A. Claret
Instituto de Astrofísica de Andalucía, CSIC, Apartado 3004, 18080 Granada, Spain
Received 12 January 2003 / Accepted 30 March 2004
Abstract
The irradiation of cool stellar atmospheres is reanalyzed
using a modified version of the UMA (Uppsala Model Atmosphere)
code which incorporates two improvements
concerning 1) the spacial resolution (18
points instead of only 6 adopted previously) and 2) adopting modern atmosphere models to simulate
hot companions. The implementation of a better spacial resolution is
shown to be essential to describe with accuracy the center to limb
variation of the specific intensities. It is also shown that
only by adopting such a resolution (or a higher one) it
is possible to detected and
quantify the limb-brightening with confidence.
As in the case of standard (non-irradiated) plane-parallel models, the Least-Squares method applied to the series expansion for the specific intensity introduced by Claret (2000) is shown to be the best procedure to derive the limb-darkening coefficients for illuminated atmospheres producing very small scattering and conserving the flux accurately.
Since UMA does not work for wavelengths shorter than 1500 Å, the introduction of hot models to simulate the external fluxes introduces severe uncertainties in the model of irradiation. These uncertainties were ignored in previous works using the same code and similar conditions of irradiation. Other limitations of the model of irradiation, such as the suitability of applying the spherical symmetry instead of the plane-parallel approximation, are also discussed.
Key words: stars: atmospheres - stars: binaries: eclipsing
The mutual irradiation of stars in close binary systems has been investigated since the first decades of the last century. From the observational point of view, Dugan (1908) and Stebbins (1911) were the first to report on the phenomenon, known generically as the reflection effect. On the other hand, Eddington (1926) and Milne (1926) initiated the theoretical study by showing that the albedo is unitary for an envelope in radiative equilibrium. Chandrasekhar, in a series of papers (1945, 1947, 1950), also investigated the physical conditions of an illuminated atmosphere and derived an exact solution in the approximation of a grey atmosphere. Geometrical improvements were introduced by Russel (1949) and Kopal (1954) to take into account the finite size of the binary components. The use of elaborated stellar atmosphere codes to investigate the limb-darkening coefficient (LDC) in irradiated atmospheres is relatively recent. Contrary to some authors' affirmation that only UMA was used to investigate the reflection effect, Buerger (1972) implemented the contribution of an external radiation field to the resulting intensities and fluxes in the ATLAS code. Rucinski (1969) found that the albedo for stars with deep convective envelopes is about 0.5 while Claret (2001), using a new numerical method which combines interior, envelope, and atmosphere models, computed the bolometric albedo as a function of the external flux, the geometry and the evolutionary status of the irradiated stars. Kirbiyik (1982) showed that the circulation currents caused by the irradiation may even penetrate into the non-irradiated hemisphere.
Concerning the limb-darkening of irradiated atmospheres, Buerger (1972) analyzed the effects of irradiation on the specific intensity distribution using an early version of the ATLAS code but, unfortunately, no quantitative results were given. Interesting papers have been published by Peraiah (1982, 1983a,b) and Peraiah & Rao (1983a,b). Although their atmosphere models were not sophisticated, they concluded that the limb-darkening for irradiated atmosphere models with spherical symmetry is different from that using the plane-parallel approximation. Giménez & Claret (1989) and Claret & Giménez (1990) used a modified version of the UMA code (Nordlund & Vaz 1990; see also Gustafsson et al. 1975) to investigate the changes in the limb-darkening of an illuminated atmosphere caused by irradiation. The authors found that, for the adopted input physics, the intensity distribution tends to be uniform under the effects of irradiation. Later, Claret & Giménez (1992) modified the code to study the albedo, the increase of the effective temperature and the change of the position of an irradiated star in the HR Diagram due to an extra radiation field. Alencar & Vaz (1999) investigated the influence of irradiation on the limb-darkening using a similar version of the same code by adopting the Flux Conservation Method to compute the LDCs, but taking into account only six emergence points.
