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3. Reduction of the data

3.1. The path of the meteor in projection on the sky

The reduction of the negatives (e.g. Whipple 1938; Ceplecha et al. 1979; Ceplecha & Borovička 1992) was performed at the Sterrewacht Leiden of the Leiden University and at the Stichting Geavanceerde Metaalkunde of the Technical University of Twente. Each set of star trails and meteor trail were routinely measured in two directions to avoid hysteresis. The meteor is measured twice per shutter break at the head of the break to minimize the influence of wakes and trains. The measurement accuracy is limited by the quality of the meteor and star trail images, not by the measuring device, a Zeiss Astrorecord X-Y measuring table, with a nominal accuracy of 0.001 mm. The image quality is determined by the quality of the camera optics, the focal length, the brightness of the meteor, the location on the negative, and by background fogging. The standard deviation of measured star positions and the calculated grid is about 30''-1' for the well focused F/2, tex2html_wrap_inline1048 cameras with 35 mm film format and the 60 mm F/4, tex2html_wrap_inline1072 cameras that were used in the small camera network. The all sky cameras, on the other hand, use 35 mm cameras with F/5.6, tex2html_wrap_inline1074 or F/2.8, tex2html_wrap_inline1076 fish-eye lenses, which results in a positional accuracy of about 3'-5'. The meteor cameras are not guided but the instant of the beginning and ending of each exposure is known with an accuracy of better than tex2html_wrap_inline1080 seconds and the time of the meteor is usually known to tex2html_wrap_inline1082 seconds, which allows for a correction in RA of the stellar trails to the instant of the meteor to an accuracy of about tex2html_wrap_inline1084 seconds (1.5'). Larger errors occur when the time of the meteor is incorrect due to a wrong identification from the visual observations. Large differences in time (more than a few minutes) do usually not result in a reasonable solution of the multistation calculation. However, extra effort is made to ensure that the correct identification is made.

The method of Turner (1907) is followed to calculate the stellar coordinates of the meteor path from the position of stars on the negative (Tadeusz 1983). The coordinates of the stars are taken from the Tirion Star Atlas 2000.0 (Sky Publishing), with 0.1'' accuracy. The method uses some 20 stars spread over the negative with a triangle of stars around the approximate plate center and adopts a coordinate grid that accounts for projection distortions from the optical system. This distortion becomes too strong for the method in the case of fish-eye images. Hence, for fish-eye images we used the method of Ceplecha outlined in the REDCON software routine (Ceplecha et al. 1979). This method makes use of seven independent constants to link the measured rectangular coordinates with the horizontal coordinates but demands that the center of the plate is near the zenith. If the number of usable stars for determining the coordinate grid is insufficient, simpler procedures with less constants are applied but with, of course, less precision.

The trajectory of the meteor in the atmosphere is calculated, subsequently, by fitting planes through the station and meteor path. The location of the stations and their altitude is known within 30 meters or better. The best plane is calculated by a least squares fit through the measured positions of the meteor path. The combination of two planes then result in a (partial) meteor path. The mean trajectory is computed from all individual paths with weight factors:
with Q the convergence angle and Wi = 1.0 for most images and Wi= 0.7 for some less accurate fish eye components.

When a meteor is recorded by more than two stations, accuracy checks on trajectory and radiant data can be made by comparing two-station solutions. In most cases errors are caused by timing errors, either in the exposure times or in the time of the meteor. These errors are reflected in the scatter in the Right Ascension (RA) of the position of the radiant in the various solutions, but do also affect the Declination. Small convergence angles between the planes affect the radiant and the geographic data of the trajectory in a negative way. It is found that for convergence angles (Q) larger than 40 degree, the error in radiant position is about tex2html_wrap_inline1100. For Q = 20 degree it is about tex2html_wrap_inline1104 and for Q = 10 degree it is about tex2html_wrap_inline1108. If the convergence angle is less than 10 degree, then the accuracy is not better than radio orbits of order tex2html_wrap_inline1110 (Sekanina 1976). As much as 41 of the 359 precisely reduced meteor orbits in our study do not lead to precise results, because they have tex2html_wrap_inline1112, but are still considered valuable in some cases due to the small number of known stream orbits, the large mass of the meteoroid, or a relatively high accuracy in a part of the orbital elements. Error estimates are given for each individual orbital element (Table 2 (click here)).


