed that the broadening of the intrinsic absorption line
profile is due to rotation (v sin i) and another mechanism that is
characterised as macroturbulence and is initially assumed to have a Gaussian distribution of velocities (with a half-width at which the profile has fallen to
of its maximum value
given by v
). Theoretical profiles are then convolved with these functions and compared to the observational data. The sum of the squares of the differences are
computed, giving a measure of the quality of the fit at each point
in (v
,
v sin i) space. The main difficulty with this
goodness-of-fit (GOF) approach is that profiles generated with different (v
,
v sin i) combinations are quite similar especially when one broadening mechanism dominates and
we return to this point in Sect. 3.2.
By contrast the Fourier transform (FT) methodology discussed by Simón-Díaz & Herrero (2006) allows the projected rotational velocity to be derived independently of any other broadening mechanism which may affect the line profile. A line profile can be considered to consist of the convolution of natural, instrumental, rotational and macroturbulent broadening profiles. As first discussed by Carroll (1933a,b) the Fourier transform of the rotational profile has zeros whose position in frequency space depend on the projected rotational velocity. The FT method for the determination of v sin i (cf. Gray 1973) is then based on the fact that in Fourier space, convolutions transform into products. Hence it is possible to determine the projected rotational velocity of a star once the position of the first of those zeroes is identified in the Fourier transform of the line profile. Although this methodology has not been widely used in the study of OB-type stars (e.g. Ebbets 1979), the recent study by Simón-Díaz & Herrero (2006) illustrates the strength of this method for determining projected rotational velocities in early type stars (see also Simón-Díaz et al. 2006).
We have used the high quality spectroscopy reported in Ryans et al.
(2002) to compare the results obtained by the two methodologies.
We note that the FT approach does not directly provide estimates
of the macroturbulence parameter (v)
and hence the comparison
is restricted to the estimates of the projected rotational velocities. In Table 1, we summarize the estimates from the two methodologies with the
goodness-of-fit results being taken directly from Ryans et al. Note that for
some lines, it was not possible to identify the zeros in the Fourier transforms
and these cases have been excluded from the comparison. Consequently the GOF
results listed in Table 1 differ slightly from those reported by
Ryans et al. The only significant change is for HD 204172 where the exclusion
of the Mg II line at 4481 Å and the Si III line at 4552 Å leads to an increase in the median from the GOF estimates.
Table 1: Comparison of the estimates of the projected rotational velocities found using goodness-of-fit (GOF) and Fourier transform (FT) techniques. For both cases, the ranges of the estimates and their medians are listed. Also tabulated are the maximum possible values of v sin i taken directly from Ryans et al. together with the number (n) of lines considered in the current comparison.
Inspection of Table 1 implies that the estimates from the
GOF technique are systematically lower than those from the FT technique with
the mean of the differences being 22
10 km s-1. However the FT
estimates are consistent with the maximum values deduced from the GOF analysis. We believe that these differences reflect the difficulty in identifying and quantifying the relatively small amount of rotational broadening in the presence of significant macroturbulent broadening.
With the GOF methodology, the profiles become effectively degenerate
for small values of v sin i with the contour maps in the
(v,
v sin i) plane having broad flat ridges along the
v sin i axis (see, for example Fig. 3 of Ryans et al. 2002).
Hence estimates of the projected
rotational velocities become sensitive to the observational uncertainties
and the assumptions adopted. For example, a Gaussian velocity distribution
has been assumed for the macroturbulence and if the distribution differed
from this, it could lead to systematic errors in the estimation of the
projected rotational velocity. Additionally the sensitivity of the GOF approach to uncertainties in the observational data (such as blending and continuum normalisation) may be the cause of the wider range of estimates found in a given star compared with the FT approach.
