International Workshop

Slides:

Advertisements

Similar presentations

THREE-DIMENSIONAL ANISOTROPIC TRANSPORT OF SOLAR ENERGETIC PARTICLES IN THE INNER HELIOSPHERE CRISM- 2011, Montpellier, 27 June – 1 July, Collaborators:

Advertisements

New Insights into the Acceleration and Transport of Cosmic Rays in the Galaxy or Some Simple Considerations J. R. Jokipii University of Arizona Presented.

S. Della Torre 1,2, P. Bobik 5, G. Boella 1,3, M.J. Boschini 1,4, C. Consolandi 1, M. Gervasi 1,3, D. Grandi 1, K. Kudela 5, F. Noventa 1,3, S. Pensotti.

Galactic and Anomalous Cosmic Rays in the Heliosheath József Kόta University of Arizona Tucson, AZ , USA Thanks to : J.R. Jokipii, J. Giacalone.

Las Cruces CRS April 21-22, 2011 F.B. McDonald 1, A.C. Cummings 2, E.C. Stone 2, B.C. Heikkila 3, N. Lal 3, W.R. Webber 4 1 Institute for Physical Science.

“Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 Diffusive Shock Acceleration of High-Energy Cosmic Rays The origin of the very-highest-energy.

Bastille Day 2000 Solar Energetic Particles Event: Ulysses observations at high heliographic latitudes M. Zhang Florida Institute of Technology.

Alfvén-cyclotron wave mode structure: linear and nonlinear behavior J. A. Araneda 1, H. Astudillo 1, and E. Marsch 2 1 Departamento de Física, Universidad.

Workshop on Stochastic Differential Equations and Statistical Inference for Markov Processes Day 1: January 19 th, Day 2: January 28 th Lahore University.

The Acceleration of Anomalous Cosmic Rays by the Heliospheric Termination Shock J. A. le Roux, V. Florinski, N. V. Pogorelov, & G. P. Zank Dept. of Physics.

Simulating the Gamma Ray Sky Andrew McLeod SASS August 12, 2009.

Shock Acceleration at an Interplanetary Shock: A Focused Transport Approach J. A. le Roux Institute of Geophysics & Planetary Physics University of California.

Practical Models of Solar Energetic Particle Transport Leon Kocharov Space Research Laboratory University of Turku, Finland Requirements.

Plasma in the Heliosheath John Richardson M.I.T. Collaborators: J. Belcher, J. Kasper, E. Stone, C. Wang.

The Injection Problem in Shock Acceleration The origin of the high-energy cosmic rays remains one of the most-important unsolved problems in astrophysics.

1 Comments A. C. Cummings, Caltech Joint ACE/SOHO/STEREO/Wind Workshop Kennebunkport, ME 8-11 June 2010.

What coronal parameters determine solar wind speed? M. Kojima, M. Tokumaru, K. Fujiki, H. Itoh and T. Murakami Solar-Terrestrial Environment Laboratory,

Computational Solid State Physics 計算物性学特論第９回 9. Transport properties I: Diffusive transport.

Solar Modulation: A Theoretical Perspective Modeling of cosmic ray charge-sign dependence in the heliosphere Marius Potgieter Unit for Space Physics North-West.

Modeling Coronal Acceleration of Solar Energetic Protons K. A. Kozarev, R. M. Evans, N. A. Schwadron, M. A. Dayeh, M. Opher, K. E. Korreck NESSC Meeting,

Cosmic Rays in the Heliosphere J. R. Jokipii University of Arizona I acknowledge helpful discussions with J. Kόta and J. GIacalone. Presented at the TeV.

Yutaka Fujita (Osaka U.) Fuijta, Takahara, Ohira, & Iwasaki, 2011, MNRAS, in press (arXiv: )

Nonlinear localization of light in disordered optical fiber arrays

02-06 Dec 2013CHPC-Cape town1 A study of the global heliospheric modulation of galactic Carbon M. D. Ngobeni, M. S. Potgieter Centre for Space Research,

Formation of Power Law Tail with Spectral Index -5 G. Gloeckler and L. A. Fisk Department of Atmospheric, Oceanic and Space Sciences University of Michigan,

P. Bobik, G. Boella, M. J. Boschini, M. Gervasi, D. Grandi, K. Kudela, S. Pensotti, P.G. Rancoita 2D Stochastic Monte Carlo to evaluate the modulation.

Lesson 4: Computer method overview

The Evolution of the Heliospheric Current Sheet and its Effects on Cosmic Ray Modulation József Kóta and J.R. Jokipii The University of Arizona Tucson,

COSPAR 2004, Paris D July 21, 2004 THE HELIOSPHERIC DIFFUSION TENSOR John W. Bieber University of Delaware, Bartol Research Institute, Newark.

