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

4. Numerical results

In the protosolar disk the gravitational forces due to the gravitational field of the Sun are largely balanced by the centrifugal forces of the orbiting medium. However, in a frame comoving with the gas of neutral molecules there is an effective gravitational field balancing the pressure gradients in the gas, i.e. the effective gravitational field is equal to the gas pressure gradient divided by the gas density. In this frame the grains are subjected to this effective gravitational field. For our model calculations we assume that there is some turbulent motion of the gas superimposed on its orbiting motion. We also assume that in a frame comoving with the orbiting gas motion the turbulent gas velocity is constant, at least within a certain distance range which is considered to be small compared to the thickness of the protosolar disk and for which we seek stationary solutions for the electric field. (See Weidenschilling & Cuzzi 1993 for likely length scales of turbulent motion).

We consider three sets of models. One consists of models for the protosolar nebula at a distance of 1 A.U. from the Sun and another is composed of models for the protosolar nebula at a distance of 5 A.U. (approximately the current radius of the Jovian orbit around the Sun). For any model belonging to either of these two sets we have assumed that the effective gravity is directed perpendicular to and towards the midplane of the protosolar nebula and that the turbulent neutral gas motion is away from it. The third set consists of models for the dust enriched subdisk. A dust enriched subdisk may be sufficiently thin that the effective gravitational force component perpendicular to the disk is weak relative to the effective gravitational force component in the radial direction. In models for a dust enriched subdisk we assume that the dominant component of the effective gravitational field is towards the Sun and the turbulent neutral motion is away from the Sun.

In all models, at the following boundary conditions hold:

At , is specified to be 0 except when we clearly state otherwise. The gas phase ion and electron number densities at are given by the solution of Eq. (1) with the derivative set to zero and Eq. (2b), while is simultaneously determined through the use of the condition that the total current is zero.

Table 1 gives the model parameters adopted for models in the three different sets considered.

Table 1. Standard model parameters

The ionization rates given in Table 1 are much lower than those usually assumed for molecular clouds if ionization is mainly provided by cosmic rays. We assume that during the formation of planets, the strong T-Tauri wind of the Sun has prevented ionizing cosmic rays from entering the solar system and that ionization is only due to decays of long-lived radioactive elements trapped in the grains (see Umebayashi & Nakano 1990). Thus, the ionization rates in Table 1 are lower limits.

The turbulent neutral velocity given in Table 1 is about 5 per cent of the sound velocity for the first column and about 11 per cent of the sound velocity for the second and third columns of the table. This is smaller than convection velocities suggested by early solar disk models but larger by about one order of magnitude for convective velocities derived from more recent analyses of convective instability modes in a rotating disk (for a review see Weidenschilling & Cuzzi, 1993). The value for the velocity of turbulent gas motion is very uncertain but could be as high as assumed by us, at least if extrinsic stirring mechanisms exist.

Since the grain fluxes and are conserved (as can be derived from Eq. (3) and according to Eq. (4)) but the bulk grain velocities as well as the grain number densities or grain mass densities may vary with distance z, we adopt standard parameters for the grain mass densities at the boundary. In Table 1, standard parameters for the ratios of grain mass densities at a distance z to neutral mass density are expressed as the corresponding adopted ratios at the boundary times respective ratios and of grain bulk velocities at the boundary to respective grain bulk velocities ocurring at distance z.

