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

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Appendix A: the timescales for the generation of the electric field and for the attainment of ionization equilibrium

We define

[EQUATION]

so that Eq. (10) is

[EQUATION]

Using (20b) and Eqs. (6) through (9) we find

[EQUATION]

with

[EQUATION]

Clearly, in the absence of charged grains the e - folding growth timescale [FORMULA] of the electric field would be given by

[EQUATION]

We also define

[EQUATION]

and

[EQUATION]

Then from (A4) to (A7)

[EQUATION]

and each of [FORMULA], [FORMULA], and [FORMULA] is an upper bound to [FORMULA].

The following estimates are based on various conditions which are discussed in Appendix C.

To estimate [FORMULA], [FORMULA], and [FORMULA] we must estimate [FORMULA], [FORMULA], [FORMULA], and [FORMULA]. We find from (1) and (2b) and evaluations of [FORMULA] 's that to a good approximation for cases that we have studied

[EQUATION]

and

[EQUATION]

and that the timescale for [FORMULA] and [FORMULA] to reach local steady state equilibrium values are less than about

[EQUATION]

[EQUATION]

respectively. [FORMULA] and [FORMULA] are the average specific mass densities inside the small and big grains.

From Eqs. (4), (5), (15), and assuming [FORMULA] as done to derive (15), we find

[EQUATION]

with

[EQUATION]

Taking only the first and second term on the right hand side of Eq. (A13) into account and neglecting all the other terms, we get in the case of local equilibrium

[EQUATION]

In fact, as will be discussed in more detail in Appendix C, (A15) is a good approximation for our 1 A.U. standard parameters and standard parameters for the dust enriched subdisk. For our 5 A.U. standard parameters the term [FORMULA] may be a significant fraction of the term [FORMULA] decreasing [FORMULA] signicantly as compared to approximation (A15), but even in this case (A15) can still be considered to be an order of magnitude estimate.

Approximating [FORMULA] and estimating [FORMULA] from Eq. (8) with [FORMULA] we find

[EQUATION]

From (3) and the evaluations of the [FORMULA] 's and [FORMULA] 's we see that for cases in which parameters differ little from the standard ones once equilibrium is reached

[EQUATION]

and small grains carrying charges with a magnitude larger than one elementary charge can be neglected. Approximation (A17) is a good approximation for our 1 A.U. standard parameters and for our standard parameters for the dust enriched subdisk. For our 5 A.U. standard parameters (A17) is less accurate but still an order of magnitude estimate. The local equilibrium distribution of small grain charges is reached on a timescale of

[EQUATION]

With the use of the local equilibrium values of [FORMULA], [FORMULA], [FORMULA], [FORMULA], and [FORMULA] and expressions for [FORMULA] and [FORMULA] evaluated under the assumption that the motions of the dust grains relative to the neutrals are very subthermal we obtain the following approximations which are reliable for parameters near the standard ones.

[EQUATION]

where A is about 0.3 when [FORMULA] and [FORMULA] and becomes larger for E and [FORMULA] becoming smaller.

[EQUATION]

and

[EQUATION]

Conditions on which approximations (A19) to (A21) are based are discussed in more detail in Appendix C. These conditions are fulfilled only if local charge equilibrium prevails. The timescale [FORMULA], being an upper bound to [FORMULA], can be approximated by (A19) if the gas phase ions and electrons have reached local equilibrium for their number densities even if the grains are not in local charge equilibrium. This condition should be fulfilled if the amount of time which has passed during the evolution of a dynamic structure is large compared to [FORMULA] as defined and approximated by (A11), or in our stationary models, for lengthscales z large compared to [FORMULA] with [FORMULA]. (For electrons we have used the approximation of local equilibrium anyway, i.e. Eq. (2b) in place of (2a)). Analogously, the timescale [FORMULA], being an upper bound to [FORMULA], can be approximated by (A20) if the big grains have reached local charge equilibrium even if the small grains are not in local charge equilibrium. This condition should be fulfilled for lengthscales z large compared to [FORMULA]. Finally, the timescale [FORMULA] can be approximated by (A21) only if the small grains have reached local charge equilibrium, which should occur only at distances large compared to [FORMULA]. In fact, because local charge equilibrium for the small grains is reached on lengthscales greater than the thickness of the protosolar nebula for many sets of parameters and because local equilibrium electric fields are usually weaker than those at which discharge is induced, the [FORMULA] 's in many storage volumes capable of powering discharges that could have formed chondrules cannot be estimated by (A8) if [FORMULA] is approximated by (A21). However, approximation (A21) for the timescale [FORMULA] is useful for the interpretation of physical processes causing the variation of the local equilibrium electric field with various parameters.

