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Improved 2001 Leonid storm predictions from a better model including cumulative non-gravitational effects on semi-major axis of particles in Leonid dust-trails

Esko Lyytinen and Markku Nissinen (URSA Astronomical Association, Meteor section)

Tom Van Flandern (Meta Research)

Abstract. It is expected that cumulative non-gravitational effect on the semi-major axis in the dust trails that produce meteor outbursts is an important factor affecting the particle spread along the orbit and therefore the apparent ZHR behaviour. In this work, we determine a numerical value of this effect from earlier observations, mainly those from the year 2000. Besides getting a better post-prediction of the course of the ZHR-curve, we also find that the observed maximum ZHR value of the 8 revolution outburst (1733) is better explained with the new model when the derived non-gravitational A2 distribution is taken into account. The model and newly derived parameter values are used to improve the predictions for the year 2001. The predicted outbursts for the year 2002 have not yet been treated in this way.

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1. Introduction

This work is a logical continuation of the work on Leonid outburst modelling presented in earlier works [1]. Similar methods used by two independent groups [2,3] are now well verified by the generally good fit of their predictions for time and peak activity of the meteor outbursts in 1999. The method calculates the position of particles released on different revolutions near the Sun, which form several narrow dust trails or trailets on subsequent return. Our most recent predictions for 2000 [1] again validate the technique, but leave some room for improvement.

The basis of our work is to assume that the ejection speeds are very small, which results from the satellite model of comets (e.g., [4]). Dust trails form as a result of particles having originally different orbital periods. In our model, the ejected particles will have different orbital periods mainly because the solar radiation pressure per particle mass differs from particle to particle.

The A2 effect includes all processes that change the meteoroid orbital period at each orbital revolution. According to our earlier study [1; EM&P], the principal cause of the a2 effect is thought to be the seasonal Yarkovsky effect. In principle, the amount of change may be different in each orbital revolution but, without better knowledge, is considered constant (within each particle) in this model. The change of orbital period itself doesn’t change the path of the particles (near perihelion), only the timing. Without planetary perturbations, this effect would not cause particle spread away from the trail center. However, the encountered planetary perturbations are slightly different as compared to the basic solution where A2=0, thus causing the spread. Typically, the effect is minor in young trails, but appears to be important with older trails. In first order, the additional spread will make the observed ZHR:s smaller, as is typical with very old central encounters of dust trails. However, for non-central encounters, this effect can also bring particles closer to Earth orbit and even increase the ZHR.

Here, we study the effect on the peak activity and the ZHR profile of various degrees of A2 effect. We go a step further than previous work, by not only fitting the time of the peak, but also the shape of the ZHR curve. The next section describes the model in detail. Results are in section 3.

2. The modelling

The basic model gives the direct spread parameters that vary in different encounters. This is a consequence of the basic ZHR model. In this model, the trail width (without the A2 effect) is proportional to the original Da multiplied by the orbit number. This requires two parameters, one corresponding the spread in the orbit plane (to the direction of the Sun) and the other normal to it. These are proportional to each other. The normal to plane parameter needed a calibration based on observations (as derived from Lorenzian half-widths for young trails). The 1999 storm data (see [5]) was used because the calibration requires a reliably observed encounter of young trail. The value arrived at is also consistent with the values derived from the storm 1966 [5] and 2 rev. encounter in 2000 [6]. The consistency with the 1966 data in [5] however gives a smaller maximum ZHR than most other sources and the derivation of width is not independent of the reached maximum value.

The A2-effect is modelled in the trail calculation program with a speed change at each perihelion. The speed change is expressed relative to the speed itself. A speed change of 10^-6 changes the orbital period by about 3/4 days each revolution. It is assumed that this speed change has a Lorentzian distribution. The A2 Lorenzian width parameter (half strength half width of the distribution) is given as millionths of the actual speed.

Figure 1. Model fits for 4 rev. (1866) and 8 rev. (1733) models added and compared with IMO compilation of Leonid Zenith Hourly Rate.

