5 Appendix

5.1 Comparison of observed and model diameter ratios

We have also compared the model stellar filter radii $R_{\rm f, m}^t$as predicted by each model-phase combination m with our measured angular stellar filter radii $R^{\rm a}_{\rm f, m}$ by comparing the observed and model diameter ratios at different wavelengths (filter f: 656/10, 673/8, 699/6, 754/6, 781/14 and 1045/9). By confronting a large variety of Mira models with the here presented narrow-bandpass observations, we may test how sensitively monochromatic radius measurements probe model structures and whether they are indeed reliable tools of Mira diagnostics. Since different models predict, at different phases, both different stellar filter radii $R_{\rm f}$and different filter CLV curves, we expect better agreement between model-predicted ratios and observed ratios (based upon the corresponding model-phase combination of CLV) for models that represent R Leo well and for model phases that are close to the observed phases than for other models and phases. Since our five models are taylored to the parameters of o Ceti which roughly has the same period and luminosity as R Leo (310 days, $6540 \pm 3010~L_{\odot}$: derived from its bolometric flux measured at phase 0.20 close to our observations and its HIPPARCOS parallax), this comparison should give some hint as to whether any of the models is a fair representation of R Leo. We must be aware, however, that model phases are close to but not identical with observed phases, and cycle-to-cycle variations may be substantial (BSW96, HSW98).

For illustration, Fig. 8 presents the observed and model ratios $R_{\rm f}$/ $R_{\rm 1045nm}$of the stellar filter radii $R_{\rm f}$ and $R_{\rm 1045nm}$as a function of wavelength $\lambda$ for all applied model series (D, E, P, M and O) and phases close to our observations (since nearly all model series consist of several cycles, a best fit selection was applied). Figure 9 displays the observed and theoretical ratios for all model-phase combinations m and all filter pairs. Figure 10 presents the distance D_m between the measured and model stellar filter radius ratios (between all possible pairs of filters) for each model-phase combination. The distance D_m is defined as

$$D_{m} := \sqrt{\frac{\sum_{i\neq j}^{N_{\rm f}} \left\vert\fr... ... f}} \left\vert\frac{R_{i, m}^t}{R_{j, m}^t}\right\vert^2 } },$$ (1)

where $N_{\rm f}$is the number of filters used and the filter designation numbers i and j range between 1 and $N_{\rm f}$ (here: $N_{\rm f}$=6). The errors of the distances D_m were estimated according to the Gaussian error propagation law.

Inspection of Figs. 8, 9 and 10 shows that from the point of view of diameter ratios, some of the model-phase combinations are acceptable as representations of the here presented observations of R Leo but none is completely satisfactory. Nearly all models are too compact in the strong TiO absorption band at 673 nm, i.e. the distance-independent ratio $(R_{673\rm nm}-R_{\rm continuum})/R_{\rm continuum}$ is too small compared with the observations. For the model-phase combinations at phases close to the observation, the P model series which exhibits the most pronounced atmospheric extension of the BSW96/HSW98 studies has in most cases (i.e., in 11 out of all 15 filter ratio combinations) model diameter ratios which are identical (within the error bars) to the measured ones. The order of ranking after the P model is: E model (6/15), D model (6/15), M model (3/15) and O model (1/15).