Ultrahighenergy galactic cosmic rays from distributed focused - PowerPoint PPT Presentation

Ultrahighenergy galactic cosmic rays from distributed focused acceleration R. Schlickeiser, S. Artmann, C. Z¨ oller Fokker-Planck . . . Institut f¨ ur Theoretische Physik Diffusion . . . Lehrstuhl IV: Weltraum- und Astrophysik Distributed . . . Ruhr-Universit¨ at Bochum, Germany Exemplary spectra . . . Summary and . . . 17 September 2010

Topics : 1. Fokker-Planck transport equation 2. Diffusion approximation 3. Distributed focused acceleration in the galaxy 4. Exemplary spectra of accelerated UHE hadrons 5. Summary and conclusions References: Fokker-Planck . . . Diffusion . . . Cosmic Ray Diffusion Approximation with Weak Adiabatic Focusing; Schlick- Distributed . . . eiser, R. & Shalchi, A., 2008, ApJ 686, 292 Exemplary spectra . . . First-order distributed Fermi acceleration of relativistic particles in nonuniform Summary and . . . magnetic fields with nonvanishing Alfvenic cross helicity turbulence; Schlick- eiser, R., 2009, Modern Phys. Lett. A 24, 1461 Cosmic ray transport in non-uniform magnetic fields: Consequences of gradient and curvature drifts; R. Schlickeiser & F. Jenko, 2010, J. Plasma Phys. 76, 317

1. Fokker-Planck transport equation The starting point for the transport of cosmic rays in magnetic field � B 0 ( z ) = e z with superposed weak electromagnetic turbulence ( δ � E, δ � B 0 ( z ) � B ) is the Fokker–Planck equation for the gyrotropic particle phase space density f 0 ( X, Y, z, p, µ, t ) per unit of magnetic line length : ∂f 0 ∂t + vµ∂f 0 ∂ ∂f 0 − v 2 L (1 − µ 2 ) ∂f 0 � ∂z − S 0 ( z, p, t ) = D x i x j (1) ∂x i ∂x j ∂µ i,j where x i ∈ [ µ, p, X, Y ] and Fokker-Planck . . . L − 1 ( z ) = − 1 dB 0 Diffusion . . . (2) B 0 dz Distributed . . . representing the adiabatic focusing of particles for spatial variations of the guide Exemplary spectra . . . field B 0 ( z ) . L > 0 for diverging guide field, L < 0 for converging guide field. Summary and . . . For uniform fields B 0 = const. → L = ∞ . We restrict our analysis to isotropic source terms S 0 ( z, p, t ) . HERE: Consequences of additional adiabatic focusing term.

2. Diffusion approximation For low-frequency MHD plasma turbulence: δ � E << δ � B so that D µµ >> D µp , D µX , D pp (3) Consequently, the gyrotropic distribution function f 0 ( X, Y, z, p, µ, t ) due to the dominating pitch-angle diffusion adjusts very quickly to a quasi-equilibrium through pitch-angle diffusion which is close to the isotropic equilibrium dis- tribution F 0 ( X, Y, z, p, t ) per unit of magnetic line length: f 0 ( X, Y, z, p, µ, t ) = F 0 ( X, Y, z, p, t ) + g 0 ( X, Y, z, p, µ, t ) (4) Fokker-Planck . . . where Diffusion . . . � 1 F 0 ( X, Y, z, p, t ) = 1 Distributed . . . dµ f 0 ( X, Y, z, p, µ, t ) , (5) Exemplary spectra . . . 2 − 1 Summary and . . . � 1 dµ g 0 ( X, Y, z, p, µ, t ) = 0 (6) − 1 and where anisotropy | g 0 | ≪ F 0 . Substituting Eq. (4) into Eq. (1), averaging over µ and using | g 0 | ≪ F 0 yields

2.1. Cosmic ray anisotropy In the weak adiabatic focusing limit the cosmic ray anisotropy �� 1 � µ dµ (1 − µ )(1 − µ 2 ) dx 1 − x 2 � v ∂F 0 g 0 ( z, p, µ, t ) = − 2 D µµ ( µ ) D µµ ( x ) 4 ∂z − 1 − 1 �� 1 � µ dµ (1 − µ ) D µp ( µ ) dxD µp ( x ) � 1 ∂F 0 + − 2 D µµ ( µ ) D µµ ( x ) 2 ∂p − 1 − 1 �� 1 � µ dµ (1 − µ ) D µX i ( µ ) dxD µX i ( x ) � 1 ∂F 0 � + − 2 , (7) D µµ ( µ ) D µµ ( x ) 2 ∂X i − 1 − 1 i =1 , 2 consists of Fokker-Planck . . . Diffusion . . . • the streaming anisotropy ( ∝ ∂F 0 /∂z ), Distributed . . . • the Compton-Getting anisotropy ( ∝ ∂F 0 /∂p ). Exemplary spectra . . . Summary and . . . • the perpendicular anisotropy ( ∝ ∂F 0 /∂X i ). In terms of the isotropic cosmic ray phase space density F = F 0 /B ( z ) and g = g 0 /B ( z ) the streaming anisotropy is modified as �� 1 � µ dµ (1 − µ )(1 − µ 2 ) dx 1 − x 2 � v � ∂F ∂z − F � g S ( z, p, µ, t ) = − 2 D µµ ( µ ) D µµ ( x ) 4 L − 1 − 1 (8)

