Free Access
Issue
A&A
Volume 531, July 2011
Article Number A124
Number of page(s) 5
Section Stellar structure and evolution
DOI https://doi.org/10.1051/0004-6361/201016230
Published online 28 June 2011

© ESO, 2011

1. Introduction

While the NASA Kepler mission (e.g. Koch et al. 2010) is dedicated to exoplanet finding, the high-quality photometric data provided by the instrument are perfectly suited for asteroseismology, specially for the study of solar-like oscillations in main-sequence and red giant stars (e.g. Chaplin et al. 2010; Bedding et al. 2010). For solar-like oscillations, the characteristics of acoustic modes, such as their frequencies, amplitudes, or lifetimes are determined by fitting the oscillation power spectrum (see, e.g., Ballot 2010, and references therein for a description of these techniques). The apparent amplitudes of modes recovered in the power spectrum – or the light curve – depend on the spectral response of the instrument. It is therefore fundamental to recover the bolometric amplitudes of modes to be able, for example, to compare the observed amplitudes to theoretical predictions.

Moreover, owing to cancellation effects, the apparent amplitude of a mode depends on its degree . Modes with degrees  > 3 are generally considered to be invisible. The knowledge of these mode visibilities is often requested for solar-like oscillation analyses. They sensitively depend on the stellar limb darkening (LD), which depends on the observed wavelengths and requests a knowledge of stellar atmospheres.

In this note, we first propose in Sect. 2 a simple bolometric correction for radial modes observed with Kepler, following an approach similar to the one developed for the Convection, Rotation, and planetary Transits (CoRoT, Baglin et al. 2006) satellite by Michel et al. (2009, hereafter M09). In Sect. 3 we give the visibilities of non-radial modes for Kepler, using LD profiles computed from ATLAS9 model atmospheres in a modified version for the convection treatment, and we discuss the use of LD laws integrated over spectral bands. Finally, we discuss in Sect. 4 the use of these correcting factors, and point out some of their limits.

2. Bolometric correction for radial modes

We derive in this part the bolometric correction for radial modes. This correction is the factor to apply to the amplitude of a radial mode observed in a given spectral band to recover the bolometric amplitude.

The fluctuations induced by radial modes identically affect the whole atmosphere, and can be considered as fluctuations δT of the effective temperature Teff. For photometric measurements, made for example with Kepler, the oscillations generate fluctuations δFK of the measured total flux FK. This flux depends on the response of the instrument. By considering fluctuations small enough to be considered as linear perturbations, and by assuming that the observed star radiates as a black body of temperature Teff, the fluctuations δFK and δT are linked through the relation: (1)where B(λ,Teff) is the Planck function, and is the transfer function of the instrument, and λ the light wavelength (see M09 for a detailed derivation). The transfer function reads , with ℰK the spectral response of the detector, and Eν = hc/λ the photon energy at the wavelength λ. The constants h and c denote the Planck constant and the light speed. The spectral response of Kepler is described in Van Cleve et al. (2009)1 and plotted in Fig. 1. For comparison purposes, the responses of the two CoRoT channels (see Auvergne et al. 2009) are also plotted.

thumbnail Fig. 1

A solid line shows the spectral response ℰK of Kepler as a function of the wavelength λ (after Van Cleve et al. 2009), whereas dotted and dashed lines show the spectral responses of CoRoT seismo- and exofield (Auvergne et al. 2009). Grey profiles indicate the spectra (in arbitrary units) of black bodies with temperatures of 7000, 6000, and 5000 K, from top to bottom.

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The fluctuation δFb of the bolometric flux Fb is simply linked with δT: (2)We then deduce (3)with the bolometric correction factor (4)We numerically computed the bolometric corrections for effective temperatures Teff within the range 4000–7500 K, covering the range of solar-like oscillating stars. Results are shown in Fig. 2. A second-order polynomial fairly fits the computed set of points. We then obtain the following law: (5)with the coefficients To = 5934 K, a1 = 1.349 × 10-4 K-1, and a2 =  −3.12 × 10-9 K-2. This approximation, also plotted in Fig. 2, gives consistent results with the numerical computations within  ~ 10-4. This error is probably negligible compared to the method error mainly induced by the departure of stellar spectra from black body emissions.

Alternatively, we express the bolometric correction as a power law (6)with α = 0.80. This approximation is consistent with the numerical computations within  ~ 10-3 over the considered temperature range.

thumbnail Fig. 2

Bolometric correction cK − bol as a function of the effective temperature Teff. Crosses are numerical calculations of Eq. (4), the line is a second-order polynomial fit of these points.

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3. Visibilities of non-radial modes

Computing the relative visibilities of non-radial modes compared to  = 0 modes requires the knowledge of the stellar limb darkening. For a monochromatic observation at the wavelength λ, the visibility V of a mode of degree reads (e.g. Dziembowski 1977; Gizon & Solanki 2003) (7)where P is the th-order Legendre polynomial and W(μ) is a weighting function depending on μ, the distance to the limb (equal to 0 at the limb and 1 at the disc centre), defined as the cosine of the angle , where M is the considered point at the stellar surface, C the centre of the star, and O the observer. The function W(μ) is identified to the LD profile gλ(μ) = Iλ(μ)/Iλ(1), where Iλ(μ) is the specific intensity at the wavelength λ.

