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

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3. The thermo-radiative mechanism

Mass loss is a dynamic phenomenon of major importance which is observed in stars of all types (Cassinelli 1979). The first theoretical 1-D description of this phenomenon was given by Parker (1958) who had predicted the solar wind as a thermally driven mass outflow from the Sun. Thermally driven winds were described self-consistently (i.e. by solving the HD equations analytically or exactly) in 2-D by Tsinganos & Vlastou (1988), Tsinganos & Sauty (1992), Lima & Priest (1993), Kakouris & Moussas (1996).

On the other hand, the thermal mechanism is considered unimportant in winds from early type stars. From the early times of radiatively driven wind models it has been pointed out that Parker's model fails to describe mass loss from early type stars because it needs an enormously hot corona ([FORMULA]). In this case, strong high energy emission and/or absence of some ions should appear in the stellar spectrum which is not consistent with observations (e.g. Weymann 1963, Lucy & Solomon 1970). This situation is discussed by Underhill (1982) (p. 242) where the author distinguishes between "truly" coronal region ([FORMULA]) and "corona-like" layer ([FORMULA]) for an area above the photosphere.

The existence of a hot corona ([FORMULA]) in early type stars and a thermal mechanism plus a radiative one (based on continuum absorption), has been proposed by Hearn (1975a,b) in order to model the atmosphere of [FORMULA] Ori which exhibits shell characteristics. The author (1975b) calculated the energy loss of a stellar corona showing that for a given surface pressure there is a surface (coronal) temperature that minimizes the energy loss. This coronal temperature is about a factor of 10 larger than the effective temperature for early type stars with high mass loss rates. Relevant works, based on observations (especially in heavy ion spectral lines with high ionization), followed in the next years investigating the possibility of a thin corona-like layer in OB stars. Lamers & Snow (1978) suggested [FORMULA] using UV satellite observations. Cassinelli, Olson & Stalio (1978) calculated the H [FORMULA] profile for [FORMULA] Ori concluding that the possible hot corona must be very thin ([FORMULA]). Olson (1978) used data for [FORMULA] Pup concluding that coronal temperatures in the range [FORMULA] are possible and the combination of UV and H [FORMULA] observations are necessary. However, Cassinelli & Olson (1979) found that the possible corona must be very thin. However, Hubeny et al. (1985) discussed the spectroscopic diagnostics of superionization in UV spectra of B stars with the use of CIV, SiIV, NV lines noting that absorption near 1550A is not CIV but a mix of FeIII lines. In this case there is not observational evidence for a hot corona in B stars. Furthermore, recent development of thermally driven stellar winds by Lima & Priest (1993) extent Parker's model in 2 - D and also relax the isothermal assumption. In that work, an example of a thermally driven solution to B stars was given.

Radiatively driven winds from early type stars use two types of line-forces which coexist with the electron scattering force: the optically thick which employes the Sobolev mechanism and first used by Castor, Abbott & Klein (1975) (CAK model) and the optically thin (Cassinelli & Castor 1973, Marlborough & Zamir 1975). The thick-line force is supposed to drive winds with high terminal velocities ([FORMULA]). The thin-line force is thought to drive winds with high mass loss rate and low ([FORMULA]) terminal velocities. In order to model winds from Be stars, de Araujo (1995) studied the wind driving transition from the optically thick to the optically thin case.

In Paper II we incorporated the thin-line radiative force in the thermally driven solution of Paper I. In that work we showed that when the thermal mechanism excites the wind solely, the temperature at the stellar surface is about [FORMULA]. By incorporating a significant radiative force close to the star the temperature at the stellar surface is reduced to [FORMULA]. In all these cases the temperature drops with distance to [FORMULA] at 100 stellar radii. The 2-D solutions of Paper II can be addressed as follows (full mathematical analysis and expressions of all flow quantities can be found in Paper II):

Paper II solution

The outflow is steady state ([FORMULA]), axisymmetric to the rotational axis ([FORMULA]) and helicoidal ([FORMULA]) (with ([FORMULA]) the usual spherical coordinates). The fluid is ideal, inviscid, non-magnetized and non-polytropic. In this case, the flow bulk velocity given by the expressions:

[EQUATION]

[EQUATION]

satisfies the governing HD equations which conserve mass and momentum:

[EQUATION]

[EQUATION]

