The basic physics behind the process of ejecting meteoroids from a cometary nucleus became straightforward as soon as a reasonably correct model for the cometary nucleus became available. Such a model for the nucleus was proposed in 1950 by Whipple [11], the so called dirty snowball model, in which dust grains were embedded in an icy matrix.

As the comet approaches the Sun, the nucleus heats up until some of the ices sublime and become gaseous. The heliocentric distance at which this occurs will depend on a number of parameters, the composition, the albedo and the rotation rate for example, but the process which follows this is independent of these details. When sublimation occurs, the gaseous material flows outwards away from the nucleus at a speed which is comparable to the mean thermal velocity of the gas molecules.

Any grains, or meteoroids not still embedded in the matrix will experience drag by the outflowing gas. The outward motion of the meteoroid will be opposed by the gravitational field of the comet nucleus and a meteoroid will escape from the cometary nucleus into inter-planetary space only if the drag force exceeds the gravitational force. Now, drag is roughly proportional to surface area while gravity depends on mass, thus smaller grains might experience a greater acceleration while gravity will win for grains over a given size. Hence there is a maximum size of meteoroid that can escape, though this size might vary from comet to comet depending on the size and activity level of the comet. The final speed achieved by any meteoroid that does escape will similarly depend on these factors as well on the grain properties. These considerations were first quantified by Whipple [12]. He obtained where a is the bulk density of the meteoroid of radius b and r the heliocentric distance in astronomical units. Rc is in kilometers and all other quantities in cgs units.

A number of authors have suggested modifications to this general formula, for example Gustafson [13] pointed out that the drag formula was incorrect if the meteoroids were non-spherical while Harris and Hughes [14] suggest that the gas outflow down a tube or cone is slightly faster than is suggested by considering the mean thermal velocities. Both these points are undoubtedly correct but the end result leads to only a slight increase in the ejection velocity. Finson and Probstein [15] produced a model for dust outflow that related the observed brightness variations along the cometary tail to the dust flow rate. The dust that causes light scattering in the tail is somewhat smaller than dust

that evolves into meteors, but nevertheless, there is no major difference between the dust velocities given by this approach and that given for example by Whipple's formula. The main conclusion, in terms of meteoroid stream formation, is that the ejection velocity is in all cases considerably less than the orbital velocity of the parent comet.

As an illustration, consider comet lP/Halley. Grains of up to a few centimeters will escape, while at 1AU, a one millimeter meteoroid would have an ejection speed of about 70ms^1. The orbital speed at 1AU is of the order of 30kms'1.

The effect of the meteoroid being ejected with a speed given by the mechanism above relative to the comet will be to produce differences between the orbit of the meteoroid and that of the comet. These changes will of course depend on the direction at which the meteoroid is ejected and the point on the cometary orbit at which the ejection takes place. There will always be a change in the specific energy E. Now, standard theory of Keplerian motion tells us that

2 a and that

where a is the semi-major axis of the orbit in Astronomical Units and P the orbital period in years. Hence we can obtain

a change in semi-major axis and period thus is an inevitable consequence of the ejection process, but since ^ is likely to be small in view of the fact that the ejection velocity is small compared to the orbital velocity, changes in a and P are also likely to be small. Observationally, it will be very difficult to detect such changes in the semi- major axis. However, changes in the orbital period are different in that their effect is cumulative. After n completed orbits, the time difference between a meteoroid and the comet passing perihelion will be nAP . For a typical situation, in about 50 orbits meteoroids will be found at all points of the orbit that is an annual stream is formed.

If there is a component of the ejection velocity in the transverse direction, then the specific angular momentum h will also be changed, we have h2 = [email protected], (5)

where p is the semi-parameter of the orbit, that is p = a(l — e2). This yields

This implies that in general there is a change in eccentricity as well. Detecting changes in the eccentricity from observations of meteors will also be very difficult.

