Fundamental properties of early-type stars

Optical Interferometry of early-type stars with PAVO@CHARA
I. Fundamental stellar properties

Abstract

We present interferometric observations of 7 main-sequence and 3 giant stars with spectral types from B2 to F6 using the PAVO beam combiner at the CHARA array. We have directly determined the angular diameters for these objects with an average precision of 2.3%. We have also computed bolometric fluxes using available photometry in the visible and infrared wavelengths, as well as space-based ultraviolet spectroscopy. Combined with precise Hipparcos parallaxes, we have derived a set of fundamental stellar properties including linear radius, luminosity and effective temperature. Fitting the latter to computed isochrone models, we have inferred masses and ages of the stars. The effective temperatures obtained are in good agreement (at a 3% level) with nearly-independent temperature estimations from spectroscopy. They validate recent sixth-order polynomial (B-V)- empirical relations (Boyajian et al., 2012a), but suggest that a more conservative third-order solution (van Belle & von Braun, 2009) could adequately describe the (V-K)- relation for main-sequence stars of spectral type A0 and later. Finally, we have compared mass values obtained combining surface gravity with inferred stellar radius (gravity mass) and as a result of the comparison of computed luminosity and temperature values with stellar evolutionary models (isochrone mass). The strong discrepancy between isochrone and gravity mass obtained for one of the observed stars,  Lyr, suggests that determination of the stellar atmosphere parameters should be revised.

keywords:
Stars: early-type - Stars: fundamental parameters - Techniques: interferometric
12

March 7, 2018

1 Introduction

Long-baseline optical interferometry (with baselines up to hundreds of meters in length) has enabled us to measure angular diameters of bright stars, with typical values of a few milliarcseconds. Combined with accurate parallax and photometry, these measurements allow direct determination of fundamental stellar properties, such as the linear radii of their photospheres or the effective surface temperature of the stars, and have become a very valuable tool to contrast observational results with stellar models of increasing complexity.

The pioneering work of Hanbury Brown et al. (1974), using the Narrabri Stellar Intensity Interferometer (NSII), provided angular sizes of 32 O- to F- type stars. Subsequently, Code et al. (1976) established the empirical temperature scale for stars of spectral type F5 and earlier by means of combining Hanbury Brown et al. (1974) diameters with multiband spectra from which they inferred bolometric fluxes. The majority of the stars observed with the NSII belonged to luminosity classes I-III, and only about one third of them were main sequence or subgiant stars (luminosity classes IV or V), since the instrument favoured observations of stars with larger diameters, given the same surface brightness. For several decades, all the early type star (between O0 and A7) diameter measurements (a total of 16) came from the NSII observations (Davis, 1997). Even in more recent years, the papers referenced in the CHARM2 catalogue (Richichi et al., 2005) contained just 24 entries corresponding to direct diameter measurements for main sequence or sub-giant stars. The advent of optical long baseline interferometry using hectometric baselines, particularly in the near infrared, has provided rapid progress in the number of main sequence and subgiant stars with direct diameter measurements. Most notably, van Belle & von Braun (2009) reported interferometric diameter measurements for 44 G-type or later main sequence stars, deriving colour-temperature relations, and Boyajian et al. (2012a, b) published results of a survey carried out on a combined sample of 77 dwarfs spanning from A to M spectral type, developing new empirical laws relating broad-band colours and effective temperature.

As a result of resolution and sensitivity constraints particularly in the near-infrared, there is a clear sample bias, as only 5 out of the 121 stars studied in these papers belong to spectral class A or earlier. The major stumbling block has been that to access significant populations of hot stars, resolutions better than 1 milliarcsecond are required. The rise of beam combiners that can operate in the visible range of the spectrum with improved sensitivity, such as PAVO@CHARA, allows routine measurements of submilliarcsecond stellar diameters (Huber et al., 2012a; White et al., 2013), and offers the possibility to extend the spectral range to earlier type stars within similar sensitivity constraints.

Obtaining precise individual properties of B and A main sequence stars is of considerable importance in stellar astrophysics, since they represent the most massive and luminous objects that can be described by models containing simplifying assumptions such as LTE physics, hydrostatic equilibrium or purely radiative envelopes, in contrast to those corresponding to more massive or evolved objects, providing a useful benchmark for stellar atmosphere models. Furthermore, the intrinsic higher surface brightness of early type (mainly B and A) stars makes them suitable calibration stars (Mozurkewich et al., 2003; Boden, 2003) for correction of instrumental and atmospheric effects in visible and near infrared interferometry. Therefore, direct measurement of submilliarcsecond angular stellar diameters has the potential to improve calibration of interferometric observations of objects with larger projected sizes.