Code - DMS sequential numbering starting with the year
Month - month
Dec. Day - day and time (UT) in decimal days
N - number of multi-station components
Stream - meteor stream identification
Mv - absolute visual magnitude
q - perihelion distance (AU)
a - semi major axis (AU)
e - eccentricity
i - inclination (Eq. 2000)
omega - tex2html_wrap_inline1126 - argument of perihelion (Eq. 2000)
Node - tex2html_wrap_inline1128 - ascending node (Eq. 2000)
pi - tex2html_wrap_inline1130 - longitude of perihelion (Eq. 2000)
tex2html_wrap_inline1132 - geocentric velocity (km/s)
tex2html_wrap_inline1134 - heliocentric velocity (km/s)
tex2html_wrap_inline1136 - apparent pre-atmospheric velocity (km/s)
V - average velocity along trajectory
tex2html_wrap_inline1140 - beginning height (km)
tex2html_wrap_inline1142 - height of brightest point on meteor track (km)
tex2html_wrap_inline1144 - end height of meteor (km)
RA obs. - apparent right ascension of radiant (2000)
[+/-] - error due to uncertainty in time of meteor
DEC obs. - apparent declination of radiant (2000)
RA Geo - geocentric right ascension of radiant (2000)
Dec Geo - geocentric declination (2000)
CosZR - cosine of zenith angle of radiant at time of meteor
tex2html_wrap_inline1148 - maximum convergence angle between planes.
Table 2: Orbital elements, encounter data and other relevant information of 359 photographic meteors. This table is published in electronic form only at the CDS. Columns give:


The initial velocity is computed by making use of a fit through all computed positions of the meteor at the head of shutter breaks (Jacchia & Whipple 1961):

with L being the geographic position of the trail, H the height, while tex2html_wrap_inline1158 and K are variables. It is found that this fit gives reliable results only for meteor trails with at least some 30 well measured breaks. In cases with less breaks (N = 15-30) a fit of the following equation is used instead (Ceplecha & Borovička 1992):
where tex2html_wrap_inline1166 is the air density at a given height (or time t of the meteor) computed from the CIRA 1961 reference atmosphere (Kallmann-Bijl et al. 1961). The final accuracy with which tex2html_wrap_inline1170 is determined is about 1% in case a sufficient number of breaks can be measured. The correction from measured velocity at the beginning of the trajectory to the pre-entry velocity before deceleration in the atmosphere varies from 0.2 km/s for fast meteors up to 1.5 km/s for some slow meteors.

Given an accurate radiant position, the uncertainty in the orbit is mainly determined by the uncertainty in the velocity determination. That demands stable rotating shutters. Before 1986, all our rotating shutters relied on the stability of the mains frequency, which is thought to be constant and accurate at 50 Hz to within 0.5%. The chopping frequency of the shutters was measured to be accurate within 0.2%. However, the rotational period of the shutters can oscillate with a variable oscillation period and an amplitude up to 4%. Such oscillations occur when the rotating shutters are perturbed by strong winds or when there are system resonances. Such oscillations occur at random. Larger than 1% errors are seldomly found when information from more than one rotating shutter is available. To improve the quality of the computed orbits, the Dutch Meteor Society introduced crystal controlled rotating shutters in 1986 at some stations. Starting in 1992, all stations were equipped with new crystal controlled rotating shutters with twice the shutter frequency, increasing the shutter speed to 50 breaks per second with a stability of 0.1% or better.

The orbital elements (J2000) follow from the true radiant position, the computed initial apparent velocity and the position of the Earth at the time of the meteor (e.g. Katasev 1957). The calculations take into account such corrections as zenith attraction, the curvature of the Earth, and diurnal aberration, and give good results even at shallow entry angles and large distances between the stations (Ceplecha et al. 1979; Ceplecha & Borovička 1992). That was demonstrated again in one case DMS 85027, a sporadic fireball, which was photographed by four Dutch stations as well as by four distant German stations of the European Network (Betlem & de Lignie 1985). Independent reduction in Leiden and Ondřejov gave good agreement between the Dutch and German stations (Ceplecha, private communication).

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