Unless the ejection took place exactly at perihelion, the changes in a and e, together with the requirement that the ejection point is on both the comet and meteoroid orbit, implies that a change in the argument of perihelion ui must also take place. Since the orbit is assumed to be Keplerian, its equation is known, and from this we can obtain

/ r , a \ r (2e + e2 cos /0 + cos /0) A p 1 - e2 A a ecos(/0 + Auj) - ecos/o = --------cos f0--(7)

2e p ze a where /o is the true anomaly of the ejection point. Though the changes in a, e and ui may each be small and indeed undetectable without very accurate observations, a combination of them can cause a change that is of fundamental importance in the observability of a meteor shower, namely the nodal distance, r^.

The nodal distances are derived from the standard equation for an ellipse with the true anomaly being taken as — u> or 7r — w, that is

Hence, we can obtain

A rjv 1 — e2 A a (e2 cos u> + cos u> — 2e) Ap

rpj 2 e a 2 e p for the first node with a similar equation for the other node.

Again, the changes in the nodal distance may appear to be small but whereas a 1% change in a, e or lo is fairly hard to detect a one percent change in rn is 0.01 AU, or about 4 times the Earth-Moon distance. This is rather a large distance when the meteoroid stream has to hit the Earth in order to produce a meteor shower.

The ejection velocity will generally also have a component perpendicular to the comet orbital plane. In consequence, the meteoroid orbital plane will be different from that of the comet. Since the line of intersection of the orbital plane with the ecliptic is defined as the line of nodes and the displacement of this from first point of Aries is defined as the longitude of the ascending node, fi, any such a velocity component will induce a change in n.

Deriving the expression for Afi is mathematically rather tedious and will not be repeated here . The derived expression is r0 sin(o; + /0) .

h sin i where Tq and /o are the heliocentric distance and the true anomaly of the ejection point, i is the inclination of the orbit, and <f> is angle between the direction of ejection and the orbital plane so that v sin 4> is the component of the ejection velocity perpendicular to the orbital plane.

Since fi measures the time at which a shower is seen, then this is also sensitive to small changes and is important in the study of meteor showers.

Hence, the effect of the initial ejection velocity is to change all the orbital parameters by a small amount, but these small changes can also produce a change in the nodal distance which is a very sensitive parameter for the production of a meteor shower. For a very young stream, perhaps one which generates a meteor storm such as the Leonids, these effects may be the dominant ones, but, as soon as the meteoroid is ejected from the immediate vicinity of the comet, it becomes an independent moving body in the Solar System and subject to all the evolutionary effects that any body is subject to.

Solar radiation falling directly on a body generates a force which is radial and depends on the strength of the incident radiation and so is proportional to the inverse square of heliocentric distance, like gravity. It can thus be regarded as weakening gravity and is usually represented by writing the effective force acting on the body as p = _G!M1-ft rl and, when numerical values for standard constants are inserted, (3 is given by (eg [16]

where as before b is the meteoroid radius in centimeters and a the relative bulk density in gcm~3. It is self-evident that meteoroids will be lost from the Solar System if (3 > 1, since the net force is then outwards. However, as Kresak [17] first pointed out, meteoroids will be lost whenever their total energy is positive. A meteoroid moving with the parent comet will have a specific energy E' given by 2£, _ ^ _ ««,[[-«

But,

so that E' is positive provided

At perihelion, r = a(l — e), and here, meteoroids for which

will be lost. This is much more restrictive limit than (3 = 1, so that larger grains are lost than is implied by the (3 = 1 limit. Taking our numerical example again, for comet lP/Halley, e = 0.964, so that meteoroids for which (3 > 0.018 will be lost. Taking a bulk density of 0.5gcm~3, meteoroids smaller than about 6 x 10_3cto will be lost from the stream.

