5 Single SPBs

We now present the results of the frequency procedure for the single stars.

   
5.1 Confirmed SPBs

5.1.1 HD 74195 - HR 3447 - HIP 42536

Figure 10: Same as Fig. 2, but for the accepted intrinsic frequencies $\nu _1$ = 0.35745(9)  $\rm {c~d^{-1}}$, $\nu _2$ = 0.35033(9)  $\rm {c~d^{-1}}$, $\nu _3$ = 0.34630(9)  $\rm {c~d^{-1}}$, and $\nu _4$ = 0.39864(9)  $\rm {c~d^{-1}}$ of HD 74195.

 

 

Table 10: Same as Table 2, but for the accepted intrinsic frequencies of HD 74195.

data-set		$\sigma$	Ai	$\phi_i$	$\sigma _{\rm res}$	$\sigma _N$
$\nu _1$ = 0.35475(9)  $\rm {c~d^{-1}}$
<v>4128	( $\rm {km~s^{-1}}$)	3.14	3.16(21)	0.48(1)
B	(mmag)	16.6	15.1(5)	0.168(5)
$H_{\rm p}$	(mmag)	15.3	18.3(9)	0.170(7)
$\nu _2$ = 0.35033(9)  $\rm {c~d^{-1}}$
<v>4128	( $\rm {km~s^{-1}}$)		2.00(23)	0.13(2)
B	(mmag)		12.2(5)	0.868(6)
$H_{\rm p}$	(mmag)		11.0(7)	0.883(13)	5.2	3.4
$\nu _3$ = 0.34630(9)  $\rm {c~d^{-1}}$
<v>4128	( $\rm {km~s^{-1}}$)		1.58(21)	0.64(2)
B	(mmag)		3.8(5)	0.22(2)
$\nu _4$ = 0.39864(9)  $\rm {c~d^{-1}}$
<v>4128	( $\rm {km~s^{-1}}$)		1.32(24)	0.35(3)	1.35	0.45
B	(mmag)		3.0(5)	0.95(3)	8.8

HD 74195 ( m_V = 3.63) is one of the seven SPB prototypes, for which five intrinsic frequencies were reported by Waelkens (1991), i.e. $\nu _{1,{\rm p}}$ = 0.35751  $\rm {c~d^{-1}}$, $\nu _{2,{\rm p}}$ = 0.35035  $\rm {c~d^{-1}}$, $\nu _{3,{\rm p}}$ = 0.39856  $\rm {c~d^{-1}}$, $\nu _{4,{\rm p}}$ = 0.36048  $\rm {c~d^{-1}}$, and $\nu _{5,{\rm p}}$ = 0.22624  $\rm {c~d^{-1}}$. Our data-sets reveal at least four intrinsic periods: $\nu _1$ = 0.35745(9)  $\rm {c~d^{-1}}$, $\nu _2$ = 0.35033(9)  $\rm {c~d^{-1}}$, $\nu _3$ = 0.34630(9)  $\rm {c~d^{-1}}$, and $\nu _4$ = 0.39864(9)  $\rm {c~d^{-1}}$ (Fig. 10). $\nu _3$ was not detected before. Variations with $\nu _1$ and $\nu _2$ are present in all our data-sets. Since strong aliasing occurs, it would not have been possible to identify $\nu _3$ and $\nu _4$ unambiguously with the Geneva data only. Indeed, evidence for the reality of $\nu _3$ and $\nu _4$ is found in the velocity variations. After prewhitening with $\nu _1$ and $\nu _2$, $\nu _3$ becomes the best candidate in <v> and <v³>. In the original <v²>-data, the peaks of $\nu _2$ and $\nu _4$ are equally strong. Variations with $\nu _1$ only become important after prewhitening with one of these two frequencies. After subsequent prewhitening with $\nu _2$ and $\nu _1$, variations with $\nu _4$ are still present. Variations with $\nu _3$ and $\nu _4$ are not found in the $H_{\rm p}$ data. After prewhitening with $\nu _1$, ..., $\nu _4$, 1.25291(9)  $\rm {c~d^{-1}}$ is the only frequency whose amplitude reaches the 1% FAP-level in the Geneva data. We propose it as a candidate frequency which still needs further observational verification.

