The comet Halley meteoroid stream: just one more model

Abstract

The present attempt to simulate the formation and evolution of the comet Halley meteoroid stream is based on a tentative physical model of dust ejection of large particles from comet Halley. Model streams consisting of 500–5000 test particles have been constructed according to the following ejection scheme. The particles are ejected from the nucleus along the cometary orbit (r <9 au) within the sunward 70° cone, and the rate of ejection has been taken as proportional to r⁻⁴. Two kinds of spherical particles have been considered: 1 and 0.001 g with density equal to 0.25 g cm⁻³. Ejections have been simulated for 1404 bc, 141 ad and 837 ad. The equations of motion have been numerically integrated using the Everhart procedure. As a result, a complicated fine structure of the comet Halley meteoroid stream, consisting not of filaments but of layers, has been revealed.

comets: individual: P/Halley, meteors, meteoroids

1 Introduction

The assumptions that the Eta Aquarid and Orionid meteoroid streams are genetically associated with comet Halley were made more than 100 years ago [see the historical information in Levin (1956) or Lovell (1954)]. They were doubted for a long time, however, because it followed from the existing toroidal model of meteoroid streams that the Eta Aquarids should be more active than the Orionids, but in reality these two showers are approximately equal in activity and width. Now it is commonly accepted that the Orionids and Eta Aquarids are twin meteor showers, i.e. manifestations of the same meteoroid stream generated by comet Halley. The shell model proposed by McIntosh & Hajduk (1983) and the numerical model by McIntosh & Jones (1988) explained many of the features of the showers and were, figuratively speaking, the first and the second iterations in the simulation process. However, as will be discussed later, some simplifications led to wrong conclusions.

Much like McIntosh & Jones (1988), I used numerical modelling to study the stream behaviour. The present attempt to simulate formation and evolution of the comet Halley meteoroid stream, however, is based on a more realistic scheme of the ejection of large particles from the comet (Ryabova 1997).

2 The Eta Aquarid and Orionid Meteoroid Streams from Observational Data

The first mentions of observations of the Eta Aquarids (74 bc) and Orionids (288 ad) are found in Chinese, Korean and Japan chronicles (Imoto & Hasegawa 1958; Zhuang 1977). The literature on the observations of these showers is quite voluminous. A nearly exhaustive list of references over the period 1900–67 was given by Hajduk (1970). A list for more recent years is presented in Table 1. The references collected in the table are restricted to instrumental observations. There are also several relatively recent reviews of the showers, but, as a rule, they are based on reviews by Hajduk (1970, 1973, 1980). In fact, in recent years nobody has succeeded in adding anything significant to his conclusions.¹ The data considered below are based mainly on these papers, as well as those of Table 1.

Table 1.

List of sources.

2.1 Period of activity

The activity of both showers lasts for about 14 d: the Eta Aquarids are at solar longitudes 37°≤λ≤ 51° and the Orionids are at 202°≤λ≤ 216°. However, there is evidence to suggest that the activity extends beyond these limits, namely from April 20 until May 23 for the Eta Aquarids, and from October 1 until November 7 for the Orionids (McKinley 1961; Cook 1973; Rendtel, Arlt & McBeath 1995).

2.2 Shape of the activity profiles of the showers

The shape of the activity² profiles changes from year to year without periodicity, but not quite without a system. In different years from one to five maxima were recorded. Sometimes a gradual shift in a maximum position can be followed over 5–10 years. This shift was ascribed to filaments in the stream (Hajduk 1970; McIntosh & Hajduk 1983). However, the activity profiles that have been obtained by long-term averaging are rather stable. Both of them (i.e. for the Eta Aquarids and Orionids) have two hill-like maxima with a small dip between them.

2.3 Variations of the stream density along the orbit

Any pronounced periodicity in the variations of the density around the orbit has not been revealed. The correlation of the activity periods with the 12-yr period of Jupiter is rather high for the Eta Aquarids but absent for the Orionids (Hajduk 1970). Hajduk (1970), however, drawing attention to this correlation, writes: ‘At first glance, the shower frequency minima correspond roughly to the Jupiter period. However it is impossible to explain the long duration of the minima and the relatively slow change of the shower activity by this phenomenon. Moreover, the agreement between the years of minimum activity and the calculated parts of the orbit under Jupiter influence is not general.’ Comparison of the mean hourly rates of the two showers in different years does not show any system. The mean activity levels of the showers are approximately equal. The observations in 1981–86 showed that the activity does not depend on the parent comet position for large particles and decreases somewhat for small particles. The stability of hourly rates for large particles is noticed by many authors.