Since the radiation may be absorbed and then re-emitted from a moving body, there can be a loss of angular momentum from the body, affecting its orbit. This effect was first mentioned by Poynting [18] and but in a relativistic frame by Robertson [19] and is now generally known as the Poynting-Robertson effect. This effect has been studied by many authors. The first to apply this to meteoroid streams was probably Wyatt and Whipple [20]. More recent accounts of this effect can be found in Hughes et al. [21] and Arter and Williams [22]. In discussing changes caused to the orbital parameters a and e, it is more convenient to use a parameter r/, rather than (3 to characterize the effects of radiation. The relationship between the two parameters is cq = [email protected](3, (16)

where c is the speed of light, r/ has a numerical value 4.4 x 1015 that of (3 in cgs units. Note that while ¡3 is dimensionless, r\ is not. Using this notation, all the authors mentioned give the following two equations, (using the same units as those used to express rj)

In order to obtain the change in a given orbit, it is necessary to specify the dimensions of the meteoroid so that the value of r/ can be obtained and then numerically integrate these equations, the latter task not being particularly difficult. However, some insight into the effect of this can be obtained without performing numerical integrations. Using the chain rule on the two above equations gives, da _ 2a(2 + 3e2)

de ~ 5e(l — e2) ' ( j an equation which can be integrated to give a(l - e2) = Ce4'5, (20)

where C is a constant of integration.

Since time has been eliminated, this equation gives no indication of how long it takes for an orbit to evolve to any given state. However an estimate of the time required to significantly change orbits can be obtained by substituting the value of a from equation (20) into equation (18), giving de _&,(! _e2)i/2

Apart from factors of general order unity, the typical time-scale of this equation is given by C2¡tj. For the case we have so far used as an example, namely a meteoroid of 1 mm radius and density 0.5gcm~3 associated with comet lP/Ha.lley, this time-scale is of order 3 x 105years. Though this is short by the standards of evolution generally in the solar system, it is a long time compared to our time-span of observation of meteor showers and is towards the top end of estimates for stream life-times. The time to significantly change the orbital parameters will also vary from stream to stream, so that the above value should be regarded as only an indication of the time scale for the Poynting-Robertson drag to be important.

Like other bodies in the Solar System, the motion of the meteoroid will be affected by the gravitational fields of all the other bodies in the system, with all the accompanying problems of accurately dealing with these perturbations that are familiar to all that have worked on orbital evolution in the Solar System. It is known since the work of Poincare in 1892, (see [23]) that no analytical solution exists to the general problem of following the orbital evolution of more than two bodies under their mutual gravitational attraction exists. Hence, following the motion of meteoroids implies some form of numerical integration of the equations of motion.

The concepts involved in considering planetary perturbations are very easy to understand though following through the consequences is somewhat harder. Each planet produces a known gravitational field. Hence, if the position and velocity of each body in the system is known at any given instant, then the force due to each body and hence the resulting acceleration can be calculated which allows a determination of the position and velocity of the body at a later time. Of course, this is only strictly true for an infinitesimal time interval and so the problem in reality is to chose a time step that is short enough to maintain a desired level of accuracy while at the same time making progress in following the evolution. The methodology described above was known and used in the mid-nineteenth century by the astronomers that calculated the orbits of comet, though, the 'computers' they used had a rather different meaning then from now. In those days it meant a low paid assistant who computed myriads of positions using hand calculators. Some of the earliest calculations on the evolution of meteoroid streams which included planetary perturbations were carried out by Newton between 1863 and 1865 ([24-26]), where he investigated the generation of Leonid meteor storms. A number of other early calculations are described in Lovell's classical text book on the subject [27]. Though some useful early results were obtained by these early calculations, it is clear that no real progress in following the evolution of a large number of meteoroids can be made by such labour intensive means and further development had to wait until the human computers were replaced by electronic ones.

The early electronic computers were also to small and slow to be able to follow a realistic number of meteoroids over realistic time-scales. In order to overcome these shortcomings, effort was spent on refining the mathematical modelling, in particular on the idea of averaging the perturbations over an orbit so that only secular effects remained. The real gain with such methods is that the whole assembly of meteoroids are replaced by one mean orbit with a consequential huge gain in effort. At first, such 'secular perturbation' methods only worked for nearly circular orbits, good for following the evolution of satellite systems and main-belt asteroids, but of little value in following the evolution of meteoroids on highly eccentric (and possibly also highly inclined) orbits. In 1947, Brouwer [28] generated a secular perturbation method that worked well even for orbits of high eccentricity (though not for values very close to unity) and this method was used by Whipple and Hamid [29] in 1950 to integrate back in time the orbit of comet 2/P Encke and the mean orbit of the Taurid meteoroid stream. They showed that 4700 years ago, both the orbits were very similar and suggested that the two were related. This was the first time that a link between a comet and a stream had been suggested based on a past similarity in orbits rather than a current similarity. This also established an age of 4700 year for the Taurid stream. Other secular schemes were also used, for example, Plavec [30] used the Gauss-Hill method to investigate the changes with time in the nodal distance of the Geminid stream.

