1.10 Theoretical Interpretation of Disc Structure

Theoretical models which have been presented include wind compressed discs (Bjorkman and Cassinelli, 1993), wind bi-stability (Lamers and Pauldrach, 1991) and viscous out-flowing discs (Lee et al. 1991, also see Porter 1999).

Each of these models has problems: the wind models (wind compression and bi-stability) cannot reproduce the Keplerian rotation in the disc as the wind conserves angular momentum leading to a rotation law similar to v 1/r (see Owocki et al. 1994). The viscous disc model requires a source of angular momentum to sustain the disc which is currently still unidentified.

1.10.1 The Wind Compressed Disc Model

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The Wind compressed disc (WCD) model (Bjorkman and Cassinelli, 1993) is such that discs form naturally around rapidly rotating B-stars by rotational focusing of the stars’ radiatively driven stellar wind toward the equatorial plane (Owocki and Cohen, 2000). A key assumption of the WCD model is that all forces, including line driven forces are strictly radial (Owocki et al., 1996). The WCD model is most easily understood through a thought experiment: consider two identical particles which are ejected from the surface of a star, at identical locations in the southern and northern hemispheres, their orbital planes are such that they include the ejection point and the centre of the star. The orbital planes will have an inclination relative to the equatorial plane of the star, and all three planes will intercept at the same place and time. The resulting collisions of a high density ejection of particles, from all over the stellar surface, forms an equatorial enhancement, or disc, around the star, see Figure 1.8 left panel. However a full calculation shows that there are problems with this model. The cause of these problems are non-radial line forces which inhibit the WCD effect.

Sobolev (1960) derived equations that enormously simplified the calculations for radiative transfer in moving atmospheres. The Sobolev approximation is that; the interaction between radiative transfer in a flowing gas is simplified to a local process.

More formally, in the limit of an infinitely narrow interaction region the optical depth, which normally contains an integration of density over distance, depends only on the local conditions at the point of absorption (Lamers and Cassinelli, 1999).

The Sobolev length, L_sob, may be written as

(1.9)

where V _th is the thermal velocity, and is the velocity gradient vector. The Sobolev optical depth, _sob, may then be written as

(1.10)

where is the cross-sectional area of interaction and is the matter density. Owocki et al. (1996) combined this work with that of the WCD model and found, through 2D hydrodynamical simulations, that the introduction of these non-radial line driving forces inhibited the formation of a disc, see Figure 1.8 middle panel. They further modified the WCD model by introducing the effects of gravity darkening and stellar oblateness, see Figure 1.8 right panel. Gravity darkening (von Zeipel, 1924) is the decrease in effective temperature of the equatorial regions of a star due to its rapid rotation (Bjorkman and Bjorkman, 1994).

The non-radial line-force arises from (i) the stellar oblateness of the stellar surface, which means that the flux vector which should be along the direction of local gravity has a poleward tilt near the star, giving the radiative force a poleward tilt (Owocki et al., 1996). (ii) Asymmetries in the line-of-sight velocity gradient. The lower effective gravity near the equator implies lower outflow speeds, the line of sight velocity gradient is therefore stronger when looking towards the equator rather than the pole when viewed from some mid-latitude location in the wind. Hence photons from the equator impart a stronger impulse than those at the pole, enhancing the the net-poleward line-driving-force (Owocki et al., 1996).

Also to be noted is that WCD models are angular momentum conserving and so the azimuthal velocity is v 1/r which is in conflict with the observed rotation law of v 1/r^1/2

1.10.2 Non-Radial Pulsations

Non-radial pulsations (NRP) are complex (Zacharias, 2001). NRPs are such that one part of the stellar photosphere moves outward whilst another part moves inward. These movements cause differential changes in temperature and pressure that occasion positive values in some areas and negative values in others.

NRPs cannot, by themselves, eject matter from the surface of a star since the pulsations are too small. Theories which invoke NRP use them to feed energy into the equatorial surface layers to accelerate them to critical velocity. It is then the centrifugal force which causes the mass loss (Balona, 2000).

Whilst a Be star as a whole does not spin at its critical velocity, it has been discussed (see Townsend, 2000a,b) that it may be possible for an equatorial belt to spin at break-up. It would not be possible to observe such a phenomena because the stellar velocity is observed as a whole. It is suggested that NRPs may be able to cause such an effect.

1.10.3 Wind Bi-Stability

Pauldrach and Puls (1990) found that when a stellar wind is fast and tenuous the Lyman continuum optical depth is less than unity. The wind is denser and slower if the optical depth is larger than unity, implying that a small change in stellar parameters can result in drastic changes in the stellar wind (Lamers and Cassinelli, 1999). This is called the bi-stability jump.

Wind bi-stability is represented by an abrupt “jump” in a star’s terminal velocity (v) from v 2.6v_esc for stars of spectral type earlier than B1 to v 1.3v_esc for those later than B1, see Figure 1.9.

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The physical process from which the bi-stability jump occurs is thus: radiation driven wind theory predicts that as a consequence of effective gravity (g_eff, a decreased gravity incurred from the radiation pressure of electron scattering) gradually decreasing, the mass loss rate gradually increases and the terminal velocity gradually decreases. There is however an abrupt jump in the observed data and models. The difference between the two data groups is that winds with low Lyman optical depth have a high ionization and those with high optical depth to Lyman photons have a lower ionization. The jump is so drastic because once the wind starts to recombine, the wind velocity decreases and the mass loss rate increases, this causes a high density in the wind and therefore more recombination. Through a series of monté-carlo simulations using hydrodynamic codes Vink et al. (1999) find that the FeIV/FeIII ionization/recombination effect at T_eff 25000K is the most dominant process causing the bi-stability jump.

Lamers and Pauldrach (1991) suggest that this effect may produce out-flowing discs in rapidly rotating early type B-stars. They suggest that the three phases of a Be star, (disc)-(no disc)-(disc re-growth), are explained by the sudden change in the stellar wind. However they must conclude by stating that only if there is some mechanism by which the mass flux at the equator of a B-star is increased more efficiently than by the reduction of gravity will the bi-stability mechanism produce out-flowing discs similar in characteristic to those observed in Be stars.

1.10.4 Viscous Discs

In viscous excretion disc models (Lee et al., 1991) angular momentum is transfered from the central star to the inner edge of the disc, although the mechanism by which this occurs is still to be determined. This angular momentum flux increases the disc’s velocity to Keplerian. A tentative suggestion has been made that the angular momentum source relates to non-radial pulsations (Osaki, 1986). The material in the disc self-interacts through viscosity and the angular momentum is conducted outwards, forming an excretion or out-flowing disc. As a natural by-product of this process the disc’s azimuthal velocity (v) is Keplerian (v R^-1/2). At present the viscous disc model is the only model that naturally yields near Keplerian discs around Be stars (Okazaki, 2001).

Viscous out-flowing discs successfully reproduce most of the attributes of the observed Be star discs (e.g., continuum excess Porter 1999, V/R variations in the emission lines Okazaki 1997), and indeed are the only discs which can do so that have been suggested to date.