For slowly rotating stars, the Fourier transform of the broadening function will have its first (and subsequent zeros) at high frequencies. Additionally if the observed stellar absorption lines are significantly broadened by macroturbulence most of their power will occur at low frequencies in the Fourier domain. Since the noise in the spectrum transforms as a "white noise'' in the Fourier space (Smith & Gray 1976), then the unambiguous identification of the zeros in the Fourier transform becomes difficult or impossible when the v sin i is low and the effect of macroturbulence is important (this is analogous to the difficulty in the GOF approach of estimating projected rotational velocities when the line profiles are dominated by macroturbulence). Indeed even for the high quality Galactic supergiant spectra it was not possible to estimate the projected rotational velocity from the Fourier transform for some spectral features in a given target. Tests undertaken by Simón-Díaz & Herrero (2006) for cases with small projected rotational velocities and significant macroturbulence indicate that the former can be overestimated by 5 to 10 km s-1. Although this would reduce the discrepancy with the estimates deduced from the GOF methodology, it would not eliminate it.
However the advantage of this methodology is that if the zeros in the Fourier Transform can be unambiguously identified the corresponding v sin i estimates should be robust and in particular they are less dependent (compared with the GOF approach) on additional assumptions or on uncertainties in the observational data. Indeed indirect evidence for this is the smaller range in estimates found from different lines in a given star using the FT methodology.
Table 2:
Median values for the macroturbulent ( v)
and projected rotational
( v sin i) velocities deduced for the SMC datasets presented in Dufton et al.
(2005) and by Trundle et al. (2004). Also listed are
estimates for the Galactic supergiants discussed by Ryans et al.
(2002). The sources of the spectral types and effective
temperatures are discussed in the text.
Given the discussion of the previous section, we concluded that the FT methodology provided the better estimates of the projected rotational velocities and adopted this approach for both our Galactic and SMC samples. These estimates are summarized in Table 2 together with spectral types and estimates of the stellar effective temperature. For the projected rotational velocities we have quoted medians as these will be less affected by any spurious measurements although very similar values would have been found if we had adopted the means of the estimates. We also undertook an analysis of the SMC dataset using the GOF methodology and will discuss this briefly in Sect. 4. Spectral types and effective temperatures were taken from McErlean et al. (1999) for the Galactic sample and Trundle et al. (2004) and Dufton et al. (2005) for the SMC targets. Note that the three studies utilised different non-LTE codes and different physical assumptions about e.g. the stellar winds and hence there may be systematic differences between the effective temperature scales. For two SMC targets, effective temperature estimates are available from both Trundle et al. and Dufton et al. and they differ by 500 K and 2000 K (in Table 2 we adopt the means of these values), which may be indicative of the magnitude of any systematic differences.
The FT technique does not provide a direct estimate of the magnitude
of other broadening mechanisms (that have here been characterised as
macroturbulence) although this will affect the overall shape
of the Fourier transform. Hence we have used the GOF technique to estimate
this quantity. Our approach was for each star to adopt the projected
rotational velocity estimate listed in Table 2 and then
to re-analyse the relevant lines to determine the macroturbulence (v)
that best fitted the profile. These estimates are
also listed in Table 2 where we have again adopted the
median of the individual estimates.
The uncertainties in our estimates of the projected rotational
velocity and macroturbulence will depend on several factors including
the quality of the observational data and the magnitude of the quantity
being estimated - for example, an estimate of a small macroturbulence
in a target with significant rotational broadening will have a relatively
high degree of uncertainty. In Table 1, we list the
range of v sin i estimates for each of our Galactic targets with a similar
spread being found for our SMC targets. These translate into a typical
uncertainty in the median value of the v sin i estimates of 10%. The
spread in the estimates of the macroturbulent velocity for a given target
is similar and implies a typical uncertainty in the medians of 10%,
apart from the smallest values (v
30 km s-1) where the uncertainty
may be larger.
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Figure 1:
Observed and theoretical profiles for the Si III line at 4552 Å in HD 206165. The former is represented by a histogram whilst the theoretical profiles assume an isotropic Gaussian velocity (dashed line) and a radial-tangential model with
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The macroturbulence has initially been assumed to be isotropic and to follow a Gaussian distribution. The good agreement between the theoretical and observed
profiles (see Fig. 1 of Ryans et al. 2002) provides some evidence in
support of this assumption. However we cannot rule out other distributions and
in particular those that are qualitatively similar to a Gaussian distribution.