One of the more popular (in terms of general usage) secular perturbation methods that were developed is the Gauss-Halphen-Goryachev method, described in detail in Hagihara [31]. This was used for example by Galibina [32] to investigate the lifetime of a number of meteoroid streams and by Babadzhanov and Obrubov [33] to investigate the changes in the longitude of the ascending node (rather than nodal distance as investigated by Plavec) of the Geminid stream. The same authors also used this method extensively during the

1980's to investigate the evolution of a number of streams (for example, [34]).

The disadvantage of the secular perturbation methods is that the averaging process, by its very nature, removes the dependence of the evolution on the true anomaly of the meteoroid. It is thus impossible to answer questions regarding any difference in behaviour between a clump of meteoroids close to the parent comet and a typical meteoroid in the stream. As computer hardware improved, the use of direct numerical integration methods became more widespread. By direct methods, we mean where the evolution of individual meteoroids, real or hypothetical, is followed rather than the evolution of an orbit. The first such investigation was probably by Hamid and Youssef [35] who in 1963 integrated the orbits of six actual Quadrautid meteoroids. In 1970, Sherbaum [36] generated a computer programme to numerically integrated the equations of motion using Cowell's method which was used by Levin et al. [37] to show that Jovian perturbations caused an increase in the width of meteoroid streams. In the same year, Kazimirchak-Polonskaya et al. [38] integrated the motion of 10 a Virginid and 5 a Capriconid meteoroids over a 100 year interval. Seven years later, the number of meteoroids integrated was still small and the time interval over which the integration was performed remained short, with Hughes et al. [39] in 1979 following the motion of 10 Quadrantid meteoroids over an interval of 200 years, using the self adjusting step-length Runge-Kutta method. This however marked the start of significant increases in both the number of meteoroids integrated and the time interval, and by 1983, Fox et al. [40] were using 500 000 meteoroids, indicating that in five years computer technology had advanced from allowing only a handful of meteoroids to be integrated to the situation where numbers to be used did not present a problem.

The direct integration methods used in meteoroid stream studies fall into two broad categories, the single step methods of which the best known is the Runge-Kutta method, (see Dormand et al. [41] for a fast version of this method) and the 'predictor-corrector' methods following Gauss (see Bulirsch and Stoer [42] for the methodology)

By the mid eighties, complex dynamical evolution was being investigated, Froeschle and Scholl [43], Wu and Williams [44] were showing that the Quadrantid stream behaved chaotically because of close encounters with Jupiter, and the proximity of mean motion resonances. A new peak in the activity profile of the Perseids, roughly coincident in time with the perihelion return of the parent comet 109P/Swift-Tuttle caused interest with models being generated for example by Williams and Wu [45] . Babadzhanov et al. [46] investigated the possibility that the break-up of comet 3D/Biela was caused when it passed through the densest part of the Leonid stream. By now, numerical integrations of models for all the major streams have been carried out. In addition to those mentioned earlier, examples of streams for which such numerical modelling exists are : the Geminids, (Gustafson [47], Williams and Wu [48]), April Lyrids (Arter and Williams [49]), t] Aquarids (Jones and Mcintosh [50]), Taurids (Steel and Asher [51]), a Monocerotids (Jenniskens and van Leeuwen [52]), 9the Giacobinids (Wu and Williams [53]) and the Leonids Asher et al. [54]). From the point of view of the discussion here, it is sufficient to say that numerical modelling has now reached a stage where it is possible to follow the evolution of given meteoroids from their formation over any time scale that appears to be of interest.

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