4 Is theory directly comparable to observations?

Comparing theoretical models with observational data, with a view to calibrating the internal parameters of the theory and to deriving physical information on the stars (e.g. the pulsation mode and the fundamental parameters) does not make sense if the models are basically wrong or if their predictions are systematically shifted because of imperfections of their input physics. We thus have to assess the reliability of the model grid, including the temperature scale, and to find a way to (partially) compensate its defects.

4.1 Static/LNA modelling

First, one may remember that the core mass-luminosity relation (CMLR) assumed in our calculations corresponds to maximum hydrogen shell luminosity (Paczynski 1970). In fact, even neglecting the very short helium flashes, the luminosity varies by a factor of two over a thermal pulse cycle, and the CMLR approximately holds over only about 25% of the interflash time (Boothroyd & Sackman 1988a, 1988b; Wagenhuber & Tuchman 1996). For a global investigation of our sample of stars, we would perhaps do better adopt an "effective'' CMLR that would be, say, 30% less luminous. Doing so, at given total mass and luminosity, the effective temperature becomes 1% higher, and the period decreases by a few %.

Our calculations also assume that the convective flux and velocity instantly adjust themselves to pulsation, i.e. that the mean eddy lifetime is about zero. In fact, it is roughly a third of the fundamental period (Ostlie & Cox 1986), which induces a significant phase lag. In order to estimate the resulting uncertainty, we have recomputed some of our models while assuming a frozen-in convection, i.e. infinite eddy lifetime. The obtained fundamental periods are shorter by 5-10%, and the first overtone longer by 1-4%. In other terms, as long as convection phase lag is concerned, the fundamental periods predicted by our model grid are probably overestimated by roughly 5%, while the first overtone is underestimated by perhaps 2%.

On the other hand, Ostlie & Cox (1986) have attempted to improve the standard LNA modelling by a horizontal opacity averaging scheme, which accounts in a simplified way for the coexistence of rising and falling convective elements in the same mass shell. Periods then increase by less than 10%. Moreover, the same authors have investigated the effects of turbulent pressure: the obtained period shifts range from +8 to +36% for the fundamental mode and from +3 to +11% for the first overtone. Turbulent viscosity and energy have negligible effects (Cox & Ostlie 1993).

Summarizing, one may expect the model grid described in Sect. 3 to underestimate the fundamental period by as much as 40% for the fundamental mode and 25% for the first overtone, and the relative shift of the latter mode is always more than a third of the former.

It is worth noting that the often quoted models of Wood (Wood 1974; Fox & Wood 1982; Wood 1990; Bessell et al. 1996; Hofmann et al. 1998; Wood et al. 1999), which include a phase-lagged convection scheme, unfortunately use an outdated equation of state. As far as we know, this is the only important difference with Tuchman's code. The EOS is thus probably the reason why Wood's linear fundamental and first overtone periods are longer by 15-70% than the ones predicted by our models, with the same opacity tables and composition regardless of our treatment of convection. The relative shift is always about the same for the two modes. Interestingly, the fundamental period shift is the same order of magnitude as that wich would result from opacity averaging and turbulent pressure together. As a consequence, Wood's fundamental pulsation models can often reach a reasonable agreement with the observations, but first overtone periods derived from this code are strongly overestimated, as also noted by Xiong et al. (1998).

Figure 2: Difference between the mean colour index and the colour at the static temperature, according to various dynamical models (see text). Left: V-K; right: J-K

4.2 Nonlinear effects

The periods of the pulsation modes predicted by LNA models correspond to small-amplitude oscillations of the static stellar envelope. The effective temperature, too, is that of the static star. In fact, the large-amplitude pulsation of an LPV is strongly nonlinear and makes the outer envelope expand. As a consequence, the mean value of the effective temperature, as well as the periods and growth-rates of the pulsation modes, do not necessarily equal the values given by a linear model of the same star.

According to the various hydrodynamical calculations performed up to now (Wood 1974; Tuchman et al. 1979; Perl & Tuchman 1990; Tuchman 1991; Ya'ari & Tuchman 1996, 1999; Bessell et al. 1996) the static effective temperature may differ from the mean $T_{\rm eff}$ of the corresponding pulsating star by plus or minus 1-5%.