We also define the normalised – or relative – visibilities , and the total visibility (8)We show (after Ballot 2010, Eq. (A.16)) that (9)Thus, we deduce the total visibility (10)For a broad band observation, the computation is not as simple. In M09 an expression for the mode visibilities is derived by following the approach of Berthomieu & Provost (1990). It is straightforward to reformulate their expression (M09, Eq. (A.18)) in the same form as Eq. (7) by considering W(μ) = WK(μ) with (11)where (12)We computed LD profiles gλ(μ) as described in Barban et al. (2003). For this, we used stellar atmosphere models computed with the ATLAS9 code2 (Kurucz 1993) in a modified version including the convective prescription of Canuto et al. (1996), known as CGM (for details, see Heiter et al. 2002). We then computed , as well as for  = 1,2 and 3, for a large range of stellar fundamental parameters: for effective temperatures Teff ∈  [4000,7500]  K, surface gravities log g ∈  [2.5,4.5] , and three metallicities3 [M/H] =  −1.0, + 0.0, and  + 1.0. Results are listed in Table 1, and are plotted in Fig. 3. As expected, it mainly depends on the temperature Teff, while the sensitivity to log g is weak. However, strong changes in metallicity have visible impacts on the results.

Table 1

Total visibility and relative visibilities for degrees  = 1,2, and 3 for different fundamental stellar parameters Teff, log g, and [M/H].

In M09, a simplification has been done by noticing4 that the product HλGλ varies slowly with λ. Moreover, is not obtained through Eq. (10), but computed as the truncated sum . As shown in Fig. 4 for log g = 4.5 and [M/H] =  + 0.0, the results of this simplification give very close results to the more complete computations.

thumbnail Fig. 3

Total visibility and relative visibilities for degrees  = 1,2, and 3 plotted as a function of the effective temperature Teff, for different surface gravities log g (plotted with different line styles), and different metallicities [M/H] (plotted with different colours).

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Even by considering this simplification, the calculations of and require one to know wavelength-dependent LD laws gλ(μ), of which computations are time-consuming. However, we find in the literature LD profiles which are already integrated over spectral bands of interest (for Kepler, see for example Sing 2010). We denote gK(μ) = IK(μ)/IK(1) the integrated LD profile where (13)We propose to use Eq. (7) with W(μ) = gK(μ) to perform approximated computations of and . Formally, this approximation is a correct simplification of the full computation for narrow-band filters only. Because the Kepler band is fairly broad, this approximation has to be verified first. We then computed the integrated LD profiles gK for our different models, and used them to calculate and . Results are shown in Fig. 4 for ( are not displayed, but their behaviours are the same as ). The obtained values for the different visibilities are satisfactory and reasonably close (better than 1%) to the full computation.

For comparison purposes, we also used the LD laws in the Kepler band published by Sing (2010). We used its non-linear 4-coefficient LD laws5 to compute the visibilities. Results for are also shown in Fig. 4 (Here again, show exactly the same behaviour). We then see noticeable differences with the results obtained with our own LD profiles, which are greater than the difference observed between our full and approximated computations. Sing (2010) has also made used of the ATLAS9 code for computing LD. The difference is mainly explained by different treatments of physical processes in the atmosphere models, especially of the convection, as already pointed out in Barban et al. (2003). We recall that Sing (2010) used ATLAS9 model atmospheres computed for mixing-length theory with a mixing-length parameter α = 1.25 and overshooting included, while in the present work we used the CGM approach for the convection treatment without overshooting (see Heiter et al. 2002, for argumentation). It shows the sensitivity of LD computations to the models and the consequences for mode-visibility calculations.

thumbnail Fig. 4

Total visibility plotted as a function of the effective temperature Teff, for log g = 4.5 and [M/H] =  +0.0. The solid line shows the complete computation, the dotted-dashed line corresponds to the simplification of M09, the dashed line shows the approximated results obtained by using integrated gK LD profiles. The red dashed line is obtained by using the LD laws published by Sing (2010).

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4. Discussion

The notations used in this note differ from those in M09. The term cK−bol of the present work is equal to 4/Rg in M09, and to Rosc/Rg. It is worth noticing that is also equal to the c factor of Kjeldsen et al. (2008). Our results could appear to be more dependent on the metallicity than the computations of M09. This is simply an impression owing to the scales and the considered variables. For example, around Teff = 6000 K, and the spread of the different curves is around 0.06, then the corresponding dispersion for Rosc is around 0.07, which is consistent with the point spread observed in Fig. 7 of M09.

The coefficients introduced here appear quite naturally with the classical analysis techniques:

  • 1.

    Using the factor cK−bol is straightforward: by multiplying the amplitude (for example in ppm) of an  = 0 mode observed in the Kepler band, one recovers its bolometric amplitude.

  • 2.

    The visibility can be used to fix a priori ratios of mode heights in power-spectrum fitting, or to be a posteriori compared to the fitted ratios if they are free parameters.

