Binary Cepheids from optical interferometry

Classical Cepheid stars have been considered since more than a century as reliable tools to estimate distances in the universe thanks to their Period-Luminosity (P-L) relationship. Moreover, they are also powerful astrophysical laboratories, providing fundamental clues for studying the pulsation and evolution of intermediate-mass stars. When in binary systems, we can investigate the age and evolution of the Cepheid, estimate the mass and distance, and constrain theoretical models. However, most of the companions are located too close to the Cepheid (1-40 mas) to be spatially resolved with a 10-meter class telescope. The only way to spatially resolve such systems is to use long-baseline interferometry. Recently, we have started a unique and long-term interferometric program that aims at detecting and characterizing physical parameters of the Cepheid companions, with as main objectives the determination of accurate masses and geometric distances.Comment: 8 pages, Proceeding of the conference "Setting a new standard in the analysis of binary stars", September 2013, Leuven, Belgiu

Extended envelopes around Galactic Cepheids IV. T Monocerotis and X Sagittarii from mid-infrared interferometry with VLTI/MIDI

Aims. We study the close environment of nearby Cepheids using high spatial resolution observations in the mid-infrared with the VLTI/MIDI instrument, a two-beam interferometric recombiner. Methods. We obtained spectra and visibilities for the classical Cepheids X Sgr and T Mon. We fitted the MIDI measurements, supplemented by B, V, J, H, K literature photometry, with the numerical transfer code DUSTY to determine the dust shell parameters. We used a typical dust composition for circumstellar environments. Results. We detect an extended dusty environment in the spectra and visibilities for both stars, although T Mon might suffer from thermal background contamination. We attribute this to the presence of a circumstellar envelope (CSE) surrounding the Cepheids. This is optically thin for X Sgr (tau(0.55microns) = 0.008), while it appears to be thicker for T Mon (tau(0.55micron) = 0.15). They are located at about 15-20 stellar radii. Following our previous work, we derived a likely period-excess relation in the VISIR PAH1 filter, f(8.6micron)[%]= 0.81(+/-0.04)P[day]. We argue that the impact of CSEs on the mid-IR period-luminosity (P-L) relation cannot be negligible because they can bias the Cepheid brightness by up to about 30 %. For the K-band P-L relation, the CSE contribution seems to be lower (< 5 %), but the sample needs to be enlarged to firmly conclude that the impact of the CSEs is negligible in this band.Comment: Accepted for publication in Astronomy and Astrophysic

Multiplicity of Galactic Cepheids from long-baseline interferometry. II. The Companion of AX Circini revealed with VLTI/PIONIER

Aims: We aim at detecting and characterizing the main-sequence companion of the Cepheid AX Cir ($P_\mathrm{orb} \sim$ 18 yrs). The long-term objective is to estimate the mass of both components and the distance to the system. Methods: We used the PIONIER combiner at the VLT Interferometer to obtain the first interferometric measurements of the short-period Cepheid AX Cir and its orbiting component. Results: The companion is resolved by PIONIER at a projected separation $\rho = 29.2 \pm 0.2$ mas and projection angle $PA = 167.6 \pm 0.3^{\circ}$. We measured $H$-band flux ratios between the companion and the Cepheid of $0.90 \pm 0.10$ % and $0.75 \pm 0.17$ %, respectively at a pulsation phase for the Cepheid $\phi = 0.24$ and 0.48. The lower contrast at $\phi = 0.48$ is due to increased brightness of the Cepheid compared to the $\phi = 0.24$. This gives an average apparent magnitude $m\mathrm{_H (comp)} = 9.06 \pm 0.24$ mag. The limb-darkened angular diameter of the Cepheid at the two pulsation phases was measured to be $\theta_\mathrm{LD} = 0.839 \pm 0.023$ mas and $\theta_\mathrm{LD} = 0.742 \pm 0.020$ mas, respectively at $\phi = 0.24$ and 0.48. A lower limit on the total mass of the system was also derived based on our measured separation, we found $M_\mathrm{T} \geq 9.7 \pm 0.6 M_\odot$.Comment: Accepted for publication in Astronomy and Astrophysic