As an example we consider anisotropic macroturbulence characterised by a radial-tangential model (Gray 1975,1992). In this model the velocity field is modelled by assuming that a fraction ()
of the material is moving radially (but with a Gaussian distribution of velocities) with the remainder moving tangentially. The methodology was developed to model
the flow of convective cells observed in the Sun. Hence it may not be
appropriate to the targets considered here and should be considered as
illustrative. As an example we have fitted the Si III line at 4552 Å in HD 206165 adopting a radial-tangential model with
= 1.0 and
= 55 km s-1 (i.e. assuming a complete radial flow). However we note that assuming that both the radial and
tangential velocities are equal (
,
=
= 55 km s-1) would have yielded an effectively identical
profile. In addition, we have also calculated a profile for an isotropic
macroturbulence with
= 35.5 km s-1. The results are shown in
Fig. 1 and both approaches are in good agreement with the
observational data, but the radial tangential profile requires a larger
macroturbulent velocity to be invoked (by approximately 20 km s-1).
Hence we conclude that it is not possible with the current dataset to
investigate the exact form or degree of the anisotropy of the macroturbulence.
However what is clear is that additional broadening is present, which is
consistent with a velocity distribution that is approximately Gaussian.
The finite spectral resolution of our observational data should not
affect our estimates of the projected rotational velocity but could
led to overestimates for the macroturbulence. Our lowest resolution
spectroscopy (R
20 000) would have a
width,
v
6 km s-1 assuming that the instrumental profile
was a Gaussian. Then if the macroturbulent velocity field
also followed a Gaussian distribution, the
estimates listed in Table 2 would represent the
instrumental profile and the actual macroturbulent velocity added in
quadrature. As an example, correcting our lowest estimate of
v
21 km s-1 for such an instrumental profile would lead to a decrease of less than 1 km s-1. For larger macroturbulent velocity estimates or better spectral resolutions, the corrections will
be smaller and hence we have not attempted to apply them.
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Figure 2: Estimates of macroturbulent ( upper figure) and projected rotational velocities ( lower figure) plotted against effective temperature. The Galactic supergiants are represented by squares and the SMC targets by circles. For the projected rotational velocities, the linear least squares fit (excluding Sk 191) is shown as a solid line, whilst the dashed line would represent the rotational velocity assuming a random distribution of projection angles. |
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Inspection of results summarized in Table 2 implies that
the amount of macroturbulent broadening in the Galactic and SMC supergiants
are comparable. To illustrate this in Fig. 2, our best estimates
of the macroturbulent velocities, v,
are plotted against
the effective temperature for both sets of supergiants.
As discussed in Sect. 3.3, the latter have been taken from
different sources and although there may be systematic differences
between these scales, we believe that this should not invalidate the
comparison.
In general, both samples show a similar trend of macroturbulence with effective temperature and would imply that this is an intrinsic property of the star. Additionally as discussed previously the effect of this broadening is significant and invalidates the use of line widths as a measure of rotational broadening for B-type supergiants. This is consistent with the analysis of IUE spectra by Howarth et al. (1997), who found significant spectral broadening in all their supergiant spectra. This they interpreted as "confirming that an important line broadening mechanism in addition to rotation must be present in these objects''.
For one SMC target, Sk 191, a relatively large projected rotational velocity is found from the FT methodology and this was confirmed by an independent GOF analysis. As can be seen from Fig. 2, our estimate of its macroturbulent velocity is compatible with the other targets having similar effective temperatures. By contrast adopting a small rotational velocity would have led to an anomalously large estimate for the macroturbulence of approximately 80 km s-1.
If the additional spectra line broadening is due to macroturbulence, the
estimated velocities, v,
are large and imply a highly
dynamic atmosphere. Additionally B-type supergiants are also found to
require significant microturbulent velocities that need to be
incorporated into the profiles of the line absorption coefficient, with
typically values being between 10 and 20 km s-1 (see, Crowther et al. 2005 and references therein). If these represent small scale
turbulence, then they and the values of macroturbulence found
here imply that the photospheres of B-type supergiants are highly
turbulent on all distance scales. In turn the adoption of a stable
photosphere in both static and unified stellar atmosphere codes must be
considered problematic although excellent agreement between theory and
observation has been found using such methods (Crowther et al. 2005;
Trundle et al. 2004; Dufton et al. 2005).
