NON EXTENSIVITY IN METEOR SHOWERS

Transcription

1 NON EXTENSIVITY IN METEOR SHOWERS Oscar Sotolongo-Costa 1, R. Gaez 2, F. Luzón 1,3,4, A. Posadas 1,3,4, Pablo Weigandt Beckann 5 1.-Cátedra Henri Poincaré de Sisteas Coplejos, Universidad de La Habana. 2.- Departaento de Física Aplicada, Universidad de Alería. 3.- Instituto Andaluz de Geofísica. Granada. Spain. 4.-Instituto de Geofísica y Astronoía, ACC; Cuba 5.- Departaento de Física, Universidad Autónoa de Chiriuí, Panaá Abstract: A eteor shower is a luinous phenoenon that takes place by the entry into the Earth s atosphere of a cascade of particles coing fro a strea intersected by our planet in its orbit. Here we investigate the possibility of a description of the ass distribution of eteoroids in eteor showers in ters of a non extensive forulation, which could shed light and give soe insight into the origin of such particles. I. Introduction Meteor Showers (MS) are flurries of eteors seeingly eanating fro spots of the sky at particular ties of the year. This happens when the Earth intersects soe of the particle swars that ove around the Sun. As to the origin of MS, the hypothesis of its coetary origin is widely accepted. Coets are celestial bodies fored by frozen dust and gas. When these objects approach their perihelion, the interaction of solar wind with its surface causes the subliation of its coponents. As a result, coets produce a long tail always in opposite direction to the Sun. The tail is ade up by a large aount of particles which eventually spread out along the entire orbit of the coet foring a eteoroid strea. When our planet intersects the orbit of the coet, eteoroids fall into Earth s atosphere at high speed ionizing it and causing a shower of luinous particles According to [1], the tail of the coet is subjected to the periodic influence of Jupiter, so that the particles of the tail for a 'eteoric pipe' that stretches continuously. The gravity forces act on the coet and its swar, so that each expulsion of particles becoes ore and ore uniue and separates fro the coet; this originates a coplex syste of eteoric fibres inside the pipe. The spatial distribution of such fibres and the way the Earth intersects one or ore of the deterines the duration and intensity of showers. The eteoroids entering the Earth s atosphere produce different luinous flares, according to their ass and the relative speed of the swar. Meteors see to radiate fro a single point in the sky. The radiating point (or just the radiant ) is caused by the effect of the perspective on the observer located on Earth s surface. The showers are usually naed after the constellation in which their radiant lies at the tie of axiu strea. Such is the origin of naes like Perseids, Leonids, etc. Apart fro the uestions raised concerning the origin of MS, features such as their size distribution are of interest to get soe clue about the processes which could have originated these particles. It is worth entioning that the study of the luinous flux of eteoric particles indicates that they are distributed following a power law [2, 3]. The above fact, the

2 existence of coplex interactions aong the eteoroids, the interaction of the coet with solar wind, its fragentation and any other phenoena present on this celestial effect constitute a very coplex field of study and speculation, posing, aong other things, the uestion of how the observation of luinous intensities in MS can give us inforation about the characteristics of MS and their generators, the coets, since MS are a anifestation of the interaction between Earth's atosphere and the coetary particles. As far as we know, no attept has been ade to deduce a functional dependence related to the ass distribution function of eteoroids, starting fro first principles. As the luinous distribution of eteoroids can be deduced fro observations, and on this basis their size distribution can be inferred, ore inforation could be obtained by observations if the size distribution function (SDF) could be linked to a given theoretical fraework. Taking the above into account, we will expound, on the basis of physical grounds, a description of SDF caused by fragentation phenoena. If the observed SDF can be fitted with theoretical expressions obtained fro specific starting hypotheses, aybe we could say soething about its possible origin. On the other hand, if there are strong suspicions that the eteoroids are the result of violent fragentation processes, we can validate our theoretical fraework. II. - Non-extensivity in fragentation As a result of developents in aterials science, cobustion technology, geology and any other fields of research, there has been an increase of interest in the uestion of object fragentation. Soe attepts have been ade to derive the fragent size distribution function fro the axiu entropy principle [5, 6], subjected to soe constraints which ainly cae fro physical considerations about the fragentation phenoena. The resulting fragent distribution function describes the distribution of sizes of the fragents in a regie in which scaling is not present. In the general field of fragentation there is a collection of papers [4-7] where a transition occurs fro a classical" distribution of fragents (e.g. log-noral or Rossin- Raler-like) to a power law distribution. This transition has not been adeuately explained in ters of any general principles, although in [8] the representation of the fragentation process, in ters of percolation on a Bethe lattice. leads to a transition to a power law in the distribution of fragent sizes. Other theoretical efforts like the study of diensional crossover in fragentation of thick clay plates and glass rods [1], and very interesting experients that show scaling in violent fragentation processes, like the burst of a fuel droplet [9] are also proofs of the interest in these phenoena. Because scaling certainly occurs when the energy of the fragentation process is high, this suggests that the traditional statistical analysis is only applicable to low energies. However, we believe that the axiu entropy principle is copletely universal and has an alost unliited range of applications. Conseuently, we would expect to be able to use in describing the transition to scaling as the energy of the fracture grows. The expression for the Boltzann-Gibbs entropy S (e.g. Shannon's for) is given by W S = k piln pi i= 1 (1)

