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Astron. Astrophys. 351, 752-758 (1999)

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2. Modelling of a theoretical stream

Meteoroid particles are most frequently released from the nucleus of their parent body when closest to the Sun, i.e. at the perihelion of the parent body. Taking this into account, we model a theoretical meteoroid stream at the moment of perihelion passage of the parent body considered.

The orbits of the particles are subsequently dispersed due to non-zero ejection velocity and gravitational as well as non-gravitational perturbances. The dispersion coming from the initial orbital velocity difference and non-gravitational forces cannot move the particles of an appropriate meteoroid stream associated with a distant parent body (such as 14P/Wolf and D/1892 T1) close to the Earth's orbit. Even if such particles cross the Earth's orbit, it is not possible to distinguish them from the sporadic meteor background. Therefore, stream particles can approach the orbit of our planet only due to quasi-systematic gravitational perturbances, which deflect a significant part of the stream from the original direction of its motion to a quasi-uniform new direction. To map the action of the gravitational perturbing forces, we study an evolution of a set of theoretical particle orbits being very adjacent to the orbit of their parent body, whereby we attempt to construct these orbits uniformly around the parent body orbit.

If the disturbances are quasi-systematic, one expects that these appear relatively soon after the release of the particles from the parent body. Otherwise, the non-gravitational forces would chaotically disperse the particle orbits and the disturbances could scarcely have a systematic character. Consequently, the dispersion of the modelled orbits has to be much smaller than the actual observed dispersion of (long lasting) meteoroid streams. Thus, the modelled set of orbits represents the most central part of the stream, not the entire stream.

The non-gravitational forces are not considered in the modelling.

To model the most central part of an investigated potential meteor stream and to identify the theoretical particles with the actual observed meteors in the catalogue, we execute the procedure consisting of the following steps:

[FORMULA] 1. Integration of parent body orbit backward up to its perihelion being most close to time [FORMULA] before the beginning of this integration. The beginning is, practically, identical to the epoch which orbital elements of the parent body in the Catalogue of Cometary Orbits are referred to. [FORMULA] is the orbital period of the parent body at the beginning. We take into account the perturbances from 8 planets (Mercury to Neptune). Their initial radius and velocity vectors are taken from the Astronomical Almanac (1983). The integration is done by using,,GAUSS-RADAU" (RA15) integration method developed by Everhart (1985).

[FORMULA] 2. Proper modelling of the theoretical stream. We consider a parent-body-centric coordinate system with the [FORMULA] plane identical to the orbital plane of the parent body at the time of the end of the integration executed in step 1 (the parent body is situated at its perihelion). The [FORMULA]axis of this coordinate system is orientated by the heliocentric perihelion velocity vector, [FORMULA], of the parent body. Every modelled particle is assumed to have the magnitude of its orbital velocity different to that of the parent body, [FORMULA], about value [FORMULA]. We assume that [FORMULA], where [FORMULA] is a constant factor. The spatial grid of modelled particles is produced calculating the components of the velocity vector in the parent-body-centric coordinate system. These components represent the differences, [FORMULA], [FORMULA], [FORMULA], between the components of heliocentric velocity vectors of [FORMULA]th particle and the parent body. In a spherical coordinate system, v, [FORMULA], [FORMULA], these differences can be given as

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA], 1, 2,..., up to the nearest integer of [FORMULA], and [FORMULA], [FORMULA],..., -1, 0, 1, 2,..., [FORMULA], [FORMULA]. Assuming a uniform distribution, [FORMULA] is constant and [FORMULA]. The uniformity of the spatial grid further requires [FORMULA]. If [FORMULA] ([FORMULA]), then [FORMULA] ([FORMULA]). Finally, [FORMULA].

In our particular case, we chose [FORMULA]. Consequently, we obtain a total number of 2578 particles. At the moment of parent body perihelion passage, the heliocentric velocity vector of [FORMULA]th particle is [FORMULA] [FORMULA] [FORMULA] and its heliocentric radius vector is identical to that of the parent body. Based on both the vectors, the appropriate orbit can be determined.

The ejection velocity of particles from the cometary nucleus is a free parameter in the above construction. Kresák & Kresáková (1987) spoke about this parameter in term of multiples ([FORMULA] in our paper) of the orbital heliocentric velocity of the nucleus. For comet Halley, they considered value [FORMULA] corresponding to [FORMULA]m s-1. Utilizing the observations of dust trails performed by the Infrared Astronomical Satellite, Sykes et al. (1989) considered the values lower than 10 (Type I trails) and about [FORMULA]m s-1 (Type II trails). The values 10 and [FORMULA]m s-1 correspond to [FORMULA] and [FORMULA], respectively. In our particular case, we consider 3 values of [FORMULA] equal to 0.0005, 0.001, and 0.002. These correspond to ejection velocities from 14P/Wolf equal to 14.7, 29.4, and 58.8 m s-1, and that from D/1892 T1 equal to 15.7, 31.4, and 62.8 m s-1, respectively.

We again note, only the most central strand of the stream is modelled in this way. E.g. for comet 14P/Wolf, the dispersion of orbits in this strand, characterized with the Southworth-Hawkins (1963) [FORMULA]discriminant, is 0.0016, 0.003, and 0.006 for [FORMULA], 0.001, and 0.002, respectively. The dispersion of orbits in an actual observed stream is that of order of [FORMULA] (Neslusan et al. 1995). So, the modelled orbits are actually much less dispersed than those of an actual observed stream.

[FORMULA] 3. Forward integration of the orbits of all modelled particles together with the orbit of the parent body itself. An evolution of all the orbits is observed making the output from the integration after elapsing every [FORMULA] for [FORMULA], 1, 2, 3,...,10.

[FORMULA] 4. Identification of orbits of the modelled particles with the orbits of actual meteors observed photographically, which are contained in the catalogue of the IAU Meteor Data Center in Lund (Lindblad 1987, 1991; Lindblad & Steel 1994). This identification is performed at each output.

We regard as similar orbits where the Southworth-Hawkins (1963) [FORMULA]discriminant is not higher than 0.24. Investigating a mutual relationship among identified orbits of actual observed meteors, we found that the limiting value [FORMULA] characterizing the dispersion of the best orbits of [FORMULA]Capricornids in the catalogue is too strict for the similarity determination. This value is in accord with the purpose of the method of selection of the meteors from the database to select only such meteors, which can definitely be assigned to the shower. The method of optimal separation of meteors (Porubcan et al. 1995) could not unfortunately be applied and limiting value of D obtained for [FORMULA]Capricornids because of their complicated structure. Since the limiting value of the [FORMULA]discriminant determined by the second (optimal) method is about 2 times higher than that determined by the first method for four studied major showers (Porubcan et al. 1995) on average, we decided to consider the value 0.12 twice as high in this paper.

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© European Southern Observatory (ESO) 1999

Online publication: November 3, 1999
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