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rays can be attenuated. At [MATH] , only [MATH] TeV photons are attenuated by the disk radiation field (with [MATH] ). The [MATH] corrections have not been included in the SED calculations, but are only important at [MATH] |
Hz. In all of our models the jet is particle-dominated ( [MATH] ) and the magnetic field is below its equipartition value. The total jet power, also divided into magnetic field and particle powers, is shown in Table . This is typical of results from modeling the observed X-ray and [MATH] -ray emission of TeV blazars wi... |
3.3. External Isotropic Radiation Field This is the case treated by Georganopoulos et al. ( 2001 , where jet electrons scatter a surrounding external isotropic radiation field |
[MATH] . By integrating eq. ( 28 ) over the angle variables, recognizing that [MATH] for this geometry, one obtains [EQUATION] where [MATH] is given by Jones’s formula (Jones, 1968 eq. ( ). Because the scattering is taking place in the stationary frame, |
[EQUATION] with [MATH] and, as before, [MATH] The limits [MATH] and [MATH] are given by [EQUATION] and [EQUATION] For the case of the cosmic microwave background radiation (CMBR), |
[EQUATION] where [MATH] is the dimensionless temperature of the blackbody radiation field, and [MATH] K. Substituting eq. ( 88 in eqs. ( 22 ) and ( 33 ) gives |
[EQUATION] Eq. ( 89 ) gives an accurate spectrum of radiation made when jet electrons with number spectrum [MATH] Compton-scatter blackbody photons (see, e.g. Tavecchio et al., 2000 ; Dermer & Atoyan, 2002 ; Böttcher et al., 2008 |
3.4. Scattered BLR Radiation Field The BLR is thought to consist of dense clouds with a specified covering factor that can be determined from AGN studies. These clouds intercept central-source radiation to produce the broad emission lines in broad-line AGN |
(Kaspi & Netzer, 1999 . The diffuse gas will also Thomson scatter the central source radiation. The scattered radiation provides an important source of target photons that jet electrons scatter to [MATH] -ray energies (Sikora et al., 1994 1997 This radiation field also attenuates [MATH] rays produced within the BLR (Bl... |
Here we calculate the angular distribution of the Thomson-scattered radiation from a shell of gas with density [EQUATION] extending from inner radius [MATH] to outer radius [MATH] (see Fig. the calculation of fluorescence atomic-line radiation differs by considering dense clouds with a volume filling factor). The shell... |
[EQUATION] when [MATH] Here [MATH] is the central source photon-production rate, assumed to radiate isotropically, and [MATH] is its spectral luminosity. |
A more accurate calculation of the Thomson-scattered photon density is obtained by integrating the expression [EQUATION] over volume, where [MATH] and [MATH] (Fig. ). Assuming that the photons are isotropically Thomson-scattered by an electron without change in energy, so that [MATH] , then |
[EQUATION] from a spherically-symmetric electron density distribution (cf. Gould, 1979 ; Böttcher & Dermer, 1995 , for a time-dependent treatment) . In the case of an isotropic, uniform surrounding medium, |
[MATH] [MATH] [MATH] and [MATH] in eq. ( 90 ), and eq. ( 93 ) gives [MATH] , noting that the integral [MATH] Thus the approximation given by eq. ( 91 ) is accurate to within a factor of [MATH] for this case. |
The angle-dependent scattered photon distribution [MATH] can be derived by imposing a [MATH] -function constraint on the angle [MATH] in eq. ( 92 (see also Donea & Protheroe, 2003 , so that |
[EQUATION] after changing variables to [MATH] . From Fig. [MATH] so that [MATH] and [MATH] The law of sines gives [MATH] , with the result |
[EQUATION] Transforming the [MATH] -function in [MATH] to a [MATH] -function in [MATH] gives, after solving, [EQUATION] where [EQUATION] |
(units of [MATH] are [MATH] ), and [EQUATION] Fig. shows the angle dependence of the scattered radiation field in the stationary frame for this idealized geometry for the parameters given in the figure caption. As can be seen, the scattered radiation field is nearly isotropic when [MATH] and starts to display increasin... |
The Compton-scattered radiation spectrum is given, in general, by eq. ( 34 ). Substituting eq. ( 96 ) for the angular distribution of the target photon source gives, for a monochromatic photon source |