Appendix B: upper bounds and the equilibrium values of the electric field

For the electric field E as a function of z we have [FORMULA] at the position z where E reaches a peak value, and also at sufficiently large distances z where the medium reaches local equilibrium. Taking [FORMULA], (A2) and (A3) give for these cases the electric field strength

[EQUATION]

For the adopted standard values of relevant parameters grains carry far more of the charge than gas phase ions or electrons do, except for a small region near the boundary [FORMULA] in z  - space. Thus, (A1) and (A2) imply that for [FORMULA]

[EQUATION]

[FORMULA] implies together with (B1) and (B2) that at positions where [FORMULA]

[EQUATION]

In fact, (B3) holds approximately whenever [FORMULA]
[FORMULA] and we will use (B3) as an approximation for the electric field variation for distances [FORMULA]. Using approximation (A19) we get [FORMULA], [FORMULA], [FORMULA], and [FORMULA] for Figs. 1, 3, 5, and 6, respectively.

According to (A8) we have [FORMULA], which yields with (B3)

[EQUATION]

where [FORMULA] is defined as

[EQUATION]

We define

[EQUATION]

For our standard parameters we have [FORMULA]. According to (A15) we have [FORMULA] in case of local charge equilibrium and for our standard parameters. Inserting (B6) and (B5) into (B4) yields

[EQUATION]

When local equilibrium obtains, and for values of the parameters near the standard ones that we have adopted [FORMULA] may be estimated by [FORMULA] as given by approximation (A21) and [FORMULA] may be estimated by (A15). Hence, (B3) yields a local equilibrium electric field strength which is roughly

[EQUATION]

(B8) holds only for a restricted range of values for the relevant parameters. For parameter values significantly different from our standard ones, [FORMULA] may not be estimated by [FORMULA] but possibly better by [FORMULA] or [FORMULA], and conditions on which (A19) to (A21) are based may not be fulfilled. Conditions on which approximations (A19) to (A21) are based and also conditions which allow the estimation of [FORMULA] by [FORMULA] are discussed in more detail in Appendix C.

Appendix C: conditions for the approximations for time scales and equilibrium values

We discuss in more detail the conditions on which approximations in Appendices A and B are based.

The motions of dust grains relative to neutrals are assumed to be very subthermal, i.e.

[EQUATION]

with d = b, sk, which is well fulfilled for our standard parameters, so that approximation

[EQUATION]

for d = b, s can be used (see Draine, 1986).

For the big grains the magnitude of the electric force is assumed to be small compared to the magnitude of the gravitational force, i.e.

[EQUATION]

or

[EQUATION]

so that with (8)

[EQUATION]

For our standard parameters and for a local equilibrium for which [FORMULA], condition (C3) is fairly well fulfilled. If local equilibrium does not obtain condition (C3) is still barely fulfilled for our standard parameters. We also assume

[EQUATION]

With (C2), (8), (9), and [FORMULA] assumption (C5) gives

[EQUATION]

or

[EQUATION]

(C7) is fulfilled in the case of a local equilibrium for which charged dust grains with [FORMULA] dominate the electrical conductivity of the medium and [FORMULA]. For [FORMULA] and [FORMULA] condition (C7) is fairly well fulfilled for our standard 1 A.U. and 5 A.U. parameters but barely fulfilled for our standard parameters for the dust enriched subdisk.