The drawback is that stars earlier than F6 often rotate rapidly (van Belle, 2012). As the rotational velocity approaches its critical value, the centrifugal force induces latitudinal temperature gradients (an effect known as gravity darkening, von Zeipel, 1924a, b; Maeder & Meynet, 2000) and, depending on the inclination of the stellar rotation axis, the apparent stellar disk may look oblate. Projected rotational velocities can be used to model the effect of rotation in the interferometric observables (Yoon et al., 2007), with the finding that effects of low rotation rates can be neglected for low spatial frequencies.

In this paper, we present results of our pilot study on a small sample of 10 nearby stars with submilliarcsecond angular size with spectral types in the B2-F6 range. Computed bolometric fluxes, together with precise Hipparcos parallaxes have been used to derive the fundamental stellar parameters linear radius, luminosity and effective temperature , with results that are in good agreement with measurements obtained using independent methods. Finally, we have estimated stellar mass and ages by means of isochrone model fitting.

2 Target sample and observations

2.1 Target sample

The stars observed in our study are extensively used as calibrators in near-infrared interferometry, mainly in observations using the MIRC (Michigan InfraRed Combiner) instrument (see, e.g. Monnier et al., 2007; Zhao et al., 2009; Che et al., 2011). In addition to their use in fundamental parameters of the observed stars (in combination with other measurements), angular diameter estimation constitutes a direct measurement of the interferometric response independent of diameter estimates based on indirect methods (see Cruzalèbes et al., 2010, and references therein) that rely on high fidelity SED templates or stellar atmosphere models.

Table 1 lists the objects observed, as well as the physical parameters describing their stellar atmospheres (, and metallicity), determined from spectroscopic and photometric observations. It also includes the measured Hipparcos parallaxes (van Leeuwen, 2007). These parallaxes correspond to distances smaller than 200 pc, with uncertainties ranging from 1% to 5% for most of the stars observed. The more distant stars  Cyg and  And have larger 17% and 11% errors at a distance of 880 and 210 pc respectively.

Notes. magnitude and colour taken from references in SIMBAD. colour computed using -band photometry taken from the General Catalogue of Photometric Data (GCPD Mermilliod et al., 1997). Stellar atmosphere parameters for every object have been averaged over the values presented inPrugniel et al. (2011),Wu et al. (2011),Pier et al. (2003),Erspamer & North (2003), Fitzpatrick & Massa (2005), Koleva & Vazdekis (2012),Blackwell & Lynas-Gray (1998),Gardiner et al. (1999),Allende Prieto & Lambert (1999),Balachandran et al. (1986),Ammons et al. (2006),Adelman et al. (2002),Gies & Lambert (1992),Nieva & Przybilla (2012),Adelman (1986),Adelman (1988),Cottrell & Sneden (1986),Hill & Landstreet (1993),Smith & Dworetsky (1993). Parallaxes from van Leeuwen (2007). Projected rotational velocities () taken from Głȩbocki & Gnaciński (2005).

Table 1: Physical parameters of the stars presented in this work. The objects are separated into main-sequence and sub giant stars (top) and giants and supergiants (bottom) in ascending temperature order.

The stars in our sample span from B2 to F6 spectral types (effective temperatures in the 21000-6400 K range). Most of them (seven) are main sequence or subgiants (luminosity classes V or IV). Two are giants ( Lyr,  And), and one is a supergiant star ( Cyg). Main-sequence stars of spectral types earlier than F6 (), exhibiting radiative envelopes, are expected to rotate rapidly (see van Belle, 2012, and references therein). As a consequence, they can show significant projected oblateness depending on the inclination of their polar axis. All the stars in our sample, with the sole exception of  And ( km/s, see Balona & Dziembowski, 1999; Głȩbocki & Gnaciński, 2005), have projected rotational velocities that are significantly less than 50% of the critical rotational velocity (see typical values for different spectral types in Tassoul, 2000) , and therefore no important deviations from projected circular shapes are expected (Frémat et al., 2005). For  And, Clark et al. (2003) reported that  And could be seen nearly equator-on, implying that the star rotates at 50% of the critical velocity, which would result in an equatorial radius less than 4% larger than the polar radius (?). Unfortunately, our observations are not sensitive to oblateness, given that we observed this object using only one baseline.

Among the stars studied, three objects ( Cyg,  Leo and  And) present some hints for the existence of companions at less than 1 degree in separation according to the Eggleton & Tokovinin (2008) compilation. Nevertheless, only one object ( And) shows observational evidence of the existence of close companions according to the Washington Double Star catalogue (WDS Mason et al., 2001), the Multiple Star Catalogue (MSC Tokovinin, 1997) and the 9 Catalogue of Spectroscopic Binary Orbits (SB9 Pourbaix et al., 2004).  And is a complex object, consisting of two components A and B(Olević & Cvetković, 2006), separated by 0.34 arcseconds (m=3.63; m=6.03). Both components have been described as spectroscopic binaries. The Aa-Ab components of the main spectroscopic binary have an estimated separation of 0.05 arcseconds. The brightest star of the pair is believed to  Cas type variable that injects material in a circumstellar shell in rapid discrete ejections (Clark et al., 2003), switching between Be- and B-type spectra in a timescale of  days. The short-term photometric variability of this object does not correlate with an enhancement in the shell emission and seems to be photosperic in origin. All these features make  And an extraordinarily difficult object to study, with conditions that might be relatively far from the simplified assumptions (circular projected shape, isothermal surface) considered throughout this paper; its continued use as an interferometric calibrator is not advised. Nevertheless, the measured angular diameter and effective temperature of  And show that the influence of companion objects is smaller than the precision of our observations.