Fig. 1 gives the electric field as a function of distance for several different models for conditions at 1 A.U. from the Sun. The upper curve, which is not marked, is for the standard parameters given in the first column of Table 1 and the boundary conditions given above. The curve marked is for standard parameters, except . The one marked is for standard parameters except and at . The curve marked is for standard parameters except that and the electron sticking coefficients for small grains is 0.3 instead of unity. The thin horizontal lines at the right hand side of the figure indicate the electric field as derived numerically for the local equilibrium (i.e. all gradients in the equations and the current J defined by Eq. (20b) are zero) for each model. Appendix A treats the timescales for the generation of the electric field and for the attainment of ionization equilibrium while Appendix B concerns upper bounds and the equilibrium values of the electric field. Results in those appendices are relevant to the understanding of the numerical results displayed in Fig. 1 and other figures. Timescales are converted to the appropriate lengthscales by multiplication with . (This is exact for the generation of electric field and is approximately true for attainment of ionization equilibrium of dust particles where , but does not necessarily hold for attainment of ionization equilibrium of ions and electrons because and deviate drastically from for large magnitudes of the electric field. Only in case of the three marked curves in Fig. 1 corresponding to , the ion and electron bulk velocities stay near due to the very low electric field strengths. Note that in case of the curve marked the time scale as given by (A12) must be multiplied by the factor since in the Appendices all sticking coefficients are assumed to be unity but lower electron sticking factors for small grains yield lower recombination rates for electrons increasing ).

Fig. 1. The Electric Field as a Function of Distance for Models Appropriate for a Position at 1 A.U. from the Sun. Unless noted otherwise all parameters have the standard values given in the first column of Table 1 and the standard boundary conditions apply. The curves marked , and are for models with . For the case at . For the case the electron sticking coefficient on small grains was taken to be 0.3 rather than unity. Dashed portions of the curve indicate that the electric field is negative. . The thin horizontal lines at the right hand side of the figure indicate the electric field for the local equilibrium

The physical processes operating to produce the electric field variations can be described as follows:

As argued in Appendix A the total current J can be decomposed into a sum of several partial currents: 1) the convection current , where is the total space charge density (Q is defined by (A1)), 2) the current due to the movement of the charged big grains as seen in the rest frame of the neutrals which would occur if the only forces to which the big grains were subjected were gravity and friction with the neutrals, 3) the current due to the movement of charged small grains as seen in the rest frame of the neutrals which would occur if the only forces to which the small grains were subjected were gravity and friction with the neutrals, and 4) the conduction current containing the response of all charged particles to the external electric field E with the time scale defined by (A4).

Right at the boundary (not shown in the figure where results are presented only for ) we have and .

In the case of the upper curve corresponding to standard parameters and standard boundary conditions, at the only charged particles are ions, electrons and few charged small grains carrying one negative elementary charge. Then, from (A3) it follows that if then producing according to Poisson's equation a negative electric field E as the medium moves to distances . However, at the same time as the medium moves upward, the big grains are charged positively in collisions with small grains and small grains colliding with big grains are charged negatively due to noninductive charge transfer. This effect results in a positive total charge density for distances z beyond but for distances not much larger than . Here, the positive charge density for big grains and the negative charge density for small grains are about of the same magnitude but since is smaller than by 4 orders of magnitude the positive charge density of big grains dominates in accordance with (A3). In the distance range we have roughly a balance between the convection current and the current of charged big grains whereas the other currents are of minor importance.

For distances significantly beyond the conduction current can no longer be neglected and eventually leads to a negative charge density at large distances. For distances near where the electric field reaches maximum and for distances beyond, variation of the electric field can be approximately described by the balance between the conduction current and the current of charged big grains as given by (B3) (see Appendix B for more details). In accordance with (A13) to (A16) the positive charge on big grains increases nearly linearly with z up to a distance and then approaches a local charge equilibrium approximated by (A15) (i.e. the gradient in (5) vanishes) at distances . In fact, the local equilibrium charge of big grains is slightly lower than given by (A15) due to electron impact not being taken into account in (A15). Most of small grains hitting big grains are discharged in the collisions if the big grains carry a charge near their local equilibrium charge because, in this case, the value of the average charge carried away by small grains is small compared to one elementary charge (see also (18)) but grains can carry only an integral number of elementary charges. A small fraction of small grains hitting the big grains carry away one negative elementary charge since is negative, and there are no small grains carrying away a charge different from either zero or one negative elementary charge, at least for electric field strengths not much larger than considered in this paper for which electrostatic polarization of grain material is negligible. In this case, small grains are charged positively by ion impact and they are charged negatively by electron impact and by collisions with big grains and they are mainly discharged by collisions with big grains. Thus, the number densities of both positively and negatively charged small grains increase significantly with distance up to about but approach local charge equilibrium (i.e. vanishing gradients in (3)) at distances much larger than that distance in accordance with (A18).

