A&A 460, 843-853 (2006)
DOI: 10.1051/0004-6361:20065853
M. Wittkowski1 - C. A. Hummel2 - J. P. Aufdenberg3 - V. Roccatagliata4
1 - European Southern Observatory, Karl-Schwarzschild-Str. 2,
85748 Garching bei München, Germany
2 -
European Southern Observatory, Casilla 19001, Santiago 19, Chile
3 -
National Optical Astronomy Observatory, 950 North Cherry Avenue,
Tucson, AZ 85719, USA
4 -
Max-Planck-Institut für Astronomie, Königsstuhl 17,
69117 Heidelberg, Germany
Received 17 June 2006 / Accepted 15 September 2006
Abstract
Context. Optical interferometry allows a measurement of the intensity profile across a stellar disc, leading to a direct test and calibration of theoretical model atmospheres as well as to a precise determination of fundamental stellar parameters.
Aims. We present a comparison of the visual and near-infrared intensity profile of the M0 giant Sagittae to plane-parallel ATLAS 9 as well as to plane-parallel and spherical PHOENIX model atmospheres.
Methods. We use previously described visual interferometric data obtained with the Navy Prototype Optical Interferometer (NPOI) in July 2000. We apply the recently developed technique of coherent integration, and thereby obtain visibility data of more spectral channels (526-852 nm) and with higher precision than before. In addition, we employ new measurements of the near-infrared K-band (2200 nm) diameter of
Sagittae obtained with the instrument VINCI at the ESO VLT Interferometer (VLTI) in 2002.
Results. The spherical PHOENIX model leads to a precise definition of the Rosseland angular diameter and a consistent high-precision diameter value for our NPOI and VLTI/VINCI data sets of
0.02 mas, with the Hipparcos parallax corresponding to
,
and with the bolometric flux corresponding to an effective temperature
55 K. Our visual visibility data close to the first minimum and in the second lobe constrain the limb-darkening effect and are generally consistent with the model atmosphere predictions. The visual closure phases exhibit a smooth transition between 0 and
.
Conclusions. The agreement between the NPOI and VINCI diameter values increases the confidence in the model atmosphere predictions from optical to near-infrared wavelengths as well as in the calibration and accuracy of both interferometric facilities. The consistent night-by-night diameter values of VINCI give additional confidence in the given uncertainties. The closure phases suggest a slight deviation from circular symmetry, which may be due to surface features, an asymmetric extended layer, or a faint unknown companion.
Key words: techniques: interferometric - stars: late-type -
stars: AGB and post-AGB - stars: fundamental
parameters - stars: atmospheres - stars: individual:
Sagittae
Cool giants on the red giant branch (RGB) and asymptotic giant branch (AGB) are very luminous and extended, have a low surface temperature, and their atmospheres can thus be rich in molecules. Cool giants are the most important source of dust formation and its delivery to the interstellar medium. The detailed structure of their extended atmospheres, including the effects from circumstellar molecular and dust layers, are still a matter of investigation and debate (cf., e.g. Scholz 1985,1998,2001; Perrin et al. 2004; Ohnaka 2004a; Ireland & Scholz 2006).
Theoretical atmosphere models predict in general the spectrum emerging from every point of a stellar disc. Optical interferometry provides the strongest observational constraint of this prediction by resolving the stellar disc. In addition, the constraints on the intensity profiles allow us to find meaningful definitions of the stellar radius and its precise measurement.
For regular cool non-pulsating giants, the centre-to-limb variation (CLV) is mainly characterised by the limb-darkening effect, which is an effect of the vertical temperature profile of the stellar atmosphere. The strength of the limb-darkening can be probed by optical interferometry in two ways (cf., e.g. Hanbury Brown et al. 1974; Quirrenbach et al. 1996; Burns et al. 1997; Hajian et al. 1998; Wittkowski et al. 2001,2004; Aufdenberg et al. 2005): (1) by measuring variations of an equivalent uniform disc diameter (i.e. the uniform disc that has the same integral flux as the true intensity profile) as a function of wavelength, and (2) by directly constraining the star's intensity profile in the second and higher lobes of the visibility function at one or several bandpasses.
