p. bobik, g. boella, m. j. boschini, m. gervasi, d. grandi, k. kudela, s. pensotti, p.g. rancoita 2d...

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

D. Grandi, K. Kudela, S. Pensotti, P.G. Rancoita

2D Stochastic Monte Carlo 2D Stochastic Monte Carlo to evaluate the modulation to evaluate the modulation

of GCRof GCRfor positive and negative for positive and negative

periodsperiods

21st ECRS Kosice - Slovakia 9-12 September 2008

Outline

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

•Conclusions and future work

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

tgr

AB

Br

ABr

2

2

0•Parker field model

•Changes after Ulysses


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 < < 90o

V = 750 km/sec, for < 30o


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 proton’s rest energy, . Elements of particle trajectory :

trajectory

)(1

)()(

3

1)sin(

sin

1)(

1 22

2

222

2VUr

rrTU

Tr

Vr

r

UK

rr

UKr

rrt

Urr

2

2

2 22 2

( )12

1 2.(1 )

rrg rr

g

d r Kr t V t R K tr dr

K Kdt R t

d r r

0

0

2T T

T T


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 for example)


Diffusion coefficients • Diffusion coeficiens :

K|| K

• Theory parameter’s

– 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)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 : 45o dominate radial diffusion

Heliopause (outer heliosphere) : 90o strong latitudinal diffusion

2 2cos .sinrrK K K

0 ( )3P

BK K K P

B

0( )K K K

K K


ParametersThe 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


http://quake.stanford.edu/~wso/gifs/helio.tiff

Tilt angle measurements

Wilcox Solar laboratory measurements


SW speed measurements

OMNIweb data browser


• 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 (pv/4), See Potgieter & Burger Astr. J., 339, 1989

Drift effects


The average drift velocity is

vd = (kTeB) = (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 rd is the radial variation with drift effect, d is the latitudinal variation of the particle, vg is the velocity of gradient drift, vd

ns is the velocity of neutral sheet drift and vθ is the velocity of curvature drift. Both vg and vd

ns are directed along er, while vθ is directed along eθ in spherical coordinates.

Drift effects

,nsd g d d

v tr r v v t cos arccos( )+arctg

r


Data – AMS01

A>0 period for estimating model results : • June 1998


Data – BESSA<0 period for estimating model results : August 2002

Evolution of BESS 98 – BESS TeV


Data – IMAX & CapriceA>0 period for estimating model results : July 1992 & August 1994

Caprice experiment (evolution of IMAX and TS93) 1994

IMAX measurements (balloon flight) 1992



Parameters of simulation for A>0 AMS, year 1998:(K)0=0.025, K0 =1.70x10-7 au2s-1, Vsw = 430 km/s, a=30°-45°, Kp(P) ~ P

Parameters of simulation for A>0 Caprice, year 1994 :(K)0=0.025, K0 =1.90x10-7 au2s-1, Vsw = 440 km/s, a ~ 10°-25°, Kp(P) ~ P

Parameters of simulation for A>0 IMAX, year 1992: (K)0=0.025, K0 =1.33x10-7 au2s-1, Vsw = 400 km/s, a~ 20°-40°, Kp(P) ~ P

Parameters of simulation for A<0 BESS, year 2002: (K)0=0.05, 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)

Estimated simulation parameters

0

1

k

03

)(

k

RRV neosservazioeliopausa

Solar wind speed: in average something like 400 km/sExpansion 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)

Idea: average parameters

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


Parameters of simulation for A>0 AMS, year 1998:(K)0=0.025, K0 =1.46x10-7 au2s-1, (1.78-1.90x10-7 ) Vsw = 430 km/s, a=50°, Kp(P) ~ P

Parameters of simulation for A>0 Caprice, year 1994 :(K)0=0.025, K0 =1.45x10-7 au2s-1, (1.65 – 2.00x10-7 ) Vsw = 500 km/s, a ~ 30°, Kp(P) ~ P

Parameters of simulation for A>0 IMAX, year 1992: (K)0=0.025, 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 A<0 BESS, year 2002 (still under

investigation..): value too big!

(K)0=0.05, K0 =0.60x10-7 au2s-1, (2.00x10-7 ) Vsw = 500 km/s, a~ 40°, Kp(P) ~ P

Fitted simulation parameters


Ou

ter

pla

nets


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 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).


p. bobik, g. boella, m. j. boschini, m. gervasi, d. grandi, k. kudela, s. pensotti, p.g. rancoita 2d...

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