Assuming that our macroturbulent velocities are due to a large scale velocity
field, we can estimate its kinetic energy. As an example we consider three relatively hot targets, HD 38771, AzV 215 and AzV 487 with atmospheric
parameters,
27 000 K and log g
2.90. The evolutionary models
of Meynet & Maeder (2000,2001) then imply an evolutionary mass of
approximately 40
.
Additionally, the corresponding TLUSTY model
atmosphere implies that the mass of a column of unit area down to a Rosseland
mean opacity of unity is 10 kg m-2. Adopting this as the extent of
the atmosphere sampled by the absorption lines implies a total atmospheric mass
of 6
1019 kg and a typical macroturbulent velocity of 60 km s-1
would then lead to a kinetic energy in the velocity field of
approximately 1029 J. It should be noted that this is only an order
of magnitude estimate as it incorporates several assumptions and
simplifications. However the bolometric luminosity of these supergiants is
approximately 6
105
or 2
1032 J s-1 and the
kinetic energy of the macroturbulence would be therefore negligible when
considering the energy balance in the atmosphere. McErlean et al.
(1998) discuss the thermal velocities and sound speed for a model with atmospheric parameters (
= 27 500 K, log g = 3.0 and a microturbulence of 12 km s-1) similar to those of the three targets considered here. For hydrogen the thermal velocity is typically 20 km s-1 and the sound
speed is 15-20 km s-1. Hence the energy of a macroturbulent velocity field
would be greater than the thermal energy of the plasma, whilst its velocity
would be supersonic. Assuming that this additional broadening is indeed due
to a large scale velocity field, its origin and nature remain unclear.
The estimates of the projected rotational velocities, summarized in Table 2, imply that the values for both the Galactic and SMC samples
are similar and this is supported by Fig. 2 where the estimates are
plotted against effective temperature. If all our targets had the same
initial mass and rotational velocity, the effective temperature would
represent their age and evolutionary status. From a comparison with
the evolutionary models of Schaller et al. (1992) for the
Galactic targets (see McErlean et al. 1999, for further details)
and Maeder & Meynet (2001) for the SMC targets (see
Trundle et al. 2004), we have estimated evolutionary masses.
These fall in the range 20 to 40 ,
apart from the SMC target,
AzV 78, which has an estimated initial mass of approximately 50
.
This significant range in masses coupled with the initial rotational
velocity of our targets being effectively unknown implies that this
figure should be considered as primarily an empirical representation.
Theoretical predictions imply that the SMC supergiants should have
larger rotational velocities (see Fig. 3 and discussion
in Sect. 4.3) and there is some marginal evidence
for this in Fig. 2. To investigate this we have independently
estimated linear least squares fits for the two samples (excluding the anomalous
SMC supergiant Sk 191 which is discussed below). The two fits are similar,
differing by 4 km s-1 at
= 12 000 K and 9 km s-1 at 28 000 K. Given the
small size of the samples and the possibilities of systematic differences
in the effective temperature scales of the two samples these differences
cannot be considered significant. Hence we have also combined the two samples
and the corresponding linear fit is shown in Fig. 2 as a solid line. The estimates of the projected rotational velocity will be
affected by the unknown angle of inclination (i). Assuming that this was
randomly orientated would lead to an average value of sin i of
and
the dashed line in the figure represents the least squares fit increased by the
reciprocal of this factor. Although the sample is small it is encouraging that
this appears to trace the upper envelope of of our estimates and would be
consistent with these stars having an angle of inclination close to 90
.
Hence
assuming that all B-type supergiants undergo the same evolution of rotational
velocity (
)
with effective temperature our best estimate for that
relationship is given by:
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Figure 3:
Predictions of the evolution of the rotational velocity of two models
with metallicities appropriate to our Galaxy (Meynet & Maeder 2003;
initial masses: 20 ![]() ![]() ![]() ![]() |
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An explanation may lie in the biases in our observation samples. For example, two supergiants were excluded from the analysis as it was not possible to identify zeros in their Fourier Transforms, which could be due to them having intrinsically low projected rotational velocities. Hence our results could be biased to larger estimates, although as our initial sample size was twenty six targets any bias would be relatively small.