2.2 Interferometry

Interferometric observations of our target sample were carried out using the PAVO (Precision Astronomical Visible Observations) beam combiner (Ireland et al., 2008), located at the CHARA Array (ten Brummelaar et al., 2005) on Mt Wilson Observatory (California, USA). The CHARA (Center for High Angular Resolution Astronomy) Array is an optical interferometer, consisting of six 1-m telescopes arranged in a Y-shaped configuration. Operating in visible and infrared wavelengths, it provides a total of 15 different baselines at different orientations with lengths in the range 34-331 m. With the longest operational baselines available in the world provided by the CHARA array, and the use of visible light (0.6-0.9 m) in PAVO, the instrumentation used in this study delivers the highest angular resolution yet achieved (0.3 mas in the visible).

The PAVO instrument (see detailed description in Ireland et al., 2008) is a pupil-plane Fizeau beam combiner optimised for sensitivity and high angular resolution. We briefly summarise the basics of the instrument here. Visible and infrared light are separated by a dichroic with a cutoff at 1m. The visible beams (up to three) enter PAVO, and are focused by a set of achromatic lenses in an image plane. The beams are passed through a 3-hole non-redundant mask that acts as a spatial filter. After going through the mask, the beams interfere and produce spatially modulated pupil-plane fringes. The fringes are formed on a lenslet array that divides the pupil in 16 independent segments, allowing an optimal usage of the multi-r apertures of the CHARA array. Finally, a prism disperses the fringes and these are re-imaged and recorded on a low-noise readout EMCCD detector. Early PAVO@CHARA results have been presented by Bazot et al. (2011), Derekas et al. (2011) and Huber et al. (2012a, b).

Observations of the objects listed in Table 1 using PAVO@CHARA were carried out in July 2010 (2-3), May 2011 (12-13), August 2011 (11), October 2011 (2-3), August 2012 (5-6) and September 2012 (7). Most of the observations were done using one baseline (two telescopes) at a time, with the only exception of  Cyg, that has also been observed in three-telescope (using the S2E2W2 triangle) mode, allowing simultaneous data collection in three baselines. The baselines used are given in Table 2. Raw interferometric data () were obtained through the use of standard procedures for PAVO@CHARA data (Ireland et al., 2008; Maestro et al., 2012). Some of the objects ( Cas, 37 And and  Leo) have been observed on only one night.

Baseline Length (m) PA (deg)
W2W1 107.93 -80.9
W2E2 156.26 +63.2
W2S2 177.45 -20.9
W2S1 210.98 -19.1
E2S2 248.13 +17.9
W1S2 249.39 -42.8
W1E2 251.34 +77.7
E1S1 330.70 +22.3
Table 2: CHARA baselines used. Position angle (PA) is measured in degrees East of North.

Gauging the point-source response of the interferometer is essential to obtain an accurate calibration of the observed sources. To this end, and according to standard practice in optical interferometry (Boden, 2003; van Belle & van Belle, 2005), we interleave observations of the science targets with others of calibration stars, in such manner that a bracket calibrator-target-calibrator is completed in 15-20 minutes. The calibration stars match, as closely as possible, the ideal point-like ( mas) source located at the smallest angular distance in the sky from the object (at an average distance of 7.5 degrees between target and calibrator). Table 3 shows all the calibrators used in our study. Expected diameters have been computed using colours (Kervella et al., 2004), dereddened according to the interstellar extinction maps presented by Drimmel et al. (2003). We have checked each calibrator in the literature for possible multiplicity or variability prior to observations. Analysis of the data obtained for HD 216523 revealed that the object is in fact a binary star, and therefore it was excluded from the list of calibrators. PAVO@CHARA sensitivity limits impose a selection bias on the calibration sources, favouring the use of distant late B to early A-type stars as calibrators, which are prone to show fast rotation and therefore non circular projected shapes (Domiciano de Souza et al., 2002; van Belle, 2012) . We circumvent this issue by choosing calibrators with low projected rotational velocities, , or accounting for an increased uncertainty in the predicted diameter that reflects deviations from the spherical shape (see Section 3). To account for some small correlated instrument systematics that persist after calibration using unresolved sources when the star has been observed only one night, we have included and additional 5% uncertainty to the visibility-squared () errors measured, based on repeated observations of the same object during several nights.

