Vi.km s

Figure 7. The relationship between the on-ecliptic heliocentric velocity component in the direction of the apex of the Earth's way and q and oj for the sda orbits.

Vi.km s

Figure 7. The relationship between the on-ecliptic heliocentric velocity component in the direction of the apex of the Earth's way and q and oj for the sda orbits.

3.2. Determination of Uncertainties

It is important to provide an estimate of the parameter uncertainties for individual orbits in order to determine the extent to which the observed shower spread may be attributed purely to such uncertainties. Uncertainties in high-level parameters, such as the orbital elements, propagate from uncertainties in the low-level direct meteor observation measurements. An orbit, as determined by amor, is derived from four directly measured parameters [1]: two receiving station time-lags (Lagi2 and Lagi3) and the elevation (v?) and azimuthal {A) angles of the echo-point. Representative uncertainties in these are 1.4 radar pulses in the time-lags, 0.5° in ^ and 2° in A. The reduction of these parameters to orbital elements consists of many steps so that a simple classical uncertainty analysis approach is therefore difficult. Two alternative methods have been tested in the current study. The first method is similar to that used by Taylor [13]: it extracts random combinations of the four parameters from Gaussian distributions having mean values based on the original measured parameter values and standard deviations based on the assumed parameter uncertainties; each of these parameter quartets are run through the orbit reduction computer program in order to produce a full set of high-level output parameters—the spread in the distributions in the latter, obtained from 30,000 such randomised simulations, is assigned to be the uncertainty in the respective parameters. The second method is found to provide equivalent results to those of the first method, however it has the advantage of a negligible computer time requirement in contrast to the rather lengthy time required for the first. Classical analytic uncertainty techniques are used; however, where normally the uncertainty in a single function is determined, here the computer reduction program is treated as that function with four input variables and many output variables. The uncertainty 6f in a function f(Vi, V2, ...Vn) is given by where the partial derivatives follow from df_ = Um f{Vk + h)~ f{Vk ~ h)

/ is a particular high-level parameter for which the uncertainty is required and V1..V4 are the low-level input parameters (Lagi2, Lagi3, and A)-, h = 10~3 is used in this equation as this is sufficiently small to give reasonable answers but large enough to avoid numerical instabilities found as h —> 0.

3.3. Shower Statistics

Calculation of the shower orbital statistics may seem straightforward but there are a number of issues which must be addressed. As noted in Section 3.1, many orbital parameters experience an apparent daily motion as they are detected by the moving Earth observation platform over time; the observed distribution in such parameters is smeared according to the perceived (generally non-uniform) distribution in time of shower activity. In the current study this problem has been removed for affected parameters by using Equation 1 together with linear least squares motion measurements. Another difficulty in dealing with Earth-detected meteors is the necessary relationship between the corresponding meteoroid orbit's q, e and w elements:

where the ± sign signifies detection at either the ascending (+) or descending (—) nodes. In obtaining mean values in these parameters one may determine the mean in e and u;, and then base the mean of q on this; alternatively one can treat these parameters independently, simply noting that Equation 4 may not be exactly obeyed for the means obtained. The latter approach is taken in the current study as the former leads to difficulties in determinations of the statistical spread in the dependent parameter.

Table 3 lists the orbital parameter statistics for the five showers under study. These parameters (apart from A0 and have been corrected where appropriate for measured daily motion. The reduction epochs used in Equation 1 for these corrections are A0 = 125°, 125°, 313°, 45°, 187° and 46° for the cap, sda, Peak 1, Peak 2, dsx and eta showers respectively. Representative uncertainties in each shower's parameters are obtained from the median measurement uncertainties calculated as discussed in Section 3.2. It is important to note that generally the representative uncertainties are very similar to the measured spread (a) in the distributions—in most cases this means that little can be learned about the physical parameter spreads apart from that they lie within these bounds. The uncertainties in the declination and ecliptic latitude angles which are directly related to the echo elevation angle are exceptions in all cases: their spreads, however, are influenced by the wavelet probe size choice and hence the "physical spread" data obtained are of limited use. The inclination angle generally has a spread exceeding that expected from uncertainty: this is one of the more stable elements measured by amor and plays a significant role in defining a shower. Exceptionally, the sda has a large ~ 10° uncertainty in i. Of the showers studied here, the eta has the largest spread in most orbital elements owing to the high geocentric speed of its constituent meteoroids. The spread in oj found in this shower is less than that expected due to the representative measurement uncertainty spread—some eta meteors have been omitted due to their distance from the shower centre in the defining 4-D space.

Owing to the very small standard errors in the mean parameters of each of the showers, we may have very good confidence in these (daily motion corrected) means. This confidence is in contrast to Lindblad [7], for example, who ignores radar meteors in his survey of the eta due to the poor mean orbits which result. While amor's strength may lie in determining high quality shower mean parameters, measuring the physical spread, as discussed above, is non-trivial and in many cases unrealistic.

The shower mean orbits were found using Southworth and Hawkin's [12] criterion of orbital similarity (Dsh) to agree favourably with several previous examples in the literature.

The sda has dissimilarities of 0.05, 0.08 and 0.06 with the mean orbits of Nilsson [8], Cook [3] and Sekanina [11] respectively. Such D$h values show close agreement, particularly those for radar based means measured by Nilsson and Sekanina which show negligible differences.

The eta shows more variation in literature comparisons, with dissimilarities of 0.10, 0.16, 0.10, 0.08, 0.07, 0.12 and 0.05 when compared with Cook [3]; Lindblad [6] photo-

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