Three-dimensional interferometric, spectrometric, and planetary views of Procyon

Three-dimensional interferometric, spectrometric, and planetary views of Procyon

A. Chiavassa L. Bigot P. Kervella A. Matter B. Lopez R. Collet Z. Magic M. Asplund Institut d’Astronomie et d’Astrophysique, Université Libre de Bruxelles, CP. 226, Boulevard du Triomphe, B-1050 Bruxelles, Belgium Université de Nice Sophia-Antipolis, Observatoire de la Côte dÕAzur, CNRS Laboratoire Lagrange, B.P. 4229, 06304 Nice Cedex 4, France LESIA, Observatoire de Paris, CNRS UMR 8109, UPMC, Université Paris Diderot, 5 place Jules Janssen, 92195 Meudon, France Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany Université de Nice Sophia-Antipolis, Observatoire de la Côte dÕAzur, CNRS Laboratoire Lagrange, B.P. 4229, 06304 Nice Cedex 4, France Centre for Star and Planet Formation, Natural History Museum of Denmark University of Copenhagen, Øster Voldgade 5-7, DKÐ1350 Copenhagen, Denmark Astronomical Observatory/Niels Bohr Institute, Juliane Maries Vej 30, DKÐ2100 Copenhagen, Denmark Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, D-85741 Garching Research School of Astronomy and Astrophysics, Australian National University, Cotter Rd., Weston Creek, ACT 2611, Australia
Received; accepted
Key Words.:
stars: atmospheres – stars: individual (Procyon) – hydrodynamics – radiative transfer – techniques: interferometric – techniques: spectroscopic – stars: planetary system

Context:Procyon is one of the brightest stars in the sky and one of our nearest neighbours. It is therefore an ideal benchmark object for stellar astrophysics studies using interferometric, spectroscopic, and asteroseismic techniques.

Aims:We used a new realistic three-dimensional (3D) radiative-hydrodynamical (RHD) model atmosphere of Procyon generated with the Stagger Code and synthetic spectra computed with the radiative transfer code Optim3D to re-analyze interferometric and spectroscopic data from the optical to the infrared. We provide synthetic interferometric observables that can be validated against observations.

Methods:We compute intensity maps from a RHD simulation in two optical filters centered at 500 and 800 nm (Mark III) and one infrared filter centered at 2.2 m (Vinci). We constructed stellar disk images accounting for the center-to-limb variations and used them to derive visibility amplitudes and closure phases. We computed also the spatially and temporally averaged synthetic spectrum from the ultraviolet to the infrared. We compare these observables to Procyon data.

Results:We study the impact of the granulation pattern on center-to-limb intensity profiles and provide limb-darkening coefficients in the optical as well as in the infrared. We show how the convective related surface structures impact the visibility curves and closure phases with clear deviations from circular symmetry from the 3rd lobe on. These deviations are detectable with current interferometers using closure phases. We derive new angular diameters at different wavelengths with two independent methods based on 3D simulations. We find mas and prove that this is confirmed by an independent asteroseismic estimation ( mas. The resulting is 6591 K (or 6556 K, depending on the bolometric flux used), which is consistent with K found with the infrared flux method. We find also a value of the surface gravity [cm/s] that is larger by 0.05 dex from literature values. Spectrophotometric comparisons with observations provide very good agreement with the spectral energy distribution and photometric colors, allowing us to conclude that the thermal gradient of the simulation matches fairly well Procyon.
Finally, we show that the granulation pattern of a planet hosting Procyon-like star has a non-negligible impact on the detection of hot Jupiters in the infrared using interferometry closure phases. It is then crucial to have a comprehensive knowledge of the host star to directly detect and characterize hot Jupiters. In this respect, RHD simulations are very important to reach this aim.


1 Introduction

Procyon ( Canis Majoris) is one of the brightest stars in the sky and one of our nearest neighbours. It is therefore an ideal target for stellar astrophysics studies.

For this reason, it has a long history of observations. bessel1844 discovered that its motion was perturbed by a invisible companion. Procyon became, after Sirius, one of the first astrometric binaries ever detected. The first orbital elements were determined by auwers1862, who showed that the period of revolution is about 40 years. The faint companion, Procyon B, was not detected visually until the end of the nineteenth century by schaeberle1896. It was one of the first detected white dwarfs (eggen65). The main component of the system is a subgiant F5 IV-V (Procyon A, HR 2943, HD 61421) which is ending its life on the Main Sequence (eggenberger05; provost06). It has a solar metallicity (griffin71; steffen85; 2002ApJ...567..544A) with an effective temperature around K (code76). One of the first radius determinations was made photometrically by gray67, who found . The binary nature of the system is a great opportunity to determine the mass of both companions. The first attempt was made by strand51 who determined the masses of Procyon A & B, by determining the parallax and orbital elements of the system. He found and , respectively. steffen85 claimed that the mass of Procyon A was too large to be compatible with its luminosity and suggested rather a mass around . More recent and more accurate determinations of the mass of Procyon A converge to a star with (girard00) or (gatewood06). The age of the system is well constrained by the white dwarf companion, whose cooling law as function of time is well established. provencal02 found an old white dwarf with an age of Gy. This determination is a strong constraint for stellar evolution models.

A way to discriminate between different masses and ages is the determination of the interferometric radius. The first attempt to measure the diameter of Procyon was done by hanbury67; 1974MNRAS.167..475H who found an angular diameter of mas. This value was confirmed later by 1991AJ....101.2207M but with a much better precision (%). More recently, 2004A&A...413..251K redetermined the angular diameter using the Vinci instrument at VLTI. They found a diameter that is even smaller mas. 2005ApJ...633..424A re-analyzed these data using hydrodynamical model atmospheres and found mas (see their Table 7). It is interesting to compare these angular diameters with the independent infrared flux method, a recent determination was performed by 2010A&A...512A..54C who derived a value of mas, which is smaller than the 2003A&A...408..681K’s result, but they agree to within 1.

