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Astron. Astrophys. 357, 1115-1122 (2000)

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5. Numerical procedures and their testing

In order to find out the most suitable procedure for numerical handling of Eq. (10), we computed several "theoretical meteors". By this term "theoretical meteor" we denote a case, when we have chosen [FORMULA] and K as a-priori-known functions of time, and then solving Eqs. (1) to (7) by Runge-Kutta method, we derived the "observed" distances along trajectory and "observed" heights as function of time. To such "observed" values, we applied then computational procedures intended to be used for application of Eq. (10) to observations. We have examined several such cases (originating from different combinations of increasing and decreasing [FORMULA] with increasing and decreasing K). This allowed us to find out the best procedures for application to really observed meteors, and also to formulate several general rules in junction with these solutions.

Our initial idea was to compute parameters for one function of time (given by one formula) applied to all points of the observed trajectory, i.e. to the observed distances along trajectory, [FORMULA], and to the observed heights, [FORMULA]. But such procedures (e.g. using interpolation polynomials) proved to be very much dependent on the chosen function. This was already found by Pecina & Ceplecha (1983) for constant [FORMULA] and K, and we only generalized this for [FORMULA] and K being variable with time. We also learned that [FORMULA] cannot be computed for the early parts of a trajectory, where velocities and decelerations are almost independent of [FORMULA]. Even a precision of [FORMULA] m used for theoretical meteors did not allow to determine [FORMULA] during the first third of the trajectory (for range of meteoroid masses we were interested in). This defines our first limitation in applying Eq. (10) even to very precise observational data: for the early parts of a trajectory we must chose only an average value of [FORMULA] corresponding to the meteoroid type. This limitation has not much influence on K during the early parts of the trajectory, because [FORMULA] and K are there almost independent. K at the early parts of the trajectory is mostly given by velocity and deceleration, and these can be derived from observed distances along trajectory as function of time.

If we would know v and [FORMULA] from observations, [FORMULA] could be computed from them, and then also K can be computed from Eq. (10) so as to fit the computed distances along the trajectory, [FORMULA], to the observed distances, [FORMULA]. We should be aware that the primary measured values on the photographic records of a meteor, are distances along the trajectory, [FORMULA], and thus fitting them to the computed values by the least squares solves our problem completely. As the best procedure we were able to find, proved to be fitting [FORMULA] to [FORMULA] for small parts of the trajectory, as small as possible from the point of view of precision. Description of this procedure follows.

We have n consecutive points with known time, [FORMULA], and, at them, we have the observed distances along the trajectory, [FORMULA], and heights, [FORMULA], where [FORMULA]. We will chose consecutive subsets of m consecutive values from [FORMULA], from [FORMULA], and so on until from [FORMULA]. We will fit [FORMULA] to [FORMULA] for each of these subsets by the least squares using a polynomial function


where [FORMULA] to [FORMULA] are constants to be determined from the respective subset of [FORMULA] values so that sum of [FORMULA] is at its minimum value.

Using Eq. (13) is equivalent to using a quadratic expression for approximating deceleration inside the short time intervals of each of the chosen subsets. We will assume that this is strictly valid only for the average time of each of the subsets: thus only [FORMULA], [FORMULA], and [FORMULA] are important for the computed distances along trajectory, and computed velocities and decelerations; the rest is important for computing standard deviations of these values. Thus for each of the above subsets, i.e for average time, [FORMULA], of each subset, we have [FORMULA] and [FORMULA] ([FORMULA]) from Eq. (13) including their standard deviations. Because generally the average times [FORMULA] of the subsets need not to coincide with the actually observed times [FORMULA] at which we have available [FORMULA], we can integrate velocities [FORMULA] using the original [FORMULA] values and determine one integration constant for the entire trajectory in order to fit these computed distances [FORMULA] to the observed distances [FORMULA] for the whole trajectory.

Now, the initial values of [FORMULA] can be computed from any two neighboring or nearby points on assumption that [FORMULA] and K are constant on a short time difference. If suffix for the first point is 1, and suffix for the second point 2, then [FORMULA] is given by ([FORMULA] is natural logarithm)


where t, v, and [FORMULA] are actually the [FORMULA], [FORMULA] and [FORMULA], i.e. the average values of each subset. For the start we can chose [FORMULA], while at the second step, when we already computed values of K from Eq. (10), we can use [FORMULA] and [FORMULA] as different values. We can improve this procedure by taking into account all possible pairs providing that [FORMULA] and [FORMULA] are separated by small time interval, and chose then the best determined values of [FORMULA] and K (with the relatively smallest standard deviations). We finish with [FORMULA] and K which best fit the observed distances along the trajectory and correspond to Eq. (10).

In solving Eq. (10) we also need to know [FORMULA], i.e. the value of K at a point, where we start the integration (values of K are only relative in this sense). In our computations we used statistical average of K for the corresponding meteoroid type as [FORMULA], and we have chosen the point at which [FORMULA] so that [FORMULA] at this point is also the statistical average for the corresponding type (types I, II, IIIA and IIIB). Numerical values of these constants are in the next section.

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000