Centenary Symposium 2012 University of Denver June 26-28, 2012 F.B. McDonald 1, W.R. Webber 2, E.C. Stone 3, A.C. Cummings 3, B.C. Heikkila 4, N. Lal 4.

THREE-DIMENSIONAL ANISOTROPIC TRANSPORT SIMULATIONS: A PARAMETER STUDY FOR THE INTERPRETATION OF MULTI-SPACECRAFT SOLAR ENERGETIC PARTICLE OBSERVATIONS.

16-20 Oct 2005SSPVSE Conference1 Galactic Cosmic Ray Composition, Spectra, and Time Variations Mark E. Wiedenbeck Jet Propulsion Laboratory, California.

Solar Energetic Particles (SEP’s) J. R. Jokipii LPL, University of Arizona Lecture 2.

Multi-spacecraft observations of solar energetic electron events during the rising phase of solar cycle 24 W. Droege 1, R. Gomez-Herrero 2, J. Kartavykh.

Probing Turbulence At and Near CME-driven shocks Using Energetic Particle Spectra Living with a Star Team meeting Sep 18th, 2008 Pasadena, CA Gang Li From.

Introduction to Space Weather Jie Zhang CSI 662 / PHYS 660 Fall, 2009 Copyright © The Heliosphere: Solar Wind Oct. 08, 2009.

08/4/2009NAS - SHINE-Suprathermal Radial Evolution (1-11 AU) of Pickup Ions and Suprathermal Ions in the Heliosphere N. A. Schwadron Boston University,

Voyager Observations of Galactic Cosmic Ray Transport in the Heliosheath and their Reacceleration at the Termination Shock F.B. McDonald 1, W.R. Webber.

Voyager SSG November 3-4, 2011 F.B. McDonald 1, A.C. Cummings 2, E.C. Stone 2, B.C. Heikkila 3, N. Lal 3, W.R. Webber 4 1 Institute for Physical Science.

GCRs & ACRs Intensities during the last Solar Minimum: Similarities and Differences J. Kόta & J.R. Jokipii University of Arizona, LPL 32 nd ICRC Beijing,

Breakout Session F: Anomalous and Galactic Cosmic Rays Rick Leske and Maher Dayeh 5 presentations…and lots of discussion.

What is the Origin of the Frequently Observed v -5 Suprathermal Charged-Particle Spectrum? J. R. Jokipii University of Arizona Presented at SHINE, Zermatt,

1 Test Particle Simulations of Solar Energetic Particle Propagation for Space Weather Mike Marsh, S. Dalla, J. Kelly & T. Laitinen University of Central.

1 Voyager Observations of Anomalous Cosmic Rays A. C. Cummings and E. C. Stone, Caltech F. B. McDonald, University of Maryland B. Heikkila and N. Lal,

Observations of spectral shapes of suprathermal H +, He + and He ++ G. Gloeckler Department of Atmospheric, Oceanic and Space Sciences University of Michigan,

Hybrid model of solar energetic particle acceleration and transport Hybrid model of solar energetic particle acceleration and transport Leon Kocharov,

Cosmic-ray acceleration by compressive plasma fluctuations in supernova shells Ming Zhang Department of Physics and Space Sciences, Florida Institute.

Acceleration of energetic particles by compressive plasma waves Ming Zhang Department of Physics and Space Sciences, Florida Institute of Technology.

Nature, Distribution and Evolution of Solar Wind Turbulence throughout the Heliosphere W. H. Matthaeus Bartol Research Institute, University of Delaware.

Solar modulation of cosmic ray positrons in a quiet heliosphere

35th International Cosmic Ray Conference

Coupled ion acceleration and

Solar Modulation Davide Grandi AMS Group-INFN Milano-Bicocca.

Importance of Pickup Ions & Suprathermal Ions in the Inner Heliosphere

Particle Acceleration at Coronal Shocks: the Effect of Large-scale Streamer-like Magnetic Field Structures Fan Guo (Los Alamos National Lab), Xiangliang.

Modeling the SEP/ESP Event of December 13, 2006

Steven R. Spangler University of Iowa

Galactic Cosmic Ray Propagation in the 3D Heliosphere

M. D Ngobeni*,1, M. S. Potgieter1

Rick Leske, A. C. Cummings, R. A. Mewaldt, and E. C. Stone

Ulysses COSPIN High Energy Telescope observations of cosmic ray and solar energetic particles intensities since its distant Jupiter flyby in 2004 R.B.

Please contact me at Matthew E. Hill

Diffusive shock acceleration: an introduction – cont.