The decay of the electric field beyond its maximum is caused by the increase of the charge on big grains and the increase of the number densities of charged small grains increasing the electrical conductivity of the medium. Whereas for small distances (i.e. for ) the electrical conductivity is dominated by gas phase ions and electrons, charged grains contribute the largest share to the electrical conductivity at the maximum of the electric field and for distances beyond of it. For distances small charged grains dominate the electrical conductivity of the medium and increase of their number densities with distance dominates the decay of the electric field accordingly. For local charge equilibrium as it should occur at distances the electric field eventually reaches its equilibrium value which has been numerically determined to be about .

The total current J is negligible compared to the current although it does not vanish for all distances in spite of at the boundary. The main reason for a finite total current is that for the upper curve in Fig. 1 (and also for the curves in Figs. 3, 4a, 5, and 6) we calculate the electron number density from (2b) rather than from (2a) as mentioned in Sect. 2 leading to (20c) rather than to (20a). That is, is conserved so that with being the number density of gas phase electrons at the boundary. For , as should be the case for , the total current J increases the electric field according to Eq. (A3). However, for distances close to and beyond the maximum in E, the total current is smaller than the sum by two orders of magnitude or more so that the electric field is only slightly affected by the finite total current there. Only for small distances, i.e. , it is found that J is larger than S, by a factor of order unity.

For the curves marked and , the boundary conditions are the same as those for the upper curve, again implying a negative space charge density at . However, in contrast to the upper curve, the big grains are charged negatively in collisions with other charged particles as the medium moves upward resulting in a negative total charge density for distances z beyond but not much larger than , and consequently in a negative electric field. Similarly as for the case of standard parameters (upper curve), the number density of charged small grains increases with distance due to ion and electron impact but here the electrical conductivity is dominated by gas phase ions and electrons for distances ranging from the boundary up to the distance where the maximum of the electric field strength occurs and is about constant in this distance range. Only at distances well beyond the maximum electric field strength (i.e. for ) do the charged small grains contribute the largest share to the electrical conductivity decreasing significantly whereas contribution of the charged big grains to is always neglibible. Since here the electric field is in the negative direction, the effect of the conduction current is to increase space charge density according to (A3) and eventually produces a positive space charge density at large distances. The magnitude of the electric field shown by the curve marked is higher than that shown by the curve marked because for the smaller electron sticking coefficients for small grains the recombination rate of electrons is smaller resulting in a larger electron number density. As a consequence electron impact on grains is increased so that big grains for which the electron sticking coefficient is still assumed to be equal to one gain a negative charge of larger magnitude implying a larger magnitude of negative space charge density according to (A3). Finally, the positive electric field at small distances for the case corresponding to the curve marked is a consequence of positive charge on big grains prescribed at . As can be seen from Fig. 1 both curves marked and end up in the same negative electric field at distances beyond so that evolution of the electric field at large distances does not depend on the boundary conditions.

The three marked curves had to be truncated beyond their maxima at about since numerical solution of Poisson's equation proved to be difficult beyond that distance. However, the local equilibrium electric fields which are expected to occur at sufficiently large distances were numerically derived also for these cases. We find a local equilibrium electric field for and for . Also here, the electrical conductivity is dominated by the charged small grains when local equilibrium obtains. For each of these curves the decay of the electric field from the respective maximum absolute values near down to the respective absolute values for local equilibrium is mainly due to the increase of number densities of charged small grains as the medium moves upward.

