ref) Takamoto+(2015), MNRAS 454, 2972.

Explosive Reconnection ofthe Double Tearing Modein Poynting-Dominated Plasmas
Makoto Takamoto

Department of Earth and Planetary Science
The University of Tokyo

collaborators:
Jérôme Pétri
Hubert Baty
(Université de Strasbourg, France)

1. Introduction!

2. Double Tearing Mode!

3. Spectrum Evolutionref) Baty+(2013), MNRAS, 436, L20.
Petri+(2015),PPCF, 57, 014034.
Takamoto+(2015), MNRAS 454, 2972.

1.1. Flares from High-Energy Astrophysical Phenomena
Rep. Prog. Phys. 77 (2014) 066901

Energy [eV]PWNe10710−8GeV
109108

Review Article

10−91035

νLν [erg s−1]νFν [erg cm−2 s−1]

1036

10−1010−11 21101022

1034102310241025

ν [Hz]ref) flux
Buehler+
ApJsix749,
Figure 9. SED at the maximum
level for(2012),
five of theCrab24.

nebula flares detected as of September 2013 (Abdo et al 2011,
Buehler et al 2012, Striani et al 2013, Mayer et al 2013). No
spectrum has been published for the low intensity flare of July 2012
(Ojha et al 2012). The blue points show the average nebula flux
values referenced in figure 2.

time scale:

8-10 hours

Peak350
MeVet al 2013).Energy:
Due to the statistical nature>of theflux variations,the definition of a flare is somewhat arbitrary. In all flares
reported to date the synchrotron component of the nebula had
a peak flux of Fns,100 > 35 × 10−7 cm−2 s−1 . Equally, thedefinition of a flare duration is not straightforward; typically,
the flux is increased with respect to the monthly average flux

spectrum:

Hayashida et al.

the <1 arcsec angular resolution achieved in these wavebands.
However, to date, despite extensive efforts a detection of theflares outside of the HE gamma-ray band remains elusive.
Strictly simultaneous observations were obtained during theSeptember 2010, April 2011 and March 2013 flares (Lobanovet al 2011, Morii et al 2011, Vittorini et al 2011, Aliu et al 2013,
Weisskopf et al 2013). Particularly dense observations with theChandra and Hubble Space Telescopes and the Keck and VLA
Observatories were carried out during the April 2011 flare. No
increased emission was detected from radio to x-rays for any
structure of the nebula above the usual levels. This finding wasvery unexpected. The inferred flux upper limits show that theSED of the HE gamma-ray flare drops steeply with decreasing
frequency, as was already suggested by the hard spectral index
of the flaring component in HE gamma-rays during the flare.
No pulsations are found in the gamma-ray emission ofFigure 15. Time-resolved broadband spectral energy distributions of 3C 279 measured in Periods A–H (as defined in Table 1) and on 2008 July 31 (MJD 54678),
coveredbyourobservational
campaigns
in 2008–2010.
X-ray, UV–optical–near-IR
data are corrected
for the Galactic absorption.
Five-digit
numbers in the panel
the flares.
Thepulsarproperties
remainunchanged
duringtheindicate MJD of the periods. For comparison, the gray open circles in the very high energy γ -ray band represent measured spectral points by MAGIC in 2006 Februaryref)
Hayashida+,(2012),
ApJ
754,114.
(Albert et al. 2008).
outbursts; no changes in the spin-down period or flux wereFurthermore,
we notethat there
are someor
opticaland γ -ray
2. (Abdo
Inefficient electron
Acceleration of the relfound inradio,
x-raysgamma-rays
et alacceleration.
2011, Morii
peaks that might well be associated with the second X-ray flare.
ativistic electrons at proton-mediated shocks is likely toHence,possible thatthe two prominent
proceedin twothe first one low-energy electrons
et alit is2011,
Buehler
et alγ -ray/optical2012).flares
The time
scale
ofsteps:thein recurrence(Periods B and D), together with the subsequent two X-ray flares
may be pre-accelerated via, for example, some collective(Periods F and G), form a sequence of four events separated byprocesses involving protons; in the second step, they mayof pulsar glitches is similar to the recurrenceof the gamma-ray
a similar time intervals. Those intervals, in turn, can be possibly
participate in the first-order Fermi acceleration process. If
determined by instabilities in the jet launching region. Here, theunder some conditions the electron–proton coupling is inefflares;
however,there
is nocorrelation
in oftime
betweendifferent
broadband
spectra during
these events
may obvious
result fromficient, the fraction
electrons
reaching the Fermi phase ofsmall changes of parameters, such as the jet direction, Lorentz
acceleration will be small. In this case the X-rays, originatthese
twoevents.
factor, and/or location and geometry of the dissipation event.
ing from lower energy electrons, are produced efficiently,

