4 Identification of evolution phase and pulsation mode

In order to compute the evolution phase and identify the pulsation modes of HR 5437, we used a standard stellar structure and evolution code which was developed by Luo (1991, 1997) and a classical adiabatic oscillation program in which the linear oscillation theory was developed by Li (1990, 1994). Henyey method, the equation of state developed by Luo (1994a, 1994b, 1997) and the latest version of the OPAL opacity tables (Iglesias & Rogers 1996) were used in the evolution code. And H and He burning for evolution was considered. First, we used Luo's code to calculate the evolution sequences of a star with 1.0-2.0 solar mass for an initial composition X=0.68, Z=0.02 and the mixing length $\alpha=1.0$. We calculated 11 evolution sequences from 1.0 to 2.0 $M_{\hbox{$\odot$}}$ in step of 0.1 $M_{\hbox{$\odot$}}$. Each evolution sequence composed of 220 evolution phases. Using Li's program (1990, 1994) we then calculated all pulsation modes of each evolution phase. Finally, we found two frequencies: 11.3357 cd^-1 with radial p mode (l=0, n=4) and 7.0235 cd^-1 with nonradial p mode (l=1, n=1) in the 140th evolution phase of the evolution sequence of a star with 1.9 $M_{\hbox{$\odot$}}$. These two calculated frequencies were the same values as the observed frequencies within the errors. The evolutionary track of a 1.9 $M_{\hbox{$\odot$}}$ star is shown in Fig. 3. The comparison between the observation of HR 5437 and theoretical calculation of a 1.9 $M_{\hbox{$\odot$}}$ star is listed in Table 3, where C is the calculated value, O is the value obtained from the observation, $\vert{\rm O}-{\rm C}\vert$ is the deviation between observation and theoretical calculation, f₁ is the first frequency, and f₂ is the second frequence. From Table 3 we can see the deviations are very small. It is only about 0.01 for the first frequency and 0.04 for the second one. l, n is the calculated order and degree of a pulsation mode, respectively. So, we tend to suggest that HR 5437 pulsates in two modes: radial p mode with l=0, n=4 and nonradial p mode with l=1, n=1.

  

Table 3: The comparison of pulsation frequencies and parameters between observation of HR 5437 and theoretical calculation of a star with 1.9 solar mass

In order to check the reliability of pulsation modes and evolution we used the absolute bolometric magnitude $M_{\rm bol}=1.25$ mag and effective temperature $T_{\rm eff}=7900$ K which were derived from the Strömgren $uvby{\rm H}_{\beta}$photometric data of HR 5437 and used some calibration formulas (Li & Jiang 1992). And, we know that the absolute bolometric magnitude of the sun $M_{\rm bol\hbox{$\odot$}}$ is 4.75 mag. Putting $M_{\rm bol}=1.25$ mag and $M_{\rm bol\hbox{$\odot$}}=4.75$ mag into the basic formula $M_{\rm bol}-M_{\rm bol\hbox{$\odot$}}=-2.5\log L/L_{\hbox{$\odot$}}$ we obtained the luminosity of HR 5437 in solar unit $\log L/L{\hbox{$\odot$}}=1.40$. And the logarithm of effective temperature equals 3.90: $\log T_{\rm eff}=3.90$. We found that the deviations between these two values and calculated values in the 140th evolution phase of the evolution sequence of a star with 1.9 $M_{\hbox{$\odot$}}$ are small. The comparison between theoretical calculation and observation is listed in Table 3. The observed position of HR 5437 on the evolution path is shown in Fig. 3 with an asterisk. It places exactly on the modeling evolution sequence curve.

  

Figure 3: The standard evolutionary track of the 1.9 $M_{\hbox{$\odot$}}$ model with an initial chemical composition X=0.68, Z=0.02. The observed position of HR 5437 is described with an asterisk. The ordinate is luminosity and the abscissa is effective temperature