In Fig. 2 results are given for the local equilibrium value of E (i.e. gradients and the total current J are zero). For each panel all input parameters but one have their standard 1 A.U. values; the value of this one input parameter varies and the dependence of the equilibrium value of E on it is explored. The question of whether local equilibrium might ever obtain arises. The scaleheight of the protosolar nebula at 1 A.U. was probably less than and results displayed in Fig. 1 (as well as in Fig. 3 and additional results shown in Fig. 4a below) are only meaningful for values of z somewhat less than that scaleheight and local equilibrium was never obtained if the assumed boundary conditions are the correct ones. However, it is possible that a closed cyclic convective flow was maintained for many turnover times, that the assumed boundary conditions were not the relevant ones, and that conditions in the flow evolved to approach local equilibrium. As described in the appendices the longest timescale in the evolution of the electrodynamic properties of our protosolar nebula models is the timescale for small grains to reach local charge equilibrium, and once a particular ion, electron, big grain, and small grain charge distribution is established the timescale for the growth of the electric field is very short compared to the turnover timescale. (In Sect. 5 we will comment on the question whether electric field strengths strong enough to induce lightning can be built up for local equilibrium.) As the numerical results given in Fig. 1 (and in Figs. 3, 5, and 6 and additional results in Fig. 4a) and considerations presented in the appendices show the largest electric field strengths obtain if the big grains have had time to charge up but the small grain charge distribution has not approached local equilibrium; thus, a particular range of lifetimes of convective cells would favour the generation of electric fields strong enough to induce discharge.

Fig. 2. Local Equilibrium Values of the Electric Field for Models Appropriate for a Position at 1 A.U. from the Sun. All independent parameters, except the one on an axis in each panel are the standard ones. In each panel, the electric field dependence on the value of the one varied independent parameter is shown. The dashed lines mark the electric field when all independent parameters have the standard values

Fig. 3. The Electric Field as a Function of Distance for a Further Model Appropriate for a Position at 1 A.U. from the Sun. Results are for cm. All other parameters have the values given in the first column of Table 1, and all boundary conditions are the standard ones. The thin horizontal line at the right hand side of the figure indicates the electric field for the local equilibrium

As is argued in Appendix B variation of the local equilibrium electric field with various relevant parameters can be approximated by (B8) if the parameters are near the standard ones. (B8) results from the balance between the conduction current and the current of charged big grains as given by (B3) if can be approximated by (A15) and can be approximated by with given by (A21). The conditions on which these approximations are based are detailed in Appendix C. The most important conditions are that 1) the electrical conductivity of the medium is dominated by charged small grains, the small grains are in local charge equilibrium and most of the charged small grains carry one positive or one negative elementary charge so that higher charge states of the small grains can be neglected and that 2) the charge on big grains is in local equilibrium and is mainly determined by noninductive charge exchange and electrostatic relaxation from collisions between big grains and small grains.

For Fig. 2a, (B8) holds for where the decrease of E with increasing is due to (see (A21)). The decrease of results from the increase of the number density of charged small grains with increasing (see(A17)) which in turn results from the increase of the number densities of gas phase ions and electrons implying an increased production rate for charged small grains.

(B8) breaks down for and for because in the lower range of charged big grains and in the higher range of it higher charge states of small grains become important for .

For Fig. 2b, (B8) holds at least for where the increase of E with increasing is due to (see (A15)).

(B8) becomes questionable for because the electric field significantly decreases and, as a consequence, becomes smaller than given by (A21). For (B8) breaks down because charged big grains dominate there, i.e. the electric field approaches as defined by (B5) where gravitational forces and electrostatic forces on the big grains balance.

For Fig. 2c, (B8) holds for the total parameter range considered where E is about constant due to (see (A15)) and so that the current of charged big grains moving downward in the rest frame of neutrals does not vary with .

For Fig. 2d, (B8) holds for where the increase of E with increasing is due to (see (A15)) and (see (A21)). The increase of with results from (see also A7), i.e. from a decrease of mobility of small grains with increasing size whereas the number density of charged small grains does not vary with (see (A17)).

(B8) breaks down for because there higher charge states of small grains become important for , and for gas phase ion and electron number densities become so high due to reduction of recombination on small grain surfaces that gas phase ions start to dominate the electical conductivity of the medium and, in addition, electron impact on big grains reduce the positive charge on them drastically so that decreases with much faster than predicted by (A15).

