The fundamental parameters of the roAp star \gamma Equulei

The fundamental parameters of the roAp star $γ$ Equulei

Key Words.:
Methods: observational - Techniques: high angular resolution - Techniques: interferometric - Stars: individual (Equ) - Stars: fundamental parameters


Context:Physical processes working in the stellar interiors as well as the evolution of stars depend on some fundamental stellar properties, such as mass, radius, luminosity, and chemical abundances. The effective temperature, the surface gravity and the mean density are useful quantities defined from these fundamental properties. Additional physical quantities, like mass loss rate, pulsation period, rotation period, and magnetic field properties are interesting for the study of peculiar evolutionary stages. A classical way to test stellar interior models is to compare the predicted and observed location of a star on theoretical evolutionary tracks in a H-R diagram. This requires the best possible determinations of stellar mass, radius, luminosity and abundances.

Aims:To directly and accurately determine its angular diameter and thus derive its fundamental parameters, we observed the well-known rapidly oscillating Ap star, Equ, using the visible spectro-interferometer VEGA installed on the optical CHARA array.

Methods:We recorded data on the W1W2 baseline of the CHARA array in the blue and in the red domains. We computed the calibrated squared visibility and derived the uniform-disk angular diameter and the limb-darkened one. We used the whole energy flux distribution, the parallax and the angular diameter to determine the luminosity and the effective temperature of the star.

Results:We obtained a limb-darkened angular diameter of 0.564  0.017 mas and deduced a radius of  = 2.20  0.12 . Without considering the multiple nature of the system, we derived a bolometric flux of erg cm s and an effective temperature of 7364  235 K, which is below the effective temperature that has been previously determined. Under the same conditions we found a luminosity of  = 12.8  1.4 . When the contribution of the closest companion to the bolometric flux is considered, we found that the effective temperature and luminosity of the primary star can be, respectively, up to  100 K and up to  0.8 L smaller than the values mentioned above.

Conclusions:For the first time, thanks to the unique capabilities of VEGA, we managed to constrain the angular diameter of a star as small as 0.564 mas with an accuracy of about 3%, and to derive its fundamental parameters. In particular the new values of the radius and effective temperature should bring further constraints on the asteroseismic modelling of the star.

1 Introduction

Rapidly oscillating Ap (roAp) stars are chemically peculiar main-sequence stars that are characterized by strong and large-scale organized magnetic fields (typically of several kG, and up to 24 kG), abundance inhomogeneities leading to spotted surfaces, small rotational speeds, and pulsations with periods of a few minutes (see, (Kochukhov 2009); (Cunha 2007), for recent reviews). roAp stars are bright, pulsate with large amplitudes and in high radial orders. Thus they are particularly well-suited for asteroseismic campaigns and they contribute in a unique way to our understanding of the structure and evolution of stars. However, to put constraints on the interior chemical composition, the mixing length parameter, and the amount of convective overshooting, asteroseismic data should be combined with high precision stellar radii ((Cunha et al. 2003); (Cunha et al. 2007)). This radius is generally estimated from the star’s luminosity and effective temperature. But systematic errors are likely to be present in this determination due to the abnormal surface layers of the Ap stars. This well known fact has been corroborated by seismic data on roAp stars ((Matthews et al. 1999)), and compromises all asteroseismic results for this class of pulsators. Using long-baseline interferometry to provide accurate angular diameters appears to be a promising approach to overcome the difficulties in deriving accurate global parameters of roAp stars, but is also very challenging due to their small angular size. In fact, except for  Cir, whose diameter is about 1 millisecond of arc (mas) ((Bruntt et al. 2008)), all roAp stars have angular diameters smaller than 1 mas. Such a small scale can be resolved only with optical or near-infrared interferometry. This was confirmed again recently by the interferometric study of the second largest (in angular size) roAp star known, namely  CrB ((Bruntt et al. 2010)).

