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Astron. Astrophys. 333, 678-686 (1998)

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4. Steady shells modeled by the 2-D thermo-radiative solution

4.1. Force balance

As deduced in Paper II, when the thermal pressure force is employed to excite and drive a wind from typical early type stars, then, a sharp temperature gradient appears close to the star which leads to [FORMULA] surface temperature. Once a radiative force contributes in that region the temperature gradient smooths leading to [FORMULA]. In isothermal models the thermal force is much less since no gradient of the sound speed exists. It was also seen that even if we adopt a strong thermal driving near the star the outflow cannot start from very low initial velocities because of the strong gravitational attraction after a certain distance. This means that the thermal force decays very rapidly with distance. By introducing a radiative force comparable to gravity (or greater than it) the previous decelerating effects and high initial velocities disappear. On the other hand, in optically thin stellar atmospheres, the radiative mechanism which is able to produce such a high force must be investigated. This is a well known subject in radiatively driven winds since the work of Lucy & Solomon (1970). Obviously, the main problem exists in astrophysical objects which posses a strong gravitational field and low luminosities without rotating close to the Keplerian-speed limit. The situation is still complicated when outflows from Be stars are studied, as well as from P Cygni type supergiants (Kuan & Kuhi 1975). Most of these stars show P Cygni profiles and there is also evidence for shell or blob ejections (Lamers 1994). These two observational features give some basis for a deceleration region in the atmosphere but the exact mechanism is not known (Kuan & Kuhi 1975, Hearn 1975, Nugis et al. 1979, Lamers et al. 1985).

In order to use the thin-line radiative force (described in the previous section) we have to choose the parameters [FORMULA] and [FORMULA] (Eq. (7)). According to CM the radial dependence of the thin radiative force is a power law:

[EQUATION]

with [FORMULA] constant. In order to create shells and not monotonously accelerating outflow solutions we have to choose [FORMULA] (Eq. (9)). Choosing [FORMULA] the line opacity is less than the continuum.

The role of the thermal force in the present work is to excite and drive the outflow in the inner acceleration region and to prevent gravitational attraction in the deceleration region up to the shell distance. An increasing with distance thin radiative force (which is less than the thermal force close to the stellar surface) contributes in the inner part and dominates in the outer. This force equals gravity at the shell distance. In this way, we fit a thermal force in order to obtain almost equilibrium conditions at the shell (across the rotational axis). The thermal driving in the inner acceleration region introduces a corona-like layer with temperature higher than the effective of the star. In present work, we restrict our analysis for early type supergiants to cases with [FORMULA] according to theoretical and observational aspects referred in Sect. 3.

4.2. Deduction of shell solutions

An approximate shell distance along the polar axis is easily evaluated neglecting the thermal and the centrifugal force. By equalizing the effective gravity and the thin radiative force we obtain:

[EQUATION]

Taking into account the centrifugal force at other co-latitudes [FORMULA], Eq. (13) becomes:

[EQUATION]

Obviously, as [FORMULA] increases the right hand term of (14) destroys the equilibrium at the shell distance and the shell disappears for [FORMULA]. With increasing [FORMULA], the rotating fluid concentrates close to the equator and consequently, the shell overlaps the polar regions reaching the equatorial area. Conclusively, it is seen that the shell distance depends on the stellar luminosity ([FORMULA]), the number and the opacity of the thin lines ([FORMULA]). The shape of the shell depends on the differential rotation of the fluid ([FORMULA]) and the rotation of the star ([FORMULA]). The exact distance and shape of the shell are found by taking into account the thermal force which is maximum at the equatorial plane.

In order to illustrate the previous results let us consider a hypothetical superluminous B1a+ type supergiant of:

[EQUATION]

[EQUATION]

and adopt a density parameter [FORMULA] and a temperature parameter [FORMULA] which correspond to a mass loss rate [FORMULA], [FORMULA], [FORMULA], [FORMULA]. We present three cases with the values of Table 1.


[TABLE]

Table 1. Parameters for the applications of Figs. 2, 3, 4


In Fig. 2a the force balance along the polar axis is shown for case 1. The outward acceleration is the sum of the thermal and radiative contributions and almost balances effective gravity at the shell distance. The equatorial centrifugal acceleration is also plotted (being negligible compared with radiative and effective gravity). As pointed out, the thermal force decays rapidly with distance. In Figs. 2b-2c the results for outflow radial velocity [FORMULA], density [FORMULA], temperature T and pressure P are shown. Solid curves are polar and dashed are equatorial. A mass concentration is formed at [FORMULA] which is the shell distance. The thermal pressure drops inside the shell (subpressure region) and as a result the fluid cools (Eq.  5). Note that the fluid temperature is about [FORMULA] at the stellar surface and reaches the effective value at [FORMULA]. Thus, the corona-like region is [FORMULA].

In Fig. 3 the steady shell solutions are illustrated for cases 1 - 3. The shell distance depends on both [FORMULA] and [FORMULA] (Eq. (13)), so, the shell appears at [FORMULA] in case 2 and at [FORMULA] in case 3.

[FIGURE] Fig. 3. Dependence of the shell distance upon parameters [FORMULA] and [FORMULA] for values of Table 1. Solid curves are polar and dashed are equatorial.

