1 Introduction

Southworth and Hawkins (1963) developed a distance function D SH —the function of the orbital similarity named by them D-criterion—an important component of the computer meteoroid stream searching algorithm. Drummond (1979, 1981) introduced its modification D D and Jopek (1993) proposed an alternative hybrid D H . All D-functions are taken to be distances in a five-dimensional space, whose coordinates are heliocentric orbital elements e, q, ω, \(\Upomega,\) i. Other variations of the original D SH function were given by (Steel et al. 1991; Asher et al. 1993), where instead five-dimensional space the authors use only three-dimensions q, e, i or a, e, i. In Valsecchi et al. (1999) the authors introduced D N function involving four quantities U, cosθ, ϕ and λ; first three borrowed from Öpik’s theory of close encounters (see Öpik 1976; Valsecchi et al. 1999): the geocentric velocity U, and the angles θ, ϕ defining the antiradiant direction in the geocentric ecliptic rotating reference frame, located at the longitude λ at the time of the meteor observation. Not long ago Neslušan (2001) introduced C-criterion based on the difference between the orbital momentum vectors (per unit mass) of two orbits.

In the present paper we introduce yet another distance function D V defined in the domain of the heliocentric orbital elements. Following direction pointed out by Neslušan we propose to use the full set of the vectorial elements. In the following sections we describe the new function, and present the results of its application to Lindblad et al. (2003) photographic meteor catalogue.

2 New Distance Function D V

We take the vector (h T, e T, E)T which consists of vectorial elements: the angular momentum vector h, the Laplace vector e and the energy constant E (see e.g. Breiter and Ratajczak (2005)). In the units AU, AU/day and the mass of the Sun, these quantities are defined by equations:

$$ {\mathbf{h}}=(h_1, h_2, h_3)^T={\mathbf{r}}\times\dot{{\mathbf{r}}} $$
(1)
$$ {\mathbf{e}}=(e_1, e_2, e_3)^T =\frac{1}{\mu}\dot{{\mathbf{r}}}\times {\mathbf{h}}-\frac{{\mathbf{r}}}{|{\mathbf{r}}|} $$
(2)
$$ E=\frac{1}{2}\dot{{\mathbf{r}}}^2-\frac{\mu}{|{\mathbf{r}}|} $$
(3)

where μ = k 2, k is the Gauss constant, whereas r = (x, y, z), \({\dot{\mathbf{r}}}=(\dot{x}, \dot{y}, \dot{z})\) are the heliocentric vectors of the position and velocity of the meteoroid. Describing the ith meteoroid by the set of vectorial elements:

$$ {\mathbf{O}}_i=({\mathbf{h}}_i^T, {\mathbf{e}}_i^T, E_i)^T=(h_{i1}, h_{i2}, h_{i3}, e_{i1}, e_{i2}, e_{i3}, E_i)^T $$

we define a new distance function, measuring dynamical similarity among two meteoroids, as:

$$ \begin{aligned} D^2_{V}=& w_{h1} (h_{i1}-h_{j1})^2+w_{h2} (h_{i2}-h_{j2})^2+ 1.5\;w_{h3} (h_{i3}-h_{j3})^2\\ &+w_{e1} (e_{i1}-e_{j1})^2+w_{e2} (e_{i2}-e_{j2})^2+ w_{e3} (e_{i3}-e_{j3})^2 + 2\; w_E (E_{i }-E_{j })^2 \end{aligned} $$
(4)

where w hk , w ek , w E are suitably defined weighting factors.

In comparison with D SH , D D , D H , inclusion of the orbital energy, due to its invariant properties, brings important advantage into D V . However inclusion of the difference between the eccentricities is a disadvantage when one of the orbits undergo significant Kozai perturbations. Also, D V slightly overestimates the differences in the orientation of two orbits, similarly to the D D criterion, as was pointed out in Jopek (1993). Despite of disadvantages mentioned above, which we meet also in D SH , D D , D H -functions, we have found that D V criterion is very useful in classification of the meteoroids. In Sect. 4 we describe the results of the cluster analysis applied with D V , D SH , D N amongst the photographic meteors taken from the IAU Meteor Data Center.

To normalize contribution of each term in D V , following Southworth and Hawkins (1963); Porubčan (1977), we propose the weights

$$ w_E=(2\sigma_E)^{-2}, \;\; \; w_{hk}=(2\sigma_{hk})^{-2}, \;\; w_{ek}=(2\sigma_{ek})^{-2}, \;\;k=1, 2, 3 $$
(5)

where σ E , σ hk , σ ek are expected standard deviations of the corresponding vectorial elements in a stream. However, due to invariant properties of E and semi-invariant character of h 3 by introducing multipliers 2 and 1.5 we have increased their influence on the resulting D-value given by definition (4).

