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

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5. Discussion

We assume that discharge occurs if the electric field strength is roughly one to several [FORMULA] (Pilipp et al. 1992) where [FORMULA] ; 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 [FORMULA] 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 [FORMULA] (see Eq. (13)) or if [FORMULA] [FORMULA] [FORMULA], which has been found to be well fulfilled for all cases presented in the present paper, so that the charge [FORMULA] 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 [FORMULA], [FORMULA] and [FORMULA] 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 [FORMULA], [FORMULA] and [FORMULA] 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 [FORMULA]. For our standard parameter [FORMULA], 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 [FORMULA] due to non-inductive charge transfer and loss of positive charge [FORMULA] due to electrostatic relaxation (see Eq. (14b)). This results in an electric field being directed into the positive z -direction at least for distances [FORMULA]. The positive charge on the big grains is only slightly reduced by impact of gas phase electrons for input parameters having the standard 1 A.U. values or the standard values for the dust enriched subdisk. For input parameters having the standard 5 A.U. values electron impact reduces the charge on big grains also only slightly for distances up to [FORMULA] but the positive charge on big grains is reduced significantly at large distances where the medium approaches local equilibrium. Additional charging processes for big grains due to the absorption of the charge [FORMULA] in a big grain - small grain collision (with [FORMULA] being the charge on the small grain before colliding with the big grain) and due to ion impact on the big grains are less important, at least for input parameters having values not considerably different from our standard values and for distances [FORMULA]. (See also Appendix C for a discussion of the relative importance of different grain charging mechanisms). Small grains are charged negatively in big grain - small grain collisions if the charge number [FORMULA] for big grains is still significantly lower than that given by Eq. (A15), (see Eqs. (18) and (A15)) and they are charged positively and negatively by collisions with gas phase ions and electrons. As [FORMULA] approaches the equilibrium value given by Eq. (A15) for large distances in the case of 1 A.U. parameters and dust enriched subdisk standard parameters then small grains are charged mainly by ion and electron impact but are discharged mainly in collisions with big grains.

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 [FORMULA] and [FORMULA] 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 E can be significantly enhanced, as compared to that derived from our standard parameters, if input parameters are varied. For example, increase of the absolute value of the free parameter [FORMULA] should result in a more effective charging of grains which should affect the generation of electric fields.

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 [FORMULA] can increase the electric field. However, there is a maximum at [FORMULA], and the electric field decreases again, if [FORMULA] (see Fig. 2b). An increase of the radius of small grains increases the electric field to a maximum value of about [FORMULA] at [FORMULA]. However, the electric field decreases drastically if [FORMULA] is increased above [FORMULA] (see Fig. 2d). Finally, variation of the radius [FORMULA] and specific mass density [FORMULA] 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 [FORMULA] have their standard values as given in the second and third column of Table 1, respectively, and where [FORMULA] is varied. These plots are not shown in the paper. We found that variation of the local equilibrium electric field with [FORMULA] 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 [FORMULA] up to a maximum of [FORMULA] at [FORMULA] but decreases rapidly with increasing [FORMULA] for [FORMULA]. The local equilibrium electric field strength is [FORMULA] at [FORMULA], 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 [FORMULA] up to a maximum of [FORMULA] at [FORMULA] but decreases rapidly with increasing [FORMULA] for [FORMULA].

Fig. 7a shows that an increase of the mass density [FORMULA] for the big grains leads to an increase in the local equilibrium electric field strength, up to [FORMULA] 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 [FORMULA] 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 [FORMULA] and [FORMULA], which were chosen as [FORMULA] and [FORMULA], we found a local equilibrium electric field strength of about [FORMULA].) 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 [FORMULA] in more detail we consider mostly negative values for this input parameter. The reason for this is that for [FORMULA] and a sufficiently large magnitude for [FORMULA] (e.g. for our standard value [FORMULA]) the electric field generated by charge separation is directed into the positive z -direction at least for distances [FORMULA] so that the ions move always into that direction and Eq. (1) can be integrated numerically along that direction for all electric field strengths. In contrast, if we choose [FORMULA] then the resulting electric field is directed into the negative z - direction yielding [FORMULA] for large electric field strengths. In this case integration of Eq. (1) along the positive z -direction becomes unstable and then we have to replace Eq. (1) by an approximation for which the term [FORMULA] is neglected. We performed also calculations for [FORMULA], with all other input parameters having their standard values (for which figures are not shown in this paper). From these calculations we find that for all input parameters (except for, of course, [FORMULA]) having their standard 1 A.U. parameters, or their standard parameters for the dust enriched subdisk, the switch of the value for [FORMULA] from [FORMULA] to [FORMULA] causes only slight differences between the curves for the variation of the absolute value of the electric field with distance. For the standard 5 A.U. parameters the deviations between corresponding curves for the absolute value of the electric field are significantly larger when the value for [FORMULA] is switched. However, both curves still agree at least within an order of magnitude. (For the distance range between [FORMULA] and [FORMULA] these deviations are in part due to a major error caused by neglecting the term [FORMULA] in Eq. (1)). We expect from these results that variation of electric field strength can be fully investigated at least within an order of magnitude if we consider only negative values for [FORMULA] but for [FORMULA] large enough to generate electric fields capable of inducing lightning.