Blazars

1010

Sept 2007
February 2009
September 2010
April 2011
March 2013

GeV TeV

The Astrophysical Journal, 754:114 (22pp), 2012 August 1< days~ GeV

A weak (and sporadically almost absent) correlation between X-rays and other spectral bands can also result fromsuch processes that preferably contribute to radiation in theX-ray band. They can be related to the following threemechanisms/scenarios.

while the γ -rays and optical radiation that involve more
relativistic electrons are not.
3. The X-rays can be also contributed by hadronic processes,
specifically by the pair cascades powered by protons losing
their energy in the photo-mesonic process (Mannheim &Biermann 1992). For this process to be efficient, it requires
extreme conditions (Sikora et al. 2009; Sikora 2011);
however, operating in the very compact central region, atdistances less than few hundred gravitational radii, it mayoccasionally dominate in the X-ray band.

5.2. The origin of the gamma-ray flares?

date, the Crab flares remain mysterious. We do not know
not Topower-law
what causes them and where they come from within the
1. Bulk-Compton process. This involves Compton-scatteringof ambient optical/UV light by the cold (non-relativistic)
electrons in the jet. This mechanism is most efficient close
to the accreting black hole where the processes responsible
for the variability of X-rays may operate independently of
those at larger distances and producing there variable non-nebula. Several ideas have been proposed,
but no definite
4.2. Broadband Spectral Energy Distribution

1.2. Energy Conversion Processes
1. Relativistic Shock
Pros: kBT/mc2 ~ Γ/3√2 >> 1 (if unmagnetized)
Cons: Magnetic field should be weakinefficient particle accelerator in most cases?
(large escaping probability)

2. Magnetic Reconnectionref) Hoshino (2008), ApJ, 672, 940.
Giacche & Kirk (2016), arXiv:1612.04282

Pros: Rapid conversion of Magnetic Field (~10L/cA)
γmc2 ~ F(σ) (increasing with σ)
Cons: current sheets are necessaryvery local heating?

1.3 1-sheet Reconnection (Plasmoid-Chain)γukB T/mc2

hot but

very locallyxx

kB T/mc2

γuf(T) T-2ref) MT (2013), ApJ 775, 50.

Can magnetic reconnection
explain very strong bursts?

1. Introduction!

2. Double Tearing Mode!

3. Spectrum Evolutionref) Baty+(2013), MNRAS, 436, L20.
Petri+(2015),PPCF, 57, 014034.
Takamoto+(2015), MNRAS 454, 2972.

2.1. Single Tearing Mode
B0

δ

-B0
inducing inﬂows

2.2. Double Tearing Mode
B0-B0

B0

Baty+(2013), MNRAS, 436, L20.
2.3. Double Tearing Mode 2 ref) Petri+(2015),PPCF,57, 014034.

Takamoto+ (2015), MNRAS, 454, 2972.

log10[pgas/pB]
B0y-B0B0x

2.4. Double Tearing Mode 3

for different Lundquist numbers: S = 200, 400, 800, 1600, 3200. A mildd a high magnetization case on the left in figure σ = 12.

