2 The atmosphere model

We use the Uppsala Model Atmosphere (UMA, Gustafsson et al. 1975; Bell et al. 1976) code, in a version by Vaz & Nordlund (1985), as described in Alencar & Vaz (1997). The code is intended for cool ($3\,500\,{\rm K}<T_{\rm eff}<8\,000\,{\rm K}$) atmospheres, and assumed hydrostatic equilibrium, a plane-parallel structure, and local thermodynamic equilibrium. Convection is modelled with the mixing length recipe.

An illuminated model is defined by its effective temperature $T_{\rm eff}$,its surface acceleration of gravity g, mixing length parameter $\alpha= l/H_P$,and chemical composition (fixed at solar abundance in this work). The external illumination is parametrized by the direction of illumination $\nu$ (the cosine of $\theta$, the incidence angle with respect to the surface normal), the effective temperature of the heating star $T_{\rm h}$,and its apparent radius $r_{\rm h}$(the ratio between the radius of the heating star and the distance from its centre to the point on the surface of the reflecting star). The geometry of illumination in a double star system is shown in Fig.1. We kept $\alpha= l/H_P$ fixed at 1.5 for the reflecting star atmosphere.

We noticed that the effect caused by illumination on the limb-darkening was rather weakly dependent on the $\log\,g$ of the model. This can be seen in Fig.2, where we show that the effect of changing the $\log g$ from 3.5 to 5.0 corresponds to only $\pm$7% and $\pm$9% of the overall effect of illuminating an atmosphere of $T_{\rm eff}=3\,697$K by $T_{\rm h}=3\,700$K and 10000K, respectively. Therefore, we did calculations only for $\log\,g=4.5$, as for this value we achieve a representative mean effect of illumination for the interval $3.5 \le \log~g \le 5.0$.We study illuminated atmospheres with line absorption (line-blanketed or ODF) in convective equilibrium. The ODF tables limit the temperature range of this study to those corresponding to (ZAMS) stars with masses ranging from $0.6\,{M}_{\odot}$ to $1.5\,{M}_{\odot}$.

  

Figure 2: Comparison between the effect of the surface gravity and of external illumination on the limb-darkening, here adjusted by the logarithmic law, see Eq.(3). The * symbols correspond to the non-illuminated model with $T_{\rm eff}=3\,697$K. The $\diamond$and $\triangle$ symbols correspond to this model illuminated by $T_{\rm h}=3\,700$K and 10000K, respectively. The dashed lines correspond to models with $\log g=3.5$, the dash-dotted ones to $\log g=4.0$, the solid lines to $\log g=4.5$ and the dotted lines to $\log g=5.0$

This same model has been used to study, in another paper and for the first time in the literature, the effect of illumination on the gravity brightening exponent $\beta$, another important parameter for LC synthesis. That work (Alencar et al. 1998) is a continuation of the study of the $\beta$ exponent using stellar atmospheres (Alencar & Vaz 1997).

2.1 The method

The atmosphere model generates intensities in six different angular directions ($\mu=0.06943$, 0.33001, 0.5, 0.66999, 0.93057 and 1.0) and 368 wavelengths ranging from 1500Å to 124000Å. Starting from these data, monochromatic, passband-specific and bolometric limb-darkening coefficients can be calculated. We will present here the results obtained using two different calculation methods. The first one, described by CG, determines the coefficients by least square-fitting of the integrated and normalized model intensities. The second method follows the procedure outlined by VH where a number of physical constraints, equal to the number of coefficients to be determined, are assumed. The limb-darkening coefficients are obtained by solving the constraint equations. Using a one parameter law, the constraint is the conservation of the total flux and for a two parameter law, the additional constraint that the mean intensity of the approximation and the atmosphere model must be equal is applied. When calculating the passband coefficients we made a convolution of the intensities with the response functions corresponding to the Strömgren filters uvby (Crawford & Barnes 1970) and the UBVRI passbands (Bessel 1983). We used in the convolution the atmospheric mean transmission by Allen (1976), the reflection curve of two aluminum coated mirrors (Allen 1976), the sensitivity of a 1P21 photo-multiplier from Kurucz (1979) and for the Strömgren filters, the sensitivity function of the SAT photometer (Florentin-Nielsen 1983, personal communication).

We calculated models with relative fluxes ($F_{{\rm rel}, \nu}=[T_{\rm h}/T_{\rm eff}]^{4}{r_{\rm h}}^2 \nu$)ranging from 0 to 2, whenever possible ($r_{\rm h} < 1$). Those values are easily found in the literature amongst many types of binary systems. Using $\nu=1$ we find $F_{\rm rel}=0.446$ for V$\,$Pup (detached, Andersen et al. 1983), 0.338 for LZCen (detached, Vaz et al. 1995), 0.386 for RYAqr (semi-detached, Helt 1987), 0.660 for DH Cep (detached, Hilditch et al. 1996), 1.35 for AT Peg (algol, Maxtedetal.1994), 2.39 for HU Tau (algol, Maxted et al. 1995) and 2.72 for TV Cas (algol, Khalesseh&Hill 1992).

