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
Fokker-Planck . . . Diffusion . . . Distributed . . . Exemplary spectra . . . Summary and . . .
Fokker-Planck . . . Diffusion . . . Distributed . . . Exemplary spectra . . . Summary and . . .
Topics:
References: Cosmic Ray Diffusion Approximation with Weak Adiabatic Focusing; Schlick- eiser, R. & Shalchi, A., 2008, ApJ 686, 292 First-order distributed Fermi acceleration of relativistic particles in nonuniform 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
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The starting point for the transport of cosmic rays in magnetic field B0(z) = B0(z) ez with superposed weak electromagnetic turbulence (δ E, δ B) is the Fokker–Planck equation for the gyrotropic particle phase space density f0(X, Y, z, p, µ, t) per unit of magnetic line length: ∂f0 ∂t + vµ∂f0 ∂z − S0(z, p, t) =
∂ ∂xi Dxixj ∂f0 ∂xj − v 2L(1 − µ2)∂f0 ∂µ (1) where xi ∈ [µ, p, X, Y ] and L−1(z) = − 1 B0 dB0 dz (2) representing the adiabatic focusing of particles for spatial variations of the guide field B0(z). L > 0 for diverging guide field, L < 0 for converging guide field. For uniform fields B0 =const. → L = ∞. We restrict our analysis to isotropic source terms S0(z, p, t). HERE: Consequences of additional adiabatic focusing term.
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For low-frequency MHD plasma turbulence: δ E << δ B so that Dµµ >> Dµp, DµX, Dpp (3) Consequently, the gyrotropic distribution function f0(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 F0(X, Y, z, p, t) per unit of magnetic line length: f0(X, Y, z, p, µ, t) = F0(X, Y, z, p, t) + g0(X, Y, z, p, µ, t) (4) where F0(X, Y, z, p, t) = 1 2 1
−1
dµ f0(X, Y, z, p, µ, t), (5) 1
−1
dµ g0(X, Y, z, p, µ, t) = 0 (6) and where anisotropy |g0| ≪ F0. Substituting Eq. (4) into Eq. (1), averaging
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2.1. Cosmic ray anisotropy
In the weak adiabatic focusing limit the cosmic ray anisotropy g0(z, p, µ, t) = 1
−1
dµ(1 − µ)(1 − µ2) Dµµ(µ) − 2 µ
−1
dx 1 − x2 Dµµ(x) v 4 ∂F0 ∂z + 1
−1
dµ(1 − µ)Dµp(µ) Dµµ(µ) − 2 µ
−1
dxDµp(x) Dµµ(x) 1 2 ∂F0 ∂p +
1
−1
dµ(1 − µ)DµXi(µ) Dµµ(µ) − 2 µ
−1
dxDµXi(x) Dµµ(x) 1 2 ∂F0 ∂Xi , (7) consists of
In terms of the isotropic cosmic ray phase space density F = F0/B(z) and g = g0/B(z) the streaming anisotropy is modified as gS(z, p, µ, t) = 1
−1
dµ(1 − µ)(1 − µ2) Dµµ(µ) − 2 µ
−1
dx 1 − x2 Dµµ(x) v 4 ∂F ∂z − F L
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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 ∂F0 ∂t − S0(z, p, t) =
∂ ∂X ∂ ∂Y ∂ ∂z
· κXX κXY −κzX κY X κY Y −κzY κzX κzY κzz
∂F0 ∂X ∂F0 ∂Y ∂F0 ∂z
+ 1 p2 ∂ ∂p
∂p
4 ∂ ∂z
∂F0 ∂p
1 4p2 ∂ ∂p
∂F0 ∂z
1 p2 ∂ ∂pp2a21 ∂F0 ∂Xi + ∂ ∂Xi a22 ∂F0 ∂p
L ∂F0 ∂z +
κzi L ∂F0 ∂Xi + v 4 a11 L ∂F0 ∂p (9) with the pitch-angle averaged transport parameters κzz = vλ 3 = v2 8 1
−1
dµ(1 − µ2)2 Dµµ(µ) , (10) κij = 1 2 1
−1
dµ[DXiXj − DXiµDµXj Dµµ(µ) ], (11)
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κzi = v 4 1
−1
dµ(1 − µ2)DXiµ Dµµ(µ) , (12) A = 1 2 1
−1
dµ
Dµµ(µ)
(13) a11 = 1
−1
dµ(1 − µ2)Dµp(µ) Dµµ(µ) , (14) a12 = 1
−1
dµ(1 − µ2)Dpµ(µ) Dµµ(µ) , (15) a21 = 1 2 1
−1
dµ
Dµµ(µ)
(16) and a22 = 1 2 1
−1
dµ
Dµµ(µ)
(17) respectively.