In Table 10, we give an overview of the characteristics of the frequencies, which have similar values. We find $\nu _2$-$\nu _3$ = 0.00403  $\rm {c~d^{-1}}$ and $\nu _1$-$\nu _2$ = 0.00712  $\rm {c~d^{-1}}$ $\approx$ 2($\nu _2$-$\nu _3$), which suggests that $\nu _1$, $\nu _2$, and $\nu _3$ are members of a multiplet with $l~\geq~2$. This suggestion would be compatible with a rotational frequency $\nu_{\rm rot}$ $\approx$ 0.004  $\rm {c~d^{-1}}$ and hence an equatorial rotation velocity of only 0.9  $\rm {km~s^{-1}}$. This is very small but not in contradiction with the moderate total line broadening (Aerts et al. 1999) as there is considerable pulsational broadening in this star.

5.1.2 HD 181558 - HR 7339 - HIP 95159

HD 181558 ( m_V = 6.24) is one of the seven SPB prototypes discussed by Waelkens (1991). The 154 data points he had at his disposal already revealed three intrinsic frequencies: $\nu _{1,{\rm p}}$ = 0.80783  $\rm {c~d^{-1}}$, $\nu _{2,{\rm p}}$ = 0.41761  $\rm {c~d^{-1}}$, and $\nu _{3,{\rm p}}$ = 0.47365  $\rm {c~d^{-1}}$. However, $\nu _{2,{\rm p}}$ and $\nu _{3,{\rm p}}$ could not be distinguished without ambiguity from their ($1-\nu$) aliases. Our data-sets of HD 181558 are all dominated by variations with $\nu _1$ = 0.80780(10)  $\rm {c~d^{-1}}$ (Fig. 11). $\nu _1$ corresponds to $\nu _{1,{\rm p}}$ as found by Waelkens (1991). This frequency already reduces 70% up to 95% of the variance in our different data-sets.

After prewhitening the data in the Geneva filters with $\nu _1$, the search algoritms lead to two aliasses, $\nu _2$ = 0.80721(10)  $\rm {c~d^{-1}}$ and $\nu _2$' = 0.80760(10)  $\rm {c~d^{-1}}$, whose amplitude is five times smaller than the amplitude $\nu _1$. Their relative importance depends on the considered filter. None of the two is found in the subset of Waelkens (1991), nor in our other data-sets. Both candidates are very close to $\nu _1$. This might indicate that the main frequency varies in time or that we are dealing with two real frequencies that were not yet separable in the data-set of Waelkens (1991). Since the frequency separation between $\nu _1$ and $\nu _2$' is very close to $1/T^{*} \approx 0.00018$  $\rm {c~d^{-1}}$ (Table 2 of Paper I), the exact value of these candidate frequencies should be treated with care (Loumos & Deeming 1978). We investigated Scargle periodograms of sub-sets composed by omitting the Geneva B data of one or more observation runs after prewhitening with $\nu _1$. The position of the peaks in the modified Scargle periodogram does not change if the time-span of the sub-set remains the same as the time-span of the original data-set, while omitting an observation run at the beginning and/or the end of the original data-set does lead to minor shifts in frequency (Fig. 12). If the time-span is small enough, the frequency peaks corresponding to $\nu _2$ and $\nu _2$' disappear. Note that all the modified Scargle periodograms shown in Fig. 12 are symmetric around $\nu _1$, although the frequency peaks corresponding to the higher-frequencies side have a somewhat lower power. If the original data-set is prewhitened with a main frequency $\nu _1$' which slightly differs from $\nu _1$, the periodogram remains symmetric around the prewhitened frequency, but the frequency peak closest to $\nu _1$ becomes higher.

Figure 11: Same as Fig. 2, but for the accepted intrinsic frequency $\nu _1$ = 0.80780(10)  $\rm {c~d^{-1}}$ of HD 181558.

Figure 12: Comparison of the Scargle periodogram of the Geneva B-filter of HD 181558 after prewhitening with $\nu _1$. 0.80721  $\rm {c~d^{-1}}$ and 0.80760  $\rm {c~d^{-1}}$ are indicated by dashed lines. The bold periodogram corresponds to the original data-set with a time-span of T* = 5544 d. Left: Periodograms corresponding to sub-sets for which T* remains the same. Right: Periodograms corresponding to sub-sets with a decreasing time-span T*from bottom to top of respectively 5321 d, 4903 d, 4569 d and 1459 d. The latter corresponds to the data-set used by Waelkens (1991).

