A&A 374, 1085-1091 (2001)
DOI: 10.1051/0004-6361:20010816
A. Cassatella1,2 - A. Altamore2 - M. Badiali1 - D. Cardini1
1 -
Istituto di Astrofisica Spaziale, CNR, Area di Ricerca Tor
Vergata, Via del Fosso del Cavaliere 100, 00133 Roma, Italy
2 -
Dipartimento di Fisica E. Amaldi, Università degli Studi Roma
Tre, Via della Vasca Navale 84, 00146 Roma, Italy
Received 28 March 2001 / Accepted 5 June 2001
Abstract
An investigation is carried out on the Wilson-Bappu effect
in the Mg II k line at 2796.34 Å. The work is based on a
selection of 230 stars observed by both the IUE and HIPPARCOS satellites, covering a wide range of spectral types (F to
M) and absolute visual magnitudes (-
). A
semi-automatic procedure is used to measure the line widths, which
applies also in the presence of strong central absorption reversal. The
Wilson-Bappu relationship here provided is considered to represent an
improvement over previous recent results for the considerably larger
data sample used, as well as for a proper consideration of the
measurement errors. No evidence has been found for a possible
dependence of the WB effect on stellar metallicity and effective
temperature.
Key words: stars: distances - stars: late-type - ultraviolet: general - line: profiles - catalogs
Several papers have been devoted to the study of the correlation between the visual absolute magnitude MV of intermediate and late-type stars, and the width of the Mg II k 2796.34 Å emission line (cf. Garcia-Alegre et al. 1981; Vladilo et al. 1987; Elgarøy 1988; Elgarøy et al. 1988; Scoville & Mena-Werth 1998, hereafter SMW; Elgaroy et al. 1999, hereafter EEL).
The width-luminosity relationship for the Mg II k line is qualitatively similar to the one found by Wilson & Bappu (1957) for the Ca II k optical line (see also recent work by Wallerstein et al. 1999), currently known as the Wilson-Bappu effect (hereafter WB). Given the similar excitation conditions of these lines, the dependence of their line widths on stellar luminosity or on other fundamental stellar parameters rests on the same physical processes. For a short review on models see Montes et al. (1994), references therein, and the recent work by Özeren et al. (1999).
In this paper we study the WB correlation for the Mg II k line using the widest possible sample of stars observed at high resolution with IUE, and having a reliable parallax determination from the HIPPARCOS experiment. More precisely, the purposes of the present paper are:
a) to refine the previous luminosity calibrations of the Mg II k line width by using an homogeneous set of measurements of a statistically significant number of stars covering also spectral types and luminosity classes poorly represented in previous investigations. Particular care has been taken to properly consider the measurement errors;
b) to clarify the problem of the possible dependence of the Mg II k line width on stellar effective temperature and metallicity discussed in previous papers.
Details about the definition of the present data sample, the observations, and the method used to measure the line widths are given in Sects. 2 and 3. The results and the conclusions are presented in Sects. 4 and 5, respectively.
A search for all stars with spectral types F, G, K and M with a known parallax from the HIPPARCOS experiment and observed at high resolution in the long wavelength camera by IUE provides 835 IUE spectra of 376 stars with a wide range of luminosity from main sequence to supergiants. From this initial sample we have purposely excluded some special classes of stars. Chromospherically active binary stars such as RS CVn and BY Dra stars were excluded since their Mg II profiles may be systematically broader than in non-active stars (Montes et al. 1994; Özeren et al. 1999). Mira variables and Cepheids were excluded because their V magnitudes and Mg II line profiles are variable and phase-dependent, and also because of the importance of shock waves associated with pulsation as a source of line excitation in these objects (see Gillet et al. 1985; Kraft 1960). Also excluded were binary stars having the Mg II lines broadened by peculiar processes (see for example the case of the M-type giant Mira Ceti, in which the emission lines arise from an optically thin disk around the white dwarf companion, as reported by Cassatella et al. 1985 and Reimers & Cassatella 1985). Finally, we have excluded rapid rotators and very active stars.
After a further selection based on data quality (see next section), we
obtained a final sample of 303 spectra of 230 stars. The sample
includes 11 F-type stars, 56 G-type stars, 133 K-type stars and 30 M-type stars. The range in absolute magnitude covered is about -
.
