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Astron. Astrophys. 329, 735-746 (1998)

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2. Equations and assumptions

We consider an electron-proton corona and assume that the gas is flowing radially without any temporal variations.

The equations for conservation of mass and momentum can be written as

[EQUATION]

and

[EQUATION]

where r is heliocentric distance, A is the cross-section of the radial flow tube, [FORMULA], where B is the magnetic field strength, [FORMULA] is the mass density, where n is the electron density and [FORMULA] is the electron (proton) mass (indices "e" and "p" are used for electrons and protons respectively), u is the flow speed, [FORMULA] is the gas pressure, where k is Boltzmann's constant, and [FORMULA] is the electron (proton) temperature, G is the gravitational constant, and [FORMULA] is the solar mass.

The two energy conservation equations can be written as

[EQUATION]

and

[EQUATION]

where [FORMULA] is the electron (proton) heat conductive flux density, [FORMULA] is the "mechanical" energy flux density from the sun, y is the fraction of this flux that is deposited in the proton gas, and [FORMULA] is the electric field. [FORMULA] is the energy transport rate per unit volume from the protons to the electrons because of Coulomb collisions, and is given by (Braginskii 1965)

[EQUATION]

where

[EQUATION]

In this study we will discuss how the structure of the corona as well as the solar wind proton flux and flow speed change in models with significant coronal proton heating. The study is performed for different amplitudes and dissipation lengths of the mechanical energy flux as well as for different flow geometries. We also change the proton heat conduction and the fraction of the energy flux going into heating the electrons. The force on the plasma associated with the propagation and damping of the energy flux is neglected.

In the collision dominated quasi-static corona a classical heat flux may be used to describe the heat conduction in the proton gas. But already in the solar wind acceleration region, where the protons are close to collisionless, this may be a significant overestimate of the actual heat flux. Olsen & Leer 1996showed that the proton heat conductive flux, found in an 8-moment fluid description, decreases rapidly in the region where the protongas becomes collisionless. This is quite close to the sun, and as an extreme case one may set [FORMULA] everywhere (cf. McKenzie et al. 1995). Such a model may be used to illustrate how the solar wind from a proton-heated corona is accelerated to high asymptotic flow speeds, but in more realistic models the electron-proton energy transfer and the heat conductive flux into the chromosphere-corona transition region should also be accounted for.

In a corona with comparable electron and proton heating, the electron gas is considerably colder than the proton gas due to the larger electron heat conductivity. A large fraction of the energy flux deposited in the electron fluid is conducted into the transition region, and the electron temperature will be relatively low. The solar wind from a corona where only the electrons are heated, will have a lower speed than the wind from a corona with proton heating. In this study we therefore consider models where most of the energy flux from the sun is deposited in the proton fluid. In these models the coronal proton temperature may be large, and the asymptotic flow speed of the solar wind should also be quite large. Models with low coronal electron density should be optimal in producing a high coronal proton temperature and high speed solar wind. We allow for heating of the electrons, and to make the model as simple as possible the spatial distribution of the electron heat input is taken to be similar to the distribution of the proton heating.

In the corona and inner solar wind, where the collision time is much shorter than the expansion time, we may use a classical expression for the electron (proton) heat flux density:

[EQUATION]

where

[EQUATION]

We have chosen to use this expression for [FORMULA] everywhere (cf. Lie-Svendsen et al. 1997), but in the region where the proton gas is collisionless [FORMULA] must be reduced (cf. Olsen & Leer 1996). Here, we replace [FORMULA] with

[EQUATION]

where h is a scale length over which the proton heat flux decreases. Notice that [FORMULA] at the inner boundary, while the heat conductive flux in the proton gas is very small for [FORMULA].

The equations for electron density, flow speed, and electron and proton temperature are solved from an inner boundary, in the upper transition region, and out to [FORMULA]. We specify the temperature at the inner boundary; the inner boundary is taken to be at a level in the upper transition region where we ensure that the electrons and protons are thermally coupled. In the present study the inner boundary is at a level where the temperature is [FORMULA].

In self consistent models of hydrogen outflow from the chromosphere, through the transition region and corona, and into interplanetary space, it is found that the transition region pressure (and the electron density in the inner corona) is determined by the heating of the transition layer and the heat conductive flux from the corona (Hansteen & Leer 1995). When there is no energy deposition in the transition region, and the solar wind enthalpy flux at the inner boundary is small compared to the radiative loss from the transition region, the pressure is, to a good approximation, proportional to the inward heat conduction flux density, [FORMULA] (Landini & Monsignori-Fossi 1973):

[EQUATION]

where C is a constant.

This approximation may be used in spherically symmetric flow, but in a rapidly expanding flow, where the solar wind is originating from a small area, the solar wind enthalpy flux at the inner boundary (where [FORMULA]) is larger than the radiative loss from the transition region. At the inner boundary we therefore have that

[EQUATION]

i.e. the inward heat flux balances the radiative loss and the increase of the solar wind enthalpy flux in the transition region. Then the electron density at the inner boundary is given by

[EQUATION]

This is the boundary condition used in the present study. The constant C is set to

[EQUATION]

Any mechanical heating of the transition region would be equivalent to a larger inward heat conduction, and therefore increased left side of Eq. (?? ). This leads to higher density, [FORMULA], or proton flux density, [FORMULA], or both.

In most of this study we will consider self consistent models, where the electron density, [FORMULA], at the inner boundary, is determined by Eq. (?? ). In contrast, in many model studies of the corona-solar wind system the density at the inner boundary is specified, independently of the inward heat flux. We will also consider such models.

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© European Southern Observatory (ESO) 1998

Online publication: December 8, 1997
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