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Astron. Astrophys. 331, 121-146 (1998)

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2. Model equations

The medium is described as consisting of five fluids corresponding to neutral gas particles ([FORMULA]), gaseous ions carrying one positive charge ([FORMULA] ions as considered by Pilipp et al., 1992), electrons, "big" grains (all of which are assumed to be spherical and identical to one another), and "small" grains (all of which are assumed to be spherical and identical to one another except for their charges). The subscripts n, i, e, b, and s are used to signify that a given parameter refers to the respective fluid. Each big grain is assumed to carry a charge [FORMULA] equal to the average charge carried by a big grain, and [FORMULA] where e is the elementary charge. The magnitude of the average charge carried by a small grain may be sufficiently small that statistical fluctuations make the assumption that each small grain carries the same charge a poor one; hence, we consider the possibility that small grains carry different charges and divide the small grain fluid into subfluids. A quantity referring to the small grain subfluid consisting of those small grains that each carry k elementary charges is identified with the double subscripts s and k.

The effective gravitational field is taken to be in the [FORMULA] direction and to be [FORMULA] with [FORMULA]. The velocity of the j th fluid is, by assumption, in the [FORMULA] direction and is written as [FORMULA] where [FORMULA] is always positive for [FORMULA] and can be negative for [FORMULA] for large electric fields. The mass density and number density of the j th fluid are [FORMULA] and [FORMULA]. [FORMULA] is constant for each fluid.

[FORMULA] is taken to be a constant, as is [FORMULA]. Steady flow is considered. The continuity equations for the ions and electrons are

[EQUATION]

and

[EQUATION]

[FORMULA] is the rate per neutral particle at which radioactive decays and penetrating cosmic rays induce ionization. [FORMULA] is the rate coefficient for species j to transfer an elementary charge to species m. The [FORMULA] 's are considered in the next section. The neglect of ion-electron recombination relative to ion-grain and electron-grain charge transfer is justified for the protosolar nebular regions in which lightning might have occurred.

For most cases considered in the present paper a strong electric field in the positive z -direction arises leading to a bulk motion of electrons in the negative z -direction, i.e. to [FORMULA]. Then the integration of Eq. (2a) along the positive z -direction was found to be unstable. For these cases (i.e. cases presented in Figs. 1, 3, 4a, 5, and 6, except for the curves in Fig. 1 marked [FORMULA], and [FORMULA]) we use the approximation

[EQUATION]

in place of (2a). Approximation (2b) should be correct if in Eq. (2a) the term [FORMULA] is negligible compared to the term [FORMULA], i.e. if the scale length for global variations along the z -direction divided by the magnitude of the electron bulk velocity is large compared to the collision time of electrons with respect to their collisions with dust particles removing the electrons from the gas phase. As may be found from estimates of corresponding electron collision times [FORMULA] as given by (A11) and (A12) in Appendix A, together with estimates of electron bulk velocities [FORMULA] for relevant electric field strengths and from global scale lengths arising from our numerical results, relation (2b) should be a good approximation.

The continuity equation for the small grains carrying k elementary charges is

[EQUATION]

[FORMULA] is the frequency at which collisions with big grains change the charge on a small grain from [FORMULA] to [FORMULA]. The [FORMULA] 's are considered in the next section. The continuity equation for the big grains is

[EQUATION]

The charge on the big grains is followed with

[EQUATION]

Inertial terms in the force equations of the charged species are negligible. For an electric field [FORMULA] the force equations of the various charged species may to a good approximation be taken to be

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[FORMULA] is the collision frequency of species j with neutrals defined such that the time-averaged frictional force on a particle of species j due to collisions with neutrals is [FORMULA]. We shall treat the [FORMULA] 's in the next section.

Neglect of the inertial terms in the force equations should be correct if the stopping length [FORMULA] for a charged particle of species j due to its friction with neutrals is small compared to the scale lengths for the global variations arising from our numerical calculations. This condition should be always well fulfilled even for big grains for the parameters considered in this paper.

The electric field is governed by Poisson's equation

[EQUATION]

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© European Southern Observatory (ESO) 1998

Online publication: February 4, 1998
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