Investigating Cepheid ℓ\ell Carinae's Cycle-to-cycle Variations via Contemporaneous Velocimetry and Interferometry

Baade-Wesselink-type (BW) techniques enable geometric distance measurements of Cepheid variable stars in the Galaxy and the Magellanic clouds. The leading uncertainties involved concern projection factors required to translate observed radial velocities (RVs) to pulsational velocities and recently discovered modulated variability. We carried out an unprecedented observational campaign involving long-baseline interferometry (VLTI/PIONIER) and spectroscopy (Euler/Coralie) to search for modulated variability in the long-period (P $\sim$ 35.5 d) Cepheid Carinae. We determine highly precise angular diameters from squared visibilities and investigate possible differences between two consecutive maximal diameters, $\Delta_{\rm{max}} \Theta$. We characterize the modulated variability along the line-of-sight using 360 high-precision RVs. Here we report tentative evidence for modulated angular variability and confirm cycle-to-cycle differences of $\ell$ Carinae's RV variability. Two successive maxima yield $\Delta_{\rm{max}} \Theta$ = 13.1 $\pm$ 0.7 (stat.) {\mu}as for uniform disk models and 22.5 $\pm$ 1.4 (stat.) {\mu}as (4% of the total angular variation) for limb-darkened models. By comparing new RVs with 2014 RVs we show modulation to vary in strength. Barring confirmation, our results suggest the optical continuum (traced by interferometry) to be differently affected by modulation than gas motions (traced by spectroscopy). This implies a previously unknown time-dependence of projection factors, which can vary by 5% between consecutive cycles of expansion and contraction. Additional interferometric data are required to confirm modulated angular diameter variations. By understanding the origin of modulated variability and monitoring its long-term behavior, we aim to improve the accuracy of BW distances and further the understanding of stellar pulsations.Comment: Accepted for publication in MNRAS. 19 pages, 13 figures, 10 table

The long-period Galactic Cepheid RS Puppis - III. A geometric distance from HST polarimetric imaging of its light echoes

As one of the most luminous Cepheids in the Milky Way, the 41.5-day RS Puppis is an analog of the long-period Cepheids used to measure extragalactic distances. An accurate distance to this star would therefore help anchor the zero-point of the bright end of the period-luminosity relation. But, at a distance of about 2 kpc, RS Pup is too far away for measuring a direct trigonometric parallax with a precision of a few percent with existing instrumentation. RS Pup is unique in being surrounded by a reflection nebula, whose brightness varies as pulses of light from the Cepheid propagate outwards. We present new polarimetric imaging of the nebula obtained with HST/ACS. The derived map of the degree of linear polarization pL allows us to reconstruct the three-dimensional structure of the dust distribution. To retrieve the scattering angle from the pL value, we consider two different polarization models, one based on a Milky Way dust mixture and one assuming Rayleigh scattering. Considering the derived dust distribution in the nebula, we adjust a model of the phase lag of the photometric variations over selected nebular features to retrieve the distance of RS Pup. We obtain a distance of 1910 +/- 80 pc (4.2%), corresponding to a parallax of 0.524 +/- 0.022 mas. The agreement between the two polarization models we considered is good, but the final uncertainty is dominated by systematics in the adopted model parameters. The distance we obtain is consistent with existing measurements from the literature, but light echoes provide a distance estimate that is not subject to the same systematic uncertainties as other estimators (e.g. the Baade-Wesselink technique). RS Pup therefore provides an important fiducial for the calibration of systematic uncertainties of the long-period Cepheid distance scale.Comment: 14 pages, 14 figures, accepted for publication in Astronomy & Astrophysic

Cepheid distances from the SpectroPhoto-Interferometry of Pulsating Stars (SPIPS) - Application to the prototypes delta Cep and eta Aql