HD SpTy ID
HD 1279 B7III -0.157 0.037 0.202 25 9
HD 1606 B7V -0.358 0.037 0.161 120 22
HD 4142 B5V -0.334 0.034 0.193 160 24
HD 10390 B9V -0.147 0.007 0.180 58 15
HD 29526 A0V -0.001 0.017 0.243 80 13
HD 29721 B9III 0.198 0.037 0.242 232 29
HD 88737 F9V 1.334 0.006 0.459 10 3
HD 89363 A0 0.117 0.015 0.150 -
HD 92825 A3V 0.155 0.008 0.352 188 18
HD 93702 A2V 0.239 0.014 0.326 208 15
HD 171301 B8IV -0.180 0.025 0.184 53 15
HD 174262 A1V 0.074 0.025 0.226 111 21
HD 174567 A0V 0.073 0.044 0.160 18 7
HD 176871 B5V -0.162 0.038 0.210 268 34
HD 179527 B8III -0.074 0.046 0.208 30 12
HD 197392 B8III -0.21 0.212 0.213 33 9
HD 204403 A5V -0.489 0.027 0.227 117 18
HD 207516 B8V -0.157 0.016 0.179 103 18
HD 211211 A2V 0.061 0.018 0.243 238 27
HD 219290 A0V -0.011 0.023 0.178 50 9
HD 222304 B9V -0.052 0.024 0.269 165 25

Notes. Calibrator not used: HD 216523 (binary star). The last column refers to the ID of the target star for which the calibrator has been used (see first column of Table 4).

Table 3: List of calibration stars used, including the spectral type, colour and extinction , the predicted diameter (in milliarcseconds) from using prescriptions of Kervella et al. (2004), as well as projected rotational velocities (, in km/s) from Głȩbocki & Gnaciński (2005). Photometry has been taken from references in SIMBAD database and the General Catalogue of Photometric Data (GCPD Mermilliod et al., 1997).

3 Fundamental stellar parameters

3.1 Angular diameters

The squared-visibility () measurements obtained for each of the stars presented in this work were fitted to the single star limb-darkened disk model (given by Hanbury Brown et al., 1974)

(1)

with

(2)

where is the visibility, is the linear limb-darkening coefficient, is the th-order Bessel function, is the projected baseline, is the angular diameter after limb-darkening correction, and is the wavelength at which the stars are observed. We use R-band linear limb-darkening coefficients interpolating within the model grid of Claret & Bloemen (2011), using the atmosphere parameters given in Table 1, and assuming a microturbulent velocity of 2 km/s. The value assumed for and its error are the result of taking the median and the 0.158 and 0.842 quantiles of the interpolated values corresponding to 10 realisations of the normally distributed values of the stellar atmosphere parameters , and [Fe/H]. The uniform disc diameters were obtained by simply assuming in Equation 3.1. Figure 1 displays the limb-darkening disc model fit to the calibrated . The observations made, as well as the uniform disc (computed assuming =0 in equation 3.1) and limb-darkened disc diameters estimated, are summarised in Table 4.

Figure 1: Squared visibility vs. spatial frequency (defined as projected baseline divided by wavelength) for all stars in our sample. Red solid lines show the fitted limb-darkened disk model. Error bars for each star have been scaled so that the reduced- equals unity.
ID Name HD N Baselines (mas) (mas) (mas)
40 Leo HD 89449 46 E2W1 0.48 0.04 0.706 0.026 0.731 0.030 0.747 0.012
7 And HD 219080 207 W1W2, E2W1, S1W2, S2W1 0.44 0.04 0.622 0.006 0.648 0.008 0.649 0.013
37 And HD 5448 115 W1W2, E2W1, S2W1 0.43 0.03 0.678 0.012 0.708 0.013 0.692 0.013
 Leo HD 97633 69 E2W1, S1E1 0.39 0.03 0.710 0.023 0.740 0.024 0.721 0.017
b Leo HD 95608 46 E2W1 0.38 0.03 0.416 0.016 0.430 0.017 0.456 0.009
 Aur HD 32630 115 W1W2, S2W1 0.26 0.03 0.444 0.011 0.453 0.012 0.539 0.010
 Cas HD 3360 46 S1E1 0.26 0.03 0.305 0.010 0.311 0.010 0.376 0.009
 Lyr HD 176437 161 W1W2, W1E2, S1W2 0.34 0.03 0.729 0.008 0.753 0.009 0.738 0.016
 Cyg HD 202850 368 W1W2, S2W1, S2E2W2 0.35 0.04 0.511 0.014 0.527 0.016 0.588 0.013
 And HD 217675 46 E2W1 0.31 0.03 0.494 0.012 0.508 0.015 0.526 0.011
Table 4: Summary of observations included in this work and measured angular diameters.