For each part of our sample, the main criteria for target selection was as follows. For the Galactic supergiants, there was a large choice of possible targets and the final selection aimed to cover a range of spectral type for both Ia and Ib luminosity classes. As such there is no clear source of bias in our selection. The VLT spectroscopy was obtained principally to investigate properties of the stellar winds, whilst the AAT targets were chosen to cover a range of photospheric N/C abundances ratios (based on the analysis of McErlean et al. 1999) but with a bias to brighter targets for ease of observation. Hence both SMC samples may be biased to the highest luminosity objects and indeed only contain two objects with luminosity class Iab or Ib. Such samples might be expected to have intrinsically small rotational velocities due to their strong winds. However this would not explain either the lack of low v sin i objects at the earliest spectra-types or the small scatter in the estimates.
However given all the caveats, discussed above the relationship in Eq. (1) has some merit in indicating the magnitude of the rotational velocities that should be found in evolutionary models of blue supergiants.
Previously, several authors have estimated projected rotational velocities and compared these with the predictions of stellar evolutionary calculations. For the A-type supergiants (Venn 1999), there appears to be reasonable agreement. However for O-type dwarfs (Bouret et al. 2003), O-type supergiants (Hiller et al. 2003; Crowther et al. 2002), B-type giants (Lennon et al. 2003) and B-type supegiants (Trundle et al. 2004), the observed values of v sin i appear to be smaller than those implied by stellar evolutionary calculations. However this comparison is complicated by comparing observed projected rotational velocities with predicted rotational velocities. Additionally it has normally been assumed that rotation was the dominant mechanism in the line broadening (with for example macroturbulence not being included) leading in effect to upper limits for the projected rotational velocities. The advantage of the current estimates are that we have attempted to distinguish between the different broadening mechanisms and thereby have been able to estimate the actual values of projected rotational velocities. In turn, this has allowed us to deduce a relationship between the mean rotational velocity and effective temperature.
In Fig. 3 we compare our results with
those predicted by evolutionary models appropriate to both Galactic
(Meynet & Maeder 2003) and SMC metallicities (Meynet & Maeder
2004). All the models illustrated have initial rotational velocities
of 300 km s-1 and have initial masses in the range of 20 to 40 .
As the initial rotational velocities of our targets are unknown (and
indeed likely to vary from target to target), this comparison should be
taken as illustrative. However all our SMC targets show
nuclearsynthetically processed material in their atmospheres
and if rotation was indeed the main mechanism for this enrichment, it would
appear that their initial rotational velocities must have been relatively
large. Also shown are the measured projected rotational velocities and
our estimate of the mean rotational velocity as a function of effective
temperature taken from Eq. (1). For the Galactic supergiants, our results
are in reasonable agreement with the predictions for the 40
model but may
be slightly lower than those for the 20
.
However given the uncertainty
in the intitial projected rotational velocities and the relatively small size
of our sample, the agreement must be considered encouraging. The evolutionary
models appropriate to our SMC sample predict larger rotational velocities -
for example at log
= 4.4 dex, the projected rotational velocities for
the 40
models are approximately 150 and 70 km s-1 at SMC and Galactic
metallicities respectively. Although the projected rotational velocity of
Sk 191 is consistent with these predictions, those for most of the other
SMC targets appear to be too small. This is confirmed by the discrepancy
between the theoretical predictions and our estimate of the rotational
velocity at early B-spectral types (
4.25), although the
agreement may be better at later spectral types. We note that tests
undertaken by Simón-Díaz & Herrero (2006) indicate that
for relatively large macroturbulences and small projected rotational velocities,
the FT methodology may overestimate the latter. However allowing for this
effect would increase the discrepancy between theory and observation
for the SMC dataset.
