It is shown that an unmagnetized nonrelativistic thermal electron-proton plasma spontaneously emits aperiodic turbulent magnetic field fluctuations of strength |δB|=3.5β(e)g(1/3)W(e)(1/2) G, where ...β(e) is the normalized thermal electron temperature, W(e) the thermal plasma energy density, and g the plasma parameter. For the unmagnetized intergalactic medium, immediately after the reionization onset, the field strengths from this mechanism are about 2×10(-16) G in cosmic voids and 2×10(-10) G in protogalaxies, both too weak to affect the dynamics of the plasma. Accounting for simultaneous viscous damping reduces these estimates to 2×10(-21) G in cosmic voids and 2×10(-12) G in protogalaxies. The shear and/or compression of the intergalactic and protogalactic medium exerted by the first supernova explosions locally amplify these seed fields and make them anisotropic, until the magnetic restoring forces affect the gas dynamics at ordered plasma betas near unity.
We revisit the susceptible-infectious-recovered/removed (SIR) model which is one of the simplest compartmental models. Many epidemological models are derivatives of this basic form. While an analytic ...solution to the SIR model is known in parametric form for the case of a time-independent infection rate, we derive an analytic solution for the more general case of a time-dependent infection rate, that is not limited to a certain range of parameter values. Our approach allows us to derive several exact analytic results characterizing all quantities, and moreover explicit, non-parametric, and accurate analytic approximants for the solution of the SIR model for time-independent infection rates. We relate all parameters of the SIR model to a measurable, usually reported quantity, namely the cumulated number of infected population and its first and second derivatives at an initial time t = 0, where data is assumed to be available. We address the question of how well the differential rate of infections is captured by the Gauss model (GM). To this end we calculate the peak height, width, and position of the bell-shaped rate analytically. We find that the SIR is captured by the GM within a range of times, which we discuss in detail. We prove that the SIR model exhibits an asymptotic behavior at large times that is different from the logistic model, while the difference between the two models still decreases with increasing reproduction factor. This part A of our work treats the original SIR model to hold at all times, while this assumption will be relaxed in part B. Relaxing this assumption allows us to formulate initial conditions incompatible with the original SIR model.
ABSTRACT The analytical theory of diffusive cosmic-ray acceleration at parallel stationary shock waves with magnetostatic turbulence is generalized to arbitrary shock speeds , including, in ...particular, relativistic speeds. This is achieved by applying the diffusion approximation to the relevant Fokker-Planck particle transport equation formulated in the mixed comoving coordinate system. In this coordinate system, the particle's momentum coordinates p and are taken in the rest frame of the streaming plasma, whereas the time and space coordinates are taken in the observer's system. For magnetostatic slab turbulence, the diffusion-convection transport equation for the isotropic (in the rest frame of the streaming plasma) part of the particle's phase space density is derived. For a step-wise shock velocity profile, the steady-state diffusion-convection transport equation is solved. For a symmetric pitch-angle scattering Fokker-Planck coefficient, , the steady-state solution is independent of the microphysical scattering details. For nonrelativistic mono-momentum particle injection at the shock, the differential number density of accelerated particles is a Lorentzian-type distribution function, which at large momenta approaches a power-law distribution function with the spectral index . For nonrelativistic ( ) shock speeds, this spectral index agrees with the known result , whereas for ultrarelativistic ( ) shock speeds the spectral index value is close to unity.
A new transport theory of cosmic rays in magnetized space plasmas with axisymmetric incompressible magnetic turbulence is developed extending the quasilinear approximation to the particle orbit. ...Arbitrary gyrophase deviations from the unperturbed spiral orbits in the uniform magnetic field are allowed. For quasi-stationary and spatially homogeneous magnetic turbulence, we derive the small Larmor radius approximation gyrophase-averaged cosmic ray Fokker-Planck coefficients. The generalized Fokker-Planck coefficients correctly reduce to their known quasilinear values in the corresponding limit. New forms of the quasilinear Fokker-Planck coefficients in axisymmetric turbulence are derived which no longer involve infinite sums of products of Bessel functions, which facilitate their numerical computation for specified turbulence field correlation tensors. The Fokker-Planck coefficients for arbitrary phase orbits of the cosmic ray particles provide strict upper limits for the perpendicular and pitch-angle Fokker-Planck coefficients, which in turn yield strict upper and lower limits for the perpendicular and parallel spatial diffusion coefficients, respectively, describing the spatial diffusion of the isotropic part of the cosmic ray phase space density. For the associated mean free paths, we find for this general case that the product of the minimum parallel mean free path with the sum of the maximum perpendicular mean free paths equals R 2 L , where RL denotes the cosmic ray gyroradius.