Errors in diameter have been estimated through model fitting of Equation 3.1 using synthetic datasets (Derekas et al., 2011; Huber et al., 2012a). These datasets are generated considering the uncertainties in: (1) the measured values for target and calibrators, (2) the adopted PAVO wavelength scale (4.5 nm), (3) the calibrator angular sizes (5%, except those cases where the calibrator is expected to show larger projected oblateness), and (4) linear limb-darkening coefficient (see Table 4). All quantities are assumed to have values that are normally distributed, using 210 simulated datasets for each diameter estimation. Possible correlation between adjacent wavelength channels is also taken into account. The median and the width (derived from the 0.158 and 0.842 quantiles) of the distribution of the fitted diameters give the adopted values for the measured diameter and its uncertainty. The results have been adjusted to assume a reduced-, compensating for underestimation of the squared-visibility error estimates (Berger et al., 2006).

Derived angular diameters show an average precision of 2.3%. Figure 2 shows the comparison between the measured limb-darkened diameters and different estimations using long baseline interferometry (CHARM2 meta-catalogue Harmanec et al., 1996; Lane et al., 2001; Vakili et al., 1997), or indirect methods: surface brightness methods and calibrations using color indices3 (JMMC catalogue Lafrasse et al., 2010; Kervella et al., 2004), and using detailed spectrophotometry in the visible and ultraviolet (Zorec et al., 2009). The overall agreement with other results is good (the average value of is 0.99 with a scatter of 0.06). data for  And does not show evidence of departure from circular shape, although this could be due to the similar orientation, projected on sky, of the baselines used. Stars of earlier spectral types , and particularly  Cyg,  Aur and  Cas, depart significantly from the range of spectral types of the stars used to infer Kervella’s -diameter relation (A- to K-type), and therefore show the larger deviations. It is worth noting the good concordance between observations and predictions made by Zorec et al. (2009).

Second lobe measurements of  Leo make possible simultaneous estimation of both diameter and linear limb-darkening coefficient for this object. As a result, we obtain = 0.7470.024 and =0.470.03. This constitutes a modest 1% increment in the estimated diameter, but a remarkable increase in the limb-darkening effect with respect to the fixed estimation (= 0.7400.024; =0.390.03). Given the spectral type (A2V), and the low projected rotational velocity ( km/s Royer et al., 2007),  Leo is likely to be a fast rotating star viewed nearly pole on. This results in a higher than expected drop in intensity near the border of the apparent stellar disc, due to the alignment of limb-darkening and gravity darkening, similar to that observed in other objects (for example, the widely studied case of the A0V fast rotator Vega Peterson et al., 2006; Aufdenberg et al., 2006; Monnier et al., 2012).

Figure 2: Fractional differences between angular diameters measured with PAVO and diameters determined using color-surface brightness relation (JMMC catalogue and Kervella relation Lafrasse et al., 2010; Kervella et al., 2004), SED fits (Zorec et al., 2009) and direct interferometric measurements collected in the CHARM2 catalogue (Richichi et al., 2005).

3.2 Bolometric fluxes

We have computed the bolometric flux, , for all the targets in our sample by fitting the observed absolute spectral energy distribution (SED) to grids of ATLAS9 model atmospheres of solar metallicity computed by Castelli & Kurucz (2003). The fit requires collection of available photometry in the Hipparcos ( bandpass), Tycho (), Johnson (), Geneva (), WBVR (WBVR) and Stromgren () phootometric systems. For the bright objects in our sample, 2MASS photometric measurements are generally saturated, and therefore are not suitable for our purpose, with the sole exception of  Leo, for which no other near-infrared colours were found. Photometry longward of near-infrared passbands was not included, as they are frequently affected by infrared excess with non-stellar origin.

Half the stars in the sample have spectral types earlier than A0, implying surface temperatures in excess of 10000 K. Therefore these objects radiate predominantly in the ultraviolet. In order to improve the fit of the emergent flux in the ultraviolet region, we have included co-added low-resolution flux-calibrated ultraviolet spectra (in the 1150-1980 Å and 1850-3350 Å ranges) retrieved from the International Ultraviolet Explorer (IUE) archive. To avoid occasional flux miscalibration close to the edges of the spectral range covered, we have considered only the 1200-1900 Å and 1900-3300 Å regions.

All photometric data have been corrected from interstellar reddening using the maps of Drimmel et al. (2003) and the extinction description presented in Fitzpatrick (1999). In the particular case of the highly reddened  Cyg, we have used the individual interstellar extinction curved presented in Wegner (2002). Photometry was calibrated in flux (using filter responses and zero points from Kornilov et al., 1996; Bessell et al., 1998; Gray, 1998; Cohen et al., 2003; Bessell & Murphy, 2012) and subsequently fitted to the grid of theoretical line-blanketed ATLAS9 spectra, interpolating in both and model parameters. The code can also estimate the reddening correction, yielding to values similar to those mentioned above. Table 5 shows the estimated bolometric fluxes, , computed as the numerical integral of the theoretical spectrum corresponding to the best model fit parameters and (also listed in Table 5). Uncertainties in the estimated parameters are computed using a synthetic population of 10 datasets accounting for errors (correlated and uncorrelated) in the photometry used and in its calibration in absolute flux.