The symbols have the usual meaning: [FORMULA] is the dimensionless radial distance ([FORMULA] is the stellar radius), [FORMULA] is the dimensionless velocity ([FORMULA], [FORMULA] is Boltzmann's constant, [FORMULA] the proton mass, [FORMULA] the temperature parameter), [FORMULA] is the dimensionless rotational velocity of the star at the equatorial plane, [FORMULA] is the fluid density. The fluid temperature is related to pressure and density by the usual equation of state:

[EQUATION]

Eq. (2) implies differential rotation for the fluid which is controlled by the parameter µ. In the force balance Eq. (4) P is the flow thermal pressure, [FORMULA] is the line radiative force and the last right hand term is the effective gravity (gravitational force reduced by the Thomson electron scattering force) where G is the gravitational constant, M is the mass of the star and [FORMULA] is the ratio of the stellar luminosity L to Eddington luminosity [FORMULA]:

[EQUATION]

where c is the speed of light and [FORMULA] the Thomson scattering cross section ([FORMULA]). In addition, we use the dimensionless parameter [FORMULA] which is the ratio of the escape velocity at the stellar surface to the velocity parameter.

The total radiative acceleration [FORMULA] is in the radial direction, because photon scattering in the atmosphere is isotropic, and proportional to the incident Eddington radiation flux [FORMULA] and absorption coefficients [FORMULA] (Mihalas 1978, p. 554): [FORMULA]. We can evaluate the force due to Thomson electron scattering separately and this force (due to continuum absorption) is merged with gravity giving the effective value of it (Eq.  4). The remaining part of [FORMULA] (i.e. [FORMULA]) is due to the line contribution. Adopting the optically thin atmosphere approximation and according to CM [FORMULA] can be expressed by line absorption opacities [FORMULA] and subsequently by Thomson cross section [FORMULA] as [FORMULA]. An average value of [FORMULA] can be determined by the quantity [FORMULA], where [FORMULA] is the average of the opacity of all thin lines, N is the number of thin lines and [FORMULA] is the local monochromatic luminosity in each line. By setting [FORMULA] the authors suggest that the number of lines and the average opacity scale with distance depending upon the local excitation and ionization equilibrium. In this way the thin line radiative acceleration is written as:

[EQUATION]

[EQUATION]

where [FORMULA] is a constant and [FORMULA].

By this formalism there is a correspondence between Paper II parameters and the works of CM and de Araujo (1995). CM introduce a power law [FORMULA]. de Araujo considers a similar parameter [FORMULA] which expresses the effect of all lines and [FORMULA]. Obviously, there is a direct correspondence of the parameters of the three works:

[EQUATION]

The parameter [FORMULA], or equivalently [FORMULA], depends on the number of thin lines and the mean line opacity and increases with them. The characteristic value:

[EQUATION]

equals the radiative force and the effective gravity at the stellar surface. [FORMULA] diverges when [FORMULA] and vanishes when [FORMULA].

[FIGURE] Fig. 2. Steady shell formation for case 1 parameters of Table 1. a force balance b outflow radial velocity c outflow density, temperature and pressure. In plots b and c solid curves are polar and dashed are equatorial. In plot a five different forces are shown. The equatorial one is the centrifugal (5) (much smaller than the others). The thermal (3) and outward (4) are polar forces. The balance of the effective gravity (1) with the total outward force (4) (i.e. thin radiative (2) plus thermal (3)) separates the stellar envelope into inner and outer acceleration regions along the polar axis. Note that the outward force almost coincides with the thermal force at the stellar surface and with the thin radiative force after 2 stellar radii.

It was shown in Paper II that an analytical self-consistent 2-D solution is obtained for any function [FORMULA] (Eq.  7) by solving Eqs. (3) - (4) and determining the functions f and [FORMULA]. It is:

[EQUATION]

[EQUATION]

where [FORMULA] is a function, [FORMULA], [FORMULA] are constants and the [FORMULA] integrals are evaluated numerically (Paper II). The integral [FORMULA] involves the function Q.

[EQUATION]

The solutions are of four types. The solution which describes subsonic outflow at the stellar surface which becomes transonic close to the star and terminates supersonic at infinity (named as Range I solution in Paper II) is applied in next section. The sonic surface is spherical and very close to the stellar surface ([FORMULA]). This is a solution with maximum radial velocity in the equatorial plane of the central object.

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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