6 Pulsational properties

 

 

Table 16: For every star, we indicate the number of detected intrinsic frequencies in the Geneva photometric data (GEN), the HIPPARCOS photometric data (HIPP) and in the velocity moments (SPEC), and the total number (TOT). + indicates that more intrinsic frequencies are present. Confirmed SPBs known before the HIPPARCOS mission are marked with $\ast$.

		star		GEN	HIPP	SPEC		TOT
SB2 $e \ne 0$	$\ast$	HD 123515		4+	4	2		4+
		HD 140873		1	1	1		1
SB1 $e \ne 0$		HD 24587		1	1	1		1
		HD 53921		1	1	1		1
	$\ast$	HD 74560		5+	3	2		5+
	$\ast$	HD 177863		2	2	1		2
SB1 e = 0		HD 92287		1	1	1		1
Single	$\ast$	HD 74195		4+	2	4		4+
	$\ast$	HD 181558		1+	1	1		1+
		HD 26326		2	2	2		3
		HD 85953		2	2	2		3
		HD 138764		1	2	2		2
		HD 215573		1	2	1		2

In Table 16, we give a summary of the results of the frequency analysis of the 13 confirmed SPBs. At least nine of them are multi-periodic. The origin of the variations of the apparent mono-periodic star HD 24587 is uncertain. In the figures of this section, the latter star is therefore indicated with a triangle while all the other considered target stars are indicated with circles.

   
6.1 K- and Q-values of the observed frequencies

Figure 17: The distribution of the K- and Q-values for the observed pulsation frequencies $\nu$.

In total, 30 of the observed intrinsic frequencies $\nu$ found in our sample are attributed to stellar pulsations. Their K-value is defined as the ratio of the horizontal to the vertical velocity amplitude. To a good approximation, it is given by $K \approx GM/4\pi^2\nu^2R^3$. In Fig. 17, we give the distribution of our "observed'' K-values, which were calculated from the mass and radius estimates given in Table 1 and from the observed frequencies. The more commonly known pulsation constant Q is expressed in days and is defined as $Q \equiv \sqrt{ \overline{\rho} / \overline{\rho}_{\odot} } / \nu$, where $\overline{\rho}$ and  $\overline{\rho}_{\odot}$ denote respectively the stellar and solar average density. The K- and Q-values are connected by the relation $Q~= \sqrt{K/74.41}$. The Q-values are also indicated in Fig. 17. Note that the large K- and Q-values for $\nu _1$ of HD 92287, for $\nu _2$ of HD 26326, and for $\nu _2$ of HD 177863 are not shown for clarity. For our sample of SPBs, the distribution of the observed K-values is centered around 40-50. This value is higher than the one found by De Cat (2002), who studied a much larger sample of (candidate) SPBs. The SPB-pulsations are clearly all dominated by the horizontal component, which is typical for high-order g-mode pulsations.

   
6.2 Intrinsic frequencies versus temperature

Figure 18: The observed frequency $\nu$ ( $\rm {c~d^{-1}}$) as a function of the effective temperature $T_{\rm eff}$ (K). HD 24587 is indicated with a triangle while all the other considered target stars are indicated with circles. The full symbols correspond to the main frequencies.

In Fig. 18, the effective temperatures $T_{\rm eff}$ of our SPBs are compared to their observed frequencies $\nu$. The observed frequencies of the coolest SPBs tend to be higher than those of the hotter stars, although the trend is only marginal. The cooler stars are situated in the lower part of the SPB instability strip, where "Maia stars'' are suggested to exist (Struve 1955). However, the observed periods in our coolest targets are still much longer than those suggested for the hypothetical Maia stars.

Such a temperature-frequency relation is not expected from theoretical excitation studies. According to the $\kappa$-mechanism, the pulsation modes in SPBs can only be excited if their periods are comparable to the thermal time-scale in the Z-bump zone. This thermal time-scale decreases when $T_{\rm eff}$ increases (Pamyatnykh 1998). Moreover, theoretical calculations show that modes with $\nu <$ 0.7 c/d are damped for stars with $\log T_{\rm eff} > 4.2$ (Dziembowski, private communication).

