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Pre-existing Turbulence, Magnetic Fields and Particle Acceleration at a Supernova Blast Wave J. R. Jokipii University of Arizona Presented at the meeting: Kinetic Modeling of Astrophysical Plasmas, Krakow, October 8, 2008

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Outline of talk Supernova blast waves and cosmic rays. Observational evidence for enhanced magnetic field behind the shock. What enhances the magnetic field? Bell instability? Enhanced B from pre-existing turbulence. Acceleration to the “knee” in the CR spectrum: results from a new global model which explicitly includes the magnetic-field angle show encouraging agreement with observations.

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Galactic cosmic-rays and SNR’s The spectrum, up to the “knee” at ~ 3x10 15 eV, is probably the result of diffusive shock acceleration at supernovae blast waves. Lagage and Cesarsky (1983) estimated the maximum energy to be less than 10 14 eV/charge –They assumed Bohm diffusion, a nominal IS magnetic field and a planar parallel shock (I will show later that the geometry is very important). However, it has been shown long ago that a higher maximum energy is attained for a larger upstream magnetic field or a quasi- perpendicular shock (or both). Consider, first, the magnetic field magnitude.

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Berezhko et al. (2004) compared X-ray observations of a supernova remnant with a model of shock acceleration of electrons (E e ~ 100 TeV) including synchrotron losses and concluded that the observations could only result if the remnant magnetic field were very strong (B > 100μG) SN 1006: Bamba et al, 2003 Berezhko et al., 2004 This analysis only constrains B behind the shock! Observational constraints on the magnetic-field magnitude

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What enhances B in the remnant? Bell and Lucek (2001) proposed that a cosmic-ray current upstream of the shock produces a J cr xB force on the plasma which drives an instability which then results in a large magnetic-field amplification. Berezhko et al., 2004 used this mechanism to explain their large field stating that “There is no alternative process, without ad hoc-assumptions, in the literature, or a new one, which we could reasonably imagine, that would amplify the MF in a collisionless shock without particle acceleration”. Nonetheless, Joe Giacalone and I have found such an alternative process – the interaction of pre- existing, large-scale turbulence with the shock.

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In situ spacecraft observations of energetic protons at a propagating shock. (from Scholer, JGR, 88, 1977, 1983) Consider in situ observations In the solar wind. Although energetic particles are often associated with shocks, The observations nearly always have anomalies. Single-spacecraft observations have seen these anomalies for decades. But they were not well understood. We have suggested that they are the result of fluctuations or turbulence. Multiple in situ observations of the same shock have also shown this. Motivation for considering upstream turbulence

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Neugebauer and Giacalone (2006), using multiple, nearby spacecraft, studied the energetic particles associated with propagating shock waves in the solar wind. They found that for the same shock, the characteristics of the energetic particles varied with distance along the shock face. The coherence scale of these random variations was the same as that of ambient, solar-wind turbulence! They concluded that the propagation of the shock through the pre-existing turbulence produced large variations in the shock, which affect the acceleration of particles.

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IGPP 2007, Honolulu Hybrid Simulations of a perpendicular shock moving through a turbulent magnetized plasma Giacalone, 2005 Magnetic field Density of Energetic Particles

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This is similar to what seen in images (from Berezhko and Volk 2005)

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The interplanetary density turbulence spectrum (Goldstein and Sicsoe, 1972). Note: (± n/n) 2 is of order unity. This is a advected spectrum, f $ wavenumber The interstellar medium is also turbulent! Again, the turbulence is of large amplitude: (± n/n) 2 is of order unity.

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The interplanetary density turbulence spectrum (Goldstein and Sicsoe, 1972). Note: (± n/n) 2 is of order unity. This is a advected spectrum, f $ wavenumber The interstellar medium is also turbulent! Again, the turbulence is of large amplitude: (± n/n) 2 is of order unity.

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Tycho supernova remnant in X-Rays. We Suggest that this Large-Scale pre-existing interstellar turbulence can account for many observations at a SN shock Large scale, upstream interstellar turbulence causes temporal and spatial fluctuations in the shock. and in the downstream flow. These are probably present at all interstellar shocks. We have addressed a number of the phenomena, and in particular, the magnetic field, quantitatively (Giacalone & Jokipii, Ap. J., 663, L41, 2007). We have found large magnetic-field amplification for strong shocks caused by the upstream turbulence, not cosmic rays. (See also Sironi and Goodman, 2007)

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The figure shows the density in a numerical simulation of a shock wave moving into a turbulent MHD plasma Approach: –We solve the MHD equations for a fluid reflected off of a rigid wall –Shock moves from right to left –The upstream medium contains turbulent density fluctuations We impose log-normal statistics, and a Kolmogorov spectrum The fluctuations are continually injected at the upstream boundary. We are currently working on the more-difficult 3-D version. 0 2L c 4L c 6L c 12L c 8L c 4L c 0 Density We used 2-D MHD simulations of a strong shock moving through pre-existing turbulence )

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Results for a very high Alfven Mach number.