3 where p i is the probability of finding the syste in the icroscopic state i, k is Boltzann's constant, and W is the total nuber of icrostates. This has been shown to be restricted to the doain of validity of Boltzann-Gibbs (BG) statistics. These statistics see to describe nature when the effective icroscopic interactions and the icroscopic eory are short ranged [8]. The process of violent fractioning, like that of droplet icro-explosions in cobustion chaber, blasting and shock fragentation with high energies and any others, leads to long-range correlations between all parts of the object being fragented. Fractioning is a paradig of non extensivity, since the fractioning object can be considered a collection of parts which, after division, have an entropy larger than that of their union i.e., if we denote by A i the parts or fragents in which the object has been divided, its entropy S obeys S ( Ai ) < i S( Ai ), defining a superextensivity in this syste. This suggests that it ay be necessary to use non-extensive statistics, instead of the BG one. This kind of theory has already been proposed by Tsallis [8], who postulated a generalized for of entropy, given by S 1 = k p ( ) 1 The integral runs over all adissible values of the agnitude x and p( xdxis ) the probability of the syste being in a state between x and x + dx. This entropy can also be expressed as S = p ( x) lp( x) dx (3) x dx (2) where the generalized logarith l ( p ) is defined in [8] as 1 p 1 l ( p) = 1 (4) where is a real nuber. It is straightforward to see that S S when 1, recovering BG statistics. Here, we try to deduce the size distribution function of eteoroids starting fro first principles of physics, i.e., axiu entropy principle with Tsallis s entropy foralis. Fracture processes characterized by a strong correlation aong all parts of the body ust be described fro the non extensive statistics [8]. Let us start fro the Tsallis s entropy for the ass distribution of the fragents in the for where M is a non-diensional ass. S = k 1 Let us ipose the noralization condition p M dm ( ) 1 (5) p( M) dm = 1 (6)

4 and a -conservation of ass in the for Mp ( M ) dm = 1 (7) Fro the axiu entropy principle, SDF can be deterined in a straightforward anner. The proble is to find the axiu entropy conditioned by the noralization of the probability distribution and the -conservation of ass already declared. Using the ethod of Lagrange ultipliers the following functional can be constructed: Its extreization leads to S ( ; α; β) = α ( ) + β ( ) k M dm (8) L p p M dm M p ( ) ( ) 1 1 = 1 + (9) p M dm a bm dm where a and b are dependent constants that can be used as adjustent paraeters. This is the ethod we will follow to find an expression for the SDF of eteoroids and contrast it with observations of MS. It is worth to say that (9) gives, in the asyptotic liit, pm ( ) M n with n= 1 ( 1). i.e, scaling is obtained in this forulation as one of its essential characteristics. III.- Method Meteor showers are subjected to visual observation, where the visual agnitude of eteoroids is registered. Fro this, the luinous flux and the ass of the particles in the shower can be deterined. In this case we are specially interested in the ass and its relation to the visual agnitude. The ass of the eteoroids can be obtained by [12]: where is the visual agnitude of the particle. M e γ (1) For a description of the resulting SDF produced in the coets for its interaction with the solar wind we analyse SDF of several MS, i.e., Leonids, Perseids and Lyrids (Table I). It is known that these MS are produced by swars eerging fro a coet. Table I MS processed and their original coets. Meteor Shower Original Coet Years # observations Leonids (LEO) Tepel Tuttle y Perseids (PER) Swift Tuttle y Lyrids (LYR) Thatcher 1861 I y Data were taken fro [13]. As atosphere restricts visualizations of low intensity eteoroids, to deterine the nuber of particles entering the atosphere a correction factor was introduced to account for the perception probability of each agnitude [14].