We evaluate the [FORMULA] 's with [FORMULA] and [FORMULA] from results given by Havnes et al. (1987). For the electrons we assume

[EQUATION]

which is equivalent with

[EQUATION]

if (C1) together with [FORMULA] is fulfilled. Condition (C8b) is fulfilled for an electric field strength [FORMULA] as is the case for our standard parameters for an electric field strength up to [FORMULA] since [FORMULA] grows sufficiently fast with the electric field strength up to [FORMULA]. A condition for the ions analogous to (C8b) is fulfilled for [FORMULA] for which [FORMULA]. This is the case for our standard parameters for local equilibrium. However, for an electric field strength relevant for inducing lightning, i.e. for [FORMULA], the condition for ions analogous to (C8b) is strongly violated since [FORMULA] for an electric field strength up to [FORMULA] and we do not require this condition.

For an estimate of [FORMULA] with j = i, e, we assume

[EQUATION]

or

[EQUATION]

at least for [FORMULA]. For the electrons conditions (C9) and (C10) are usually fulfilled. For the ions and for [FORMULA] they are usually well fulfilled only for the 1 A.U. standard parameters but may be barely fulfilled for our standard 5 A.U. and dust enriched subdisk parameters due to the low ion temperature there. For the big grains we proceed from the assumption

[EQUATION]

or

[EQUATION]

We find from our numerical results that condition (C12) is well fulfilled for our standard parameters, at least for distances [FORMULA]. We assume that

[EQUATION]

[EQUATION]

which is consistent with assumption (C9) and [FORMULA] and with our standard parameters. For local equilibrium, (1) and (C14) yield (A9), and (2b) and (C13) yield (A10).

We now consider the conditions under which we can neglect the terms [FORMULA] and [FORMULA] in (A13) compared to [FORMULA].

From (A10) and evaluating [FORMULA] and [FORMULA] with (C8a) through (C12) we get

[EQUATION]

with

[EQUATION]

and

[EQUATION]

According to our calculations of the electron temperature, [FORMULA] increases with E from [FORMULA] at [FORMULA] up to [FORMULA] at [FORMULA], implying with (C17) a variation of [FORMULA] with E. For local equilibrium we have [FORMULA] for our standard 1 A.U. parameters, [FORMULA] for our standard 5 A.U. parameters, and [FORMULA] for our standard parameters for the dust enriched subdisk so that we find from (C17) [FORMULA] ; [FORMULA] ; and [FORMULA] for our standard 1 A.U., 5 A.U., and dust enriched subdisk parameters, respectively. Taking [FORMULA] as derived from approximation (A15), we find that the factor [FORMULA] in (C15) is 0.15; 0.47; and 0.44, respectively.

Requiring

[EQUATION]

we get with (C1), (C4), (C5), definition (A14) and approximation (C15) that

[EQUATION]

Requiring

[EQUATION]

we get with (C1) through (C7) and with (C15) that

[EQUATION]

In addition, in (A13) we can neglect [FORMULA] compared to [FORMULA] if

[EQUATION]

With (A9) we get that

[EQUATION]

With (C11) and (C23) we estimate that

[EQUATION]

If

[EQUATION]

which with (C15) is equivalent with

[EQUATION]

then condition (C22) is fulfilled. From (C17) and [FORMULA] we see that for parameters with values near the standard ones (C26) is fulfilled if [FORMULA] can be approximated by (A15) which is justified below.

In order to arrive at approximation (A15) we neglect the term [FORMULA] in (A13), requiring that

[EQUATION]

Using definition (A14), approximation (C15) and assumptions (C1) through (C7) we get from (C27) that

[EQUATION]

From our numerical results we find that conditions (C19), (C21) and (C26) are fairly well fulfilled for our standard parameters, at least for distances [FORMULA]. The left hand sides are usually lower than the corresponding right hand sides by an order of magnitude or more, except for (C19) in the case of the standard 1 A.U. parameters (corresponding to the upper curve in Fig. 1) where E is lower than the right hand side of (C19) only by a factor 3 near the peak of the electric field and except for (C21) in the case of the standard parameters for the dust enriched subdisk (corresponding to Fig. 6) where the left hand side is lower than the right hand side only by a factor less than 2 for a large distance range. Condition (C28) is well fulfilled for our standard 1 A.U. and dust enriched subdisk parameters. For our standard 5 A.U. parameters it is also well fulfilled for distances up to [FORMULA] and somewhat beyond where the medium has not yet reached charge equilibrium but is only barely fulfilled for local equilibrium.

