A&A 444, L29-L32 (2005)
DOI: 10.1051/0004-6361:200500203
S. S. Hasan^{1} - J.-P. Zahn^{2} - J. Christensen-Dalsgaard^{3}
1 - Indian Institute of Astrophysics, Koramangala, Bangalore-560034, India
2 - LUTH, Observatoire de Paris, 92195 Meudon, France
3 - Institut for Fysik og Astronomi, Aarhus Universitet, 8000 Aarhus C,
Denmark
Received 28 July 2005 / Accepted 22 October 2005
Abstract
Aims. We suggest that high-order g modes can be used as a probe of the internal magnetic field of SPB (slowly pulsating B) stars. The idea is based on earlier work by the authors hich analytically investigated the effect of a vertical magnetic field on p and g modes in a plane-parallel isothermal stratified atmosphere. It was found that even a weak field can significantly shift the g-mode frequencies - the effect increases with mode order.
Methods. In the present study we adopt the classical perturbative approach to estimate the internal field of a 4 solar mass SPB star by looking at its effect on a low-degree (l=1) and high-order (n=20) g mode with a period of about 1.5 d.
Results. We find that a polar field strength of about 110 kG on the edge of the convective core is required to produce a frequency shift of 1%. Frequency splittings of that order have been observed in several SPB variables, in some cases clearly too small to be ascribed to rotation. We suggest that they may be due to a poloidal field with a strength of order 100 kG, buried in the deep interior of the star.
Key words: stars: magnetic fields - stars: variables: general - stars: oscillations
In an earlier paper Hasan and Christensen-Dalsgaard (1992) analytically determined the frequency shift of p and g modes in an isothermal plasma due to a homogeneous vertical magnetic field. Using the full MHD equations, they found that even a weak field (more precisely when , where is the ratio of gas to magnetic pressure) can produce a significant shift of g-mode frequencies, while the effect on the p-mode spectrum is comparatively small. In principle this means that g-mode frequencies offer a diagnostic to probe the internal field of stars in which g modes have been observed on the stellar surface.
Extensive observation campaigns have uncovered the existence of a class of variable stars known as slowly pulsating B (SPB) stars which are multiperiodic typically over a time scale of days (Waelkens 1991; De Cat et al. 2005 and references therein). These pulsations have been identified with low degree l (typically l=1 and 2) g modes of high order, that are excited by the mechanism in the metal opacity bump at a temperature of about K (Dziembowski et al. 1993). These modes often occur in multiplets with closely spaced periods (with a typical separation of 1%). In some cases this separation can clearly not be due to rotational splitting, which would yield much larger spacings, as was pointed out by De Cat & Aerts (2002). In this letter we propose that such frequency splittings are due to the presence of a magnetic field. If this hypothesis is correct, then the splitting of frequencies can be used to estimate the field strength in the interior of SPB stars.
(1) |
Let us consider a poloidal axisymmetric field of the form:
(2) |
(3) |
(7) |
(8) |
(9) |
Finally, let us recall that the rotational splitting for g modes of high order is given by
Figure 1: Horizontal component of the eigenfunction (scaled such that at the surface) as a function of normalized radius (r/R) for a g_{20}^{1} mode with a period of 1.5 d for a 4 SPB star of 94 Myr. The inset shows an enlargement of the region close to the edge of the convective core, where the eigenfunction displays strong oscillations due to the presence of a steep composition gradient. | |
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Table 1: Frequency and periods for g modes of different radial orders and degree l=1, for a 4 star with an age of 94 Myr.
We first evaluate the frequency shift due to a magnetic field for the g_{20}^{1} mode (i.e. a g mode of radial order 20 and degree l=1), which has a period of 1.5 d (a typical period for a SPB star). The horizontal eigenfunction ( ) of this mode is shown in Fig. 1: the radial component is normalized to unity on the surface r=R of the star.