[EQUATION] the [MATH] flux [EQUATION] In this expression, [EQUATION] (compare eq. ). Substituting eq. ( 96 ) into eq. ( 20 for a monochromatic photon source, eq. ( 99 ), gives |
[EQUATION] for the opacity of a photon with measured energy [MATH] emitted outward along the jet axis at height [MATH] . The [MATH] opacity vanishes when [MATH] due to the [MATH] pair-production threshold. |
3.4.1 Thomson-Scattered Isotropic Monochromatic Radiation Field Substituting [MATH] for an isotropic monochromatic radiation field in eq. ( 35 gives, with [MATH] for the Thomson regime away from the endpoints of the spectrum, |
[EQUATION] We consider Compton up-scattering, i.e., [MATH] for the power-law electron spectrum given by eq. ( 47 ). The Thomson approximation is only valid far from the endpoints. The asymptotes for the Thomson-scattered spectrum of a surrounding isotropic, monochromatic radiation field therefore becomes |
[EQUATION] and [EQUATION] Note the different beaming factors in the two asymptotes. Eq. ( 105 ) agrees with the Thomson expression derived by Dermer et al. ( 1997 , eq. (22), to within factors of order unity. |
3.4.2 Numerical Calculations We now present calculations of the SED of FSRQ blazars, including the external Compton-scattering component formed by jet electrons that scatter target photons which themselves were previously scattered by BLR material. This EC BLR component is not found in conventional synchrotron/SSC mode... |
For the simplified shell geometry depicted in Fig. we calculate the external Compton scattering component using eq. ( 100 ), and calculate the [MATH] opacity using eq. ( 102 ). To simplify the calculations, the spectrum of the radiation scattered by the BLR is assumed to be monochromatic with energy [MATH] eV, correspo... |
There is very little [MATH] absorption by the accretion-disk radiation or the BLR radiation when the jet is found outside the BLR. On the other hand, when the jet is within the BLR, there can be significant [MATH] opacity. This is shown in Fig. 11 , where we use the standard parameters for the jet and a BLR with [MATH]... |
The effect of changing the radial gradient of BLR scattering material is shown in Fig. 12 . For a steeper density gradient, [MATH] and a constant Thomson depth, the material is concentrated near the inner edge of the BLR at [MATH] , so the changes in the [MATH] opacity are most dramatic when [MATH] due to geometric eff... |
4. Discussion and Summary We have presented accurate expressions for modeling synchrotron and Compton-scattered radiation from the jets of AGN that include target radiation fields from the accretion disk and BLR. This extends our technique for modeling synchrotron and SSC emission (Finke et al., 2008 to include externa... |
In the results presented here (Figs. and 10 12 ) we have chosen parameters for demonstration purposes that give an exaggerated EC component, particularly a high |
[MATH] . Lowering the disk luminosity lowers the radiation considerably, as seen in Fig. . The disk radiation field in FSRQ blazars can be seen when the nonthermal blazar radiation is in a low state, as in the cases of 3C 279 (Pian et al., 1999 , 3C 454.3 |
(Raiteri et al., 2007 and, most clearly, 3C 273 (e.g., von Montigny et al., 1997 . These observations can be used to assign the accretion-disk luminosity when modeling a specific blazar, though the disk brightness could also vary during the flaring epoch. |
The models presented here do, however, have limitations. They do not yet include enhancements from secondary cascade radiation initiated by |
[MATH] pairs formed by [MATH] rays interacting with lower energy radiation from the disk and the BLR. For the parameters considered here, this would not make a significant difference in the calculated SEDs because the energy flux of the attenuated radiation is a small fraction of the escaping flux. But even in this cas... |
Spectral features result from [MATH] absorption by BLR radiation, as seen in Fig. 12 . By assuming hard primary [MATH] -ray emission components, |
Aharonian et al. ( 2008 argue that hard intrinsic spectra from blazars such as 1ES 1101-232 (Aharonian et al., 2007b could be formed through [MATH] attenuation. If the primary TeV radiation originates from an underlying jetted electron distribution, then a consistent model requires that a |