Another great particularity of Procyon is the presence of oscillations that are due to trapped acoustic modes. The first claims of a detection of an excess power were made by gelly86; gelly88 and brown91 who found a mean large spacing between consecutive acoustic modes of about and , respectively. Martic99 made a clear detection using the ELODIE fiber échelle spectrograph with a strong excess power around and confirmed the results of brown91. They determined the frequency spacing to be . Later, eggenberger04 and martic04 made the first identifications of individual frequencies of spherical harmonic degrees with mean large spacings of (eggenberger04) and (martic04). More recent observations have been made from the ground during single or multisite campaigns (mosser08; arentoft08; bedding10) or from space on board of the MOST satellite (matthews04; guenther08), which considerably improved the precision in frequencies within .

The stellar evolution model of Procyon was made by hartmann75, who used the astrometric mass, photometry, and the angular diameter of 1974MNRAS.167..475H to constrain the model. More realistic modeling, especially with better equation-of-states, came with guenther93, who showed the importance of diffusion of elements. The values of individual p-modes frequencies considerably constrained the determinations of the fundamental parameters (barban99; dimauro01; eggenberger05; provost06; bonanno07; guenther08). eggenberger05 and provost06 made a realistic stellar evolution modeling to constrain simultaneously the location in the HR diagram and the large frequency separations of eggenberger04 and martic04, respectively. They disagreed on the derived mass. eggenberger05 found a mass that agrees well with the astrometric value derived by girard00, whereas provost06 found a mass that corresponds to the more recent value of gatewood06. Part of the difference in the two stellar evolution models comes from the slightly different large separation (which increases with the mass), and partly due to the difference in the stellar evolution codes themselves. We emphasize that the best stellar model of provost06 that fits asteroseismic and spectro-photometric data has an age ( Gy) that is consistent with the age of the white dwarf. Indeed, the astrometric mass of Procyon B implies that the mass of its progenitor was about , which has lived about My on the Main Sequence. Therefore, we can safely conclude that the age of the system must be at least 2 Gy. The larger mass of eggenberger05 corresponds to a younger ( Gy) system, incompatible with the result of provencal02.

The atmospheric parameters () and the interferometric radius, that are used to define the stellar evolution model and the analysis of frequencies, strongly depend on the realism of the atmosphere and the exactness of the temperature gradient in the surface layers. In the case of F-stars, these gradients are strongly modified by the convective transport which is more vigorous than in the Sun. This is clearly seen in line bisectors (gray81; dravins87), which are two or three times larger than in the Sun. The higher convective velocities are due to the higher stellar luminosity and smaller densities. This strong effect of convection must be taken into account for stellar physics diagnostics. Realistic 3D time-dependent hydrodynamical simulations of the surface layer of Procyon were performed by several authors (atroshchenko89; nordlund90; 2002ApJ...567..544A), who showed that the 3D effects in such F-star can bring significant differences for line profile formation and abundance analysis (2002ApJ...567..544A). nelson80 and nordlund90 showed that the surface defined at is not flat but rather “corrugated” due to the large fluctuations and the high contrast of granulation. These hydrodynamical simulations can reproduce with success the line shifts, asymmetries, and in particular the observed bisectors of various lines (2002ApJ...567..544A, FeI and FeII). 2002ApJ...567..544A also showed that the 3D limb darkening law significantly differ 1D law, up to , leading to a correction of K, which is non-negligible for precise stellar evolution modeling. 2005ApJ...633..424A made 3D models using the COBOLD code (2002AN....323..213F; 2011arXiv1110.6844F) to calculate limb darkened intensity profiles to analyze visibility curves obtained by the Vinci instrument (2003A&A...408..681K) and the Mark III. Their 3D analysis led to a smaller radius by mas than that of 2003A&A...408..681K, who used 1D limb darkened law. 2005ApJ...633..424A also showed that 3D models better reproduce the spectral energy distribution in the UV, whereas 1D model are unable.

Regarding the importance of a realistic hydrodynamical modeling of the atmosphere of Procyon, we re-analyze the interferometric and spectroscopic data at the different wavelengths of 2005ApJ...633..424A using up-to-date line and continuum opacities (2008A&A...486..951G) to derive a new radius. We propose a solution that agrees well with the astrometric, asteroseismologic, infrared flux method, and interferometric data. We also explore the impact of convection-related surface structures on the closure phases and assess how the direct search of planets by interferometry may be affected by the host star surface structures.

2 Three-dimensional radiative-hydrodynamical approach

2.1 Procyon simulation

The convective surface of Procyon is modeled using the Stagger Code (2009LRSP....6....2N, Nordlund & Galsgaard1111995, In a local box located around the optical surface , the code solves the full set of hydrodynamical equations for the conservation of mass, momentum, and energy coupled to an accurate treatment of the radiative transfer. The equations are solved on a staggered mesh where the thermodynamical scalar variables (density, internal energy, and temperature) are cell centered, while the fluxes are defined on the cell faces. This scheme has several numerical advantages to simulate surface convection. It is robust against shocks and ensures conservation of the thermodynamic variables. The domain of simulation contains the entropy minimum located at the surface and is extended deep enough to have a flat entropy profile at the bottom. The code uses periodic boundary conditions horizontally and open boundaries vertically. At the bottom of the simulation, the inflows have constant entropy and pressure. The outflows are not constrained and are free to pass through the boundary. The code is based on a sixth order explicit finite difference scheme and fifth order for interpolation. Numerical viscosity of the Rytchmeyer & Morton type is used to stabilize the code. The corresponding adjustable parameters are chosen to minimize the viscosity and are not adjusted to fit the observables. We used a realistic equation-of-state that accounts for ionization, recombination, and dissociation (MHD) and continuous (Trampedach et al. private communication) and line opacities (2008A&A...486..951G). An accurate treatment of the transfer is needed to get a correct temperature gradient of the transition region between optically thin and thick layers. The transfer equation is solved using a Feautrier-like scheme along several inclined rays (one vertical, eight inclined) through each grid point. The wavelength dependence of the radiative transfer is taken into account using opacity bins 1982A&A...107....1N. The numerical resolution is . The geometrical sizes are 22 Mm 22 Mm horizontally and 17 Mm vertically. The horizontal dimensions of the box are defined to include at each time step a sufficient number of granules and the vertical one is chosen to ensure that the entropy profile is flat at the bottom. The equations of magnetohydrodynamics are not computed for this model. The stellar parameters corresponding to our RHD model (Table 1) are K, [cm/s], and a solar chemical composition (asplund09). The uncertainty in represents the fluctuations with time around the mean value. These parameters roughly correspond to those of Procyon. The exact values of the parameters do not influence the limb darkening (2005ApJ...633..424A; 2011A&A...534L...3B).