We calculated the numerator of Eq. (6) with different functional forms b(x) for the magnetic field. When the field is constant throughout the star, the main contribution to that integral comes from above , where has its largest amplitude. But when the field is buried below that depth, or when it tapers off as with q>1, its main contribution originates from a small region just above the convective core at , where there is a steep gradient in the molecular weight in this evolved star. When we choose , the results depend little on q; those presented hereafter were obtained with q=3.
For this g_{20}^{1} mode = s^{-2}. Using C_{1,1}-C_{1,0} = 1/5, one finds . From Eq. (10), one deduces that in order to produce a 1% frequency shift in , the polar field just above the convective core has to be 110 kG.
For comparison, we consider now the effect of the field on a g mode of order 10 and the same degree l = 1. The horizontal component of the eigenfunction for this mode is shown in Fig. 2. In this case, 1100 kG, which is an order or magnitude larger than for the n=20 mode with the same degree.
Tables 2 and 3 give the splitting coefficient for l=1and l=2 g modes of various orders, and the corresponding polar field strength required to produce a 1% frequency shift. In Table 3 we separate the contributions due to C_{20}-C_{22} and C_{21}-C_{22} terms which produce different frequency shifts. Note how rapidly increases with radial order n.
Figure 2: Same as in Fig. 1, but for the g_{10}^{1} mode. The inset shows an enlargement of the region close to the convective core. | |
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Table 2: Splitting constant () and value of the polar field ( ), at the edge of the convective core for l=1 g modes that would produce a frequency splitting of , for a 4 star with an age of 94 Myr.
Table 3: Splitting coefficient () and value of the polar field ( , at the edge of the convective core for l=2 g modes corresponding to different transitions that would produce a frequency splitting of , for a 4 star with an age of 94 Myr.
We concentrated on g modes of low degree and high order, which are typically excited in SPB stars. We found that a polar field of 110 kG in the vicinity of the convective core causes a splitting of 1% for a g^{1}_{20} mode. For a simple dipolar configuration of the magnetic field, this would translate into a 120 G polar field on the surface of the star; we quote this figure only for illustration purpose, since there is no way to deduce the surface field from the deep field. We have checked that the rest of the star contributes little to the splitting, provided that the field is buried below the depth of about x=0.80, or that it tapers off at a faster rate than . In such a situation we find that high-order g-modes can be be used to probe the deep interior field. This result is not sensitive to the precise field configuration.
With moderate- to low-order modes, the diagnostic is much less sensitive. For a l=2, n=10 g mode with a period of 1.1 d, a polar field of about 2 MG would is required to produce a frequency splitting of 1%.
So far only one SPB star has been detected with a magnetic field. Recently, Neiner et al. (2003) have reported the discovery of a field on the SPB star Cas, in which a non-radial pulsation with a period P=1.56 d was detected. A field strength for the time-averaged line of sight polar component of 330^{+120}_{-65} G was inferred. If this field is the visible part of a deeply rooted magnetic field, it could also leave a signature in the splitting of high-order g modes. We should emphasize that according to our results, a field of order 100 kG at the edge of the convective core would be required to produce a 1% splitting in typical g modes.
We considered here only a purely poloidal configuration, similar to many earlier papers quoted in Sect. 1. Such a configuration is known to be unstable. As was shown by Tayler and collaborators (Tayler 1973; Pitts & Tayler 1985), and as was illustrated recently by the numerical simulations of Braithwaite & Spruit (2004), the configurations which are likely to resist non-axisymmetric MHD instabilities are combinations of large-scale toroidal and poloidal fields of about equal strength. Taking this into account is unlikely to alter our conclusions concerning the detectability of the deep magnetic field, because the frequency splitting would be much more sensitive to the poloidal than to the toroidal component in such a combined field.
Acknowledgements
We thank the referee for comments clarifying the role of the global properties of the magnetic field. S.S. Hasan and J.-P. Zahn are grateful to the Indo-French Centre for the Promotion of Advanced Research, New Delhi for supporting this project through grant number 2504-3. We thank M. J. Thompson for useful discussions. J. Christensen-Dalsgaard acknowledges the hospitality of the Indian Institute of Astrophysics, Bangalore, and the High Altitude Observatory, Boulder, CO, USA during this project.