The change for the shell shape is shown in Fig. 4. Plot (c) corresponds to case 1 while in the other plots [FORMULA] is different. For small [FORMULA] ([FORMULA], plot (a)) the shell is restricted at polar regions and looks like a double blob. For large [FORMULA] ([FORMULA], plot (d)) the shell reaches the equator.

[FIGURE] Fig. 4. Dependence of shell shape upon differential rotation parameter [FORMULA] for case 1 of Table 1 shown by selected logarithmic density ([FORMULA]) contours. a [FORMULA], b [FORMULA], c [FORMULA], d [FORMULA]

4.3. M supergiants

Outflows from late type M5 supergiants have lower terminal velocities ([FORMULA]) compared with early type stars. There is an uncertainty for their radii, but, let us consider the following values for a typical M5 supergiant:

[EQUATION]

[EQUATION]

A steady shell at [FORMULA] appears (Fig. 5) using a density parameter [FORMULA] and a temperature parameter [FORMULA] (which correspond to mass loss rate [FORMULA], [FORMULA], [FORMULA]), [FORMULA] (critical value [FORMULA]), [FORMULA], [FORMULA], [FORMULA]. There is observational evidence for chromospheric emission from late type supergiants seen in H, CaII and K spectral lines (Cassinelli 1979 and references therein). So, the temperature could reach values of the order [FORMULA] at the stellar surface (Fig. 5b).

[FIGURE] Fig. 5. Steady shell solution for a hypothetical M5 supergiant a radial velocity b density and temperature. Solid curves are polar and dashed are equatorial.

4.4. P Cygni

The envelope of hypergiant P Cygni has been extensively studied with the use of observations. According to Lamers (1986) we adopt the parameters:

[EQUATION]

[EQUATION]

In order to apply the solution we set the density parameter [FORMULA] and the temperature parameter [FORMULA] (which correspond to a mass loss rate [FORMULA], [FORMULA], [FORMULA]). The critical value of the line opacity is [FORMULA] and we choose the value [FORMULA]. According to Lamers (1986) the total outward acceleration seems to scale as [FORMULA]. In the analysis of Sect. 4.2 (Fig. 2a) it was shown that the total outward acceleration in the outer acceleration region is approximately equal with the thin radiative force. So, we set [FORMULA]. We present two cases: (1) where [FORMULA], [FORMULA], (2) where [FORMULA], [FORMULA]. In both cases a steady shell is formed at [FORMULA] (Fig. 6). In Fig. 6c the meridional projection of the shell for case 2 is illustrated by contours of the density logarithm ([FORMULA]). The asymptotic outflow velocity at [FORMULA] is about [FORMULA].

[FIGURE] Fig. 6. Steady shell solution for hypergiant P Cygni (Sect. 4.4). a radial velocity b density and temperature c selected logarithmic density ([FORMULA]) contours for case 2. A steady shell is formed at [FORMULA] when [FORMULA] and the radiative force scales as [FORMULA].

The shell mass for case 2 (Fig. 6c) is evaluated numerically by considering an inner [FORMULA] and an outer [FORMULA] radius for the shell. In this case [FORMULA] and [FORMULA], so, the thickness of the shell is [FORMULA]. The shell exists for colatitudes [FORMULA]. The density in that region is bounded by [FORMULA]. An approximate numerical integration gives:

[EQUATION]

It is also found that across the rotational axis the fluid spends [FORMULA] to travel up to [FORMULA] and passes [FORMULA] at the shell region.

The ejection of shells from the hypergiant P Cygni has been extensively investigated by Lamers et al. (1985) who found stable and variable shell components in the observed spectrum. Lamers (1986) also studied the acceleration in the envelope of this star. He suggested that the radiative force is mainly based on the large number of thin lines in the spectrum and the total outward force barely exceeds gravity, scaling as [FORMULA]. The mechanism described in Sect. 2 has also similarities with the work of Nugis et al. (1979) who proposed a three zone envelope in order to explain observations of P Cygni. In this way they introduced ADA - models because of acceleration - deceleration - acceleration of the flow in each zone respectively. They suggest radiative acceleration and gravitational deceleration for the outer zones but they correlate the inner acceleration and excitation of the flow with possible stellar pulsations. For P Cygni they found that inner acceleration occurs up to [FORMULA] and then the flow is decelerated up to [FORMULA]. Lamers et al. (1985) argue that this model explains their observations because it implies high velocities ([FORMULA]) not observed.

In this work, just using the thin radiative force as described by CM scaling as [FORMULA] (in accordance with Lamers (1986) for the stellar post - shell envelope) we find a three - zone envelope similar to Nugis et al. (1979). The inner acceleration is up to [FORMULA], and the deceleration is up to the shell distance [FORMULA]. Afterwards the flow is accelerated by the thin radiative force. The present velocities are less than [FORMULA] (Fig. 6). The main difference with Nugis et al. (1979) is that the inner acceleration is thermo - radiative and steady (not related with possible stellar pulsations and time dependent HD). We evaluate a shell mass of [FORMULA] very close to the estimated by Lamers et al. (1986) ([FORMULA]).

We must note that the main purpose of this article is to study the mechanism of steady shell creation. So, the application to P Cygni shown in Fig. 6 is just indicative for shell formation in its envelope. It is seen that the mass loss rate and the shell mass are underestimated (observed [FORMULA]). This divergence can be removed by redefining the parameter [FORMULA]. Furthermore, the shell distance, shape and the terminal velocity can change as described in Sect. 4.2.

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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