Solar
Modulation
Davide Grandi
AMS Group-INFN
Milano-Bicocca
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
The heliosphere
the region of influence of
the solar magnetic field...
11th ICATPP, Como 5-9 October 2009
the region of influence of
the solar magnetic field...
The heliosphere
La physique d'AMS, Annecy 9-10 March 2010
Field polarity
Configuration for A>0
Configuration for A<0
Solar cycle period is approx 11 years
11th ICATPP, Como 5-9 October 2009
Magnetic field
generated from the
Sun
Field lines “frozen” in the plasma
created by the solar corona
adiabatic expansion
11th ICATPP, Como 5-9 October 2009
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)
11th ICATPP, Como 5-9 October 2009
Wavy
Neutral Sheet
Solar Wind and Magnetic
Field
Latitudinal Dependence
11th ICATPP, Como 5-9 October 2009
Solar Wind
Low Solar Activity
High Solar Activity
VSW  V0 1  cos( ) 
Modulazione solare dei RCG
L’effetto del trasporto dei RC è una DIMINUZIONE del flusso di RC dall’esterno
verso l’interno della eliosfera
Modulazione Solare
Il Flusso Integrale dei Raggi Comici
diminuisce avvicinandosi al Sole
Flusso integrale misurato dalle sonde Voyager 1 (V1)
Voyager 2 (V2) e Pioneer 10 (P10)
Introduzione
Mod. Parker
Propag. RC Nostro Modello
Risultati & Conclusioni
Campo Magnetico Interplanetario
Le linee di campo si deformano secondo una “spirale di
Archimede”
L’eliosfera si divide in due emisferi a
La rotazione del sole è differenziale
polarità opposta divisi a uno strato neutro di
corrente
Il Sole Ruota
Introduzione
Mod. Parker
Propag. RC Nostro Modello
Risultati & Conclusioni
Propagazione di RC in Eliosfera
Force Field
Risoluzione analitica dell’equazione di Parker monodimensionale.
Ha come unico parametro il fattore di modulazione 
Nymmik
Modello empirico, richiede come unico
parametro il numero di smoothed sunspot
number per calcolare il potenziale effettivo
modulazione nei mesi precedenti la data
di osservazione
Introduzione
Mod. Parker
Propag. RC Nostro Modello
Risultati & Conclusioni
Parker’s FP Equation
The global effect on CR si given by:
Magnetic irregularities on a small scale
Diffusion
Magnetic field gradients on larger domains
Magnetic Drift
Solar Wind expansion
Convection
11th ICATPP, Como 5-9 October 2009
Parker’s Equation
CR propagation in the heliosphere is decribed by:
U is density number of CR for unit interval of energy
Is essentially a Fokker-Planck equation
Convective/Drift
Term
Diffusive Term
31st ICRC Lodz 7-15 July 2009
Diffusione e Moti di Deriva
Interazione della particella cosmica con il mezzo interplanetario
Continui URTI che causano una variazione del percorso
Processo di Random Walk
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
11th ICATPP, Como 5-9 October 2009
Magnetic Drift
using the Guiding Center approximation
11th ICATPP, Como 5-9 October 2009
Deriva Magnetica
La deriva magnetica è legata alla componente anti-simmetrica del tensore di
diffusione
Introduzione
Mod. Parker
Propag. RC Nostro Modello
Risultati & Conclusioni
Different Solar polarities....
11th ICATPP, Como 5-9 October 2009
Drift model: WNS vs. PM
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
Wavy Neutral Sheet Model (1995)
Magnetic drift
equation is solved for
the approximation
<<1 rad  30°
31st ICRC Lodz 7-15 July 2009
Neutral Sheet Drift
2D Approximation
Potgieter & Moraal (1985)
Burger & Potgieter (1989)
Wavy Neutral Sheet - Hattingh & Burger (1995)
N
Ordinary Drift
2R 