It was found that pulsating giants as well as supergiants may exhibit more complex intensity profiles at near- and mid-infrared wavelengths, showing Gaussian-shaped intensity profiles, tail-like extensions to a photospheric intensity profile, and multiple components, such as a photosphere plus a circumstellar shell (cf., e.g. Woodruff et al. 2004; Ohnaka 2004a,b; Perrin et al. 2004,2005; Fedele et al. 2005). Additionally, observed intensity profiles might be affected by dust shells (e.g. Ohnaka et al. 2005; Ireland & Scholz 2006) or horizontal surface inhomogeneities (e.g. Burns et al. 1997).
In Wittkowski et al. (2001, hereafter Paper I), we
used the Navy Prototype Optical Interferometer (NPOI,
Armstrong et al. 1998), used the method of baseline
bootstrapping (cf. Hajian et al. 1998), and developed
improved methods of compensation of noise and detection bias terms,
in order to obtain precise visual visibility measurements in the second lobe
of the visibility function for three cool giants.
We found agreement with predictions by plane-parallel
ATLAS 9 (1993) model atmospheres within the obtained
wavelength range and precision.
Thereby, the strength of the
limb-darkening effect and the stars' fundamental parameters were constrained.
Aufdenberg & Hauschildt (2003) compared one of the NPOI
observations of Sagittae from Paper I
to a spherical PHOENIX (1999) model atmosphere
and found agreement.
In Wittkowski et al. (2004, hereafter Paper II), we
directly measured the limb-darkening effect of the M4 giant
Phoenicis
using the ESO Very Large Telescope Interferometer (VLTI) in the near-infrared
K-band, confronted the observations with predictions by independently
constructed ATLAS 9 and PHOENIX model atmospheres, and found
agreement with all considered models.
Recently, Hummel et al. (2003) developed the method of coherent integration and its application to NPOI data in order to increase the precision of visibility measurements. This method was recently applied by Peterson et al. (2006a,b) to NPOI observations of Altair and Vega.
Here, we reanalyse the NPOI data of the M0 giant Sagittae (HR 7635, HD 189319), the
brightest of the targets in Paper I, using the newly developed method
of coherent integration.
We obtain visibility data with higher precision than in Paper I, and - due
to the lower noise - are able to make use of more spectral channels toward
the blue end of NPOI's wavelength range. Now, the wider wavelength range
covers 526-852 nm, compared to 649-852 nm in Paper I.
We thus also increase our maximum spatial resolution from
3.3 mas to
2.7 mas, which gives important additional
visibility data in the second lobe that are sensitive to the limb-darkening
effect.
In addition, we observed
Sagittae with the ESO Very Large
Telescope Interferometer (VLTI) and its K-band instrument VINCI, in order
to compare results derived from different interferometric facilities, and
to probe the consistency of the wavelength-independent Rosseland diameter
from visual to near-infrared wavelengths.
The cool giant Sagittae does not appear
in the Combined General Catalogue of Variable
Stars (Samus et al. 2004), indicating that it lacks strong
photometric variability. Thus, it is a good target for the purpose of
calibrating model atmospheres and deriving high-precision fundamental
parameters. The spectral type has been listed as K5-M0 III by
Morgan & Keenan (1973), and been revised to M0 III by
Keenan & McNeil (1989).
Wisniewski & Morrison (private communication) confirm by means of optical
echelle spectra recently obtained at Ritter Observatory
that
Sagittae's
spectrum closely resembles that of the MK standard
UMa (M0 III).
We determine the bolometric flux of
Sagittae
to
10-9 W/m2 by means of
a spline fit and integration of the narrow-band spectrophotometric data
by Alekseeva et al. (1997) covering 405 nm to 1080 nm
complemented by broadband photometry shortward and longward of
this range from the 13-colour photometry by Johnson et al. (1975).
The values for
of
10-9 W/m2and
10-9 W/m2 by Alonso et al. (1999)
and Mozurkewich et al. (2003), respectively, are derived
from broad-band photometry alone and likely overestimate
because of a too sparse sampling of the visual spectrum including the TiO band heads and other features. The limb-darkened angular diameter of
Sagittae has been determined
in Paper I to be 6.18
0.07 mas, based on a comparison of NPOI visibility data to ATLAS 9 model atmospheres. This value corresponds
to a limb-darkened radius of 56
4
,
derived with the
Hipparcos parallax of 11.90
0.71 mas (Perryman & ESA 1997).
