The nearby eclipsing stellar system \delta Velorum

# The nearby eclipsing stellar system δ Velorum

III. Self-consistent fundamental parameters and distance1
###### Key Words.:
Stars: individual: (HD 74956, Vel); Stars: binaries: eclipsing; Stars: early-type; Stars: Rotation; Techniques: high angular resolution; Techniques: interferometric
2

## Abstract

Context:The triple stellar system  Vel (composed of two A-type and one F-type main sequence stars) is particularly interesting as it contains one of the nearest and brightest eclipsing binaries. It therefore presents a unique opportunity to determine independently the physical properties of the three components of the system, as well as its distance.

Aims:We aim at determining the fundamental parameters (masses, radii, luminosities, rotational velocities) of the three components of  Vel, as well as the parallax of the system, independently from the existing Hipparcos measurement.

Methods:We determined dynamical masses from high-precision astrometry of the orbits of Aab-B and Aa-Ab using adaptive optics (VLT/NACO) and optical interferometry (VLTI/AMBER). The main component is an eclipsing binary composed of two early A-type stars in rapid rotation. We modeled the photometric and radial velocity measurements of the eclipsing pair Aa-Ab using a self consistent method based on physical parameters (mass, radius, luminosity, rotational velocity).

Results:From our self-consistent modeling of the primary and secondary components of the  Vel A eclipsing pair, we derive their fundamental parameters with a typical accuracy of 1%. We find that they have similar masses, respectively and . The physical parameters of the tertiary component ( Vel B) are also estimated, although to a lower accuracy. We obtain a parallax  mas for the system, in satisfactory agreement () with the Hipparcos value ( mas).

Conclusions:The physical parameters we derive represent a consistent set of constraints for the evolutionary modeling of this system. The agreement of the parallax we measure with the Hipparcos value to a 1% accuracy is also an interesting confirmation of the true accuracy of these two independent measurements.

## 1 Introduction

Early-type main sequence stars exhibit a number of peculiarities usually not encountered in cooler stars: fast rotation, debris disks, enhanced surface metallicities (Am), magnetic fields and rapid oscillations (Ap and roAp stars), etc. Although stellar structure and evolution models are now rather successful in reproducing the observed physical properties of most A-type stars, the observational constraints on these models remain relatively weak, occasionally leading to surprising discoveries. An example is provided by the recent interferometric observations of the A0V benchmark star Vega, that confirmed that Vega, as previously shown by Gulliver, Hill & Adelman ((1994)), is a pole-on fast rotator near critical velocity (Aufdenberg et al. (2006)). The same interferometric observations showed that Vega harbors a hot debris disk within within 8 AU from the star (Absil et al. (2006)).

$δ$ Vel (HD 74956, HIP 41913, GJ 321.3, GJ 9278) is a bright multiple star including at least three identified components, and is among our closest stellar neighbors, with a revised Hipparcos parallax of  mas (van Leeuwen (2007)). This object has many observational peculiarities. Firstly, it was discovered only in 1997 that  Vel hosts one of the brightest of all known eclipsing binaries (Otero et al. (2000)), with a remarkably long orbital period ( days). This eclipsing binary is also one of the very few that are easily observable with the naked eye (). The eclipsing pair was first resolved using optical interferometry by Kellerer et al. ((2007)). Secondly,  Vel is known to have a moderate thermal infrared excess (e.g. Aumann (1985), Su et al. (2006)), and Spitzer observations revealed a spectacular bow shock caused by the motion of  Vel in a dense interstellar cloud (Gáspár et al. (2008)). The presence of interstellar material was also reported by Hempel & Schmitt ((2003)), who observed two red-shifted absorbing components in absorption in the Ca II K line, of probable interstellar origin. In Paper I of the present series, Kervella et al. ((2009)) confirmed that the infrared excess is essentially emitted by the bow shock, and not warm circumstellar material located close to the stars. In the framework of a search for resolved emission due to debris disks, Moerchen et al. ((2010)) obtained thermal infrared images of  Vel using the Gemini South telescope and the T-ReCS instrument, and detected a marginally resolved emission at m.

In Paper II, Pribulla et al. ((2011)) used a combination of high-resolution spectroscopy and photometric observations (from the SMEI instrument, attached to the Coriolis satellite) to derive an accurate orbital solution for the eclipsing binary  Vel A, and estimate the physical parameters of  Vel Aa and Ab. They identified that the two eclipsing components are fast rotating stars, with respective masses of and ( accuracy), and estimated the mass of  Vel B to be .

