P. Bobik, G. Boella, M. J. Boschini, M. Gervasi,

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2D Stochastic Monte Carlo to evaluate the
modulation of GCRfor positive and negative
periods

• P. Bobik, G. Boella, M. J. Boschini, M. Gervasi,
• D. Grandi, K. Kudela, S. Pensotti, P.G. Rancoita

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Outline

• Stochastic MC approach
• Ulysses mission
• Description of the model
• Diffusion coefficients
• Parameters solar wind velocity Vsw and tilt
  angle a
• Drift effects
• Data sets AMS-01, IMAX, Caprice, BESS
• Estimated and fitted parameters
• Idea averaged values for K0
• Conclusions and future work

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The Heliosphere Stochastic MC approach

• 2D model
• radius and heliolatitude
• The input is Local Interstellar Spectrum of
  protons (LIS) Burgers model
• particles are generated at 100AU
• they are forward traced from outside to 1 AU,
  outside the heliosphere they are killed

• Parker field model
• Changes after Ulysses

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Ulysses mission

• The Ulysses Mission is the first spacecraft to
  explore interplanetary space at high solar
  latitudes (launch oct. 1990 ? 1994 southern and
  1995 north high latitudes).
• Previous models estimate V const
  400 km/s
• After Ulysses mission change dependence solar
  wind speed V on heliolatitude ?
• 2D model
• V 400 (1 cos ?) km/s,
• for 30o lt ? lt 90o
• V 750 km/sec, for ? lt 30o

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Description of model
Heliospheric stochastic simulation is based on
equation for GCR transport in Heliosphere
(without drift terms) where ? is the
heliolatitude, U is the cosmic ray number density
per unit interval of kinetic energy T (per
nucleon), r is radial distance and V solar wind
velocity,T0 is protons rest energy,
. Elements of particle trajectory trajectory
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Diffusion coefficients

• Krr radial diffusion coefficient, Rg is Gaussian
  distributed
• Random number with unit variance, ?t is time step
  of
• calculation, Kp(P) is function of rigidity in GV,
  (K?)0 is
• ratio between perpendicular and parallel dif.
  coeff.
• Particles from generated initial spectrum are
  traced with steps ?r, ??, ?t
• From 102 to 104 or 105 trajectories per second
  for good enough spectrum at 1 AU we need
  calculation in order 108 trajectories (in our
  case 5x 108 trajectories for every condition in
  Vsw and a for example)

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Diffusion coefficients

• Diffusion coeficiens
• K ?? K?
  
• Theory parameters
• Kp(P) describe dependence of diffusion tensor
  from rigidity, in GV
• Kp(P) P1/2 to P
• Kp(P) P2/3, P0,68 etc. P0.78, P1
• quasi linear approach Kp(P) P
• (K?)0 is ratio between perpendicular and
  pralalell diffusion coeficients
• (K?)0 0.01 0.05
• (K?)00.025 J. Giacalone, J.R.Jokipii, The
  Astrophysical Journal, 520, 204,1999
• Some authors (K?)00.05 (maybe better for Alt0)

• Earth ??45o
• dominate radial diffusion
• Heliopause (outer heliosphere)
• ? ? 90o ? strong latitudinal
• diffusion

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Parameters

• The model is time dependent due to variation of
  measured values Tilt angle ? and solar wind
  velocity
• Experimental (measured) values
• Tilt angle key parameter of model describe a
  level of the solar activity (Expected lower GCR
  flux for solar maxima)
• Solar wind speed

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Tilt angle measurements
Wilcox Solar laboratory measurements
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SW speed measurements
OMNIweb data browser
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Drift effects

• Parker model allows an analytical solution for
  drift velocities
• Drift effects are included through analytical
  effective drift velocities
• Gradient drift
• Curvature drift
• Neutral sheet drift
• In our case the drift is averaged over a solar
  rotation, total volume limited by the titl angle
  is called neutral sheet region.
• In this region the gradient and curvature drift
  contributes are (matematically) decreased
  according to the dominant NS drift (in ecliptic
  plane their contribution is nearly zero)
• See Hatting Burger Adv. Sp. Res. 9, 1995
• Drift velocity is locally unlimitedspatially
  averaged max value (pv/4), See Potgieter Burger
  Astr. J., 339, 1989