Figure 3: SED fits for the stars in our sample. Continuous line represent the best model fit to the optical and infrared broadband photometric data (red crosses). Horizontal error bars stand for the effective width of the corresponding broadband filter. The ultraviolet co-added low resolution IUE spectra are plotted as blue dots.

Figure 3 display plots of the resulting SED fits for the stars in the sample studied. For all the stars, the overall agreement between observed and theoretical fluxes is excellent, both in the ultraviolet and visible/near-infared regions. The best fit physical parameters and are also in excellent agreement with spectroscopic determinations, with the only exception of  Lyr, that shows a value smaller than the one derived from spectroscopy. Temperature measurements of  Lyr are widely spread in the range 9346 K-12715 K (Ammons et al., 2006; Koleva & Vazdekis, 2012), with associated  g values in the interval , that appear abnormally high for a giant star. Despite the SED fit method followed should not be considered accurate enough in terms of estimation, the large discrepancy observed seems compatible with a lower expected surface gravity for a giant star than the spectroscopic values (see Table 1). The atmospheric parameters of this object will be discussed again in Section 3.6.

Star E(B-V) Spectrophotometry Model Fit
Name (mag) / T (K) (c.g.s.) (10erg cm s)
40 Leo 0.004 0.001 82 1/1 5812 126 4.29 0.29 30.9 0.9
7 And 0.006 0.001 67 3/- 7024 354 4.00 0.17 41 2
37 And 0.004 0.001 43 1/1 7577 291 4.00 1.06 80 5
 Leo 0.005 0.001 65 2/1 9147 47 3.55 0.27 147 4
b Leo 0.005 0.001 50 3/4 8759 37 3.83 0.35 51.0 1.2
 Aur 0.014 0.001 85 11/40 16212 63 4.40 0.30 831 11
 Cas 0.04 0.04 54 1/4 18562 151 4.02 0.28 693 20
 Lyr 0.017 0.004 83 1/1 9640 60 2.64 0.35 215 4
 Cyg 0.13 0.05 78 1/1 9990 0 1.90 0.06 133 3
 And 0.049 0.022 45 13/12 13812 45 3.17 0.10 383 5
Table 5: Estimated bolometric fluxes from SED fits of ATLAS9 grids of stellar atmosphere models computed by Castelli & Kurucz (2003), using optical and near-infrared photometry () and low-resolution IUE ultraviolet spectra in short (1200-1900Å) and long (1900-3300Å). The table includes the reddening used for each star, as well as the best fit and parameter values.

3.3 Luminosities, temperatures and linear radii

The combination of the derived values for stellar angular diameters and bolometric fluxes with the Hipparcos parallaxes provides us with estimations of linear radii, luminosities and effective temperatures for the stars in our sample. Measured angular limb-darkened diameters are transformed into linear radius for each star using the Hipparcos (van Leeuwen, 2007) parallaxes. The absolute luminosity is computed from the known distance to the object and the bolometric flux by solving

(3)

Finally, the combination of angular diameter with the estimated bolometric flux allows us to measure the effective temperature of a star, defined according to

(4)

The values obtained are summarised in Table 6. Figure 4 display the results in a Hertzprung-Russell diagram.

Figure 4: Hertzprung-Russell diagram containing luminosity and effective temperature derived for the stars in our study (see Table 6). Solar-metallicity PARSEC (Bressan et al., 2012) evolutionary tracks for masses in the range 1-10  in steps of 1  are shown as grey dotted lines.
Star
40 Leo 1.68 0.07 4.4 0.9 6450 140
7 And 1.71 0.02 7.8 0.6 7380 90
37 And 3.03 0.11 40 3 8320 150
 Leo 4.03 0.10 118 5 9480 120
 Leo 1.80 0.07 24.1 1.4 9540 180
 Aur 3.64 0.10 1450 70 18660 230
 Cas 6.1 0.3 7200 900 21500 400
 Lyr 15.4 0.8 2430 190 10330 80
 Cyg 50 9 33000 8000 10940 180
 And 11.5 1.3 5300 900 14540 170
Table 6: Linear radii, luminosities and effective temperatures.

3.4 Effective temperature

Figure 5 displays a comparison of the derived effective temperature values derived from our measurements of and and values determined using spectroscopy listed in Table 1. There is an excellent agreement between both sets of measurements, specially for stars of spectral types A0 or later. The average and median deviation are and respectively, with a scatter of . For main-sequence stars, the scatter reduces to .

Given the effective temperatures in excess of 10000 K of stars with spectral types earlier than A0, a significant fraction of the total emergent flux is radiated in the ultraviolet region, where systematics caused by limitations of plane-parallel stellar atmosphere models are conspicuous, especially in the case of giants or supergiants. The physical parameters describing the model stellar atmosphere best fit do not necessarily represent the best description of the actual physical parameters of the star. Nonetheless, the dependency of makes the determination of effective temperature to remain robust in spite of the simplified assumptions contained in the stellar atmosphere models used.