2.4 Rates and incident flux density

It should be remarked here that rate profiles, contrary to incident flux density profiles, can give a misrepresented picture on account of observational selectivity. For radar observations, for example, there are three essential factors of selectivity (Bel'kovich 1971). First, the minimal detectable electron line density and, consequently, rates depend on the distance to the trail and on the antenna gain for each direction. Secondly, for a definite direction observed rates depend on the radiant zenith angle and on the meteor velocity. The state of the ionosphere is also important. Hajduk (1973), in particular, pointed out that the proportion of long-duration echoes (i.e. large particles) is different in both showers, but it can be explained by ionospheric conditions at the time of the observations: the mean radiant of the Orionids culminates about 1 hour before sunrise, and the Eta Aquarid radiant culminates 3 hours after sunrise. So all conclusions about stream structure based on rates should be taken with caution.

I managed to find only one study of flux density variations with time for the meteor showers in question. Kolomiets & Milyutchenko (1995) (see also Kolomiyets et al. 2002) analysed radar data obtained in Kharkov in 1986 and found that the activity curve for the Eta Aquarids has two distinct maxima, while the curve for the Orionids has two weak maxima and, possibly, a third (the plot terminates as activity increases at λ= 218°) — but this is only one year.

2.5 Density of meteoroids

Verniani (1967) estimated the particle density for Orionid meteoroids to be ρ= 0.25 g cm⁻³ (from data for four meteors), and for the Eta Aquarids to be ρ≈ 0.6 g cm⁻³ (one meteor). Tokhtas'ev (1982) calculated the density for Orionid meteoroids from 23 meteors and it was 0.25 g cm⁻³. The dust particle density in the vicinity of the nucleus of comet Halley was estimated to be 0.35 g cm⁻³ for submicron particles (Krasnopolskij et al. 1986).

3 Earlier Models of the Comet Halley Meteoroid Stream

3.1 The shell model

The first model, which explained many of the features of the Orionid and Eta Aquarid meteor showers, was proposed by McIntosh & Hajduk (1983). The authors postulated that meteoroids merely exist in the orbits where the comet was many revolutions ago, and constructed a model of the stream named the ‘shell model’ or, later, the ‘ribbon-like model’. It differed dramatically from the toroidal model dominating at that time.

The shell model was the first to explain the fact, believed to be an incongruity, that the Orionids and Eta Aquarids have nearly equal activity even though the Earth approaches the cometary orbit at a minimum distance of 0.07 au in May (the Eta Aquarids) but 0.15 au in October (the Orionids). The existence of numerous peaks of activity was explained by Kozai's libration cycles, i.e. the ‘ribbon’ is a set of several ‘ribbons’ shifted towards each other.

3.2 The numerical model

The second model was constructed by McIntosh & Jones (1988) a few years later. Here the orbital motion of up to 500 test particles was numerically integrated from 1404 bc to 1986.

The ejection velocity of particles was taken according to the formula developed by Whipple (1951). Velocity vectors were distributed isotropically. The starting position of particles is not completely clear. The authors noted that ‘The relative starting positions of particles were based on an assumed number density distribution proportional to the flux of solar radiation. This generates a distribution of particles very much concentrated near perihelion.’ However, their plots show that ejection from the parent comet probably was modelled at perihelion (see Section 4.2). This model seems to be realistic regarding the gross behaviour of the stream, but it is not so regarding the fine structure of the stream, which will be the subject of the following study.

The paper of McIntosh & Jones (1988) was not the last word on this point. Wu & Williams (1993) presented a model like that of McIntosh & Jones (1988): 40 test particles ejected from the parent comet at perihelion at 1404 bc were numerically integrated 3400 yr forward. The authors' intention, however, was not to study the evolution of the structure of the stream, but to compare the orbits of observed and model Eta Aquarid and Orionid streams.

4 The Model

4.1 Reference orbit

The orbital motion of comet Halley has been studied by many authors. A detailed retrospective journey into the history of the topic can be found in Yeomans (1977) and Yeomans & Kiang (1981). An extensive body of literature has evolved in connection with the last cometary approach: see for example Brady (1982), Landgraf (1986), Batrakov et al. (1986) and Sitarski & Ziolkowski (1987).