[MATH] -ray spectrum formed by Compton-scattering processes arises from the same radiation field responsible for [MATH] absorption. As our calculations show, soft Compton-scattered TeV spectra are formed due to Klein-Nishina effects in scattering, whether from the accretion disk or from photons scattered by the BLR. Th... |
and 12 ), which would easily be detected with Fermi or a hard primary [MATH] -ray spectrum must originate from other processes. Moreover, if this explanation was correct, then blazars with stronger broad emission lines should have harder VHE [MATH] -ray spectra than weak-lined BL Lacs. |
Cascade radiation induced by ultra-relativistic hadrons could inject high-energy leptons to form a hard radiation component, though a sufficiently dense target field for efficient photohadronic losses will itself severely attenuate the TeV radiation |
(Atoyan & Dermer, 2003 . Depending on the underlying acceleration model, a distinctive hadronic signature could appear at [MATH] -ray energies only, though one would expect that both electrons and protons would be accelerated synchronously. |
Our calculations also do not as yet include absorption by the diffuse extragalactic background light (EBL), which would be important for EGRET [MATH] -ray FSRQs, which have a broad redshift distribution with a mean value [MATH] , larger than the mean value [MATH] for BL Lacs (Mukherjee et al., 1997 . In cases where the... |
[MATH] , detected from 80 to [MATH] GeV energies with MAGIC (MAGIC Collaboration, 2008 , EBL effects cannot be neglected. For the models studied here, the soft calculated [MATH] -ray spectra cannot in any case account for the measured, let alone intrinsic, spectrum of 3C 279. Based on the simultaneous optical – X-ray –... |
In conclusion, we have derived expressions to model FSRQ blazars that self-consistently include [MATH] attenuation on the same target photons that are Compton-scattered by the relativistic electrons. By assuming that the lower energy, radio – UV emission is nonthermal synchrotron radiation, then the underlying electron... |
[MATH] -ray emission spectrum from SSC and external Compton scattering processes from this electron distribution. Separately, one can determine whether the inferred electron distribution can be derived from specific acceleration scenarios. Complementary to the technique of injecting electron spectra and cooling, this m... |
We thank the referee for asking us to address the potential importance of higher-order SSC fluxes, and for other helpful comments. The work of J.D.F. is supported by NASA Swift Guest Investigator Grant DPR-NNG05ED411 and NASA GLAST Science Investigation DPR-S-1563-Y, which also supported summer research by H.K. at NRL,... |
# Source: arxiv 0808.3195 # Title: Oblique Ion Two-Stream Instability in the Foot Region of a Collisionless Shock # Sections: all # Downloaded: 2026-03-02T07:57:58.091558+00:00 |
Oblique Ion Two-Stream Instability in the Foot Region of a Collisionless Shock Abstract Electrostatic behavior of a collisionless plasma in the foot region of high Mach number perpendicular shocks is investigated through the two-dimensional linear analysis and electrostatic particle-in-cell (PIC) simulation. The simula... |
To investigate nonlinear behavior of the ion two-stream instability, we have made PIC simulations for the shock foot region in which the initial state satisfies the Buneman instability condition. In the first phase, electrostatic waves grow two-dimensionally by the Buneman instability to heat electrons. In the second p... |
The present result implies that overheating problem of electrons for shocks in supernova remnants is resolved by considering ion two-stream instability propagating highly obliquely to the shock normal and that multi-dimensional analysis is crucial to understand the particle heating and acceleration processes in shocks. |
supernova remnants – shock waves – plasmas – instabililes – cosmic rays – acceleration of particle Introduction The discovery of thermal and synchrotron X-rays from young supernova remnants (SNRs) provides the evidence that electrons are heated up to a few keV and that a portion of them are accelerated to highly relati... |