Equ (HD201601 ; A9p ;  = 4.7 ;  = 27.55  0.62 mas ((van Leeuwen 2007)) ; v  10 km/s (Uesugi & Fukuda 1970)) is one of the brightest objects of the class of roAp stars with a period of about 12.3 min (Martinez et al. 1996) in brightness as well as in radial velocity. Despite photometry and spectroscopy of its oscillations obtained over the past 25 years, the pulsation frequency spectrum of Equ has remained poorly understood. High-precision photometry with the MOST satellite has led to unique mode identifications based on a best model ((Gruberbauer et al. 2008)) using a mass of 1.74 0.03 M, an effective temperature of log  = 3.882  0.011 and a luminosity of log  = 1.10  0.03 (Kochukhov & Bagnulo (2006)). As regards to abundance inhomogeneities, Ryabchikova et al. (2002) considered the following stellar parameters ( = 7700 K, log 4.2, [M/H] = +0.5) to compute synthetic spectra and presented the evidence for abundance stratification in the atmosphere of Equ: Ca, Cr, Fe, Ba, Si, Na seem to be overabundant in deeper atmospheric layers, but normal to underabundant in the upper layers, which according to the authors agrees well with diffusion theory for Ca and Cr, developed for cool magnetic stars with a weak mass loss of about 2.5 M/yr. Pr and Nd from the rare earth elements have an opposite profile since their abundance is more than 6 dex higher in the upper layers than in the deeper atmospheric ones. Such abundance inhomogeneities clearly lead to a patchy surface, a redistribution of the stellar flux, and a complex atmospheric structure, resulting in biased photometric and spectroscopic determinations of the effective temperature.

Guided by these considerations, we have observed Equ with a spectro-interferometer operating at optical wavelengths, the VEGA spectrograph ((Mourard et al. 2009)) installed at the CHARA Array ((ten Brummelaar et al. 2005)). The unique combination of the visible spectral range of VEGA and the long baselines of CHARA has allowed us to record accurate squared visibilities at high spatial frequencies (Sect. 2). To derive the fundamental parameters of Equ, calibrated spectra have been processed to estimate the bolometric flux and to determine the effective temperature (Sect. 3). Finally, we can set the star Equ in the HR diagram and discuss the derived fundamental parameters (Sect. 4).

2 Interferometric observations and data processing

2.1 Data

Data were collected at the CHARA Array with the VEGA spectropolarimeter recording spectrally dispersed fringes at visible wavelengths thanks to two photon-counting detectors. Two telescopes along the W1W2 baseline were combined. Observations were performed between 570 and 750 nm (according to the detector) at the medium spectral resolution of VEGA (R = 5000). Observations of Equ were sandwiched with those of a nearby calibration star (HD 195810). The observation log is given in Table 1.

Date UT (h) Star B (m) PA ()
2008-07-29 5.59 HD 195810 78.9 106.6
2008-07-29 6.08 Equ 76.2 106.4
2008-07-29 6.41 HD 195810 92.3 101.9
2008-08-03 8.64 HD 195810 107.3 93.0
2008-08-03 8.98 Equ 107.8 93.8
2008-08-03 9.31 HD 195810 103.7 91.0
2008-08-05 7.68 HD 195810 107.3 108.8
2008-08-05 8.14 Equ 106.7 95.8
2008-08-05 8.63 HD 195810 106.9 92.6
Table 1: Journal of Equ observations on July 29, and August 3 and 5, 2008.

Each set of data was composed of observations following a calibrator-star-calibrator sequence, with 10 files of 3000 short exposures of 15 ms per observation. Each data set was processed in 60 files of 500 short exposures using the estimator and the VEGA data reduction pipeline detailed in Mourard et al. (2009). The spectral separation between the two detectors is fixed by the optical design and equals about 170 nm in the medium spectral resolution. The red detector was centered around 750 nm on July 29 and around 640 nm on August 3 and 5. The blue detector was centered around 590 nm on July, 29 and around 470 nm on August 3 and 5. The bluer the wavelength, the more stringent the requirements on seeing. As a consequence the blue data on August 3 and 5 did not have a sufficient signal-to-noise ratio and squared visibilities could not be processed. All the squared visibilities are calibrated using an uniform-disk angular diameter of 0.29 0.02 mas in the V and R bands for the calibrator HD 195810. This value is determined from the limb-darkened angular diameter provided by SearchCal2(Table LABEL:tab:V2).