In spite of this recent progress, uncertainties remain on the fundamental parameters of the different components of the system, in particular on their exact masses. Taking advantage of the availability of NACO astrometry of the  Vel A-B pair, and new interferometric observations from the VLTI/AMBER instrument, we propose here to revisit the system along two directions. In Sect. 2, we describe our new VLTI/AMBER interferometric data, as well as our re-analysis of the spectroscopic and photometric data previously used in Paper II. Sect. 2.2 is dedicated to the description of our self-consistent model, and the derivation of an improved orbital solution, physical parameters as well as an independent distance. In Sect. 3 we employ NACO astrometry of the visual  Vel A-B binary to obtain an improved orbital solution. Compared to our work presented in Paper II, this new analysis result is a clearer view and better confidence in the derived fundamental parameters of the system (for all three components), thanks to the redundant nature of our data and our independent determination of the distance.

## 2 The orbit and parameters of δ Vel Aa and Ab

### 2.1 Observations and data analysis

#### Interferometry

AMBER (Petrov et al. (2007)), the three-telescope beam combiner of the VLTI, has the proper angular resolution to resolve the Aa-Ab pair. This instrument combines simultaneously 3 ATs (Auxiliary Telescopes) or 3 UTs (Unit Telescopes) of the VLT and operates in the near infrared (H and K band). It has a choice of spectral resolutions of , or . For this study, we had data in low resolution (H+K bands at ) and medium resolution (H or K band, at ). We used baselines of the order of 100 m in order to obtain spatial resolution in the milli-arcsecond regime. These interferometric data have been collected in a dedicated program (ESO program 076.D-0782), as well as during Guaranteed Time (GTO) from Arcetri Observatory. We present here a reduction of these data.

We reduced the data using the AMBER reduction package amdlib3 (Chelli et al. (2009), Tatulli et al. (2007)) and performed the calibration using stellar calibrators chosen in the catalogue by Mérand et al. ((2005)) and a custom software which estimates and interpolates the transfer function of the instrument. For each night, we derived the separation of Aa and Ab using a map as a function of the separation vector (two parameters). The other parameters, such as flux ratio or individual diameters, were set using simple hypothesis and their choice did not affect significantly our final estimated angular separations. The resulting separation vectors are listed in Table 1 in coordinate towards East and North (which correspond to the and axes in the projected baselines map). The error bars on this vectors were estimated in the map.

#### Photometry

We used the photometric data from the SMEI satellite, presented in Paper II. The available quantity is the relative flux normalized to the value outside of the eclipses, since there was no absolute calibration of SMEI data available. We corrected for the presence of the B component which is in the field of view of SMEI. From our model of B, the expected flux ratio between B and Aab is 7.5% in the SMEI bandpass. The transmission of the instrument has a triangular shape that peaks at 700 nm, with a quantum efficiency around 47%, and falls to at 430 nm towards the blue, and 1025 nm towards the red (Spreckley & Stevens (2008)). We removed this contribution which, if not taken into account, would result in an underestimation of the depth of the eclipses. We also incorporated to our photometric dataset the photometric measurement we derived of Aab in the K band (Paper I). We use this value in our fit as the only constraint in term of absolute photometry.

#### Spectroscopy

The observables we derived from the visible spectroscopic data are the broadening functions (BF) presented in paper II (see this reference for more explanations). These functions contain a lot of information: not only do they contain the radial velocities that result from the orbital motion, but also the broadening due to the stellar rotation and the flux ratio in the considered band.

From the observed BF, it is possible to derive the from the two components. After a few experimentations using the stellar surface model we are going to present later, we found that the following ad-hoc function parameterizes well the BF for a star seen from the equator:

 BF(v)BF(v0)=(1−cos([1−(v−v0vsini)2]×π/2))α (1)

where is the only parameter constraining the gravity darkening and is the velocity offset. The function is defined for only, and its value is 0 otherwise. Using this analytical model and a global fit, we estimated for each component and the radial velocities for each epoch (see Fig. 1 and 2 for the quality of the fit). We found to be  km/s and  km/s for Aa and Ab respectively. Incidentally, we find and . The rotation rate value is relatively independent of the actual gravity darkening (parameterized here by ) since it is set by the width of the broadening function, not its shape.

For our fit of the orbit and the stellar parameters, we do not use the center-to-limb darkening we derive here from the broadening functions. The surface brightness distribution is constrained by the photometric profil of the eclipses. However, we will check a posteriori the agreement between our best fit model and the limb-darkening derived in the analytical BF by modeling the BF from our model. See Sect. 2.2.3 and, more specifically, Fig. 7.

