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

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3. Momentum transfer and charge transfer coefficients

The temperatures, [FORMULA], of the various gaseous fluids are required to evaluate the momentum and charge transfer coefficients. As did Pilipp & Hartquist (1994) in a study of shocks in interstellar clouds, we neglect inertial terms in the equations governing the ion and electron temperatures. If we assume that the only energy transfer to or from the ions occurs via collisions between ions and neutrals that are elastic, the net energy transfer rate (e.g. Draine 1986) to or from the ions is zero if

[EQUATION]

which we will assume gives [FORMULA] (e.g. Pilipp & Hartquist 1994). [FORMULA] is Boltzmann's constant. If we equate the rate of energy transfer (e.g. Draine 1986) to electrons that occurs via collisions with neutrals that are elastic to the rate that electrons lose energy via the collisionally induced excitation of [FORMULA] (e.g. Draine et al. 1983) we obtain [FORMULA] from

[EQUATION]

with

[EQUATION]

[FORMULA] (e.g. Massey 1969, p. 772). If (12a) yields [FORMULA] then we set [FORMULA] assuming that additional cooling processes for the electrons operate limiting the electron temperature accordingly. We specify [FORMULA].

The momentum transfer rates between the gaseous neutral fluid and each of the other fluids is evaluated as described by Draine (1986). The momentum transfer rates between neutral gas particles and grains are appropriate for both totally elastic scattering with specular reflection and for totally inelastic scattering.

Expressions for the [FORMULA] 's where j = i, e and d = b, sk are obtained straightforwardly from results given elsewhere (Havnes et al. 1987). They depend on the grain radius [FORMULA], the grain charge, [FORMULA], and [FORMULA] ; we assume the sticking coefficients of electrons and ions with all grains to be unity except when we clearly state otherwise. Note that because the [FORMULA] 's in this paper are the rate coefficients for the transfer of an elementary charge one obtains the [FORMULA] 's by multiplying the appropriate collision rate coefficients by -1.

We now focus on [FORMULA]. We assume that [FORMULA], [FORMULA], and [FORMULA] and consider the collision of a small grain carrying k elementary charges with a big grain. [FORMULA] is defined as the angle with respect to [FORMULA] of a vector from the center of the big grain to a point on the big grain's surface. If [FORMULA] then the small grain can strike the big grain only at points at which [FORMULA], and if [FORMULA] the small grains strike the big grains surface only at points at which at which [FORMULA]. Usually for the cases considered in this study we have [FORMULA]. However, if the small grains attain sufficiently negative charges and a large enough electric field obtains [FORMULA]. In the following we derive the charge transfer coefficient [FORMULA] and the charge transfer frequency [FORMULA] for [FORMULA], but the derivation of the corresponding coefficients for cases in which [FORMULA] is analogous. The surface charge density of the big grain before the collision is assumed to be

[EQUATION]

[FORMULA] is the radius of big grains. The change, [FORMULA], in the charge of a big grain in a collision with a small grain is assumed to be on average

[EQUATION]

[FORMULA] and [FORMULA] are introduced as free parameters describing the charge transfer in a big grain - small grain collision, and [FORMULA] is the radius of small grains. If we set [FORMULA] and [FORMULA] (Latham & Mason 1962) Eq. (14a) would give the Elster-Geitel charge transfer taking place in a collision between a big perfectly conducting sphere and a small perfectly conducting sphere. In fact, the actual amount of charge transferred in such a collision depends on the ratio of the collision-contact time to the electrical relaxation time. In the real case, this number is so small as to make the inductive charge transfer negligible. In any event, as will become clear, inductive charge transfer is wholly inadequate for producing lightning - even in the idealized case of instantaneous electrical relaxation as is assumed here - and charge transfer must be dominated by assumed noninductive processes if lightning is to be possible.

[FORMULA] is the change of the electrostatic potential on a small grain due to noninductive charge transfer; various possible noninductive charging mechanisms are reviewed by Morfill et al. (1993). For example, the average charge transferred from a small grain per collision by non-inductive processes is one elementary charge for [FORMULA] and for a small grain radius [FORMULA].

Clearly, since [FORMULA]

[EQUATION]

an approximation which we use everywhere in place of (14a). Thus,

[EQUATION]

We now consider the [FORMULA] 's. From the requirement of charge conservation it follows that if a small grain carrying the charge [FORMULA] hits a big grain at an angle [FORMULA] the charge that the small grain carries away is on the average (as averaged over many collisions at the same angle [FORMULA]) equal to [FORMULA]. However, in a particular big grain - small grain collision the small grain carries away an integral number of elementary charges rather than this average charge. In order to account for this fact we assume that the number of elementary charges that the small grain carries away is either k or ([FORMULA]) where the integral number k is defined by [FORMULA] and where the probability for carrying away k elementary charges is [FORMULA] and the probability for carrying away [FORMULA] elementary charges is [FORMULA].

In detail we proceed as follows: We define

[EQUATION]

and

[EQUATION]

We define [FORMULA] such that

[EQUATION]

We also define

[EQUATION]

From (13) and (14b) it follows that [FORMULA] with k' being the number of elementary charges which the small grain carries immediately before the collision with the big grain. Our assumptions for the number, k, of elementary charges which the small grain carries away after the collision can also be stated as follows:

[FORMULA] If a small grain collides with a big grain the number, k, of elementary charges that the small grain carries away lies between [FORMULA] and [FORMULA].

[FORMULA] If [FORMULA] then there are two intervals in the angular range [FORMULA], defined by the grid points given by (17a) to (17c), where the small grains may carry away the charge [FORMULA] after collision. If the collision angle [FORMULA] is between [FORMULA] and [FORMULA], then the probability that the small grain carries away a charge k e is assumend to be [FORMULA] ; if the collision angle [FORMULA] is between [FORMULA] and [FORMULA], then the probability that the small grain carries away a charge k e is assumed to be [FORMULA]. For [FORMULA] outside of these two intervals the probability for carrying away a charge k e by the small grain is assumed to vanish.

[FORMULA] If [FORMULA], then there is at most one interval of finite measure in the [FORMULA] range where the small grain may carry away a charge k e after collision. That is, for [FORMULA] the corresponding probability is assumed to be [FORMULA], but to vanish for [FORMULA]. For intervals of zero measure given by [FORMULA] with [FORMULA] we assume that the small grain carries away the charge [FORMULA] with the probability [FORMULA] where [FORMULA].

[FORMULA] Analogously, if [FORMULA], there may be again at most one interval of finite measure in the [FORMULA] - range where the small grain may carry away the charge k e after collision. That is, for [FORMULA], the corresponding probability is assumed to be [FORMULA] but to vanish for [FORMULA]. For intervals of zero measure given by [FORMULA] with [FORMULA] we assume that the small grain carries away the charge [FORMULA] with the probability [FORMULA] where [FORMULA].

Thus, with [FORMULA],

[EQUATION]

where [FORMULA] is the Kronecker [FORMULA], yielding for [FORMULA]

[EQUATION]

where (17a) and (17c) together with [FORMULA] and [FORMULA] have been taken into account. For [FORMULA] or [FORMULA] we have

[EQUATION]

Use of Eqs. (1), (2a), (3) through (5) and (15) and (19a), (19b) yields

[EQUATION]

where J, the current, is given by

[EQUATION]

Replacing (2a) by (2b) we get

[EQUATION]

rather than (20a). Of course, [FORMULA] is also a constant.

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© European Southern Observatory (ESO) 1998

Online publication: February 4, 1998
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