UT (h) B (m) (nm)
6.08 76.1 745.0 0.84 0.02
6.08 76.2 582.5 0.72 0.02
8.98 107.6 640.0 0.62 0.04
8.14 106.7 640.0 0.61 0.05
Table 2: Calibrated squared visibilities of Equ. Each visibility point corresponds to the average on the 60 blocks of 500 frames.

2.2 Angular diameter determination

Equ is the brightest component of a multiple system. The closest component lies at 1.25”  0.04”, it has a magnitude difference with the primary star of = 4 and a position angle of PA = 264.6  1.3 ((Fabricius et al. 2002)). The entrance slit of the spectrograph (height=4” and width=0.2” for these observations) will affect the transmission of the companion flux. Taking into account the seeing during the observations (about 1”), the field rotation during the hour angle range of our observations ([-30 ; 0]), the position angle of the companion, we determine the throughput efficiency of the VEGA spectrograph slit for this companion. This efficiency varies from 10% for the longer baselines (around 107 m) to 30% for the smaller ones (around 80 m). We use the Visibility Modeling Tool (VMT)3 to build a composite model including the companion of Equ. For the longer baselines, the resulting modulation in the visibility is below 2%, which is 3 or 4 times below our accuracy on squared visibilities. We thus neglected the influence of the companion and interpreted our visibility data points in terms of angular diameter (Fig. 1). We performed model fitting with LITpro4. This fitting engine is based on a modified Levenberg-Marquardt algorithm combined with the trust regions method ((Tallon-Bosc et al. 2008)). The software provides a user-expandable set of geometrical elementary models of the object, combinable as building blocks. The fit of the visibility curve versus spatial frequency leads to a uniform-disk angular diameter of 0.540 0.016 mas for Equ. We used the tables of Diaz-Cordoves et al. (1995) to determine the linear limb-darkening coefficient in the R band for 4.0  4.5 and 7500 K  7750 K. By fixing this limb-darkening coefficient, LITPRO provides a limb-darkened angular diameter in the R band of  = 0.564  0.017 mas with a reduced of 0.37.

Figure 1: Squared visibility versus spatial frequency for Equ obtained with the VEGA observations. The solid line represents the uniform-disk best model.

3 Bolometric flux and effective temperature

The effective temperature, , of a star can be obtained through the relation,


where stands for the Stefan-Boltzmann constant ( erg cm s K), for the limb-darkened angular diameter, and is the star’s bolometric flux given by,


Thus, the effective temperature of Equ can be computed if we know its angular diameter and its bolometric flux. The angular diameter of Equ was derived in Sect. 2. To compute the bolometric flux we need a single spectrum that covers the whole wavelength range. This spectrum was obtained by combining photometric and spectroscopic data of Equ available in the literature, together with ATLAS9 Kurucz models, in the way explained below.

3.1 Data

We collected two rebinned high resolution spectra ( = 18000 at  = 1400 Å,  = 13000 at  = 2600 Å) from the Sky Survey Telescope obtained at the IUE “Newly Extracted Spectra” (INES) data archive5, covering the wavelength range [1850 Å  ; 3350 Å]. The two spectra were obtained with the Long Wavelength Prime camera and the large aperture of 10” 20” (Table 3). Based on the quality flag listed in the IUE spectra ((Garhart et al. 1997)) we removed all bad pixels from the data, and we also removed the points with negative flux. The mean of the two spectra was then computed to obtain one single spectrum of Equ in the range 1850 Å  3350 Å.

Image Date Starting time Exposure time
Number (UT) (s)
06874 08/10/1985 18:55:04 599.531
09159 23/09/1986 20:41:13 539.730
Table 3: UV spectra obtained with IUE.

We collected two spectra for Equ in the visible, one from Burnashev (1985), which is a spectrum from Kharitonov et al. (1978) reduced to the uniform spectrophotometric system of the “Chilean Catalogue”, and one from Kharitonov et al. (1988). We verified that the latter was in better agreement with the Johnson ((Morel & Magnenat 1978)) and the Geneva ((Rufener 1988)) photometry than the other spectrum. To convert from Johnson and Geneva magnitudes to fluxes we used the calibrations given by Johnson (1966) and Rufener & Nicolet (1988), respectively.