A major effect of increase of [FORMULA] (as compared to its standard value) is that [FORMULA] reaches a higher value (see Eqs. (5) and (15) and note that [FORMULA], or see corresponding Eqs. (A13), (A15), (A16)). In addition, increase of [FORMULA] results also in an increase of the absolute value of the negative charge [FORMULA] carried away on the average by the small grains in big grain - small grain collisions if [FORMULA] 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 E at the position z where E reaches a peak value and probably also at distances beyond this position can be approximately derived from a balance between the conduction current [FORMULA] and the current [FORMULA] of the charged big grains (see Eq. (B3)), at least for input parameters not strongly different from our standard ones. Assuming now, for simplicity, that increase of [FORMULA] increases [FORMULA] but does not affect the particle number densities [FORMULA], the absolute value of the current [FORMULA] is increased but simultaneously [FORMULA] is decreased according to Eq. (A4). If [FORMULA] dominate the electrical conductivity [FORMULA] of the medium rather than the charged big grains then the electric field increases with increasing [FORMULA]. In contrast, for large charge numbers [FORMULA] for which [FORMULA] becomes dominated by the charged big grains we have [FORMULA] (see Eq. (A4)) resulting in a decrease of the electric field with increasing [FORMULA] as described by Eq. (B5). In fact, as mentioned above, also the absolute value of the average charge [FORMULA] carried away by the small grains may become larger if [FORMULA] is increased, so that the contribution of the charged small grains to the electrical conductivity increases, leading to a decrease of [FORMULA] in addition to that caused by the increase of [FORMULA]. This tends to reduce the peak electric field. We find from numerical calculations for the standard 1 A.U. parameters, but varying [FORMULA], that the maximum value of [FORMULA] obtained is largest when [FORMULA] has approximately its standard value.

Similar results are found from numerical calculations for input parameters (excepting [FORMULA]) 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 [FORMULA] for the small dust grains is varied and all other input parameters have their standard values for the respective models. For [FORMULA] having a value larger than its standard value the increase of [FORMULA] with distance z is lower and [FORMULA] reaches a smaller value for local equilibrium at large distances (where [FORMULA]) if non-inductive charge transfer processes dominate charging of big grains (see Eqs. (A13), (A15), and (A16)). Then, the absolute value of the current [FORMULA] is decreased. Simultaneously, increase of [FORMULA] results in a decrease of the mobility of small grains due to [FORMULA] which tends to reduce the contribution of charged small grains to the electrical conductivity of the medium. Also the contribution of the charged big grains to the electrical conductivity is reduced if [FORMULA] is reduced. On the other hand, increase of [FORMULA] reduces the surface area per unit mass and thus the absorption of free electrons and ions. As a consequence [FORMULA] and [FORMULA] increase, raising the contribution of gas phase ions and electrons to the electrical conductivity. Numerical results show that for the standard 1 A.U. parameters (except [FORMULA] which is varied) the peak electric field increases from [FORMULA] for the standard value [FORMULA] to about twice of this value for [FORMULA] (compare Figs. 1 and 3) mainly due to the decrease of the mobility of small grains and to the decrease of the charge on the big grains. As [FORMULA] is increased further, up to [FORMULA], the peak electric field decreases to a value an order of magnitude smaller than that for the standard case. This is mainly due to the strong increase of gas phase ion and electron number densities and the resultant increase in conductivity (Figures for the latter results are not shown).

