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*Astron. Astrophys. 357, 1115-1122 (2000)*
## 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
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 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,
, and to the observed heights,
. 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 and *K*, and
we only generalized this for and
*K* being variable with time. We also learned that
cannot be computed for the early
parts of a trajectory, where velocities and decelerations are almost
independent of . Even a precision of
m used for theoretical meteors
did not allow to determine 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
corresponding to the meteoroid type.
This limitation has not much influence on *K* during the early
parts of the trajectory, because 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
from observations, 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,
, to the observed distances,
. We should be aware that the primary
measured values on the photographic records of a meteor, are distances
along the trajectory, , 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 to
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,
, and, at them, we have the observed
distances along the trajectory, , and
heights, , where
. We will chose consecutive subsets
of *m* consecutive values from ,
from , and so on until from
. We will fit
to
for each of these subsets by the least squares using a polynomial
function
where to
are constants to be determined from
the respective subset of values so
that sum of 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
, ,
and 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,
, of each subset, we have
and
()
from Eq. (13) including their standard deviations. Because generally
the average times of the subsets
need not to coincide with the actually observed times
at which we have available
, we can integrate velocities
using the original
values and determine one integration
constant for the entire trajectory in order to fit these computed
distances to the observed distances
for the whole trajectory.
Now, the initial values of can be
computed from any two neighboring or nearby points on assumption that
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 is given by
( is natural logarithm)
where *t*, *v*, and are
actually the ,
and
, i.e. the average values of each
subset. For the start we can chose ,
while at the second step, when we already computed values of *K*
from Eq. (10), we can use and
as different values. We can improve
this procedure by taking into account all possible pairs providing
that and
are separated by small time
interval, and chose then the best determined values of
and *K* (with the relatively
smallest standard deviations). We finish with
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
, 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
, and we have chosen the point at
which so that
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.
© European Southern Observatory (ESO) 2000
Online publication: June 5, 2000
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