Table 7 compares derived in our study with results presented in Zorec et al. (2009) for a subset of the 5 overlapping objects in both samples, with 9000 K. The agreement of both sets of measurements is excellent. Small disagreements are likely due to the simultaneous estimation of the photosphere diameter and the identification of model temperature and effective temperature made in Zorec et al. (2009). In our case, including a direct measurement of the limb-darkened diameter, assumed to represent the photosphere diameter and estimating exclusively from the estimation of the total amount of radiated energy reduce considerably the model dependence of our results, that is limited to the use of theoretical spectra to fit the observed fluxes.

Figure 5: (Top) Comparison of effective temperature estimations derived in this paper with respect to spectroscopic determinations (see Table 1). (Bottom) Relative difference between both estimations of effective temperature.
Star
 Leo 9480 120 9180 290
 Aur 18660 230 17940 1070
 Cas 21500 400 21850 1390
 Lyr 10330 80 10000 350
 Cyg 10940 180 11170 450
Table 7: Comparison of derived effective temperatures with results presented in Zorec et al. (2009).

3.5 Color-effective temperature relations

Whereas the empirical temperature scale for giant stars seems to be firmly established with uncertainties under 2.5% (Code et al., 1976; Underhill et al., 1979; van Belle et al., 1999), the same cannot be said for main sequence stars (luminosity classes IV-V). For those stars in our sample with 10000 K (spectral types A0 or later), we have contrasted the computed effective temperatures with empirical colour-temperature relations presented in Boyajian et al. (2012a) and van Belle & von Braun (2009) using () and () colours. Figures 6 and 7 display effective temperature versus () and () colours for the stars in our sample lying in the temperature range considered, as well as the results presented in Boyajian et al. (2012a), fitted to a sixth-order polynomial in () and () respectively. Figure 6 adds the empirical relation found by van Belle & von Braun (2009), using a third-order polynomial. Most of the stars in both previous studies are cooler than 7000 K, and both empirical relations nearly overlap in that range of temperatures. The same does not apply for earlier spectral types, as the two fits differ quite significantly between 6500 K and 8500 K, where there were no stars in their samples. Three of the stars in our study (37 And, 7 And and 40 Leo) have temperatures within this range. Their location in the -() diagram (Figure 6) shows strong agreement with the more conservative third-order polynomial relation presented by van Belle & von Braun (2009). On the other hand, our data is fully consistent with the -() sixth-order polynomial relation of Boyajian et al. (2012a) (Figure 7). Observational effects induced by ubiquitous fast rotation among stars earlier than F6 could explain the disagreement with Boyajian et al. (2012a) sixth-order () polynomial relation. Most of the A-type stars studied in that paper show high projected rotational velocities (), compatible with fast rotation seen at a lower inclination angle (closer to be equator-on) with respect to the same spectral type stars contained in our sample. Further accurate estimations of for a larger sample of main sequence stars earlier than F5 will significantly improve the empirical temperature scale for these stars.

Figure 6: Plot of the effective temperature versus color for the main sequence stars in the studied sample with below 10000 K. The solid line represents the empirical relation found in Boyajian et al. (2012a) by means of a sixth-order polynomial fit to a sample of 44 A- to G-type stars (small black dots). Triple dot-dash line represents the third-order polynomial fit presented in van Belle & von Braun (2009).
Figure 7: Effective temperature versus color for the main sequence stars in the observed sample with below 10000 K. Data from Boyajian et al. (2012a) (small black dots), as well as empirical relation based on a sixth-order polynomial fit (solid line) are also included.

3.6 Masses and ages

We have estimated the stellar masses () and ages for our target sample by fitting the inferred luminosity and effective temperature values to PARSEC4 isochrones (Bressan et al., 2012). Figure 8 illustrates the method using 7 And as an example. For each object in our sample, we have computed a grid of isochrones equally spaced in ( in years) in the range 1 Myr-10 Gyr, and adopt the best fit to the luminosity and effective temperature computed values. We have assumed solar metallicity for all the stars in the sample. Uncertainties are derived by considering solutions at 1- separation in both luminosity and temperature dimensions. Results obtained are displayed in Table 8.

Figure 8: Luminosity-temperature diagram showing the isochrones generated for 7 And.
Star
40 Leo 1.35 0.06 2630 210
7 And 1.6 0.1 1120 30
37 And 2.21 0.09 724 21
 Leo 2.8 0.1 407 12
 Leo 2.11 0.06 195 15
 Aur 5.6 0.1 41 6
 Cas 8.96 0.13 22.9 1.2
 Lyr 5.76 0.13 74.8 5.1
 Cyg 11.2 0.2 19.1 0.6
 And 6.5 0.5 52 9
Table 8: Isochrone masses and ages.