The most abundant are giant stars (about
50%) and main sequence stars (25%). The rest of the sample is
represented by Class I, II and IV stars, about equally distributed.
Representative examples which show the different typologies of the Mg II k profiles are given in Fig. 1. One can easily notice that the profiles differ significantly from case to case. The combined effects of the width of the emission component, and of the strength and wavelength position of the absorption cause, quite frequently, the widths of the emission to be not directly measurable on the line profile itself.
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Figure 1: Typical profiles of the Mg II k line. The vertical marks define the left and right wings of the emission line. The horizontal line indicates the FWHM of the profile. |
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We have then adopted a procedure which allows line widths to be
evaluated in the large majority of cases. It is based on the
reconstruction of the emission profile by fitting the observed
portions of the emission line wings with a Gaussian function. Let the
wavelength boundaries of the violet and red wings to be fitted be
to
and
to
,
respectively. These wavelengths were determined manually from
individual spectra and are indicated, in the order, with vertical bars
in the examples of Fig. 1. The spectra were re-binned
to increase the number of data points by a factor of 5, and slightly
smoothed. After subtraction of the local continuum and normalization
to the line peak emission, all spectra, irrespective of the typology
of the profile were fitted with a Gaussian function. Line widths were
defined as the full width of the fitted profile measured at half
maximum of the observed line peak.
This procedure is
strictly needed for a correct evaluation of the line widths in
the presence of strongly mutilated profiles, and is also useful in
improving the accuracy of the line width measurements when dealing with
noisy spectra. In the normal case of "well behaved'' profiles such as c),
d) and e) in Fig. 1, in which a direct measurement of
the line width can be obtained from the spectra, the procedure
provides values which are fully consistent, within the errors, with the
ones obtained through the traditional methods.
Examples of extreme, although quite frequent cases in which a direct measurement using traditional methods cannot be done, are the profiles a) and b) of Fig. 1, in which the central absorption divides the profile in two unequal parts, the flux peak of one of them being less than 50% of the other, or in profiles of the type shown in panel f), in which one of the wings is mutilated by superimposed absorption (on the red wing in this case).
It should be stressed that, according to the numerical simulations by Cheng et al. (1997), the fluxes and profiles of chromospheric resonance lines depend on the temperature structure and dynamics of the atmosphere, which may inevitably lead to line profiles that are rather different from Gaussian. Our choice to use a Gaussian profile is just an empirical approach, having only to improve the accuracy of the line width measurements through a reliable reconstruction of the emission line wings. This procedure is particularly useful in the presence of severe intrinsic or interstellar line absorption.
As a general rule, the results of the fit were rejected whenever the peak intensity of the fitted profile exceeded the peak of the Mg II k observed profile by more than a given amount, somewhat arbitrarily fixed to 180%. Also rejected were poor quality results where the relative error of the fit was larger than 10% (rms), and the results from strongly mutilated and asymmetric line profiles with a deep "central'' absorption dividing the profile in two halves, with one having a peak intensity less than 10% of the other (i.e. similar to the limiting case a) of Fig. 1).
As for the errors on measured line widths, we have estimated it to be approximatively equal to half of the sampling interval of the spectra, corresponding to 3.64 kms-1, for line profiles as in Figs. 1c-e, and twice this value for the other typologies.
The measured line widths W need to be corrected for instrumental
broadening. Assuming the IUE Point Spread Function for high
resolution spectra to be a Gaussian with a full width at half maximum
b=18 kms-1 at 2800 Å (Evans & Imhoff 1985), the true width
W0 can be written as:
For each of the 230 stars, the following information has been extracted from the Hipparcos Catalogue (ESA 1997): the Johnson Vmagnitude, the B-V colour and the corresponding error, the parallax with its error, the spectral type and luminosity class, and information on variability or duplicity. In a few cases (suitably labeled in Table A, described below) the B-V colour index, its error, and the spectral type were taken from the Strasbourgh Data Center because the values contained in the HIPPARCOS catalogue were not considered reliable enough.
Table A contains, for each object of
the final sample, the HIP and HD numbers, the spectral type and
luminosity class, the absolute magnitude MV and its error
,
the B-V colour index with the corresponding error
,
the IUE images used, the logarithm of the
Mg II k line width
with its error
.
