3 Comparison of the observations with Mira star models

In this section we compare our monochromatic radius observations with monochromatic radii predicted by Mira models. Since the wavelength dependence of the stellar radius sensitively depends on the model-predicted structure of the Mira atmosphere (Bessell et al. 1989, BSW96, HSW98), this comparison should give some hint whether any of the models is a fair representation of R Leo. We discuss the linear radii and the pulsation mode obtained by adopting R Leo's HIPPARCOS parallax and the effective temperature derived from the 1045/9 measurement and the bolometric flux.

All Mira star models used in this paper are from BSW96 (D and E series) and from HSW98 (P, M and O series). They are meant as possible representations of the prototype Mira variable o Ceti, and hence have periods Pvery close to the 332 day period of this star; they differ in pulsation mode, assumed mass M and assumed luminosity L; and the BSW96 models differ from the (more advanced) HSW98 models with respect to the pulsation modelling technique. Solar abundances were assumed for all models. The five models represent stars pulsating in the fundamental mode (f; D, P and M models) or in the first-overtone mode (o; E and O models). Table 5 lists the properties of these Mira model series ($R_{\rm p}$ = Rosseland radius, i.e., distance from the non-pulsating "parent star's" center at which the Rosseland optical depth $\tau_{\rm Ross}$ equals unity; $T_{\rm eff} \propto (L/R_{\rm p}^2)^{1/4}$ = effective temperature). We compare predictions of these models at different phases and cycles with our observations (Table 6; arbitrary numbering of model cycles). The correlation of bolometric model phases with visual phases was taken from Lockwood & Wing (1971) and Lockwood (1972) who discuss the observed light curves of Mira variables in their 104 filter (1035 nm/13 nm) which closely matches the bolometric light curve.

  

 

Table 5: Properties of Mira model series (see text).

Series	Mode	P/day	$M/M_{\odot}$	$L/L_{\odot}$	$R_{\rm p}/R_{\odot}$	$T_{\rm eff}$
D	f	330	1.0	3470	236	2900
E	o	328	1.0	6310	366	2700
P	f	332	1.0	3470	241	2860
M	f	332	1.2	3470	260	2750
O	o	320	2.0	5830	503	2250

 

 

Table 6: Link between the 27 abscissa values (model-phase combinations m) in Figs. 6, 7, 9 and 10, and the models. Additionally the variability phase $\phi _{\rm vis}$, the Rosseland radius R and the 1045 nm radius $R_{1045{\rm nm}}$in units of the parent star radius $R_{\rm p}$, and the effective temperature $T_{\rm eff}(R)$associated to the Rosseland radius are given.

Model	cycle+ $\phi _{\rm vis}$	$R/R_{\rm p}$	$R_{1045}/R_{\rm p}$	$T_{\rm eff}(R)$	Abscissa
D27360	0 + 0.8	0.90	0.90	3050	1
D27520	1 + 0.0	1.04	1.04	3020	2
D27600	1 + 0.2	1.09	1.10	3010	3
D27760	1 + 0.5	0.91	0.90	2710	4
D28320	1 + 0.8	0.90	0.90	3050	5
D28760	2 + 0.0	1.04	1.05	3030	6
D28847	2 + 0.2	1.09	1.09	3000	7
D28960	2 + 0.5	0.91	0.90	2690	8
E8300	0 + 0.83	1.16	1.07	2330	9
E8380	1 + 0.0	1.09	1.09	2620	10
E8460	1 + 0.1	1.12	1.11	2760	11
E8560	1 + 0.21	1.17	1.15	2610	12
P71800	0 + 0.5	1.20	0.90	2160	13
P73200	1 + 0.0	1.03	1.04	3130	14
P73600	1 + 0.5	1.49	0.85	1930	15
P74200	2 + 0.0	1.04	1.04	3060	16
P74600	2 + 0.5	1.17	0.91	2200	17
P75800	3 + 0.0	1.13	1.14	3060	18
P76200	3 + 0.5	1.13	0.81	2270	19
P77000	4 + 0.0	1.17	1.16	2870	20
M96400	0 + 0.5	0.93	0.84	2310	21
M97600	1 + 0.0	1.19	1.18	2750	22
M97800	1 + 0.5	0.88	0.83	2460	23
M98800	2 + 0.0	1.23	1.20	2650	24
O64210	0 + 0.5	1.12	1.00	2050	25
O64530	0 + 0.8	0.93	0.91	2150	26
O64700	1 + 0.0	1.05	1.01	2310	27