2.2 Spherical tiling models, intensity maps, and spectra

Figure 1: Intensity maps in the Mark 500 nm filter for the RHD simulation for different inclination angles . The intensity ranges from to  erg cm s Å.

The computational domain of each simulation represents only a small portion of the stellar surface. To obtain an image of the whole stellar disk, we employ the same tiling method explained in 2010A&A...524A..93C. For this purpose, we used the 3D pure-LTE radiative transfer code Optim3D (2009A&A...506.1351C) to compute intensity maps from the snapshots of the RHD simulation of Table 1 for different inclinations with respect to the vertical, =[1.000, 0.989, 0.978, 0.946, 0.913, 0.861, 0.809, 0.739, 0.669, 0.584, 0.500, 0.404, 0.309, 0.206, 0.104] (Fig. 1) and for a representative series of simulation’s snapshots: we chose 25 snapshots taken at regular intervals and covering 1 h of stellar time, which corresponds to 5 p-modes. Optim3D takes into account the Doppler shifts due to convective motions and the radiative transfer equation is solved monochromatically using pre-tabulated extinction coefficients as functions of temperature, density, and wavelength (with a resolving power of ). The lookup tables were computed for the same chemical composition as the RHD simulation (i.e. asplund09) and using the same extensive atomic and molecular opacity data as the latest generation of MARCS models (2008A&A...486..951G).
Then, we used the synthetic images to map onto spherical surfaces accounting for distortions especially at high latitudes and longitudes cropping the square-shaped intensity maps when defining the spherical tiles. Moreover, we selected intensity maps computed from random snapshots in the simulation time-series: this process avoided the assumption of periodic boundary conditions resulting in a tiled spherical surface globally displaying an artifactual periodic granulation pattern.

Based on the stellar radius estimates and on the sizes of the simulationsÕ domains (Table 1), we required 215 tiles to cover half a circumference from side to side on the sphere (number of tiles , where 22.0 is the horizontal dimension of the numerical box in Mm and the radius of the star). To produce the final stellar disk images, we performed an orthographic projection of the tiled spheres on a plane perpendicular to the line- of-sight (). The orthographic projection returned images of the globe in which distortions are greatest toward the rim of the hemisphere where distances are compressed (2010A&A...524A..93C).

222Horizontally and temporal average and standard deviation of the emergent effective temperatures [Fe/H] -dimensions -resolution
[cm/s] [Mm] [grid points] []
65123332011JPhCS.328a2003C 0.0444Chemical composition by asplund09 4.0 240240240 2.055555Angular diameter of 5.443 mas (2004A&A...413..251K) converted into linear radius with Eq. (7)
Table 1: 3D simulation of Procyon used in this work.

In this work, we computed a synthetic stellar disk image for interferometric spectral bands used in 2005ApJ...633..424A: (i) the Mark III (1988A&A...193..357S) centered at 500 and 800 nm, and (ii) Vinci (2003SPIE.4838..858K) centered at 2.2 m (Fig. 2). The Mark III sensitivity curves are assumed to be Gaussian with central wavelengths 500 and 800 nm, each of them with a FWHM of 20 nm (1991AJ....101.2207M). We produced a number of synthetic stellar disk images corresponding to different wavelengths in the filters with a spectral resolving power of 20 000. Figure 3 shows the resulting synthetic stellar disk images averaged over each passband.
With Optim3D, we computed also the spectra, which are normalized to the filter transmission as: , where is the intensity and is the transmission curve of the filter at a certain wavelength. The spectra in Fig. 2 have been computed along rays of four -angles [0.88, 0.65, 0.55, 0.34] and four -angles [, , , ], after which we performed a disk integration and a temporal average over all selected snapshots.

Figure 2: A spatially and temporally averaged synthetic spectrum in the optical wavelength region with the Mark 500 nm filter (red: top left), Mark 800 nm (green: top right), and Vinci infrared filter (blue: bottom).
Figure 3: Top row: synthetic stellar disk images of the RHD simulation. The intensity range is [0.3 - 1.0], [0.3 - 5.2], and [0.3 - 0.9] erg cm s Å for Mark 500 nm, Mark 800 nm, and Vinci filters, respectively. Bottom row: enlargements of the images above.

2.3 Three-dimensional limb-darkening

Figure 3 shows irregular stellar surfaces with numerous convective related surface structures. There are pronounced center-to-limb variations in the Mark 500 nm and Mark 800 nm filters while these are less noticeable in the Vinci filter. This is manly due to the different Planck functions in the optical range and in the infrared region.