From the references listed above, only Brady (1982) and Yeomans & Kiang (1981) give osculating orbital elements for dates before 240 bc. The ephemeris by Yeomans & Kiang (1981) has been taken here as a reference one, because it has the smallest ‘observed minus calculated’ (O−C) value for perihelion passages.

I have chosen four epochs to model particle ejections: 1910 ad, 837 ad, 141 ad and 1404 bc. The last date is the earliest for which an osculating orbit can be computed without additional observations prior to it. In 141 and 837 ad the perihelion passage observations were unusually accurate. Cometary orbital elements were considered constant during the revolution, where a model stream was generated.

4.2 Ejection of particles

For the comet Halley meteoroid stream, unique data are available on the ejection of particles from the parent comet. In 1986 during a recurrent approach to the Sun, an extensive research programme including ground and in situ measurements of the dust particles was developed.

Collecting all accessible literature data for the dust production rate, the ejection velocity distribution, the mass distribution and physical characteristics of particles, the shape of the nucleus etc., I have made an attempt to construct a model of dust ejection from the nucleus of comet Halley for comparatively large dust particles (>10⁻⁹ g). As a base and test for the adequacy of the model, the variations in the particle flux density along the Vega-1 trajectory measured by the onboard Photon instrument (Anisimov et al. 1987) have been used. A probabilistic method was proposed to model a stream of particles ejected from a cometary nucleus (or other source) into a spacecraft trajectory. All details of this study may be found in Ryabova (1997).

The main results related to the present model are the following. The mean ejection velocity is equal to about 8 per cent of the velocity c_w calculated using the formula of Whipple (1951), and the upper limit is 20 per cent. The ejection of the particles occurs mostly within the cone 70°± 5° oriented toward the Sun. The nucleus of comet Halley has a prolate irregular shape, but functions for the distribution of the ejection velocity in direction for a spherical nucleus turned out to be good approximations, maybe because the nucleus is involved in a rapid and complicated rotational motion.

4.3 The method and the initial parameters

The essence of the method described below is the following. On a reference orbit, points of particle ejection are chosen and ejection velocity vectors are generated according to some assumed distributions. The orbital motion of the test particles is numerically integrated until the present epoch. The model stream structure, mainly its cross-sections, can then be analysed. Let us consider the model components.

The ejection points on a reference orbit are taken to be distributed proportionally to r⁻⁴, where r is the distance from the Sun (see the review of dust production rate in Ryabova (1997)). The ejection velocity, unless otherwise indicated, is taken to be 0.1 c_w. To calculate c_w the classical formula (Whipple 1951) is used:  

where 1/n is the fraction of solar radiation used for sublimation of the cometary ices, ρ and s are the particle density in g cm⁻³ and the radius in cm, r is in au, and ρ_c and R_c are the nucleus density in g cm⁻³ and the radius in km. In the model 1/n= 1, R_c= 10 km and ρ_c= 1 g cm⁻³.

The ejection velocity should exceed the escape velocity ≈2 m s⁻¹. The appearance of the dust coma has been noticed as far out as r= 9 au (Wyckoff et al. 1985). This value has been taken as the limit for the beginning/end of sublimation. The directions of the vectors have been distributed uniformly in the sunlit hemisphere of the nucleus. Two masses m of particles have been considered: 1 and 0.001 g (spherical with density 0.25 g cm⁻³). The equations for the orbital motions of the particles have been numerically integrated by an Everhart procedure of 19th order with variable stepsize from the moment of release until 1986 October 20. Perturbations by Jupiter and Saturn plus radiation pressure (RP) were taken into account (McIntosh & Jones 1988). For one control model, perturbations by all planets and the Moon have been considered. Planetary coordinates have been taken from the Jet Propulsion Laboratory Planetary Development Ephemeris — DE406. It is found that 500 test particles are quite enough to understand the gross behaviour of the stream, and this is in agreement with the conclusions of McIntosh & Jones (1988), but to study fine structure we need about 5000 particles.

4.4 The initial configuration of orbits

Let us consider initial structure of model bunches of orbits. Fig. 1(a) shows a distribution of the density of orbits over the model stream cross-section at the initial stage, i.e. just after formation of the initial bunch of orbits. A pronounced cross-like structure is recognizable in the density distribution. The reason for its occurrence is that the orbital characteristics of the particles ejected when the parent comet is approaching perihelion and when it is moving away from perihelion differ. To put this another way, this structure appears only when ejections occur on a relatively long arc of a cometary orbit, including the perihelion. Note that the cross is weakly distinguishable in the case of isotropic dust ejection, but in the case of an ejection from the sunlit hemisphere of the nucleus, as in Figs 1(a) and b, it is clearly defined.