Figure 4: Linear Rosseland observed and model radii. The observed linear Rosseland radii are based on the 1045/9 measurements for each model/phase combination; the model radii are calculated for the transmission curve of each filter used; Table 6 gives the link between the abscissa values and the models and their phases.

In this paper we use the conventional stellar radius definition where the monochromatic radius $R_{\lambda}$ of a star at wavelength $\lambda$ is given by the distance from the star's center at which the optical depth equals unity ( $\tau_{\lambda}=1$). In analogy, the photospheric stellar radius R is given by the distance from the star's center at which the Rosseland optical depth equals unity ( $\tau_{\rm Ross}=1$). This radius has the advantage of agreeing well (see Table 6 and the discussion in HSW98 for deviations sometimes occurring in very cool stars) with measurable near-infrared continuum radii and with the standard boundary radius of pulsation models with parameter $T_{\rm eff} \propto (L/R^2)^{1/4}$.

Figure 5: Linear observed and model stellar filter radii $R_{\rm f}$versus $\lambda$for all five models and phases close to our observations. The linear observed stellar filter radii $R_{\rm f}$ are derived from the measured visibility by using the visibilities of the model CLVs as fit functions, as described in the text. The plotted model 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). For each measurement two different symmetric error bars are plotted. The larger error bars contain both errors, the parallax error and also the speckle error. The smaller error bars (inner error bars) contain only the speckle error.

For each of our six filters (656/10, 673/8, 699/6, 754/6, 781/14 and 1045/9) we calculated the theoretical CLVs corresponding to the above mentioned five Mira models at different phases and cycles. The stellar radius for filter transmission ${f}_{\lambda}$is the intensity and filter weighted radius $R_{\rm f} = \int R_{\lambda}\,I_{\lambda}\,{f}_{\lambda}\,{\rm d}\lambda\,/\,\int I_{\lambda}\,{f}_{\lambda}\,{\rm d}\lambda$which we call stellar filter radius $R_{\rm f}$after the definition of Scholz & Takeda (1987). In this equation $R_{\lambda}$ denotes the above monochromatic $\tau_{\lambda}=1$ radius, $I_{\lambda}$ the central intensity spectrum and ${f}_{\lambda}$ the transmission of the filter. Owing to the chosen positions and narrow widths of our filters f, the $R_{\rm f}$ radii are almost monochromatic $\tau_{\lambda}=1$ radii if the molecular line structure of the TiO bands is neglected.

The observed angular stellar filter radius $R^{\rm a}_{\rm f, m}$ of R Leo corresponding to a certain filter f and model-phase combination m, was derived by a least-squares fit between the azimuthally averaged measured visibility and the visibility of the corresponding theoretical CLV. For a detailed description of the visibility fitting procedure we refer to HS98.

In the following subsections we apply CLVs predicted from all five models at phases both near our R Leo observations (phase 0.20) and, for comparison, also at other phases.

3.1 Comparison of linear observed and model radii

Since R Leo is located in our neighborhood, the radii of the Mira star can be directly compared with the predictions of Mira star models. Linear stellar R Leo radii can be obtained if we use its HIPPARCOS parallax of $9.87 \pm 2.07$ mas (ESA 1997, Whitelock & Feast 2000; see discussion in this paper on the parallax errors in the case of stars with large extended shapes).

