## 2. TheoryNovikov 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 , where
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 ;
- 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 - 1) the observed heights of beginning, maximum light, and of end of meteor events as published by Hawkins & Southworth (1958);
- 2) the observed dependence of the height of maximum luminosity on the velocity of meteoroids as evidenced by the data of Jacchia et al. (1967);
- 3) the observed shift of the maximum light towards the beginning of the atmospheric trajectory of meteors as inferred by Hawkins & Southworth (1958), Jacchia et al. (1967), and Babadzhanov & Novikov (1987).
Therefore, we will neglect the deceleration when deriving formulae
for light intensity, where stands for the rate of evaporation of
meteoroid mass, represents the luminous
efficiency, the ionization efficiency, (both
considered as a function of velocity, Now we will derive the formula for needed
for the evaluation of represent the luminosity produced at time where denotes the mass of PMB at time . 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 with representing Heaviside's unit step function, denoting the lifetime of PMB to complete fragmentation, being the lifetime of the fragment having been released at the instant of beginning of fragmentation, , where is the lifetime of a fragment which was detached at the end of fragmentation until its full evaporation. can be considered the time of termination of the whole meteor event. Let represent the electron line density produced by one fragment. Summing up contributions of all fragments we derive the following ionization equation By substituting for from (2) and for from (3) into (4) and for from (5) into (6), and comparing the formulae obtained in this way with (1) we can see that the expression for attains the form common to both light and ionization. When changing in (7) the integration variable into we get where ; stands for the atmospheric density at the height of termination of the whole meteor event, , represents atmospheric density at the height of disappearance of fragments which were detached at the height of beginning of QCF, with being the corresponding atmospheric density, represents the atmospheric density at the height of termination of fragmentation of PMB, designates the atmospheric density at any height, . The quantities and will be specified below. The equations for and have been derived by Lebedinets (1980). They read where stand for the heat transfer
coefficient, the shape-density coefficient and the bulk density of PMB
respectively, and are analogous quantities
valid for fragments, where In general (e. g. Novikov et al. 1993).
This functional dependence was inferred for large fireballs. However,
we deal here with fainter meteors whose can be
kept constant. With our assumption of constant meteoroid velocity,
where we have introduced the auxiliary quantities 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, , for which 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 and is the energy of fragmentation valid for this fast type. The first term in (13) corresponds to the fact that within the interval 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 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 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 The second type of the QCF, the slow one, labelled by the subscript , for which , 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 and Here stands for the energy of fragmentation for the slow type. The first term in (17) describes the situation when within the interval 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 , 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 It can easily be seen that the exchange of the former form of QCF for the latter one can simply be performed by substituting and . Two pairs of parameters and 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, . 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). © European Southern Observatory (ESO) 1998 Online publication: December 8, 1997 |