## 3. The thermo-radiative mechanismMass 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 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 (). 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 () and "corona-like" layer () for an area above the photosphere. The existence of a hot corona () 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 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 (). The thin-line force is thought to drive winds with high mass loss rate and low () 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 . By incorporating a significant radiative force close to the star the temperature at the stellar surface is reduced to . In all these cases the temperature drops with distance to 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 solutionThe outflow is steady state (), axisymmetric to the rotational axis () and helicoidal () (with () 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: satisfies the governing HD equations which conserve mass and momentum: The symbols have the usual meaning: is the dimensionless radial distance ( is the stellar radius), is the dimensionless velocity (, is Boltzmann's constant, the proton mass, the temperature parameter), is the dimensionless rotational velocity of the star at the equatorial plane, is the fluid density. The fluid temperature is related to pressure and density by the usual equation of state: Eq. (2) implies differential rotation for the fluid which is
controlled by the parameter where The total radiative acceleration is in the
radial direction, because photon scattering in the atmosphere is
isotropic, and proportional to the incident Eddington radiation flux
and absorption coefficients
(Mihalas 1978, p. 554): .
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 (i.e. ) is due to
the line contribution. Adopting the where is a constant and . 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 . de Araujo considers a similar parameter which expresses the effect of all lines and . Obviously, there is a direct correspondence of the parameters of the three works: The parameter , or equivalently , depends on the number of thin lines and the mean line opacity and increases with them. The characteristic value: equals the radiative force and the effective gravity at the stellar surface. diverges when and vanishes when .
It was shown in Paper II that an analytical self-consistent 2-D
solution is obtained for any function
(Eq. 7) by solving Eqs. (3) - (4) and determining the
functions where is a function, ,
are constants and the
integrals are evaluated numerically (Paper II). The integral
involves the function 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 (). This is a solution with maximum radial velocity in the equatorial plane of the central object. © European Southern Observatory (ESO) 1998 Online publication: April 20, 1998 |