The absolute magnitudes were computed without any
allowance for interstellar reddening. The errors in MV have been
derived from the errors in trigonometric parallaxes assuming a typical
error of
0.01 mag in the observed visual magnitudes.
The MV vs. B-V diagram of our sample of stars is shown in
Fig. 2. One can easily recognize that main sequence
stars with
,
as well as giant and supergiant stars, are
well represented in our sample.
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Figure 2: Colour-magnitude diagram for the 230 stars in our sample. |
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To determine the coefficients a and b of the Mg II k Wilson-Bappu
relationship
We considered regression solutions which take into account the
measurement errors in only one of the variables, as often done in the
literature, or in both variables.
Unweighted regression lines were also computed, although only for
comparison purposes. In any case, the correlation coefficient between
the MV and
data is r=-0.94.
The results obtained in the different cases are reported in Table 1. Column 1 contains the label assigned to the fit. In
Col. 2 we specify the variable taken as independent. The variables
for which the errors were taken into account are indicated in Col. 3. In particular, fits A and B were done without taking into
account the measurement errors; in fits C and D only the errors on
MV or
were considered, respectively; in the
fits E and F the errors on both variables were
taken into account. The regression coefficients a and b together
with the corresponding errors are given in Cols. 4 and 5. They are
consistently given in the format appearing in Eq. (2),
ever a switch of variables has been performed. In that
case, the uncertainties on the coefficients have been accordingly
propagated.
As shown in Table 1, the a and b coefficients
derived by switching the independent variable (
or MV),
vary significantly if allowance is made for the measurement errors
in one variable only. This is the case of fit C, in which the
independent variable is
and only the errors on MV are
taken into account, and fit D, in which the independent variable is
MV and only the errors in
are taken into account. On the
contrary, fits E and F, in which the errors on both variables are
considered, provide fully consistent coefficients, irrespective of the
variable considered as independent, as expected (Akritas & Bershady 1996).
The results in Table 1 show the important role played by the method used to analyse the data. This point is further discussed in Sect. 4.4.
We consider fit E or, equivalently, fit F in Table 1 as the most representative of the WB relationship,
since the measurement errors on both variables were taken into
account. In conclusion, the proposed WB relationship for the
Mg II k line is:
FIT | I.V. | ![]() |
a | b |
A | ![]() |
none |
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B | MV | none |
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C | ![]() |
MV |
![]() |
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D | MV | ![]() |
![]() |
![]() |
E | ![]() |
both |
![]() |
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F | MV | both |
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Notes: Column 2, labeled I.V., specifies the variable, or MV, taken as "independent variable''. Column 3, labeled
,
indicates for which of the variables the measurement errors were taken
into account. The coefficients of the linear fit
,
together with the corresponding errors are given in Cols. 4
and 5. The correlation coefficient for all fits is r= -0.94.
![]() |
Figure 3: The Wilson-Bappu effect in the Mg II k line for the 230 stars in our sample. The regression line labeled E corresponds to the proposed relationship in Eq. (3), obtained by taking into account the measurement errors on both variables. For comparison, the regression line C (see Table 1) is also provided, which has been obtained from the same data by taking into account only the errors on MV. |
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We investigated the possible dependence of
on stellar
effective temperature, an item which has been addressed in previous
recent analyses and is still under discussion.
Figure 4 shows our measurements of
plotted as a
function of the B-V colour index, taken to be a temperature
indicator. This representation is obviously equivalent to the
colour-magnitude diagram in Fig. 2 by virtue of the
Wilson-Bappu linear relationship. The figure shows, in particular,
that
is a decreasing function of B-V for main sequence
stars (indicated as black points), while it is an increasing function
of B-V for giants and supergiants (gray points on the upper part of
the diagram). Therefore, the weak inverse relationship between line
widths and stellar temperature found by SMW (see their Fig. 5) simply
reflects the distribution of their sample of stars in the HR diagram,
which are mostly giants and supergiants.
An effective way to face the problem is to concentrate on stars in
restricted ranges of MV (so that any dependence on luminosity is
taken out), and to compare their distribution in B-V and in
.
To start, we selected the stars having
(31 stars). The results, reported in Fig. 5,
show clearly that B-V is distributed over a wide range of values
(
), while the distribution in
has a
very prominent peak around
.