 

 

Table 11: Same as Table 2, but for the accepted intrinsic frequencies of HD 181558.

data-set		$\sigma$	Ai	$\phi_i$	$\sigma _{\rm res}$	$\sigma _N$
$\nu _1$ = 0.80780(10)  $\rm {c~d^{-1}}$
<v>4130	( $\rm {km~s^{-1}}$)	5.31	6.64(21)	0.259(6)	0.85	0.18
B	(mmag)	25.2	31.9(8)	0.979(4)	10.2
$H_{\rm p}$	(mmag)	17.7	24.9(19)	0.924(10)	9.2	6.2

Although we cannot decide upon the reality of $\nu _2$ and $\nu _2$', we did continue the frequency analysis in order to be able to test the significance of $\nu _{2,{\rm p}}$ and $\nu _{3,{\rm p}}$. Two different prewhitening schemes were applied: (1) After prewhitening with $\nu _2$, $\nu _3$ = 0.80182(10)  $\rm {c~d^{-1}}$ and $\nu _4$ = 0.58617(10)  $\rm {c~d^{-1}}$ are found. They are aliasses of respectively $\nu _2$' and $\nu _{2,{\rm p}}$. (2) After prewhitening with $\nu _2$', $\nu _3$' = 0.80724(10)  $\rm {c~d^{-1}}$ $\approx$ $\nu _2$ and $\nu _4$' = 0.58615(10)  $\rm {c~d^{-1}}$$\approx$ $\nu _4$ are found. In both cases, indications for 0.47356(10)  $\rm {c~d^{-1}}$ are found afterwards. Both sets of frequencies, {$\nu _1$, $\nu _2$, $\nu _3$, $\nu _4$} and {$\nu _1$, $\nu _2$', $\nu _3$', $\nu _4$'}, reduce the variance by the same amount. With our current data-set, we cannot decide upon the reality of $\nu _2$ nor $\nu _2$', and hence upon the reality of the other candidates.

After prewhitening the $H_{\rm p}$ data with $\nu _1$, the residual standard deviation $\sigma _{\rm res}$ = 9.2 mmag is still large and points towards multi-periodicity. However, no other intrinsic frequencies can be determined without ambiguity. The Scargle periodograms of the velocity moments after prewhitening with $\nu _1$ do not contain amplitudes reaching the 1% FAP-level. An overview of the characteristics of the accepted intrinsic frequency is given in Table 11.

   
5.2 HIPPARCOS SPBs

5.2.1 HD 26326 - HR 1288 - HIP 19398

Figure 13: Same as Fig. 2, but for the 3 accepted intrinsic frequencies $\nu _1$ = 0.5338(8)  $\rm {c~d^{-1}}$, $\nu _2$ = 0.1723(8)  $\rm {c~d^{-1}}$, and $\nu _3$ = 0.7626(10)  $\rm {c~d^{-1}}$ of HD 26326.

Before the launch of the HIPPARCOS mission, HD 26326 ( m_V = 5.43) was known as a suspected variable. Thanks to our data-sets, the photometric and spectroscopic variability of HD 26326 is beyond any doubt. Most of our data-sets of HD 26326 lead to the same first intrinsic frequency: $\nu _1$ = 0.5338(8)  $\rm {c~d^{-1}}$ (Fig. 13, upper panel). Only for <v²>, the presence of $\nu _1$ is not clear, but no other frequency is present anyway.

After prewhitening the $H_{\rm p}$ data with $\nu _1$, many peaks in the modified Scargle periodogram still exceed the 1% FAP-level (Fig. 13, middle panel). The three best overall candidates resulting from the periodograms and $\theta$-statistics are 1.4458(8)  $\rm {c~d^{-1}}$, 0.1723(8)  $\rm {c~d^{-1}}$ and 0.5090(8)  $\rm {c~d^{-1}}$, but it is hard to decide upon their reality. Combining these results with those of <v>, however, allows us to conclude that $\nu _2$ = 0.1723(8)  $\rm {c~d^{-1}}$ is the true second intrinsic frequency since the amplitudes of <v> are maximised by a peak at $\nu _2$ (Fig. 13, middle panel). In the Geneva photometry, no evidence for $\nu _2$ is found. We therefore accept $\nu _2$ as an intrinsic frequency in the $H_{\rm p}$ and spectroscopic data only. After additional prewhitening, no other intrinsic frequencies could be found in either of the data-sets.