These values of angular diameter, absolute radius, and bolometric flux
constrain the effective temperature
to
3768 K
70 K, and the
luminosity to
2.75
0.10. Placing
Sagittae
on the Hertzsprung Russel diagram using these values, and comparing to
stellar evolutionary tracks by Girardi et al. (2000) as
in Paper II (Fig. 1 of Paper II) we can estimate a mass
of M= 1.3
0.4
,
and thus a surface gravity
of
= 1.06
0.22. These values are used as an a priori estimate
for our analysis and will be refined in the conclusions.
We reanalyse the visual
Sagittae data in Paper I obtained
with NPOI on July 21, 2000.
The centre (C), east (E) and west (W) siderostats of
the astrometric sub-array of NPOI were used to obtain baselines
of ground length 18.9 m (CE), 22.2 m (CW), and 37.5 m (EW).
The data were recorded in 32 spectral channels of equal width
in wavenumber and covering the band from
450 nm to 850 nm.
Due to low photon count rates only the 10 reddest channels could be
used in Paper I (covering 649 nm to 852 nm).
We reanalyse the same raw data using the newly developed
coherent integration algorithm as first described by
Hummel et al. (2003).
The details of our new analysis are described below.
The benefits of the new analysis include an improved
signal-to-noise-ratio (SNR) of the visibility data on the long EW baseline,
as well as a much improved SNR of the triple amplitudes and phases.
These improvements enable us to use the 20 reddest channels in the present
paper, now covering 526 nm to 852 nm.
Therefore, in the so-called incoherent analysis, as used in Paper I,
the bin counts of each 2 ms sample would be Fourier transformed, and an unbiased estimate for the squared visibility amplitude derived as
follows (see Shao et al. 1988):
The signal to noise ratio SNR of the squared visibility estimator is as
follows (Shao et al. 1988):
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For the observations described here on Sagittae, the
importance of coherent integration follows from the very small
visibility amplitudes measured on the long EW baseline since it
samples the second lobe of the Fourier transform of the stellar
disc brightness profile. This baseline is therefore most sensitive
to stellar limb darkening, the focus of this work.
While the low visibility amplitudes on this baseline would prevent
a precise determination of the group delay, the following paragraph
describes how to obtain this estimate in a different way.
In the case described here, the EW baseline is boot-strapped by the CE and CW baselines. Therefore, the fringe delay on the EW baseline is equal to the difference of the delay between the other two baselines, and can thus be computed in this way without using the measurement on the EW baseline itself.
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Figure 1: Incoherent integration. Off-fringe squared visibility amplitude, i.e. the visibility bias that remains for data off the fringe packet and that is compensated after the Z2 compensation (see text for more details). This bias can be described by a power law as a function of photon rate P. As an example, the residual bias is shown for the four reddest channels on the EW baseline. Data are 2 ms incoherent integrations from July 22. Channels 1 though 4 use symbols plus, star, diamond, and triangle, respectively. Power-law fit coefficients are given for each channel, in the same order as shown on the corresponding plot for the coherent analysis in Fig. 2. |
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Figure 2: Coherent integration. As Fig. 1, but for 200 ms coherent integrations as used in this paper. For reasons of comparison, the bin counts are renormalised to 2 ms intervals. It is an additional benefit of the coherent integration that this residual bias shown here is clearly reduced compared to the incoherent integration in Fig. 1. |
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Figure 3:
Squared visibility amplitudes of ![]() |
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Figure 4: As Fig. 3, but showing the squared visibility amplitudes on the NPOI CW baseline. |
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Figure 5: As Fig. 3, but showing the squared visibility amplitudes on the NPOI CE baseline. |
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Figure 6: As Fig. 3, but showing the NPOI triple amplitudes. |
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Figure 7:
As Fig. 3, but showing the
NPOI closure phases. Note that the slope of the model flip from 0 to ![]() |
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As a first characterisation of our NPOI data, we use models of
a uniform disc (UD, I=1 for
,
I=0 otherwise),
and a fully darkened disc (FDD,
).
