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Fouka M.,Research Center in Astronomy | Ouichaoui S.,University of Science and Technology Houari Boumediene
Monthly Notices of the Royal Astronomical Society | Year: 2014

In several high-energy astrophysical sites, shocks are assumed to produce a power-law distribution of accelerated charged particles (e.g., electrons, protons) and to generate mild to strong magnetic fields, which favours synchrotron emission. For such environments and conditions, we have performed and present here four practical formulas, with different levels of accuracy, for fitting the synchrotron spectral power radiated by a pure power-law particle distribution, with isotropic pitch angle distribution. The first three ones can be useful compared to the fourth one, because of their simplicity, in the case of particle distribution with no high-energy cutoff. However, the fourth formula, the more accurate one, can be of great interest for astrophysical applications (even though it is more complicated) for the more general case of a power-law distribution with high-energy sharp cutoff. The latter is derived for index p within the range, 1 < p < 6, with maximum relative error of less than 0.5 per cent in the case of infinite energy range and of less than 8.2 per cent in the more general case of particle energy with sharp cutoff. The latter is expressed in terms of parameters as functions of index p. These parameters have been fitted with adopting the Levenberg-Marquardt algorithm in log-log scale. According to these formulas, initially derived for the total spectral power, we have then derived (1) the degree of polarization, (2) cooling spectra for a broken power-law distribution and (3) synchrotron self-absorption spectra for both the broken and non-broken power-law distributions. The proposed expressions are relevant for non-thermal astrophysical sources in the sense that by using them one avoids usually complicated and long CPU time calculations, without performing any integration. © 2014 The Authors. Published by Oxford University Press on behalf of the Royal Astronomical Society.

Fouka M.,Research Center in Astronomy | Fouka M.,University of Science and Technology Houari Boumediene | Ouichaoui S.,University of Science and Technology Houari Boumediene
Astrophysical Journal | Year: 2011

We have calculated the inverse Compton (IC) integrated spectral power within the Thomson limit for a monoenergetic isotropic photon field upscattered off highly relativistic electrons assuming an isotropic power-law distribution of the latter, N(γ) = Cγ-p, with Lorentz parameter values γ1 < γ < γ2. Our interest was essentially focused on the case of a finite energy range (finite γ2) possibly having realistic applications in high-energy astrophysical sites, mainly relativistic shock regions. To this end, we have defined and derived a dimensionless parametric function, Fp (z 1, η), with variables z 1 = ε1/ 4γ2 1ε and η = γ2/ γ1. This result was used to derive the IC-integrated spectral power for an upscattered blackbody (BB) photon field using a dimensionless parametric function, Wp (ξ, η), with variable ξ = ε1/4γ2 1 kT. Asymptotic forms of this function have been derived for three energy ranges, i.e., ξ ≪ 1, 1 ≪ ξ ≪ η2, and ξ ≫ η2. Then, a characteristic value, ηc(p, ε) with ε ≪ 1, of parameter η was defined such that the middle range asymptotic form of W p (ξ, η) could be valid and good when η ≳ ηc(p, ε), by deriving an approximate expression of this particular value for ε = 10-3. The resulting spectra featured by a high-energy cutoff in the case of low values of the ratio η can be discussed at least for a population of short gamma-ray bursts (GRBs), those best described by the cutoff power-law model with a low-energy spectral index, α 0. Furthermore, it is suggested that for GRB spectra with α < -1/2 pertaining to the prompt emission phase, the IC is a likely emission mechanism for both monoenergetic and BB photon fields if one assumes that the former photon field could exist specifically in the GRB environment. Various suitable astrophysical applications are presented and discussed. © 2011. The American Astronomical Society. All rights reserved.

Zouaoui E.,University of Science and Technology Houari Boumediene | Fouka M.,Research Center in Astronomy | Ouichaoui S.,University of Science and Technology Houari Boumediene
AIP Conference Proceedings | Year: 2012

In this paper, we present and compare the classical, generic and realistic models describing the evolution of a relativistic blast wave of the GRB fireball. Indeed, we have shown that while the classical model holds only in the afterglow relativistic phase, the generic and the realistic models are valid for the whole evolution of the blast wave, i.e., from the relativistic phase to the non relativistic one, consistently with Sedov's solution for an adiabatic expansion. We are especially interested here in the case of an evolved radiation efficiency, taking into account the synchrotron emission as a basic radiation mechanism. This case is compared to that of constant radiation efficiency, simply identified to be the conversion coefficient. © 2012 American Institute of Physics.

Fouka M.,Research Center in Astronomy | Ouichaoui S.,University of Science and Technology Houari Boumediene
Research in Astronomy and Astrophysics | Year: 2013

Accurate fitting formulae to the synchrotron function, F(x), and its complementary function, G(x), are performed and presented. The corresponding relative errors are less than 0.26% and 0.035% for F(x) and G(x), respectively. To this end we have, first, fitted the modified Bessel functions, K 5/3(x) and K2/3(x). For all the fitted functions, the general fit expression is the same, and is based on the well known asymptotic forms for low and large values of x for each function. It consists of multiplying each asymptotic form by a function that tends to unity or zero for low and large values of x. Simple formulae are suggested in this paper, depending on adjustable parameters. The latter have been determined by adopting the Levenberg-Marquardt algorithm. The proposed formulae should be of great utility and simplicity for computing spectral powers and the degree of polarization for synchrotron radiation, both for laboratory and astrophysical applications. © 2013 National Astronomical Observatories of Chinese Academy of Sciences and IOP Publishing Ltd.

