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Interstellar Pickup Ions and Injection Problem for Anomalous Cosmic Rays: Theoretical Aspect

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Interstellar Pickup Ions and Injection Problem for Anomalous Cosmic Rays: Theoretical Aspect — 8 — Interstellar Pickup Ions and Injection Problem for Anomalous Cosmic Rays: Theoretical Aspect Sergey Chalov1 Institute for Problems in Mechanics Moscow, Russia Abstract. The physical properties of pickup ions originating in the so- lar wind due to ionization of interstellar atoms penetrating the helio- sphere are considered here. The equation for the anisotropic velocity distribution function of the ions is derived and some particular solutions illustrating their transport in the heliosphere are presented. Special at- tention is concentrated on stochastic acceleration of pickup ions in the solar wind and shock-drift acceleration at the termination shock. The injection problem for anomalous cosmic rays at the termination shock is discussed. 8.1 Introduction The local interstellar medium (LISM) surrounding the Solar System is a partly ionized medium consisting mainly of neutral components. It has become evident in recent years that the interstellar atoms have a pronounced effect on the global structure of the extensive region separating the supersonic solar wind and LISM and on the physical processes operating in the heliosphere. Apart from the fact that the position and shape of the heliospheric termination shock and heliopause are significantly determined by the action of the atoms, they give rise to a peculiar hot population of pickup ions in the solar wind. Due to large mean free paths, which are comparable with the dimensions of the heliosphere, interstellar atoms can penetrate close to the Sun. In the heliosphere some portion of the atoms is ionized through charge exchange with protons or through the processes of photoionization and electron impact ionization. According to current knowledge, the speed of the interstellar wind relative to the Sun is about 26 km/s. The speeds of interstellar atoms in the heliosphere can deviate from this value due to the thermal velocity spread, due to acceleration by solar gravity (near the Sun), or due to the charge exchange process in the disturbed interstellar 1in The Physics of the Heliospheric Boundaries, V. Izmodenov and R. Kallenbach (eds.), ISSI Scientific Report No. 5, pp. 245 - 282, ESA-ESTEC, Paris 2006 246 8. Pickup Ions and Anomalous Cosmic Rays medium between the heliopause and bow shock. But in any case, this speed is much smaller compared with the solar wind speed ranging from 350 km/s up to 750 km/s depending on heliolatitude or solar activity. Therefore, we can conclude that the speed of newly created ions in the solar wind frame approximately equals the local solar wind speed VSW . The ions are then picked up by the solar wind electric and magnetic fields, so that they have a ring velocity distribution in the solar wind frame: the pickup ions gyrate around magnetic field lines and stream along the lines with a speed VSW | cos ξ|, where ξ is the initial pitch angle. The angle ξ is determined by the local configuration of the interplanetary magnetic field B: cos ξ = −B · VSW/ (BVSW ). Thus the initial streaming velocity of newly created pickup ions is parallel or antiparallel to B depending on the polarity of the solar magnetic field, but in any case it is directed towards the Sun in the solar wind rest frame. This ring velocity distribution of pickup ions is highly unstable, as was shown by Wu and Davidson (1972), and for rather short times (as compared with the solar wind convection time) evolves to a nearly isotropic distribution, while free kinetic energy of the initial distribution is realized in the form of Alfv´enwaves. Direct observations of pickup helium (M¨obiuset al., 1985) and pickup hydrogen (Gloeckler et al., 1993) made it apparent that the velocity distributions of pickup ions differ in significant ways from the velocity distributions of primary solar wind ions. In consequence of their specific features the pickup ions have been proposed by Fisk et al. (1974) as a seed population for anomalous cosmic rays (ACRs). Originally, it was believed that stochastic acceleration by solar wind turbulence is the principal acceleration mechanism producing ACRs (e.g., Fisk, 1976). Later Pesses et al. (1981) proposed an alternative model in which the anomalous com- ponents are originated at the solar wind termination shock through the diffusive acceleration process. At present, this idea of ACR origin is commonly accepted. 