The parallax of pulsation, and its implementations such as the Baade-Wesselink method and the infrared surface bright- ness technique, is an elegant method to determine distances of pulsating stars in a quasi-geometrical way. However, these classical implementations in general only use a subset of the available observational data. Freedman & Madore (2010) suggested a more physical approach in the implementation of the parallax of pulsation in order to treat all available data. We present a global and model-based parallax-of-pulsation method that enables including any type of observational data in a consistent model fit, the SpectroPhoto-Interferometric modeling of Pulsating Stars (SPIPS). We implemented a simple model consisting of a pulsating sphere with a varying effective temperature and a combina- tion of atmospheric model grids to globally fit radial velocities, spectroscopic data, and interferometric angular diameters. We also parametrized (and adjusted) the reddening and the contribution of the circumstellar envelopes in the near-infrared photometric and interferometric measurements. We show the successful application of the method to two stars: delta Cep and eta Aql. The agreement of all data fitted by a single model confirms the validity of the method. Derived parameters are compatible with publish values, but with a higher level of confidence. The SPIPS algorithm combines all the available observables (radial velocimetry, interferometry, and photometry) to estimate the physical parameters of the star (ratio distance/ p-factor, Teff, presence of infrared excess, color excess, etc). The statistical precision is improved (compared to other methods) thanks to the large number of data taken into account, the accuracy is improved by using consistent physical modeling and the reliability of the derived parameters is strengthened thanks to the redundancy in the data.Comment: 10 pages, 4 figures, A&A in pres

Observational calibration of the projection factor of Cepheids

Context. The distance to pulsating stars is classically estimated using the parallax-of-pulsation (PoP) method, which combines spec- troscopic radial velocity (RV) measurements and angular diameter (AD) estimates to derive the distance of the star. A particularly important application of this method is the determination of Cepheid distances in view of the calibration of their distance scale. How- ever, the conversion of radial to pulsational velocities in the PoP method relies on a poorly calibrated parameter, the projection factor (p-factor). Aims. We aim to measure empirically the value of the p-factors of a homogeneous sample of nine bright Galactic Cepheids for which trigonometric parallaxes were measured with the Hubble Space Telescope (HST) Fine Guidance Sensor by Benedict et al. (2007). Methods. We use the SPIPS algorithm, a robust implementation of the PoP method that combines photometry, interferometry, and radial velocity measurements in a global modeling of the pulsation of the star. We obtained new interferometric angular diameter mea- surements using the PIONIER instrument at the Very Large Telescope Interferometer (VLTI), completed by data from the literature. Using the known distance as an input, we derive the value of the p-factor of the nine stars of our sample and study its dependence with the pulsation period. Results. We find the following p-factors: p = 1.20 ± 0.12 for RT Aur, p = 1.48 ± 0.18 for T Vul, p = 1.14 ± 0.10 for FF Aql, p = 1.31 ± 0.19 for Y Sgr, p = 1.39 ± 0.09 for X Sgr, p = 1.35 ± 0.13 for W Sgr, p = 1.36 ± 0.08 for β Dor, p = 1.41 ± 0.10 for ζ Gem, and p = 1.23 ± 0.12 for ` Car. Conclusions. The values of the p-factors that we obtain are consistently close to p = 1.324 ± 0.024. We observe some dispersion around this average value, but the observed distribution is statistically consistent with a constant value of the p-factor as a function of the pulsation period (χ2 = 0.669). The error budget of our determination of the p-factor values is presently dominated by the uncertainty on the parallax, a limitation that will soon be waived by Gaia

Wages, Employment and Futures Markets

This paper places the competitive firm under output price uncertainty in a standard efficiency wage model, wherein the work effort of labor depends on the wage rate set by the firm. Irrespective of the availability of a commodity futures market, we show that the Solow condition holds in that the equilibrium effort-wage elasticity is unity. The optimal wage rate is preference-free and independent of the underlying output price uncertainty under the efficiency wage hypothesis. Furthermore, we show that the introduction of the commodity futures market induces the firm to hire more labor and thereby produce more output if the firm is sufficiently risk averse. (JEL: D21; J31)link_to_OA_fulltex