In the present paper, we try to improve the treatment of the intensity distribution of an irradiated atmosphere concerning limb-darkening. In particular, we introduce a non-linear law which is able to describe with accuracy the intensity distribution and that simultaneously leads to the flux conservation with high accuracy (see Claret 2000). In Sect. 2 we summarize some aspects of the physics of the irradiation. Section 3 is dedicated to a preliminary comparison between the two usual methods to obtain the LDC: the Least-Squares (LSM) and the the flux conservation (FCM) method. The discussion of the results and the limitations of the present model of irradiation are presented in Sects. 4 and 5.
The implementation of irradiation in stellar atmosphere codes
has been discussed in some of the papers quoted in the
Introduction, so here we only review its basic aspects. A
modification at the upper boundary condition is needed to consider the
presence of an infalling radiation field: the intensity at the optical
depth
= 0 will depend on the nature of the external flux as
well as on the angle of irradiation. As a boundary condition,
the bottom of both hemispheres - the standard and the irradiated one -
should present the same physical conditions since they represent the
same star. Trial models, taking into account the external fluxes, are
generated and their pressure-temperature relationships are then
compared to those of the non-irradiated model. The
correct irradiated model is obtained when the boundary condition
is fulfilled. This is done by iteration. With the present version of
the UMA code we can simulate the irradiation of a cool atmosphere by
a cool or moderately hot companion.
The relation between the bolometric albedo A, the effective temperature of
the model without irradiation
and the effective temperature of
the irradiated model Th is
![]() |
(1) |
In a previous paper (Claret & Giménez 1990) as well as in the
paper by Alencar & Vaz (1999), only six points of the emergence
angle
were used which is, of course, a very important
limitation. In the present work we will show that the
inclusion of extra
points is essential to draw more definitive
conclusions. One expects that the behavior of a perturbed
atmosphere be more complex than that of a standard one and
therefore, a more accurate sampling of the disk is needed.
A second modification with respect to the mentioned above papers
is the introduction of new atmosphere models to simulate hot
irradiating stars. Until now the models by Kurucz (1979) were adopted.
Here we adopt the updated ones provided by the same author (Kurucz
1993). The Kurucz models are computed for a more extensive set of
wavelengths than in UMA. As commented in Claret & Giménez (1992),
the interpolation in the ATLAS fluxes to be used by the UMA code as
source stars may introduce some loss of flux in both extremes of the
spectrum, but mainly in the ultraviolet where the UMA set does
not have many points (i.e.
(UMA) = 1533 Å, while
(ATLAS) = 91 Å). In this way, systematic errors are
introduced in the effective temperatures T* of the irradiating star
if those models are used to simulate irradiating stars. An additional
complication arises from the imbalance of energy in that
spectral region. The repercussion of these limitations and
modifications will be analyzed in the next sections.
Recently Claret (2000) introduced an approximation for the LDC of
LTE atmospheres for ATLAS (Kurucz 2000) and PHOENIX models (Allard &
Hauschildt 1995; Allard et al. 1997; Hauschildt et al. 1997a,b). Both codes use the plane-parallel approximation.
The LDC law is written as a series:
![]() |
(2) |
where I(1) is the specific intensity at the center of the disk,
=
,
and
is the angle between the line of sight and the
emergent flux. Before the introduction of Eq. (2), the most
commonly adopted approximations
were:
Linear
![]() |
(3) |
Quadratic
![]() |
(4) |
Square root
![]() |
(5) |
Logarithmic
![]() |
(6) |
A good limb-darkening law must be able to reproduce very accurately
the intensity distribution across the stellar disk and must lead
to flux conservation. In this way, a compromise between an accurate
description of the intensities and the flux calculation is required.
Such requirements are only possible through the Least-Squares
Method by using an adequate law.
The LDC law must be really adequate. If very simple laws
are used, the fluxes resulting from the LSM are not conserved
within acceptable numerical uncertainties. Opposite to this, the LDC
as derived from the FCM, once obtained, cannot be adjusted to improve
the corresponding s.
