## 5. DiscussionWe assume that discharge occurs if the electric field strength is roughly one to several (Pilipp et al. 1992) where ; then a nonnegligible fraction of electrons have energies exceeding the ionization potential of the neutral gas. (However, the critical field strength for breakdown is uncertain, and Gibbard et al. 1997 advocate a higher value.) Inspection of the results presented in the figures in the previous section shows that discharges might have occurred in the protosolar nebula; however, according to Fig. 1 discharges occurred only if processes other than gas phase ion and electron collisions with grains and the Elster-Geitel mechanism also operated to charge the grains and local equilibrium did not obtain even if the ionization rate was low compared to the rates that obtain in interstellar clouds and approached the rate provided by the decay of trapped in grains (e.g. Umebayashi & Nakano 1990). Though charging mechanisms other than the Elster-Geitel effect may well have operated, the conclusion that lightning's existence in the protosolar nebula depended on the functioning of mechanisms that remain unidentified is an unsatisfying one. We find from our calculations that polarization of the grains' surface charge distribution by the external electric field has a negligible effect on the charge transfer process. In fact, electrostatic polarization of the grain material produces only a small change of the surface charge density of big grains if (see Eq. (13)) or if , which has been found to be well fulfilled for all cases presented in the present paper, so that the charge transferred to a big grain in a big grain - small grain collision is not significantly affected by polarization (see Eqs. (13) and (14b)). Thus, we conclude that the Elster-Geitel mechanism is unimportant for charging of the grains in the protosolar nebula. More important grain charging mechanisms result from impact of gas phase ions and electrons on the grains as well as electrostatic relaxation in big grain - small grain collisions which are the dominant grain charging mechanisms for the cases shown by the curves marked , and in Fig. 1. However, as mentioned above, grain charging which is effective
enough to generate electric fields capable of inducing lightning could
occur only if additional charging mechanisms operate. Therefore, most
of our results presented in this paper, except for the curves marked
, and
in Fig. 1, are based on the assumption that in
big grain - small grain collisions non-inductive charge transfer
processes occur which we describe by the free parameter
. For our standard parameter
, big grains are charged positively as they
move upward and reach a positive equilibrium charge at large distances
which is mainly determined by the balance between gain of positive
charge due to non-inductive charge transfer
and loss of positive charge due to
electrostatic relaxation (see Eq. (14b)). This results in an electric
field being directed into the positive It should be mentioned that chondrule precursors were not spheres but were probably irregular fractal aggregates, whereas for our models we assume that the dust particles were spherical. Charges on such irregular bodies would produce highly nonuniform electric fields in the immediate neighborhood of the dust particles so that charge transfer would be unlike that expected for uniform spheres. However, for our model calculations we treat the non-inductive grain - grain charge transfer, which is mainly responsible for the generation of strong global electric fields, by the two free parameters and which allow us to vary the charge on the big grains. The uncertainties which the assumption of spherical dust grains add to our treatment is therefore included in the uncertainties for the choice of these free parameters. The question arises whether the strength of the global electric
field Figs. 2 and 4b show that, at least for a medium in local equilibrium, variation of several relevant parameters which could significantly deviate from our standard parameters given in the first column of Table 1 allow only a limited increase of the electric field. As expected, increase of the ionization rate decreases the electric field (see Fig. 2a), so that choosing the minimum ionization rate allowed by physical considerations should yield the maximum electric field strength. Increase of the absolute value of the parameter can increase the electric field. However, there is a maximum at , and the electric field decreases again, if (see Fig. 2b). An increase of the radius of small grains increases the electric field to a maximum value of about at . However, the electric field decreases drastically if is increased above (see Fig. 2d). Finally, variation of the radius and specific mass density of big grains has only minor effects on the local equilibrium values of the electric field strength (see Figs. 2c and 4b). Analogous to the plot presented in Fig. 2d, we also produced plots for the variation of the equilibrium electric field for the protosolar nebula at 5 A.U. distance from the Sun and for the dust enriched central disk where all input parameters but have their standard values as given in the second and third column of Table 1, respectively, and where is varied. These plots are not shown in the paper. We found that variation of the local equilibrium electric field with for the models corresponding to 5 A.U. distance from the Sun and to the dust enriched central disk is largely similar to that shown in Fig. 2d: In the case of the protosolar nebula at 5 A.U., the local equilibrium electric field strength increases