The 1045 nm model CLV visibility was fitted to the measured 1045 nm visibility of R Leo yielding the linear monochromatic $\tau_{\lambda}=1$ radius R₁₀₄₅ and, as predicted by the specific model, the associated linear Rosseland radius R (Fig. 4). We see that the derived linear Rosseland radius values are nearly the same for all model-phase combinations, that is they depend very little on differences of CLVs of our models. The average measured R₁₀₄₅ radius is $395 \pm 92~R_{\odot}$ ( $18.2 \pm 1.9$ mas), and the average Rosseland radius is $423 \pm 99~R_{\odot}$ ( $19.4 \pm 2.0$ mas) (average over all model-phase combinations). The Rosseland radii of the E model series at all available phases (0.83, 1.0, 1.1, 1.21), of the M model series at near-maximum phases (1.0, 2.0), and of the O model at near-maximum phases (0.8, 1.0) are very close or close (within the error bars) to the Rosseland radii derived from the measured 1045 nm visibility at phase 0.20. All other model-phase combinations yield large differences (larger than the error bars).

The 1045 nm continuum model radius R₁₀₄₅ and the Rosseland model radius Raveraged over all available E model phases is 404 $R_{\odot}$ and 416 $R_{\odot}$, respectively, and the measured value (using the E model CLVs) is $399 \pm 93~R_{\odot}$ and $411 \pm 96~R_{\odot}$, respectively. The E model at phase 1.21 (closest to our observed phase 0.20) has $R_{1045} = 420~R_{\odot}$( $R = 431~R_{\odot}$), and data reduction by means of this CLV yields for R Leo the observed value of $R_{1045} = 407 \pm 95~R_{\odot}$( $R = 417 \pm 97~R_{\odot}$). Hence, the difference between model and star is smaller than the error bar.

Figure 6: Linear observed and model stellar filter radii $R_{\rm f}$for all 27 model-phase combinations m. The stellar filter radii $R_{\rm f}$are given for the six filters 656/9 (top left), 673/8 (top right), 699/6 (middle left), 754/5 (middle right), 782/13 (bottom left), and 1045/9 (bottom right). Table 6 gives the link between the abscissa values and the models and their phases. The large error bars of $\sim$20% are basically due to the parallax error; the speckle measurements with the optical filters have error bars of $\sim$5%, and with the near infrared filter $1045/9 \sim 10$% (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: model radii calculated for the transmission curve of each filter used).

Figure 5 presents all linear observed and model stellar filter radii $R_{\rm f}$as a function of wavelength for all five models and phases close to our observations (in the case of model series with several cycles the best fitting model is shown).

In Fig. 6, we compare our observed linear stellar filter radii with the model radii of all 27 model-phase combinations m. Clearly the three fundamental-mode model stars are systematically too small (by $\sim$20 to 50%), whereas most model radii of the overtone E and O series coincide within the error bars with the measured values. We see that the TiO forming layers of the Mira atmosphere may extend as far as 4 to 6 AU from the star's center. We also see from the figure that the model atmospheres are systematically more compact than the observed Mira atmosphere, i.e. the distance-independent ratio $(R_{\rm strong-TiO}-R_{\rm continuum})/R_{\rm continuum}$ is too small in the here considered models (see Appendix for details).

3.2 Pulsation mode

After the period-radius relation of Miras from Feast (1996) a fundamental mode pulsator with a period of 310 days (R Leo) and with a mass ranging between 1.0 and 1.5 $M_{\odot}$should have a linear photospheric radius ranging from 220-270 $R_{\odot}$, and a first overtone pulsator should have a linear photospheric radius ranging from 390-450 $R_{\odot}$. Therefore, the measured linear Rosseland radius of 417  $R_{\odot} \pm 97~R_{\odot}$(derived from the 1045 nm-visibility measurement and the E-model at phase close to our observation) places R Leo among the first-overtone pulsators. The question why the period-radius relation of M-type Miras indicates first-overtone pulsation whereas MACHO observations (Wood et al. 1999) and pulsation velocities (Scholz & Wood 2000) favor fundamental mode pulsation, remains open.