8.2 Anisotropic transport of pickup ions in the so- lar wind The general kinetic equation for the velocity distribution function of charged particles f (t, x, v0) can be written as ∂f ∂f F0 ∂f + v0 · + · = Sˆf + Q. (8.1) ∂t ∂x m ∂v0 In Equation (8.1) v0 and x are the velocity vector and vector in configuration space in the inertial frame, m is the mass of a particle, F0 is the external large- scale electromagnetic force, Sfˆ is the scattering operator applied to the function f describing the effect of stochastic wave-particle interactions, and Q is the source term. 8.2.1 Kinetic equation for the velocity distribution function As regards pickup ions, we would point to the next two facts. The gyrora- dius and gyroperiod of pickup ions are generally much smaller than other spatial and time scales, respectively. Thus their velocity distribution can be considered 8.2. Anisotropic transport of pickup ions in the solar wind 247 as gyrotropic. Furthermore, the velocity distribution is highly anisotropic in the inertial frame since the thermal velocities of pickup ions are comparable with the solar wind speed, while in the solar wind rest frame the anisotropy is much smaller. Therefore it is more convenient to consider the distribution function in the moving frame. According to Skilling (1971) and Isenberg (1997) the transport equation de- scribing the phase-space evolution of the gyrotropic velocity distribution function f (t, x, v, µ) of pickup ions in a background plasma moving at a velocity U (t, x) can be written in the following form (see Appendix A): ∂f ∂f 1 − 3µ2 ∂U 1 − µ2 ∂U + (U + vµb ) + b b j − i − ∂t i i ∂x 2 i j ∂x 2 ∂x i i i µb ∂U ∂U ∂f 1 − µ2 ∂b ∂U − i i + U i v + v i + µ i − v ∂t j ∂x ∂v 2 ∂x ∂x j i i ∂U 2b ∂U ∂U ∂f −3µb b j − i i + U i = Sˆf + Q (t, x, v, µ) , (8.2) i j ∂x v ∂t j ∂x ∂µ i j where v and µ = cos ξ are the speed and cosine of the particle pitch angle ξ in the solar wind rest frame, b is the unit vector of the large-scale magnetic field. The general form of the scattering operator is (see, e.g., Schlickeiser, 1989) ∂ ∂f ∂ ∂f 1 ∂ ∂f Sfˆ = D + D + v2D ∂µ µµ ∂µ ∂µ µv ∂v v2 ∂v µv ∂µ 1 ∂ ∂f + v2D . (8.3) v2 ∂v vv ∂v Operator Sˆ describes the pitch-angle scattering and energy diffusion of charged particles. The Fokker-Planck diffusion coefficients Dµµ, Dµv, and Dvv in Equa- tion (8.3) can be calculated in the frame of the quasi-linear theory of resonant wave-particle interactions. We will consider further a spherically symmetric solar wind with the spatially independent radial speed VSW . The large-scale interplanetary magnetic field is generally well described by the Parker model (Parker, 1963): 2 2 rE rEΩ Br = BrE , Bϑ = 0 ,Bϕ = −BrE sin ϑ , (8.4) r rVSW where the radial component of the magnetic field BrE does not depend on helio- latitude [see, e.g., Ulysses data in Balogh et al., 1995; Smith and Balogh, 1995), −6 −1 Ω = 2.7 × 10 s , rE = 1 AU, and ϑ is the polar angle (colatitude) in the helio- centric spherical coordinate system (r, ϑ, ϕ). If we introduce the variable χ = cos ψ, where ψ(r, ϑ) is the angle between the vectors VSW and B, we obtain: 2 1/2 br = χ, bϕ = −sign(χ) 1 − χ . (8.5) Then in the case of the outward pointing magnetic field −1/2 2 2 χ (r, ϑ) = 1 + (Ωr/VSW) sin ϑ . (8.6) h i 248 8. Pickup Ions and Anomalous Cosmic Rays The general transport equation (8.2) then becomes 1/2 ∂f ∂f vµ 1 − χ2 ∂f + (VSW + vµχ) − + ∂t ∂r r sin θ ∂ϕ 1 − 3µ2 1 − χ2 1 − µ2 ∂f + − V v + 2 r r SW ∂v 1 − µ2 v d 2V 1 − χ2 ∂f + r2χ + SW µ − 3µV = 2 r2 dr r SW r ∂µ = Sˆf + Q (t, r, ϑ, ϕ, v, µ) . (8.7) The second and third terms on the left-hand side of Equation (8.7) describe the convective motion of particles with the solar wind and their streaming along the magnetic field lines due to anisotropy of the pitch-angle distribution. The forth term corresponds to adiabatic cooling in the expanding solar wind, and the last term to adiabatic focusing in the nonuniform magnetic field. 8.2.2 Fokker-Planck diffusion coefficients In collisionless plasma the interaction of charged particles with electromagnetic waves is determined by the resonance condition ω − kkvk = nΩg , (8.8) where ω is the wave frequency, kk is the projection of the wave number vector k on B, vk is the velocity of a particle along the magnetic field, and Ωg is the gy- rofrequency. In the frame of the quasi-linear theory, the Fokker-Planck diffusion coefficients are derived by expanding the collisionless Vlasov equation for the phase space density of particles to second order in perturbated quantities and ensemble averaging over the statistical properties of the electromagnetic waves [e.g., Kennel and Engelmann, 1996). Due to their particular importance in space plasma we will consider here only Alfv´enwaves propagating along the magnetic field (the slab com- ponent of solar wind turbulence).
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