Many of the binary systems above also have temperatures that fall in our selected ranges: RY Aqr ($T_{\rm eff}=4\,500$K and $T_{\rm h}=7\,600$K), AT Peg (4898K and 8395K), HU Tau (5495K and 12022K) and TV Cas (5248K and 10471K).

As with $T_{\rm h}$ close to $T_{\rm eff}$ we could not get large values of $F_{\rm rel}$ with reasonable values of $r_{\rm h}$, we studied models mostly ranging in the interval $7\,000\,{\rm K}<T_{\rm h}<12\,000\,{\rm K}$.The illuminating non-grey fluxes with $T_{\rm h}\le 7\,000$K were generated with the UMA code, while for higher $T_{\rm h}$ we took the fluxes from Kurucz (1979). We did calculate some models with $T_{\rm h}$ close to $T_{\rm eff}$($T_{\rm h}=3700$K, 4600K, 5500K, 6700K) and the results are in agreement with our studies with higher $T_{\rm h}$ so that our results do apply for lower heating temperatures, also. However, we advise caution in using those approximations extended to outside the limits proposed in this work.

In Sect.2.2 we give the results obtained with line-blanketed atmospheres illuminated by line-blanketed fluxes, showing differences in the calculations with the two distinct methods.

2.2 Results

  

Figure 3: Adjusted limb-darkening laws for a non-illuminated model and 4 illuminated models (varying $r_{\rm h}$). The asterisks represent the model results

Figure 3 shows that the external illumination strongly affects the limb-darkening laws and coefficients of illuminated atmospheres as compared with the non-illuminated ones. These models are approximately equivalent to those calculated by CG and $\nu$ has a mean value between vertical and grazing incidence. In order to be consistent we will show the results for the Strömgren u filter, but the effects are similar for the other calculations done, bolometric, monochromatic and for all the Strömgren filters and the UBVRI filters of the Johnson-Morgan system, as well. As we increase the amount of incident energy, for example by increasing the apparent radius while keeping the other parameters fixed, the law for non-illuminated models no longer represents the calculated intensities, showing that non-illuminated coefficients yield wrong limb-darkening laws when used with illuminated stars.

  

Figure 4: The run of the temperature with the optical depth (Rosseland mean) for the models shown in Fig.3

  

Figure 5: $I(\mu)/I(1)$ vs. $\mu$ curves for the Strömgren u filter and various models, see the text for explanation. The meaning of the different linestyles and symbols is the same as in Fig.3

A first interesting result from Fig.3 is that, depending on the set of parameters chosen, we observe an effect of limb brightening instead of darkening. This can be understood as follows: when looking at a stellar disk we normally see the limb darker than the center because, despite of seeing the same optical depth in both, it corresponds to deeper, and consequently hotter, layers in the center than in the limb. In a illuminated star, the temperature distribution in the atmosphere becomes more and more homogeneous as we increase the illumination flux, until it reaches a state where there is no center-to-limb darkening and the star's brightness is the same throughout the stellar disk. This can be seen in Fig.4, where we show the temperature structure for the models shown in Fig.3. The larger the amount of infalling flux the higher the temperature of the external layers and the larger the region where it stays approximately constant.

  

Figure 6: $F^{\prime}/F_{\rm m}$ calculated with the coefficients determined by the CG method. Models used: $\nu=0.07$, $T_{\rm h}=7000$ K

If we keep increasing $F_{\rm rel}$, by changing either $r_{\rm h}$, $T_{\rm h}$ or $\nu$, we reach a situation where the most external layers are hotter than the internal ones. As we see more external layers looking at the limb than looking at the center, we have a limb brightening effect. In Fig.5 we illustrate that effect in three panels, each with four $I(\mu)/I(1)$ vs. $\mu$ curves, varying the many parameters. The effect is, though, dependent not only on the amount of the infalling flux. The first panel of Fig.5 shows models that reach limb brightening with constant $T_{\rm h}$ and incident direction, but changing $F_{\rm rel}$ through changes in $r_{\rm h}$. Panels 2 and 3 show that limb brightening can also be achieved with constant $F_{\rm rel}$ (= 0.5), either by keeping $T_{\rm h}$constant and using different pairs of $r_{\rm h}$ and $\nu$, or by fixing $\nu$ and changing $r_{\rm h}$ and $T_{\rm h}$.We can also notice in Figs.3 and 5 that the linear law (straight lines) is a poor approximation for the limb-darkening of illuminated atmospheres, while for $T_{\rm eff}\approx 5\,500$K it fits well the non-illuminated case (Fig.3). One should mention here that, although the models of Fig.3 are similar to those calculated by CG, they do not report any evidence for the limb brightening effect in their work.

