4 Application to Hipparcos

4.1 Detection of spurious periods

When using usual period search methods, the well known problem of aliasing can produce spurious periods. On the basis of the phase diagram only, these periods can not be rejected. However, the variogram can be used as a complementary method. It can be very helpful to rule out certain periods, or to increase the level of confidence in others. The accurate determination of the period is left to the usual period search algorithm.

  

Figure 6: Phase diagram for the star HIP 023743. Above the data is folded with the period p1=3.348 days. Below, with the period p2=80.9 days

  

Figure 7: Estimated variogram $\hat\gamma_{\rm Q}$ of the star HIP 023743

Here we present an example of such a situation. The star HIP 023743 has in its power spectrum two possible periods: p₁=3.348 days and p₂=80.9 days. The associated phase diagrams are presented for these two possibilities in Fig. 6. Both seem to be reasonable. However, with the variogram, the short period can be eliminated. As we see in Fig. 7, there is no evidence of variability signature for the periods shorter than 3 days which could explain the dispersion of the data. Figure 6 shows that the phase diagrams both have the same amplitude of approximately 0.07. Figure 7 does not reveal any such amplitude around p₁=3.348, whereas it does around p₂=80.9. This case is also instructive because the minimum of the variogram does not exactly coincide with the period p₂=80.9 days. This is due to the irregularity and scarcity of the pairwised differences near h=80.

4.2 Estimation of measurement noise

The precision on the Hipparcos magnitudes is strongly variable. It can be interesting to know its dependence with respect to the magnitude. As already said, the nugget value can be viewed as a good estimator of the noise. Only the very short pairwised time differences are used (h=20 minutes or h=108 minutes). The underlying assumption is that very few variable stars have detectable variability at lags around h=20 minutes.

In fact the law (cf. Fig. 8) is very close to the one obtained by Eyer & Grenon (1997). For a given magnitude, the dispersion can be high and the degradation is very rapid (exponential). Furthermore there are some outliers which are caused by some problems of the satellite like light pollution from companions, mispointing, or by rapid oscillations of stars.

  

Figure 8: Estimated measurement noise by short-time differences using $\hat\gamma_{\rm Q}$

Based on Fig. 8, a set of potentially interesting stars was selected on which a high frequency analysis has been performed. As a result we found three periodic variable stars (HIP 071119, HIP 044025 and HIP 029055) which are not in the published catalogues (Grenon et al. 1997 and van Leeuwen et al. 1997).

The method permits also to compare our estimated noise with the quoted errors $\epsilon_i$, furnished in the Hipparcos photometric data base. For a given star we define a mean error by $\epsilon_{\rm M} = \sqrt(\sum \epsilon_i^2/n)$, and a highly robust noise estimator $\epsilon(h)=\sqrt{\hat \gamma_{\rm Q}(h)}$,with h=20 min. In order to confirm the adequacy of the method, a simulation is carried out on the 118204 stars. The magnitudes m(t_i) are drawn from a Gaussian distribution with constant mean and variance $\epsilon_i^2$. Then, the estimator $\epsilon(h)$ is calculated on the simulated m(t_i) for each star.

  

Figure 9: Simulated measurement noise using the highly robust noise estimator $\epsilon(h=20)$ versus the mean error $\epsilon_{\rm M}$. The grey data cloud represents the 118204 stars of Hipparcos, and the black filled circles denote the moving median. The graph shows that $\epsilon_{\rm M}$ and $\epsilon(h=20)$ are reasonable estimators for the noise

  

Figure 10: Estimated measurement noise of the Hipparcos data using the highly robust noise estimator $\epsilon(h=20)$ versus the mean error $\epsilon_{\rm M}$. The grey data cloud represents the 118204 stars of Hipparcos, and the black filled circles denote the moving median. The graph shows a bias of 9% for the slope

  

Figure 11: Estimated variogram $\hat\gamma_{\rm Q}$ of the star HIP 111771

  

Figure 12: Estimated variogram $\hat\gamma_{\rm Q}$ of the star HIP 060998

We compare the two estimators $\epsilon_{\rm M}$ and $\epsilon(h)$in Fig. 9 for the simulated data; a moving median is computed and is represented by the filled circles, indicating a reasonable match between the two noise estimators. In Fig. 10, the estimated noise of the Hipparcos real data is represented following the same method. The slope of the moving medians is 1.09 indicating a 9% bias probably due to $\epsilon_i$.It seems that for small $\epsilon_{\rm M}$ the $\epsilon_i$ can be overestimated whereas for large $\epsilon_{\rm M}$ the $\epsilon_i$ are underestimated in the Hipparcos data base. As a consequence, when doing a photometric variability analysis, some bright variable stars might be taken as constant, and some constant faint stars might be taken as variable.

  

Figure 13: Fourier power spectrum of the star HIP 060998. Although a characteristic time-scale can be given in the variogram (Fig. 12), the power spectrum is noisy

  

Figure 14: Estimated variogram $\hat\gamma_{\rm Q}$ of the star HIP 052507

4.3 Example of three stars from the Main Mission

Three examples illustrate the use of the wave variogram. We have a sequence of giant stars, located in the same region of the HR diagram, but with different properties: HIP 111771 is the hottest star (M2III type) with a pseudo-period of approximately 70 days (cf. Fig. 11), and an amplitude (peak to peak) of 0.61 Hp mag; HIP 060998 (M4III type) has a small amplitude of 0.23 Hp mag, and a shorter period of about 35 days (cf. Fig. 12). For a comparison, the Fourier power spectrum of the latter star is presented in Fig. 13, no clear information can be deduced from it. HIP 052507 (M5III type) has a very long time-scale variation (clearly seen in the light curve), with a shorter one superimposed. The latter is revealed by the variogram in Fig. 14, with a period of about 85 days, and an amplitude of 0.53 Hp mag.