The fate of relativistic pair beams produced in the intergalactic medium by very-high energy emission from blazars remains controversial in the literature. The possible role of resonance beam plasma instability has been studied both analytically and numerically but no consensus has been reached. In this paper, we thoroughly analyze the development of this type of instability. This analysis takes into account that a highly relativistic beam loses energy only due to interactions with the plasma waves propagating within the opening angle of the beam (we call them parallel waves), whereas excitation of oblique waves results merely in an angular spreading of the beam, which reduces the instability growth rate. For parallel waves, the growth rate is a few times larger than for oblique ones, so they grow faster than oblique waves and drain energy from the beam before it expands. However, the specific property of extragalactic beams is that they are extraordinarily narrow; the opening angle is only Δθ ∼ 10−6 − 10−5. In this case, the width of the resonance for parallel waves, ∝Δθ2, is too small for them to grow in realistic conditions. We perform both analytical estimates and numerical simulations in the quasilinear regime. These show that for extragalactic beams, the growth of the waves is incapable of taking a significant portion of the beam’s energy. This type of instability could at best lead to an expansion of the beam by some factor but the beam’s energy remains nearly intact.
Modeling the propagation of radiative heat-waves in optically thick material using a diffusive approximation is a well-known problem. In optically thin material, classic methods, such as classic diffusion or classic P1, yield the wrong heat wave propagation behavior, and higher order approximation might be required, making the solution harder to obtain. The asymptotic P1 approximation [Heizler, NSE 166, 17 (2010)] yields the correct particle velocity but fails to model the correct behavior in highly anisotropic media, such as problems that involve sharp boundary between media or strong sources. However, the solution for the two-region Milne problem of two adjacent halfspaces divided by a sharp boundary, yields a discontinuity in the asymptotic solutions, that makes it possible to solve steady-state problems, especially in neutronics. In this work we expand the timedependent asymptotic P1 approximation to a highly anisotropic media, using the discontinuity jump conditions of the energy density, yielding a modified discontinuous P1 equations in general geometry. We introduce numerical solutions for two fundamental benchmarks in plane symmetry. The results thus obtained are more accurate than those attained by other methods, such as Flux-Limiters or Variable Eddington Factor. * avnerco@gmail.com † highzlers@walla.co.il deterministic methods, and they are both exact when N → ∞ [13]. Alternatively, a statistical implicit Monte Carlo (IMC) approach can also be used [14], which is exact when the number of particles (histories) goes to infinity. Although these three methods approach the exact solution, their application requires extensive numerical calculations that might be difficult to carry out, especially in multi-dimensions. Hence, there is an extensive body of literature dealing with the search for approximate models which will be relatively easy to simulate, and yet produce solutions that are close to the exact problem (for example, see [15,16]).The classical (Eddington) diffusion theory, as a specific case of the P 1 is relatively easy to solve and is commonly used [1,4,13]. The diffusion equation is parabolic, and thus yields infinite particle velocities. The full P 1 equations, that give rise to the Telegrapher's equation, has a hyperbolic form, but with an incorrect finite velocity, c/ √ 3 [17]. Possible solutions, such as flux-limiters (FL) solution (in the form of a non-linear diffusion notation), or Variable Eddington Factor (VEF) approximations (in the form of full P 1 equations), yielding a gradient-dependent nonlinear diffusion coefficients (or a gradient-dependent Eddington factor), are harder to solve, especially in multi-dimensions [15,16,[18][19][20][21][22][23][24].In previous work, Heizler [17] offered a modified P 1 approximation, based on the asymptotic derivation (both in space and time), the asymptotic P 1 approximation (or the asymptotic Telegrapher's equation approximation) [17]. In steady state, it tends to the well-known asymptotic diffusion approximation [13,25,26]. This approximation shar...
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