Line widths appear then to
be substantially insensitive to the colour index, i.e. to the stellar
effective temperature, at least in the above range of MV. Tests
made with stars lying in different ranges of MV show a pattern
which is similar to the one in Fig. 5, but the results
cannot be considered as equally conclusive due to the paucity of stars
available, and/or to the narrow B-V range covered.
![]() |
Figure 4:
Dependence of line width ![]() |
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Using a different approach, previously followed by EEL, we investigated whether a different width-luminosity relationship applies to different spectral types. In Fig. 6, we compare our results of the width-magnitude relationships for G, K and M stars (solid lines) with the regression line corresponding to the full set of stars in our sample (Eq. (3); dashed line). The corresponding coefficients together with the number of stars used are given in Table 2. The fits were done by taking into account the errors on both variables. No attempt has been made to fit F-type star's data, which are too few for a statistical approach. In any case, not even better statistics could help in view of the intrinsically small range of MV covered by F-type stars.
As it appears from Table 2, the coefficients for K-type stars (133 objects) are very similar to the ones in Eq. (3), as expected, given that these stars represent more than 50% of the entire sample.
The poorest statistics are those of M-type stars (30 objects), which
are also unevenly distributed in MV (see the gap
). Given these limitations, the WB coefficients
are strongly dependent on the stars considered as input, so that the
differences appearing from the table are not significant enough.
As for G-type stars (56 objects), we find a significantly flatter
slope than for K-type stars (133 objects). In particular, the line
widths of the G-type stars in Fig. 6 with
are broader than one would expect from the WB relation in
Eq. (3). A similar trend has been reported by EEL for a
sample of 22 G-type stars. In this regard, it is interesting to
mention that Wallerstein et al. (1999), in their study of the WB
effect in the Ca II k line, also found a group of luminous G-type
stars having substantially broader Ca II lines than predicted
by the WB relation. Whether the apparently different WB relation for
G-type stars should be regarded as a true temperature effect as
claimed by EEL, or is due to other causes (including selection
effects), cannot be clarified at this stage.
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Figure 5:
Histogram of B-V (top) and ![]() ![]() |
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Sp. Type | N | a | b | r |
G-stars | 56 |
![]() |
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-0.91 |
K-stars | 133 |
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-0.93 |
M-stars | 30 |
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![]() |
-0.96 |
Notes: Columns 1 to 5 give, in order, the spectral type considered, the number of stars used, the regression coefficients with the corresponding errors, and the linear correlation coefficient.
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Figure 6: The Wilson-Bappu relationship for different spectral types. |
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Figure 7:
The departure of the observed line width ![]() |
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We investigated possible departures from the Wilson-Bappu
relationship in Eq. (3) due to a dependence of the
Mg II k line width on metallicity. To this purpose, we looked
for [Fe/H] literature data and found information for 156 stars of
our sample. For 12 stars, the [Fe/H] values are from Carney et al. (1994), and for the remaining stars from Cayrel de Strobel et al. (1997). This latter being a compilation from several sources, more
than one determination was in general available for a given star. In
these cases, the most recent one has been adopted, and assumed to be the
most reliable. In Fig. 7 we plot as a function of
[Fe/H] the difference
between the measured line
width
and the value expected from the Wilson-Bappu relationship
in Eq. (3). It clearly appears from the figure that, in
spite of the quite large range in metallicity, there is no
trend for the line widths to show any systematic deviation. The
correlation coefficient between [Fe/H] and the deviations
from Eq. (3) is indeed very poor:
r=0.16.
As a further confirmation of this point, we divided the 156 stars into
four groups according to their metallicity. For each group we made a
linear fit of MV as a function of
(allowing for the
errors on both variables). The metallicity groups were defined as
follows:
Fe/H
(23 stars),
Fe/H
(56 stars),
Fe/H
(54 stars), and
Fe/H] > +0.1 (23 stars). Slightly different slopes and
intercepts of the WB relationship were found, but no dependence on
metallicity was found.
The results of the present study are compared with the literature values of EEL and SMW in Table 3 which gives, for each author, the number of stars used, the coefficients a and b(with their errors, if available), and the range of MV values from which the WB relationship has been derived. Reference is made only to papers making use of the same definition of line width (FWHM in this case), and based on precise HIPPARCOS parallaxes.
