Issue |
A&A
Volume 506, Number 3, November II 2009
|
|
---|---|---|
Page(s) | 1335 - 1340 | |
Section | Stellar atmospheres | |
DOI | https://doi.org/10.1051/0004-6361/200912423 | |
Published online | 03 September 2009 |
A&A 506, 1335-1340 (2009)
Does the HD 209458 planetary system pose a challenge to the stellar atmosphere models?
A. Claret
Instituto de Astrofísica de Andalucía, CSIC, Apartado 3004, 18080 Granada, Spain
Received 4 May 2009 / Accepted 8 July 2009
Abstract
Aims. There are very few comparisons between theoretical and
empirical limb-darkening coefficients (LDCs). To develop this scenario,
we analyse the case of the HD 209458 planetary system for which
10 passband measurements of linear and quadratic LDCs are
available.
Methods. The empirical data for HD 209458 were derived from the
light curves obtained with the Hubble Space Telescope (HST). Since the
corresponding effective wavelengths differ from those usually adopted,
we computed monochromatic calculations to gain some insight into the
problem. The theoretical LDCs were computed by adopting the
least-squares method (LSM) but with a larger numerical resolution
(100
points instead of 11). The flux conservation method (FCM) was also
applied. The plane-parallel stellar atmosphere model ATLAS and the
spherically symmetrical PHOENIX model were adopted in the calculations.
Results. We found systematic disagreement between the
theoretical and empirical LDCs for the linear case. Disagreements are
also found when we compared the quadratic LDCs with observations. Even
taking into account uncertainties in the metallicity, micro-turbulent
velocity, and effective temperature in the calculation of the
theoretical LDCs, the corresponding shifted curves cannot match the
empirical data. It seems that the current atmosphere models are unable
to explain the specific intensity distribution of HD 209458.
Key words: stars: atmospheres - planetary systems
1 Introduction
Stellar physics still exhibits severe discrepancies when we compare
its theoretical predictions with the available observational data. Some
of these disagreements occur in studies of eclipsing binaries. For
example, the system DI Her, whose apsidal-motion rate seems
to present a problem for General Relativity (see for example, Claret 1998), or the case of low mass binaries (0.7-1.1 ), whose observed radii are larger than those predicted by evolutionary models (Popper 1997; Clausen et al. 1999).
The comparison of eclipsing binary data with theoretical predictions
involving atmosphere models has been limited because of the scarcity of
information about limb and gravity-darkening. In a previous paper, we
presented theoretical limb-darkening coefficients (LDCs) which have
been calculated using standard least-squares method (LSM) for 100
points (Claret 2008). These LDCs were compared with previous calculations of LDCs based on only 11
points that we had previously published (Claret 2000) and with those derived by adopting the r-integration
method. In terms of flux conservation, the average improvement is
around one order of magnitude. With these specific calculations, we
have compared theoretical with the few available empirical linear LDCs
measured for nine eclipsing binaries. This comparison highlighted some
discrepancies between the theoretical and observational data. These
disagreements may be related to intrinsic problems in either the ATLAS
models or the numerical methods adopted (including the chosen
functions) to derive the theoretical LDCs. However, it may also
indicate that the measurements are not accurate enough or exhibit
systematic effects. At the true level of accuracy and given that the
empirical LDCs are only available for 3 filters, it is impossible
to identify more precisely the causes of this discrepancy.
As mentioned, the data from these nine eclipsing binaries are scarce and rather disperse. The observational data are only in B V bands and for the R band,
only one empirical determination is available. This limits the quality
of our conclusions significantly, although a dependence on the
effective temperature was detected. On the other hand, Southworth (2008, 2009)
presented a comprehensive and homogeneous determination of light curves
and physical properties of fourteen transiting extrasolar planetary
systems. For one parent star (HD 209458), the linear and quadratic
LDCs were directly determined from the HST light curves. The advantage
of this determination is that the number of wavelengths used (10)
is higher than those often obtained for eclipsing binaries. The covered
range in
is 320 up to 980 nm. An extra and important advantage is that
the reflection effect which in ordinary eclipsing binaries can
complicate the derivation of the LDCs, is negligible in these cases. In
this paper, we compare more deeply the empirical LDCs for
HD 209458 with theoretical calculations by adopting different
numerical methods and atmosphere models.
2 Numerical methods and atmosphere models
In the present paper, we compute the theoretical LDCs by adopting two numerical methods: the
standard LSM with 100 points,
and the flux conservation method (FCM). The laws which most frequently
used in limb-darkening studies are: the linear (coefficient
), the quadratic (coefficients
and
), the square root, the logarithmic, and a law with five terms introduced by Claret (2000).
These laws are well known and not reproduced here. The LDCs are often
computed for some determined passbands but here we compute the
monochromatic values to obtain a clearer insight into the problem. We
adopt the plane-parallel atmosphere models ATLAS (Kurucz 2000) and the spherically symmetrical LDCs presented in Claret & Hauschildt (2003).