0

Burst phasey0=3L, S=3200
1Max(Vx)

0.1

σ=0.2
σ=0.4
σ=0.6
σ=1.2
σ=12
σ=120

0.01

1200

0.001

01000

2000

2-sheet

3000

4000

5000

6000t / (L/c)

linearphasefor runs usingdifferent
magnetizations: σ. The Lundquist number is S = 200 oninteraction

PoyintingFlux
B2⌘'ParticleEnergyFlux
4⇡wc2 2

1102.5. DTM Temperature dependence
-2-1

10100

11010210ε [eV]310410105610

log10kBT/mc2, σ=120

S=320010000Θ ≡ (kB T / m c2)

1000
100
101
7.5 σ0.915

0.1
0.1

110σσ

100

1000Figure 5. Top: The energy spectrum with various magnetization parameter . Bottom: maximum temperature with respect
2 toB parameter. In both case, S = 3200 isbackground magnetization
assumed, and the pulsar parameters
are r = 50rL and W = 2300.2
2k T
B' 7.5 = 7.5
mc4⇡wclinear in σequilibrium due to the MHD approximation, the Lorentz factor in Equation (4) can be expressed for an ultra-relativistic
plasma equation of state as: ⇠ 3kB T /mc2 ⌘ 3⇥. For latera large volumeis heated(12.6. DTM Burst Timescaleϵ̄sync ∼ 1.73 × 1= 0.156

where B̄0 = B0 /the fluid comovi
Lorentz factor oftion into the puls

burst occurs

when 2 sheets approachessuﬃciently
Figure 9. toThe geometry
temperature profiles log10 [kB T/mc2 ] during the double tearinsensitive
ing burst phase of the run SB6. The top and bottom panels show just before
and after the burst phase, respectively.

sheet width⌧burst⇠plasma
50 gives very narrow reconnection
the high-Lundquist
numberc
sheets in which several plasmoids can be observed(see also ourPaper II). The plasmoid region becomes very dense and also very hotwith temperature as high as kB T ! σ mc2 because of the compression´ϵsync,L =%W (1 −where ϑ is the anand the observer
unit of light velothat this blob is pδ W ≃ 2% W , the pϵsync,L ∼ 0.311 eV

The energy spec
over the numeric
ber density of em
panel of Fig. 10
the magnetization

1. Introduction!

2. Double Tearing Mode!

3. Spectrum Evolutionref) Baty+(2013), MNRAS, 436, L20.
Petri+(2015),PPCF, 57, 014034.
Takamoto+(2015), MNRAS 454, 2972.

our numerical results. The typical photon energy for the3.1. Thermal
Synchrotron
Spectrumynchrotron
emissioncan be writtenas:3✏sync =
22 B

Bq

2mc ,

(4)

where B is the magnetic field strength
measured*Lorentzas
transform(Γbulk) in the
3 2*γ~3 k2
BT/mc
rame where ✏sync is detected, Bq =m c /e~ is the critica*kBT/mc2 ~ 7.5 σmagnetic field, m the electron mass, e its electric charge,◆⇣◆ the reduced
he typical Lorentz factor of✓theelectrons~⌘2 ✓ andB̄
bulk✏sync,Lab Assuming⇠ 0.88[keV](1)is2
22Planck constant.
thesynchrotronradiation
101010 [G]mainly emitted by thepairplasma
which
isin
localtherma(for thermal particles)

non-thermal
tail >1–8
the above valuec 2015 RAS, MNRAS
000,

Next, we investigate the synchrotron energy spectrum
5our numerical results. The typical photon energy fo
synchrotron emission can be written as:

Thermal synchrotron radiation from DTM

3.2. Applications for Crab GeV Flare
S=32003 2B
mc2 ,Takamoto, Pétri,
2 & BBaty
q108✏sync =

σ=0.2
6
σ=1.2
σ=12
is themagnetic
fieldthestrengthasthemeasured
σ=120where B
where# is theangle between
velocity ofreconnectini