As seen from Fig. 2d the local equilibrium electric field strength is larger for than for . Thus, Fig. 3 shows results for the dependence of the electric field when . All other parameters have the values given in the first column of Table 1, and all boundary conditions are the standard ones. Also here, the thin horizontal line at the right hand side of the figure indicates the electric field for the local equilibrium.

It should be mentioned that for Fig. 3 the ratio of the total current J to the sum S is much larger than for the upper curve in Fig. 1 for distances close to and beyond the maximum electric field, so that E is more strongly affected by the total current there. This difference results mainly from the fact that if the small particles are larger, the total current is increased as a result of the increase of both and , and that the current of the charged big grains is lowered as a result of the decrease of . Our numerical results show that the total current J is never larger than half the current and is significantly smaller than the sum S for distances . The ratio shows a local maximum of about 0.55 at the distance and is about 0.3 at , where E reaches its maximum. That is, there is a moderate effect of J on E, increasing the maximum electric field strength by a factor of about 1.3. In summary, we conclude that increasing the radius of small grains from the standard value of to increases the maximum electric field strength roughly by a factor 2. At large distances, i.e. , the electric field shown in Fig. 3 approaches its local equilibrium value as calculated by setting gradients and J equal to zero (indicated by the thin horizontal line at the right hand side of the figure).

Fig. 4 gives results for the electric field E for models appropriate for a position at 1 A.U. from the Sun, when the mean specific mass density of the big grains is varied but their mass is kept constant at . Very low mean specific mass densities correspond to very porous grains. Again, all other input parameters have their standard values given in the first column of Table 1, and all boundary conditions are the standard ones. Fig. 4a gives the electric field as a function of distance for specific mass densities of and of as indicated at the corresponding curves. The upper curve, which is not marked, is the same as the upper curve in Fig. 1, corresponding to standard 1 A.U. parameters (Table 1). The thin horizontal lines at the right hand side of the figure indicate the respective electric field for the local equilibrium.

Fig. 4. The Electric Field for Further Models Appropriate for a Position at 1 A.U. from the Sun. The density of the material of which the big grains is composed, , is varied and the radius of the big grains is assumed to be . All other parameters have the values given in the first column of Table 1. a  The electric field as a function of distance. The curves marked , , are for models for which the specific mass density of big grains has the indicated value. The upper curve which is not marked is for the standard parameter , (i.e. the same curve as the upper curve in Fig. 1) and is given for comparison. The thin horizontal lines at the right hand side of the figure indicate the respective electric field for the local equilibrium. b  Local equilibrium values of the electric field where , is the independent variable. As in Fig. 2 the dashed line marks the electric field for the standard choice of the independent parameter which is

Fig. 4b gives results for the local equilibrium value of E for a larger range of values for the parameter . For Fig. 4b, (B8) holds nearly for the total parameter range considered where E is about constant due to similar reasons as for Fig. 2c, except for where reduction of by charged big grains becomes significant.

From Fig. 4a it can be seen that the electric field strength is generally reduced when decreases. This reduction of the electric field strength is predominantly a transient effect which becomes less significant for large distances where the medium approaches local equilibrium in accordance with Fig. 4b. As can be seen from Eq. (A16) in Appendix A, the time scale , which is the time scale for big grains to reach their equilibrium charge (as given by (A15)) increases with decreasing when simultaneously (= constant grain mass). In addition, the absolute value of the average charge (with ) carried away by the small grains in collisions with big grains increases when is reduced relative to its equilibrium value (see Eq. (18)), resulting in a larger number of charged small grains and a higher electrical conductivity. These are the two prime effects responsible for the decrease of the electric field during the transient phase.

As for the upper curve in Fig. 1 the total current is negligible at least for distances and affects the electric field by a factor of only order unity for smaller distances.

Fig. 5 displays the electric field as a function of distance for the conditions that obtained in the protosolar nebula at what is presently Jupiter's orbit (5 A.U.). The curve is for the standard 5 A.U. parameters listed in Table 1 and the standard boundary conditions. The scale height of the nebula was probably less than . The thin horizontal line at the right hand side of the figure again indicates the electric field for the local equilibrium.

