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

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3. Application of the technique

To demonstrate the application of the derived formulae, we shall show the determination of the parameters of QCF, [FORMULA], and [FORMULA], 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, [FORMULA], 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 [FORMULA], and the light curve [FORMULA]. Defining the auxiliary function

[EQUATION]

we arrive with the help of (1) and (16) to its form useful for further computations

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

Radar observations yield the ionization curve [FORMULA]. Defining in analogy with the above case the normalized electron line density

[EQUATION]

and inserting into (1) from (16) we can see that [FORMULA] corresponds to [FORMULA] 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 [FORMULA]. Constructing within the interval [FORMULA] the dependence [FORMULA] as a function of [FORMULA] we can establish the interval within which [FORMULA]. It can easily be seen that [FORMULA] is maximum on the first part of the meteor trajectory inside the interval [FORMULA]. This implies that the first interval corresponds to the rising part as well as to some portion of the declining part of the [FORMULA] curve. On the other hand, the second interval, [FORMULA], and the third one, [FORMULA], correspond only to the declining part of the [FORMULA] curve. It can be proved from (22) that the following constraint holds true within the second interval

[EQUATION]

Thus, knowing [FORMULA] and determining [FORMULA], we are able to determine [FORMULA]. We would like to point out here that when constructing [FORMULA] as a function of [FORMULA], we can also determine [FORMULA] and [FORMULA] as bounds of the interval within which [FORMULA]. [Indeed, [FORMULA] as a function of [FORMULA] possesses a region where it is largely constant.] The limits of this interval correspond to [FORMULA] and [FORMULA], respectively. Hence, we find [FORMULA] from the formula for [FORMULA], and consequently, also the height of beginning, [FORMULA]. On the other hand, we find [FORMULA] from the formula for [FORMULA]. Determining [FORMULA], [FORMULA], [FORMULA], we find [FORMULA], and finally, also the height of the end of the whole meteor event, [FORMULA].

When the light curve [FORMULA] is known from observations, we can construct [FORMULA] from that data. When using the above described method, we can determine not only the parameters [FORMULA] and [FORMULA] but also [FORMULA] and [FORMULA] and consequently, the theoretical heights [FORMULA] and [FORMULA]. From the parameters of fast QCF, we can easily find the parameters of slow QCF, [FORMULA] and [FORMULA]. 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 [FORMULA] by solid lines corresponding to the following two models. The left y -axis gives the magnitude, [FORMULA], while the right y -axis represents [FORMULA] as evaluated from (21). The x -axis represents the height expressed in km. Figs. 3 and 4 show the curve [FORMULA] as a function of [FORMULA] along the meteor trajectory valid for model 1 and 2, respectively. The x -axis definition is the same as in the previous case. In constructing this picture, the parameters of PMB and fragments belonging to model task 1 were chosen as follows (for the numerical value of [FORMULA], see Lebedinets 1986): [FORMULA] g, [FORMULA] km/s, [FORMULA], [FORMULA] g/ [FORMULA], [FORMULA] g/ [FORMULA], [FORMULA] erg/g, [FORMULA] km, [FORMULA] g, [FORMULA] erg/g, [FORMULA], [FORMULA], [FORMULA] km. With these parameters, the light curve reaches a maximum within the interval [FORMULA]. The results of processing the light curve corresponding to model 1 are listed in Table 1.


[FIGURE] Fig. 1. Theoretical light curves. The left y-axis gives the intensity of light in stellar magnitudes, [FORMULA], while the right one gives [FORMULA]. The solid line corresponds to [FORMULA] and the dashed to [FORMULA]. The x-axis gives the height above the ground expressed in km. This case corresponds to the model No. 1.

[FIGURE] Fig. 2. The same as in Fig. 1 but for model No. 2.

[FIGURE] Fig. 3. The natural logarithm, [FORMULA] vs. height, h, expressed in km. Model No. 1 case.

[FIGURE] Fig. 4. The same as Fig. 3 but for model No. 2.

[TABLE]

Table 1. Results of data processing using the first method.


Fig. 2 is analogous to Fig. 1, but with the following model parameters: [FORMULA] g, [FORMULA] km/s, [FORMULA], [FORMULA] g/ [FORMULA], [FORMULA] g/ [FORMULA], [FORMULA] erg/g, [FORMULA] km, [FORMULA] g, while the values of Q, H, [FORMULA], [FORMULA], A, and [FORMULA] are the same as above. For the numerical value of [FORMULA], see again Lebedinets (1986). These data produced the maximum of the light curve within the interval [FORMULA]. The results of processing the light data of model 2 are listed in Table 1, as well. In solving the models, we adopted Öpik's b - model (1955) of the luminous efficiency, [FORMULA]. It can be seen from Table 1 that the output results concerning [FORMULA], [FORMULA], [FORMULA], [FORMULA], and [FORMULA], coincide relatively well with the input values. The same holds also true with respect to [FORMULA]. However, we can also see that the method does not yield a sufficiently correct estimate of [FORMULA].

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

[EQUATION]

But on the other hand, according to Eq. (22) we can write for the second interval of light curve

[EQUATION]

Comparison of Eq. (25) with Eq. (26) yields

[EQUATION]

[EQUATION]

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 [FORMULA]

[EQUATION]

Thus it can easily be shown that, having defined the following auxiliary quantities

[EQUATION]

the solution of Eq. (28) reads

[EQUATION]

Now we can construct, according to Eq. (29), the dependence of [FORMULA] on the height, h, from the whole curve [FORMULA]. We will accept only those values of [FORMULA] for which the subsequent values differ from the previous ones by less than the chosen accuracy, i. e. [FORMULA] must hold true within the prescribed interval. We can compute [FORMULA], [FORMULA], [FORMULA], and consequently, [FORMULA], the theoretical height of disappearance of the whole meteor event, from (27). Then we find [FORMULA] from the relation [FORMULA]. Since [FORMULA] holds true, we find also the theoretical height of the beginning of the QCF of the meteoroid, [FORMULA]. The results of the solution of the example processed by our second procedure are listed in Table 2. They indicate that the output parameters [FORMULA], [FORMULA], [FORMULA], and [FORMULA] coincide again relatively well with the input values. As for the parameter [FORMULA], we again see that even our second method is not capable of determining it correctly. We conclude that this method is not sufficiently sensitive to it.


[TABLE]

Table 2. Results of data processing using the second method.


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).

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© European Southern Observatory (ESO) 1998

Online publication: December 8, 1997
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