Whereas variation of the input parameters [FORMULA] or [FORMULA] 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 [FORMULA] of big grains have their standard values and [FORMULA] is varied. In case of local equilibrium the electric field is not affected significantly if the input parameter [FORMULA] 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 [FORMULA] is increased. The magnitude of the current [FORMULA] increases as [FORMULA] increases with distance z according to Eq. (A13) reaching an equilibrium value at large distances which does not significantly vary with [FORMULA] if the local equilibrium value of [FORMULA] can be approximated by Eq. (A15). However, the time scale [FORMULA] or the length scale [FORMULA] for the big grains to reach charge equilibrium decreases with increasing value for [FORMULA] as [FORMULA] (see (A16)), so that the magnitude of the current [FORMULA] of charged big grains increases faster with distance z towards its equilibrium value if [FORMULA] is increased. On the other hand, small grains tend to gain less charge [FORMULA] in collisions with big grains if [FORMULA] is nearer to the approximate equilibrium value given by (A15) (see Eqs. (18) and (A15)), contributing less to the electrical conductivity of the medium. Also charged big grains may contribute less to the electrical conductivity if [FORMULA] is increased. In fact, numerical calculations which we performed (but for which we do not show figures in this paper) give a peak electric field that is [FORMULA], [FORMULA], and [FORMULA], if all input parameters but [FORMULA] have their standard 1 A.U., standard 5 A.U. and standard dust enriched subdisk values, respectively, and [FORMULA]. (These values are to be compared with the peak electric field strengths of [FORMULA], [FORMULA], and [FORMULA] for corresponding standard values shown by the upper curve in Fig. 1 and by the curves in Figs. 5 and 6, respectively.) If big grains are the precursors of chondrules then we expect the mass [FORMULA] to be about [FORMULA] implying [FORMULA] to be of the order of [FORMULA] if [FORMULA] has about its standard value of [FORMULA]. Nevertheless, the calculations do indicate that the electric field may have been significantly larger than derived from our standard parameters if a non-negligible fraction of big grains with [FORMULA] were present in addition to the precursors of chondrules.

If all input parameters but [FORMULA] and [FORMULA] have their standard values and [FORMULA] and [FORMULA] are varied simultaneously such that [FORMULA] is always [FORMULA] then [FORMULA] and [FORMULA] increase with [FORMULA] as [FORMULA] (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 [FORMULA] 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 [FORMULA] 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, [FORMULA], that can be heated sufficiently to melt meteoritic material is

[EQUATION]

where [FORMULA] is the energy released per molecule in the discharge channel required for chondrule formation, [FORMULA] is Boltzmann's constant and [FORMULA]. 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 [FORMULA] times the timescale to build the field up to breakdown strength, [FORMULA]. The requirement that this time was comparable to or less than the lifetime of the protosolar nebula, [FORMULA], gives that

[EQUATION]

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 [FORMULA] 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 [FORMULA] must be about [FORMULA] or larger. Much higher values for [FORMULA] 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 [FORMULA] and [FORMULA] for the medium not affected by lightning and with [FORMULA], [FORMULA] ranging from [FORMULA] to [FORMULA] but [FORMULA] for the initial state of the fully ionized medium within the discharge channel. Taking into account a dissociation energy of [FORMULA] for the [FORMULA] molecules, an ionization energy of [FORMULA] for the H atoms and requiring [FORMULA] to [FORMULA], the energy [FORMULA] released per molecule ranges from 47 eV to 92 eV. These values are two orders of magnitude larger than [FORMULA]. 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 [FORMULA] per molecule necessary to melt dust particles may be even smaller than [FORMULA] if the solid mass contains only a small fraction of the total mass. Thus the value for [FORMULA] is rather uncertain.

The upper curve in Fig. 1 shows a maximum of [FORMULA] at [FORMULA], and then a slow decay of E with increasing z where the electric field reaches half of its maximum value at [FORMULA]. The time scale [FORMULA] as defined by (A4) was numerically determined to be [FORMULA] at [FORMULA] and [FORMULA] at [FORMULA]. According to condition (23) the time scale [FORMULA] to build up the electric field up to break down field strength [FORMULA] should be smaller than [FORMULA] if we set [FORMULA] and smaller than [FORMULA] if we set [FORMULA]. Thus, we see from [FORMULA] that if lightning were to occur as described in this paper, then condition (23) should have been satisfied for our standard 1 A.U. parameters given in Table 1 if [FORMULA] is not significantly larger than [FORMULA]. In case of [FORMULA] as resulting from the assumptions by Horanyi et al. (1995) only one percent or less of the volume where the electric field is built up to break down strength is sufficiently heated.

Pilipp et al. (1992) derived an estimate for the lower limit of the lengthscale L of a region in which the electric field is strong enough to induce lightning if lightning could give rise to sufficient heating in the discharge channels for melting chondrules. According to their arguments the maximum energy released per molecule in the discharge channel during discharge is given by

[EQUATION]

where w is the width of the lightning bolt and where the length of the lightning bolt is assumed to be L. Since the thickness of a lightning bolt in the Earth's atmosphere is typically few thousand electron mean free paths Pilipp et al. assumed that also the width w of a discharge channel in the protosolar nebula is of the same magnitude, i.e. [FORMULA], if the lightning bolt has reached a length larger than [FORMULA]. Here, we use a similar estimate for the value of w. Requiring that [FORMULA] we get from (24)

[EQUATION]

Condition (25) yields [FORMULA] if we set [FORMULA] and [FORMULA]. From [FORMULA] we see that in this case also condition (25) is satisfied for [FORMULA].

