The total intensities of the Ly- and O VI lines reported in this chapter have been calculated for different configurations of coronal holes and for the quiet unstructured corona, using the plasma parameters obtained by two types of solar wind model, a two--fluid model developed by Esser et al. (1986) and a one--fluid radiative energy balance model developed by Withbroe (1988). Here we describe the principal characteristics of these models.
The plasma parameters chosen at the coronal base for Esser et al. (1986) model are given in Table 2 for each region, while the density velocity and temperature profiles are shown in Figures 3.1 and 3.2. The density profile for the coronal hole (Figure 3.1a, solid line) is compared to the lowest density profile derived by Lallement et al.(1986) for the coronal hole described by Munro and Jackson (1977). These are the lowest coronal hole densities that we could find in the literature, and the hole must therefore be considered rather extreme. The corresponding velocities (dashed line in Figure 3.1a) are compared to velocities derived from Doppler dimming by Strachan (1990) (triangles), and velocities derived from Interplanetary Scintillation measurements by Tyler et al. (1981) (square), and from measurements of Doppler shifts in Mg X 625 by Rottman et al. (1982) (circle). Note that there are even fewer velocity measurements in the inner corona than density measurements. In section 3.2.3 we have calculated the line intensities for two cases, one with the velocity shown in Figure 3.1a, and one with a velocity that is approximately constant below 3.5 . Electron and proton thermal temperatures are shown in Figure 3.1b (solid line and dashed line respectively), as none of these temperatures have been measured before there is not much to compare them with. The proton thermal temperature in Figure 3.1b falls off slightly slower than adiabatically due to the coupling to the electrons, the electron temperature is assumed to be given by:
This is the expression for a conductively dominated electron gas where the conduction is given by the Spitzer (1962) formula (), and the temperature vanishes at . The electron and proton thermal temperatures at the coronal base are set equal . This electron temperature profile is only an estimate. The values at the coronal base (Table 2) are inside observed limits. The effective temperature, which is also shown in Figure 3.1b (dotted line), is given by:
where C is a constant which was chosen to be 1 (e.g., Esser et al. 1986; Maltby 1968), and is the Alfvén wave velocity amplitude, k the Boltzman constant, m the mass of the scattering particles. This expression applies only if the line profile is Gaussian, and the only source of non-thermal broadening is due to Alfvén waves (see also Esser, 1990, and references therein.
The Alfvén wave velocity amplitude is calculated from the expression (Hollweg 1973; Jacques 1977)
3.1 Esser et al. (1986) model. 3.2 Withbroe (1988) model.
Figure 3.1. Assumed plasma parameters for the coronal hole,
(a) outflow velocity solid line, and density (dashed line). (b)
electron, proton and effective proton temperature (solid line,
dashed line and dotted line respectively). Symbols refer to observed
quantities (see text for details).
Figure 3.2. Assumed plasma parameters for the low speed solar wind.
Symbols as in Figure 3.1.
Here is the Alfvén Mach number, , is the Alfvén speed, the mass density, u the flow speed, and n the electron density (), both are taken from Figure 3.1a.
The calculated effective temperature is compared in Figure 3.1b to temperatures derived from Lyman- line width observations (dots) by Withbroe et al. (1985 and 1986). The assumed wave velocity amplitude at the coronal base, and magnetic field strength are also inside observational limits (e.g., Hollweg et al. 1982). The magnetic field strength given in Table 2 has to be multiplied by the expansion factor of the coronal hole which was taken to be 7.26 (e.g., Munro and Jackson 1977).
The calculated velocity at 1 AU for the region considered in Figure 3.1 is about 1370 and the mass flux approximately . This is the lowest mass flux value given by Withbroe (1988).
The density as a function of heliocentric distance in the quiet unstructured corona is plotted in Figure 3.2a (solid line), and compared to densities derived from polarization brightness observations by Saito (1970, dots) for the equatorial direction, and by Allen (1973, crosses) for solar maximum. For the outflow velocity of this region (dashed line in Figure 3.2a) we have only an upper observational limit derived by Withbroe et al. (1982a) from Ly- Doppler dimming. In Figure 3.2b we have plotted the proton, electron and effective temperatures. The proton temperature again falls off less steeply than adiabatically, the electron temperature is given by Equation (4), and the effective temperature by Equation (5). For comparison we have also plotted the temperatures derived from EUV intensity measurements and Ly- line width measurements (see Withbroe 1988, and references therein) and, at 3 , a temperature derived from charge state measurements (Bame et al. 1974). It is not known whether the temperatures determined from the optical measurements correspond to a magnetically open or closed region. Comparing to other temperature measurements it seems more likely that this was a closed region, and we have therefore chosen temperatures that are significantly lower for the intensity calculations. Note that this increases the ion equilibrium values of the O VI ions.
TABLE 2. Parameter Values for Esser et al. (1986) Model
(Calculated line intensities in ).
Spadaro and Ventura (1993), by adopting a one--fluid radiative energy balance model developed by Withbroe (1988), calculated the intensities of the O VI and lines expected from an equatorial and a polar coronal hole at solar minimum, when observed on the plane of the sky in the 1.2 -- 8 range of heliocentric distance. They also considered the contribution from the quiet unstructured corona surrounding tha coronal holes. The Withbroe (1988) model describes the coronal holes as radial flow tubes with a cross--sectional area which has a non--radial increase with the heliocentric distance. The non--radial expansion factor is given by Kopp and Holzer (1976).
The radial profiles of the coronal electron densities, temperatures and outflow speeds inside the structures considered are plotted in Figure 3.3 (see Withbroe 1988, for a comparison of these profiles with observations). The boundaries of the two coronal holes are shown in Figure 3.4. Note that the polar hole is much broader than the equatorial hole and is similar to the coronal hole described by Munro and Jackson (1977).
Figure 3.3. Densities (top), temperatures (centre) and flow velocities (bottom) of the considered coronal models. Solid curves are relative to the equatorial hole at the solar minimum, dot curves to the polar coronal hole at the solar minimum, and dashed curves to the quiet unstructured corona (from Withbroe 1988).
Figure 3.4. Geometry of the simulated observations of the two coronal holes surrounded by the quiet corona (from Spadaro and Ventura 1993).
The O VI kinetic temperature and outflow velocity have been assumed to be equal to the corresponding plasma values. The atomic parameters are those reported in Table 1, as well as the intensity and profile of the chromospheric exciting lines. The effect of non--thermal motions, such as the Alfvén waves, on the simulated line profiles (e.g, Esser 1990) has not been considered.
The contribution from the quiet corona has also been calculated by arbitrarily setting the outflow velocity equal to zero and leaving the other physical parameters unchanged, in order to consider tha case of the coronal hole surrounded by a static atmosphere, such as that confined in magnetic closed regions. This is only a rough estimate of the contribution from the static quiet corona. A better evaluation requires more detailed models.