For the infrared, we collected the photometric data available in the literature. The calibrated observational photometric fluxes that we considered in this study are given in Table 4.

Band Flux Source Calibration
(Å) ()
I 9000 15.53 1 a
J 12500 5.949 2 b
H 16500 2.420 2 b
K 22000 0.912 2 b
L 36000 0.140 2 b
M 48000 0.0512 2 b
J 12350 6.090 3 c
H 16620 2.584 3 c
K 21590 1.067 3 c

Source references: (1) Morel & Magnenat (1978); (2) Groote & Kaufmann (1983); (3)  Cutri et al. (2003).

Calibration references: (a) Johnson (1966); (b) Wamsteker (1981); (c) Cohen et al. (2003).

Table 4: Calibrated photometric infrared fluxes for Equ.
Figure 2: The whole spectrum obtained for Equ. Black line corresponds to the average of the IUE spectra and to the (Kharitonov et al. 1988)’s spectrum. For wavelengths  Å and  Å , the figure shows the curve obtained using the interpolation method (dark grey line), the Kurucz model that best fits the spectroscopy in the visible and the photometry in the infrared when models are calibrated with the star’s magnitude (grey line) and when models are calibrated with the relation (light grey line). The Geneva and infrared photometry from Table 4 (circles) and Johnson UBVRI photometry (triangles) are overplotted to the spectrum.

3.2 and determination

The spectrum of Equ was obtained by combining the averaged IUE spectrum between 1854 Å  and 3220 Å, the Kharitonov’s (1988) spectrum from 3225 Å to 7375 Å, and, for wavelengths  Å and  Å we considered two cases: (1) we used the synthetic spectrum for the Kurucz model that best fitted both the star’s spectrum in the visible and the star’s photometry in the infrared and, (2) we performed a linear extrapolation between 506 Å and 1854 Å, considering zero flux at 506 Å, a second linear interpolation to the infrared fluxes between 7390 Å and 48000 Å, and a third linear extrapolation from 48000 Å and 1.6  Å considering zero flux at 1.6  Å. In case (1), when searching for the best Kurucz model we intentionally disregarded the data in the UV, because Kurucz models are particularly unsuitable for modeling that region of the spectra of roAp stars. To find the Kurucz model that best fitted the data in the visible and infrared we ran a grid of models, with different effective temperatures, surface gravities, and metallicities. Since Kurucz models needed to be calibrated (they give the flux of the star, not the value observed on Earth), we tried two different calibrations, namely: (i) the star’s magnitude in the band, , (ii) the relation , where is the radius and the distance to the star. For the we used the limb-darkened angular diameter determined in the previous section. The final spectra obtained for Equ with the two different calibration methods and with the interpolation method are plotted in Fig. 2. The bolometric flux, , was then computed from the integral of the spectrum of the star, through Eq. 2 and the effective temperature, , was determined using Eq. 1 (Table 5).

Calibration method (erg cm s) (K)
10 7351 229
10 7381 234
Interpolation 10 7361 235
Table 5: Bolometric flux and effective temperature obtained for Equ, using three different methods (see text for details).

The uncertainties in the three values of the bolometric flux given in Table 5 were estimated by considering an uncertainty of on the total flux from the combined IUE spectrum ((González-Riestra et al. 2001)), an uncertainty of on the total flux of the low resolution spectrum from Kharitonov et al. (1988), an uncertainty of on the total flux derived from the Kurucz model, and an uncertainty of on the total flux derived from the interpolation. The latter two are somewhat arbitrary. Our attitude was one of being conservative enough to guarantee that the uncertainty in the total flux was not underestimated due to the difficulty in establishing these two values. The corresponding absolute errors were then combined to derive the errors in the flux which are shown in Table 5. Combining these with the uncertainty in the angular diameter, we derived the uncertainty in the individual values of the effective temperature. As a final result we take the mean of the three values and consider the uncertainty to be the largest of the three uncertainties. Thus, the flux and effective temperature adopted for Equ are, respectively, ( erg cm s and 7364  235 K. If, instead, we took for the effective temperature an uncertainty such as to enclose the three uncertainties, the result would be  = 7364  250 K.