3 The HD 209458 planetary system: comparison with theoretical predictions of LDCs
HD 209458 was the first transiting extrasolar planet discovered (Charbonneau et al. 2000). The mass of the parent star is 1.165
0.033
,
its effective temperature is 6117
50 K, its log g = 4.368
0.005, and its metallicity is slightly higher than the solar value ([Fe/H] = 0.02
0.05) (Santos et al. 2004; Southworth 2009).
Since we consider the error bar in the effective temperature to be too
low, we assume here an error of 250 K, which seems to be more
realistic. The derived age is 2.3
+0.7-0.6 Gyr by using stellar evolutionary models (Southworth 2009).
3.1 The linear approach
Basically, there are four factors that could influence the comparison between the theoretical and empirical LDCs. The first is the degeneracy in the fitting of the transit light curves, which may cause systematic errors in the LCDs. The information contained in transit light curves is not optimal, and the LDCs of two-parameter laws are usually correlated (see for example Fig. 2 in Southworth 2008). However, this does not necessarily cause a systematic error. In the case of the linear law, the corresponding LDCs are not affected by this degeneracy. We can consider that empirical LDCs obtained by Southworth (2008) are free from the degeneracy in the fitting of the transit light curves. The second factor is related to the kind of function used to fit the stellar limb-darkening. As mentioned, Southworth (2008) obtained empirical linear and quadratic LDCs. We focussed our attention on these two laws. The linear law, in general, can be ruled out by very high quality observations (recall also that the limb-darkening is a non-linear phenomenon). In addition, for the linear case, Southworth (2008) found that the orbital inclination is correlated with the wavelength of observation, which clearly highlights the inadequacy of the linear law. For the quadratic case, no correlation was found. The bi-parametric laws (quadratic, cubic, root-square, and logarithmic) produce almost identical fit qualities (Southworth, personal communication). For completeness, we investigate the linear and the quadratic cases here by adopting the ATLAS and PHOENIX models and considering two numerical methods: the LSM and the FCM. The third factor that could influence the comparison between the theoretical and empirical LDCs is the size of the error bars in the effective temperature, log g, and metallicity may also affect the comparison between theoretical and empirical LDCs. We carried out a detailed comparison between theory and observations by considering the mentioned error bars. Finally, and perhaps the most important factor, we should that consider are the limitations of the stellar atmosphere models in predicting the specific intensity distribution.
![]() |
Figure 1:
Comparison between empirical linear LDCs for HD 209458 (error bars) and theoretical predictions (continuous line): a) theoretical values from Claret (2000) using 11 |
Open with DEXTER |
![]() |
Figure 2:
Comparison between empirical linear LDCs for HD 209458 (error bars) and theoretical predictions (continuous line). a) Theoretical values computed using 100 |
Open with DEXTER |
![]() |
Figure 3:
Comparison between empirical quadratic LDCs for HD 209458 (error bars) and theoretical predictions (continuous line): a) theoretical values computed using 100 |
Open with DEXTER |
![]() |
Figure 4:
Effects of uncertainties in both
|
Open with DEXTER |
![]() |
Figure 5:
Comparison between empirical quadratic LDCs for HD 209458 (error bars) and theoretical predictions (continuous line): a) theoretical values computed using 100 |
Open with DEXTER |
Southworth (2008) compared the
empirical linear LDCs for HD 209458 (the corresponding error bars
come from the Monte Carlo simulations) with those derived from the
atmosphere models. As a theoretical support, he adopted the LDCs
computed for the ATLAS model (SLOAN photometric system). However, a
more complete comparison with theoretical predictions was not carried
out because this comparison was beyond the scope of that paper. Here we
compare the LDCs by using more theoretical tools. In the first panel of
Fig. 1
we show the same comparison as that by Southworth for the linear law.
In that figure, we adopted monochromatic calculations instead of taking
into account the filters transmission to gain some deeper insight given
that the shortest effective wavelength of the photometric bands usually
adopted is longer than 320 nm (the lower limit for the
HST passbands). We note that there are differences between the
theoretical and empirical LDCs that are consistent with Southworth's
comparison; the theoretical LDCs are systematically larger than the
empirical ones. For shorter wavelengths, the differences are small and
we can even accept that, within the errors, the theoretical values fit
the observations. However, for wavelengths longer than 400 nm, the
differences increase. Is this indicative of a problem with the
atmosphere models at longer wavelengths? Before trying to answer this
question we explore the numerical methods used to derive the
theoretical LDCs. The second panel of Fig. 1 shows the same comparison but for when the number of points
are increased to 100 (instead of 11). The same systematic
effect is still detected, although on a small scale.