Figure 4: Radially averaged intensity profiles (black line) derived from the synthetic stellar disk images of Fig. 3. The gray areas denote the one sigma spatial fluctuations with respect to the averaged intensity profile. The dashed horizontal line is an uniform disk intensity profile. The intensity is normalized to the mean intensity at disk center and the radius is normalized to the radius given in Table 1. The blue dotted-dashed line in central and right panels is the average intensity profile of Mark 500 nm. The red dotted-dashed line in left panel is the continuum only averaged intensity profile in Mark 500 nm.

We derived azimuthally averaged (i.e., averaged over different angles) intensity profiles for every synthetic stellar disk image from the simulation (Fig. 4). Using the method described in 2009A&A...506.1351C; 2010A&A...524A..93C, the profiles were constructed using rings regularly spaced in for (i.e. ), with the angle between the line of sight and the vertical direction. The standard deviation of the average intensity, , was computed within each ring, the parameter is connected to the impact parameter through the relationship , where is the stellar radius reported in Table 1. The total number of rings is 20, we ensured that this number is enough to have a good characterization of the intensity profiles.
Figure 4 displays a steeper center-to-limb variation for the optical region, as already visible in the disk images, with fluctuations of in the Mark III 500 nm filter down to and in the Mark III 800 nm and Vinci filters, respectively. We tested the impact of spectral lines in the Mark 500 nm filter, for which the effects of lines are stronger, computing a synthetic disk image considering only the continuum opacities. Its averaged intensity profiles (red line in left panel of Fig. 4) is very similar to the one computed with spectral lines (black line) with differences lower than for and between at the limb (). It is also interesting to notice that the continuum profile tends to be closer to the uniform disk profile (dashed line in the figure), as well as the Mark 800 nm and Vinci filters with respect to the intensity profile of the Mark 500 nm.

Figure 5: Center-to-limb fits (red dashed line) obtained using Eq. (1) with for the 3D azimuthally average intensity profile (solid line). A full limb-darkening (dash-dotted line), a partial limb-darkening (triple dot-dashed line) are also shown. = , with the angle between the line of sight and the vertical direction.

We used the following limb-darkening law (2009A&A...506.1351C) to fit the averaged profiles:


In this equation, is the intensity, are the limb-darkening coefficients, and their number. We performed a Levenberg-Marquardt least-square minimization to fit all the radially averaged profiles of Fig. 4 using this law and weighting the fit by , due to 3D fluctuations (gray areas of Fig. 4) We varied the order and found that provides the optimal solution with very minor improvements to the minimization using . This was already found by 2010A&A...524A..93C for K giants. Figure 5 shows the fits in the different filters used in this work, while Table 2 reports the limb-darkening coefficients from the fits. The figure illustrates that the average intensity profiles from a RHD simulation is largely different from the full limb-darkening () and partial limb-darkening () profiles as well as the power law profile that does not even provide an appreciable fit to the radially average profile. We therefore discourage the use of these simple laws.

Mark 500 nm 1.000 0.066 0.421
Mark 800 nm 1.000 0.041 0.236
Vinci 1.000 0.016 0.188
Table 2: Limb-darkening coefficients from the law described in text for RHD simulation of Table 1.

2.4 Visibility curves

The synthetic disk images are used to derive interferometric observables. For this purpose, we used the method described in 2009A&A...506.1351C to calculate the discrete complex Fourier transform for each image. The visibility, , is defined as the modulus , of the Fourier transform normalized by the value of the modulus at the origin of the frequency plane, , with the phase , where and are the imaginary and real parts of the complex number , respectively. In relatively broad filters, such as Vinci, several spatial frequencies are simultaneously observed by the interferometer. This effect is called bandwidth smearing. 2003A&A...408..681K; 2003A&A...404.1087K show that this effect is negligible for squared visibilities larger than 40 but it is important for spatial frequencies close to the first minimum of the visibility function. To account for this effect, we computed the squared visibilities as proposed by 2004A&A...413..711W


where is the squared visibility at wavelength , and are the filter wavelength limits, is the transmission curve of the filter, and is the flux at wavelength (see Fig. 2). has been computed from the disk images that already include in the calculations.
In our case we follow these steps:

  1. we generate a synthetic stellar disk image at different wavelengths of the filters;

  2. we compute the visibility curves for 36 different cuts through the centers of the stellar disk images;

  3. we apply Eq. (2) to obtain the averaged squared visibilities.

Figure 6 (top panels) shows the visibility curves computed with the Eq. (2) and for 36 different cuts through the centers of synthetic disk images. This is equivalent to generating different realizations of the stellar disk images with intensity maps computed for different sets of randomly selected snapshots. A theoretical spatial frequency scale expressed in units of inverse solar radii (R) is used. The conversion between visibilities expressed in the latter scale and in the more usual “arcseconds” scale is given by


where 214.9 is the astronomical unit expressed in solar radius, and is the distance of the observed star. The spatial frequency in arcsec (i.e, ) is related to the baseline (i.e., ) in meters by


where is the wavelength in .
The first null point of the visibility is mostly sensitive to the radial extension of the observed object (e.g. 2001ARA&A..39..353Q, and 2010A&A...515A..12C, 2010A&A...515A..12C for an application to RHD simulations), while the first null point and the second lobe of the visibility curves are sensitive to the limb-darkening (1974MNRAS.167..475H). Since we want to concentrate on the small scale structure of the surface, the visibility curves of Fig. 6 are plotted longward of the first null point. They show increasing fluctuations with spatial frequencies due to deviations from the circular symmetry relative to uniform disk visibility. This dispersion is clearly stronger in the optical filter at 500 nm and appreciable from longward of the top of third lobe. Moreover, it is also noticeable that the synthetic visibilities are systematically lower than the uniform disk with a weaker divergence for the Vinci filter. This is due to a the non-negligible center-to-limb effect visible in Figs. 3 and 4.