In order to find approximation (A17) we consider an estimate for the frequencies [FORMULA]. From (A13), (C18), (C20), and (C22) we get for local equilibrium that

[EQUATION]

(C29) gives a somewhat more accurate estimate for [FORMULA] than (A15) where the term [FORMULA] decreases [FORMULA] as compared to approximation (A15) due to electron impact on the big grains. In addition, we require that

[EQUATION]

which with (C29) we see is fulfilled if (C18) is fulfilled. Requirement (C30) implies that we can neglect electrostatic polarization of grain material in (16b), (17b), and (18) so that (16a), (16b), and (18) yield with (C30)

[EQUATION]

From (C31), (16a), (C29), (C15), (A14), and (C1) through (C7),

[EQUATION]

That is, for parameters near the standard ones and for local equilibrium the average charge carried away by a small grain in a collision with a big grain is negative and its magnitude is small compared to one elementary charge.

We require now

[EQUATION]

which yields together with (C1) through (C4) that

[EQUATION]

Condition (C34), which is well fulfilled for our standard 1 A.U. and dust enriched subdisk parameters and barely fulfilled for our standard 5 A.U. parameters, implies with (C32) that

[EQUATION]

Although (C34) is only barely fulfilled for our standard 5 A.U. parameters condition (C35) holds also in this case.

From (C31), (C35) and definitions of [FORMULA] given in Sect. 3 we get [FORMULA], [FORMULA], and with (17a) and (17c) [FORMULA], [FORMULA]. Then, we get from (19b) immediately that for [FORMULA] or [FORMULA]

[EQUATION]

and from (19a) with (16a), (C29), (C30), (A14), and (C5)

[EQUATION]

[EQUATION]

From (C37) and (C38) we get that

[EQUATION]

Eq. (3) gives for local equilibrium and for [FORMULA] or [FORMULA] together with (C36), (C38) and (C39)

[EQUATION]

where [FORMULA] is the Kronecker [FORMULA] and where [FORMULA]. With (A9), (A10), and (C13) we get that

[EQUATION]

implying with (C33)

[EQUATION]

Finally, we assume that

[EQUATION]

and

[EQUATION]

Inserting (A9), (A10), (C13), (C42) to (C44) into (C40) we get that

[EQUATION]

(C43), (C44), and (C45) yield (A17).

Approximation (A19) is based on (A9) and (A10) and neglect of [FORMULA] compared to [FORMULA] which is correct within an order of magnitude for [FORMULA] according to (C16). (A19) should be valid for a wide range of parameter values within an order of magnitude, at least for electric field strengths E varying from 0.3 to [FORMULA] and [FORMULA] where A varies from about 0.8 to 0.3.

Besides the conditions detailed above on which (A19) to (A21) are based, Eq. (B8) is based also on the assumption [FORMULA] which holds only if the following conditions are both fulfilled

[EQUATION]

[EQUATION]

or, with the use of (A19) to (A21),

[EQUATION]

and

[EQUATION]

For example, for local equilibrium and for our standard parameter values except for [FORMULA] which is varied, condition (C48) is only valid if [FORMULA], [FORMULA], and [FORMULA] for the parameter values given in column 1, column 2 and column 3 of Table 1, respectively. For larger values of [FORMULA], [FORMULA] in (B3) should be approximated by [FORMULA] as given by (A20) yielding a local equilibrium electric field strength given by (B7) if " [FORMULA] " is replaced by " [FORMULA] ".


[TABLE]

Table 2. List of symbols



[TABLE]

Table 2. (continued)



[TABLE]

Table 2. (continued)


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© European Southern Observatory (ESO) 1998

Online publication: February 4, 1998
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