Figure 2.The 2000, 4 rev.(1866) ZHR-change with A2-distribution width parameter (half strength half width). Best fit around 5 to 7; data from [8,7].

Figure 3. The 2000, 8 rev. (1733) ZHR-change with A2-distribution width parameter.

Best fit 3.2; data from [7]

3. Results

3.1. The 2000 encounters

On November 17/18, 2000, Earth passed at some distance from the dust trailets ejected in 1932, 1733 and 1866. The second (1733) trail encounter was observed from Europe, the first and third from the United States. The last quarter Moon disturbed the observations, which are not as reliable as at other times. The A2-modeling is done for the older trails (8 and 4 rev.) only. Arlt et al. [7] gathered the data submitted to IMO and found peak rates of about ZHR = 270 and 450, respectively, shown in Fig. 1. In this work, the IMO data are approximated at 0.01 deg spacing. This is also true for the comparison figure 1. We needed values that correspond the values that we have in our A2 model, for least squares fits. Different fits are shown in Table 1. The fits with derived A2-width parameter are fits with the A2 model. There are only two free parameters in it, namely the A2 width parameter and the other that corresponds to the strength (linearly). The solar longitudes for these fits correspond the highest point in the modelled course at the fit, except with the 8-revolution encounter. With this the model clearly gives two peaks. The other fits are typical ("free" Lorenz) fits.

Figures 2 and 3 show how the 1866 (4 rev.) and 1733 (8 rev.) ZHR-curves behave when altering the A2-distribution-width. For the 1866 trail, the effect of A2 shifts the peak time of the outburst earlier in time and causes a flat-topped profile. For the 1733 trail, we find a large difference between cases with A2 (width parameter) = 0 and 1, where the peak activity quickly drops. This encounter without an A2 effect would have happened with a short piece of trail. However, the piece will disperse effectively even with a small amount of A2 effect. The increase of the A2-parameter gives a rise to a second maximum earlier in time (see figure 3).

In the A2-parameter fits, some share from the other nearby peak was subtracted, but no yearly background was subtracted.

Table 1: Results from curve fits (those with derived A2 half width parameters are with the fits by means of model parameters, the others are "free" Lorenz fits):

It is expected that the A2 parameter value increase with the original Da (a= orbital semi-major axis or "mean" distance). It is not expected that the dependency truly is a simple power law over a wide range of Da. However, in lack of a better knowledge, a simple power law dependency of (A2 width-parameter from Da) is assumed in the predictions. In the 2000 4 rev. data more weight is given to the [6] and [8] data than to [7], because of better fit with the model curve and their mutually better consistency. We accept power 0.7 and expect the curve go trough the 2000 8 rev. data point.

The trail profile of the 2000 encounter with the 1932 trail (2 rev.) is not well recorded in the IMO compilation [7], but a more accurate profile (figure 4) was measured from aircraft. Figure 4 [6] also shows the fit to similar data from the 1733 and 1866 trails. The 1932 trail is the best example of a large passing distance with no expected A2 effect. In general, with such a two-revolution trail the apparent effects of the A2 effect are small and indistinguishable within the original Da distribution. The effects typically increase quite rapidly from around 4 rev. up. It is also very much dependent on each individual case, reflecting close encounters with Jupiter. Indeed, we find a generally good fit to the1932 dust trail data even with no A2 effect. A good fit is obtained for a Lorenz width value consistent with the 1999 storm, rather than an increased width with rE-rD. We conclude that much of the spreading perpendicular to Earth's orbit (in the comet orbit plane) occurs as an accumulation in later revolutions.

The year 2000 data, was also applied to update the course of the function fn(Da) in the basic model. The function fn gives the ZHR of a one revolution encounter at exact hit. A relatively good fit with the earlier approximation was achieved, although the 4 revolution IMO data gives a point somewhat (almost by a factor of 2) below the general course. (The fitted McLeod data has an agreement with the general course to better than 10%.) The earlier known 11 rev. encounter in 1903 and the newly found 8 rev. encounter in 1868 were also used for this purpose. In addition, a re-treatment of the 1866 storm was done according to the A2 model. The fn values in the smaller Da range (below 0.1 au) were increased because of new data points. The result is shown in figure 5. In the constructed numeric model (fit-curve), the 1999 storm data point was given more weigh than the other points nearby.