The bottom panel of Fig. 6 displays the one visibility fluctuations, , with respect to the average value (). The dispersion increases with spatial frequency. This is due to the small scale structure on the model stellar disk (see e.g. 2010A&A...515A..12C). The dispersion is stronger in the case of the Mark III 500 nm filter with respect to the redder filters.
Figure 4 shows small differences between the averaged intensity profiles of the Mark III 500 nm filter with and without considering the spectral lines in our calculations. Therefore, we computed visibility curves in both cases and found that the visibility fluctuations are indistinguishable (Fig. 6, bottom panel). While the molecular absorption can cause a strong difference in stellar surface appearance (and consequentially also on the visibility curves) in the case of cool evolved stars (e.g., the contribution of HO to the radius measurement for red supergiants stars, 2010A&A...515A..12C), this is not the case for Procyon where the atomic lines are not strong enough to cause an appreciable effect and also the atmosphere is very compact.
The bottom panel of Fig. 6 also shows that, on the top of the second lobe ( R), the fluctuations are of the order of of the average value for Mark III 500 nm filter and for Mark III 800 nm and Vinci filters. From the top of the third lobe ( R) on, the fluctuations are of the average value in the optical region, which is larger of the instrumental error of VEGA on CHARA (1, 2009A&A...508.1073M).
It must be noted that our method of constructing realizations of stellar disk images inevitably introduces some discontinuities between neighbouring tiles by randomly selecting temporal snapshots and by cropping intensity maps at high latitudes and longitudes. However, 2010A&A...524A..93C proved that for the signal artificially introduced in to the visibility curves is largely weaker than the signal due to the inhomogeneities of the stellar surface.

3 Fundamental parameters

3.1 Multiwavelength angular diameter fits in the Mark III and Vinci filters

We used the data from the Mark III interferometer at 500 and 800 nm (1991AJ....101.2207M) plus additional measurements at 500 nm reported in 2005ApJ...633..424A. It has to be noted that 1991AJ....101.2207M do not provide a calibrator list even thought they claim that the calibrators are all smaller that the science star. The Mark III 500 nm filter has points on the first and the second lobe while the Mark III 800 nm has limited baselines in the first lobe. For the Mark III 500 nm filter, we used only the data with baselines larger than 20 meters because at shorter baselines the error bars are of the squared visibility. We also used Vinci data from the Very Large Telescope Interferometer (2004A&A...425.1161K; 2005ApJ...633..424A). The wavelength of these observations is around 2.2 m and the baselines go up to 66 m, significantly lower in the first lobe of the Procyon visibility curve.

Figure 6: Top panels: visibility curves () from the Procyon simulation in the filters computed in this work. The visibilities are computed for 36 different azimuth angles 5 apart (thin gray lines). The dotted line is an uniform disk scaled to match the same radius, i.e. the first null in visibility curve. The visibilities are displayed only longward of the first null visibility point. A logarithmic scale is used on y-axis. Bottom panel: visibility fluctuations () with respect to the visibility average value () as a function of spatial frequencies for all the considered filters.

We used two independent methods to determine the angular diameters:

  1. the synthetic visibilities of Fig. 6 obtained from the spherical tiling method described in this work and scaled to absolute model dimensions (using Eq. (3) and (4)) to match the interferometric observations;

  2. the method described in 2005ApJ...633..424A; 2006A&A...446..635B; 2011A&A...534L...3B, that computes the normalized fringe visibilities using the van Cittert-Zernike theorem.

Method (1) is a the direct consequence of the Fourier Transform of the synthetic disk images (Fig. 3): the visibilities vary for different cuts through the centers of synthetic disk images (i.e., the position of the first null changes its position). Method (2) is based on the integration of the spatial average intensity profile. For both methods, we used a Levenberg-Marquardt least-square minimization.
Table 3 reports the best-fit angular diameters. While for the uniform disk there is a significantly wide range of values among the different filters, the angular diameters from the RHD simulation are closer and overlap with the uncertainties. The two independent methods used in Table 3 show a clear tendency of the optical diameters to appear smaller than in the infrared (). The possible explanation for the different radius between Mark III and Vinci are: (i) the RHD model used is not fully appropriate to the observations, for instance the center-to-limb variation may not be correct because, for instance, the points on the second lobe (sensitive to the center-to-limb variation) of Mark 500 nm filter do not match very well (Fig. 7, top panel); (ii) the data of Mark III may present problems (e.g., calibrator or systematics).
In addition to this, we also conclude that the two methods give consistent results and can be used without distinction to perform angular diameter fits at least for dwarf stars. If fact, in the case of cool evolved stars with low surface gravity (), the surface asymmetry may strongly impact the shape of the star and the radius depends on orientation of the projected baseline (2009A&A...506.1351C).

The angular diameters found for the Vinci filter (Table 3) are in fairly good agreement with 2005ApJ...633..424A who found 5.403 0.006 mas. However, is smaller than: (i) 5.448 0.03 mas (2004A&A...413..251K), based on the fit of the Vinci with baseline points of Fig. 7 lower than 22 meters; (ii) 5.48 0.05 mas (2002ApJ...567..544A), obtained with Eq. 7 from the linear radius of 2.071 0.020 ; (iii) 5.50 0.17 mas (1991AJ....101.2207M). Our is also close to 5.326 0.068 mas found by 2010A&A...512A..54C with the infrared flux method.

Transforming the Vinci angular diameter to linear radius (Eq. 7), we found 2.019 . This value is compared to what we chose as a reference in Table 1: 2.055 . The difference is 0.023 , which is 16 Mm (i.e., 0.7 times the numerical box of RHD simulation, 22 mM, Table 1, 4th column). We checked that this has a negligible impact on the observables found in this work using the spherical tiling method described in Section 2.2.
The astrometric mass ( gatewood06, used in Section 3.3) combined with our interferometric diameter leads to a new gravity [cm/s], which is larger by dex than the value derived in 2002ApJ...567..544A. The contribution of the revised Hipparcos parallax is 0.01 dex in the change.

