1. 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 of GCR for positive and negative periods

2. 21 st ECRS Kosice - Slovakia 9-12 September 2008 Outline Stochastic MC approach Ulysses mission Description of the model Diffusion coefficients Parameters: solar wind velocity V sw and tilt angle  Drift effects Data sets: AMS-01, IMAX, Caprice, BESS Estimated and fitted parameters Idea: averaged values for K 0 Conclusions and future work

3. The Heliosphere: Stochastic MC approach 2D –model radius and heliolatitude The input is Local Interstellar Spectrum of protons (LIS) : Burger’s 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 21 st ECRS Kosice - Slovakia 9-12 September 2008

4. 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 30 o <  < 90 o V = 750 km/sec, for  < 30 o 21 st ECRS Kosice - Slovakia 9-12 September 2008

5. 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,T 0 is proton’s rest energy,. Elements of particle trajectory : trajectory 21 st ECRS Kosice - Slovakia 9-12 September 2008

6. Diffusion coefficients K rr radial diffusion coefficient, R g is Gaussian distributed Random number with unit variance,  t is time step of calculation, K p (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 10 2 to 10 4 or 10 5 trajectories per second – for good enough spectrum at 1 AU we need calculation in order 10 8 trajectories (in our case 5x 10 8 trajectories for every condition in V sw and  for example) 21 st ECRS Kosice - Slovakia 9-12 September 2008

7. Diffusion coefficients Diffusion coeficiens : K ||  K  Theory parameter’s –K p (P) describe dependence of diffusion tensor from rigidity, in GV K p (P) ~ P 1/2 to P K p (P) ~ P 2/3, P 0,68 etc. P 0.78, P 1 quasi linear approach: K p (P) ~ P –(K  ) 0 is ratio between perpendicular and pralalell diffusion coeficients (K  ) 0 = 0.01 – 0.05 (K  ) 0 =0.025 [J. Giacalone, J.R.Jokipii, The Astrophysical Journal, 520, 204,1999] Some authors: (K  ) 0 =0.05 (maybe “better” for A<0) Earth :  45 o  dominate radial diffusion Heliopause (outer heliosphere) :   90 o  strong latitudinal diffusion 21 st ECRS Kosice - Slovakia 9-12 September 2008

8. 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 21 st ECRS Kosice - Slovakia 9-12 September 2008

9. Tilt angle measurements Wilcox Solar laboratory measurements 21 st ECRS Kosice - Slovakia 9-12 September 2008

10. SW speed measurements OMNIweb data browser 21 st ECRS Kosice - Slovakia 9-12 September 2008

11. 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 unlimited…spatially averaged max value ( p v/4), See Potgieter & Burger Astr. J., 339, 1989 Drift effects 21 st ECRS Kosice - Slovakia 9-12 September 2008

12. The average drift velocity is v d =  (k T e B ) =  (  P/3B) Where P is the CR particle’s 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  r d is the radial variation with drift effect,  d is the latitudinal variation of the particle, v g is the velocity of gradient drift, v d ns is the velocity of neutral sheet drift and v θ is the velocity of curvature drift. Both v g and v d ns are directed along e r, while v θ is directed along e θ in spherical coordinates. Drift effects 21 st ECRS Kosice - Slovakia 9-12 September 2008

13. Data – AMS01 A>0 period for estimating model results : June 1998 21 st ECRS Kosice - Slovakia 9-12 September 2008

14. Data – BESS A<0 period for estimating model results : August 2002 Evolution of BESS 98 – BESS TeV 21 st ECRS Kosice - Slovakia 9-12 September 2008

15. Data – IMAX & Caprice A>0 period for estimating model results : July 1992 & August 1994 Caprice experiment (evolution of IMAX and TS93) 1994 IMAX measurements (balloon flight) 1992 21 st ECRS Kosice - Slovakia 9-12 September 2008

16. Parameters of simulation for A>0 AMS, year 1998: (K  ) 0 =0.025, K 0 =1.70x10 -7 au 2 s -1, Vsw = 430 km/s, a =30°-45°, K p (P) ~ P Parameters of simulation for A>0 Caprice, year 1994 : (K  ) 0 =0.025, K 0 =1.90x10 -7 au 2 s -1, Vsw = 440 km/s, a ~ 10°-25°, K p (P) ~ P Parameters of simulation for A>0 IMAX, year 1992: (K  ) 0 =0.025, K 0 =1.33x10 -7 au 2 s -1, Vsw = 400 km/s, a ~ 20°-40°, K p (P) ~ P Parameters of simulation for A<0 BESS, year 2002: (K  ) 0 =0.05, K 0 =0.88x10 -7 au 2 s -1, Vsw = 420 km/s, a ~ 35°-50°, K p (P) ~ P K 0 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) Estimated simulation parameters

17. 21 st ECRS Kosice - Slovakia 9-12 September 2008

21. 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 K 0 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) Idea: average parameters At first approximation: different value of K 0 back in time –Average value (12 months) centered in the period considered –Average value (12 months) back 1 year from the period considered 21 st ECRS Kosice - Slovakia 9-12 September 2008

22. Parameters of simulation for A>0 AMS, year 1998: (K  ) 0 =0.025, K 0 =1.46x10 -7 au 2 s -1, (1.78-1.90x10 -7 ) Vsw = 430 km/s, a =50°, K p (P) ~ P Parameters of simulation for A>0 Caprice, year 1994 : (K  ) 0 =0.025, K 0 =1.45x10 -7 au 2 s -1, (1.65 – 2.00x10 -7 ) Vsw = 500 km/s, a ~ 30°, K p (P) ~ P Parameters of simulation for A>0 IMAX, year 1992: (K  ) 0 =0.025, K 0 =1.20x10 -7 au 2 s -1, (1.14-1.25 x10 -7 ) Vsw = 410 km/s, a ~ 55°, K p (P) ~ P Parameters of simulation for A<0 BESS, year 2002 (still under investigation..): value too big! (K  ) 0 =0.05, K 0 =0.60x10 -7 au 2 s -1, (2.00x10 -7 ) Vsw = 500 km/s, a ~ 40°, K p (P) ~ P Fitted simulation parameters 21 st ECRS Kosice - Slovakia 9-12 September 2008

26. Outer planets 21 st ECRS Kosice - Slovakia 9-12 September 2008

27. 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 K 0 and Kp are able to reproduce the measured data for protons and A>0. Still needed a deeper knowledge on diffusion coefficients and accurate data for different phases of the solar activity (A<0). 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). 21 st ECRS Kosice - Slovakia 9-12 September 2008