A multi-island mechanism for particle acceleration during magnetic reconnection in

solar flares

J. F. Drake and M. SwisdakUniversity of Maryland

Some energetic particle observations in impulsive flares

• In solar flares energetic electrons up to MeVs and ions up to GeVs are measured

• A significant fraction of the released magnetic energy appears in the form of energetic electrons and ions (Lin and Hudson ‘76, Emslie et al ‘05)

– Must explain such extraordinary efficiency and link particle energy gain to magnetic energy release

– The energetic electron beta can approach unity in over-the-limb observations (Krucker et al 2010)

• Correlation between > 300keV energetic electrons and > 30 MeV ions (Shih et al 2009)

– Common acceleration mechanism?

• In impulsive flares see enhancements of high M/Q ions (e.g., Mason ‘07)

Impulsive flare timescales

• Hard x-ray and radio fluxes– 2002 July 23 X-class flare– Onset of 10’s of seconds– Duration of 100’s of

seconds.– Bursts of 100keV x-rays

occur on time scales as short as 100ms (Vilmer et al 1996)

RHESSI and NoRH Data

(White et al., 2003)

RHESSI observations

• July 23 γ-ray flare• Holman, et al., 2003• Double power-law fit

with spectral indices: 1.5 (34-126 keV) 2.5 (126-300 keV)

– Typically see soft-hard-soft spectral evolution

Energetic electron and ion correlation

• > 300keV x-ray fluence (electrons) correlated with 2.23 MeV neutron capture line (> 30 MeV protons)

• Acceleration mechanisms of electrons and protons linked?

Shih et al 2008

RHESSI occulted flare observations

• Observations of a December 31, 2007, flare– All electrons in the flaring region are part of the energetic

component (10keV to several MeV)– The pressure of the energetic electrons approaches that of the

magnetic field.– Remarkable!

30-50keV

17GHz

Krucker et al 2010

Impulsive flare energetic ion abundance enhancement

• During impulsive flares see heavy ion abundances enhanced over coronal values

• Enhancement linked to Q/M

∝QM

⎛⎝⎜

⎞⎠⎟

−3.26

Mason, 2007

Main Points

• A single x-line model of flares can not explain the observations– Parallel electric fields are too localized around the x-line to explain the

large numbers of energetic electrons (Drake et al 2005, Egedal et al 2009)– Magnetic energy release does not take place at the x-line but downstream

as the magnetic field line relaxes its tension• Instead magnetic energy release takes place in regions with volume

filling magnetic islands (Tajima and Shibata 1997, Onofri et al 2006, Drake et al 2006)

• Electron acceleration is dominated by first order Fermi acceleration in contracting islands (Kliem 1994, Drake et al 2006)– Electron energy gain is linked to magnetic energy release

Main Points (cont.)

• Ion acceleration takes place in two steps– Ions entering reconnection exhausts act like “pickup” particles and gain a

thermal velocity equal to the Alfven speed • Abundance enhancements of high M/Q ions occur because during guide-field

reconnection (typical for the sun) they more easily behave like “pickup” particles than protons (Drake et al 2009)

– Super-Alfvenic ions, like electrons, gain energy through Fermi reflection in contracting islands (Drake et al 2010)

• Links acceleration mechanism of electrons and ions

• Both ions and electrons gain energy through island contraction (increasing the parallel energy) until reaching the marginal firehose condition– At the marginal firehose limit reconnection is throttled– The energetic particle beta approaches unity (e.g., Krucker et al 2010)

• Balance of Fermi drive and convective loss leads to powerlaw distributions with spectral indices with a lower limit of 1.5 in a low-beta system

Electron acceleration by the parallel reconnection electric field

• Parallel electric fields during reconnection are typically highly localized near the x-line and along separatrices

• A single x-line model can not explain the large numbers of electrons seen in flares

– Parallel electric fields are too spatially localized to be a significant source of large numbers of energetic electrons

– The electron flux would produce currents that exceed the coronal fields by orders of magnitude

– Finally, the x-line is not where magnetic energy is released.

E||

A multi-island acceleration model

• Narrow current layers spawn multiple magnetic islands in reconnection with a guide field– Must abandon the classical single x-line picture!!

Multi-island reconnection

• Give up single x-line model • How are electrons and ions accelerated in a multi-island

environment?– Fermi reflection in contracting magnetic islands increase the parallel

particle energy

– Rate of energy gain independent of particle mass Thermal protons are not fast enough to bounce since are sub-Alfvenic -- need seed heating mechanism

uin

CAx

CAx

dεPdt

~2εPcALx

Seeding super-Alfvenic ions through pickup in reconnection exhausts

• Ions moving from upstream cross a narrow boundary layer into the Alfvenic reconnection exhaust

• The ion can then act like a classic “pick-up” particle, where it gains an effective thermal velocity equal to the Alfvenic outflow Ti ~ micA

2 ~ 10-100keV/nucleon.• Energy proportional to mass (Fujimoto and Nakamura, 1994; Drake

et al 2009)

What about the numbers problem?