710N(ε)106

3blob and✏sync
the observerline ofsight,
is2 cthevelocityof thframe whereis detected,
Bq = Wm/e~is thec*Lorentz
transform
wind
in themunitof light
velocity,
andis 2the“Dopplerfac
Wmagnetic
field,
theelectronmass,e itselectriccha*γ~3kBT/mc
tor ”. If we assume that this blob is pointing towards Eart
*B=B
L (rL/r) and ~ the re
the typical
Lorentz factor of theelectrons
in the most favorable case, W ' 2 W , the peak energy oPlanck theconstant.
Assuming
the synchrotron radiati
photons willbe✓ ◆ 1 which✓◆ ✓is in local◆ thmainly emitted by the pair plasmarB̄
B

105
104
103
102

✏sync,L ⇠ 0.311[eV] ⇥2101 -2
1010-110000Θ ≡ (kB T / m c2)

1000
100
101100101

102
ε [eV]

103

ε[eV]

104

LrLc 2015
RAS,MNRAS 000, 1–85610

B̄0100[T](7The energy spectrum can be obtained by integrating
Equa
10Rtion (7) over the numerical spatial domain, dV n✏sync
where n is the number density of emitting particles in thS=3200
laboratory frame . The top panel of Figure 5 shows the energy spectra with several value for the magnetization parameter . Typical values for the Crab pulsar are: r = 50r
and W = 300 3 . This panel clearly shows the energy isync,Ltheir body part increases with -parameter, approximatel
100 times increase as -parameter increases by one order o(when σ~105, r~50r
L, B~0.1B
magnitude.
This can0)be understood from the bottom paneof Figure 5. The panel is the maximum temperature durin
the explosive phase in terms of the -parameter. It show
the maximum temperature increases linearly with theparameter. Since the photon energy depends on the temperature as T 2 , as indicated in Equation (7), the resultin

ε~ 300[MeV]3.3. Temporal Evolution of Photon Spectrum(when σ~120, r~50rL, B~0.1B0)

3.4. Time Scale
sheet width⌧burst ⇠ 50c
Sheet width:
if MHD (density is sufficiently high)basically no criterion(~ √(η/Δt) if highly collisional)

if collisionless plasma (low density)
sheet width ~ min{Gyro radius, skin depth}(maximum energy will be limited by synchrotron cooling)3.5. Scenario
Striped WindDTMCrabDTM

Lorentz transformationr ~ 50 rL

allows to avoid
acceleration energy limitSummaryref) Takamoto+(2015), MNRAS 454, 2972.

• We investigated a candidate of the strong flare origin
considering Double Tearing Mode (DTM).
!

• Double tearing mode (DTM) is a plasma instability
resulting in a sudden energy release.
!

• The thermal synchrotron obtained by the simulations
gives an energy spectrum resembling to the observations.
!

• The high Lorentz factor of the wind allows us to explain
energy flux naturally.

O Ma Ke

1.4 1-sheet Reconnection (Turbulent Reconnection)ref) MT+, (2015), ApJ 815, 16.
kB T/mc2γuTurbulent
sheet

more broadbutmore moderateand steadykB T/mc2γu2.4. Double Tearing Mode 3

strongly
thermalized

drivenreconnection

log10[kB T/mc2]

1.6. Crab Flares 1Iref)Buehler+ (2012), ApJ 749, 24.
Buehler+ (2014),RPR, 77,066901.

Rep. Prog. Phys. 77 (2014) 066901⪆100MeV

1% ofLspindown
the <1arcsec angular rEnergy [eV]10−8107108

1091010to date, desp(~5 However,1038erg/s)

