*Astron. Astrophys. 331, 121-146 (1998)*
## 2. Model equations
The medium is described as consisting of five fluids corresponding
to neutral gas particles (), gaseous ions
carrying one positive charge ( 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
equal to the average charge carried by a big
grain, and 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
direction and to be with
. The velocity of the *j* th fluid is, by
assumption, in the direction and is written as
where is always positive
for and can be negative for
for large electric fields. The mass density and
number density of the *j* th fluid are and
. is constant for each
fluid.
is taken to be a constant, as is
. Steady flow is considered. The continuity
equations for the ions and electrons are
and
is the rate per neutral particle at which
radioactive decays and penetrating cosmic rays induce ionization.
is the rate coefficient for species *j* to
transfer an elementary charge to species *m*. The
'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
. 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 , and
) we use the approximation
in place of (2a). Approximation (2b) should be correct if in Eq.
(2a) the term is negligible compared to the
term , 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
as given by (A11) and (A12) in Appendix A,
together with estimates of electron bulk velocities
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
is the frequency at which collisions with
big grains change the charge on a small grain from
to . The
's are considered in the next section. The
continuity equation for the big grains is
The charge on the big grains is followed with
Inertial terms in the force equations of the charged species are
negligible. For an electric field the force
equations of the various charged species may to a good approximation
be taken to be
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 . We shall treat the
's in the next section.
Neglect of the inertial terms in the force equations should be
correct if the stopping length 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
© European Southern Observatory (ESO) 1998
Online publication: February 4, 1998
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