Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
PARTICLE ACCELERATION BY DIRECT
ELECTRIC FIELDS IN AN ACTIVE REGION
MODELLED BY A CELLULAR AUTOMATON
Cyril Dauphin – Nicole Vilmer
Anastasios Anastasiadis
1- CA model
2- Acceleration model
3- particle energy distributions
4- X-ray and gamma ray fluxes
Particle acceleration by direct electric field in an active region modelled by a CA model
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
Introduction
We use a cellular automaton (CA) model
to mimic the energy release process
(Vlahos et al, 1995 ...)
CA can reproduce statistical properties of
solar flares (i.e. for the all sun)
Frequency distributions of, e.g., flares
energy  power law
Hudson, 1991 Crosby et al, 1993 …
 no characteristic scale
 Simple rules can model the system
Aschwanden et al,2000
Particle acceleration by direct electric field in an active region modelled by a CA model
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
Introduction
Can we use a CA model to mimic the energy release process in an active
region ?
Current sheet can have a fractal structure (Yankov, 1996)
Extrapolation of magnetic field shows the complexity of an active region
Hughes et al, 2003: solar flare can be
reproduced by cascades of
reconnecting magnetic loops which
evolve in space and time in a SOC
state
Hughes et al, 2003
Particle acceleration by direct electric field in an active region modelled by a CA model
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
Introduction
Random foot-points motion => buildup of magnetic discontinuities in
the corona
photosphere
photosphere
- E
photosphere
energy
Log(dN/dE)
photosphere
Log(energy)
Particle acceleration by direct electric field in an active region modelled by a CA model
time
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
CA model
t0
Is the evolution of magnetic discontinuities
based on Self Organized Criticality system ?
 assumption (Lu et Hamilton, 1993;
Vlahos, 1995; McIntosh et al, 1992 …)
We use a CA model based on the SOC concept
(Vlahos et al, 1995; Georgoulis et al, 1998)
Basic rules:
-3D cubic grid: Bi=B(x,y,z) at each grid
point
- At each time step Bi(t+1)=Bi(t)+Bi(t)
and prob(Bi)=Bi-5/3
- if (Bi-1/6∑Bj)>Bcr i~Bi2
Curvature of B at the point i = dBi
Particle acceleration by direct electric field in an active region modelled by a CA model
i
t1>t0
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
CA model
6
Magnetic field evolution: Bi (t  1)  Bi (t )  dBi (t )
Isliker et al, 1998
7
- Link to diffusion
1
Bi  j (t  1)  Bi  j (t )  dBi  j (t )
7
B
   ( v  B )   2 B
t
prob(Bi (t ))  (Bi (t )) 5 / 3 to mimic the turbulent motion
of the magnetic loops foot
points
Espagnet et al,1993
Particle acceleration by direct electric field in an active region modelled by a CA model
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
CA model
i(t)~B2(t) Energy
released
time series
 power law distribution:
E~ -1.6
Particle acceleration by direct electric field in an active region modelled by a CA model
i(t)~B2(t)
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
Acceleration model
We have one model of energy release process in an active region
We want to accelerate particle
 each magnetic energy release process RCS (reconnecting current
sheet; observed in tokamak (Crocker at al, 2003) and in laboratory )
We have to make the link between the energy release
process and the acceleration process.
One of the first step in this sense Anastasiadis et al, 2004
Particle acceleration by direct electric field in an active region modelled by a CA model
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
Acceleration model
Inflow vin
We equate the magnetic energy flux
P
1

2
0
vin B bl
B┴
y
∆le
B//
∆lp
E0
B//
z
P  NÝe eE 0  le  n e  NÝP eE 0  l p  n p
with NÝ 4lbv in n

E0 
B02
4 e( l e
ne
ne  l p
np
np )
Particle acceleration by direct electric field in an active region modelled by a CA model
a
x
l
to the particle energy gain per
unit time
B0
b
Inflow vin
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
Acceleration model
Electric field
Particle energy gain (electron, proton and heavy
ions)
   eE0  l  We use a simple approach of the acceleration by direct
electric field =random([0,1]) efficiency of the acceleration
2
0
B
e  
4 (ne  n p )
B02
 p  
4 (ne  n p )
B02
i  Z i
4 (ne  n p )
CA Model
OR :X-CA (hybrid simulation)
OR: MHD simulation ; extrapolation

i 
l p
le
li
le
np
ne
ni
ne
RCS (direct electric field)
Particle acceleration by direct electric field in an active region modelled by a CA model
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
Acceleration model
Inflow vin
B┴
We use 3 assumptions for B//
y
B// = 0 (Speiser, 1965)
 E0
  2mc 
 B
2




2
∆le
B//
∆lp
E0
B//
z
me
Protons gain more energy than electrons
Particle acceleration by direct electric field in an active region modelled by a CA model
a
x
l
mp
B0
b
Inflow vin
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
Acceleration model
Inflow vin
B// >Bmag(e-,ions) (Litvinenko, 1996)
electrons and ions are magnetized
=> follow the magnetic field lines
Bmag ( e, p ) 
B//

eaE0
B
y
∆le
mc 2 E0 B0
eaB
B┴
B0
B//
∆lp
a
E0
x
l
B//
z
b
Inflow vin
  i  1
Same energy gain for all particle
B┴
B0
Electron trajectory
Dauphin & Vilmer
Particle acceleration by direct electric field in an active region modelled by a CA model
B//
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
Acceleration model
Inflow vin
B┴
(e-)
B// >Bmag
only the electrons are
magnetized
me