νLν [erg s−1]νFν [erg cm−2 s−1]

flares outside of the H
Strictly simultaneous o
September 2010, April1036
et al 2011, Morii et al 210−9
Weisskopf et al 2013).
Chandra and Hubble Sp
Observatories were car1035
increased emission wa−1010structure of the nebula
Sept 2007very unexpected. TheFebruary 2009
September 2010
SED of the HE gamma
April 20111034
frequency, as was alrea
March 2013of the flaring compone
10−11 211010221023
10241025No pulsations areν [Hz]
the flares. The pulsar pnoﬂare
case
Figure 9. SED at the maximum flux level for five of the six Craboutbursts; no changes
nebula flares detected as of September 2013 (Abdo et al 2011,
found in radio, x-rays oBuehler et al 2012, Striani et al 2013, Mayer et al 2013). Noet al 2011, Buehler et alspectrum has been published for the low intensity flare of July 2012of pulsar glitches is sim(Ojha et al 2012). The blue points show the average nebula flux
values referenced in figure 2.
flares; however, there is1.5. Crab Flares 1
Rep. Prog. Phys. 77 (2014) 066901

ref)Buehler+ (2012), ApJ 749, 24.
Buehler+ (2014),RPR, 77,066901.

2011 April Crab Flare
Review Article~ 8 hours

Figure 10. Integral flux above 100 MeV from the direction of the Crab as a function of time during the 2011 April flare, reproduced from
Buehler et al (2012). The points represent the sum of the nebula and pulsar fluxes. The dotted line indicates the sum of the 33-monthaverage fluxes from the inverse Compton nebula and the pulsar, which are stable over time. The dashed line shows the flux of the averagesynchrotron nebula summed to the latter. The red vertical lines indicate time intervals where the flux remains constant within statisticaluncertainties. The time windows are enumerated at the top of the panel. The corresponding flux is shown by the red marker below eachnumber. The SED for each of the time windows is shown in figure 11. Reproduced by permission of the AAS.

~9 days

Strong time variability in gamma-ray (~100MeV)!!

1.7. Theoretical Models and Difficulties 1ref) Guilbert+(1983), MNRAS,205
Uzdensky+(2011),
2 −1 ApJL, 737

Assumption:τsyn ∝ (γB )τB2) ∝ (γ/B).
in MHD plasma (B2 > E

acceleration time > Larmor period
2 −1τsyn ∝ (γB )(Minimum 2timescale)(γ B)Max = const.,
2−1τB ∝ (γ/B).
τsyn ∝ (γB )
2ϵ∝
γB
sync
acceleration time < synchrotron coolingtimeτB ∝ (γ/B).
2
maximumradiation energy(γ B)Max = const.,~ 160MeV(γ 2 B)Max=2 const., < εobs~375 MeV

ϵsync ∝ γ B

3 2B
ϵsync = γmc2 ∝ γ 2 B
2 Bq

(1)
(2)

(3)
(4)

1.8. Theoretical Models and Difficulties 2
observed duration of the flares:

τflare ~ 10 hours
if pulsar origin:P~33[ms]<< τflare

if PWN origin :1[pc]/c ~ 1yr >> τflare3.2. Radiation Energy Flux & Duration
no beaming case
(PWN)

beaming case
(Wind)(E×B)c(E×B)cobserver
Photons by DT

Photons by DT

only a part of photons

nearly all the photons

can be observed.

can be observed.

only a partof the emitted
should be converted.

Hence more than 1% of Poynting energy

photons can be observed1%of1%energy
conversion
At mostPoynting energy
is suﬃcient!!

conversion is suﬃcient.

(L ~ 0.01 Lspindown)1.9. Reconnection in PWN

X-line = dissipation region

E2>B2ref) Uzdensky+(2011), ApJL, 737
Cerutti+(2013), ApJ
Cerutti+(2014), ApJ, 782

B0E-B0Condition: a very long and coherent sheet
very weak perturbation on particles
weak guide fieldNecessary Conditions of DTM
Necessary Condition:

B0y0 ≾ 6 L(otherwise, interaction of 2 sheets-B0becomes too weak)

2L2y0

B0

Observed Sheet-Thickness in Wind
No. 2, 2010

gamma-ray pulseref) Petri+(2005),ApJL, 627, 37.
Abdo+(2010), ApJ, 708, 1254.