### 2.2 Global fit

#### Self consistent model

In order to extract the fundamental parameters (masses, radii, surface temperatures, semi-major axis, etc.) from the observational data, we propose a self consistent modeling centered around the use of physical quantities: we model the system using two stars whose characteristics are computed based on their radii, total luminosities and mass.

To illustrate the advantage of this approach, we can consider that in order to model the eclipses, we could use an ad-hoc model based on fractional radii (ratio to the semi-major axis) and brightness ratio, but this would not lead directly to the fundamental parameters of the system such as effective temperatures or luminosities. Our approach uses radii, masses and luminosities: we get the fractional radii by self consistency between the semi-major axis based on Newton’s form of the Kepler’s law (from the masses and the period of the orbit) and the measured apparent semi-major axis (constrained by interferometric separation vectors). The brightness ratio arises from the luminosity and radii, and the photospheric models we use to model the surface of the stars.

Our stellar surface model also includes stellar rotation. We model the appearance of the star using a model developed contemporarily and similar to the one used in Aufdenberg et al. ((2006)) to model the interferometric visibilities of the star Vega. To compute the photometry, in particular during the eclipses, we generated synthetic images and integrated them to derive the light curves. The parameters we use are:

• total mass of the system (Aa+Ab);

• the fractional mass of Aa to the total mass of the eclipsing system (Aa+Ab);

• the physical radii of each component;

• the absolute luminosity of each component;

• the of each component, to parameterize the rotation;

In addition, we have the usual 7 parameters for the visual orbit:

• the period;

• the date of passage at periastron;

• the eccentricity;

• three angles: inclination, and ;

• the apparent semi-major axis (in milliarcseconds)

The only parameterization of the physics of the two components is contained in the luminosity: the model we use of the stellar surface is the Roche approximation, which only uses the mass, radius, luminosity and rotational velocity. Once the shape of the surface is computed, we link the local surface gravity and local effective temperature using the von Zeipel theory (von Zeipel (1924) and Aufdenberg et al. (2006) for more details). The luminosity is constrained using several mechanisms: through the absolute photometry of the system, but also through the surface brightness that sets the depth of the eclipses.

The advantages of using the apparent semi-major axis compared to the physical quantity are simple. First of all, it is directly related to one of our observables: the interferometric separation vector. Secondly, we already have the semi-major axis by combining the Kepler’s third law, the period and the total mass. Combined with the angular semi-major axis, we derive a distance independently from the Hipparcos value (a brief discussion is presented in Sect. 2.3). The distance is derived internally in our model and used to extract a model apparent magnitude in the K band, which is used as one of the constraints as we mentioned before.

It is to be noted that we make the following assumptions:

• the stars have their rotation axis perpendicular to the plane of the orbit.

• we use a Von Zeipel gravity darkening coefficient of in , where and are the effective temperature and gravity at the surface. Even if recent observational constraints from Monnier et al. ((2007)) suggest , we chose to use von-Zeipel’s classical value since it does not lead to qualitatively nor quantitatively different results in our case.

#### Orbital and stellar derived parameters

The result of the global fit is in excellent agreement with all the observed data we used for the fit: the relative photometry of the eclipses (Fig. 3 and 4), the radial velocities (Fig. 5) and the separation vectors (Fig. 6). The corresponding orbital parameters and stellar parameters are presented in Tables 2 and 3.

#### a posteriori verifications

Our model is rather simple and does not take into account one aspect: the heating of one star by the other’s radiation. This effect could be important in our case in different conditions. The first one is if one star was heated by the other one and developed a bright spot on its surface. We can discard this possibility because of the absence of photometric variations outside the eclipses. The second case where the heating by the other star can be a problem is if this contribution is enough to actually modify the temperature structure of the stellar photosphere. We can check a posteriori that the radiation received from the other star is of the order of 3% of the radiation emitted (assuming similar surface brightness and a 2.5  star seen at a distance of 89 ). Approximating the surface as a blackbody, it corresponds to an increase of temperature of less than 1% (). We can thus assume that our hypothesis that locally the photosphere can be approximated by a ATLAS model is correct.

We can also check the consistency of our model beyond the fit of the data we presented. In particular, we can compare the predicted broadening functions based on our model and the broadening functions we observe since we did not implement the direct fit of broadening function to our model. Doing so (Fig. 7), the comparison is very satisfactory, even though the wings of the data (represented by the analytical function fit to the data in thick gray line) seems deeper than for the model. In other words, the gravity darkening of the model is slightly underestimated, but only by a small fraction, considering that a model without gravity darkening (small gray dots on Fig. 7) produces a very strong disagreement.

