5 Physical parameters

The most important physical parameters of V784 Cas were determined using the mean Strömgren colours and the accurate Hipparcos parallax. Unfortunately, there is no $\beta$ measurement for this star in the literature, therefore, we could not follow the "standard'' procedures of Strömgren photometric calibrations (e.g. Zhou et al. 2001a,b). The parallax is 9.81 $\pm$ 0.75 mas (ESA 1997) that corresponds to a distance of 102 ⁺⁸_-7 pc. Consequently, the reddening can be neglected even in this low galactic latitude region ( $b=-1.39^\circ$). The distance and mean V-brightness result in $M_{\rm V}=$ 1 $.\!\!^{\rm m}$61 $\pm$ 0 $.\!\!^{\rm m}$15. The spectral type (or, equivalently, the mean B-V= 0 $.\!\!^{\rm m}$33) implies a bolometric correction of $BC=-0\hbox{$.\!\!^{\rm m}$ }11$ (Carroll & Ostlie 1996). Thus, the bolometric absolute magnitude is $M_{\rm bol}=1\hbox{$.\!\!^{\rm m}$ }50$ $\pm$ 0 $.\!\!^{\rm m}$15 ( $L/L_\odot \approx$20), which is in accordance with the expected absolute magnitude of an evolved early F-type star. The atmospheric parameters were estimated from the mean Johnson and Strömgren colours ( $\langle b-y \rangle=0\hbox{$.\!\!^{\rm m}$ }20$, $\langle m_1 \rangle=0\hbox{$.\!\!^{\rm m}$ }17$, $\langle c_1 \rangle=0\hbox{$.\!\!^{\rm m}$ }74$) and synthetic colour grids of Kurucz (1993). (We note, that the observed uvby colours are in very good agreement with those of listed in the SIMBAD database: $\langle b-y \rangle=0\hbox{$.\!\!^{\rm m}$ }208$, $\langle m_1 \rangle=0\hbox{$.\!\!^{\rm m}$ }204$, $\langle c_1 \rangle=0\hbox{$.\!\!^{\rm m}$ }735$ - Olsen 1983). The results are: $\langle T_{\rm eff} \rangle=$ 7100 $\pm$ 100 K and ${\rm log}~g=$ 3.8 $\pm$ 0.1. For the given luminosity and corresponding solar values ( $M_{\rm bol, \odot}= 4\hbox{$.\!\!^{\rm m}$ }75$, $T_{\rm eff, \odot}=$ 5770 K, Allen 1976), the calculated stellar radius is 2.9 $\pm$ 0.3 $R_\odot$, while the estimated mass is 2.0 $\pm$ 0.8 $M_\odot$. Again, we find such stellar parameters that are typical for evolved main-sequence or giant stars (see, e.g., Appendix E in Carroll & Ostlie 1996). In summary, therefore, we adopt

$$\begin{eqnarray*}&& M_{\rm V}=1\hbox{$.\!\!^{\rm m}$ }61\pm0\hbox{$.\!\!^{\rm m}... ...\rm dex} \\ && R/R_\odot=2.9\pm0.3 \\ && M/M_\odot=2.0\pm0.8. \end{eqnarray*}$$

We can draw some constraints on the stellar mass and age using the evolutionary tracks from Claret (1995) for solar abundances. They are plotted in Fig. 9 with the corresponding position of V784 Cas in the ${\rm log}~T_{\rm eff}-{\rm log}~L$ plane. Two models with ${\rm log}~M[M_\odot]=$ 0.25-0.30 are closest to the stellar error box and they result in an evolutionary mass of $1.89\pm0.11~M_\odot$. The corresponding ages range from 1.03 to 1.65 Gyr.

Figure 8: Asymmetric metallic line profiles of V784 Cas. The bottom spectrum is shown for comparison. The labels of the variable star spectra mean $Hel.JD_{\rm obs}-2451480$.

The given parameters permit calculation of the pulsational constants of the determined frequencies. The equation

$$\begin{eqnarray*}{\rm log}~Q=-6.456+{\rm log}~P+0.5{\rm log}~g+0.1 M_{\rm bol}+{\rm log}~T_{\rm eff} \end{eqnarray*}$$

was used in terms of four observables (Breger et al. 1993). For the adopted set of four frequencies we have calculated the pulsational constants listed in Table 4 (assuming 20% uncertainty). We have also determined amplitude and phase relations for the given frequencies utilizing Strömgren data obtained in 2001. By fixing the frequencies but allowing their amplitudes and phases to vary for different wavebands, we could draw some contraints on the pulsation modes (see a description of the discrimination procedure in Garrido et al. 1990). For the given parameters, it seems to be clearly established the radial character of f₁ and the non-radial character of f₅ (possibly l=2). We can say nothing definitive for f₂ and f₃because errors are higher than values, but the small values seem to indicate l=1modes. We note, that when the rotation is high enough the mode identification becomes unclear (see Daszynska-Daszkiewicz et al. 2002).

The parameters and pulsation pattern outlined above suggest V784 Cas to be an evolved $\delta$ Scuti-type variable star with a mixture of radial plus non-radial modes. The star is presently about 1.4 mag brighter than main sequence stars of the same spectral type (e.g. Carroll & Ostlie 1996) and its position above the main-sequence does not contradict the luminosity class III determined by Gray et al. (2001). The star is located on the HR diagram about halfway between the theoretical Blue Edge for radial overtones and the empirical Red Edge (see Fig. 1 in Breger & Pamyatnykh 1998). Furthermore, comparing V784 Cas with the evolved Am stars in Fig. 5 of Rodríguez & Breger (2001), its position is also in agreement with the weak Am nature suggested spectroscopically by Gray et al. (2001). The temperature, surface gravity, mass and luminosity give a consistent picture compared with the standard evolutionary models used by Breger & Pamyatnykh (1998). The determined physical parameters place V784 Cas in that region where no fast evolutionary period changes are expected. Therefore, the slight phase shift of $\approx$-0.1 between the Hipparcos data and our observations (see Fig. 1) may indicate either more and yet undetected pulsational frequencies or non-evolutionary period change due to, for instance, light-time effect in a binary system (Kiss & Szatmáry 1995).

Figure 9: Evolutionary tracks (Claret 1995) and location of V784 Cas in the HR diagram (thick box).