In the Geneva data, the best candidates for a second intrinsic frequency are the aliasses 0.2397(10)  $\rm {c~d^{-1}}$, 0.7629(10)  $\rm {c~d^{-1}}$ and 1.7632(10)  $\rm {c~d^{-1}}$. We do note that 0.7638(8)  $\rm {c~d^{-1}}$ was one amongst many frequencies with an amplitude exceeding the 1% FAP-level while searching for the second frequency in the $H_{\rm p}$ magnitudes. Given the frequency resolution and the totally different alias patterns of both data-sets, we conclude that $\nu _3$ = 0.7629(10)  $\rm {c~d^{-1}}$ is the true second intrinsic frequency in the Geneva photometry. However, further observational verification is desirable. After prewhitening with $\nu _3$, there are no longer frequencies with an amplitude reaching the 1% FAP-level in the residual Geneva variations. In Table 12, we give an overview of the characteristics of the accepted intrinsic frequencies in the different data-sets of HD 26326.

 

 

Table 12: Same as Table 2, but for the accepted intrinsic frequencies of HD 26326.

data-set		$\sigma$	Ai	$\phi_i$	$\sigma _{\rm res}$	$\sigma _N$
$\nu _1$ = 0.5338(8)  $\rm {c~d^{-1}}$
<v>4128	( $\rm {km~s^{-1}}$)	2.04	1.80(22)	0.36(2)
B	(mmag)	12.2	8.9(12)	0.99(2)
$H_{\rm p}$	(mmag)	11.9	10.8(11)	0.09(2)
$\nu _2$ = 0.1723(8)  $\rm {c~d^{-1}}$
<v>4128	( $\rm {km~s^{-1}}$)		1.49(22)	0.03(2)	1.21	0.43
$H_{\rm p}$	(mmag)		6.1(11)	0.67(3)	6.5	5.0
$\nu _3$ = 0.7629(10)  $\rm {c~d^{-1}}$
B	(mmag)		8.0(11)	0.26(2)	9.2

5.2.2 HD 85953 - HR 3924 - HIP 48527

Figure 14: Same as Fig. 2, but for the 3 accepted intrinsic frequencies $\nu _1$ = 0.2663(6)  $\rm {c~d^{-1}}$, $\nu _2$ = 0.2189(6)  $\rm {c~d^{-1}}$, and $\nu _3$ = 0.2353(7)  $\rm {c~d^{-1}}$ of HD 85953.

HD 85953 ( m_V = 5.93) is one of our target stars for which no spectroscopic variations were detected before the start of our project. Most of our data-sets of HD 85953 agree upon the first frequency: $\nu _1$ = 0.2663(6)  $\rm {c~d^{-1}}$ (Fig. 14, upper panel). In the variations of <v>, also two p-type candidate frequencies, 8.2859(6)  $\rm {c~d^{-1}}$ and 9.2890(6)  $\rm {c~d^{-1}}$, were found. Note that the latter is rather close to the $\beta$ Cephei-type frequency reported by Jakate (1979). These candidates, however, are both alias frequencies of $\nu _1$, from which we are tempted to conclude that the variability reported by Jakate was misinterpreted by him in terms of a p-mode. However, HD 85953 is situated within the common part of the instability domains of the $\beta$ Cephei stars and the SPBs (Waelkens et al. 1998). Since our current data-set is not suited for the detection of p-mode variability, we can not exclude that p- and g-modes are simultaneously excited in this star.

As $\nu _2$ = 0.2189(6)  $\rm {c~d^{-1}}$ is found as a candidate in the residual variations of the both the $H_{\rm p}$ magnitude and the Geneva we accept it as second frequency although the amplitude of the corresponding variations is low. After prewhitening with $\nu _2$, no additional frequencies can be detected in the photometric observations. For the HIPPARCOS photometry, $\sigma$ is reduced to the $\sigma _N$ level (Table 13).