2   

r 

er
NS drift
Transition Function that emulate the effect of a wavy
neutral sheet
S
Drift model: PM
Minimo Solare
Potgieter Moraal model (1985)
Transition function, is
NS term, is maximum
0 on the ecliptic plane
on the ecliptic
and ± 1 at the poles Massimo Solare
where the term f(θ), is
11th ICATPP, Como 5-9 October 2009
Polar field corrections
Counting rate (1/s)
~16%
Ulysses
(orbita
polare
solare)
IMP8
(1AU)
[Heber 1998]
11th ICATPP, Como 5-9 October 2009
Polar field corrections
11th ICATPP, Como 5-9 October 2009
Polar field corrections
31st ICRC Lodz 7-15 July 2009
Dynamic parameters
Sun magnetic field in not constant in the Heliosphere
100 AU
Magnetic
perturbations
move with the
solar wind
11th ICATPP, Como 5-9 October 2009
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
11th ICATPP, Como 5-9 October 2009
Cosmic Rays moduated spectra
BESS
High Solar Activity
IMAX
Medium Solar Activity
CAPRICE
Low Solar Activity
A>0
A<0
30°
AMS-01
Low SOlar ACtivity
11th ICATPP, Como 5-9 October 2009
Cosmic Rays moduated spectra -IMAX
Menn et al. 2000
11th ICATPP, Como 5-9 October 2009
Cosmic Rays moduated spectra - BESS
Shikaze et al.2007
11th ICATPP, Como 5-9 October 2009
Cosmic Rays moduated spectra - Caprice
Boezio et. al. 1999
11th ICATPP, Como 5-9 October 2009
Cosmic Rays moduated spectra – AMS 01
Alcaraz et. al. 1998
11th ICATPP, Como 5-9 October 2009
Cosmic Rays moduated spectra
11th ICATPP,
Como
5-9 7-15
October
31st ICRC
Lodz
July2009
2009
Cosmic Rays moduated spectra
11th ICATPP,
Como
5-97-15
October
31st ICRC
Lodz
July 2009
Cosmic Rays moduated spectra
11th ICATPP,
Como
5-9 7-15
October
31st ICRC
Lodz
July2009
2009
Cosmic Rays moduated spectra
11th ICATPP,
Como
5-97-15
October
31st ICRC
Lodz
July 2009
AMS-02 measurements
We estimated the expected GCR flux for
the AMS-02 mission
Estimated Sunspot Numbers
A<0
A>0
A<0
11th ICATPP, Como 5-9 October 2009
Predictions for AMS-02
11th ICATPP, Como 5-9 October 2009
Relation between Solar Activity and Tilt angle
11th ICATPP, Como 5-9 October 2009
Relation between Solar Activity and Tilt angle
11th ICATPP, Como 5-9 October 2009
Conclusions
We realized a 2D Stochastic Montecarlo to evaluate the CR
modulation in the Heliosphere
We introduced the JK modification for the polar field and used the
PM as NS Drift models, suitable for different solar conditions
We introduced a dynamic approach to the use of parameters in
order to reproduce the real physical process
 We reproduced the proton CR flux for different experiments
(AMS, Caprice, BESS and IMAX) in different solar polarities for
medium, high and low solar activity
 We used our 2D Montecarlo to predict the CR flux that AMS-02
will measure on the ISS from 2010 to 2012 (maximum), this will also
help a better tuning and small corrections
We are able to modulate different kind of particles (antiprotons,
nuclei, electrons etc.)
We are investigating also a more strict connection between the tilt
angle and the solar activity

Thank you for your
attention!
Polarity/Charge dependence
31st ICRC Lodz 7-15 July 2009
Modulation
Rate of flux in two consecutive period with similar solar
activity
Variation between two consecutive minimum
(it change the Field polarity)
Boella et. Al. 2001
There is a strong dependence of the modulation from the
polarity of the field
Raggi Cosmici in Eliosfera
La rotazione differenziale del Sole causa
una divisione dell’eliosfera in 2 regioni
divise da uno strato neutro di corrente
Lo strato neutro di corrente oscilla entro
un certo angolo con l’eclittica
Angolo di Tilt
Raggi Cosmici in Eliosfera
Il modello
comprende
Deriva magnetica dovuta a
curvatura e gradiente dell’IMF
Deriva dovuta a allo
strato neutro di
corrente
Il modello dipende dalla polarità
del campo magnetico solare e
dalla carica delle particelle (in
figura positive)