Xi Luo1, Ming Zhang1, Hamid K. Rassoul1, and N.V. Pogorelov2

Galactic Diffuse Emission for DC2

Conveners: M. A. Dayeh (SwRI), R. Bucik (MPS/UG), and C. Salem (UCB)

Jakobus le Roux (1,2) & Gary Webb (1)

Mariette Hitge, Adri Burger

ACE Reveals Isotopic Composition of Interstellar Material

Presentation transcript:

Stochastic Differential Equation Approach to Cosmic Ray Propagation and Acceleration International Workshop New Perspectives on Cosmic Rays in the Heliosphere 14:00,March 25, 2010 Ming Zhang Florida Institute of Technology

Brownian Motion Experiment (1824) 100 Particle simulation

Properties of Brownian Motion Particle density distribution: Average distance: Diffusion:

Einstein Theory of Brownian Motion for steady state H O 2 Pollen Langevin Equation: (poorly defined) Stochastic differential equation:

Fokker-Planck Equation For a stochastic process: (Ito) The probability density to find the process at a given time and location follows a Fokker-Planck equation: (Time forward) (Time backward) diffusion coefficient is:

Solve second-order partial differential equations Advantages: (1) Straight-forward, (2) get solution for full space Disadvantages: (1) Limited to source or initial value problems, (2) Low statistics in high dimensions.

Advantages: (1) Efficient if no need for global solution (2) No statistics problem in high dimension Disadvantage: Global solution too computational intense

Numerical solution to stochastic differential equation Stochastic differential equation is similar to ordinary differential equation Euler integration scheme works: error cancellation due to stochastics Range-Kutta scheme to speed up calculation. Simulation in high-dimensions needs little computer resource Flexible geometry

Cosmic ray transport is most likely a diffusion process in phase space Cosmic ray transport is most likely a diffusion process in phase space. Stochastic differential equation can be applied to the following studies of cosmic rays: Solar modulation of cosmic rays Cosmic ray propagation through interstellar medium with nuclear interaction network Diffusive shock acceleration Solar (cosmic ray) energetic particle transport

1. Solar Modulation of Cosmic Ray

Cosmic ray transport mechanisms in the heliosphere Drift Vd Solar wind Convection Vsw Adiabatic deceleration ISM Inward diffusion

Stochastic method to solve modulation spectra Backward trajectory of particles (pe2) Interstellar Spectrum fism(p) (pe1) all starting at (x,p,t) Modulated spectrum

Ulysses observation of a north-south asymmetry of the heliosphere (Simpson, Zhang, Bame, 1996) -80 -60 -40 -20 20 40 60 80 Ulysses Latitude 1.00 0.95 0.90 0.85 0.80 0.75 Ulysses/IMP-8 >90 MeV p E>90 MeV GCR

Modeling the asymmetry Heliosphere Solar wind Bnorth Bsouth

Distribution of particle (1 GeV/c) intensity on a sphere at 1 AU (3000 particle calculation)

Latitudinal distribution of particles when entering the heliosphere (all particles arrive at north pole 1 AU)

Model of the Heliosphere and GMIR propagation HCS tilt 45o Polarity: negative SW speed 400km/s TS compression 3.3 GMIR coverage +- 45o Lat GMIR shock speed: 600 km/s in SW 400 km/s in sheath GMIR shock compression 3 in SW 1.5 in sheath

Cosmic ray modulation in 3-d MHD model heliosphere (Ball et al. 2005) MHD model heliosphere provided by Linde et al. (1998)

Ball et al. 2005

From Florinski and Pogorelov (2009) MHD heliosphere including interaction with neutral ISM

Add Second-order Fermi Acceleration Enhanced Fermi acceleration in the heliosheath: Compression of plasma increases Strong scattering reduces

2. Cosmic Ray Propagation in Interstellar Medium Elemental Abundance

(p+p —> p0 —> g and Inverse Compton)

Diffusion Model for cosmic ray propagation in the interstellar medium with halo At least 97 isotopes need to be considered. For each nuclear isotope: Matrix representation (with an assumption that all species have the same diffusion coefficient with a function of energy per nucleon) Nuclear reaction matrix

Stochastic solution to diffusion equation with source problem Integration is carried along stochastic trajectories described by stochastic differential equation

Results with input of measured interstellar gas distribution Elemental contribution to 12C and 10B observed at the solar system Spectrum of B/C ratio at the solar system

3. Diffusive Shock Acceleration Nearly isotropic distribution Time-forward stochastic simulation:

Transport equation for particles with large anisotropy Backward Stochastic differential equation solver

From McKibben et al. (2003)

SEP Propagation Model includes: Pitch angle diffusion focusing, Streaming Convection Perpendicular diffusion Adiabatic cooling (pitch-angle dependent)

Stochastic differential equation vs Stochastic differential equation vs. Fokker-Planck partial differential equation Monte-Carlo simulation with SDE can be used to solve Fokker-Planck equation. Track stochastic trajectories to look into the details of particle transport. Problems in high dimensions do not necessarily increase computation demand. Programming is extremely simple and less numerical instability. Statistics in some problems may require special consideration. Global solutions to Fokker-Planck equation need too much computation resources. Cannot handle non-linear problems.