Note that the distance [FORMULA] as derived from our stationary model depends on the bulk velocity [FORMULA] which in case of our standard 1 A.U. parameters is about 5% of the sound velocity [FORMULA]. Since the time scales for the variation of the electric field should not significantly depend on the turbulent gas velocity [FORMULA] (see also Appendix A for relevant time scales) and since the time scales are converted to appropriate length scales by multiplication with [FORMULA] for the electric field we expect that the distance [FORMULA] varies approximately as [FORMULA]. That is, for [FORMULA] not significantly larger than [FORMULA] condition (25) is satisfied already for moderate turbulent velocities [FORMULA], whereas for the large turbulent velocities assumed by our standard parameters condition (25) is fulfilled even if [FORMULA] is as large as [FORMULA].

Similar conclusions can be drawn from Fig. 3 and corresponding time scales [FORMULA]. Here, condition (23) is again fairly well fulfilled and condition (25) is fulfilled probably for even lower turbulent velocities [FORMULA] than in case of the upper curve in Fig. 1 if [FORMULA] is assumed.

As mentioned above, numerical calculations performed by us (but for which figures are not shown) for which all input parameters but [FORMULA] have the standard 1 A.U. values and [FORMULA], give a maximum electric field [FORMULA] with respect to variation with distance z. In this case, conditions (23) and (25) are fulfilled for even higher values of [FORMULA] and/or probably lower turbulent gas velocities. On the other hand, we find from curves shown in Fig. 4a and corresponding time scales [FORMULA] that the maximum electric field [FORMULA] decreases with decreasing specific mass density [FORMULA] but constant mass [FORMULA] for the big grains and that conditions (23) and (25) can be fulfilled only if [FORMULA] decreases with decreasing value for [FORMULA].

For Fig. 5, the maximum of the electric field strength of about [FORMULA] occurs at [FORMULA], and the electric field strength decays to [FORMULA] at [FORMULA]. The time scale [FORMULA] was determined to be [FORMULA] at [FORMULA] and [FORMULA] at [FORMULA]. For [FORMULA] as is the case for our standard 5 A.U. parameters, condition (23) yields [FORMULA] if we set [FORMULA] and [FORMULA] if we set [FORMULA]. That is, from [FORMULA] condition (23) fails by an order of magnitude for [FORMULA]. Condition (25) requires [FORMULA] for [FORMULA] and [FORMULA] which is barely fulfilled for [FORMULA].

From numerical calculations for which all input parameters but [FORMULA] have their standard 5 A.U. values and [FORMULA] (mentioned above) we found that [FORMULA] and that both conditions (23) and (25) can be fulfilled if [FORMULA] is of the order of [FORMULA].

For Fig. 6, the maximum of the electric field strength of about [FORMULA] occurs at [FORMULA], and the electric field strength decays to [FORMULA] at [FORMULA]. The time scale [FORMULA] was determined to be [FORMULA] at [FORMULA] and [FORMULA] at [FORMULA]. For [FORMULA] as is the case for our standard parameters for the dust enriched subdisk condition (23) yields [FORMULA] if we set [FORMULA] and [FORMULA] if we set [FORMULA]. Thus, condition (23) should be fulfilled at least if [FORMULA] is not significantly larger than [FORMULA]. However, condition (25) requiring [FORMULA] for [FORMULA] and for [FORMULA] fails by two orders of magnitude for [FORMULA]. At large distances, say at [FORMULA], the electric field has decreased to about [FORMULA], where the time scale [FORMULA] was determined to be [FORMULA] and where conditions (23) and (25) require [FORMULA] and [FORMULA] respectively. Again, condition (23) is fairly well fulfilled whereas condition (25) fails at least by a factor 5 for [FORMULA].

From numerical calculations for which all input parameters but [FORMULA] have their standard values for the dust enriched subdisk and [FORMULA] (mentioned above) we found that [FORMULA] and that condition (23) can easily be fulfilled for [FORMULA]. Condition (25) fails by more than an order of magnitude if [FORMULA] and if the lenth [FORMULA] between the position [FORMULA] where the electric field reaches its peak value and the position [FORMULA] beyond [FORMULA] 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 [FORMULA] from position [FORMULA] where [FORMULA] and position [FORMULA] beyond [FORMULA] where the electric field has decreased to [FORMULA] is compared with the right hand side of relation (25) then condition (25) is fairly well fulfilled for [FORMULA].

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Online publication: February 4, 1998
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