Using the linear radii estimated from combined measurements of angular diameter and parallax and the spectrocospic determination of , we can estimate the so-called gravity mass (van Belle et al., 2007) of each object according to

(5)

where is the gravitational constant, and and stand for the mass and linear radius of the star considered. Figure 9 displays the relation between isochrone and gravitational masses. Comparison of masses determined by both methods show a remarkable good agreement for all the objects in the sample except  Lyr. The reason for the large disagreement in that case is most likely related to the spectroscopic determination of , since values of , and reported by different authors (Balachandran et al., 1986; Prugniel et al., 2011; Koleva & Vazdekis, 2012, respectively) are clearly discrepant with the values expected for a B9 giant, which combined with our linear radius estimation result in unphysical gravity mass values of 27 , 41  and 111  (55  for the mean value =3.81). However, the value of derived from our SED fit yields to a gravity mass value of 4.4 , in better agreement with the value obtained.

Figure 9: Comparison of mass estimations using isochrone fit () and the gravitational mass , computed using spectroscopic surface gravity and linear stellar radii (Equation 5). The dashed line represents identity.  Lyr (55 , 5.760.13 ) is not shown in the plot.

4 Conclusions

We have presented results of a pilot study on a sample of 10 stars (7 main sequence or sub giants and 3 supergiants or giants) with spectral types between B2 and F6. For all the objects in the sample, we have measured submilliarcsecond angular diameters and bolometric fluxes with an average precision of 2.3% and 2.1% respectively. Combined with Hipparcos parallaxes, we have derived fundamental stellar parameters, linear radii, effective temperatures, and luminosities with 8%, 3% and 5% average relative uncertainties. Finally, we have fitted PARSEC isochrones to the values of temperature and luminosity found in order to obtain estimates of mass and age for every star in the sample. Our findings can be summarised as follows:

  1. Measured diameters show generally good agreement with predictions based on colour relations or SED fits, as well as with previous interferometric measurements. data points beyond the first null for  Leo enable simultaneous estimation of diameter and linear limb-darkening coefficient . Whereas the derived diameter is less than 1% larger than the fixed limb darkening estimation, the estimated value for exceeds by 20% the value interpolated in Claret & Bloemen (2011) using stellar atmosphere parameters. This enhanced limb darkening, together with the star’s low projected rotational velocity, suggests that  Leo is in fact a fast rotating star viewed nearly pole-on.

  2. We derived bolometric fluxes from SED fits of theoretical ATLAS9 stellar atmosphere models to optical and near-infrared photometric data, as well as including flux calibrated IUE ultraviolet spectra, accounting for the effects of interstellar reddening. The mean absolute deviation between derived effective temperatures and estimates using spectroscopy is 3%. The agreement is excellent for main sequence stars, and only for two of the B giants,  Lyr, and  Cyg and the main-squence star  Aur, the discrepancy is as large as 7%. Comparison with Zorec et al. (2009), for the subset of overlapping objects from our sample reveals an excellent agreement in the results of both studies, despite some subtle differences in the modelling assumptions.

  3. Our derived effective temperatures for A and F type stars confirm the sixth-order polynomial () color-temperature relation presented in Boyajian et al. (2012a). However, there is a clear disagreement with their sixth-order ()-temperature relation, and our data clearly favours the () cubic polynomial presented by van Belle & von Braun (2009). Different inclination angles of the polar axis between Boyajian’s sample of A and early F stars with respect to ours, together with the effects of fast rotation in these stars might explain the observed discrepancy. New observations of main-sequence stars in this range of temperature, correcting the scarcity of available data should provide the basis for a more detailed empirical colour-temperature relation.

  4. The PARSEC model isochrone fit in the temperature-luminosity plane provides mass values that are consistent with the estimated gravitational masses computed from spectroscopically determined , showing larger discrepancies for larger mass values. The unphysical result obtained for  Lyr gravitational mass, together with the large scatter in temperature estimations using spectroscopy questions the accuracy of  Lyr atmosphere parameters, and suggests strong degeneracy between and . Additionally, a estimation based on the SED fit to spectrophotometric data seems to give a gravity mass that is more consistent with the value obtained by means of the isochrone fitting.

Acknowledgements

This research has made use of the SIMBAD database and the VizieR catalogue access tool, operated at CDS, Strasbourg, France. Some of the data presented in this paper were obtained from the Multimission Archive at the Space Telescope Science Institute (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NAG5-7584 and by other grants and contracts. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. The CHARA Array is funded by the National Science Foundation through NSF grant AST-0606958, by Georgia State University through the College of Arts and Sciences, and by the W.M. Keck Foundation. We acknowledge the support of the Australian Research Council. V.M. is supported by an International Denison Postgraduate Award.

Footnotes

  1. pagerange: Optical Interferometry of early-type stars with PAVO@CHARA I. Fundamental stellar propertiesLABEL:lastpage
  2. pubyear: 2013
  3. http://cdsarc.u-strasbg.fr/viz-bin/Cat?II/300
  4. http://stev.oapd.inaf.it/cmd

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