with up to a maximum of at but decreases rapidly with increasing for . The local equilibrium electric field strength is at , i.e. if all input parameters have their standard values. In the case of the dust enriched central disk, the local equilibrium electric field strength increases with up to a maximum of at but decreases rapidly with increasing for . Fig. 7a shows that an increase of the mass density for the big grains leads to an increase in the local equilibrium electric field strength, up to for a dust enriched central disk. In contrast, Fig. 7b shows that the local equilibrium electric field strength in this central plane does not vary strongly if the mass density for the small grains is varied around its standard value within a wide range. These results indicate that in most cases high electric field strengths capable of inducing lightning are not reached for local equilibrium conditions, although local equilibrium electric field strengths approaching the breakdown field have been found for a narrow range of parameters. (For example, for standard parameters appropriate for a distance of 1 A.U. as given in the first column of Table 1, except for the input parameters and , which were chosen as and , we found a local equilibrium electric field strength of about .) If the small grains are far from local charge equilibrium, however, so that the electrical conductivity of the medium is much lower than that in local equilibrium, sufficiently high field strengths can be generated more easily. We investigated also whether the electric field strength can be significantly increased by variation of input parameters if local equilibrium did not obtain. In order to discuss the dependence of the electric field strength
on in more detail we consider mostly negative
values for this input parameter. The reason for this is that for
and a sufficiently large magnitude for
(e.g. for our standard value
) the electric field generated by charge
separation is directed into the positive A major effect of increase of (as compared to its standard value) is that reaches a higher value (see Eqs. (5) and (15) and note that , or see corresponding Eqs. (A13), (A15), (A16)). In addition, increase of results also in an increase of the absolute value of the negative charge carried away on the average by the small grains in big grain - small grain collisions if is still well below the value given by (A15) (see Eqs. (18) and (A15)). Finally, as a consequence of these changes of grain charges several additional mutual interactions as described by the equations in Sects. 2 and 3 between grains, gas phase ions, gas phase electrons, and the electric field change the ionization structure of the medium. As discussed in Appendix B, the electric field Similar results are found from numerical calculations for input parameters (excepting ) at the standard 5 A.U. values, as well as the standard values for the dust enriched subdisk. The corresponding peak electric field shown in Figs. 5 and 6, respectively, is near or only slightly below the maximum possible. Analogous results are derived if the radius
for the small dust grains is varied and all other input parameters
have their standard values for the respective models. For
having a value larger than its standard value
the increase of with distance Whereas variation of the input parameters or
leads to similar conclusions for the electric
field variation in case of local equilibrium (as shown in Figs. 2b and
2d) and for the variation of the peak electric field with respect to
distance (if local equilibrium does not obtain), significant diffences
arise if all input parameters but the radius of
big grains have their standard values and is
varied. In case of local equilibrium the electric field is not
affected significantly if the input parameter
is varied within a wide range of values (see Fig. 2c). In contrast, if
the medium is not in local equilibrium then several transient effects
allow the build up of a peak electric field, which is significantly
increased compared to the standard value, when
is increased. The magnitude of the current
increases as increases with distance If all input parameters but and have their standard values and and are varied simultaneously such that is always then and increase with as (see Eq. (A16)) resulting in transient effects with respect to grain charging which are opposite to those occurring if all input parameters have their standard values but only is varied. This results in a peak electric field being smaller than that observed for our standard values although the electric field shows only a slight dependence on in case of local equilibrium as can be seen from Fig. 4. We may use the timescales required for the establishment of an electric field strong enough to induce discharge to consider whether discharge might have been sufficiently frequent to have affected a significant amount of material in the protosolar nebula as required by the ubiquity of chondrules (e.g. Morfill et al. 1993). A rough estimate of the maximum possible fraction of the volume of a region in which the electric field is at breakdown strength, , that can be heated sufficiently to melt meteoritic material is where is the energy released per molecule in the discharge channel required for chondrule formation, is Boltzmann's constant and . The amount of time required for lightning to heat a substantial fraction of the volume in any region in the disk to the degree required for chondrule formation would be times the timescale to build the field up to breakdown strength, . The requirement that this time was comparable to or less