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

2. Outline The heliosphere Sun’s Magnetic Field , Polarity and Activity
Solar Wind and Neutral Sheet
Solar modulation of GCR
Diffusion, Convection, Energy Loss, Drift
The basic: Parker Model
Force Field Approx.
Our 2D Stochastic Monte Carlo
JK modif. of polar field
Drift model: WNS & PM
Dynamic parameters
Comparison with data & Prediction for AMS-02
Conclusions
La physique d'AMS, Annecy 9-10 March 2010

3. The heliosphere the region of influence of the solar magnetic field...
La physique d'AMS, Annecy 9-10 March 2010

4. Magnetic field generated from the Sun
Field lines “frozen” in the plasma created by the solar corona adiabatic expansion
La physique d'AMS, Annecy 9-10 March 2010

5. Field polarity Configuration for A>0 Configuration for A<0
Archimedean Spiral
La physique d'AMS, Annecy 9-10 March 2010

6. Solar Activity A>0 A<0 A<0 A>0 A<0 A>0
The solar activity is related to:
- Sunspot number
(<10 minimum; >100 maximum)
- Wavy Neutral Sheet opening/tilt angle
(10° minimum ; >75° maximum)
La physique d'AMS, Annecy 9-10 March 2010

7. Solar Wind and Magnetic Field Wavy Neutral Sheet
Latitudinal Dependence
La physique d'AMS, Annecy 9-10 March 2010

8. Solar Wind La physique d'AMS, Annecy 9-10 March 2010
High Solar Activity
Low Solar Activity
La physique d'AMS, Annecy 9-10 March 2010

9. Solar Modulation
Cr transport effect is a flux DECREASE moving from the outer to the inner Heliosphere
La physique d'AMS, Annecy 9-10 March 2010

10. Parker’s FP Equation The global effect on CR si given by:
Diffusion
Magnetic irregularities on a small scale
Magnetic Drift
Magnetic field gradients on larger domains
Convection
Solar Wind expansion
CR propagation in the heliosphere is decribed by:
U is density number of CR for unit interval of energy
La physique d'AMS, Annecy 9-10 March 2010

11. Force Field
Analitical solution of Parker’s equation in the monodimensional approx.
The only parameter is the modulation potential 
Assumptions:
V constant with r
K constant with r
Steady state (no sources)
No convection
No drift (K simmetric)
La physique d'AMS, Annecy 9-10 March 2010

12. Convective/Drift term
Stochastic 2D Montecarlo
Parker’s equation, in the 2D (radius and co-latitude) approximation, is mathematically equivalent to the following set of stochastic differential equations
Convective/Drift term
Diffusive term
La physique d'AMS, Annecy 9-10 March 2010

13. Diffusion and Drift Particles generated at 100 AU
Interact with the Interplanetary Medium
Random Walk Process
Down to 1 AU
La physique d'AMS, Annecy 9-10 March 2010

14. Magnetic Drift using the Guiding Center approximation
La physique d'AMS, Annecy 9-10 March 2010

15. Neutral Sheet Drift 2D Approximation
Potgieter & Moraal (1985)
Burger & Potgieter (1989)
Wavy Neutral Sheet - Hattingh & Burger (1995) er N S
Ordinary Drift
NS drift
Transition Function that emulate the effect of a wavy neutral sheet
La physique d'AMS, Annecy 9-10 March 2010

16. Drift model: PM where the term f(θ), is Potgieter Moraal model (1985)
Minimo Solare
Potgieter Moraal model (1985)
Transition function, is 0 on the ecliptic plane and ± 1 at the poles
NS term, is maximum on the ecliptic
Massimo Solare
where the term f(θ), is
La physique d'AMS, Annecy 9-10 March 2010

17. (orbita polare solare)
Polar Field corrections
Ulysses
(orbita polare solare)
IMP8
(1AU)
Counting rate (1/s)
~16%
[Heber 1998]
La physique d'AMS, Annecy 9-10 March 2010

18. Polar Field corrections
La physique d'AMS, Annecy 9-10 March 2010

19. Sun magnetic field in not constant in the Heliosphere
Dynamic parameters
Sun magnetic field in not constant in the Heliosphere
100 AU
Magnetic perturbations move with the solar wind
La physique d'AMS, Annecy 9-10 March 2010

20. Dynamic parameters
At a first approximation we can divide the heliosphere in different regions
The time needed for a magnetic perturbation to reach
the external limit of the heliosphere (100AU) is roughly:
months
In every sector we consider solar condition
of a period x-months before the data taking
La physique d'AMS, Annecy 9-10 March 2010

21. Cosmic Rays moduated spectra
La physique d'AMS, Annecy 9-10 March 2010
31st ICRC Lodz 7-15 July 2009

22. Cosmic Rays moduated spectra
La physique d'AMS, Annecy 9-10 March 2010
31st ICRC Lodz 7-15 July 2009

23. Cosmic Rays moduated spectra
La physique d'AMS, Annecy 9-10 March 2010

24. Cosmic Rays moduated spectra
La physique d'AMS, Annecy 9-10 March 2010

25. AMS-02 measurements La physique d'AMS, Annecy 9-10 March 2010
We estimated the expected GCR flux for the AMS-02 mission
Estimated Sunspot Numbers
La physique d'AMS, Annecy 9-10 March 2010

26. Predictions for AMS-02
La physique d'AMS, Annecy 9-10 March 2010

27. Relation between Solar Activity and Tilt angle
La physique d'AMS, Annecy 9-10 March 2010

28. Summary
GCRs entering the Heliosphere undergo diffusion, convection, magnetic drift and adiabatic energy loss
Parker in 1965 gave the first description of Solar Modulation with his famous FP like equation
First simmetric approximation is called Force Field and depends on just one parameter, the modulation potenzial 
The heliosphere is much more complex: polarity of the field changes every 11 years, magnetic drift plays an important role depending also on particle’s charge, and solar Wind is not constant
Using a set of SDEs that solves Parker’s equation we realized a 2D Stochastic Montecarlo for GCRs modulation with JK modification for the polar field and the PM as NS Drift, for different solar conditions. With a dynamic approach we reproduced the proton CR flux for different experiments (AMS, Caprice, BESS and IMAX) and predicted the CR flux that AMS-02 will measure on the ISS
We are working on a connection between tilt angle and solar activity and different particles (antiprotons, nuclei, electrons etc.)

29. Thank you for your attention!

31. Modulation
Rate of flux in two consecutive period with similar solar activity
Boella et. Al. 2001
Variation between two consecutive minimum
(it change the Field polarity)
There is a strong dependence of the modulation from the polarity of the field