Investigating the possible causes of this problem, we realized that we can reproduce the more pronounced darkening of the equator compared to the pole by tilting the star to 10 degrees from the plane of the sky: this makes the pole more in line of sight of the observer and hence increases the contrast between the pole and the equator.

Forcing the inclination of the spin of the stars to be 80 degrees instead of 90 degrees does not change dramatically the fundamental parameters of the star estimated from our fit. One of the reasons is that it changes the sin(i) by only 1.5%, the actual rotational velocity is mostly unaffected. The fit converges as well as in the case of aligned spins, with fundamental parameters within error bars of the one estimated in the case we presented in the main part of this work.

In conclusion of the analysis of the broadening function prediction of our model compared to the observed one, we find a confirmation of the consistency of our model with the data. This comparison may also indicate that our model underestimate slightly the gravity darkening, or, alternatively, that the stars have their rotational axis tilted on the order of 10 degrees toward the observer, which does not impact qualitatively nor quantitatively the fit of the data that led to the estimation of the fundamental parameters presented earlier.

Another test is to compare the predicted wide band photometry with the observed ones. In the literature, there is a handful of data available, for the combined AB or A and B separately. Doing so, we see (Table 4) that the bluest magnitudes are not reproduced well. Assuming a excess of , the difference is nicely explained using an ISM extinction law as presented in Kervella et al. ((2004)).

### 2.3 Distance

Our parameterization allows us to derive the distance as the ratio between of the semi-major axis of the eclipsing component (from the total mass, the period and Kepler’s third law) and its apparent semi-major axis (from the interferometric observations). Such a distance estimate is particularly interesting as it is purely geometrical, and independent of the Hipparcos measurement. From our model, we obtain a parallax of  mas for  Vel.

Comparing this value to the  mas revised Hipparcos parallax7 obtained by van Leeuwen ((2007)) shows a good agreement of the two measurements, within . This confirmation of the true accuracy of these independent measurements, at the 1% level, shows that the Hipparcos measurement was not disturbed by the binary nature of  Vel A. This somewhat surprising result is due to the similar brightness ratio and mass ratio of the  Vel A pair. This results in a very small apparent displacement of the center of light of the Aab system during the orbit, with respect to the center of gravity of the two stars. Using our model of the eclipsing system, we computed the expected photocenter displacement during an full orbit. We find that the peak-to-peak photocenter displacement is of the order of one milliarcsecond, which is much smaller than the apparent astrometric shift due to the parallax. The binarity of the system therefore did not bias significantly the Hipparcos parallax measurement, neither did the low brightness of the B component. The observations of the photocenter displacement through high-precision differential astrometry with the VLT/NACO instrument will be the subject of a future article.

## 3 The orbit and parameters of δ Vel B

### 3.1 Astrometric data

The binarity of  Vel was discovered by S. I. Bailey in 1894 from Arequipa, Peru (and independently by Innes (1895)). Over more than one century, the separation between  Vel A and B has been decreasing at a rate which nicely matches the progression of the angular resolution of the successive generations of imaging instruments (visual observations, photography, electronic devices). This progression allowed a relatively regular tracing of the visual orbit of the pair, down to the sub-arcsecond separations that occur around the periastron passage. With the advent of speckle interferometry (Tango et al. (1979)) and the Hipparcos satellite (ESA (1997)) the accuracy of the measured relative positions improved significantly. In Paper I, we present in details the new data we obtained with the Very Large Telescope, using both the K-band adaptive optic system VLT/NACO (Rousset et al. (2003); Lenzen et al. (1998)) and the N-band camera VLT/VISIR (Lagage et al. (2004)). Thanks to the large aperture of the telescopes and the diffraction-limited angular resolution, these observations provide us with new high-precision astrometry of the A-B pair. The resulting separations of  Vel B relatively to A are presented in Table 5. For the conversion of the separation measured in pixels to angular separations, we adopted the pixel scale of  mas/pixel (Masciadri et al. (2003)) for NACO and  mas/pixel for VISIR. The assumed NACO plate scale is in good agreement with the calibration by Neuhäuser et al. ((2008)), who demonstrated that this figure is stable over a period of at least 3 years. The VISIR plate scale uncertainty is set arbitrarily to , although it is probably better in reality. The angular separation was only for the epoch of our observations.

In addition to these new astrometric measurements, we also take advantage of the historical astrometric positions assembled by Argyle et al. ((2002)) in his Table 5, that includes 17 epochs between 1895 and 1999. It is to be noted that these authors used the two speckle interferometry epochs from Tango et al. ((1979)) with a different definition for the projection angle, leading to an apparent inconsistency with the other measurements. Transforming the Tango et al. projection angle using , these two data points become much more consistent with the other epochs and observing techniques.