On the other hand, some recent calculations (Ya'ari & Tuchman 1996, 1999) have shown that the period of the nonlinear fundamental mode may, after thermal relaxation, be shorter than the LNA value by as much as 35% (depending, at least, on the luminosity). But models based on Wood's code (Wood 1995; Bessell et al. 1996; Hofmann et al. 1998) predict only a small increase of the fundamental period, even after spontaneous thermal relaxation (models P, M and O of Hofmann et al. 1998). As stated above, the main difference between these two families of models are the treatment of time-dependent convection and the equation of state, which both have important thermal effects. As phase-lagged convection tends to increase the nonadiabaticity of the pulsation, it is likely that the EOS is the main cause of the more "quiet'' behaviour of Wood's models (as long as the numerical scheme is not at stake).

4.3 Outer boundary

Ya'ari & Tuchman (1996) report that their nonlinear results appear basically insensitive to the various outer boundary conditions that they have tried. However, all abovementioned models neglect the fact that dust condensates in the levitating circumstellar layers and, as an effect of radiation pressure, generates a significant stellar wind, ranging from 10^-8 to $10^{-6}~M_\odot/$yr for Miras and semi-regulars (Jura 1986; Jura 1988; Jura et al. 1993; Kerschbaum & Hron 1992). The physics of this phenomenon was extensively described by Fleischer et al. (1992) and Höfner & Dorfi (1997).

Due to the extension of the envelope, the outgoing waves are only partially reflected in the photospheric region, and this does not occur at a fixed level but at the sonic point, which depends on the wind. Pijpers (1993) has shown, in the case of polytropic AGB star models, that the adiabatic fundamental period may increase by more than a factor 5 if a mass-loss rate of $10^{-6}~M_\odot$/yr is assumed. It would be surprising if such a large shift of the adiabatic periods had no effect on the nonadiabatic ones. Even though polytropic models are just a rough approximation, one may also expect some effects of this kind in real stars.

4.4 Temperature scale

The nonlinear shape of the temperature-colour relations is another source of difficulty: it may generate a significant mismatch between the mean colour index and the colour corresponding to the mean temperature, depending on the latter and on the pulsation amplitude.

It is thus clear that a linear model having the fundamental parameters of a given LPV (in particular its static temperature), the correct mixing length, and predicting the correct period, will nevertheless disagree with the observed colours by a significant amount. In order to estimate the overall colour mismatch ( $<V-K>-(V-K)_{\rm stat}$, $<J-K>-(J-K)_{\rm stat}$) that may be expected we have applied the above-defined temperature-colour relations to simulated temperature oscillations based on the dynamical models of Tuchman et al. (1979), Bessell et al. (1989), Bessell et al. (1996), Ya'ari & Tuchman (1996) and Hofman et al. (1998), which have very diverse fundamental parameters and amplitudes (usually 1 $M_\odot$, but up to 6 $M_\odot$ in Tuchman et al. (1979); 1800-35000 $L_\odot$; 2250-3500 K (static); 15-40 ${\rm km\,s}^{-1}$). The results are shown in Fig. 2. The mean shifts and standard deviations are -0.15 and 0.95 for V-K (but 4/5 of the models lie within 0.7 of the mean), and -0.01 and 0.05 for J-K. No obvious correlation with any fundamental parameter or with the pulsation period or amplitude was found. It is likely that the colour mismatch depends on a non-trivial combination of these parameters.

4.5 Consequences

All this discussion leads us to conclude that no linear model grid can be expected to directly fit the observations, and that the existing nonlinear models are not yet able to provide a reliable grid, or even an a priori correction of the linear models. As a consequence, linear models must be complemented by additional free parameters, to be added prior to comparison with the observations. As a first-order approximation, these parameters can be some systematic corrections of the colour ( $\Delta(V-K)$ or $\Delta (J-K)$) and, for each pulsation mode, of the period ( $\Delta\log P$). Before trying to interpret the results of Paper I, these parameters and the mixing length have to be calibrated by fitting the models to independent data, namely the LPVs observed in globular clusters and in the LMC.

Figure 3: The $\Delta (J-K)$ colour correction parameter (dashed line), the mixing-length parameter $\alpha /2-1$ (dotted line) and the mass discrepancy of the fundamental mode in SMC-like clusters $\delta M$ (solid line) as a function of the period correction parameter $\Delta\log P_0$