Mark 500 nm 5.313 0.03 5.324 0.03 5.012 0.03
Mark 800 nm 5.375 0.06 5.370 0.06 5.208 0.06
Vinci 5.382666Since the resulting diameters are very similar, we adopted an average angular diameter of 5.390 for all the calculations in the article 0.03 5.397 0.03 5.326 0.03
Table 3: Best angular diameters, in mas, for RHD simulation with Method (1) () and Method (2) () described in Section 3.1; and for the uniform disk. The error is one with respect to the average value.
Figure 7: Best matching synthetic visibility from Fig. 6 for the Mark 500, Mark 800, and Vinci data. The stellar disk maps in each filter have been scaled to match the radius of the star. For the Mark 500 nm filter, we used only the data with baselines larger than 20 meters.

3.2 The effective temperature

The effective temperature is determined from the bolometric flux and the angular diameter in the Vinci filter (average between the two methods in Table 3, = 5.390 mas) by the relation


where stands for the Stefan-Boltzmann constant. Our derived diameter being smaller by % than 1991AJ....101.2207M or 2004A&A...413..251K, the derived effective temperature has to be larger by % compared to 2002ApJ...567..544A or 2005ApJ...633..424A, who used the previous mentioned references. There are several sources for the bolometric flux leading to slightly different . The values of (fuhrmann97) and (2005ApJ...633..424A) lead to K and K, respectively.

Our new 3D returns a value closer to K obtained by 2010A&A...512A..54C than the old derived value of K (2005ApJ...633..424A).
The influence of the uncertainties in the selected fundamental parameters (,,) of our RHD model atmosphere has a negligible impact on the limb-darkening and therefore on the derived angular diameter and . This was tested for HD 49933 (2011A&A...534L...3B), a star similar to Procyon.

3.3 Asteroseismic independent determination of the radius

The radius of the star can be derived from its oscillation spectrum. The frequency of the maximum in the power spectrum is generally assumed to scale with the acoustic cut-off frequency of the star (e.g. brown91; kjeldsen11), therefore . Then, it is straightforward to derive the radius


The validity of such scaling relation has been verified on large asteroseismic surveys (e.g. bedding03; verner11). The value of derived from photometry is accurately determined, (arentoft08). The solar value of is taken from belkacem11. We emphasize that the dependence on in Eq. 6 is weak, therefore the derived radius is not very sensitive to the selected value of the effective temperature. We use the value of K derived in Section 3.2, since it is closer to the infrared flux method determination. Since Procyon is a binary star, the mass can by determined by the astrometric orbital elements and the third Kepler’s law. However, the derived value is subject to debate. Indeed, girard00 found a mass of , whereas gatewood06 found . As discussed in the introduction, we prefer to keep the value of gatewood06 since stellar evolution models that use this mass, are in agreement with the age of the white dwarf companion. Using these values, we found a radius of . We can translate this radius into angular diameter using the relation


with the solar angular radius arcsecs (chollet99) and the parallax mas (vanleeuwen07). This radius agrees well with our interferometric value within error bars.

4 Spectrophotometry and Photometry

The radiative transfer code Optim3D includes all the up-to-date molecular and atomic line opacities. This allows for very realistic spectral synthetic RHD simulations for wavelengths from the ultraviolet to the far infrared. It is then possible to compute realistic synthetic colors and compare them to the observations. Figure 8 shows the spectral energy distribution (SED) computed along rays of four -angles [0.88, 0.65, 0.55, 0.34] and four -angles [, , , ], and after a temporal average over all selected snapshots (like in Fig. 2). The shape of SED reflects the mean thermal gradient of the simulations.

Figure 8: Spatially and temporally average synthetic spectral energy distribution from the UV to the near infrared. Red curves correspond to 1990PASP..102.1181B for the filters .

We used the prescriptions by 1990PASP..102.1181B to compute the color indexes for the filters . Table 4 displays the comparison of the synthetic colors with the observed ones. The difference is very small.

RHD simulation 0.419 0.255 0.252 0.507
Observation 0.420 0.245 0.245 0.490
Table 4: Photometric colors for RHD simulation of Table 1 and for the corresponding observations of 1990A&AS...83..357B

We used the absolute spectrophotometry measurements at ultraviolet and visual wavelengths collected by 2005ApJ...633..424A to compare with the synthetic SED. The data come from: (i) the Goddard High Resolution Spectrograph (GHRS) data sets Z2VS0105P ( PI A. Boesgaard), Z17X020CT, Z17X020AT, Z17X0208T (PI J. Linsky) from 136 and 160 nm; (ii) the International Ultraviolet Explorer (IUE) 1999A&AS..139..183R from 170 to 306 nm; (iii) the Hubble Space Telescope imaging Spectrograph (STIS) from 220 to 410 nm; (iv) and the visual and near-infrared wavelength data from 1992A&AS...92....1G. The data from the GHRS are far superior to any other measurements below 160 nm because the continuum drops by more than a factor of 100 here, too much for the limited dynamic range of IUE. The flux was estimated by computing the mean flux between the emission lines in each spectrum incorporating the flux uncertainties provided with each calibrated data set (2005ApJ...633..424A).

Figure 9: Comparisons between synthetic SED (black) and spectrophotometric measurements at ultraviolet and visual wavelengths (green dots with error bars in left column plots and red line). Top row: the models and data are binned to 2 nm resolution in the UV for clarity and to 5 nm in the visual to match the resolution of the observed spectrophotometry of 1992A&AS...92....1G. Bottom row: same as above with the synthetic SED and the observations at higher spectral resolution. The synthetic SEDs are scaled in absolute flux using an angular diameter of 5.390 mas found from the fit of Vinci filter observations.