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Drift effects

• The average drift velocity is
•  
• vd ?? (kTeB) ?? (?P/3B)
•  
• Where P is the CR particles rigidity. In the
  Parker spiral field, gradient, curvature and
  drift along the neutral sheet are added to the
  previous formulas to calculate a position of a
  test particle during a time step ?t
•  
•  
• Where ?rd is the radial variation with drift
  effect, ??d is the latitudinal variation of the
  particle, vg is the velocity of gradient drift,
  vdns is the velocity of neutral sheet drift and
  v? is the velocity of curvature drift. Both vg
  and vdns are directed along er, while v? is
  directed along e? in spherical coordinates.

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Data AMS01

• Agt0 period for estimating
• model results
• June 1998

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Data BESS
Alt0 period for estimating model results
August 2002
Evolution of BESS 98 BESS TeV
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Data IMAX Caprice
Agt0 period for estimating model results July
1992 August 1994
IMAX measurements (balloon flight) 1992
Caprice experiment (evolution of IMAX and TS93)
1994
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Estimated simulation parameters
Parameters of simulation for Agt0 AMS, year
1998 (K?)00.025, K0 1.70x10-7 au2s-1, Vsw
430 km/s, a30-45, Kp(P) P Parameters of
simulation for Agt0 Caprice, year 1994
(K?)00.025, K0 1.90x10-7 au2s-1, Vsw 440
km/s, a 10-25, Kp(P) P Parameters of
simulation for Agt0 IMAX, year 1992
(K?)00.025, K0 1.33x10-7 au2s-1, Vsw 400
km/s, a 20-40, Kp(P) P Parameters of
simulation for Alt0 BESS, year 2002 (K?)00.05,
K0 0.88x10-7 au2s-1, Vsw 420 km/s, a
35-50, Kp(P) P K0 values from Moskalenko et
al. Secondary antiprotons and propagation of
cosmic rays in the galaxy and helisphere(2002)
and Usoskin et al. Cosmic ray modulation,
monthly reconstruction (2005)
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Idea average parameters
Solar wind speed in average something like 400
km/s Expansion time roughly 1.2 years (magnetic
field frozen into the solar wind) Why use just
fixed K0 value? Maybe better (in the forward
tracing approach) consider a larger
period? Particles lifetime (time spent in the
heliosphere) between some days (-10 GeV) and a
little bit more than a month (-100/200 MeV)

• At first approximation different value of K0
  back in time
• Average value (12 months) centered in the period
  considered
• Average value (12 months) back 1 year from the
  period considered

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Fitted simulation parameters
Parameters of simulation for Agt0 AMS, year
1998 (K?)00.025, K0 1.46x10-7 au2s-1,
(1.78-1.90x10-7 ) Vsw 430 km/s, a50, Kp(P)
P Parameters of simulation for Agt0 Caprice,
year 1994 (K?)00.025, K0 1.45x10-7 au2s-1,
(1.65 2.00x10-7 ) Vsw 500 km/s, a 30,
Kp(P) P Parameters of simulation for Agt0
IMAX, year 1992 (K?)00.025, K0 1.20x10-7
au2s-1 , (1.14-1.25 x10-7 ) Vsw 410 km/s, a
55, Kp(P) P Parameters of
simulation for Alt0 BESS, year 2002 (still under
investigation..) value too
big! (K?)00.05, K0 0.60x10-7 au2s-1,
(2.00x10-7 ) Vsw 500 km/s, a 40, Kp(P)
P
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Outer planets
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Conclusions

• 2D stochastic MC model particles propagation
  across the heliosphere with drift effects.
• Proton spectra predicted by model are decreasing
  with increasing tilt angles and solar wind speed
• The model is able to reproduce measured values at
  1AU for different periods.
• Different combination of parameters as K0 and Kp
  are able to reproduce the measured data for
  protons and Agt0.
• Still needed a deeper knowledge on diffusion
  coefficients and accurate data for different
  phases of the solar activity (Alt0).
• We are studying the difference in diffusion
  tensor between value expected by force field
  model (F) and best fit values average values
  back in the past?
• Final answer will come from theory and also from
  future measurements (for example the PAMELA and
  future AMS-02 measurements).

21st ECRS Kosice -
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