### 3.2 Orbital elements

We adjusted the orbital parameters of the  Vel A-B pair to the whole sample of astrometric data, and the result is presented graphically in Fig. 8. The corresponding orbital elements are listed in Table 6. It should be noted that thanks to a semi-major axis twice more precise, and a period ten time more precise, the total mass value derived from based Kepler’s third law is significantly improved, which becomes limited by our parallax estimation of .

 M(Aab+B)=6.15±0.15orbit±0.17parallaxM⊙ (2)

### 3.3 Physical properties of δ Vel B

We used a spectral energy distribution (hereafter SED) fit to the photometric data (Table 7) corrected for reddening assuming . We used a carefully interpolated grid of ATLAS models (e.g. Kurucz (2005)) in order to derive the angular diameter and effective temperature of  Vel B. We find a photospheric limb darkened angular diameter of  mas and an effective temperature of  K. Based on our distance estimate, we can derive the physical radius to be and thus a luminosity of . Assuming the star is on the main sequence we can infer, based on the mass-luminosity relation by Torres, Andersen & Giménez ((2010)), that  Vel B has a spectral type F7.5V and a mass of:

 Mphotometric(B)=1.35±0.1M⊙ (3)

These parameters estimated using an independent method are comparable to the values we obtained in Paper I.

We can compare this mass estimate with the value we can compute from the mass of A ( from table 2) and A+B ( from table 6. This leads to:

 Mdynamic(B)=M(A+B)−M(A)=1.46±0.23M⊙ (4)

which is consistent with the photometric estimate (Eq. 3)

## 4 Conclusion

We presented a self consistent model of the triple stellar system  Vel. Our model reproduces photometric, spectroscopic and interferometric data of the eclipsing pair Aa-Ab. We determined the orbital (Table 2) and fundamental stellar parameters (Table 3) of the three components of the system. The physical properties of the eclipsing components are surprisingly similar to the A0V benchmark star Vega. Thanks to the resolution of the system using the AMBER instrument, we also independently determine the distance to the system ( mas, or  pc), as well as the interstellar reddening value towards this nearby system, with . The combination of two fast rotating A stars of slightly different masses and a late F star, all coeval and with accurately measured fundamental parameters, will likely make of  Vel a cornerstone for the study of early-type main sequence stars.

###### Acknowledgements.
T.P. acknowledges support from the European Union in the FP6 MC ToK project MTKD-CT-2006-042514. This work has partially been supported by VEGA project 2/0094/11. This work also received the support of PHASE, the high angular resolution partnership between ONERA, Observatoire de Paris, CNRS and University Denis Diderot Paris 7. This research has made use of the AMBER data reduction package of the Jean-Marie Mariotti Center11. This research took advantage of the SIMBAD and VIZIER databases at the CDS, Strasbourg (France), and NASA’s Astrophysics Data System Bibliographic Services.

### Footnotes

1. thanks: Based on observations made with ESO telescopes at Paranal Observatory, under ESO programs 076.D-0782(B), 081.D-0109(B), 081.D-0109(C), 282.D-5006(A) and Arcetri GTO program 084.C-0170(C)
2. offprints: A. Mérand
3. For simplicity, error bars have been estimated along E and N directions even though the true error is an ellipse with orientation dicted by the map. It turns out the ellipses have small flattening, so the approximation is relevant.
4. The last four lines present the agreement as reduced .
5. Parameters for Vega are adapted from Aufdenberg et al. ((2006)). The references for the metallicity are Gray et al. ((2006)) and Castelli & Kurucz ((1994)).
6. The magnitudes predicted by the best fit model are listed together with the observed magnitudes (brackets). In bold is the K magnitude of Aab which was used to constrain our fit. The B column comes from the fit presented in Sect. 3. The last two columns are the observed differences and the expected extinction for , which best fits the observed colors. Observed magnitudes are from Morel & Magnenat ((1978)) , Paper I and Skrutskie et al. ((2006))
7. The revised Hipparcos parallax is consistent within with the original Hipparcos reduction ( mas; ESA (1997)), and with the ground-based parallax of this star ( mas; Van Altena et al. (1995)).
8. measurements from NACO have symbol , whereas measurements from VISIR have . The angles are all expressed in milliarcseconds.
9. The total mass (last line) is computed using Kepler’s third law, assuming the revised parallax  mas.
10. The third column is the reddening correction corresponding to . The model (last column) is for an angular diameter of 0.53 mas and effective temperature of 6600K.
11. Available at http://www.jmmc.fr/amberdrs

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