Figure 9 (top row) shows that our synthetic SED matches fairly well the observations from the UV to the near-IR region. This result can be compared to the already good agreement found with composite RHD model done with COBOLD code in 2005ApJ...633..424A. Moreover, the bottom row of the Figure shows that at higher spectral resolution, the agreement is even more remarkable. RHD models match also the observations between 136 and 160 nm, that are supposed to form at depths beneath its chromosphere (2005ApJ...633..424A) and thus being affected by the convective-related surface structures. It should also be noted that the ultraviolet SED longward of 160 nm may be impacted by non-LTE treatment of iron-group elements. 2005ApJ...618..926S found that non-LTE models for the Sun have up to 20 more near-UV flux relative to LTE models. It is not possible to determine if this difference is also present in Procyon because either the RHD model calculation and the post-processing calculations have been done with LTE approximation. The ultraviolet SED may also be affected by scattering at these wavelengths. This effect is also not included in our calculations. However, 2010A&A...517A..49H demonstrated that, in RHD simulations, the scattering does not have a significant impact on the photospheric temperature structure in the line forming region for a main sequence star.
We conclude that the mean thermal gradient of the simulation, reflected by the spectral energy distribution, is in very good agreement with Procyon.

5 Closure phases and perspectives for hot Jupiter detection

Interferometry has the potential for direct detection and characterization of extrasolar planets. It has been claimed that differential interferometry could be used to obtain spectroscopic information, planetary mass, and orbit inclination of extrasolar planets around nearby stars: 2000SPIE.4006..269S; 2000SPIE.4006..407L; 2004SPIE.5491..551J; 2008SPIE.7013E..91R; 2008SPIE.7013E..45Z; 2008PASP..120..617V; 2010A&A...515A..69M; 2011A&A...535A..68A; 2011PASP..123..964Z. However, current interferometers lack sufficient accuracy for such a detection. When observing a star with a faint companion, their fringe patterns add up incoherently and the presence of a planet causes a slight change in the phases and, consequently, the closure phases.This difference can be measured with a temporal survey and should be corrected with the intrinsic closure phases of the host star.
In this framework, theoretical predictions of the closure phases of the host stars are crucial. Closure phase between three (or more) telescopes is the sum of all phase differences: this procedure removes the atmospheric contribution, leaving the phase information of the object visibility unaltered. The major biases or systematic errors of closure phases come from non-closed triangles introduced in the measurement process, which in principle, can be precisely calibrated. Therefore, it is a good observable for stable and precise measurements (e.g. 2007NewAR..51..604M). The closure phase thus offers an important complementary piece of information, revealing asymmetries and inhomogeneities of stellar disk images.

5.1 Closure phases from the star alone

Figure 10: Scatter plot of closure phases of 2000 random baseline triangles with a maximum linear extension of 155 m at 500 nm, 250 m at 800 nm, and 680 m at 2.2 m, respectively. A parallax of 284.56 mas (vanleeuwen07) and an apparent radius of 5.390 mas have been used.

Figure 10 displays deviations from the axisymmetric case (zero or ) that are particularly occurring in optical filters. There is a correlation between Fig. 10 and Fig. 6 because the scatter of closure phases increases with spatial frequencies as for the visibilities: smaller structures need large baselines to be resolved. Moreover, it is visible that the closure phase signal becomes important for frequencies larger than  R (top of the third visibility lobe, i.e., from Fig. 6). These predictions can be constrained by the level of asymmetry and inhomogeneity of stellar disks by accumulating observations on closure phase at short and long baselines.
Observing dwarf stars at high spatial resolution is thus crucial to characterize the granulation pattern using closure phases. This requires observations at high spatial frequencies (from 3rd lobe on) and especially in the optical range. In fact, considering that the maximum baseline of CHARA is 331 meters (2005ApJ...628..453T), for a star like Procyon with a parallax of 284.56 mas (vanleeuwen07), a baseline of 55 (240) meters is necessary to probe the third lobe at 0.5 (2.2) m. For comparison, the nominal measurement error with CHARA array is 0.3 with a peak of performance of 0.1 for a shorter triangle 2008SPIE.7013E..45Z (to be compared with closure phases value of Fig. 10).

5.2 Closure phases from the hosting star plus the hot Jupiter companion

Among the detected exoplanets, the direct detection and the characterization of their atmospheres appears currently within reach for very close planet-star system (0.1AU) and for planets with temperatures 1000 K, implying their infrared flux is of their host stars. Since the bulk of the energy from hot Jupiters emerges from the near-infrared between 1-3 m (2008ApJ...678.1436B), interferometry in the near infrared band (like in the range of the Vinci filter centered at m) can provide measurements capable to detect and characterize the planets. Detecting hot Jupiters from Earth is endeavoring because it is challenging and at the limits of current performance of interferometry and, once these conditions are met, the signal from the host stars must also be known in detail. As visible in the synthetic stellar disk images of Fig. 3, the granulation is a non-negligible aspect of the surface of dwarf stars and has an important signal in the closure phases (Fig. 10).

To estimate the impact of the granulation noise on the planet detection, we used the RHD simulation of Procyon and added a virtual companion to the star. The modelling of the flux of an irradiated planet requires careful attention on the radiative transfer conditions related to the stellar irradiation and therefore we used the models of 2001ApJ...556..885B. In particular, we used spectra of hot irradiated extrasolar planet around a star with about the same spectral type of Procyon, a mass of 1 Jupiter mass, and an intrinsic temperature 1000 K. We assumed a radius of 1.2 Jupiter radii and various orbital distances [0.1, 0.25, 0.5, 1.0] AU following 2010A&A...515A..69M. The atmospheric composition of the hot Jupiter is identical to the two models of 2001ApJ...556..357A where: (1) dust (particles and grains) remains in the upper atmosphere and (2) where dust has been removed from the upper atmosphere by condensation and gravitational settling.

