Romanian Report in Phic, Vol. 60, No., P. 7 80, 008 ESTIMATIONS OF THE COLLISION INTEGRAL IN THE CHAOTIC GUN EFFECT GHEORGHE DUMITRESCU Gr. Sc. Ind. Toma N. Socolecu Ploieti, Romania, meditatie@ahoo.com (Received Augut 4, 007) Abtract. The aim of thi paper i to etimate the colliion time integral of the kinetic equation decribing a tochatic mechanim of acceleration. Thi mechanim, namel the chaotic gun effect, might be an efficient mechanim of injection. The efficienc of the mechanim can be expreed uing the colliion integral of the charged particle moving in a turbulent electromagnetic field. Ke word: colliion integral, chaotic motion, correlation tenor, correlation length, pectral index.. INTRODUCTION The kinetic equation for a charged particle moving in a magnetic field with mall cale inhomogeneitie and alo with trong aniotrop comprie a colliion integral in the right-hand part of the equation. For a while there were ome attempt to tud the behavior of the charged particle in uch tpe of field a mentioned above. The uual wa to reolve the problem wa to approach the proce to a diffuion mechanim []. Some author claimed the neceit of the trong aniotrop in order to explain the injection of the charged particle acro the hock front occurring in upernovae, active galactic nuclei, etc. That i wh it i important to invetigate ome theoretical olution of the kinetic equation for trong aniotrop. Melnikov tranformed the mall-cale colliion integral in an approximation of a weak regular magnetic field uing unequal parallel and perpendicular correlation length to the regular magnetic field []. He advocated hi approximation on the bai of the great number of tpe of diturbance of a random magnetic field and alo of the fractal tructure of the field. Hence, for thi aumption he elected the tenor part of the correlation tenor (contained in the Paper preented at the Annual Scientific Conference, Facult of Phic, Univerit of Bucharet, June, 007, Bucharet, Romania.
7 Gheorghe Dumitrecu colliion integral) in an iotropic form. Uing a diffuion approximation in the kinetic equation, Melnikov found that the form of the patial diffuion tenor alo remain the ame a in the iotropic cae. The onl one changing i the formula for the mean free path. The mean free path lightl differ from it value in iotropic cae at L L and at L L. Here L and L are the radii of the charged particle along the perpendicular and parallel direction to the regular magnetic field. But, the mean free path tend to the infinit in a harpl aniotropic mall cale random magnetic field at L L. In thi cae random magnetic tructure are flattened in the direction of a regular field. To date the mot accepted mechanim able to accelerate charged particle to high energie in atrophical environment i the diffuive hock acceleration mechanim. But in one of our previou paper we have adopted a pecial mechanim of acceleration which dipla trong aniotrop of the charged particle [6]. Such trong aniotrop ma be explained b the aniotrop of the mall cale inhomogeneitie of the magnetic field. The fundamental aumption of the theor of diffuive hock acceleration i that accelerated particle diffue in pace, i.e. that the particle flux i proportional to the gradient of the particle denit (Fick law). Charged particle deflected b fluctuation in the electromagnetic field obe thi relation onl if their velocitie are ditributed almot iotropicall. More preciel, the theor emplo an expanion in the ratio of the plama peed in the hock frame to the particle peed, and the velocit aniotrop i conidered to be mall in the firt order. At a hock front, the downtream plama peed i of the ame order a the thermal peed of the ion in the plama, o that the theor of diffuive hock acceleration doe not appl to particle whoe energ i everal time le than the thermal energ. The quetion of how particle might be accelerated from the thermal pool up to an energ where the can be aumed to diffue i referred to a the injection problem, and cannot be treated within the framework of the diffuive acceleration theor When the aniotrop i ver large Fick law doe not work. The aniotrop of the magnetic field ma be obtained b different correlation length and indice of the power pectrum along two direction. The aim of thi paper i to derive and to etimate the colliion integral for the pecial cae of trong aniotrop and which cannot be reduced to the iotropic cae. In the econd ection we preent the correlation tenor of the econd order of the turbulent magnetic field. Then in the third ection we make ome etimation of the mall-cale colliion integral. Some final dicuion in the paper point out a few other problem concerning our derivation.