(Leroy, 1983 . The plasma in the foot region consists of incident ions and electrons and reflected ions and returning ions which are made from reflected ions and move to the shock after a gyration. As for the electron heating machanism, Papadopoulos ( 1988 proposed that when the Mach number is larger than [MATH] , inci... |
(Stage et al., 2006 . This discrepancy has been an open issue to be resolved for a long time. On the other hand, Shimada & Hoshino ( 2000 and Hoshino & Shimada ( 2002 performed one-dimensional full particle-in-cell (PIC) simulations to investigate the electron acceleration at perpendicular shocks. Their simulation solv... |
In this paper, we study the time evolution of electrostatic collisionless plasma instabilities in the foot region by making linear analysis and by performing two-dimensional electrostatic PIC simulation. We perform simulations with a higher resolution, a larger simulation box and a longer simulation time than in Ohira ... |
and [MATH] are the drift velocity of reflected protons and the magnetic field, respectively. Because reflected ions have a large free energy, we expect that more energy is transported to electrons through collective instabilities with the realistic mass ratio in the foot region. The drift velocity is not large enough t... |
In §2 we perform linear analysis for two-dimensional electrostatic modes. In §3 we describe the initial setting of the PIC simulations and numerical results, followed by a discussion in §4. |
Linear analysis In this section, we perform linear analysis for two-dimensional electrostatic modes in unmagnetized plasmas with beams. In the foot region of perpendicular shocks of SNRs, we regard that there are several beams with a finite temperature and that their relative velocities are much smaller than the light ... |
[EQUATION] is relatively small, where [MATH] and [MATH] are electron cyclotron frequency, electron plasma frequency, and electron number density, respectively. So plasma oscillations are hardly changed by the magnetic field in the foot region of shocks of SNRs. When we consider spatial scale smaller than the gyroradius... |
We define such that the [MATH] -direction is shock normal direction and the [MATH] -direction is the direction that is perpendicular to shock normal and wave vectors are on the [MATH] plane. For unmagnetized collisionless plasmas, the electrostatic dispersion relation reads as |
[EQUATION] where the subscript s represents particle species, here electrons, ions and beam ions, [MATH] is the plasma frequency of the particle species s and [MATH] is the normalized distribution function of the particle species s, |
[EQUATION] where [MATH] and [MATH] is the thermal velocity and drift velocity of the particle species s, respectively. To make equation (2) simpler, we use new coordinates [MATH] and [MATH] as |
[EQUATION] Then, equations (2) and (3) become [EQUATION] and [EQUATION] Finally, we substitute equation (7) into equation (6), and we obtain |
[EQUATION] where [MATH] is the plasma dispersion function and it can be numerically solved (Watanabe, 1991 We here present results of the linear analysis about two cases of plasma conditions and discuss on three kinds of plasma instabilities. |
2.1 Buneman instability We first consider the situation in which there are three beams, incident protons, incident electrons and reflected protons and the temperatures of all plasma beams are low, typically around 1 eV. We thus we neglect the contribution of returning ions in the dispersion relation, for simplicity. As... |
[MATH] , where [MATH] is the shock velocity. Hence, a typical velocity ratio is [MATH] . We assume that the proton reflection ratio is [MATH] |
The growth rate obtained by solving the linear dispersion relation is displayed in Figure 1(a). In this condition, the most unstable mode is the Buneman instability. The Buneman instability is caused by the resonance between the electron plasma oscillation of the upstream electrons and proton plasma oscillation of the ... |
[MATH] and the color contours show the growth rate normalized by [MATH] , where only the growth rate of growing modes is shown. As is seen, the growth rate of obliquely propagating modes is as large as that of modes parallel to the beam direction. This feature of the Buneman instability can be well understood in the co... |
[EQUATION] and the dispersion relation (2) is reduced to [EQUATION] Because [MATH] does not appear in equation (10), the growth rate of electrostatic instabilities in the cold limit does not depend on |