Fouka M.,Research Center in Astronomy | Ouichaoui S.,University of Science and Technology Houari Boumediene
Advances in Space Research | Year: 2011

We have analysed a sample of 328 time-integrated GRB prompt emission spectra taken via the Konus instrument on board the US GGS-Wind spacecraft between 2002 and 2004 using a couple of two-components models, Cut-off Power Law (CPL) + Power Law (PL) and blackbody (BB) + PL. The spectra show clear deviation from the Band function. The PL term is interpreted as the low energy tail of a nonthermal emission mechanism. The distributions of corresponding index β give values β < -2/3 consistent with synchrotron and synchrotron self-Compton mechanisms. The distribution of low energy index α associated with the CPL term shows clear discordance with synchrotron models for 31.4% of the analysed GRBs with values exceeding that for the line of death, α = -2/3. Then, a set of nonthermal radiation mechanisms producing harder slopes, i.e., α > -2/3, are presented and discussed. For the remaining majority (68.6%) of GRBs with CPL index α < -2/3, we show that optically thin synchrotron produced by a power law electron distribution of type, N(γ) ∼ γ-p, γ1 < γ < γ2, for finite energy range (γ2 ≠ ∞) is a likely emission mechanism with α ∼-(p + 1)/2 in the frequency range ν1 ≪ ν ≪ ν2 (where ν2 = η2ν1 with η = γ2/γ1), such that for p > 1/3, one gets α < -2/3. We also show that corresponding spectra in terms of F ν and νFν functions are peaked around frequency ν2 instead of ν1, respectively for p < 1 and p < 3. Besides, thermal emission is examined taking a single Planck function for fitting the low energy range. It can be interpreted as an early emission from the GRB fireball photosphere with observed mean temperature, kT′ ∼ 16.8 keV. Furthermore, we have performed a statistical comparison between the CPL + PL and BB + PL models finding comparable χ2-values for an important fraction of GRBs, which makes it difficult to distinguish which model and specific radiation mechanism (possible thermal or nonthermal γ-ray emissions) are best suitable for describing the reported data. Therefore, additional information for those bursts, such as γ-ray polarization, would be highly desirable in future determinations of GRBs observational data. © 2010 COSPAR. Published by Elsevier Ltd. All rights reserved.

Baddari K.,University of Boumerdès | Makdeche S.,University of Boumerdès | Bellalem F.,Research Center in Astronomy
Acta Geophysica | Year: 2013

Based on the moment magnitude scale, a probabilistic model was developed to predict the occurrences of strong earthquakes in the seismoactive area of Zemmouri, Algeria. Firstly, the distributions of earthquake magnitudes Mi were described using the distribution function F0(m), which adjusts the magnitudes considered as independent random variables. Secondly, the obtained result, i. e., the distribution function F0(m) of the variables Mi was used to deduce the distribution functions G(x) and H(y) of the variables Yi = Log M0,i and Zi = M0,i, where (Yi)i and (Zi)i are independent. Thirdly, some forecast for moments of the future earthquakes in the studied area is given. © 2012 Versita Warsaw and Springer-Verlag Wien.

Fouka M.,Research Center in Astronomy | Fouka M.,H+ Technology | Ouichaoui S.,H+ Technology
Astrophysical Journal | Year: 2011

We have derived asymptotic forms for the degree of polarization of the optically thin synchrotron and for synchrotron self-absorption (SSA) spectra assuming a power-law particle distribution of the form N(γ) ∼ γ-p with γ1 < γ < γ2, especially for a finite high-energy limit, γ2, in the case of an arbitrary pitch angle. The new results inferred concern more especially the high-frequency range x ≫ η2 with parameter η = γ2/γ 1. The calculated SSA spectra concern instantaneous photon emission where cooling effects are not considered. They have been obtained by also ignoring likely effects such as Comptonization, pair creation and annihilation, as well as magnetic photon splitting. To that aim, in addition to the two usual absorption frequencies, a third possible one has been derived and expressed in terms of the Lambert W function based on the analytical asymptotic form of the absorption coefficient, αν, for the high-frequency range ν ≫ ν2 (with ν2 the synchrotron frequency corresponding to γ2). We have shown that the latter frequency may not have realistic applications in astrophysics, except in the case of an adequate set of parameters allowing one to neglect Comptonization effects. More detailed calculations and discussions are presented. © 2011. The American Astronomical Society. All rights reserved.

Fouka M.,Research Center in Astronomy | Ouichaoui S.,H+ Technology
AIP Conference Proceedings | Year: 2010

Synchrotron emission behind relativistic magnetic internal-external shocks in gammaray bursts cosmological explosions is assumed to be the basic emission mechanism for prompt and afterglow emissions. Inverse Compton from relativistic electrons can also have appreciable effects by upscattering initial synchrotron or blackbody photons or other photons fields up to GeV-TeV energies. For extreme physical conditions such as high magnetic fields (e.g., B > 105 Gauss) self-absorption is not negligible and can hardly affect spectra at least for the low energy range. In this paper we present calculations of the synchrotron power, Pν, and their asymptotic forms, generated by a power law relativistic electron distribution of type Ne(γ) =Cγ-p with γ1 <γ<γ 2, especially for finite values of the higher limit γ2. For this aim we defined the dimensionless parametric function Zp(x,η) with x = ν/ν1 and η= γ2/γ1, so that Pν∝Z p(ν/ν1,η), with ν1 = (3/4π)γ12qBsinθ/mc (θ being the pitch angle). Asymptotic forms of this later are derived for three different frequency ranges, i.e., x≪1, 1≪x≪ν2 and x≫ν2. These results are then used to calculate the absorption coefficient, αν, and the source function, Sν, together with their asymptotic forms through the dimensionless parametric functions Hp(x,η) and Yp(x,η), respectively. Further calculation details are also presented and discussed. © 2010 Institute of Physics.

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