Here, I is the intensity,
(or
)
the cosine of the angle between the line of sight and the normal
of the surface element of the star (R the stellar radius, r the
distance from the centre of the disc).
Monochromatic synthetic visibility values V were obtained
for the UD and FDD cases, and subsequently integrated over the bandpass
of each NPOI spectral channel (covering frequencies
to
) as
Note that in the case of NPOI the monochromatic visibility amplitudes are integrated before building the square, while for VLTI/VINCI the monochromatic squared visibility amplitudes are integrated (cf. Paper II). The reason for the difference is that the data processing of NPOI first integrates the photons on the APDs and the squared visibility is computed from the already integrated bin counts, while for VLTI/VINCI first the full powerspectrum is computed and integrated thereafter. This leads to noticeable differences, in particular around the minima of the visibility function.
Best fitting angular diameters
are derived from a least square optimisation. The resulting
values are shown in Table 1 together with the
reduced
values. The number of degrees of freedom is
644 (7 observations times 19 spectral channels times (3 squared visibility
plus 1 triple amplitude plus 1 closure phase), minus 21 values flagged
for quality).
The formal errors of the obtained diameter values are of the order of
0.01 mas, and are small compared to
calibration uncertainties that are estimated to
1%
0.06 mas.
Total errors are thus
0.06 mas.
Table 1:
Fit results of our NPOI data to models of a uniform disc (UD)
and a fully darkened (FDD) disc. The formal errors of the diameter values
are 0.01 mas, additional calibration uncertainties are
0.06 mas, total errors thus
0.06 mas.
Figures 3-7 show the obtained NPOI squared visibility amplitudes on baselines EW, CE, CW, the NPOI triple amplitudes, and the NPOI closure phases, respectively. Also shown are the model atmosphere predictions as described below in Sect. 4.
The gain with respect to Paper I in the signal-to-noise ratio and in the number of usable spectral channels is thanks to the method of coherent integration, as can be seen by comparing these Figs. to the results based on incoherent averaging (Fig. 3 of Paper I).
The results (Figs. 3-7 and Table 1) show that the visual intensity profile of Sge
is limb-darkened, clearly closer to a FDD model than to a UD model, while
both of these simple descriptions do not provide a very good representation
of our data. A detailed comparison of our visibility data to model atmosphere predictions
is discussed below in Sect. 4.
The near-infrared K-band interferometric data of Sagittae were
obtained with the ESO Very Large Telescope Interferometer
(VLTI, Glindemann et al. 2003), the
instrument VINCI (Kervella et al. 2003), and the two VLTI test siderostats on June 28, July 8, July 11, July 15, August 8, September 12, and September 18, 2002. These data are public commissioning data released from the
VLTI
.
The VLTI stations E0 and G1 forming a ground baseline length of 66 m were
used for all our observations.
The observations were repeated during 7 different nights spread over
more than 2 months in order to compute the night-to-night variation of
the obtained diameter and thereby to estimate the calibration uncertainty
caused by different atmospheric and possibly instrumental conditions.
All data were obtained as series of typically 100 or 500 interferograms
with a scan length of 250
m and a fringe frequency of 295 Hz.
The stars 56 Aquilae and 31 Orionis were used as primary
calibration stars and were observed in each of our observation nights
close in time to the
Sagittae observations. A number of additional calibration stars
observed during these nights were used as secondary calibrators
for
Sagittae. The characteristics of all calibration stars used
are taken from Bordé et al. (2002,
based on Cohen et al. 1999) and are listed in
Table 2.
We computed mean coherence factors for each series of interferograms using the VINCI data reduction software (version 3.0) by Kervella et al. (2004) employing the results based on the wavelets power spectral density. The calibration of the visibility values was performed as in Paper II using a weighted average of the transfer function values obtained during the night.
Table 3 shows the observational details together
with the resulting calibrated squared visibility amplitudes for each
series of Sagittae interferograms. The listed errors include
the scatter of the coherence factors of the single scans, the errors of
the adopted diameter values of the calibration stars, and the variation
of the obtained transfer function during each observing night.
