## 4. Numerical resultsIn 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.
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
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
The physical processes operating to produce the electric field variations can be described as follows: As argued in Appendix A the total current 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 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 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 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 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
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 (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 (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 For Fig. 2d, (B8) holds for where the
increase of (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 Fig. 4 gives results for the electric field
Fig. 4b gives results for the local equilibrium value of 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.
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 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
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.
For Fig. 7a, (B8) is fulfilled when roughly
, where the increase of (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 (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 |