To estimate the standard deviations (5), we simulated formation and the dynamical evolution for several meteoroid streams: the Perseids, Leonids, Orionids and Geminids. The particles ejection model, their evolution were slightly different to those used by Williams and Wu (1993, 1994). Next, having the corresponding distributions, the averaged standard deviation of each vectorial element has been found, and the resulting values, on several epochs are given in Table 1. As can be noticed, for a given epoch, dispersions of the vectorial element are not the same, they differ up to 4–5 orders. In searching of the meteoroid streams we applied the set of weight coefficients corresponding to the epoch 4,000 years after the stream formation.

Table 1 The mean values of standard deviations of components of the vectorial elements of the typical meteoroid stream and the sporadic background
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3 The Meteor Data and the Stream Searching Method Used in this Study

We used 4,097 photographic meteors extracted from the computer files \({\tt geo2003.dat}\) and \({\tt orb2003.dat}\) downloaded from the IAU Meteor Data Center (Lindblad et al. 2003). Before being used for the classification, the available 4,581 meteor data were examined to check their internal consistency by the method slightly different to those described in Jopek et al. (2003). The test failed 306 times, and all this data as well as the orbits with e  > 1.1 were rejected. The meteor data were analysed using the distance functions D SH , D N and D V . First, the set of 4,097 meteors was pre-classified (single neighbour linking technique, D SH function and a rough estimate of the threshold) obtaining the sporadic sample of 2,699 meteors. Using this sample, for each distance function the threshold of the dynamical similarity was found by the method similar to (Jopek and Froeschlé (1997); Jopek et al. (1999, 2003), see Table 2. Next, alike as in Jopek et al. (2003) we processed all 4,097 meteors accepting all streams of nine or more members detected with the reliability level 99%.

Table 2 The values of thresholds D c,M and their uncertainties applied in the meteoroid association tests amongst 4,097 meteoroids
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4 Results of the Classifications With D SH , D N , D V

Using the function D SH we detected 14 streams, combining 36% of the 4,097 orbits; in the search made using D N , 17 streams were detected and the stream component included 46% of the sample; in case of the D V we obtained 12 streams forming 36.2% of the sample.

The main results of all searches are summarized in Table 3. In general D V seems to work more similarly to D SH rather then to D N . With D N function (based on two 3-body secular semi-invariants), more streams and more members of the given stream have been identified.

Table 3 Meteoroid streams detected in three searches
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In Table 3 the best agreement of the results we see for the Geminids, Leonids, Lyrids, December Monocerotids, Perseids and Quadrantids, which were detected in all searches. However with D SH we found 24 Leonids, of which only 16 were found and D V , plus an additional one only.

Using D N function, the Orionids and η Aquariids were identified, practically, as two separate groups of 67 and 15 members. With D V and D SH the Orionids and η Aquariids form a single group of 56 and 72 members, respectively. The N-branch of α Capricornids was identified in all searches, the S-branch members of this stream significantly less numerous, were found only with D N and D V . Opposite results we have in case of δ Aquariids: with D SH both N, S branches were found as two separated streams of 13 and 36 members. With D N function, both branches formed one group of 84 members, with D V we identified only 34 members of the Southern branch. Also, with D SH , the \(\kappa\) Cygnids was identified as two groups of 15 and 20 members. With D N and D V functions only one stream was found consisting of 56 and 41 members.

The most complex result we observed in case of the Taurids. Using D N we have found main group of 271 members and a second small one of 21 members. The main group contains many N and S Taurids as well as quite a lot of χ Orionids, as already flagged in the original IAU 2003 catalogue. Also with D SH we identified two groups of Taurids, the main one of 152 members and small one of 14 members. The main group found with D N included all Taurids detected by D SH and all but four members identified with D V . Using D V , one group of Taurids was found, it included 138 members of the main group found with D SH and 11 members of the second group detected with D SH -function.

The last rows of Table 3 lists four streams detected with D N only: the Virginids, σ Hydrids, ω Piscids and Cassiopeiids.

5 Conclusions

The new D V -function proved to be useful in the classification of the IAU2003 photographic meteoroids. In comparison with D SH and D N -function, for major streams the results agree very well. For minor, and near-ecliptical streams the results of the identification may differ markedly. As was expected, the distance functions D V , D N , D SH are not mutually equivalent. However, the main results obtained with the D V function are more similar to those obtained by D SH than by D N criterion. In this study we begin initial investigations of the new D-criterion and the authors are aware that several points need future study e.g. the final shape of the D V -function and suitable choice of the weighting coefficients.