Equation (2) presents several advantages with respect to others approximations:
The main problems inherent to any limb-darkening law that adopts the FCM are:
Before dealing directly with irradiated models, we have checked
the modifications we have introduced in the code to increase the
number of the emergent angles. We performed calculations for several
models adopting 18
points (without irradiation) and after
comparison with other atmosphere models concluded that the agreement
between the respective models was acceptable, and perfectly within
the expected differences due to the different input physics and
numerical methods used in the respective codes.
We performed bolometric, monochromatic and passband (
)
calculations of LDCs for some irradiated models. The
models were selected in such a way that cold (3500 K) and moderately
cold models (6500 K) are irradiated by moderately cold (7000 K) and
moderately hot source stars (15 000 K). The simulations were carried out
adopting some combinations of the above values. However, there are
two implicit limitations to the present theoretical model of
irradiation. The first one is related to the way the external
flux is introduced in the atmosphere equations. The functions
and
are expanded in Taylor series to obtain
.
This
approximation limits the magnitude of the external bolometric flux.
But there is another constraint to the application of the present
model of irradiation which is related mainly to the ultraviolet
flux of the irradiating star. The ratio of the ultraviolet fluxes
depends strongly on the ratio (
.
As UMA is
specifically developed for cool stars, one must use other codes to
simulate a hot irradiating star, as for example those derived from
ATLAS. This not only affects the effective temperature of the
irradiating star as discussed previously but also the ultraviolet
imbalance since the shorter wavelength in UMA is 1533 Å. Even if we
adopt a UMA model for the irradiating star to avoid
this imbalance, an additional imbalance remains since the
incident energy with wavelengths shorter than 1533 Å is not taken
into account. Therefore, the resulting specific intensities in the
ultraviolet are less reliable than those corresponding to other
spectral regions. In addition, in some cases, the monochromatic
ultraviolet flux of the irradiating stars can be a few orders of
magnitude larger than that of the irradiated star. This means that
not only the bolometric flux ratios must be used as a check of
the applicability of the perturbed equations but also the
monochromatic ones.
The specific intensity for a generic band a is given by
![]() |
(7) |
Until now, works on irradiated atmospheres adopting the UMA code
considered only 6
points with
and
,
that is, the limb was not well covered (in some modern codes the
minimum value of
is between 0.005-0.001). This is one of the
reasons why reliable evidence of limb-brightening in Claret &
Giménez (1990) was not detected and it was only reported as a
tendency of the intensity distribution of an irradiated atmosphere to
become uniform. The irradiated atmospheres presented in
Figs. 3-6 by Claret & Giménez (1990) are still limb-darkened,
although an increase of the specific intensity near the limb can
easily be seen. The limbs of those irradiated models are brighter
than those of non-irradiated ones with the same net flux, but the
same also happens to the other parts of the disk which leads to a more
uniform intensity distribution. Brueger (1972) found a similar effect
(see his Fig. 3). On the other hand, limb-brightening is a phenomenon
that appears even in standard atmosphere models (in the sense that no
infalling radiation is present). Wade & Rucinski (1985) and Claret
(2000) detected it for some models generated with ATLAS (and also
with PHOENIX) but this is probably caused by temperature inversions
which are not realistic.
To detect real limb-brightening in irradiated atmospheres a
good sampling of the stellar disk is necessary. The limb in
particular must be well covered. Let us investigate how the number
and position of the emergence angles affect the intensity
distribution. Figure 1 shows the intensity distribution for irradiated
models with
K, log g = 3.5, T* = 12 000 K, log g* = 4.0,
for r = 0.10, 0.20, 0.30 (filter b). The continuous line indicates the new actual intensities
computed at 18
points while the asterisks indicate the 6
points adopted previously. The inclusion of the 12 extra points is
determinant to decide whether there is a darkening or
brightening towards the limb. As we have mentioned in the last
paragraph, the previous low
resolution explains why such a
phenomenon was not detected in the Claret & Giménez (1990) paper
for similar conditions of irradiation. This reveals that the
analysis of the intensity distribution based on a few
points -
whatever the numerical method to derive them - should be taken with
extreme caution. Some results by Alencar & Vaz (1999), who used only
6
points, are interpreted as limb-brightening; see for
example the model for rh=0.26 in their Fig. 3. The
limb-brightening they detected in this case is more due to the bad FCM fitting (specially for the smaller
s) than to the
location of the points themselves. This tendency is one of the
consequences of the failure of the FCM predictions particularly for
regions near the border of the disk, as already noted by Claret
(2000) (Fig. 6, first frame; see also Fig. 3 of the present work).