Fig. 5. The Electric Field as a Function of Distance for a Model Appropriate for a Position at 5 A.U. from the Sun. All parameters have the values given in the second column of Table 1. The boundary conditions are the standard ones. The thin horizontal line at the right hand side of the figure indicates the electric field for the local equilibrium

Here, the effect of the total current on the electric field is again small at least for distances although it is larger than for the conditions of Figs. 1 and 4a. It is of order unity for smaller distances. The ratio of the total current J to the sum S shows a local maximum of about 0.2 at the distance and it is about 0.1 at where E reaches its maximum so that the current J should increase the maximum electric field by a factor 1.1.

The curve in Fig. 6 gives the electric field as a function of distance for the standard parameters for the dust enriched subdisk in Table 1 and standard boundary conditions. The radial extent of the subdisk may have been as much as almost , but its thickness was probably about or less (Dubrulle et al. 1995). The thin horizontal line at the right hand side of the figure, as usual, indicates the electric field for the local equilibrium. Here, the effect of the total current J on the electric field is negligible everywhere.

Fig. 6. The Electric Field as a Function of Distance for a Model Appropriate for a Dusty Subdisk. All parameters have the values given in the third column of Table 1. The boundary conditions are the standard ones. The thin horizontal line at the right hand side of the figure indicates the electric field for the local equilibrium

The physical processes operating to produce the electric field variations as shown in Fig. 3, by the two lower curves in Fig. 4a, and in Figs. 5 and 6 are qualitatively similar to those discussed for the upper curve in Fig. 1 although quantitative details are different. As is the upper curve in Fig. 1, the curves in Figs. 3, 4a and 6 are plotted up to distances well beyond (i.e. well beyond for Figs. 1, 3, 4a and for Fig. 6) where the electric field almost reaches local equilibrium. However, the curve shown in Fig. 5 is not plotted up to a distance beyond , which here is , because numerical solution proved to become difficult beyond . Therefore, the curve is still significantly above the local equilibrium value of the electric field at the largest distances shown in the figure.

Fig. 7 shows the local equilibrium electric field for models appropriate for a dusty subdisk. The ratio is the independent variable in panel a, and the ratio is the independent variable in panel b. is assumed. All other parameters have the standard values in column 3 of Table 1.

Fig. 7. Local Equilibrium Values of the Electric Field for Models Appropriate for a Dusty Subdisk. In panel a is the independent variable; in panel b is the independent variable. For results in both panels . All other parameters have the standard values given in the third column of Table 1. Just as in Figs. 2 and 4b the dashed lines indicate the result obtained when all input parameters have their standard values

For Fig. 7a, (B8) is fulfilled when roughly , where the increase of E with increasing results largely from the increase of the current of charged big grains as (for ) and, in addition, when from the increase of . The increase of is caused by the increase of the rate of discharging for small grains due to collisions with big grains with increasing decreasing the number density of charged small grains in spite of the fact that also increases with . However, the increase of with increasing as predicted by (A21) becomes neglibible at as, at the same time, increases nearly as . In fact, even slightly decreases with increasing at probably because there the decrease of with the increase of the electric field results in a decrease of the discharging rate for small grains as compared to what is expected if electric forces are neglected for determining .

(B8) breaks down for and where the electric field as shown in Fig. 7a is larger in the first range and smaller in the second range than that estimated by (B8). For very small values of the assumptions for estimating the number densities of charged small grains by (A17) break down, mainly because here small grains are discharged also significantly by ion and electron impact in addition to being discharged by collisions with big grains. For large numbers of charged big grains dominate , i.e. the electric field approaches as defined by (B5).

For Fig. 7b, (B8) is fulfilled when , where the variation of E observed in the figure results from the variation of with as given by (A21) and the assumed variation of with whereas is roughly constant.

(B8) breaks down at because there number densities of gas phase ions and electrons become so large due to their small recombination rates that the positive charge on big grains is drastically reduced by electron impact and, at gas phase ions and electrons dominate the electrical conductivity of the medium.

© European Southern Observatory (ESO) 1998

Online publication: February 4, 1998
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