## 3. Momentum transfer and charge transfer coefficientsThe temperatures, , 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 which we will assume gives (e.g. Pilipp & Hartquist 1994). 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 (e.g. Draine et al. 1983) we obtain from with (e.g. Massey 1969, p. 772). If (12a) yields then we set assuming that additional cooling processes for the electrons operate limiting the electron temperature accordingly. We specify . 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 's where We now focus on . We assume that
, , and
and consider the collision of a small grain
carrying is the radius of big grains. The change, , in the charge of a big grain in a collision with a small grain is assumed to be on average and are introduced as free parameters describing the charge transfer in a big grain - small grain collision, and is the radius of small grains. If we set and (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. 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 and for a small grain radius . an approximation which we use everywhere in place of (14a). Thus, We now consider the 's. From the requirement
of charge conservation it follows that if a small grain carrying the
charge hits a big grain at an angle
the charge that the small grain carries away is
on the average (as averaged over many collisions at the same angle
) equal to . 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
In detail we proceed as follows: We define From (13) and (14b) it follows that with
If a small grain collides with a big grain
the number, If then there are
two intervals in the angular range , defined by
the grid points given by (17a) to (17c), where the small grains may
carry away the charge after collision. If the
collision angle is between
and , then the
probability that the small grain carries away a charge If , then there is
at most one interval of finite measure in the
range where the small grain may carry away a charge Analogously, if ,
there may be again at most one interval of finite measure in the
- range where the small grain may carry away
the charge Thus, with , where is the Kronecker , yielding for where (17a) and (17c) together with and have been taken into account. For or we have Use of Eqs. (1), (2a), (3) through (5) and (15) and (19a), (19b) yields where rather than (20a). Of course, is also a constant. © European Southern Observatory (ESO) 1998 Online publication: February 4, 1998 |