Therefore, there is an additional criticism for the use of the FCM: in some cases it predicts limb-brightening where it does not
exist while in other cases it fails to detect limb-brightening
when it is present. We will see in the next Section how this
method introduces systematic and severe errors in the
matching of the intensities of an irradiated atmosphere.
![]() |
Figure 1:
The effect of irradiation on the specific intensity
distribution for models with log g = 3.5,
![]() |
Open with DEXTER |
Although the versions of the code used in the previous papers are
similar to the present version, there are a few basic
differences. First, the irradiating models are not updated
since the fluxes given by Kurucz (1979) were adopted. We introduce
more recent models by Kurucz for the irradiating sources
to homogenize the sample of the external radiation fields.
Alencar & Vaz (1999) do not mention the systematic errors introduced
in the effective temperature if an ATLAS model is used to simulate the
irradiating star, nor do they mention the imbalance of energy in
the ultraviolet. As we have seen in the last section, another
important and serious limitation is that the calculations
presented by these authors (as well as those presented in our
previous papers on irradiation) were performed using only 6 points over the disk.
The capability of the numerical methods to derive the LDC and of the implementation of our modifications can be checked by inspecting the properties of some irradiated models. With these models, it will be possible to test under which conditions real limb-brightening appears and how the LDCs computed following the LSM and FCM prescriptions describe the intensity distribution. The characteristics of such models are given in Table 1.
Table 1: Irradiated model parameters.
![]() |
Figure 2: Monochromatic specific intensities for some irradiated models. The full lines represent the actual intensities while the asterisks denote the fitting obtained using the LSM-Eq. (2) prescription. The FCM calculations were carried out adopting the logarithmic approach (crosses). The irradiation parameters are indicated in the figure. |
Open with DEXTER |
![]() |
Figure 3:
The relative difference
![]() ![]() |
Open with DEXTER |
In order to evaluate the goodness of the matches, we introduce the
merit function
![]() |
(8) |
The function ,
as defined in Eq. (8), is a kind of average of
the differences between the models and the fits over the entire
stellar disk. The behavior of the intensity distribution near the
border of the disk is crucial to compute the light variations
when the stellar disk is scanned during eclipses. Therefore, it is
also interesting to study how the LSM and FCM results compare
near the limb. In Fig. 3 we show the relative differences in the
intensities obtained using the LSM-Eq. (2) prescription and those
following from the FCM (logarithmic case) as a function of the
wavelength. Again, the quality of the LSM fitting is superior to the
case of the FCM whatever the adopted LDC law approximation. For
the LSM fitting the relative differences of the intensity near the
border of the disk are almost zero. On the contrary, for the
FCM, the differences are large, especially at shorter
s.
![]() |
Figure 4: The specific intensity distribution for irradiated models in the case of filter b. The conditions of irradiation are indicated inside each frame. Continuous lines denote the actual intensities, while asterisks and crosses represent the Eq. (2)-LSM and FCM (logarithmic) matches, respectively. |
Open with DEXTER |
Let us now examine the behavior of the passband intensities of the
irradiated models. Figure 4 displays how the intensities are
distributed over the disk for the conditions of irradiation indicated
in the figure. The vertical bars indicate the position of the points used by Alencar & Vaz (1999). Again, if only 6 points are
used, important information on the intensity distribution is
lost, especially when
< 0.4. Furthermore, it should be
pointed out that even if we consider 18
points, the FCM
will not give a good fitting, as indicated by the crosses.
