Superadditivity With Mixed Letter States

Monthly Notices of the Royal Astronomical Society: Letters

MacDonald

2006

We examine the stability of ℓ= 1 and ℓ= 2 g modes in a pair of nitrogen‐rich Wolf–Rayet stellar models characterized by differing hydrogen abundances. We find that modes with intermediate radial orders are destabilized by a κ mechanism operating on an opacity bump at an envelope temperature of log T≈ 6.25. This ‘deep opacity bump’ is due primarily to L‐shell bound–free transitions of iron. Periods of the unstable modes span ∼11–21 h in the model containing some hydrogen, and ∼3–12 h in the hydrogen‐depleted model. Based on the latter finding, we suggest that self‐excited g modes may be the source of the 9.8‐h periodic variation of WR 123 recently reported by Lefèvre et al.

Section: S Ta B I L I T Y a Na Ly S I Smentioning

confidence: 99%

Excitation of g modes in Wolf—Rayet stars by a deep opacity bump

Monthly Notices of the Royal Astronomical Society: Letters

MacDonald

2006

“…A classification scheme suitable for fully trapped modes was presented by Scuflaire (1974) and Osaki (1975), which consists of considering the so-called`phase path' of solutions in the (v, w) plane, where the phase variables v and w are calculated from jl lm ac 1v 2 2 V g j 1a2 r 2 y 1 31 and w rgr 1a2 jc 1v 2 2 A*j 1a2 r 2 y 2 ; 32 these definitions have been adapted, for use within the tradition approximation, from the equivalent zero-rotation ones (e.g. Shibahashi 1979) through the usual trivial replacement of ll 1) with l lm .…”

Section: Modal Classificationmentioning

confidence: 99%

“…By following the phase path corresponding to a pair of eigenfunctions y 1,2 from the origin to the surface, the number of clockwise and anti-clockwise crossings of the axis v 0 are enumerated. Denoting these two integers by N p and N g , respectively, their differencẽ n N p 2 N g Y 33 may be used as an unambiguous index, conserved during evolutionary changes to the stellar structure (Osaki 1975) and over changes in the harmonic degree l , for labelling each mode of a star. In unevolved stellar models, positive and negative n Ä correspond to p-and g modes, respectively, whilst n 0 corresponds to the f-mode.…”

Section: Modal Classificationmentioning

confidence: 99%

Surface trapping and leakage of low-frequency g modes in rotating early-type stars - II. Global analysis

Monthly Notices of the Royal Astronomical Society

2002

A global analysis of the surface trapping of low‐frequency non‐radial g modes in rotating early‐type stars is undertaken within the Cowling, adiabatic and traditional approximations. The dimensionless pulsation equations governing these modes are reviewed, and the boundary conditions necessary for solution of the equations are considered; in particular, an outer mechanical boundary condition, which does not enforce complete wave trapping at the stellar surface, is derived and discussed in detail. The pulsation equations are solved for a 7‐M⊙ model star over a range of rotation rates, using a numerical approach. The results of the calculations confirm the findings of the preceding paper in the series: modes with eigenfrequencies below a cut‐off cannot be fully trapped within the star, and exhibit leakage in the form of outwardly propagating waves at the surface. The damping rates resulting from leakage are calculated for such ‘virtual’ modes, and found to be appreciably larger than typical growth rates associated with opacity‐driven pulsation. Furthermore, it is demonstrated that the surface perturbations generated by virtual modes are significantly changed from those caused by fully trapped modes; the latter result suggests differences in the line‐profile variations exhibited by these two types of mode. The findings are discussed in the context of the 53 Per, SPB and pulsating Be classes of variable star. Whilst wave leakage will probably not occur for overstable g modes in the 53 Per and slowly rotating SPB stars, the adoption of the new outer mechanical boundary condition may still affect the pulsational stability of these systems. Wave leakage for overstable modes remains a possibility in Be stars and the more rapidly rotating SPB stars.

“…To derive a local dispersion relation for the pulsation equations (3±4), it is useful first to place the equations in a canonical form similar to that introduced by Osaki (1975) for the non-rotating case. By defining the two new eigenfunctions,…”

Section: I S P E R S I O N R E L At I O Nmentioning

confidence: 99%

“…Note that h(r) is always positive, so that the original eigenfunctions j r and p H everywhere share the same sign as j Ä and h Ä , respectively. Qualitative solution of these canonical equations is accomplished using the same method as Osaki (1975), namely, by assuming that the coefficients on the right-hand sides are independent of r. Such an assumption will be valid if the characteristic variation scale of the solutions is much smaller than that of the coefficients. Then, local solutions of the form j Yh , expik r rY 10 lead to a dispersion relation for the radial wavenumber k r ,…”

Section: I S P E R S I O N R E L At I O Nmentioning

confidence: 99%

Surface trapping and leakage of low-frequency g modes in rotating early-type stars--I. Qualitative analysis

Monthly Notices of the Royal Astronomical Society

2000

A qualitative study of the surface trapping of low‐frequency non‐radial g modes in rotating early‐type stars is undertaken within the Cowling, adiabatic and traditional approximations. A dispersion relation describing the local character of waves in a rotating star is derived; this dispersion relation is then used to construct propagation diagrams for a 7‐M⊙ stellar model, which show the location and extent of wave trapping zones inside the star. It is demonstrated that, at frequencies below a cut‐off, waves cannot be fully trapped within the star, and will leak through the surface. Expressions for the cut‐off frequency are derived in both the non‐rotating and rotating cases; it is found from these expressions that the cut‐off frequency increases with the rotation rate for all but prograde sectoral modes. While waves below the cut‐off cannot be reflected at the stellar surface, the presence of a sub‐surface convective region in the stellar model, owing to He ii ionization, means that they can become partially trapped within the star. The energy leakage associated with such waves, which are assigned the moniker virtual modes owing to their discrete eigenfrequencies, means that stability analyses which disregard their existence (by assuming perfect reflection at the stellar surface) may be in error. The results are of possible relevance to the 53 Per and SPB classes of variable star, which exhibit pulsation frequencies of the same order of magnitude as the cut‐off frequencies found for the stellar model. It is suggested that observations either of an upper limit on variability periods (corresponding to the cut‐off), or of line‐profile variations owing to virtual modes, may permit asteroseismological studies of the outer layers of these systems.