As a first characterisation of the K-band stellar angular diameter,
we compute the equivalent UD ()
and FDD (
)
diameters,
as for our NPOI data. The broad-band squared visibility
amplitudes for the VINCI bandpass (K-band) are computed as
Note that in the case of VLTI/VINCI the squared visibility amplitudes are integrated (Eq. (4)), while in the case of NPOI the visibility amplitudes have first to be integrated and squared thereafter (Eq. (3)), see the note in Sect. 2.3.
Table 4 lists the obtained diameter values for our VINCI data.
Figure 8 shows our obtained VINCI squared visibility
amplitudes of Sge together with the best-fitting models
with parameters listed below in Table 5.
Since our VINCI data cover only one bandpass and only data of the first lobe of the visibility function, it is - contrary to our NPOI data -
not feasible to constrain the limb-darkening effect solely based
on these VINCI data. This is also reflected by equal
values
obtained for UD and FDD models as well as by the virtually identical model
visibility curves in Fig. 8.
The increased equivalent UD diameter with respect to the shorter NPOI wavelengths is consistent with the general trend of decreasing strength of the limb-darkening effect with increasing wavelength. A detailed comparison of our data to model atmospheres follows below in Sect. 4.
This confirms that our diameter value
is reliable and that our estimate of uncertainties
is realistic.
The obtained high-precision (0.3%) UD and FDD diameter values
of
0.02 mas
and
0.02 mas can thus be used without
further uncertainties.
Additional possible systematic errors that are constant over time scales larger than covered by our analysis, i.e. about 2 months, can not be ruled out. Such systematic errors could in principle be related to the calibration of the interferometric array and the instrument, such as the calibration of the baseline length or the effective wavelength. Such uncertainties are not expected to represent a considerable source of error.
We compare our measured visibility data to predictions by theoretical
model atmospheres in order to calibrate and test these models, and to
derive fundamental stellar parameters of Sge.
We use plane-parallel ATLAS 9 (Kurucz 1993)
as well as plane-parallel and spherical PHOENIX
(Hauschildt et al. 1999) model atmospheres to calculate
synthetic visibility data, as done in Papers I and II.
We refer to the descriptions in Papers I and II for more details on the
employed model atmosphere files and their use.
Differences between ATLAS 9 and PHOENIX models include
different opacity tables, a different sampling of the model wavelengths,
a different sampling of the angles ( values),
and convective overshooting that is taken into account for ATLAS 9,
but not for PHOENIX.
The most important stellar input parameters for the
plane-parallel models are effective
temperature
and surface gravity
,
and for the spherical PHOENIX models in addition the mass M. We use solar chemical abundance, as
appropriate for local cool giants. The values of
,
,
and M are already well constrained for
Sagittae, as outlined
in the Introduction (Sect. 1), namely
3768 K,
1.06,
1.3
.
The closest model of the ATLAS 9 grid is the one for
3750 K and
= 1.0 (see Papers I and II for details
on the model file used).
We have constructed a corresponding plane-parallel PHOENIX model
atmosphere with parameters
3750 K,
= 1, as well
as a spherical PHOENIX model atmosphere
with parameters
3750 K,
= 1.0,
and M= 1.3 (see Paper II for details on the model files).
We take into full account the bandpasses of our observations by integrating
the synthetic visibility data of monochromatic intensity profiles for
each spectral channel of NPOI and for the K-bandpass of VLTI/VINCI.
This ensures that the synthetic visibility values fully resemble the
true bandpasses used for the observations and that they include the
model-predicted effects from atomic lines and molecular bands for each
of our spectral channels. Monochromatic visibility values at frequency are calculated as (cf. Davis et al. 2000; Eq. (6) from Paper I;
Eq. (1) from Paper II)
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Table 2:
Characteristics of the stars that were used as calibration stars
for our VLTI/VINCI observations of Sagittae. Listed are the spectral
type, the K-band magnitude, the uniform-disc diameter and its error, and
the effective temperature, all from Bordé et al. (2002; based
on Cohen et al. 1999).
Table 3:
Details of our VLTI/VINCI observations of Sagittae
(date and time of observation, spatial frequency, azimuth angle of the
projected baseline (E of N)), together with the measured squared
visibility amplitudes and their errors. The last column denotes the
number of successfully processed interferograms for each series.