A comparison is also possible with the models of Heger & Langer (2000). Unfortunately they do not explicitly tabulate rotational velocities as their models evolve from the main sequence. However their calculations imply that massive stars should evolve to higher effective temperature after their red supergiant stage (the so called "blue loops''). Heger & Langer predict that such objects should have typical rotational velocities of 50 km s-1 and that their maximum effective temperature should be approximately 15 000 K. For the late B-type objects, our estimated rotational velocities are smaller but compatible with these predictions.
The theoretical predictions for objects evolving from the main sequence also
imply that the amount of processed material mixed to the surface should
depend on the initial stellar rotational velocity. For example Maeder
& Meynet (2001) considered 20
evolutionary models with
an SMC metallicity and initial rotational velocities ranging from 0 to
400 km s-1. At the end of the hydrogen burning phase, these models showed
changes in their N/C abundance ratio, which were negligible for the
non-rotating model but increased to a factor of approximately six for
the fastest rotating model. More recently Meynet & Maeder (2004)
have discussed models with different initial metallicities. For models
with significant rotational velocities, they find increases of approximately
a factor of ten in the nitrogen abundances as the
stars evolve into blue supergiants.
Additionally Howarth & Smith (2001) have discussed
the projected rotational velocities for ON and O-type main sequence stars.
The cumulative probability function of the former implied that they
were rotating more quickly with a Kolmogorov-Smirnov test indicating
that the differences were statistically significant. Hence for these O-type
stars (that will evolve into B-type supergiants), such a correlation
between nitrogen abundances and rotational velocities appears to be
present.
We have searched for such a correlation in our SMC sample (we do not have reliable nitrogen abundances for our Galactic targets) by plotting projected rotational velocities against nitrogen abundances in Fig. 4. The latter have been taken directly from Dufton et al. (2005) and Trundle et al. (2004), although as discussed in Dufton et al. (2005), for two stars in common there was a difference of approximately 0.15 dex in the nitrogen abundance estimates. Tests showed that allowing for this offset made no substantial difference and hence it has not been included in the figure. No positive correlation of nitrogen abundance with projected rotational velocity is observed. Indeed there is a suggestion that our most nitrogen enriched targets have the lowest values of v sin i. However our sample size is relatively small and hence we cannot preclude the possibilty that a correlation has been masked by different angles of inclination and differences in spectral type.
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Figure 4: Projected rotational velocities plotted against nitrogen abundances for the SMC targets. The cross (+) is for the target, AzV 487, for which only an upper limit for the nitrogen abundance could be estimated. |
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Although our targets normally have relatively small rotational velocities, there is one exception in the SMC supergiant Sk 191. We believe that the result for this star is robust as it is based on two different spectroscopic datasets and is supported by analyses using both the GOF and FT methodologies. Additionally whilst the metal line profiles of the other supergiants appear to be approximately Gaussian, those of Sk 191 have a profile closer to that characterising rotation (see, for example, Gray 1992). However the estimated atmospheric parameters and chemical composition of Sk 191 are consistent with those of the other targets in our SMC sample and hence we are unable to explain the cause of this enhanced rotation.
Hence we would appear to have a situation where effectively all our SMC supergiants have processed material at their surface with most of them having relatively small rotational
velocities. Recently, consideration has been given to the effects of
magnetic fields (Maeder & Meynet 2003,2004,2005). These
would appear to suppress mixing of nucleosynthetically
processed material to the stellar surface. However when magnetic and
thermal instabilities are also considered,
these favour the chemical transport of elements but also lead to larger
rotational velocities during the main sequence stage of, for example,
a 15
model considered by Maeder & Meynet (2005).
Hence although magnetic fields may influence the presence of
nucleosynthetically processed material at the surface of B-type
supergiants, current models do not appear to explain the relatively
low rotational velocity estimates found here.
Our principle conclusions are as follows:
Acknowledgements
We are grateful to the staff of the Anglo-Australian Telescope and the the European Southern Observatory for their assistance and for financial support from the UK Particle Physics and Astronomy Research Council. We would like to thank Prof. I.D. Howarth for useful comments and especially for alerting us to possible observational biases. S.S. and C.T. are grateful for the support of the Spanish Ministerio de Educación y Ciencia through the project AYA2004-08271-C02-01.