  • 3.

    In some cases, specially for stars with low signal-to-noise ratios (e.g. Mathur et al. 2010), the power density spectrum is convolved by a box car with a width equal to the large separation Δ. The smoothed spectrum is then used to recover the power maximum Pm. Assuming that one mode of all degrees is present in the interval Δ and they all have a similar intrinsic amplitude (hypothesis of energy equipartition), we recover the maximum amplitude of radial modes in the Kepler band as , and, then, the bolometric maximal radial-mode amplitude is .

At this point, it is important to stress the limits of the kind of approach we used. As already mentioned, our bolometric corrections ignore the departure of stellar surface emissions from black body spectra. As a consequence, the effects of photospheric absorption lines and bands are neglected, as are the effects of interstellar reddening. By using a classical reddening law (e.g. Schild 1977), the bolometric correction cK−bol is increased by less than 2% for a reddening E(B − V) = 0.14 – typical for the open cluster NGC6811 – and is negligible for nearby stars.

Second, the LD laws are only models, which are also, in essence, approximations. It is nowadays possible to test modelled LD, at least linear approximations of LD, with various kinds of observations, such as interferometry (e.g. Burns et al. 1997), light curves of eclipsing binaries (e.g. Popper 1984), planetary transits (e.g. Barge et al. 2008), or microlensing (e.g. Fouqué et al. 2010). Besides, Sing (2010) has shown discrepancies between its modelled LD and the LD deduced from planetary transits observed by CoRoT. Nevertheless, new 3D atmosphere modelling should provide better stellar LD prediction (e.g. Bigot et al. 2006; Chiavassa et al. 2009).

Last, even considering the LD profiles are perfectly known, some other assumptions could be not fulfilled. In our approach, the physics of modes inside the atmosphere is simplified: particularly, non-adiabatic effects are neglected, and the stellar photosphere is assumed to be thin enough to render the variations of mode amplitudes with the altitude in the atmosphere negligible. For the Sun, whose surface is resolved, we have a direct measurement of the LD at different wavelengths. It is then possible to compute visibilities from the real LD and to compare them to the ratios of observed mode amplitudes: Salabert et al. (2011) have shown differences between the computed visibilities and the observations of the helioseismic VIRGO (Variability of solar IRradiance and Gravity Oscillation) instrument. The lack of the previously mentioned physical effects in the modelled visibilities can be made responsible to explain the discrepancies. However, it is also possible that the intrinsic amplitudes of modes are different. Other stars have shown unexpected high apparent amplitudes of  = 3 modes (e.g. Deheuvels et al. 2010). It is still unclear whether this is a consequence of incorrect computations of mode visibilities, or a signature of the over-excitation of some modes. These two examples show us how important it is to improve in the near future the computation of mode visibilities, by considering more realistic physics of acoustic modes in the stellar atmospheres, to be able to recover intrinsic mode amplitudes.


3

We recall that [X/Y] means log (X/Y)star − log (X/Y), X and Y being chemical element abundances in number of atoms per unit volume.

4

This simplification was also made to obtain Eq. (1), see M09.

5

The improved 3-coefficient LD laws was also tested and yielded very close results. For a fair comparison, we also fitted our LD profiles with 4-coefficient laws and used them to compute visibilities: our results were almost unchanged.

Acknowledgments

We acknowledge E. Michel for useful comments, as well as the International Space Science Institute (ISSI) for supporting the asteroFLAG international team 6, where this work was started. We are very grateful to R. Kurucz for making available and open to the scientific community his ATLAS9 code. This research made use of the VizieR catalogue access tool, operated at the CDS, Strasbourg, France.

References

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All Tables

Table 1

Total visibility and relative visibilities for degrees  = 1,2, and 3 for different fundamental stellar parameters Teff, log g, and [M/H].

All Figures

thumbnail Fig. 1

A solid line shows the spectral response ℰK of Kepler as a function of the wavelength λ (after Van Cleve et al. 2009), whereas dotted and dashed lines show the spectral responses of CoRoT seismo- and exofield (Auvergne et al. 2009). Grey profiles indicate the spectra (in arbitrary units) of black bodies with temperatures of 7000, 6000, and 5000 K, from top to bottom.

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In the text
thumbnail Fig. 2

Bolometric correction cK − bol as a function of the effective temperature Teff. Crosses are numerical calculations of Eq. (4), the line is a second-order polynomial fit of these points.

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In the text
thumbnail Fig. 3

Total visibility and relative visibilities for degrees  = 1,2, and 3 plotted as a function of the effective temperature Teff, for different surface gravities log g (plotted with different line styles), and different metallicities [M/H] (plotted with different colours).

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In the text
thumbnail Fig. 4

Total visibility plotted as a function of the effective temperature Teff, for log g = 4.5 and [M/H] =  +0.0. The solid line shows the complete computation, the dotted-dashed line corresponds to the simplification of M09, the dashed line shows the approximated results obtained by using integrated gK LD profiles. The red dashed line is obtained by using the LD laws published by Sing (2010).

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In the text

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