[MATH] . Therefore, in the cold limit, excited waves have any [MATH] and the structure of electrostatic potential to the [MATH] -direction is strongly disordered and loses coherence to the [MATH] direction. This feature is very important to negate the electron surfing acceleration mechanism (Ohira & Takahara, 2007 . Th... |
[EQUATION] In reality, because of a finite temperature, the modes with large wavenumbers are suppressed to grow. This is seen in Figure 1(a); the growth rate for large [MATH] decreases. The dispersion relation of the Buneman instability depends on [MATH] and the number density ratio. When [MATH] is small, modes that ha... |
[MATH] in the present plasma condition and the boundary between growing and damping region in Figure 1(a) is about [MATH] , as is consistent with the present consideration. This result implies that in SNRs condition, oblique modes propagating to the beam direction, i.e., to the shock normal, can grow as strongly as the... |
2.2 Ion two-stream instability and ion acoustic instability After the upstream electrons are heated by the Buneman instability, the thermal velocity of electrons increases up to |
[MATH] while the thermal velocity of protons is roughly the same as the initial one, so that the situation of [MATH] is realized. The other parameters are the same as in the low temperature case described in the previous subsection. In this condition, while the Buneman instability is stabilized, other types of instabil... |
(Cargill & Papadopoulos, 1988 , the ion two-stream instability that has not been well noticed in the literature becomes unstable. The ion two-stream instability is caused by the resonance of the ion plasma oscillation of the upstream plasma and that of reflected ions in the situation |
[MATH] . Thus, it occurs concurrently with the ion acoustic instability which is caused by the resonance between ion acoustic waves of the reflected protons and electron plasma oscillation of the upstream plasma, where the former modes are mediated by the presence of the hot upstream electrons. |
The numerical results of the growth rate for ion two-stream instability and ion acoustic instability are shown in Figures 2(a) and (b), respectively. In Figure 2, [MATH] [MATH] and the growth rate are normalized in the same way as in Figure 1, and the color contours show the growth rate of the growing modes. Note that ... |
[MATH] , ion two-stream instabily grows at much smaller [MATH] . It is also noted that ion two-stream instability is seen for low but finite values of [MATH] , In the present conditions, the wavenumber corresponding to the ion Debye length is |
[MATH] while that to the electron Debye length is [MATH] . Ion plasma oscillation exists between these two wavenumbers. For wavenumbers lower than [MATH] it reduces to the ion acoustic mode while for wavenumbers higher than |
[MATH] it damps by thermal motions of ions. It should be noted that in Figure 2(a), ion two-stream modes of parallel propagation to the beam direction are only weakly growing, but that highly oblique modes grow very fast and the maximum growth rate is larger than that of the ion acoustic instability by a factor of a fe... |
[MATH] . For propagation parallel to the beam direction, this is imcompatible with the wavenumber condition and the growth rate is very small. In contrast, when the wave has a large [MATH] , both the wavenumber condition and the resonance condition are fulfilled simultaneously and a larger growth rate is obtained. The ... |
To understand the ion two-stream instability through the dispersion relation, we consider a situation where the proton temperature is zero and the electron temperature is very high and [MATH] |
(if the electron drift velocity is much smaller than thermal velocity, we do not need the final condition). Then, we can approximate as |
[MATH] and [MATH] and the dispersion relation becomes [EQUATION] The second term represents the Debye shielding effect of hot electrons and becomes small for [MATH] as discussed above. It is seen that for [MATH] , the dispersion relation has the same form as equation (10) and we obtain the maximum growth rate of the io... |
[EQUATION] Because the upstream plasma stays in foot region by about the proton gyro-period [MATH] and because [MATH] , ion two-stream instability can grow enough in the foot region. |