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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: va11 4L ∂F0 ∂p = VAH 3L p∂F0 ∂p For positive values of the product a11L > 0 it represents a continuous momen- tum loss term, whereas for negative values a11L < 0 it represents a first-order Fermi-type acceleration term. The focusing length L(z) is positive for a diverg- 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 a11 depend sensitively on the cross helicity and magnetic helicity of the magnetic field turbulence.
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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 µM = p2
⊥
2mγB(z) = pv(1 − µ2) 2B(z) = const. (18) in a slowly varying guide magnetic field L ≫ rg, with the particles’ gyroradius rg and the focusing length L−1 = −
1 B(z) dB(z) dz ;
(2) if the physical system contains magnetohydrodynamic plasma waves such as Alfven waves, whose magnetic field component is much larger than their electric field component, the quickest particle-wave interaction process is pitch-angle scattering, so that the charged particle distribution function is isotropised on a very short time scale τiso ≪ 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 < pv >0 = B(z) B(0) = exp(−z/L) (19) where in the last step we assume an exponentially varying guide magnetic field.
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If the intensities of forward (with parallel phase speed +VA) moving Alfven waves (I+) and backward (with parallel phase speed −VA) 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 VN = HVA 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 + VNt = HVAt so that according to Eq. (19) the particle momentum < pv >z (t) =< pv >0 exp
L2 t
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- tum acceleration rate ˙ p/p = −HVA/L which apart from a factor 3 agrees with the exact rate. This novel distributed 1st order Fermi acceleration process operates in all cosmic sources with HL < 0.
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Figure 1: Conditions for 1st-order distributed Fermi acceleration in diverg- ing (a) and converging (b) guide magnetic fields. In diverging magnetic fields a net negative (h = −Hc < 0) cross helicity state
regions of stronger field stength. In converging magnetic fields a net positive (Hc > 0) cross helicity state of Alfven waves also convects the average particle to regions of stronger field stength. In both cases the conservation of the pitch-angle averaged mag- netic moment of the particle requires the increase of the particle momentum.
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Radio continuum surveys of our own and external disk galaxies (for review see Sofue et al. 1986) reveal large-scale spatial variations of the galactic guide magnetic field perpendicular to the galactic plane suggesting the exponential variation B(z) = B0 exp(−z/zb) with values of zb of a few hundred parsecs. . For the exponentially diverging galactic guide magnetic field the focusing length L = zb is positive and constant, Negative values of the Alfvenic corss helicity Hc will therefore lead to distributed first-order Fermi acceleration of cosmic rays provided that the acceleration rate dominates all continous momentum loss processes of cosmic ray particles. In axisymmetric isospectral undamped slab Alfvenic turbulence with equal po- larisation states of f- and b-Alfven waves the steady-state focused diffusion transport equation for F0 = FB(z) is κzz ∂2F ∂z2 − κzz L + VAH ∂F ∂z + 1 p2 ∂ ∂p
∂F ∂p + HVA 3L p3F + p2 ˙ plossF
Tc = −S(z, p) (21)
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3.1. Rigidity restriction
The transport terms involving the focusing length L result from the conventional guiding center equations of motion (Northrop 1963) which are valid for focusing length L ≫ rg much larger than the gyroradii rg of cosmic ray particles which limits the cosmic ray particle rigidity to values p
Z ≪ eBL