Generally, for a given atmosphere model, the FCM produces smaller linear LDCs than those computed by using the LSM (Claret 2008). The third panel of Fig. 1
compares the LDCs for HD 209458 evaluated by adopting the FCM. The
disagreements clearly decrease at all wavelengths. However, as has been
shown in several papers, the FCM does not represent well the specific
intensities and produces very high values of s.
By definition, this method preserves integrally the flux. This should
be an advantage but in contrast, if we analyse the situation in more
detail we see that it is a disadvantage. The limb-darkening is clearly
a non-linear phenomenon. So, the FCM is unable to find the best
function because it always preserves the flux. In contrast, the LSM
produces large
s and
Fs (the ratio of the true flux at the wavelength
and that using a given limb-darkening law) when we adopt the linear
approximation and provides a robust means of differentiating between
the functions chosen to fit the stellar limb-darkening. For a more
detailed discussion of this subject, we refer the readers to Claret (2000).
The observational error bars in the effective temperature of the parent star need to be considered when comparing with theoretical predictions. In the first panel of Fig. 2 we show the theoretical LDCs evaluated by taking into account error bars of 250 K. Even considering these error bars, the shifted curves of the theoretical LDCs are incapable of matching the observations. On the other hand, the chemical composition is not well determined in most of the stellar systems. We computed LDCs by varying the metal content by 0.50 dex and the results are shown in the second panel of Fig. 2. Although the theoretical LDCs with [m/H] = -0.50 somewhat improve the comparison mainly at shorter wavelengths, they are unable to fit the observational data satisfactorily.
Another parameter that can be explored is the micro-turbulent velocity
but even by increasing this to 8 km s-1, the empirical LDCs remain smaller than the resulting theoretical ones (third panel of Fig. 2). Finally, in the fourth panel of Fig. 2, we show the calculations considering the spherical PHOENIX models (Claret & Hauschildt 2003). There is no large difference from the panels 1 and 2 of Fig. 1,
but the discrepancies are slightly smaller. This may indicate that
PHOENIX models can represent the atmosphere of HD 209458 better
than the ATLAS models. We note that for this model we adopted the U B V R I J H K passbands instead of monochromatic calculations.
3.2 The quadratic approach
Southworth (2008) also
obtained LDCs for the quadratic law case. As in the previous case, the
Monte Carlo simulations were carried out to estimate the error
bars. Figure 3 shows the comparison
between the empirical quadratic LDCs and the theoretical calculations for the ATLAS
model, with [m/H] = 0.00, = 2 km s-1.
The comparison highlights some differences with respect to the linear
case. In the last panel, we found a systematic disagreement with
wavelength. For the quadratic approximation, the theoretical linear
components a(
)
are larger than the empirical ones for wavelengths longer than 800 nm and smaller than the observed for
shorter than 600 nm (first panel). Only for the interval
700-800 nm do the theoretical linear components seem to fit the
observations. A more conspicuous difference is found when we
compare the theoretical and empirical quadratic component b(
)
(second panel of Fig. 3). The
theoretical b(
)
is almost independent of wavelength, while the empirical values
exhibits a pronounced dependence, being negative at shorter
wavelengths. As pointed out by Southworth (2008),
this characteristic is not present in any theoretical predictions that
he adopted. To check whether this also holds for other numerical
methods, we also compare the
)
and b(
)
obtained by using the FCM with the observed ones (panels 3 and 4, Fig. 3). The theoretical LDCs generated by using the FCM are unable to fit the observed
)
and b(
). Finally, we compare the observations with the predictions from the spherical models (panels 5 and 6, Fig. 3). The aspects of panels 1-6 of Fig. 3 are similar, and it seems that only
if we consider a ``rotation'' of the theoretical linear and quadratic components around
=
550 nm would it be possible to reproduce the observed quadratic
LDCs. Even if we take into account the effects of uncertainties
in
and in [m/H], the general aspect of Fig. 3 does not change much. Figure 4
shows these comparisons. We note that while the theoretical linear
components that take account of the uncertainties in both
and [m/H]
(panels 1 and 3) improve the comparison, the quadratic
components are not affected significantly by the error bars.
Since the previous comparisons indicate that the atmosphere model with the observed
and log g
is unable to reproduce the empirical LDCs of HD 209458, we decided
to search for a model that could match the measurements more
successfully to compare its physical properties with the observed
and log g. The ATLAS model that matches the quadratic LDCs more closely is that with
= 5250 K, log g = 4.5, [m/H] = 0.00, and
= 2 km s-1 and is represented in Fig. 5 with the data for HD 209458. The agreement can be considered to be acceptable only for
s
shorter than 450-500 nm, but for longer wavelengths the
disagreement is clear and similar to that displayed in the first and
second panels of Fig. 1 for the linear component. The theoretical quadratic components marginally fit the observations for