The effective wavelength for our
Sagittae observations
is
2.19
m. For each date of observation, we list the
equivalent uniform disc (UD) diameter obtained from only the data of
the specific night. Using all data together, we obtain an equivalent
UD diameter of
0.02 mas, or an equivalent
FDD diameter of
0.02 mas.
Table 4: Fit results of our VINCI data to UD and FDD models.
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Figure 8:
Measured ![]() |
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Broad-band visibility values integrated over the bandpasses of our NPOI spectral channels and VINCI sensitivity function are calculated using Eqs. (3) and (4), respectively.
We calculate the best fitting limb-darkened (0% intensity)
angular diameter
as described above for each
of these three model atmospheres (plane-parallel ATLAS 9,
plane-parallel PHOENIX, spherical PHOENIX models)
and for each of our two data sets (NPOI and VLTI/VINCI).
For each model fit, we treat
as the only
free parameter, and use all NPOI visibility data (squared visibility
amplitudes, triple amplitudes, and closure phases), a total of 644 data points, simultaneously. The fit is a standard least-square fit, and optimises the total
value of all 644 NPOI data points.
As discussed in Sect. 3.4 of Paper II, models based on plane-parallel
geometry are optically thick from all viewing angles, the intensity
steeply dropping to 0 directly at the stellar limb. A plane-parallel
model has, by definition, an atmosphere with an negligible thickness
relative to the stellar radius.
Therefore, since any depth in such an atmosphere has a radius
effectively equal to the stellar radius, a Rosseland diameter
in such a geometry is equivalent to the limb-darkened (or 0% intensity) diameter
.
Intensity profiles based on atmosphere models with spherical geometry,
exhibit an inflection point and steepest decrease at radii smaller than
the outermost model radius. The Rosseland mean optical depth increases
slowly for increasing angles .
Here, the ratio of the Rosseland diameter
and
the 0% intensity diameter
differs from unity, and
this ratio
is model-dependent and can be
derived from the structure of the model atmosphere. This value depends
on the definition of the outermost radius R0 of the model. R0 of
the spherical PHOENIX model used here is given by the
standard boundary conditions, which are a continuum optical depth
of 1e-6 at 1.2
m and an outer gas pressure of 1e-4 dynes/cm2(see Paper II).
Table 5:
Results for the fit of ATLAS 9 and PHOENIX model
atmospheres to our interferometric VLTI/VINCI and NPOI data sets
of Sagittae.
The corresponding synthetic visibility data are compared to the measured data in Figs. 3 to 7 for our NPOI data set and in Fig. 8 for our VLTI/VINCI data set.
The plane-parallel PHOENIX model leaves the near-infrared
VLTI/VINCI diameter quasi unchanged (6.09 mas compared to 6.08 mas)
with respect to the plane-parallel ATLAS model, while it results in a smaller (by 1
)
visual NPOI diameter compared to the ATLAS model. A comparison
of these models' temperature structures reveals that the ATLAS model exhibits a steeper temperature gradient relative to the
plane-parallel PHOENIX near Rosseland optical depth unity. This
steeper gradient leads to stronger limb darkening at NPOI wavelengths
and consequently a larger best angular diameter. The shallower
temperature gradient of the plane-parallel PHOENIX model leads
to a better agreement between the NPOI and VLTI/VINCI diameters.
Finally, the spherical PHOENIX model leads to a Rosseland
angular diameter of
0.06 mas for the NPOI data set and
0.02 mas for the VLTI/VINCI data set. The larger best-fit diameters for the plane-parallel PHOENIX model compared to the spherical PHOENIX model appears to be
due to model geometry. The agreement of NPOI and VLTI/VINCI data sets
within their 1
error bars gives confidence in both, the atmosphere
models and the accuracy of the results from NPOI and VLTI/VINCI.
The weighted mean of the NPOI and VLTI/VINCI results is
0.02 mas.
These differences are most evident in (1) a lower second maximum of the measured visibility function with respect to the model prediction on the EW baseline (Fig. 3), and (2) a flattened measured visibility function with respect to the model predictions at the blue end on the CW baseline (Fig. 4). It is not yet clear if and by how far these deviations of measured and synthetic visibility functions indicate different details of the limb-darkening effect at visual spectral channels, or if they are caused by additional calibration uncertainties that are not included in the error bars. In particular the flattening of the measured visibility function at the bluest spectral channels on the CW baseline can most likely be explained by additional calibration uncertainties of our NPOI data, as the instrumental transfer function for these data exhibits a drop which may not be fully compensated.