mp
y
∆le
B//
∆lp
E0
a
x
l
B//
Zme
i 
mi
z
b
Inflow vin
We have the particle energy gain for 3 different RCS configurations
For each particle (e- or ions) we select a value of  and we select a value
of B2free from the energy release time series
2
0
B
e  
4 (ne  n p )
 p  
2
0
B
4 (ne  n p )
B02
i  Z i
4 (ne  n p )
Particle acceleration by direct electric field in an active region modelled by a CA model
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
Particle distribution
x
Log(dN/dE)
y
Particle energy
Equivalent to a Levy Walk  powerful events govern the particle trajectory
time
z
?
Log(Particle energy)
Particle trajectory
We calculate the particle energy distribution for 106 particles from a
maxwellian distribution (T=106 K)
We normalize the electric field in
- For electrons, protons, and heavy ions the case B// large to the Dreicer
electric field.
- For the 3 different configuration of RCS => free parameter
Particle acceleration by direct electric field in an active region modelled by a CA model
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
Particle distribution
Example for electron
Emin
Nmax
Particle acceleration by direct electric field in an active region modelled by a CA model
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
Particle distribution
Example for proton
Emin
Nmax
Particle acceleration by direct electric field in an active region modelled by a CA model
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
Particle distribution
Example of alpha
energy distribution
No difference
between the two
spectra in
energy/nucleon for
the case Bsmall
Emin
Nmax
Particle acceleration by direct electric field in an active region modelled by a CA model
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
Bsmall
Bmiddle
gamma-ray flux
Conclusions
Energy contained
Energy contained in
accelerated particles
(for 1 arcsec3)
Particle acceleration by direct electric field in an active region modelled by a CA model
Blarge
Btotal
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
X-Ray flux
Thick target approach
Cases observed:
- Bsmall, Emin=1000ED
- Emin =10ED
Emin
Nmax
Particle acceleration by direct electric field in an active region modelled by a CA model
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Gamma Ray flux
We compute the gamma ray ratio calculated in the thick target
approximation
12C+p
24Mg+p
12C+
24Mg+
16O+p
 4.438 MeV
28Si+p
16O+
16O+p
28Si+
 1.364 MeV
 1.779 MeV
 6.129 MeV
16O+
20Ne+p
 1.634 MeV
20Ne+
Particle acceleration by direct electric field in an active region modelled by a CA model
Conclusions
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
Gamma Ray flux
gamma ray ratio (proton+alpha)
4
C/O
N1000 ED100 photosphere
N1000 ED1000 photosphere
Si/O Ne/O Mg/O
3
2,5
2
1,5
1
0,5
0
5
4
3
2
1
0
C/O, Si/O, Ne/O, Mg/O
3,5
N100 ED100 photosphere
N100 ED1000 photosphere
N1000 ED100 corona
N1000 ED1000 corona
N100 ED100 corona
N100 ED1000 corona
Observed
Abundance
of the
ambient
plasma
energy of the gamma ray
Particle acceleration by direct electric field in an active region modelled by a CA model
corona
photosphere
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
Gamma Ray flux
Observations from Share and Murphy, 1995
4,5
4
4
3,5
3,5
3
gamma(Ne/0)
gamma(C/O)
3
2,5
2
1,5
2,5
2
1,5
1
1
0,5
0,5
0
0
0
5
10
15
20
0
5
10
flare
Average=1.06
15
flare
Average=1.44
Particle acceleration by direct electric field in an active region modelled by a CA model
20
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
Gamma Ray flux
Observations from Share and Murphy, 1995
1,4
2,5
1,2
2
1,5
gamma(Si/O)
gamma(Mg/O)
1
1
0,5
0,8
0,6
0,4
0,2
0
0
0
0
5
10
15
20
5
10
-0,2
flare
Average=1
flare
Average=0.5
Particle acceleration by direct electric field in an active region modelled by a CA model
15
20
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
Gamma Ray flux
gamma ray ratio (proton+alpha)
Si/O and Mg/O
correspond to the
coronal
abundance
4
C/O
3
2,5
2
1,5
1
0,5
0
5
4
3
2
1
0
energy of the gamma ray
Particle acceleration by direct electric field in an active region modelled by a CA model
C/O, Si/O, Ne/O, Mg/O
3,5
C/O in
agreement with
the ratio deduced
by using a
photospheric
abundance
Problem with
Ne/O
N1000 ED100 photosphere
N1000 ED1000 photosphere
Si/O Ne/O Mg/O
N100 ED100 photosphere
N100 ED1000 photosphere
N1000 ED100 corona
N1000 ED1000 corona
N100 ED100 corona
N100 ED1000 corona
Observed
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
Conclusions
We investigate particle acceleration due to interaction with many RCS.
The magnetic energy release distribution is given by a power law
- particle energy distributions wander from a power law with the
increase of the interaction number and strongly depend on the
considered RCS configuration
Spectral index of the particle distribution is function of the considered
energy range
This implies different X-ray spectra and gamma ray line fluence ratio;
in most cases X-ray spectra are too flat compared to observations. This is
mainly due to the spectral index of the magnetic energy released
distribution which is -1.6.
 Observed gamma ray lines fluence ratio can be reproduced except for
Neon
Particle acceleration by direct electric field in an active region modelled by a CA model
Introduction
CA model
Acceleration model
Particle distribution
X-ray flux
gamma-ray flux
Conclusions
Conclusions
=> This implies different X-ray spectra and gamma ray line fluence ratio
Energy contained in electron and proton strongly depends on the RCS
configuration -> see observations
With a volume of 102-103 arcsec3, it is possible to obtain enough energy in
electron and proton to reproduce most of the observations
Particle acceleration by direct electric field in an active region modelled by a CA model