Figure 1.

Cross-sections of the model comet Halley stream of 5000 particles at the descending node just after formation. The abscissa axis lies in the orbital plane of the parent comet (orbit of 1404 bc), is normal to its velocity vector, and is directed away from the Sun. The numbers along the axes are the lengths (in au) of the sides of the corresponding rectangles enclosing the cross-section. m= 0.001 g. (a) Three equidensities and the densest part (black) for ejections from the solar hemisphere of the nucleus around the cometary orbit, c=0.1 c_w. (b) The same model stream and the same cross-section as in (a), but here the different colours show the cells (rectangular elements) through which the orbits of particles pass: particles ejected before perihelion passage (blue), after perihelion passage (red), and both before and after perihelion passage (green). (c) Three equidensities and the densest part (black) for isotropic ejection at perihelion with c=c_w.

So the orbits of pre-perihelion and after-perihelion particles occupy a somewhat different space, and that is illustrated by Fig. 1(b). If the ejection takes place at perihelion with a fixed ejection velocity, and if it is isotropic, then the orbital density distribution takes on quite another appearance (Fig. 1c). In the process of evolution this ‘ellipse’ becomes distorted, but still recognizable. Judging from McIntosh & Jones (1988, figs 6–10), similar ‘ellipses’ probably were initial configurations in their model. So the linear features in the distribution of nodes, ascribed by McIntosh & Jones to the effect of perturbing forces, are probably a result of the chosen simplified scheme of ejection.

5 Discussion

5.1 The stream generated in 1404 bc

In fact, ejection of particles proceeds during several years before and after the corresponding perihelion passage, but for the sake of brevity, expressions like ‘1404 bc ejection’ or ‘1404 bc stream’ will be used hereinafter. Fig. 2 shows a section of two model streams in the ecliptic plane.³ These streams were generated with different ejection velocities. It is of interest that the factor of 10 increase of ejection velocities does not result in a 10-fold stretching of the stream. If ejection velocities are small the Earth does not meet the stream at its descending node. The trend in the change of the nodes of comet Halley (see, for example, McIntosh & Jones 1988; Fig. 3) implies that meteoroids with m > 0.001 g of the currently observed Orionids were generated before 1404 bc. Such an inference has already been made by McIntosh & Jones (1988), but only for large (1 g) particles.

Figure 2.

Distributions of ecliptic-plane crossings of two model streams of 500 particles ejected from the comet about 1404 bc. The particle mass is 0.001 g. The ejection velocities were 0.1 c_w (black dots) and c_w (grey dots).

Figure 3.

Sections of the model stream of 5000 particles generated about 1404 bc in the ecliptic plane at the descending [panels (a) and (b)] and ascending [panels (c) and (d)] nodes. The designations for (a) and (c) are the same as in Fig. 1(a) (squares for the densest part), and those for (b) and (d) are as in Fig. 1(b) (only here blue = open circle, red = filled circle and green = plus symbol). Perturbations by Jupiter and Saturn plus RP have been taken into account, and m= 0.001 g.

Fig. 3 presents maps of orbital density distributions in the ecliptic-plane section. There are noticeable dense filaments in the stream, especially for the Eta Aquarids (Fig. 3a). This is the same ‘cross’ as in Fig. 1(a), but distorted in the process of evolution. The filaments consist of orbits generated either before or after perihelion ejections, as is illustrated by Figs 3(b) and (d). This model has been calculated considering perturbations by Jupiter and Saturn, as well as RP. For comparison, the orbital motion of the same initial bunch of 5000 orbits has been integrated taking into account all the major planets and the Moon, plus RP. The result is in Fig. 4. The model stream structure certainly becomes more complicated, mainly because of numerous encounters of particles with planets, but the pre- and post-perihelion filaments are still visible. Regarding the gross behaviour of the stream, the use of the simplified scheme of perturbations leads to some decrease of the stream width and to some displacement in space, but these changes can hardly be called dramatic.

Figure 4.

As in Fig. 3, but here perturbations by all planets and the Moon plus RP have been taken into account.

All calculations were performed on a PC Pentium III/800, and in the first case (Jupiter + Saturn + RP) they took about a week. In the second case (all planets + RP) the computer worked for a month.