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The enhanced B downstream of a shock moving through density turbulence (without cosmic-ray excited waves!) is due primarily to the induced downstream vorticity and is quite robust. Interpretation:

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The growth time of the fluctuations in the downstream flow may be estimated to be ¼ U 1 /L c If r sh is the ratio of densities or flow velocities at the shock, we have for the spatial scale of magnetic-field amplification: ¼ U1/r sh ¼ L c /(r s h U s } For typical parameters, this gives ¼ parsec. Note that Ellison and Blondin (2001) assume r > 4 (due to efficient particle acceleration). If this is the case, the distance above may be shorter.

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Particle Acceleration Our turbulent amplification does not increase B upstream of the shock, as does the Bell mechanism. Hence, since particle acceleration depends primarily on the upstream field, we cannot reach the knee at a quasi-parallel or Bohm shock. However, it is well-established that a planar perpendicular shock can get to the knee using the approach of Lagage and Cesarsky.

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The magnetic-field shock-normal angle A SNR blast waves moves into a B with a preferred direction –The angle between B and shock normal varies from 90 deg to 0 deg. The physics of acceleration at parallel and perpendicular shocks is different Parallel shocks slow Perpendicular shocks fast (K ┴ < K ║ ) for a given time interval or size of shock, a perpendicular shock will yield a larger maximum energy than a parallel shock.

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Parallel shock Perpendicular shock The acceleration time depends on and ? << k The rate of change of the maximum energy is much larger for quasi-perpendicular shocks. Hence, for any given situation, a perpendicular shock will yield a larger maximum energy than a parallel shock. So, reaching the knee is not a problem. But the shock is not planar. We should consider a spherical shock, with a varying magnetic-field angle. For a modified Planar Sedov Blast wave, we have the maximum energies:

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B ? or

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Relation to the Observed Morphology of Supernova Remnants Rothenflug, et al, in 2004, pointed out that observations rule out a “barrel” shape and suggested that that consequently ruled out acceleration at the perpendicular shock. However, it is by no means clear that acceleration at the perpendicular shock leads to a barrel shape. Need global modeling. We have looked into this problem (Giacalone, Jokipii, Kota, Bobik, to be submitted after this meeting.)

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The Parker Transport Equation: ) Diffusion ) Convection w. plasma ) Grad & Curvature Drift ) Energy change – shock accel ) Source Solve Parker’s equation in this situation including the shock, with a variable shock-normal angle, This gives diffusive shock acceleration in this geometry

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We have solved Parker’s transport equation for a spherical, modified Sedov blast wave propagating in a uniform magnetic field. We used two completely independent and different codes—finite-different ADI and stochastic integration and found the same results. For reasons of computational economy, we started the particles at the shock at an energy of 3 x 10 12 eV, with injection rate constant with shock radius and angle. Reducing the injection where the shock was quasi-perpendicular did not change the results qualitatively. We considered various values of the ratio of perpendicular to parallel diffusion ranging from.01 to 1, with parallel mean free paths at the highest energy in the range from 2 to 10, in a nominal 5 x 10 -6 gauss interstellar magnetic field. We considered a range of downstream magnetic field values to account for possible downstream magnetic-field amplification. The following results are quite robust. We find that acceleration to the knee is possible. Moreover, and unexpectedly, the accelerated particles, although being accelerated at the equator, migrate to the polar regions, where the shock is quasi-parallel.

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Our interpretation of the simulations is that the accelerated particles tend to stay on the original field line, which stays behind as the shock moves out. The observed radiation comes from these lower-energy particles. Most of the results presented here are from the stochastic integration code, which has a wider dynamical range and allows particles to be tracked

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Histogram of the starting shock-normal angle for two final energies.

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Energy spectrum summed over all particles.

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Conclusions Large-scale, broadband upstream turbulence plays a significant role in astrophysical shocks and the transport/acceleration of energetic particles. Pre-existing fluid density fluctuations affect significantly the downstream magnetic field. In a very strong shock, large field amplification may occur. This may help explain observations of magnetic-field enhancements at supernova blast waves. This is a robust result. Acceleration of cosmic rays at perpendicular shocks is a possibility. This would enable acceleration to the knee without upstream magnetic field amplification. The Rothenflug observational constraints can be satisfied, with significant acceleration at the perpendicular shock. This is a possible alternative scenario to the Bell upstream magnetic-field amplification scenario.

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