• To explain the large numbers of accelerated electrons in flares the contracting island region must be macroscopic Island region

Parker’s equation and Fermi acceleration in contracting islands

€

∂F∂t

+∇ • uF −∇ • κ • ∇F −13∇ • u( )

∂∂v

vF = 0

w

⇒

Fermi acceleration in contracting islands

€

μ =mv⊥2 /B

w

⇒

€

W =12

mv02 2L

3L0

+L0

2

3L2

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Linking magnetic energy release to particle energy gain

• A key flare observation is that energetic particle energy gain is linked to the released magnetic energy. Why?

• Island contraction continues until the firehose marginal stability condition is reached. At this point

– The rate of particle energy gain equals the rate of release of magnetic energy

– This establishes the key linkage between particle and magnetic energy seen in flares

• Magnetic energy released depends on β0

– Low β0

– High β0

€

dWB

dt+

dW p

dt= 0

€

ΔWB ~ −WB0

€

ΔWB ~ −WB0/ β 0

Particle acceleration in a periodic magnetic field• Simulations of particle acceleration in a realistic sheared magnetic field is

intrinsically 3-D -- Not feasible in PIC model• Treat a system with periodic reversals as a test bed

– Can study for the first time particle acceleration in a true multi-island environment with a PIC model

• Low initial beta with a strong guide field to mimic the corona

Jez

Jez

Development of pressure anisotropy

€

β⊥

€

β||

• The parallel pressure increases faster the perpendicular pressure consistent with the Fermi mechanism.• System approaches the marginal firehose condition at late time• Shuts off reconnection since magnetic fields have no tension at the firehose marginal limit

• e.g., Alfven wave

€

ω 2 = k||2cA

2 1−12

β || +12

β⊥ ⎛ ⎝

⎞ ⎠

Firehose condition

• Within islands and along separatrices bump against the firehose condition

– Not as clear as in anti-parallel case (Drake et al 2010)

– This condition limits island contraction

– Controls particle spectra

• Self-consistency is crucial in exploring particle acceleration

€

β|| − β⊥− 2

Jez

Electron and ion energy spectra

• Both ions and electrons gain energy

– Electrons gain energy first and then saturate. Why?

• A key feature is that the rate of energy gain of particles increases with energy

– first order Fermi

• Ongoing study of acceleration mechanisms

ions

electrons

€

dεdt

∝ ε

E

f(E)

1-D Model equations• Rate of energy gain: first order Fermi

• Model equation for the omni-directional distribution function

• Above the source energy this is an equidimensional equation– powerlaw solutions

€

˙ v =dvdt

=1τ h

1−4πpB2

⎛ ⎝

⎞ ⎠

1/ 2

v

€

∂F∂t

+∂∂v

˙ v F = −1

τ L

F − F0(v)[ ]€

t h =cA

Lw

−1

€

tL =cA

L

−1

€

F(v, t) = 4πv 2 f (v, t)

Distributions and spectral indices• Exact steady state solutions for F(v)

• Spectral index

• Since the island sizes are much smaller than the system size,

– spectral index controlled by marginal firehose condition

€

F(v) = γ −1( )v −2γ ds0

v

∫ s2γ −1F0(s)

€

(2γ −1) 1−4πp0

B2

2γ −12γ − 3

⎛ ⎝ ⎜

⎞ ⎠ ⎟1/ 2

=τ h

τ L

€

γ =1.5 +β 0

2

€

t h << τ L

Magnetic islands release energetic electrons in bursts

• Secondary magnetic island merging with a large-scale field releases energetic electrons in narrow filaments– Source of millisecond bursts of energetic electrons?– Can analysis of the time structure of energetic electrons reveal

information about the size distribution of magnetic islands?

Te

Conclusions

• The single x-line model can not explain the large number of energetic electrons seen in flares

• Magnetic reconnection with a guide field as in the corona naturally leads to a multi-x-line configuration– High energy particle production during magnetic reconnection involves

the interaction with many magnetic islands

• Electron acceleration is dominated by a Fermi-like reflection in contracting magnetic islands

• Ion interaction with the reconnection exhaust seeds them to super-Alfvenic velocities.– Ions that act as pickup particles as they enter reconnection exhausts gain

most energy• M/Q threshold for pickup behavior• Yields abundance enhancement of high M/Q ions

Conclusions (cont.)

• Efficient heating of super-Alfvenic ions through magnetic island contraction

• Balance of contraction drive and convective loss yields powerlaw solutions for all species– Spectral indices are controlled by the approach to firehose stability

• M/Q threshold for pickup behavior is a possible explanatation of impulsive flare heavy ion abundance enhancements – This hypothesis can be tested with PIC simulations

Mirror and firehose conditions

• Within islands bump against the firehose condition

– This condition limits island contraction

– Controls particle spectra

• In current layers and along separatrices bump against mirror mode limit

• Self-consistency is crucial in exploring particle acceleration

firehose

mirror

Pickup threshold: guide field

• Protons and alpha particles remain adiabatic (μ is conserved)• Only particles that behave like pickup particles gain

significant energy threshold for pickup behavior

mi=16mp

t

μi

Ey

Bz0 = 5.0

€

viy

Δ≈0.1cApxρsp

>Ωi ⇒μ i

Ziμ p

>10cps

cApxβpx

€

ΔT⊥ =12

micAx2

€

ΔT|| = 0