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Astron. Astrophys. 353, 797-812 (2000)

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4. Calculation of the initial conditions

In this section we describe the procedure by which we have calculated the initial positions and velocities of the bolides, needed to numerically integrate their orbits. Tables 1 and 2 summarize all starting data; in Table 1 the number of significant digits is consistent with the expected accuracy (apart from the apparition times [FORMULA], whose error is at most a few minutes). The errors in the orbital elements are caused by the limited accuracy of both the observations and the models used to reconstruct the trajectory prior to atmospheric entry (the corresponding errors for the coordinates of the radiant points and the entry velocity are given in Table 2). In all cases, these errors are small enough that a qualitative dynamical study such as performed in this paper is meaningful.

To fulfil the requirements of our long-term integration software, the data reported in Table 1 had to be:

  • complemented by the moment of perihelion passage;

  • transformed into rectangular coordinates;

  • integrated to a common initial epoch, in our case JD 2440000.5.

All these steps have been carried out by the same method as described in Jopek et al. (1995), to which we refer for further details. Following the first two steps, the rectangular coordinates of Table 3 give the state vectors of the orbits listed in Table 1. The corresponding final state vectors at the common osculating epoch JD2440000.5 and the equivalent set of the orbital elements are listed in Tables 4 and 5, respectively.


[TABLE]

Table 3. Rectangular coordinates of 26 bolides in the 1950.0 heliocentric ecliptic reference frame. [FORMULA] is the corresponding epoch in Ephemeris or TDT Julian days. x, y, z are given in AU, [FORMULA],[FORMULA],[FORMULA] in AU/day.



[TABLE]

Table 4. Rectangular coordinates of the 26 bolides at the common epoch JD 2440000.5. Reference frame: barycentric, ecliptic 1950.0 (x, y, z in AU; [FORMULA], [FORMULA], [FORMULA] in AU/day).



[TABLE]

Table 5. Osculating elements of the 26 bolides at the common epoch JD 2440000.5. Reference frame: heliocentric, ecliptic 2000.0.


Since the orbital elements listed in Table 1 certainly include considerable observational and model errors, we may wonder whether the results of the numerical integration procedure are sensitive to any small change in the values of the initial state vectors. Therefore, exactly in the same way as in our earlier paper (Jopek et al. 1995), we have estimated the propagation of these uncertainties to our final values for the coordinates and velocities used as initial conditions for the long-term integrations. Table 6 shows the maximum differences between the final coordinates of the test particles corresponding to each bolide. In general, the sensitivity does not appear very strong: for most of the orbits the differences are of the order of [FORMULA] AU and [FORMULA] AU/day, i.e. much larger than the initial "noise", but small enough that the long-term integrations to be discussed in Sect. 5 can still be seen as representative for the real population of small bodies hitting the Earth. The largest instability appears to be associated with bodies nos. 4, 5, 10, 11, 19 and 25.


[TABLE]

Table 6. Estimate of the sensitivity of the numerical integration to the initial coordinates. The Table gives the maximum differences between the final coordinates of the test particles associated to each bolide ([FORMULA], [FORMULA], [FORMULA] in AU; [FORMULA], [FORMULA], [FORMULA] in AU/day).


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© European Southern Observatory (ESO) 2000

Online publication: December 17, 1999
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