c = 1.1 · 1018(B/4 µG)(L/300 pc) eV c (22) Note: for Fe-ions (Z = 28) rigidity upper limit corresponds to total energy limit Etot,Fe ≪ 3.0 · 1019(B/4 µG)(L/300 pc) eV (23) Applying the diffusion approximation to cosmic rays at energies above 1016 eV is justified by the small measured anisotropies at these energies (Antoni et al. 2004, H¨
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3.2. Conditions for distributed first-order Fermi acceleration
For negative cross helicity values 0 > H = −h, h > 0, the term L = 1 p2 ∂ ∂p HVA 3L p3F
p2 ∂ ∂p hVA 3L p3F
in the transport equation (21) describes first-order Fermi acceleration with the acceleration rate ˙ pacc = a1p, a1 = hVA 3L = 3 · 10−14 b4h L300n1/2
−3
s−1 (25) provided that this rate is larger than the particle’s loss rate ˙ pacc > ˙ ploss (26) 3.2.1. Hadrons For hadrons a1 > bπngas is always larger than the pion production loss rate. The Coulomb and ionisation losses then define a threshold momentum value pc, given by a1 = bIngas/p3
c so that
pc = Z2bIngas a1 1/3 = 0.17Z2/3 ngasL300 b4h 1/3 n1/6
−3
GeV c (27)
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whose value depends only weakly on the assumed interstellar gas parameters. All hadrons with momenta above pc will undergo this first-order acceleration process up to momenta determined by condition (22) which covers about 10
3.2.2. Electrons For electrons the acceleration condition (26) can only be fulfilled in the narrow momentum range (ATIC and PAMELA excess!) pc1 ≤ p ≤ pc2 (28) with pc1 ≃ δ1ngas a1 = 10−2 ngasL300n1/2
−3
b4h GeV/c (29) and pc2 ≃ a1 δ3ngas = 300 b4h ngasL300n1/2
−3
GeV/c (30) Obviously, we have identified a distributed first-order Fermi accel- eration process that preferentially accelerates relativistic hadrons
are accelerated over 4 orders of magnitude in momentum values from 0.01 to 300 GeV/c.
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4.1. Neglect of constant convection speed and momentum diffusion
Introduce the dimensionless momentum variable x =
p mpc = pt Ampc.
With nominal interstellar parameter values the focused convection speed (α = A/Z) κ0 L αηxη = 5.7 · 105 (αx)η (1 − σ2)L300 cm/s (31) is larger than the constant convection speed hVA = 2.75 · 107h cm/s for x > 1.1 · 105[hL300(1 − σ2)]3/α, (32) where we adopt a value of η = 1/3. For momenta p > 1.1 · 105Ampc/α ≃ 105/α GeV/c, we may neglect the constant convection speed as compared to the focused convection speed. For a degenerate cross helicity value h = −H = 1, which provides the maximum focused acceleration rate (25), momentum diffusion does not occur (AD = 0). The transport equation (21) then reduces to κ0 ∂ ∂z ∂F ∂z − F L
a1 αηx2+η ∂ ∂x
F αηxηTc = −S1(z)S2(p) αηxη (33)
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4.2. UHE hadrons
We assume that cosmic ray point sources in the Galaxy inject hadrons with the power law distribution S2(x) = S2x−k with the same spectral index k for xL ≤ x ≤ xM up to some maximum momentum pmax = mpcxM of the order
higher momenta x > xM at the position of the solar system (z = 0) then is F(z = 0, x > xM) =
∞
WnHn(x) (34) with weighting constants Wn and the momentum-dependent function Hn(x) = x−(3+ψ) exp
xc(n) η , (35) with ψ(A > 1) = 1 a1Tc = 0.167A0.7 L300ngasn1/2
−3
b4h , (36) and the characteristic dimensionless momentum xc(n) = η Λn 1/η (37) The lowest eigenfunction H1(x) extends to the highest momenta.
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In terms of the characteristic momentum (for η = 1/3) xc = xc(1) = (3Λ1)−3 = 104 α (38) we obtain H1(x) = W1x−s exp
x xc η , (39) which is a power-law with a slowly decreasing exponential function. The power law spectral index (Θ denotes the spep function) s = 3 + ψ = 3 + A0.7 6 Θ[A − 1] (40) depends on the cosmic ray species. Because cosmic ray protons undergo no fragmentation or spallation losses ψ = 0, their power law index sprotons = 3 is the smallest. Heavier cosmic ray hadrons have a larger index s(A) = 3 + A0.7 6 , (41) because the spallation and fragmentation time scale decreases with mass num-
number A. In Table 1 we calculate the phase space density spectral index values for cosmic ray hadrons of different mass number.