3.3 Effective temperature

Effective temperatures of R Leo were derived from its angular Rosseland radii $R^{\rm a}_{\rm m}$ (derived from the 1045/9-observation with all 27 above discussed models) and its bolometric flux using the relation $T_{\rm eff} = 2341~{\rm K} \times (F_{\rm bol}/\Phi^2)^{1/4}$ where $F_{\rm bol}$ is the apparent bolometric flux in units of 10^-8 ergcm^-2s^-1and $\Phi = 2~R^{\rm a}_{\rm m}$ is the apparent angular photospheric diameter in mas. Figure 7 (top) displays the angular Rosseland radii obtained from the 1045/9 observation by fitting all 27 theoretical model-phase CLV visibilities to the measured 1045/9 visibility. The average measured angular Rosseland radius is 19.4 mas (average over all model-phase combinations). Figure 7 (top) also shows that inaccuracies caused by adopting incorrect continuum limb-darkening from inadequate models rarely exceed $\sim$10% (i.e. 5% in $T_{\rm eff}$).

Figure 7: Top: Angular Rosseland radii (in mas) derived for each model-phase combination m from the 1045/9 measurements. Middle: Linear Rosseland observed and model radii. The observed linear Rosseland radii were derived for each model-phase combination m from the 1045/9 observations. Bottom: Effective model temperatures for all model-phase combinations m and effective temperatures derived from the above angular Rosseland radii (top) and the the bolometric flux of R Leo measured around phase 0.20. Table 6 gives the link between the abscissa values and the models and their phases (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 values).

For cool stars such as LPV's, where most of the luminosity is emitted at near-infrared wavelengths, a convenient method for calculating bolometric magnitudes is to use a blackbody function to interpolate between photometric measurements in the J, H, K and L bands. For estimating the bolometric flux we have used JHKL-flux measurements from P. Whitelock's (1997, private communication) R Leo observations of April 6, 1996. These observations were carried out at nearly the same variability phase (0.20) and cycle as our 1045/9 observation of April 4, 1996. The bolometric magnitude of R Leo, calculated by P. Whitelock (1997, private communication) yields $m_{\rm bol} = 0.24 \pm 0.20$ and, assuming that a zero magnitude star has a flux of $2795 \times 10^{-8}$ ergcm^-2s^-1, yields $F_{\rm bol} =(2240.69 \pm 412.30) \times 10^{-8}$ ergcm^-2s^-1. As a by-product we determined the luminosity of R Leo near maximum phase 0.20 from its bolometric magnitude and HIPPARCOS parallax to $6540 \pm 3010~L_{\odot}$, which is close to the near-maximum luminosity of the E and O model series (BSW96, HSW98).

Figure 7 (bottom) presents $T_{\rm eff}$ values derived for each of the 27 model-phase combinations from the bolometric flux and the angular Rosseland radii $R^{\rm a}_{\rm m}$ measured at near-maximum phase 0.20. Fortunately, this value depends little on the model-phase combination CLV and is about $2590 \pm 180$ K (average over all model-phase combinations). The $T_{\rm eff}$ values of the E model series at all near-maximum phases (1.0, 1.1, 1.21) and of the M model series at all near-maximum phases (1.0, 2.0) are very close (within the error bars) to the $T_{\rm eff}$ values derived from the 1045 nm visibility observation and the JHKL photometric measurement performed at the near-maximum phase 0.20. All other models (i.e. D, P and O) yield at all available near-maximum phases large differences between theoretical and measured $T_{\rm eff}$ values. The closest E model with phase 1.21 has an effective temperature of 2610 K. Application of its CLV and $m_{\rm bol} = 0.24 \pm 0.20$ (measured at phase $\sim$0.20) to our 1045/9 observation yields for R Leo an angular Rosseland radius of 19.1 mas $\pm$ 2.0 mas and an effective temperature of 2600 K $\pm$ 180 K. The average theoretical effective temperature of all near-maximum phases (1.0, 2.0) of the M model is 2700 K. The average measured effective temperature derived from the 1045/9 observation by application of the M model CLVs at the near-maximum phases 1.0 and 2.0 has a value of 2650 K $\pm$ 180 K.