s shorter
than 800 nm but fail to match them at longer wavelengths. In
addition, the effective temperature of this exploratory model is too
low compared with the observational one, even taking into account the
error bars adopted here (
250 K).
4 Final remarks
From the above comparisons, we can conclude that the current stellar atmosphere models are unable to predict the measured LDCs for HD 209458. The unique case that seems to fit the observations marginally is the linear approach (Sect. 3.1) when we adopt the FCM. However, the limb-darkening is clearly a non-linear phenomenon and this law is then inadequate. For FCM, this method, which preserves the flux integrally, is inadequate because it does not describe well the specific intensities. As is already known, for a given atmosphere model, the FCM provides smaller linear LDCs than those computed by adopting the LSM. This is probably the reason why the linear theoretical LDCs computed by using the FCM seem to reproduced more successfully the empirical values (third panel, Fig. 1). When we apply the FCM to the quadratic law, the corresponding LDCs also do not match the observations (panels 3 and 4, Fig. 3). In this way, the exact flux conservation condition seems to be disconnected from the goodness of the fitting.
The disagreements are also present when we adopt the quadratic law. We have explored the effects of the error bars in the effective temperature, metallicity, and micro-turbulent velocity in the prediction of the theoretical LDCs. The shifted curves are not enough to explain the measured values.
In addition to the different numerical methods, we have tested two sets of stellar atmosphere models: ATLAS and PHOENIX. Both models, for the direct comparison with the empirical LDCs, provide similar theoretical predictions and none is able to match the observations. This indicates that the current atmosphere models need updating and that probably, given the effective temperature of HD 209458, these modifications should involved the process of convective transport of energy and related to the opacities tables.
The impact of these disagreements on the derived masses and on the
radii of transiting extrasolar planets can be large, achieving 3
to 5
in the case of the ratio of the radii. New high-accuracy light curves
are necessary for the transiting extrasolar planetary systems. The two
alternatives, including the LDC as fitted parameters and taking them
directly from the stellar atmosphere models, should be compared. On the
other hand, other fields of the stellar astrophysics may also be
affected by the inability of the stellar atmosphere models to reproduce
correctly the intensity distribution, at least for the range of
effective temperature studied here. It would desirable that the impact
of the disagreement to be investigated and confirmed (or not) in other
independent fields of astrophysics.
I would like to thank J. Southworth, J. V. Clausen, M. A. Valverde, V. Costa, G. Torres and an anonymous referee for useful discussions. The Spanish MEC (AYA2006-06375) is gratefully acknowledged for its support during this work.,
References
- Charbonneau, D., Brown, T. M., Latham, D. W., & Mayor, M. 2000, ApJ, 529, L45 [NASA ADS] [CrossRef]
- Claret, A. 1998, A&A, 330, 533 [NASA ADS]
- Claret, A. 2000, A&A, 363, 1081 [NASA ADS]
- Claret, A. 2008, A&A, 482, 259 [NASA ADS] [CrossRef] [EDP Sciences]
- Claret, A., & Hauschildt, P. H. 2003, A&A, 412, 241 [NASA ADS] [CrossRef] [EDP Sciences]
- Clausen, J. V., Baraffe, I., Claret, A., & VandenBerg, D. A. 1999, ASP Conf. Ser. 173, ed. A. Giménez, E. F. Guinan, & B. Montesinos, 265
- Kurucz, R. L. 2000, private communication
- Popper, D. M. 1997, AJ, 114, 1195 [NASA ADS] [CrossRef]
- Santos, N. C., Israelian, G., & Mayor, M. 2004, A&A, 415, 1153 [NASA ADS] [CrossRef] [EDP Sciences]
- Southworth, J. 2008, MNRAS, 386, 1644 [NASA ADS] [CrossRef]
- Southworth, J. 2009, MNRAS, 394, 272 [NASA ADS] [CrossRef]
All Figures
![]() |
Figure 1:
Comparison between empirical linear LDCs for HD 209458 (error bars) and theoretical predictions (continuous line): a) theoretical values from Claret (2000) using 11 |
Open with DEXTER | |
In the text |
![]() |
Figure 2:
Comparison between empirical linear LDCs for HD 209458 (error bars) and theoretical predictions (continuous line). a) Theoretical values computed using 100 |
Open with DEXTER | |
In the text |
![]() |
Figure 3:
Comparison between empirical quadratic LDCs for HD 209458 (error bars) and theoretical predictions (continuous line): a) theoretical values computed using 100 |
Open with DEXTER | |
In the text |
![]() |
Figure 4:
Effects of uncertainties in both
|
Open with DEXTER | |
In the text |
![]() |
Figure 5:
Comparison between empirical quadratic LDCs for HD 209458 (error bars) and theoretical predictions (continuous line): a) theoretical values computed using 100 |
Open with DEXTER | |
In the text |
Copyright ESO 2009
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