As a final check of the LSM-Eq. (2) prescription, we plot in
Figs. 5 and 6 the monochromatic and passband ratios between the
actual flux and the flux computed using our new limb-darkening
law (F'/F). By inspecting both figures we can conclude that the flux
obtained using our new approximation is conserved to a high
degree of accuracy, as required. This is a consequence of the
goodness-of-the-fitting and is not imposed a priori. In the
ultraviolet, the irradiated models are very perturbed and less
reliable, so that the fitting at those points is less good (about 0.5 percent of deviation). Even so, the new law is able to derive the
flux with an acceptable accuracy. Note that in Fig. 5, for a fixed
irradiation condition but for increasing
the
deviations decrease, partly due to the smaller ultraviolet ratio
and partly due to the decrease of the relative importance
of the ultraviolet loss. The largest scatter is found only at
the 9-10 first wavelengths (asterisks in Fig. 5); for larger
s, abs[F'-F] is smaller than 0.001. In this way, the two
imposed constraints are fulfilled for any limb-darkening law: the
intensity distribution is accurately described over the entire disk
and simultaneously, the resulting flux is conserved within a small
tolerance limit.
The new LDC law we have introduced is able to match the
intensity distribution of the irradiated models much better than those
that make use of the FCM. We have found that the ratio between the
actual fluxes and those computed with this new law is very close to 1,
which indicates a very good accuracy. Although the global results
seem to be very reliable, some cautionary remarks should be
made. The general model of irradiation presents some
limitations that are not considered by other authors. For example,
the set of wavelengths in the UMA code is very limited with
respect to modern models not only concerning the number of
points, but also concerning the inferior and superior limits of
and the distribution of the points. As we have seen, one of
the consequences of these limitations is that if we use more
modern models to simulate the infalling radiation, the flux is
not correctly taken into account due to the loss of information
at shorter wavelengths. There is, however, an important
additional problem: as the
for UMA is 1533 Å, the
code is not able to balance correctly the infalling radiation in this
spectral region. It is true that if we use an UMA model as the source
star, the problem of the wavelength set no longer raised, but the
loss of information in the far ultraviolet still remains. Even if the
radial relative flux F*r is small, the ratio of the monochromatic
fluxes may be very high in the ultraviolet. We emphasize
this point since the satellite observations in the ultraviolet
are important to determine stellar parameters in close binary systems
for which the irradiation effect is notorious. All this puts severe
limits on the results obtained for shorter wavelengths.
![]() |
Figure 5: The monochromatic F'/F test for irradiated models. |
Open with DEXTER |
![]() |
Figure 6: The passband F'/F test for some irradiated models. |
Open with DEXTER |
Another significant improvement which can be introduced are wider and more recent opacities tables since UMA calculations are limited to a narrow range of effective temperatures.
A third point is the probable presence of circulation currents. Their presence and how much they can penetrate into the "dark" hemisphere may change the intensity distribution as well as other atmospheric parameters.
A final question is whether the intensities near the stellar limb are reliable. The effects of sphericity should be investigated as well. Preliminary results by Peraiah & Rao (1983) suggest that some changes in the intensity distribution may occur when spherical symmetry is used instead of the plane-parallel approximation when an external flux is considered. Results based on modern models that apply spherical symmetry indeed indicate that the profile of the intensities is very different from that derived from the plane-parallel approach: the fall of the intensities near the limb are steeper in the case of spherical geometry (Orosz & Hauschildt 2000; Claret & Hauschildt 2003).
An estimate of the importance of some of the above limitations on the LDC of irradiated atmospheres is currently under investigation. Also, we are preparing an extensive grid of irradiated models for reliable orbital parameters which will be published elsewhere.
Acknowledgements
I am grateful to B. Willems and J. Orosz, who helped to improve the paper. The Spanish DGYCIT (PB98-0499) is gratefully acknowledged for its support during the development of this work.