As far as we are aware, this oblique unstable mode has not been considered up to now. We expect that this instability heats ions and that much affects subsequent electron heating processes. |
Simulation To perform two-dimensional simulations with real proton electron mass ratio, we confine our attention to the foot region through a proper modeling instead of solving the whole shock structure. Our simulation box is taken to be at rest in the upstream frame of reference, i.e., that of incident protons and ele... |
3.1 Setting We define the [MATH] -direction as the shock normal pointing to the shock front, and thus the reflected protons move in the [MATH] -direction and returning protons move in the [MATH] -direction. The magnetic field is taken to be spatially homogeneous pointing in the [MATH] -direction and we solve the partic... |
[MATH] . In addition, reflected and returning protons have an extra drift velocity in the [MATH] -direction of [MATH] [MATH] ). The number densities of each population are taken as |
[MATH] and [MATH] , where subscripts e, p, ref and ret represent upstream electrons, upstream protons, reflected protons and returning protons, respectively (see Figure 3). These parameters are typical of young SNRs and satisfy the charge neutrality and a vanishing current. |
We employ the periodic boundary condition both in the [MATH] - and [MATH] -directions. The electric field is solved by the Poisson equation. We have examined two cases of the background magnetic field, 0 and 90 [MATH] G ( [MATH] and 0.03), where |
[MATH] is the electron cyclotron frequency. We sometimes refer the former and latter cases to the unmagnetized and magnetized cases, respectively. |
The size of the simulation box to the [MATH] - and [MATH] -directions is taken to be [MATH] and [MATH] with a total of 2048 [MATH] 512 cells, where |
[MATH] is the wavelength of the most unstable mode of the Buneman instability. Thus, the length of each cell [MATH] is 3 times the initial electron Debye length. The number of macroparticles is taken so that initially each cell includes 96 electrons and 96 total protons. The time step [MATH] is taken as [MATH] |
and the simulation is followed until [MATH] or [MATH] where [MATH] corresponds to the time scale the upstream plasma stays in the foot region. |
The differences from previous simulations (Ohira & Takahara, 2007 are as follows. First, the initial temperature is lower than the previous one [MATH] This is a more realistic one because the typical temperature of the interstellar matter is about 1eV. Secondly, we add returning proton beam in order that the total curr... |
3.2 Results Although we have performed simulations for two cases of the magnetic field strength ( [MATH] ), the results turn out to be almost the same. Hence, we present the results of the unmagnetized case and add those of the magnetized case when necessary. |
First, we discuss the time development of the electric field. The evolution of the spatially averaged energy density of the electric field is shown in Figure 4. The solid and dashed curves show the |
[MATH] - and [MATH] -components, respectively, and bold and thin curves are unmagnetized and magnetized cases, respectively. In the first stage for [MATH] , the Buneman instability occurs and the electric field to both directions grow. After they attain peak values, they continue to decay till around |
[MATH] . Then, after [MATH] , the [MATH] -component of the electric field starts to grow again while the [MATH] -component continues to decay. This feature is due to the ion two-stream instability as discussed below. |
It should be noted that in the first stage of the Buneman instability [MATH] ), the [MATH] -component of electric field is larger than the [MATH] -component. This is different from our previous result. As discussed in §2, this is because the temperature is lower and the waves with a larger obliqueness grow faster compa... |
[MATH] In contrast to Cargill & Papadopoulos ( 1988 , after the electrostatic waves caused by the Buneman instability decay at about [MATH] only the [MATH] -component of the electric field grows and oscillates after the amplitude saturates. In contrast, the [MATH] -component continues to decay. Of course, this feature ... |
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