3 Colliion integral in the chaotic gun effect 73. THE FORM OF THE SMALL-SCALE COLLISION INTEGRAL The motion of a relativitic charged particle in a magnetic field with mall cale inhomogeneitie ha the kinetic equation of the form f f f v F(,) r t 0 t r p where F(r, t) i the total force acting on the charged particle and f( rp,, t) the ditribution function. The equation can be averaged b the ue of the method of Vedenov et al. [] f f v ( ò ) f d ec T ( ( ), ) t ò r ò r 0 K ( ( ), ) ec r ò e S S ò () p p p p f ( rr( ), pp( ), t) () where T (r, t) = B t (r, t )B t (r, t ), r = r r (3) K (r, t) = E t (r, t )E t (r, t ), t = t t (4) S (r, t) = E t (r, t )B t (r, t ) (5) are the correlation tenor of the econd order, and eb0c = (6) eb0 e 0 c, B c v ò (7) p ebb e b c, B c v ò (8) p Here B 0 i the trength of the uniform magnetic field along the z axi, B b i the trength of the magnetic inhomogeneitie, e i the charge and i the energ of the particle. The above kinetic equation () ma decribe the charged particle behavior in the chaotic gun effect. Thi i a mechanim of tochatic acceleration in a random magnetic field with mall inhomogeneitie. The random magnetic field B b lie
74 Gheorghe Dumitrecu 4 along z axi, while the random electric field E b i along the axi. Both are produced b the accelerated charged particle in their own nchrotron emiion. A the numerical imulation of the chaotic gun effect have diplaed, at ever complet gration of the charged particle, the particle receive a kick of energ when it reache the wave vector direction. Hence the particle i deflected trictl along the x direction perpendicular to the uniform magnetic field B 0. The acceleration take place along the wave vector direction and there i no angular deflection. That i wh we will omit in the equation () all the term containing the ò operator. The ò operator i the angular variation of the velocit direction. The equation () become impler [6] f/ tv f/ r 0 0 d e K ( (), ) r f ( rr(), pp(), t) p p After ome manipulation the right-hand part of the equation (9) turn out to be f d e K ( ( ), ) p r p ecet kmin [ kmin, v] K [ kmin, v ] f d { [ K ( a) a ] a p E ec t kmin, [ kmin, v] K [ kmin, v ] f d { [ K ( b) b ] b p ec Bt kmin, (9)
5 Colliion integral in the chaotic gun effect 75 f [ kmin, v] K [ kmin, v ] [, ( ) ] d K a a a p, ec Bt kmin, [ k v ] K [ k v ] min, min, d [ K b, ( b) b ] p, where we ued a correlation tenor of the magnetic field K (k, ) f E t kmin, kmin, / / k min, k, kmin, k, where and are the pectral indice toward axi and an other direction perpendicular to axi, repectivel,, with the following propertie: (i) when k (or k ) 0 while k (or k ) remain contant, then T (k, )~ cont. (ii) when k (or k ) while k (or k ) remain contant, then z z z T (k, ) k (or T (k, ) kz ), i.e., the caling in both and direction are not the ame. (iii) >; a i gamma function and K z i the modified Beel function of the econd order. For intance, numerical experiment howed trong aniotrop along axi for k min 00, k min 400, 0.4, 0.75 [3]. We ued in the equation(0) the following variable (0) () akmin, (v ) and bkmin, (v ) () In the right hand of the equation (0) we can perform the following derivation
76 Gheorghe Dumitrecu 6 f c f c ik f p v f c f c ik f p v (3) (4) Therefore the equation (0) ma be rewritten a 4 ece t ik Mf [ kmin, v] K [ kmin, v ] kmin, d[ a K ( a) a K ( )] a f kmin, d[ b K ( b) b K ( )] b f 4 E ec t ik [ kmin, v] K [ kmin, v ] kmin, d[ b K ( b) b K ( )] b f kmin, d[ a K ( a) a K ( )] a f The right-hand part of the equation (9) wa labeled Mf equation (5). (5) in the
7 Colliion integral in the chaotic gun effect 77 The equation (5) dipla the complex dependence of colliion integral on the trength of the magnetic field, on the correlation length and on the pectral indice of the power of the random magnetic field. On the other hand, one mut mention that we recover the exponential dependence of the colliion integral on time and pace through the modified Beel function of the econd order. 3. SOME ESTIMATIONS OF THE COLLISION INTEGRAL The choice of different magnitude of the correlation length and of the pectral indice lead to a trong aniotrop of the colliion integral a we will how further. To do thi we will adopt /, 3/, k min 0.5 k min. A proper tud of the behavior of the colliion integral require patial interval of the order of the correlation length. Therefore we will perform a computation involving k min, v. In our paper we propoed a model of a random nchrotron emiion of moving charged particle, thi emiion being alo the random magnetic field that accelerate them. Hence one can adopt for relativitic particle the mall cone of the emiion which lead to the ratio of the velocitie v (6) v where v and v are the velocitie along axi, repectivel along a perpendicular direction to the axi. According to the aumption made above the colliion integral become 0.6ec 4E t ik Mf [ k v / min, ] / [ min, v ] K k / / min / 3/ 0.3 k d[ a K ( a) a K ( a)] f 0.089 7 kmin d 3 f.7.7 4 t 0.79ecE ik.7 0.089k min, d 3 7 f. 7.7 / / min, / 3/ 0.3 k d[ a K ( a) a K ( a)] f (7)
78 Gheorghe Dumitrecu 8 A final form of the equation (7) ma be obtained if one take into account the form of the modified Beel function of the econd order K / exp( a) ( a) (8) a ( a)exp( a) K3/, ( a) (9) a3/ But for it phical ignificance it i more intereting to evaluate the ratio of the two component of the integral colliion Mf / Mf 0,8 (0) Thi ratio depend exponentiall on the Lorentz factor a Mf / Mf 0.494 exp( 0.5c v 0.75 0.5c v 0.5 The equation (0) etablihe a relation between the two component of the colliion integral. In order to accomplih our tak, one can alo etimate the colliion integral b fitting the theoretical pectrum of the nchrotron emiion () in d [ kv Nf / i ] e k,, P K 4 3 e e K 4 3 e c d Nf / i c k,, () co d [ kv Nf / ik,, ] d Nf / ik,, to the data[6], where and Mf / i k,, Mf / i k,, Nf / i k,, (3) i the colliion integral where we formall replaced f b and pointed out it dependence on and k. In the equation () K e i the contant of the power law pectrum dn Ke d (4) of the number of the particle. We performed a viual fitting to the averaged pectrum of Mkn 50, oberved in April 997, uing the function where x x 0.008 x x 0.008 f x log[ 0 in(0 ) 0 co(0 )] (5) log x 0 Hz.