The second velocity moment <v²> is the only data-set for which no signature of $\nu _1$ is found. For both Si II lines, <v²> is dominated by $\nu _3$ = 0.2353(7)  $\rm {c~d^{-1}}$, which also turns out to be the second intrinsic frequency in <v> (Fig. 14, lower panel). $\nu _3$ is not found in the variations of the third velocity moment <v³>. After having prewhitened <v> with $\nu _1$ and $\nu _3$, <v²> with $\nu _3$, and <v³> with $\nu _1$, there are (almost) no peaks in the modified Scargle periodogram exceeding the 1% FAP-level and  $\sigma _{\rm res}$ has reached the $\sigma _N$ level (Table 13).

In total, three intrinsic frequencies are found for HD 85953. An overview of their characteristics is given in Table 13. The observed frequency spacings $\nu _3$ - $\nu _2$ = 0.0164  $\rm {c~d^{-1}}$ and $\nu _1$ - $\nu _3$ = 0.0310  $\rm {c~d^{-1}}$ $\approx$ 2($\nu _3$ - $\nu _2$) suggest that these frequencies are members of a multiplet with $l~\geq~2$. These observed frequency spacings are indeed found if we assume a rotational frequency $\nu_{\rm rot}$ $\approx$ 0.02  $\rm {c~d^{-1}}$, which leads to an equatorial rotation velocity of about 5  $\rm {km~s^{-1}}$.

 

 

Table 13: Same as Table 2, but for the accepted intrinsic frequencies of HD 85953.

data-set		$\sigma$	Ai	$\phi_i$	$\sigma _{\rm res}$	$\sigma _N$
$\nu _1$ = 0.2663(6)  $\rm {c~d^{-1}}$
<v>4128	( $\rm {km~s^{-1}}$)	2.35	2.87(21)	0.31(2)
B	(mmag)	10.3	10.1(8)	0.97(1)
$H_{\rm p}$	(mmag)	11.7	11.2(7)	0.96(1)
$\nu _2$ = 0.2189(6)  $\rm {c~d^{-1}}$
B	(mmag)		3.2(8)	0.92(4)	7.2
$H_{\rm p}$	(mmag)		6.6(8)	0.96(2)	5.9	5.9
$\nu _3$ = 0.2353(7)  $\rm {c~d^{-1}}$
<v>4128	( $\rm {km~s^{-1}}$)		1.09(19)	0.90(3)	1.12	1.1

5.2.3 HD 138764 - HR 5780 - HIP 76243

HD 138764 ( m_V = 5.15) is a bright member of the Upper Scorpius subgroup of the Scorpio-Centaurus OB-association for which Bidelman (1965) already detected asymmetries in Si II profiles. Thanks to the photometric measurements of the HIPPARCOS mission, the photometric variability of HD 138764 is beyond any doubt now. All our data-sets of HD 138764 agree upon the first intrinsic frequency $\nu _1$ = 0.7944(9)  $\rm {c~d^{-1}}$ (Fig. 15, upper panel). Already 80% of the variance is reduced by $\nu _1$ alone.

After prewhitening the velocity moments with $\nu _1$, the periodograms and $\theta$-statistics of <v> (Fig. 15, lower panel) and <v³> both point towards $\nu _2$' = 0.6735(9)  $\rm {c~d^{-1}}$. For <v²>, $\nu _2$ = 0.6372(9)  $\rm {c~d^{-1}}$ is found, together with other candidate frequencies. In <v³>, $\nu _2$ also reaches the 1% FAP-level. $\nu _2$ and $\nu _2$' are each other's alias frequencies. After having prewhitened the $H_{\rm p}$ data, the frequency search algorithms lead to $\nu _2$ (Fig. 15, lower panel) while no signature of $\nu _2$' is found. In the residual Geneva data, there are no longer frequencies with an amplitude reaching the 1% FAP-level. We therefore conclude that $\nu _2$ is the second physical frequency in the velocity moments and in the HIPPARCOS magnitudes. After additionally prewhitening the data-sets with $\nu _2$, no more intrinsic frequencies could be determined without ambiguity. In Table 14, we give of the characteristics of the accepted intrinsic frequencies.

Figure 15: Same as Fig. 2, but for the 2 accepted intrinsic frequencies $\nu _1$ = 0.7944(9)  $\rm {c~d^{-1}}$, and $\nu _2$ = 0.6372(9)  $\rm {c~d^{-1}}$ of HD 138764.