The methods by CG and VH, used here to calculate the coefficients, differ from each other in some important aspects. While CG emphasize the point of using coefficients that best match the relation $I(\mu)/I(1)$ vs. $\mu$, even if the total flux is not perfectly conserved, VH says that the physical constraints of his method are introduced in order to avoid non-physical coefficients. In non-illuminated atmospheres CG find that the total emergent flux obtained by integrating the intensities calculated from the adjusted coefficients ($F^{\prime}$) does not differ more than 2% from the total flux obtained from the model intensities ($F_{\rm m}$), and that when choosing a non-linear law, the difference vanishes. We calculated the relation ($F^{\prime}/F_{\rm m}$) for all our illuminated models, and the result for a set of models is shown in Fig.6. As noticed, the linear law is not a good approximation when dealing with illuminated atmospheres. We find that the mean difference between the fluxes is around 4% (with a maximum of 26% in some models) in the case of a linear law and less than 1% (however with a maximum of 6%) for the non-linear ones. These results show that for an illuminated atmosphere the CG method should be used with care, as, for some models, even choosing a non-linear law, the flux is not conserved. Due to that we decided to adopt the VH method in our calculations of all the results presented in Tables 1 and 2.

Tables 1 and 2 are a sample of the tables with the monochromatic, bolometric and passband-specific limb-darkening coefficients than can be accessed electronically. We calculated coefficients for the linear, quadratic, cubic, logarithmic and square root laws, for stars with $3700\,{\rm K}\le T_{\rm eff}\le 6700\,{\rm K}$ heated by a companion with $7000\,{\rm K}\le T_{\rm h}\le 12000\,{\rm K}$, with relative fluxes ranging from 0.0 to 2.0 and the illumination direction, $\nu$, varying from 0.07 to 0.97. The parameter Q in Table 1 has the same meaning as in VH. It is a quality factor that shows how good is the fitting to the model intensities, the smaller the Q the better the fitting:

$$Q_{\lambda}=\sqrt{\frac{\sum_{i=1}^{6}[R_{\lambda}(\mu_{i})-\widehat{R_{\lambda}}(\mu_{i})]^2}{6-m}}$$ (5)

where m=1 for a one-parameter law (linear law) and m=2 for a two parameter law (non-linear laws). $R_{\lambda}(\mu)$ stands for the ratio $I_{\lambda}(\mu)/I_{\lambda}(1)$ determined with the atmosphere model and $\widehat{R_{\lambda}}(\mu)$ for the same ratio determined with the limb-darkening approximation. When determining the bolometric coefficient, Q was calculated as:

$$Q=\frac{\int_{\lambda_{i}}^{\lambda_{f}}F(\lambda)Q(\lambda)}{\int_{\lambda_{i}}^{\lambda_{f}}F(\lambda)}$$ (6)

where $F(\lambda)$ is the monochromatic flux, ${\lambda}_{i}$ is the shortest and ${\lambda}_{f}$ the longest wavelength used in the UMA code.

 

Table 1: Sample of the table with the bolometric and monochromatic limb-darkening coefficients

 

Table 2: Sample of the table with the passband-specific limb-darkening coefficients. We calculated the coefficients for the Strömgren uvby filters and for the Johnson-Morgan UBVRI filters. B2 and B3 are the response function of the B filter with and without the earth's atmosphere transmission function, respectively (see Buser 1978)

 

Table 3: Sample of the table with the polynomial adjusted coefficients for the Strömgren u filter. See Eqs. (2), (8)-(11)

 

Table 4: Linear and quadratic limb-darkening coefficients for non-illuminated models. Our calculations were made with $T_{\rm eff}=6700$K while Claret&Giménez (1990) used $T_{\rm eff}=6730$K, both with $\log g=4.5$.For the quadratic law, at the top is the linear coefficient and below it the non-linear one. Here CG stands for the results by Claret&Giménez (1990), CG$^{\prime}$ and VH$^{\prime}$correspond to our calculations using the methods by CG and VH and, respectively, and, finally, VH correspond to the values by Van Hamme (1993), bi-linearly interpolated for $T_{\rm eff}=6700$K and $\log g=4.5$

