## 3. Application of the techniqueTo demonstrate the application of the derived formulae, we shall show the determination of the parameters of QCF, , and , characterizing PMB and debris of its disintegration, from the observed light and/or ionization curve. We will confine ourselves to the case of the variability of the light curve along the meteoroid atmospheric trajectory, i.e. we will consider the light intensity to be a function of the atmospheric density, , even though we will derive also the formula for the electron line density. It is sufficient to analyze only one kind of QCF, e. g. the fast one. Observations can yield , and the light curve . Defining the auxiliary function we arrive with the help of (1) and (16) to its form useful for further computations Radar observations yield the ionization curve . Defining in analogy with the above case the normalized electron line density and inserting into (1) from (16) we can see that corresponds to defined by (22). Thus, we have justified the decision to confine ourselves only to the case of the light curve to show how to use our approach in practice. When employing the formulae of numerical differentiation, we can determine the second derivative . Constructing within the interval the dependence as a function of we can establish the interval within which . It can easily be seen that is maximum on the first part of the meteor trajectory inside the interval . This implies that the first interval corresponds to the rising part as well as to some portion of the declining part of the curve. On the other hand, the second interval, , and the third one, , correspond only to the declining part of the curve. It can be proved from (22) that the following constraint holds true within the second interval Thus, knowing and determining , we are able to determine . We would like to point out here that when constructing as a function of , we can also determine and as bounds of the interval within which . [Indeed, as a function of possesses a region where it is largely constant.] The limits of this interval correspond to and , respectively. Hence, we find from the formula for , and consequently, also the height of beginning, . On the other hand, we find from the formula for . Determining , , , we find , and finally, also the height of the end of the whole meteor event, . When the light curve is known from observations, we can construct from that data. When using the above described method, we can determine not only the parameters and but also and and consequently, the theoretical heights and . From the parameters of fast QCF, we can easily find the parameters of slow QCF, and . The kind of QCF that occurs in each particular case depends on the type of meteoroid matter causing the meteor event. We will illustrate the capability of the method developed here for
the determination of the parameters of PMB and fragments in the
following example. Figs. 1 and 2 show the theoretical light
curve by solid lines corresponding to the
following two models. The left
Fig. 2 is analogous to Fig. 1, but with the following
model parameters: g,
km/s, , g/
, g/
, erg/g,
km, g, while the
values of Unfortunately, the results of numerical differentiation are very sensitive to imperfections in input data and consequently this procedure has to be applied with care. As a consequence, we will put forward another method of light curve analysis. The principle of this method consists in the following. Eq. (24) implies that But on the other hand, according to Eq. (22) we can write for the second interval of light curve Comparison of Eq. (25) with Eq. (26) yields We need to know values of light intensity in at least three points in order to be able to construct the following set of linear equations for finding the coefficients Thus it can easily be shown that, having defined the following auxiliary quantities the solution of Eq. (28) reads Now we can construct, according to Eq. (29), the
dependence of on the height,
The bulk densities of meteoroids resulting from the application of the above methods to a larger amount of observational data were published by Novikov et al. (1996a). Application of the method to radar observations has been performed by Novikov & Zhdanov (1997). © European Southern Observatory (ESO) 1998 Online publication: December 8, 1997 |