9 Colliion integral in the chaotic gun effect 79 Fig. Our model (dotted line) fitting BeppoSAX (aterik), RXTE (quare) and HEGRA (open circle) obervation of April 7, 997 (MJD 50545) and the HEGRA 997 time-averaged pectrum caled according to the detection rate of thi da (full circle) [7]. In the cae choen in Fig. ( = 5), a crude etimation obtained b numerical imulation lead to K.76 0 e for energ in the range 05 ev. Thi correpond alo to / 5.6 erg d Nf i k,,. (6) Therefore, if we chooe, which i the cae for patial e-folding, then Mf / i 0 5. k,, 4. DISCUSSIONS AND CONCLUSIONS Although the pectrum of the nchrotron emiion inferred in our model i not a trictl power law pectrum, till at high frequencie, it i a likel one. A we how in the preent paper the time colliion integral of the equation (9) pla an important role in order to etablih the hape of the pectrum. Adopting the appropiate Lorentz factor in the colliion integral one can fit the preent data concerning gamma ra emiion of TeV blazar. Thi lead u to etimate the colliion integral of a certain ource and hence, uing our model, to etimate the efficienc of the injection. If one can prove the efficienc of the injection, then our model might be a olution to the iue of the electromagnetic cenario of producing gamma ra burt or high energ gamma ra in blazar. The derivation and the etimation made to the colliion integral in thi paper proved the trong aniotrop of the mean free path when auming the anotropic
80 Gheorghe Dumitrecu 0 correlation tenor of the random magnetic field. We applied our computation to the chaotic gun effect [6]. Here the particle are accelerated b uual free-pace tranvere mode and our reult matche thoe cae where the atrophical plama can be neglected. Thi might be for intance the cae of the plama compound of low energ electron emitting in the radio frequencie domain and of high energ electron emitting in the gamma ra frequencie domain. The code ued b Argri and Ciubotariu emphaized an unphical behavior of the emitted radiation if one uppoed that the ame radiation i the engine of the acceleration: a linear polarization of the tranvere field. The preent paper prove the wa to fit the analtical picture of the chaotic gun to the phical behavior of the emiion of a charged particle. We allow the emiion toward two direction perpendicular to the magnetic field and alo to each other. But the emiion are of different trength and one of them i much larger than the other one. The etimation performed in thi paper provide one of the factor of the ditribution function of the charged particle involved in a chaotic motion. Thi function can contribute to computing the efficienc of the injection of the charged particle acro the hock front of atrophical plama. The efficienc i the ratio between the particle which cro the hock front and thoe which attempt to cro it. The latter one obe the regular motion in the magnetic field. The are at thermal equilibrium or are accelerated b the chaotic gun effect, but do not cro the hock front. Hence a formal expreion of the efficienc of injection i n fk, dk n f dk n f d k, f k, dk f dk f d k, where n i the averaged number of particle in unit volume, fk, i the ditribution function which i the olution of the equation (9) and f the Maxwell ditribution. Our reult doe not take into account the reconnection proce and the aborption proce due to pair production which ma affect our reult. (7) REFERENCES. I. N. Toptgin, Comic ra in interplanetar magnetic field, D. Reidel Publihing Compan, Doordrecht, Holland, p. 0 34, p. 84 9, p. 36 (985).. Yu. P. Mel nikov, 9th International Comic Ra Conference Pune) 00, 0 04 (005). 3. Y.-P. Zhao, G.-C. Wang, T.-M. Lu, Diffraction from aniotropic random rough urface, Ph. Rev. B, Vol. 58, No. (998). 4. J. Argri, C. Ciubotariu, Chao, Soliton and Fractal,, 00 (000). 5. V. L. Ginzburg, S. I. Srovatkii, Comic magnetobremtrahlung, ARA&A 3 79 (965). 6. G. Dumitrecu, Atrophical Chaotic Gun Effect, ubmitted to A&A. 7. H. Krawczink et. al., atro-ph/0049 A&A.