2.2. Diffusion-convection transport equation Diffusion-convection equation for the isotropic part of the cosmic ray phase space distribution per unit of magnetic line length in the weak ( | L | >> λ ) adiabatic focusing limit is ∂F 0 ∂t − S 0 ( z, p, t ) = ∂ ∂F 0       κ XX κ XY − κ zX ∂X ∂X 1 ∂ � p 2 A∂F 0 �  ·  + ∂ ∂F 0 κ Y X κ Y Y − κ zY     ∂Y ∂Y p 2 ∂p ∂p ∂ ∂F 0 κ zX κ zY κ zz ∂z ∂z � 1 + v ∂ � ∂F 0 � 1 ∂ � ∂F 0 � ∂ ∂F 0 ∂ ∂F 0 � Fokker-Planck . . . p 2 va 12 � ∂pp 2 a 21 a 11 − + + a 22 4 p 2 p 2 4 ∂z ∂p ∂p ∂z ∂X i ∂X i ∂p Diffusion . . . i =1 , 2 Distributed . . . + κ zz ∂F 0 κ zi ∂F 0 + v a 11 ∂F 0 � ∂z + (9) Exemplary spectra . . . L L ∂X i 4 L ∂p i =1 , 2 Summary and . . . with the pitch-angle averaged transport parameters � 1 3 = v 2 dµ (1 − µ 2 ) 2 κ zz = vλ D µµ ( µ ) , (10) 8 − 1 � 1 dµ [ D X i X j − D X i µ D µX j κ ij = 1 ] , (11) 2 D µµ ( µ ) − 1

� 1 dµ (1 − µ 2 ) D X i µ κ zi = v , (12) 4 D µµ ( µ ) − 1 � 1 A = 1 D pp ( µ ) − D µp ( µ ) D pµ ( µ ) � � dµ , (13) 2 D µµ ( µ ) − 1 � 1 dµ (1 − µ 2 ) D µp ( µ ) a 11 = , (14) D µµ ( µ ) − 1 � 1 dµ (1 − µ 2 ) D pµ ( µ ) a 12 = , (15) D µµ ( µ ) − 1 Fokker-Planck . . . � 1 a 21 = 1 D pX i ( µ ) − D pµ ( µ ) D µX i � � Diffusion . . . dµ , (16) 2 D µµ ( µ ) − 1 Distributed . . . and Exemplary spectra . . . Summary and . . . � 1 a 22 = 1 D X i p ( µ ) − D µp ( µ ) D X i µ � � dµ , (17) 2 D µµ ( µ ) − 1 respectively.

2.3. Focused acceleration due to weak adiabatic focusing Adiabatic focusing gives rise to the last three terms in Eq. (9) that represent convective transport terms parallel to the guide field, perpendicular to the guide field and in momentum space, respectively. In the limit L → ∞ of negligible adiabatic focusing these three new terms vanish. The convective term along the guide field has been derived before by Earl (1976) and Kunstmann (1979); the other two are new. The respective convec- tive speeds depend on the ratio of the corresponding diffusion coefficients or adiabatic deceleration rate to the focusing length. Particularly interesting is the new convection term in momentum space: va 11 ∂F 0 ∂p = V A H 3 L p∂F 0 Fokker-Planck . . . 4 L ∂p Diffusion . . . For positive values of the product a 11 L > 0 it represents a continuous momen- Distributed . . . tum loss term, whereas for negative values a 11 L < 0 it represents a first-order Exemplary spectra . . . Fermi-type acceleration term. The focusing length L ( z ) is positive for a diverg- Summary and . . . ing guide magnetic field (see Eq. (2)) and negative for a converging guide field. On the other hand, the absolute value and the sign of the deceleration rate a 11 depend sensitively on the cross helicity and magnetic helicity of the magnetic field turbulence.

2.4. Physics explanation of the new 1st-order Fermi acceleration New 1st-order Fermi acceleration mechanism is closely related to two effects: (1) within the adiabatic guiding-center approximation of the transport of charged particles (e. g. Boyd and Sanderson 1969, Rossi and Olbert 1970), the mag- netic moment of charged particles is an adiabatic invariant 2 mγB ( z ) = pv (1 − µ 2 ) p 2 ⊥ µ M = = const. (18) 2 B ( z ) in a slowly varying guide magnetic field L ≫ r g , with the particles’ gyroradius r g and the focusing length L − 1 = − dB ( z ) 1 dz ; B ( z ) Fokker-Planck . . . (2) if the physical system contains magnetohydrodynamic plasma waves such as Diffusion . . . Alfven waves, whose magnetic field component is much larger than their electric Distributed . . . field component, the quickest particle-wave interaction process is pitch-angle Exemplary spectra . . . scattering, so that the charged particle distribution function is isotropised on a Summary and . . . very short time scale τ iso ≪ L/v . Averaging the magnetic moment (18) with respect to the cosine of pitch-angle µ then yields for the respective quantities at the two positions z = 0 and z < pv > z = B ( z ) B (0) = exp( − z/L ) (19) < pv > 0 where in the last step we assume an exponentially varying guide magnetic field.

If the intensities of forward (with parallel phase speed + V A ) moving Alfven waves ( I + ) and backward (with parallel phase speed − V A ) moving Alfven waves ( I − ) is different, the resulting net cross helicity of Alfven waves H = ( I + − I − ) / ( I + + I − ) results in a net convection speed V N = HV A of charged particles as each Alfvenic wave mode isotropises the particles in its rest frame. As a consequence, the average particle position convects as z = 0 + V N t = HV A t so that according to Eq. (19) the particle momentum � � − HLV A < pv > z ( t ) = < pv > 0 exp t (20) L 2 increases exponentially with time if HL < 0 or decreases exponentially with time if HL < 0 . For relativistic particles ( v ≃ c ) Eq. (20) implies the momen- Fokker-Planck . . . tum acceleration rate ˙ p/p = − HV A /L which apart from a factor 3 agrees with Diffusion . . . the exact rate. Distributed . . . This novel distributed 1st order Fermi acceleration process operates in all cosmic Exemplary spectra . . . sources with HL < 0 . Summary and . . .