than the lifetime of the protosolar nebula, , gives that A discharge event does not alter the flow or charge distribution throughout very much of the storage volume from which the discharge draws electric energy; hence, once flow and charge structures are set up in a storage volume, discharge should repeatedly reoccur at intervals of about as defined in Appendix A. For a model of lightning strokes where only a small fraction of neutrals is ionized within the discharge channel and where the total energy released by lightning heats gas and dust to the same temperature within the channel the energy must be about or larger. Much higher values for result from assumptions by Horanyi et al. (1995) who performed model calculations for the expansion, cooling and recombination of an initially fully ionized discharge channel and for the energy flux reaching the surface of an embedded dust grain and its subsequent heating. They found that short duration melting and rapid cooling of dust particles can occur in lightning discharges if lightning occurs in the solar nebula where and for the medium not affected by lightning and with , ranging from to but for the initial state of the fully ionized medium within the discharge channel. Taking into account a dissociation energy of for the molecules, an ionization energy of for the H atoms and requiring to , the energy released per molecule ranges from 47 eV to 92 eV. These values are two orders of magnitude larger than . On the other hand, solid material may also be molten by radiation pulses such as considered by Eisenhour & Buseck (1995) where radiation may be produced by lightning and may be predominantly absorbed by dust grains within and in the neighbourhood of lightning strokes. In this case the energy input per molecule necessary to melt dust particles may be even smaller than if the solid mass contains only a small fraction of the total mass. Thus the value for is rather uncertain. The upper curve in Fig. 1 shows a maximum of
at , and then a slow
decay of Pilipp et al. (1992) derived an estimate for the lower limit of the
lengthscale where Condition (25) yields if we set and . From we see that in this case also condition (25) is satisfied for . Note that the distance as derived from our stationary model depends on the bulk velocity which in case of our standard 1 A.U. parameters is about 5% of the sound velocity . Since the time scales for the variation of the electric field should not significantly depend on the turbulent gas velocity (see also Appendix A for relevant time scales) and since the time scales are converted to appropriate length scales by multiplication with for the electric field we expect that the distance varies approximately as . That is, for not significantly larger than condition (25) is satisfied already for moderate turbulent velocities , whereas for the large turbulent velocities assumed by our standard parameters condition (25) is fulfilled even if is as large as . Similar conclusions can be drawn from Fig. 3 and corresponding time scales . Here, condition (23) is again fairly well fulfilled and condition (25) is fulfilled probably for even lower turbulent velocities than in case of the upper curve in Fig. 1 if is assumed. As mentioned above, numerical calculations performed by us (but for
which figures are not shown) for which all input parameters but
have the standard 1 A.U. values and
, give a maximum electric field
with respect to variation with distance
For Fig. 5, the maximum of the electric field strength of about occurs at , and the electric field strength decays to at . The time scale was determined to be at and at . For as is the case for our standard 5 A.U. parameters, condition (23) yields if we set and if we set . That is, from condition (23) fails by an order of magnitude for . Condition (25) requires for and which is barely fulfilled for . From numerical calculations for which all input parameters but have their standard 5 A.U. values and (mentioned above) we found that and that both conditions (23) and (25) can be fulfilled if is of the order of . For Fig. 6, the maximum of the electric field strength of about occurs at , and the electric field strength decays to at . The time scale was determined to be at and at . For as is the case for our standard parameters for the dust enriched subdisk condition (23) yields if we set and if we set . Thus, condition (23) should be fulfilled at least if is not significantly larger than . However, condition (25) requiring for and for fails by two orders of magnitude for . At large distances, say at , the electric field has decreased to about , where the time scale was determined to be and where conditions (23) and (25) require and respectively. Again, condition (23) is fairly well fulfilled whereas condition (25) fails at least by a factor 5 for . From numerical calculations for which all input parameters but have their standard values for the dust enriched subdisk and (mentioned above) we found that and that condition (23) can easily be fulfilled for . Condition (25) fails by more than an order of magnitude if and if the lenth between the position where the electric field reaches its peak value and the position beyond where the electric field has decreased to half of its peak value is compared with the right hand side of relation (25). However, if the distance range from position where and position beyond where the electric field has decreased to is compared with the right hand side of relation (25) then condition (25) is fairly well fulfilled for . © European Southern Observatory (ESO) 1998 Online publication: February 4, 1998 |