5 Rotational modulation model

5.1 A Bp star

We searched the literature for chemical inhomogeneities in this star. Hiltner et al. (1969) indeed classified HD 131120 as a Bp star and reported spectral peculiarities similar to those found in 3 Sco and HD 144334. These latter stars appear to be He-weak Si stars (see Garrison 1967; Norris 1971). The average equivalent width of the He 4121 Å line measured in our data is about 68.9 mÅ. By comparing this value with the ones for normal B2.5 stars in Didelon (1982), we can confirm that HD 131120 is a He-weak star. We note that, in SIMBAD, we find a MK spectral type of B7, which is misleading as it relies on He line strengths. The average equivalent width of the Si 4128 Å line is about 95.7 mÅ which is quite normal for a B2.5 star.

The chemically peculiar Bp stars show monoperiodic variations. Where light and line-profile variations are present, the same frequency is found in both data sets. The observed periods range from 1 to 20 days in the majority of cases. The periods show an inverse correlation with the projected rotational velocity. All these characteristics indicate that the variations of Bp stars are due to rotational modulation. Moreover, up to now, the variability of He-weak stars is explained by the rotation of the star in the presence of a non homogeneous distribution of helium on the stellar surface.

5.2 Model with two spots

We compared the line-profile observations of HD 131120 with a rotational modulation model by using a code kindly put at our disposal by Dr. L. Balona. This code calculates line-profile variations for a spotted star.

The following parameters are needed to construct line-profile variations caused by a circular spot: the equatorial and polar radii $R_{\rm e}$ and $R_{\rm p}$, the equatorial and polar fluxes $F_{\rm e}$ and $F_{\rm p}$, the projected rotational velocity $v_{\Omega }$, the angle of inclination i, the linear limb-darkening coefficient u, the intrinsic line-profile width in the photosphere $\sigma_{i}$, the longitude (relative to some arbitrary epoch) $\lambda$, the latitude $\beta$, the spot radius in degrees $\gamma$, the flux from the spot relative to the photosphere F, the intrinsic line-profile width in the spot $\sigma_{\rm s}$. We take the equatorial and polar radii $R_{\rm e}=R_{\rm p}=3.6~R_{\odot}$, the equatorial and polar fluxes $F_{\rm e}=F_{\rm p}=1$ and the linear limb-darkening coefficient u=0.36. The other parameters are free parameters.

As HD 131120 appears to be a He-weak star, it was important to test the rotational modulation model also on the He 4121 Å line. First we computed the first three moments of this line. The frequency search leads to f₁=0.6375 c/d and f₂=2f₁=1.275 c/d for $\langle v^1\rangle$ and $\langle v^3\rangle$. No frequency can be found for $\langle v^2\rangle$. The frequency f₁ and its first harmonic f₂ reduce the standard deviation by 53% for $\langle v^1$$\rangle$ and by about 37% for $\langle v^3\rangle$. A phase diagram for the frequency f₁=0.6375 c/d for $\langle v^1\rangle$, $\langle v^2\rangle$ and $\langle v^3\rangle$ is shown in Fig. 5. Figure 6 shows that the EW also varies with the same frequency. The relative EW variation of the HeI line is about 16%.

Figure 5: Phase diagram of the first three moments of the He 4121 Å line. We show the observed values (dots), the fit using 0.6375 c/d and its first harmonic (solid line), and the best spot model (dashed line).

Figure 6: Phase diagram of the equivalent width of the He 4121 Å line, which is expressed in Å. The comparison with the best spot model is represented in dashed line.

We point out that the first moment of the SiII 4128 Å line and of the He 4121 Å line do not have the same form. This observation is clearly an argument against a NRP model. Morever, they are not in phase. This is not compatible with a NRP model.

Such variations can be reproduced by a spot model if the difference of longitude between a spot of Si and a spot of He is $180^{\circ}$. They can also appear if for one spot silicon is overabundant and helium is underabundant or silicon is underabundant and helium is overabundant.

A non-sinusoidal first moment with f and 2f can be obtained if we consider two spots of the same element, which have a longitude difference of $180^{\circ}$. Two spots give 13 free parameters, which leads to an enormous computational time. In order to reproduce the form of the first moment, we decided to test the two following cases:

$$\begin{eqnarray*}\left\{\begin{array}{lllll} \lambda_1 = \lambda_2 + 180^{\circ}... ...neq F_2,\\ \sigma_{\rm s1} =\sigma_{\rm s2}, \end{array}\right. \end{eqnarray*}$$

and

$$\begin{eqnarray*}\left\{\begin{array}{lllll} \lambda_1 = \lambda_2 + 180^{\circ}... ...1 = F_2,\\ \sigma_{\rm s1} =\sigma_{\rm s2}, \end{array}\right. \end{eqnarray*}$$

where the indices 1 and 2 are respectively for the first spot and the second spot. We compared such a spot model to both the He line and the Si line. We again determine the parameters for which the theoretical profiles best fit the observations by minimizing the standard deviation $\Sigma$ in the intensity over all profiles as done for the fitting with BRUCE. Then the parameters were varied around the best set of parameters in order to refine the solution.

For the He line, the following parameters lead to a good fit, as we can see in Fig.7:

$$\begin{eqnarray*}\left\{\begin{array}{llllllll} \lambda_1 = 90^{\circ}, \ \lambd... ...1},\\ \Sigma = 0.0022~{\rm continuum~units}. \end{array}\right. \end{eqnarray*}$$

Figure 7: Observed line profiles (dots) of the He 4121 Å line (left) and of the SiII 4128 Å line (right) averaged over phase bins of 0.05 and theoretical line profiles (full lines) for the spot model with two spots.

In Fig.5 the first three moments of the theoretical profiles shown in Fig.7 are compared to the ones of the observed profiles of He. In Fig.6 the variation of the theoretical EW is compared to the observed EW variation. It results that the behaviour of the observed moments is well reproduced by the theoretical moments.

For the Si line, we also obtain a good fitting with these parameters (see Fig.7):

$$\begin{eqnarray*}\left\{\begin{array}{llllllll} \lambda_1 = 310^{\circ}, \ \lamb... ...1},\\ \Sigma = 0.0026~{\rm continuum~units}. \end{array}\right. \end{eqnarray*}$$

In Fig.2 the first three moments of the theoretical profiles are compared to those of the observed profiles of Si and in Fig.3 we show the comparison between the observed and theoretical EW variations of the Si line. Again we find a good agreement between observed moments and theoretical moments. We note that the amplitude of the theoretical second moment is very small, which is compatible with the fact that the observed second moment is very noisy.

We point out that the models for the He line and for the Si line are compatible since they have the same value for the parameters $v_{\Omega},\ i,\ \sigma_{i}$. We also point out that we end up with a model for which helium is underabundant in the spots while silicon is overabundant there. This naturally explains the very weak He line and the very strong Si line.

It would be interesting to compute the variation of the luminosity of HD 131120 in order to compare it to the observed photometric amplitudes. As we do not know if the star presents additional non homogeneous distributions of other elements on the stellar surface, we cannot compute this variation as long as we do not have a complete view of the chemical abundances.