Two-Fluid Model of the Solar Wind

This theoretical study of the solar wind takes account of the fact that the collisional energy-exchange rate between electrons and protons is sufficiently slow that the electrons and protons can have quite dif- ferent temperatures. The basic equations comprise a continuity equation, an equation of motion, and two heat equations, one for each species-electrons and protons. Each heat equation takes account of thermal conductivity in that species and energy exchange with the other species. In order to determine the region of applicability of these fluid-type equations, the electron-electron, and proton-proton energy/ momentum relaxation rates are compared with the local expansion rate. These comparisons indicate that, for the model investigated, the use of fluid equations is justifiable out to about 1O~ Ro. The conditions imposed on the solution of these coupled equations are that the electron (or proton) density and the electron and proton temperatures should have specified values at the "base" (inner boundary) of the model, that there should be a subsonic-supersonic flow transition, and that the electron and proton temperatures should tend to zero as the heliocentric radius tends to infinity. The effects of solar rotation, viscosity, non-thermal heat sources (e.g., wave dissipation), and magnetic field are neglect- ed, and the flow pattern is required to be steady and spherically symmetrical These requirements appear to determine the model uniquely. The equations are solved numerically using iterative procedures It is found that a good fit to the Blackwell model of the electron density of the outer corona is ob- tained, over the range 2-20 Ro, for the following choice of number density n (electrons or protons), electron temperature Te, and proton temperature T~ at the base, the radius of which is taken to be Ro: no = 3 X 1O~ cm3, T60 = T~0 = 2 X 106 ° K. If the base is regarded as being 2 Ro (the radius at which agreement with the Blackwell model begins), the values would be no = 1.5 X 106 cm3; Te0 1 5 X 106 ° K, T~0 = 1.2 X 106 ° K. This model has the following values for these variables and for the velocity v at Earth's orbit: ~E = 15 cm3, VE 250 km sec', TeE = 3 5 X 10~ °K, T~E = 4 4 X 1O~ ° K. These results for ~B, VE, TPE fit most nearly the observed characteristics of the solar wind at geomagneti- cally quiet times, although the density is somewhat high, the velocity is somewhat low, and the proton temperature is quite low. It is believed that the discrepancy between this model and Blackwell data between the base of the corona and 2 Ro is to be attributed to coronal heating, and that departures of the solar-wind charac- teristics near Earth from those of this model are also to be attributed to heating by a flux of non-thermal energy; additional heating inside the radius at which the flow becomes supersonic, which is at 7.1 Ro, will produce primarily an increase in solar-wind velocity, whereas additional heating outside 7.1 Ro will produce primarily an increase in solar-wind temperature. Data are presented which show the effect upon the characteristics of this model at Earth's orbit of a change in density or in electron or proton temperature at the base. The density increases sharply with the base electron temperature, even more sharply with the base proton temperature, and approximately linearly with the base density. The velocity varies almost linearly with the base electron temperature, and is insensitive to the other base quantities The proton temperature varies approximately linearly with the base density and electron temperature, but increases sharply with the base proton temperature The electron temperature increases with the base electron temperature but varies inversely with the base density and proton temperature; these variations are not stron