Figure 5: Function fn(Da );(The 1866and 1999 data points have been moved to the "new data group" after an applied A2 model.)

The 1866 passage during the 1999 encounter is interesting because of its large Da involved (Table 1). The nominal miss distance rE-rD is about +0.0016 au. Interestingly enough, the observed maximum time was significantly earlier than predicted. We now find that the A2 effect with assumed Lorenz shape and parameter values will bring particles closer to the Earth's orbit at the observed time (solar longitude 235.8-235.9). The effect on the flux rate is not large (Fig. 6a) because the perturbation is not quite strong enough. In order to bring the particles sufficiently far in, one has to introduce an additional amount of the particle's direct spread. We introduced a very small additional direct effect to rD directly proportional to the A2 value. That does result in an activity curve much as observed (Fig. 6b). The direction of the increase is as expected from non symmetric reflections on dust grains (assuming the A2 effect is caused by the seasonal Yarkovsky effect). Even the amount of dependency is so small that it could very well be true. However, by trying the same model with the 2000 4 and 8 r. trails the fit-results did not improve. Indeed, the effect is thought to be important only for very large A2 values and other explanations may account for the particles being closer to Earth's orbit. Given the limited amount of justification for such additional complication, we will not include this additional direct effect in our ongoing work for the time being.

Figure 6a: 1999 encounter of the 1866 trail. The A2 effect introduces a feature at 235.8-235.9. The A2 width parameter is 3.7.

Figure 6b: As Fig. 6a in case of an additional slight dependency on A2 value in the direct spread. The feature is much enhanced and in good agreement with observations. The A2-width parameter is 3.7 and the additional rD dependency ("shift") is 0.000008 au times the (particle) individual A2 value (in millionths).

 4. Predictions for the year 2001

In November 2001, Earth is about to encounter several dust trails ranging from 4 to 11 revolutions old. The encounters of trails 4, 7, 9, 10, and 11 were treated with the A2-model for improving the predictions of ZHR courses and maximum values. The Da values are known from the earlier work [1] and the assumed direct proportionality gave values for this parameter to be used in the predictions. The results are shown in Table 2 and in figures 7 and 8.

Figure 7. The predictions for the year 2001 (stacked bars).

 Figure 8. The 2001, 7 rev. ZHR-change with A2-distribution width parameter.

A parameter value close to 4 is used for the 7 rev. encounter prediction. The strongest storm peak of the 4 rev. (1866) encounter is only very little affected by the A2-effect. Typically, central encounters with young trails are very little affected. The modelling shifts the 4 rev. maximum only about five minutes earlier and keeps the maximum ZHR practically unchanged. The peak is anticipated at 18:14 UT (Nov. 18).

This study resulted in an increase in the fn(Da)course by a factor of about two in the small Da range that corresponds the encounters with the 10 and 11 revolution trails. The most remarkable result, however, is a marked asymmetry in the profiles. This asymmetry may be detected as a tail in the storm profile after the main peak. The 9 rev. encounter is predicted to be somewhat shifted towards the 4 rev. storm and occurs almost simultaneously and inseparably. With the 9 rev. encounter the fn increase from the earlier is about 1.75-fold, giving a maximum ZHR of about 2600.

A further weakness of this model is that an old trail encounter, when computed according to this model, will have particles at considerably different Da values. Because the A2-distribution parameter itself is dependent of this Da, one value (corresponding to just one Da value) A2-distribution is not the best possible.

We thank Peter Jenniskens for the unpublished data and figures that he submitted for this study and valuable suggestions for this publication.

[9] Arlt R.,Rubio L. B.,Brown P., and Gyssens M., Bulletin 15 of the International Leonid Watch: First Global Analysis of the 1999 Leonid Storm, WGN, the Journal of IMO 27:6, 286-295 (1999).

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