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Table 1: Phase space density spectral index s for different cosmic ray hadrons for η = 1/3 Particle mass number A spectral index s(A) Protons 1 3.0 α-particles 4 3.44 C 12 3.95 O 16 4.16 Fe 52 5.65 In Fig. 2 we show the differential number density momentum spectra N(x) = 4πx2H1(x), normalized at xM, for η = 1/3 for protons and ions: N(x) N(xM) = x xM 2−s(A) exp
x − xM xc 1/3 (42)
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Figure 2: Normalized differential number density momentum spectrum of cosmic ray protons (blue full curve), cosmic ray Fe (dashed green curve) from distributed focused acceleration in comparison with p−3-power law (dot-dashed red curve). It is remarkable how well the slowly decreasing exponential variation matches a p−3-power law spectrum over a large momentum range.
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tra from a p−3-power law spectrum for protons, α-particles and iron, illustrating again the good match of the slowly decreasing exponential variation. Figure 3: Deviations from a p−3-power law of the normalized differential number density momentum spectrum of cosmic ray protons (blue full curve), cosmic ray Helium (dashed green curve) and cosmic ray Fe (dot-dashed red curve).
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4.3. Power law solutions for η = 0
The resulting solutions are extremely sensitive to the value of η = 2 − q, describing the momentum dependence of the spatial diffusion coefficient. q is the spectral index of the power spectrum of magnetic field fluctuations. At large wavenumbers (corresponding to small spatial turbulence scales), the Kolmogorov spectral index (q = 1/3) is well established. However, at large spatial turbulence scales q is very uncertain. For q = 2, so that η = 0, the transport equation (33) becomes κ0 ∂ ∂z ∂F ∂z − F L
x2 ∂ ∂x
Tc = −S1(z)S2(p), (43) providing for the hadron momentum spectrum at the position of the solar system F(z = 0, x > xM) =
∞
WnGn(x) (44) with weighting constants Wn and the power-law Gn(x) = x−(3+ψ+Λn) x dx0 x2+ψ+Λn) S2(x0) (45) Assuming again that cosmic ray point sources in the Galaxy inject hadrons with the power law distribution S2(x) = S2x−k with the same spectral index k for xL ≤ x ≤ xM up to some maximum momentum pmax = mpcxM we find
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Gn(xL < x ≤ xM) = S2x−k 3 + ψ + Λn − k
(46) and Gn(x > xM) = S2x−(3+ψ+Λn) 3 + ψ + Λn − k
M
− x3+ψ+Λn−k
L
In order to match the observed spectral index sp = 4.8 of cosmic ray protons a value of Λ1 = 1.8 is required. For higher-metallicity cosmic rays (A > 1) the equilibrium spectral indices are automatically steeper s(A) = sp + ψ(A) = 4.8 + A0.7 6 (48) due to the influence of the additional spallation and fragmentation losses (see Table 2).
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Table 2: Phase space density spectral index s for different cosmic ray hadrons for η = 0 Particle mass number A spectral index s(A) Protons 1 4.8 α-particles 4 5.24 C 12 5.75 O 16 5.96 Fe 52 7.45
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limit gives rise to three new convective terms (parallel to the guide field, perpendicular to the guide field and in momentum space) in the diffusion- convection transport equation of cosmic rays.
vection term represents a continuous momentum loss term, whereas for negative values a11L < 0 it represents a first-order Fermi-type acceler- ation term. The focusing length L(z) is positive for a diverging guide magnetic field and negative for a converging guide field. The absolute value and the sign of the deceleration rate a11 depend sensitively on the cross helicity Hc and magnetic helicity of the magnetic field turbulence.
where the product HcL < 0 is negative.
cosmic ray electrons which have larger continous momentum loss pro-
momentum spectra with slowly decreasing exponentials result that closely match a simple p−3-power law spectrum. For q = 2 power law spectra re-