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Astron. Astrophys. 329, 769-775 (1998)

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2. Theory

Novikov et al. (1984a) have made the following assumptions when constructing the theory of QCF:

  • 1) the only mechanism of ablation of PMB is QCF;
  • 2) the only mechanism of ablation of fragments is evaporation. This implies that the contribution to light and ionization curves comes from atoms and molecules of meteoroid matter which evaporated directly from the surface of fragments and not of PMB (this contribution is proportional to [FORMULA], where [FORMULA] represents the specific energy of QCF and Q the same quantity related to evaporation);
  • 3) initial masses of all fragments are the same and equal to [FORMULA] ;
  • 4) deceleration of both the PMB and fragments is negligibly small;
  • 5) the basic formulae of the physical theory of meteors ( e. g. Bronshten 1983) are valid for both PMB and fragments.

We shall accept these assumptions, too. Contrary to Novikov et al. (1984a), where the formula for fast QCF is erroneous, we shall derive below the correct one (Eq. (16)).

The most general theory of light and ionization taking into consideration the QCF has to consider also deceleration both of PMB and fragmentation products. Elements of such a theory have been put forward by McCrosky (1958) and a more detailed theory has been published by Babadzhanov et al. (1987). But there exist many observational results demonstrating that both luminosity and ionization can also be explained without taking into account the deceleration. These are

Therefore, we will neglect the deceleration when deriving formulae for light intensity, I, as well as for electron line density, [FORMULA], characterizing the ionization state. Under these conditions the following formulae are valid (e. g. Bronshten 1983)

[EQUATION]

where [FORMULA] stands for the rate of evaporation of meteoroid mass, [FORMULA] represents the luminous efficiency, [FORMULA] the ionization efficiency, (both considered as a function of velocity, v). The mass of evaporated meteoric particle is designated as [FORMULA]. We have adopted also the assumption of the dependence of atmospheric density on the height within the meteor zone, h, in the form [FORMULA] where h is expressed in km (e. g. Poole & Nicholson 1975) (scale height of 6 km).

Now we will derive the formula for [FORMULA] needed for the evaluation of I and [FORMULA]. Let

[EQUATION]

represent the luminosity produced at time t by an individual fragment detached from PMB at the instant [FORMULA]. The function [FORMULA] is the mass of the fragment. Then all fragments which detached at time [FORMULA] and later (where [FORMULA] stands for the lifetime of individual fragment counted from [FORMULA] until its full evaporation), will contribute to ablation of PMB and, consequently, also to light and ionization at any instant t. The number of fragments detached from PMB per unit time can be expressed as

[EQUATION]

where [FORMULA] denotes the mass of PMB at time [FORMULA]. Taking into consideration the lifetime of fragments and of PMB, and summing up the contributions of all fragments we get the following formula for the meteor luminosity

[EQUATION]

with [FORMULA] representing Heaviside's unit step function, [FORMULA] denoting the lifetime of PMB to complete fragmentation, [FORMULA] being the lifetime of the fragment having been released at the instant of beginning of fragmentation, [FORMULA], where [FORMULA] is the lifetime of a fragment which was detached at the end of fragmentation until its full evaporation. [FORMULA] can be considered the time of termination of the whole meteor event. Let

[EQUATION]

represent the electron line density produced by one fragment. Summing up contributions of all fragments we derive the following ionization equation

[EQUATION]

By substituting for [FORMULA] from (2) and for [FORMULA] from (3) into (4) and for [FORMULA] from (5) into (6), and comparing the formulae obtained in this way with (1) we can see that the expression for [FORMULA] attains the form

[EQUATION]

common to both light and ionization. When changing in (7) the integration variable [FORMULA] into [FORMULA] we get

[EQUATION]

where [FORMULA] ; [FORMULA] stands for the atmospheric density at the height of termination of the whole meteor event, [FORMULA], [FORMULA] represents atmospheric density at the height [FORMULA] of disappearance of fragments which were detached at the height of beginning of QCF, [FORMULA] with [FORMULA] being the corresponding atmospheric density, [FORMULA] represents the atmospheric density at the height [FORMULA] of termination of fragmentation of PMB, [FORMULA] designates the atmospheric density at any height, [FORMULA]. The quantities [FORMULA] and [FORMULA] will be specified below.