As has been noted in Section 4.2, the best agreement with experimental data was obtained using the model where ejection of large particles took place in a Sun-oriented 70°± 5° cone. Fig. 5 shows how a cross-section of the model stream changes if ejections occur in such a cone, and not over the whole sunward nucleus hemisphere. As we can see, this factor influences the section width, but not its length. As expected, the filaments became more distinct.

Figure 5.

The same sections as in Figs 3(a) and (c). Different colours designate the cells through which the orbits of particles pass: particles with ejection velocity vectors within a Sun-oriented 70° cone (red), beyond the cone (blue), and both in and out of the cone (green).

The model stream of particles with m= 1 g is shifted noticeably in space relative to the stream of particles with m= 0.001 g (see Fig. 6). It is more compact, because the ejection velocity of larger particles is lower. It is of interest that, in the case of the model Eta Aquarids, the Earth first meets small particles, then passes a region with small and large particles and, finally, meets small particles again. If the model Orionids were ‘observable’ at the Earth, then we would first observe only small particles and later a mixture of particles. Unfortunately the existing data on the mass distribution in these showers do not allow one to verify this hypothesis.

Figure 6.

Distributions of ecliptic-plane crossings of two model streams of 500 particles ejected from the comet about 1404 bc. Particle masses are 1 g (black dots) and 0.001 g (grey dots).

Yeomans & Kiang (1981) noted that ‘computed times of perihelion passage⁴ back to 1404 bc are not likely to be in error by more than a month’. To clarify what effect the error in perihelion passage time has upon the gross structure of the model stream, the following two models were considered. In the reference osculating orbit by Yeomans & Kiang, the moment of perihelion passage was shifted by ±120 d. The orbital evolution of 500 particles with m= 0.001 g was calculated with the use of a simplified scheme of perturbations (Jupiter + Saturn + RP). No visible distinctions in the two model streams were found.

5.2 Comparative analysis of model streams generated in different epochs

The stream or, to put it better, the swarm of meteoroids generated in the approach of 1910 cannot be observed at the Earth now, because it did not stretch enough around the orbit. So it is not involved in the ensuing analysis.

The gross behaviour of model streams generated in 141 ad and 837 ad is qualitatively the same as that of the 1404 bc stream considered above, but there are distinctions in fine structure (Fig. 7). The configurations of the densest structures in the stream of 837 ad are less chaotic, probably owing to the low age of the stream. Filaments are more prominent. Strictly speaking, they are not filaments but layers, and the spatial structure of the comet Halley streams resembles a puff-pastry pie. Comparing the locations of the cross-sections for all three model streams (Fig. 8), we may conclude that Eta Aquarid meteoroids observed at the present time had presumably been released from the comet not later than 837 ad. The relative location of the cross-sections in Fig. 8 suggests that the Earth, in its motion through the Eta Aquarids, meets a mixture of meteoroids generated at various epochs. In the other node, however, streams of different ages are more differentiated.

Figure 7.

Maps of orbital density distributions: cross-sections in the ecliptic plane for 5000 model particle orbits. The epochs of generation and nodes are noted on the panels. The designations are as in Fig. 3(a); m= 0.001 g.

Figure 8.

Distributions of ecliptic-plane crossings of three model streams of 500 particles ejected from the comet about 1404 bc, 141 ad and 837 ad (the younger the stream, the lighter the dots); m= 0.001 g.

The width of the model Eta Aquarids is 14°–15° of the Earth orbital arc or, to put it another way, 14–15 d (Fig. 8a). The upper limit of the width estimated from observations is 34° (see Section 2.1). Taking into account that the width of the model stream is fully provided by dispersion of the 1404 bc stream, we may conclude that the observed Eta Aquarids contain meteoroids generated in approaches before 1404 bc.

6 Conclusions

In this paper the emphasis has been on the fine structure of the comet Halley meteoroid stream and its two meteor showers. The filamentary structure of the stream is caused by the initial orbital distribution of meteoroids: ejections before and after perihelion form two branches (or filaments). The filaments generated during many cometary returns to the Sun are superimposed. Numerous approaches of meteoroids to planets make the stream structure complicated and rather chaotic. The simulation has shown that ejections after 837 ad could not contribute to present observations of the Eta Aquarid meteor shower. The mass distributions in the Orionids and Eta Aquarids probably are dissimilar.

Acknowledgments

I thank Dr C. Trayner for his help in transforming my Russian English to British English.

1

We do not discuss the mass distribution index here.

2

Hereafter, ‘activity’ means rates unless otherwise specified.

3

All of the following plots show various sections in 1986.

4

The authors mean the interval from 141 ad until 1404 bc.