3 Pulsational model calculations

Oscillations in hot sdB stars have now been well established by observations (see O'Donoghue et al. 1998 and references therein). Potentially, this allows analysis of the internal structure of sdB stars by comparison of the observed frequencies with those corresponding to stellar models with different physical assumptions. Some work has already been done in this direction (see, e.g., Charpinet et al. 1997; Billeres et al. 1998; or Ulla et al. 1999).

Here we compare the frequency peaks reported in the previous section and those found by Piccioni et al. (2000) with theoretical frequencies based on stellar structure models compatible with the surface parameters of PG 0856+121. We have computed stellar structure models of different masses, suitable for the sdB star PG 0856+121. The equation of state, opacity and nuclear reactions are briefly described in Jiménez & MacDonald (1996). An additional change is the use of OPAL95 opacity tables (Iglesias & Rogers 1996). These models have helium cores and thin H-rich envelopes. A summary of the models here considered is given in Table 2. To produce surface abundances similar to those in PG 0856+121, we have included gravitational settling and element diffusion (Iben & MacDonald 1985) in models 4, 6 and 8. The envelope compositions for models 1 and 2 are X=0.71, Y=0.29, Z=0.0001and for models 3, 5 and 7 X=0.60, Y=0.38, Z=0.02. For the models with diffusion the initial envelope composition is also X=0.60, Y=0.38, Z=0.02. Gravitational settling causes helium and heavy elements to quickly sink below the photosphere. The outer layers are then pure hydrogen. The fact that $n({\rm He})/n({\rm H}) = 0.01$ in PG 0856+121 and other sdBs can be explained by the presence of a wind that counters the effects of gravitational settling. The wind mass loss rates required to do this are quite small (10^-15- $10^{-14}~M_\odot$/year) and completely undetectable up to date with ordinary techniques and instrumentation.

 

 

Table 2: Properties of sdB models 1 through 8, suitable for PG 0856+121

Model	Mass	log g	Envlp. Mass	Radius	Central $\rho$	$T_{\rm eff}$	Luminosity
nr.	$M_{\odot}$		$M_{\odot}$	x109 cm	x104 gr/cm3	x104 K	$L_\odot$
1	0.3999	5.843	3.18 10-3	8.724	3.329	2.641	6.891
2	0.4517	5.644	6.46 10-3	11.69	2.550	2.642	12.42
3	0.5563	5.401	6.1 10-3	17.12	1.699	2.651	26.94
4	0.5563	5.340	5.0 10-3	18.37	1.699	2.560	26.97
5	0.7941	4.862	2.43 10-2	38.05	0.941	2.696	142.4
6	0.7941	4.821	2.07 10-2	39.88	0.941	2.644	144.7
7	1.0608	4.390	5.06 10-2	75.75	0.611	2.629	510.6
8	1.0608	4.312	4.53 10-2	82.83	0.612	2.541	532.7

The models eigenfrequencies were computed in the adiabatic approximation, using the code developed by Christensen-Dalsgaard (see Christensen-Dalsgaard & Berthomieu 1991). For models with similar internal structure, as in the present case, the dynamical time scale $t_{\rm dyn}=(R^3/GM)^{1/2}$ dominates the variation of the oscillation frequencies. Thus it is convenient to compare our results in terms of dimensionless frequencies $\sigma$, defined by

$$\sigma \equiv \left(\frac{R^3}{GM}\right)^{1/2} \omega \; ,$$ (1)

where $\omega$ is the angular oscillation frequency and the other symbols have their conventional meanings.

In Fig. 4 we show the theoretical dimensionless frequencies, computed by using Eq. (1). Only modes with $\ell\leq 2$ are considered since modes of higher degree can hardly be observed for point-like stars. For the model with $0.4~M_{\odot}$, the 6 p-mode frequencies shown in Fig. 4 correspond to the fundamental and first overtone of the $\ell =0,1,2$degrees. For other models, the p-mode spectrum is more complex due to the presence of g-like modes. As it can be seen in the figure, the dimensionless frequencies decrease with mass and increase when diffusion is considered.

Figure 4: Dimensionless frequencies for PG 0856+121. Only 5 models, of those in Table 2, are shown for clarity

The dimensionless frequencies corresponding to the observational periods are usually estimated by expressing $\sigma$ in terms of $T_{\rm eff}$, $\log g$ and the luminosity L of the stars. However, for this kind of star it seems better to use some estimate of the mass rather than of L. In fact, the distance quoted in Table 1 was obtained by assuming the canonical mass for sdB stars. The relations are

$$\sigma= \left( \frac{L}{4\pi\sigma_{\rm SB}}\right)^{1/4} \fr... ...g^{1/2}T_{\rm eff}} = \left(\frac{GM}{g^3}\right)^{1/4} \omega$$ (2)

where $\sigma_{\rm SB}$ is the Stefan-Boltzmann constant.

In particular, for PG 0856+121, we have used the value $\log g = 5.73 \pm 0.15$ (Saffer et al. 1994). For the mass we use the canonical value $0.5\pm 0.1~M_{\odot}$ (Saffer et al. 1994). In addition, the observational frequencies have associated errors, but they are negligible when compared to those of $\log g$ and M. The resulting dimensionless frequencies for the observed frequency peaks reported here are shown at the top of Fig. 4 as horizontal continuous lines. Use of the results of Piccioni et al. (2000) instead of those shown in the figure, does not change the following discussion. The horizontal dashed lines in Fig. 4 were obtained by assuming the same uncertainty in $\log g$ as quoted above but for a mass range of $0.4~M_{\odot} < M < 1~M_{\odot}$. This allows us to explore the possibility that PG 0856+121 has a mass substantially larger than the canonical one.

In the previous analysis the frequency splittings caused by rotation have been neglected. Although we do not know the rotational velocity of this particular star, we shall consider the value 90 kms^-1 as an upper limit. This follows from the work by Saffer et al. (1994; and a private communication), who measured this quantity for about 50 sdB stars and none was found to be rotating faster than about 90 kms^-1. Then, by using the stellar radii given in Table 2, it can be seen that the value of the rotational frequency is, at most, $5\%$ that of the observational frequencies and, hence, first order corrections for the frequency splitting will be enough for our purposes. By using the values of M and R in Table 2, a rotational frequency splitting $\beta_{\rm nl} m \sigma_{\rm rot}$ (here m is the azimuthal order, $\sigma_{\rm rot}$ the dimensionless rotational angular frequency and the parameter $0\leq\beta_{\rm nl}\leq 1$) smaller than 0.25 is found for modes with $\ell\leq 2$. Considering this additional uncertainty in Fig. 4 also does not change the conclusions given below. From Fig. 4 it follows that the peak at 3.3 mHz can be either a g mode or a p mode. Since in other sdB stars only p modes are detected (see e.g. Billeres et al. 1998; Koen et al. 1998b) in agreement with the theoretical expectations of Fontaine et al. (1998) for the EC14026 stars, the latter possibility can be considered with preference. In this case, and assuming the canonical mass, the peak at 3.3 mHz would be a fundamental p-mode with degree $\ell=0,1$, or 2. On the other hand, if any of the peaks at 1.9 and 2.1 mHz are real and the photometric value of $\log g$ is correctly determined, from Fig. 4 it follows that these peaks must be g-modes of low order. It is important to note that this conclusion does not depend on the details of the model structure, but only on the stellar parameters. The basic reason is that the dimensionless frequencies $\sigma$ of the p-modes are, in a first approximation, independent of such details.