Figure 11: Synthetic stellar disk images (yellow-orange scale) of the simulation in Vinci filter (the intensity range is [0.3 - 0.9] erg cm s Å) together with a hot Jupiter (pink) at a distance of 0.1 AU for a star with a parallax of 284.56 mas (vanleeuwen07). The intensity of the hot Jupiter is 11 times lower than the maximum stellar surface intensity.

Figure 11 shows the geometrical configuration of the star-planet system for a particular distance. First we average the exoplanetary spectrum in the range of the Mark III 800 nm and Vinci filters (we had no data for the exoplanet spectrum in the range of Mark III 500 nm filter), and then we used this intensity for the stellar companion in the Fig. 11 . The intensity of the planet is stronger in the infrared with respect to the optical. The ratio between the stellar intensity at its center (i.e., ), or , and the planet integrated intensity in the same filter, , is:


for distances [0.1, 0.25, 0.5, 1.0] AU, respectively.

We computed the closure phases from the image of Fig. 11 and for similar systems corresponding to the Vinci and Mark III 800 nm filters and for the distances reported above. These closure phases were compared with the resulting phases computed for exactly the same triangles but for a system without the presence of a planet. Figure 12 displays the absolute differences between closure phases with and without the presence of a hot Jupiter. Setting as a reference the closure phase nominal error of CHARA (0.3) and also 1 (horizontal lines in the plot), only the Vinci filter gives differences that should be detectable on the third lobe () while there is no signature on the second lobe (). The absolute difference increases as a function of spatial frequency. It is indistinguishable for all the planet’ distances for frequencies larger than 0.8.

The purpose of this Section was to show the alteration of the signal due to the granulation noise in the detection of a hot Jupiter around Procyon-like stars using closure phases. So far, no companions have been detected around Procyon except for a white dwarf astrometric companion detected already in the 19th century (see Section 1).

The studies of hot Jupiter atmospheres will reveal their composition, structure, dynamics, and planet formation processes. It is then very important to have a complete knowledge of the host star to reach these aims.

Figure 12: Absolute differences of the closure phases computed for the 2000 random baseline triangles in the case of a system without the presence of a hot Jupiter companion () and a system star-planet (, see text and Fig. 11). The horizontal solid lines indicate the 1.0 and 0.3 degrees and the vertical dashed line the spatial frequency corresponding to the end of third visibility lobe. The black, blue, green, and red colors denote the planet-star distance of [0.1, 0.25, 0.5, 1.0] AU, respectively. The maximum linear extension, the apparent radius and the parallax are the same as Fig. 10.

6 Conclusions

We have provided new predictions of interferometric and spectroscopic observables for Procyon, based on RHD hydrodynamical simulation, that affect the fundamental parameter determination of the star and are important for the detection hot Jupiter exoplanets.

We have studied the impact of the granulation pattern on the center-to-limb intensity profiles and provided limb-darkening coefficients in the optical as well as in the infrared. We showed that synthetic visibility curves from RHD simulation are systematically lower than uniform disk and this effect is stronger in the optical filters. In addition to this, visibilities display fluctuations increasing with spatial frequencies (i.e., departure from circular symmetry) that becomes on the top of third lobe in Mark III 500 nm filter. However, in the Mark III 800 nm and Vinci filter the dispersion is much weaker.

We have derived new angular diameters at different wavelengths with two independent methods based on the RHD simulation. The angular diameter in the Vinci filter is mas, which leads to an effective temperature of K or K, depending on the bolometric flux considered. This value is now more consistent with K from the infrared flux method (2010A&A...512A..54C, and Casagrande private communication).
Using an independent estimation of the radius from asteroseismology, we found mas. This radius agrees well with our interferometric value within the error bars.
Eventually, the combination of the astrometric mass and our new interferometric diameter leads to a new gravity [cm/s], which is larger by dex than the value derived in 2002ApJ...567..544A.

We provided accurate comparison of synthetic spectrum from a RHD simulation to observations from the ultraviolet to the infrared. The photometric colors are very similar to the observations while it is difficult to conclude on the infrared colors because of the saturation of the observations. Also the comparison with the absolute spectrophotometric measurements, collected in 2005ApJ...633..424A, is in agreement. We conclude that the mean thermal gradient of the simulation, reflected by the spectral energy distribution, is in very good agreement with Procyon.

The convective related surface structures impact also the signal of the closure phases that show departures from symmetry at about the same spatial frequencies of visibility curves. We concluded that closure phases not equal to 0 or may be detected with today interferometers such as CHARA in the visible filters where the baselines are long enough to get to the second/third lobes.
We estimated the impact of the granulation noise on the hot Jupiter detection using closure phases around stars with the same spectral type of Procyon. We used the synthetic stellar disks obtained from RHD in the infrared and optical filters and added a virtual companion to the star based on real integrated spectra of irradiated extrasolar planet. Then, we computed the closure phases for planet-star system and star only and found that there is a non-negligible and detectable contamination to the signal of the hot Jupiter due to the granulation from spatial frequencies longward of the third lobe. This is valid only for the infrared where the energy from the hot Jupiters is stronger. It is thus very important to have a comprehensive knowledge of the host star to detect and characterize hot Jupiters, and RHD simulations are very important to reach this aim. In a forthcoming paper, we will extend this analysis to solar type stars and K giants across the HR diagram.

The authors thank a lot J. Aufdenberg for his help and the enlightening discussions. A.C. is supported in part by an Action de recherche concertée (ARC) grant from the Direction générale de l’Enseignement non obligatoire et de la Recherche scientifique - Direction de la Recherche scientifique - Communauté française de Belgique. A.C. is also supported by the F.R.S.-FNRS FRFC grant 2.4513.11. We thank the Rechenzentrum Garching (RZG) for providing the computational resources necessary for this work. This research received the support of PHASE, the high angular resolution partnership between ONERA, Observatoire de Paris, CNRS and University Denis Diderot Paris 7.


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