FERMI-LAT OBSERVATIONS OF THE CRAB PULSAR AND NEBULA
1800

1257

400
350

1600

1403001202501400

100200150

12008010060

C o u n ts

50

1000

00.96 0.97 0.98 0.99

11.01 1.02

0.36 0.37 0.38 0.39 0.4 0.41 0.42sheet thickness:

800~0.1

600

2 πrL

400

2000sheet separation:
2 πrL00.2

0.4

0.6

0.8

11.2

1.4

1.6

1.8

2PhaseFigure 1. Light curve obtained with photons above 100 MeV within an energy-dependent circular region, as described in Section 4.1. The light curve profile is binned
to 0.01 of pulsar phase. Insets show the pulse shapes near the peaks, binned to 0.002 in phase. The radio light curve (red line) is overlaid (arbitrary units). The main
peak of the radio pulse seen at 1.4 GHz is at phase 0. Two cycles are shown.
(A color version of this figure is available in the online journal.)

all events between 100 MeV and 300 GeV, two clear peaks
P1 and P2 can be seen at phases φ1 = 0.9915 ± 0.0005 andφ = 0.3894 ± 0.0022, respectively. Hence, the peaks areof 1.4 µs, which is significantly smaller than the 21.1 µs
accuracy of the overall timing solution, and therefore leads toan error of 0.0006 in phase on the γ -radio delay.

3.3. Cut-off energy and Constraints on Crab Parametersmaximum energy:
DTM dynamical time ~ radiation cooling time!
"!
"−2 !
"2τ̄syncΓB̄r
W=9.4 × 105
Θ−1τ̄dyn300
50rL
0.05B̄0!
"−2 !
"−1 !
"−1B̄Ll/2πrLP×.
100[T]
0.05
33[ms]
τ̄dyn ≃ 50l/c = 25 [s]ϵsync,L ∼ 311[MeV]!

Θ106# α $ !Γ " !
W

0.05

"2 !

r50rL

300

"−1 !

P33[ms]B̄
0.05B̄0"

(1)

,"!

(2)

BL
100[T]"(1)

Next, we investigate the synchrotron energy spectrum
our numerical results. The typical photon energy fo
synchrotron emission can be written as:

DTM in Blazar

Thermal synchrotron radiation from DTM

S=320010141013m=0.2
m=1.2m=12m=1201011
1010
109
108 510108

3 2Bmc2 ,2S=3200
Bq

σ=0.2
where B is thestrength
as measured i
σ=1.2
107 magnetic field*Lorentz transform
σ=122 3frame where10✏6sync is detected,
Bq k=
mc2 /e~ is the cσ=120*γ~3BT/mc
magnetic field,m the electron
mass,
*B=B
L / Γe its electric cha
105
the typical Lorentz
factor of theelectrons*N(ε)=∫ nε dVand ~ the re104
Planck constant.
Assuming the synchrotron radiati103
mainly emitted
by the pair plasma which is in local th210N(ε)

1012✏sync =106107108✓◆2 ⇣ ⌘ ✓2101⇥ 3-11⌫Lab,sync⇥010810[Hz]
10-2 '107.53
1010210c2015
RAS,MNRAS000,
1–8101011121314ε[eV]1010101010109i [Hz]

ν[Hz]

1041

B0 6
105
10100[mG]◆

S=320010000
1000

νsync,L ~ 109[Hz](when σ~1, Γ~ 5, B~0.1[G])Θ ≡ (kB T / m c2)

N(i)

1015

5100
10kBT/mc2 ~ 10σ

1

7.5 σ

0.1
0.1

Figure 4. The temperature profiles log [k T /mc2 ] during the110σσ

0.915

1001000

Figure 5. Top: The energy spectrum with various magnetiza-