Figure 3: The position of Equ in the Hertzsprung-Russell diagram. The constraints on the fundamental parameters are indicated by the 1-error box (log , log (/L)) and the diagonal lines (radius). The box in solid lines corresponds to the results derived when ignoring the presence of the companion star. The box in dashed lines corresponds to the results derived after subtracting from the total bolometric flux the maximum contribution expected from the companion (see text for details). The box in dotted lines corresponds to the fundamental parameters derived by Kochukhov & Bagnulo (2006) and used by Gruberbauer et al. (2008) in the asteroseismic modelling of Equ.

3.3 Contamination by the companion star

In fact, since Equ is a multiple system and the distance between the primary (hereafter, Equ A) and the secondary (hereafter, Equ B) is 1.25”, the bolometric flux of Equ determined in Sect. 3 contains the contribution of both components. Given its magnitude, one may anticipate that the contribution of Equ B to the total flux will be small. Although the data available in the literature for this component is very limited, we used them to estimate the impact of Equ B’s contribution on our determination of the effective temperature of Equ A.

We collected the magnitudes  = 9.85  0.03 and  = 8.69  0.03 of Equ B from Fabricius et al. (2002) and determined a value for its effective temperature using the color- calibration from Ramírez & Meléndez (2005). This was done assuming three different arbitrary values and uncertainties for the metallicity, namely , and dex. The values found for the effective temperature were  = 4570, 4686 and 4833 K, respectively, with an uncertainty of K (Ramírez & Meléndez (2005)). The metallicity, the effective temperature, and the absolute V-band magnitude were used to estimate log , using theoretical isochrones from Girardi et al. (2000)6. For the three values of metallicities and mentioned above, we found log  = 4.58, 4.53, and 4.51, respectively. With these parameters we computed three Kurucz models and calibrated each of them in three different ways: (i) using the H magnitude ((Perryman et al. 1997)), (ii) using the magnitude, and (iii) using the magnitude. To convert from Hipparcos/Tycho magnitudes into fluxes we used the zero points from Bessel & Castelli (private communication). The maximum flux found for Equ B through the procedure described above was 0.19, which corresponds to 6% of the total flux. This implies that the effective temperature of Equ A determined in the previous section may be in excess by up to 111 K due to the contamination introduced by this companion star.

4 Discussion

4.1 Position in the HR-diagram

We derive the radius of Equ thanks to the formula:


where stands for the limb-darkened angular diameter (in mas), for the stellar radius (in solar radius, ), and for the distance (in parsec). We obtain  = 2.20  0.12 .

We use the bolometric flux and the parallax to determine the Equ’s luminosity from the relation:


where stands for the conversion factor from parsecs to meters. We obtain  = 12.8  1.4 and can set Equ in the HR diagram (Fig. 3).

Recently, seismic data of  Equ obtained with the Canadian-led satellite MOST have been modeled by Gruberbauer et al. (2008) based on the fundamental parameters coming from Kochukhov & Bagnulo (2006) and using a grid of pulsation models including the effect of the magnetic field. A comparison of the HR diagram error-box considered by the authors (see dotted-line box in Figure 3) and our uncertainty regions shows that the regions are considerably different. In fact, even if we do not account for the contribution of the companion, we obtain a lower effective temperature with log  = 3.867 0.014 to be compared to log  = 3.882 0.011 from Gruberbauer et al. (2008). This discrepancy between the uncertainty regions increases if the companion contribution is taken into account. In that case, the overlap between the two regions is very small.

