4 Discussion and conclusion

From our speckle observations of R Leo at near maximum phase (phase 0.20) we derived UD diameters of

• 75.6$\pm$3.7 mas at the strong TiO absorption band (673 nm/8 nm);
• 60.6$\pm$3.0 mas, 52.5$\pm$2.5 mas and 55.0$\pm$2.7 mas
  at the moderate TiO absorption bands (656 nm/10 nm, 699 nm/6 nm and 781 nm/14 nm);
• 48.7$\pm$2.3 mas at the weak TiO absorption band (754 nm/6 nm); and
• 37.9$\pm$4.0 mas in the continuum at 1.04 $\mu$m (1045 nm/9 nm).

R Leo shows no significant asymmetry of its shape in all bands observed, i.e. in the continuum at 1.04 $\mu$m and in the TiO absorption bands showing the upper atmosphere (see Sect. 2.2). Contrary to R Leo, R Cas shows a significant asymmetry of its shape in all TiO absorption band filters (Weigelt et al. 1996; Hofmann et al. 2000a). On the other hand Lattanzi et al. (1997; November 1995 at cycle+phase of -1+0.71) and Tuthill et al. (1999; January 1992 at cycle+phase of -5+0.27, June 1993 at cycle+phase of -4+0.88) detected a weak asymmetry of the shape of R Leo, too.

Tuthill et al. (1994) derived UD diameters of $45 \pm 2.0$ mas and $43 \pm 2.0$ mas at 833 nm/41 nm and 902 nm/50 nm, respectively and derived Rosseland diameters of $37.4 \pm 2.0$ mas and $39.0 \pm 2.0$ mas, respectively. These measurements agree well with our angular Rosseland diameter of $38.4 \pm 4.0$ mas at phase 0.20 (April 4, 1996). Haniff et al. (1995) reported on R Leo observations through their 700 nm/10 nm filter comparable to our 699 nm/6 nm filter. They derived an UD diameter of $64.2 \pm 5.7$ mas at cycle+phase of -4+0.88which is approximately 20% larger than our near-maximum phase (0.20) 699 nm/6 nm UD diameter of $52.5 \pm 2.5$ mas. This difference might be explained by UD diameter variations with the variability phase reported by Burns et al. (1998). Di Giacomo et al. (1991) derived an UD diameter of $33 \pm 1.3$ mas from their lunar occultation measurements (May 1990, cycle+phase of -7+0.2) in the Br$\gamma$ line of atomic hydrogen at 2.16 $\mu$m. Tej et al. (1999) obtained UD diameters of $39 \pm 3$ mas and $34 \pm 2$ mas from their lunar occultation observations through a narrowband filter at 3.36 $\mu$m (December 1997, cycle+phase of 2+0.17) and a broadband filter at 2.2 $\mu$m (March 1998, cycle+phase of 2+0.44), respectively. K-band observations with the IOTA interferometer by Perrin et al. (1999) yielded UD diameters of $28.18 \pm 0.05$ mas at cycle+phase of 0+0.24 (April 17-18, 1996) and $30.68 \pm 0.05$ mas at cycle+phase of 1+0.28 (March 3-10, 1997). Note that the 1996 measurement of Perrin et al. was done at nearly the same phase of the same pulsation cycle as the here presented observations.

The difference between our 1045 nm/9 nm UD diameter (at cycle+phase of 0+0.20) and the K-Band UD diameter of Perrin et al. (1999; at cycle+phase of 0+0.24) is larger than expected and may indicate that there exists an additional near-infrared extinction, not included in the BSW96 and HSW98 models, which blankets the 1 $\mu$m region more strongly than the K bandpass. No such opacity source is known so far, but Bedding et al. (2001) noticed that dust particles condensating in the uppermost atmospheric layers may produce this type of effect by generating a two-component appearance of the CLV which is more pronounced at shorter $\lambda$. Danchi et al. (1994) claim that R Leo belongs to a class of stars whose inner dust-shell radii are very close to the photosphere (3 to 5 photospheric radii), i.e. dust might be formed in the uppermost atmospheric layers. Therefore, we cannot exclude that our measured radius has to be scaled to the true-continuum radius resulting in a smaller stellar radius and in a higher effective temperature. A similar 1 $\mu$m vs. K-band discrepancy was reported for the Mira variable R Cas (Weigelt et al. 2000).

Figure 8: Observed and model radius ratios $R_{\rm f}$/ $R_{\rm 1045nm}$ of stellar filter radii $R_{\rm f}$ and $R_{\rm 1045nm}$as a function of wavelength for all five models and for model phases close to our observations. The plotted theoretical stellar filter radius curve $R_{\rm f}(\lambda )$ is derived from the monochromatic one by convolution with a rectangular-shaped function with a bandwidth of 6 nm (=bandwidth of our optical filters with the narrowest width).

Figure 9: Observed and model radius ratios Ri/Rjof stellar filter radii Ri and Rj (i and j denote filters). The 15 plots show all possible filter combinations. Table 6 gives the link between the abscissa values and the models and their phases.

Figure 10: Normalized distance Dm between measured and model diameter ratio vectors (see text; diamonds: observations reduced with model CLVs with phases close to the phase of the observation; crosses: observations reduced with model CLVs with phases far from the phase of the observation; squares: theoretical model radii). Table 6 gives the link between the abscissa values and the 27 model-phase combinations m.

When measured visibility data are reduced with limb-darkening profiles predicted by recent Mira models (BSW96, HSW98), we find that strong-TiO $\tau_{\lambda}=1$ diameters depend substantially on the adopted model, whereas the continuum diameter does not. Since these models are taylored to the parameters of o Ceti which has nearly the same period and luminosity as R Leo (310 days, $6540 \pm 3010~L_{\odot}$), they should predict also quantitatively the basic properties of R Leo. The predictions of the E model series are in good agreement with (i) the stellar filter radii measured through five of six filters (however, the stellar filter radii measured in the strong TiO absorption band at 673 nm are about 50% larger than the model-predicted values, i.e. the models of the E series as well as the other models considered here are systematically too compact; see Appendix); (ii) the measured Rosseland radius and the derived pulsation mode; and (iii) the measured effective temperature of 2590 K $\pm$ 180 K at near-maximum phase 0.20. We obtain a Rosseland $\tau_{\rm Ross} =$ 1 radius of $R = 417~R_{\odot}$(based on the E-model at phase 1.21 close to the phase of our observations) with an accuracy of about 23% (the error of the HIPPARCOS parallax of $\sim$20% is the largest fraction of the total error; the speckle error is $\sim$10%).

Acknowledgements

We thank P. Whitelock for sending us her JHKL photometric observations of R Leo and for calculating the bolometric flux.