5 Here we start fro the idea that ass distribution in MS ay exhibit siilar characteristics as fragent distribution functions eerging fro fragentation processes with high energy. As entioned above, fragentation processes have been recently forulated on the ground of Tsallis non extensive statistics. Fragentation can occur as a violent process, where long range interactions are present in the fragenting body, so that the extensive statistics of Boltzann-Gibbs cannot be applied. Tsallis foralis gives, as we already have seen, the ass distribution function (8). Now we use the relation (1) to get dm d γ = (11) ce where c and γ are constants. Taking into account that we finally get p( d ) = pm ( ) dm (12) 1 γ γ p ( ) d= μe k ρe + 1 d (13) This is the expression we re going to use to copare with the data already entioned. It sees worthless to be concerned about the eaning of the specific values of the fitting constants, since what really atters in this case is the physical considerations leading to the above followed ethod. Naely, the fact that the fragents foring the coet and detached by solar wind coe fro violent processes of fragentation. As a result, their ass distribution obeys (8) and visual agnitude distribution corresponds to (13). IV. - Application of the odel to MS In figures 1 to 3 the dots show the data of the visual agnitude of the MS Leonids, Perseids and Lyrids. The fitting curves correspond to euation (13). In all cases, the agreeent with observational data is pretty good. The results of the fitting highlights the general character of non-extensivity for the phenoenon of MS. We can assue that eteoric particles exhibit the sae characteristics as when they were in the interior of their carrier coet, i.e, assue that the observed ass distribution is the sae as that of the solid aterial foring the coet s nucleus. According to this hypothesis the genesis of such particles occurred before the 'birth' of the coet. They are liberated fro the coet s nucleus when it approaches the Sun, foring a coplex swar of particles along the coet s orbit.

6 .7 Leonids (1995).7 Leonids (1996) p ().3 p () Leonids (1998).7 Leonids (1997) p ().15 p () Leonids (1999).6 Leonids (21) p ().3.2 p () Fig.1 Density distribution of eteoroids by agnitude in Leonids for and 21; tting was ade with euation (13).

7 .6 Perseids (1995).7 Perseids (1996) p ().3 p () Perseids (1998).5 Perseids (1997) p ().3.2 p () Perseids (1999).7 Perseids (21) p ().3 p () Fig.2 Density distribution of eteoroids by agnitude in Perseids for and 21; tting was ade with euation (13).

8 .8 Lyrids (1995).5 Lyrids (1996) p ().3 p () Lyrids (1998).7 Lyrids (1997) p ().4 p () Lyrids (1999).8 Lyrids (21) p ().1 p () Fig.3 Density distribution of eteoroids by agnitude in Lyrids for and 21; tting was ade with euation (13).

9 The particles foring the coet are the residuals of the original processes of the Solar Syste. In an early phase, solid bodies of different sizes were fored. It is widely accepted that during such processes the interactions aong these bodies led to freuent and violent collisions. We assue that such early processes of fragentation gave origin to the size distribution we presently observe. The fitting with (13) leads to think that such processes were characterized by long range correlations, i.e, violent breaking phenoena. V. - Conclusions First of all, we see once again that non-extensive statistics using Tsallis' entropy can be applied, in a satisfactory fashion, to phenoena related to violent fragentation processes. If we use Shannon Boltzann's entropy subject to the conditions S = p( M)ln( p( M)) dm pm ( ) dm= 1 and Mp( M) dm = M we get, by eans of Lagrange variational ethod, and, because of (13), λm p( M) dm = Ce dm γ ( ηe + γ) p( d ) e d This relation, in general, does not fit with the whole data, showing that Boltzann's Statistics fails in this case. It is safe to conclude that in MS the universal character of fragentation processes is confired, as well as the generality of the non extensive entropy foralis. Nevertheless, old swars, due to the loss of their particles along the orbit for any years, can lead to deviations fro the power law distribution. This is due to the Poynting effect: the drift of the saller particles by solar wind. Older swars are, therefore, ore affected than newer ones. The antiuity of the swars intersecting terrestrial orbit should have influence on the ass distribution function of MS. Conseuently, future works could be addressed to the uestion of the relationship between the age of the swar and the ass distribution of the corresponding MS. Acknowledgeents We wish to acknowledge the collaboration of Dr. E. Weigandt for helpful suggestions. One of us (O.S.) also wishes to acknowledge the war hospitality of all the people related to UNACHI during his tie as visiting professor..

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