As regards to luminosity, our calculation shows that for  Equ (as well as for  Cir) the contributions of the uncertainties in the bolometric flux and parallax to the uncertainty in are comparable. This is quite different from the results obtained by Kochukhov & Bagnulo (2006) who found that the dominant contribution to the uncertainty in comes from the parallax. The authors mentioned that the bolometric flux that was adopted in their work was that for normal stars. When dealing with peculiar stars, like Ap stars, it may be more adequate to properly compute the bolometric flux. However, it is precisely the difficulty in obtaining the full spectrum of the star that increases the uncertainty in the computed bolometric flux and, hence, in the luminosity and effective temperature. That is well illustrated by the following fact: if the somewhat arbitrary 20 uncertainties adopted in our work for the total fluxes derived from the Kurucz model and from the interpolation, were replaced by 5 uncertainties, we would obtain formal uncertainties in and comparable and smaller, respectively, to those quoted by Kochukhov & Bagnulo (2006).

4.2 Bias due to stellar features

We use the whole spectral energy density to determine the bolometric flux. We then deduce the effective temperature from this bolometric flux and the angular diameter. The determination of the angular diameter is based on visibility measurements that are directly linked to the Fourier Transform of the object intensity distribution. For a single circular star, the visibility curve as a function of spatial frequency B/ (where B stands for the interferometric baseline and for the operating wavelength) is related to the first Bessel function, and contains an ever decreasing series of lobes, separated by nulls, as one observes with an increasing angular resolution. As a rule of thumb, the first lobe of the visibility curve (see Fig. 1 for an example) is sensitive to the size of the object only. As an example, for a star whose angular diameter equals 0.56 mas like Equ, the difference in squared visibility between a uniform-disk and a limb-darkened one is of the order of 0.5% in the first lobe. The following lobes are sensitive to limb darkening and atmospheric structure but consist of very low visibilities. Finally, departure from circular symmetry (due to stellar spots from instance) requires either interferometric imaging by more than two telescopes or measurement close to the null. As a consequence, our interferometric data collected in the first part of the first lobe are only sensitive to the size of the target and cannot be used to study the potential complex structure of the atmosphere.

5 Conclusion

Thanks to the unique capabilities of VEGA/CHARA, we present an accurate measurement of the limb-darkened angular diameter of a target as small as 0.564 0.017 mas. In combination with our estimate of the bolometric flux based on the whole spectral energy density, we determine the effective temperature of  Equ A. Without considering the contribution of the closest companion star ( Equ B) to the bolometric flux, we found an effective temperature 7364  235 K, which is below the effective temperature that has been previously determined. An estimate of that contribution leads to the conclusion that the above value may still be in excess by up to about 110 K, which increases further the discrepancy between the literature values for the effective temperature of  Equ A and the value derived here. The impact on the seismic analysis of considering the new values of the radius and effective temperature should be considered in future modeling of this star.

More generally, this study illustrates the advantages of optical long-baseline interferometry for providing direct and accurate angular diameter measurements and motivates observations of other main-sequence stars to bring constraints on their evolutionary state and their internal structures. Within this context, the operation of VEGA in the visible is very complementary to the similar interferometric studies performed in the infrared range since it allows to study spectral types ranging from B to late-M and thus it opens the new window of the early spectral types ((Mourard et al. 2009)).

Another promising issue would be to use longer interferometric baselines to be sensitive to the stellar spots and bring constraints on the stellar surface features.

VEGA is a collaboration between CHARA and OCA/LAOG/CRAL/LESIA that has been supported by the French programs PNPS and ASHRA, by INSU and by the Région PACA. The project has obviously taken benefit from the strong support of the OCA and CHARA technical teams. The CHARA Array is operated with support from the National Science Foundation through grant AST-0908253, the W. M. Keck Foundation, the NASA Exoplanet Science Institute, and from Georgia State University. This work was partially supported by the projects PTDC/CTE-AST/098754/2008 and PTDC/CTE-AST/66181/2006, and the grant SFRH / BD / 41213 / 2007 funded by FCT/MCTES, Portugal. MC is supported by a Ciência 2007 contract, funded by FCT/MCTES(Portugal) and POPH/FSE (EC). This research has made use of the SearchCal and LITPRO services of the Jean-Marie Mariotti Center, and of CDS Astronomical Databases SIMBAD and VIZIER.


  1. offprints:
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