ABSTRACT The diffusive propagation of nonrelativistic cosmic ray (CR) protons undergoing energy losses by ionization in a dense homogeneous infinitely extended interstellar molecular cloud (MC) is ...investigated. The steady-state transport equation for the differential number density of nonrelativistic CR protons is solved with the boundary condition that at the edge of cloud it agrees with the interstellar CR number density. It is shown that giant interstellar MCs with column depths much greater than about cm−2 are an efficient sink of nonrelativistic CRs. At small penetration depths the CRs lose energy by ionizing and heating the molecular gas, whereas at large penetration depths they are collectively dissipated by the streaming instability, which transfers one-half of the energy density of the incoming interstellar nonrelativistic CRs to Alfvénic magnetic field turbulence.
This paper presents the original anisotropy that TeV cosmic rays (CRs) have in the local interstellar medium. This anisotropy is obtained using a method of flux mapping with the Liouville theorem and ...a magnetohydrodynamic (MHD) heliosphere model of the electromagnetic field to remove the particle propagation effects hidden in the measurements made by the Tibet ASγ experiment at Earth. The original interstellar anisotropy turns out to be almost a pure dipole, which results from a diffusion flow of CRs escaping along the local interstellar magnetic field into the northern Galactic halo. The observed anisotropy maps at Earth appear quite complex because the heliosphere distorts the dipole anisotropy, generating a significant amount of high-order multipoles, while interstellar magnetic field fluctuations contribute to some weak anisotropy on small angular scales. It is found that the density gradient of these CRs points approximately toward Vela in the Local Bubble, providing experimental evidence to show that the local supernova is making a special contribution to the TeV CRs we see at Earth. This special contribution will keep growing in the future tens of thousand years. The original anisotropy also reveals that the CRs spread from the source primarily along the interstellar magnetic field, while experiencing a nearly isotropic pitch-angle scattering process caused by interstellar turbulence.
The broad-band spectral energy distributions (SEDs) of blazars exhibit two broad spectral components which in leptonic emission models are attributed to synchrotron radiation and synchrotron ...self-Compton (SSC) radiation of relativistic electrons. During high-state phases, the high-frequency SSC component dominates over the low-frequency synchrotron component implying that the inverse Compton SSC losses of electrons are at least equal or greater than the synchrotron losses of electrons. The linear synchrotron cooling, included routinely in radiation models of blazars, then has to be replaced by the SSC cooling. It is shown that the SSC energy-loss rate of electrons calculated in the Thomson limit (SST cooling) depends on an energy integral of the actual electron spectrum, reflecting the dependence of the energy density of the target synchrotron photons on the differential electron energy spectrum. The dependence of the SST loss rate on the initial kinetic energy of injected electrons is a collective effect completely different from the linear synchrotron cooling case. For the illustrative case of instantaneous injection of monoenergetic particles, we solve the non-linear kinetic equation for the intrinsic temporal evolution of the relativistic particles under SST cooling and compare the solution with the standard linear synchrotron cooling solution. For standard blazar emission region parameters, we find that under SST cooling electrons cool much more rapidly than in linear synchrotron cooling. The different cooling behaviour of electrons implies differences for the optically thin synchrotron and SSC radiation intensity and fluence energy spectra. The first difference concerns the duration of the SSC flare at different scattered photon energies. At large SSC photon energies, the SST cooled duration time is much smaller than the synchrotron cooled duration time, whereas at small SSC photon energies the SST cooled flare lasts much longer than the synchrotron cooled flare. Secondly, strong differences in the spectral behaviour of the total SSC fluence occur. At low scattered photon energies, the synchrotron cooled total SSC fluence exhibits a flatter (∝k−1/4s) power-law behaviour than the SST cooled total SSC fluence (∝k−3/4s). Thirdly, the SST cooled total synchrotron fluence exhibits a steeper (by Δα= 1) power-law behaviour FT(ε) ∝ε−3/2 than the synchrotron cooled total synchrotron fluence FS(ε) ∝ε−1/2 below the same exponential cut-off energy. These predictions of spectral behaviour with time and frequency provide conclusive tests for the presence or absence of linear synchrotron cooling or non-linear SST cooling in flaring non-thermal sources.
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