The obtained diameter values in Table 5 are not affected by possible calibration uncertainties since the best-fitting diameter for any given model atmosphere is mostly constrained by the position of the first minimum and the global shape of the visibility curve.
At optical wavelengths including all our NPOI spectral channels, TiO absorption bands are very important for the modelling of atmospheres of cool giants. It has been shown that the use of different line list combinations of TiO and H2O leads to significantly different model structures and spectra, in particular in the optical where TiO bands are important (Allard et al. 2000). A possible explanation for differences between our visibility data and the model predictions could thus also be mismatching opacity tables for the TiO bands and/or a mismatching spatial structure of the layers where TiO molecules reside.
In order to estimate the effect of a lower model strength of the limb-darkening
effect on the obtained diameter value, we used a spherical PHOENIX model
with
K instead of our favourite model with
K (other parameters unchanged). The height of the
second maximum of the visibility function at a wavelength of 600 nm
is reduced from
0.0085 (see the lower right panel of Fig. 3)
to
0.0070. The obtained best-fitting diameter value for the NPOI data
set changes from
0.06 mas to
0.06 mas, and is unchanged for
the VLTI/VINCI data set. This shows that an imperfect modelling of the
strength of the limb-darkening effect within our uncertainties
does not have a significant effect on our obtained diameter values for
Sge.
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Figure 9:
Flux of ![]() ![]() |
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We have compared our visual Sagittae NPOI visibility data
for 19 spectral channels with central wavelengths between 526 nm to 852 nm
as well as our near-infrared K-band VLTI/VINCI visibility data
with effective wavelength 2.19
m to a plane-parallel ATLAS 9, a plane-parallel PHOENIX, and a spherical PHOENIX model atmosphere. The stellar parameters
effective temperature
,
surface gravity
,
and M of the model atmospheres used were fixed a-priori based on
previous information on this star.
The spherical geometry of the PHOENIX model enables us to precisely define the Rosseland radius of the star with respect to the outermost model layer and thus the 0% intensity diameter. This model leads to consistent Rosseland angular diameters for our NPOI and VLTI/VINCI data sets. This agreement increases the confidence in the model atmosphere predictions from optical to near-infrared wavelengths as well as in the calibration and accuracy of both interferometric facilities. In addition, the consistent angular diameter derived from our VLTI/VINCI data on a night-by-night basis over a total range of about 2 months increases confidence in the given calibration uncertainties.
Table 6:
Revised fundamental parameters of the M0 giant Sagittae
based on the analysis of this paper. For the details of the calculation,
see the text.
The Rosseland angular diameter of Sagittae of
0.02 mas, based on the comparison
of our NPOI and VLTI/VINCI data to the spherical PHOENIX model, corresponds to a Rosseland linear radius
of
,
derived with the
Hipparcos parallax of
0.71 mas. The error of the
Rosseland linear radius is dominated by the uncertainty of the parallax,
not by the precision of our interferometric measurement.
With the bolometric
flux
10-9 W/m2
(Sect. 1) and the Rosseland angular diameter,
the effective temperature is constrained to
55 K. Again, the major contribution to this
error originates from the uncertainty in
and not
from our interferometric measurement.
The Rosseland linear radius and
result in a luminosity of
0.08. Placing
Sagittae
on the Hertzsprung Russel diagram using these values, and comparing to
stellar evolutionary tracks by Girardi et al. (2000) as
in Paper II (Fig. 1 of Paper II) we can estimate a mass
of M=1.4
,
and thus a surface gravity of
0.2.
Table 6 summarises our revised values of
Sagittae's
fundamental parameters.
The closure phases show a smooth transition from 0 to rather than
a sharp flip, which could be due to a small deviation from
circular symmetry of the well resolved stellar disc due to surface
features such as spots, an asymmetric extended molecular layer,
or a faint companion.
Acknowledgements
This work was performed in part under contract with the Jet Propulsion Laboratory (JPL) funded by NASA through a Michelson Fellowship Program (JPA).