When calculating with the CG method we do not determine the monochromatic limb-darkening coefficients, so Q cannot be obtained by integrating them. In order to be able to compare the results from the CG and VH methods, we used in Table 2, the following similar definition:

$$Q_{\rm filter}=\sqrt{\frac{\sum_{i=1}^{6}[R_{\rm filter}(\mu_{i})-\widehat{R_{\rm filter}}(\mu_{i})]^2}{6-m}}\cdot$$ (7)

Although only the VH results are presented in Table 2, we performed calculations with both methods. In general, the CG method gives a smaller Q than VH, representing a better fit. That was an expected result as CG does not require any constraint to the fitting, the coefficients being calculated directly by applying the least squares method to the integrated intensities obtained with the atmosphere model. As VH applies the constraint of total flux conservation the fittings are often worse, but always physically correct. We observe that the worse the fitting with VH method the more the CG coefficients fail in conserving the total flux ($\vert(F^{\prime}/F_{\rm m})\vert \gt 1$).

As already noticed by Claret & Giménez (1990), the limb-darkening coefficients of an illuminated atmosphere strongly depend on many parameters, making it difficult to find a simple function that describes the effect. We propose, for the passband specific coefficients, a solution to easily account for the effect in LC synthesis programs, parametrizing with polynomials the results obtained. In Eq.(8) K represents $x_{\lambda}$ or $y_{\lambda}$of Eqs.(1) to (4) for the different limb-darkening laws:

$$K=a_{\scriptscriptstyle 0}(t) + {\sum_{n=1}^3} a_{\scriptscriptstyle n}(t, \nu, t_{\rm h})F^n$$ (8)

$$a_{\scriptscriptstyle n}={\sum_{m=0}^3} b_{\scriptscriptstyle nm}(\nu,t_{\rm h})t^m$$ (9)

$$b_{\scriptscriptstyle nm}={\sum_{l=0}^3} c_{\scriptscriptstyle nml}(t_{\rm h})\nu^l$$ (10)

$${\rm and}~~c_{\scriptscriptstyle nml}={\sum_{k=0}^2} d_{\scriptscriptstyle nmlk}{t_{\rm h}^k},$$ (11)

with $F=F_{\rm rel}$, $t=T_{\rm eff}\ 10^{-3}$and $t_h=T_{\rm h}\ 10^{-3}.$

In Table 3 we give a sample of the adjusted polynomial coefficients obtained from the data in Table 2 and in Fig.7 we show the adjusted polynomial surface for one chosen model. In the case of a non-linear law, we give in Table 3 the linear adjusted polynomial coefficients (cnm0_L, cnm1_L, cnm2_L, cnm3_L) followed by the non-linear ones (cnm0_NL, cnm1_NL, cnm2_NL, cnm3_NL). A complete version of Table 3 with the adjusted polynomial coefficients for the 9 chosen photometric filters (see Sect. 2.1) can be accessed in electronic form at the CDS.

  

Figure 7: Polynomial fittings to the quadratic law coefficients. Strömgren u filter, $\nu=0.97$, $T_{\rm h}=7000$ K. The x axis corresponds to $F_{\rm rel}$ and the y to $T_{\rm eff}$. The theoretical grid was calculated using the coefficients in Table 3

2.3 A control

It is important to compare limb-darkening coefficients calculated from different atmosphere models and computational methods. Claret & Giménez (1990) have also used the UMA code in their calculations, and our results can be easily compared. In Table 4 we show the coefficients for the linear and quadratic laws for non-illuminated atmospheres, obtained by Claret & Giménez (1990), Van Hamme (1993, only for the linear law, because VH did not include the quadratic law in his study) and by us, using both VH and CG methods. As can be noticed, our values are very similar when using the same method but show a little expected difference with those determined by the VH method. Actually, Díaz-Cordovés & Giménez (1992) had already found that differences between tabulations by different works are normally due to the computational method rather than to the adopted model atmospheres themselves. This makes us confident that our calculations are correct for the non-illuminated case and that our results probably would be similar if we used another atmosphere model, instead of UMA. Note that the calculations with VH's method yield values somewhat systematically smaller for the linear law and that the limb-darkening coefficients given in VH (using ATLAS) are the smallest in each passband.

This control in the case of illuminated atmospheres is more problematic. To the best of our knowledge only CG worked on the effect of external irradiation on the limb-darkening coefficients using numerical atmosphere models (UMA in that case), but their results are only presented in a qualitative way, making impossible a numerical comparison. We do not know of any other work on the effect of external illumination on the limb-darkening coefficients with models other than UMA.

However, if the theoretical approximations used are similar to those used in this work we expect that the results will be similar to ours, irrespective of the atmosphere model used in the study. And we are confident that the results derived here are better, even used as a first order approximation, than the use of constant limb-darkening coefficients overall in the analysis of eclipsing binary LC.