Reference | N | a | b | MV range |
Eq. (3) (Fit E) | 230 |
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-5.4 9.0 |
Fit C | 230 |
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-5.4 9.0 |
EEL 1999 | 65 |
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-5.1 12.3 |
SMW 1998 | 92 | 28.63 | -13.53 | -2.2 7.5 |
Notes: EEL and SMW stand for Elgarøy et al. (1999) and Scoville & Mena-Werth (1998), respectively. Column 2 provides the number of stars used by the different authors. Fits E (the proposed WB relationship) and C are the same as in Table 1.
Since our sample contains 84 stars in common with SMW and 34 in common with EEL, it is useful to make a direct comparison between the line width measurements.
As it appears from the top panel of Fig. 8, our measurements of W0 are in a good agreement with SMW's values for the broadest lines, but there is an increasing discrepancy for narrower and narrower lines, whose widths are systematically overestimated by SMW, as would happen if their measurements were not corrected for the instrumental width (Eq. (1)), an item which is not specified in their paper. Compared with ours, the SMW's measurements of W are about 8% higher, on average. In spite of this discrepancy, one can see from Table 3 that SMW's fit is very close to our fit C, but differs strongly from the others. This might suggest that it has been obtained by weighting only on MV, as fit C.
The agreement with EEL's measurements of W (Fig. 8, bottom panel) is substantially better, amounting to 2%, if 4 discrepant data points are excluded (note that for such a comparison, we have averaged the two determinations of W obtained by EEL with different methods - see their Table 1). As one can easily appreciate from Table 3, our best estimate of the WB effect in Eq. (3) is compatible with EEL's fit, at least within their large error bars. Unfortunately, no information about the treatment of measurement errors is given by EEL. The only authors who declared to have taken the measurement errors on both variables into account are Vladilo et al. (1987), but their results cannot be compared with ours due to their much more limited range in MV (-1 to 7). We should note also that the values of MV here adopted coincide with those of SMW, while there is, on average, a systematic difference of 0.12 mag, of unknown origin, between EEL's values of MV and ours.
In conclusion, given the substantially larger amount of observations analysed in this work, and the proper way to take the measurement errors into account, we consider Eq. (3) to adequately represents the WB effect for the Mg II k line, at least within the observational limits set by the IUE and HIPPARCOS experiments.
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Figure 8: Comparison of our line width measurements with those of Scoville & Mena-Werth (1998; labeled SMW) and of Elgarøy et al. (1999; labeled EEL). |
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In this paper we provide an accurate Wilson-Bappu relationship for
the Mg II k line (see Eq. (3)) based on IUE
and HIPPARCOS observations, which can be used to estimate
stellar distances over a wide range of absolute magnitudes:
.
The sample of stars (230 objects) is considerably
larger than the ones used in previous works.
It has been shown that the coefficients of the WB relationship critically depend upon the choice of the method of analysis. In our case, different fitting algorithms have been tested and the one taking into account the measurement errors on both variables has been adopted as the most suitable.
In spite of the more accurate method and of the wider sample of stars
used, we have not found any observational confirmation to the claimed
dependence of the WB effect on effective temperature. On the
contrary, we have found arguments against such an effect, such as the one
provided in the histograms in Fig. 5, which show that the
Mg II line width is substantially the same for stars in a
narrow range of MV (
), in spite of the
wide range of B-V values. In any case, on the basis of theoretical
considerations, the temperature effect is expected to be small
(Reimers 1973; Özeren et al. 1999).
We also investigated possible systematic deviations caused by differences in metallicity. Using metallicity data for 156 stars in our sample, we came to the conclusion that such an effect is not present, at least within the accuracy of the measurements.
Given the difficulty of disentangling the metallicity and temperature effects using the present sample of field stars, it would be important to extend the analysis to homogeneous populations of stars such as galactic clusters. In general, observations by newly proposed space missions (e.g. GAIA) will hopefully enrich our knowledge on parallaxes of very large numbers of stars and clarify the problems left open.
Acknowledgements
We would like to thank Prof. M. J. Fernández-Figueroa and Drs. Anna Marenzi, Vittoria Caloi, and R. González-Riestra for useful comments and advice.