The equations for [FORMULA] and [FORMULA] have been derived by Lebedinets (1980). They read

[EQUATION]

where [FORMULA] stand for the heat transfer coefficient, the shape-density coefficient and the bulk density of PMB respectively, and [FORMULA] are analogous quantities valid for fragments, Q denotes the energy of evaporation, while [FORMULA] is the energy of fragmentation. Using [FORMULA] as the new independent variable converts (9) into the alternative form:

[EQUATION]

where H is the constant scale height used in the dependence [FORMULA] with [FORMULA] representing atmospheric density at the height of beginning of fragmentation, and [FORMULA], [FORMULA] being zenith distance of the radiant.

In general [FORMULA] (e. g. Novikov et al. 1993). This functional dependence was inferred for large fireballs. However, we deal here with fainter meteors whose [FORMULA] can be kept constant. With our assumption of constant meteoroid velocity, v, the solution of (10) reads

[EQUATION]

where we have introduced the auxiliary quantities

[EQUATION]

It is necessary to distinguish two kinds of QCF (see e. g. Novikov et al. 1984a,b). The first one, fast, labelled by the subscript, [FORMULA], for which [FORMULA] and the PMB is completely fragmented before the fragments released at the height of beginning of QCF can evaporate. Inserting (11) into (8) we get

[EQUATION]

[EQUATION]

[EQUATION]

where now

[EQUATION]

and

[EQUATION]

[EQUATION]

[FORMULA] is the energy of fragmentation valid for this fast type. The first term in (13) corresponds to the fact that within the interval [FORMULA] the PMB disintegrates due to fragmentation into fragments which further evaporate. But no fragment evaporates completely. The second term describes the situation when the PMB has already completely fragmented within the span [FORMULA] and the fragments can be found in various states of evaporation, although no fragment completely disappears. And finally, the third term describes the circumstance that within the interval [FORMULA] the evaporation of fragments still occurs but some fragments have already evaporated completely. On substituting (14) into (13) and carrying out the integration we arrive at

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

The second type of the QCF, the slow one, labelled by the subscript [FORMULA], for which [FORMULA], concerns those fragments that have detached at the height of the beginning of fragmentation and disappeared due to evaporation before the PMB completely fragmented. Carrying out the same procedure as in the case of fast fragmentation we get

[EQUATION]

[EQUATION]

[EQUATION]

where now

[EQUATION]

and

[EQUATION]

[EQUATION]

Here [FORMULA] stands for the energy of fragmentation for the slow type. The first term in (17) describes the situation when within the interval [FORMULA] the creation of fragments and their evaporation takes place. The second term shows that fragments still form due to fragmentation, and that they still evaporate. And finally, the third term corresponds to the situation when PMB has already fragmented completely within [FORMULA], the evaporation of fragments still continues and some portion of them has already completely evaporated. On substituting (18) into (17) and performing the integration we obtain

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

It can easily be seen that the exchange of the former form of QCF for the latter one can simply be performed by substituting [FORMULA] and [FORMULA]. Two pairs of parameters [FORMULA] and [FORMULA] can be inferred from observations of meteors. The observed light curve can result from either kind of fragmentation, the fast or the slow one which corresponds to two different types of meteoroid material. These kinds differ by bulk densities of PMB and fragments, by their initial masses, and also by their energies of fragmentation, [FORMULA]. Thus, the possible ambiguity in the determination of physical parameters of PMB and fragments can be overcome by considering these differences.

We have now reached the part of our goals mentioned in the introduction concerning the derivation of the basic formulae of our approach. In order to illustrate the capability of the method to obtain the correct values of the basic parameters of PMB and fragments and to show how to apply it in practice, we shall use it to solve some simulated cases. We will construct a theoretical light curve and then will subject it to our approach in order to recover its original input parameters. Since we will arrive at the conclusion that the same basic function applies to both light and ionization cases, we will confine ourselves only to the processing of the light curve. Processing of the ionization curve would be quit analogous. This was carried out by Novikov & Zhdanov (1997).

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© European Southern Observatory (ESO) 1998

Online publication: December 8, 1997
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