 

 

Table 14: Same as Table 2, but for the accepted intrinsic frequencies of HD 138764.

data-set		$\sigma$	Ai	$\phi_i$	$\sigma _{\rm res}$	$\sigma _N$
$\nu _1$ = 0.7944(9)  $\rm {c~d^{-1}}$
<v>4130	( $\rm {km~s^{-1}}$)	3.04	3.64(22)	0.52(1)
B	(mmag)	17.6	20.4(14)	0.22(1)	9.4
$H_{\rm p}$	(mmag)	17.1	21.4(8)	0.153(7)
$\nu _2$ = 0.6372(9)  $\rm {c~d^{-1}}$
<v>4130	( $\rm {km~s^{-1}}$)		0.80(20)	0.42(5)	1.20	0.42
$H_{\rm p}$	(mmag)		9.6(9)	0.27(1)	5.3	4.4

5.2.4 HD 215573 - HR 8663 - HIP 112781

Figure 16: Same as Fig. 2, but for the 2 accepted intrinsic frequencies $\nu _1$ = 0.5439(6)  $\rm {c~d^{-1}}$, and $\nu _2$ = 0.5654(6)  $\rm {c~d^{-1}}$ of HD 215573.

HD 215573 ( m_V = 5.35) is one of the few normal mid-B type stars with very sharp spectral lines. It is the only star in our sample for which no obvious common main frequency is found in our different data-sets. The data in the $H_{\rm p}$ filter are dominated by two frequencies with a similar amplitude: $\nu _1$ = 0.5439(6)  $\rm {c~d^{-1}}$ and $\nu _2$ = 0.5654(6)  $\rm {c~d^{-1}}$ (Fig. 16, upper panel). After prewhitening with $\nu _1$, $\nu _2$ is found (and vice versa). For the variations in the Geneva photometry, 0.4293(9)  $\rm {c~d^{-1}}$, and 0.5684(9)  $\rm {c~d^{-1}}$ are two of the best candidates (Fig. 16, lower panel). Both are aliasses of $\nu _2$. The frequency resolution of the 43 spectra of HD 215573 is much worse than for the photometric data-sets. For <v> (Fig. 16, lower panel) and <v³>, the best candidate for the first intrinsic frequency is 1.521(3)  $\rm {c~d^{-1}}$, which is also an alias frequency of $\nu _2$. $\nu _2$ itself also exceeds the 1% FAP-level. Although no signature of $\nu _2$, nor of its alias frequencies is found in <v²>, the observed alias frequency patterns lead to the conclusion that it is commonly present in the photometric and spectroscopic data-sets. After prewhitening with $\nu _2$, the residual standard deviations of the data-sets are still quite large (Table 15). However, no other intrinsic frequencies could be determined without ambiguity. No signature of $\nu _1$ is found in the velocity moments, nor in the Geneva photometry.

In Table 15, an overview of the characteristics of the accepted intrinsic frequencies in the different data-sets of HD 215573 is given. The observed frequency spacing between $\nu _1$ and $\nu _2$ is small, which may point towards membership of frequency multiplet. For a triplet, this would imply a rotational frequency $\nu_{\rm rot}$ $\approx$ 0.04  $\rm {c~d^{-1}}$and an equatorial rotation velocity of some 6  $\rm {km~s^{-1}}$, which is not incompatible with the small total line broadening (Aerts et al. 1999).

 

 

Table 15: Same as Table 2, but for the accepted intrinsic frequencies of HD 215573.

data-set		$\sigma$	Ai	$\phi_i$	$\sigma _{\rm res}$	$\sigma _N$
$\nu _1$ = 0.5439(6)  $\rm {c~d^{-1}}$
$H_{\rm p}$	(mmag)	13.2	11.5(10)	0.68(1)
$\nu _2$ = 0.5654(6)  $\rm {c~d^{-1}}$
<v>4130	( $\rm {km~s^{-1}}$)	2.18	2.03(35)	0.86(3)	1.61	0.27
B	(mmag)	17.1	16.3(22)	0.63(2)	12.1
$H_{\rm p}$	(mmag)		10.8(9)	0.59(1)	7.1	5.1