   
6.3 Photometric versus radial velocity amplitude

Figure 19: Comparison of the amplitudes of variations for the observed pulsation frequencies: Geneva B versus <v> (top), HIPPARCOS $H_{\rm p}$ versus <v> (middle), and HIPPARCOS $H_{\rm p}$ versus Geneva B (bottom). The amplitudes for HD 24587 are marked by open triangles, those of HD 92287 and HD 181558 by full stars, and the others by full circles. The error bars denote the standard errors of the amplitudes. The full line denotes in the top and middle panel a least-squares fit the considered amplitudes (HD 92287 and HD 181558 excluded), and in the bottom panel the bisector.

In Fig. 19, the observed amplitudes of the variations for all the observed intrinsic frequencies in the different kinds of data-sets are compared.

In the top and middle panel, the radial velocity amplitude AA is compared respectively to the photometric amplitude $A_{\rm B}$ and $A_{\rm H_{\rm p}}$. For most of the observed intrinsic frequencies, a clear linear relation is found between the photometric and spectroscopic amplitudes. These linear relations are represented by the full lines denoting a least-squares linear fit (HD 92287 and HD 181558 excluded). The slopes of the fits in the upper and middle panel of Fig. 19 are respectively 5.4(8) and 4.3(7) mmag s km^-1. The one frequency for which the observed amplitude ratio is significantly smaller than for the others corresponds to $\nu _1$ of HD 92287 (lower star). This might indicate a high-degree mode for which one expects small photometric, but large line profile variations. Also, we recall that this star is one of the ellipsoidal variables and so is deformed due to the strong tidal forces. The observed light-to-velocity amplitude ratios are small for all our observed pulsation frequencies (Fig. 19). This seems to contradict theoretical expectations. Indeed, theoretical 5 $M_{\odot }$ SPB models at different effective temperatures only reveal small light-to-velocity amplitude ratios for short pulsation periods (Dziembowski, private communication).

In the lower panel of Fig. 19, the amplitudes of the photometric data-sets are compared. The full line corresponds to an amplitude ratio $A_{H_{\rm p}}$/A_B = 1. Only the main frequency of HD 181558 significantly deviates from the full line (upper star).

   
6.4 Phase behaviour

In Fig. 20, the observed phases of the variations for all the observed intrinsic frequencies in the different kinds of data-sets are compared. No phase lag was observed for the variations of the SPB prototypes in the data of the seven filters of the Geneva photometry (Waelkens 1991). We confirm this result for our sample, which is much larger. Phase differences in different photometric filters and/or in different colours are therefore not a good constraint to identify the modes in SPBs.

In the top and middle panel of Fig. 20, the radial velocity phases $\phi_{AA}$ are compared respectively to the photometric phases $\phi_{B}$ and $\phi_{H_{\rm p}}$. A full line corresponding to a phase lag is $\Delta\phi= 0.25$ drawn. The phases of the observed pulsation variations fall within a narrow range around this line. Again, such a light-to-velocity phase behaviour is only expected theoretically for short pulsation periods (Dziembowski, private communication).

In the lower panel of Fig. 20, the photometric phases $\phi_{B}$ and $\phi_{H_{\rm p}}$ are compared. A full line is drawn for $\Delta\phi= 0$. Given the absence of phase lags between the variations in the different Geneva magnitudes, we do not expect phase lags with the HIPPARCOS magnitudes either. This is generally the case for the observed variations in our SPB sample.

We stress that the frequencies displayed in Figs. 18-20 are the observed intrinsic frequencies and not the intrinsic frequencies in the corotating frame which are considered in theoretical models. Therefore, our observational results are not necessarily in contradiction with the theory.