Relativistic Magnetic Reconnection in Kerr Spacetime

0.33 MB·

by  Felipe A. Asenjo

Accepted for publication in Physical Review Letters
Relativistic Magnetic Reconnection in Kerr Spacetime
Felipe A. Asenjo1,∗ and Luca Comisso2,†
1Facultad de Ingenier´ıa y Ciencias, Universidad Adolfo Iba´n˜ez, Santiago 7941169, Chile.
2Department of Astrophysical Sciences and Princeton Plasma Physics Laboratory,
Princeton University, Princeton, NJ 08544, USA.
The magnetic reconnection process is analyzed for relativistic magnetohydrodynamical plasmas
around rotating black holes. A simple generalization of the Sweet-Parker model is used as a first
approximation to the problem. The reconnection rate, as well as other important properties of
the reconnection layer, have been calculated taking into account the effect of spacetime curvature.
Azimuthal and radial current sheet configurations in the equatorial plane of the black hole have
7
1 been studied, and the case of small black hole rotation rate has been analyzed. For the azimuthal
0 configuration,itisfoundthattheblackholerotationdecreasesthereconnectionrate. Ontheother
2 hand, in the radial configuration, it is the gravitational force created by the black hole mass that
decreases the reconnection rate. These results establish a fundamental interaction between gravity
n
and magnetic reconnection in astrophysical contexts.
a
J

3
Introduction.-Therapidconversionofmagneticenergy field of a rotating black hole on the magnetic reconnec-
1
into plasma particle energy through the process of mag- tionprocess. Ultimately,ourgoalistodeterminehowthe

netic reconnection is of paramount importance in many reconnection rate, and other properties of the reconnec-
]
E astrophysical processes [1]. Magnetospheric substorms, tion layer, are modified by spacetime curvature effects.
H coronal mass ejections, stellar and gamma-ray flares are Governing equations.- We consider a plasma governed
. just a few examples of pheneomena in which magnetic by the equations of General Relativistic Magnetohy-
h
reconnection is thought to play a crucial role. drodynamics (GRMHD) [25, 26], which are composed
p
- In recent years, significant work has been undertaken by the continuity equation ∇ν(ρUν) = 0, the energy-
o
to better understand magnetic reconnection in magnet- momentum equation
r
t ically dominated environments, where relativistic effects
s ∇ (hUµUν)=−∇µp+JνFµ , (1)
a become significant [2]. This has led to the generalization ν ν
[
of the classical Sweet-Parker and Petschek reconnection and the resistive Ohm’s law

1 models to the special relativistic regime [3, 4]. Further-
v more, many numerical campaigns have been devoted to UνFµν =η(Jµ−ρ(cid:48)eUµ). (2)
9 the investigation of the reconnection rate [5–12] and the
6 particle acceleration [13–20] in this regime. Relativistic Here, ∇ν denotes the covariant derivative associated to
6 the connections of the curved spacetime, Uµ and Jµ are
reconnection has been found to be a very efficient mech-
3 the four-velocity and four-current density, respectively,
0 anism of magnetic energy conversion and particle accel- while Fµν is the electromagnetic field tensor. Further-
. eration, making it a primary candidate to explain non-
1 more, ρ, h and p are the mass density, enthalpy density,
0 thermal emissions from pulsar wind nebulae, gamma-ray andpressureoftheplasma. Finally,η istheelectricalre-
7 bursts, and active galactic nuclei.
sistivity and ρ(cid:48) =−UνJ is the charge density observed
1 e ν
While special relativistic effects on the reconnection by the local center-of-mass frame of the plasma. The
:
v process are starting to become clear, the effects related description of the dynamics is then completed when the
Xi to the spacetime curvature are far less explored and fluidequationsarecomplementedbyMaxwell’sequations
r a detailed understanding is lacking. General relativis- ∇νFµν = Jµ and ∇νF∗µν = 0, where F∗µν is the dual
a tic magnetohydrodynamic simulations have repeatedly of the electromagnetic field tensor.
shown the formation of reconnection layers in proxim- A very effective representation of these equations can
ity of blacks holes [21–24], where spacetime curvature be obtained by writing them in the 3+1 formalism [27].
effects can be important. However, the difficulties re- In this case, the curvature effects are displayed explicity
lated to the spatial and temporal resolution of typical in a set of vectorial GRMHD equations. In this formula-
reconnection processes have been such to prevent their tion, the line element can be written as
thorough study.
3
A step forward in the comprehension of magnetic re- ds2 =g dxµdxν =−α2dt2+(cid:88)(h dxi−αβidt)2, (3)
µν i
connectionincurvedspacetimecouldbetakenbystudy-
i=1
ing simple generalizations of known theoretical models.
ItisthepurposeofthisLettertodevelopsuchtheoretical whereα=[h2+(cid:80)3 (h ω )2]1/2isthelapsefunctionand
0 i=1 i i
study considering the contribution of the gravitational βi =h ω /αistheshiftvector,whileh2 =−g ,h2 =g
i i 0 00 i ii
2
and h2ω =−g =−g are the non-zero components of
i i i0 0i
the metric. Notice that the shift vector can also be writ-
2δ
(cid:112)
ten as βi = ± α2−h2/α, depending on the direction
0
of the black hole rotation. Using the 3+1 formalism, we
can isolate the effects of the spacetime rotation, which
is useful for studying plasmas around rotating compact
objects. It is also particularly convenient to introduce
a locally nonrotating frame, the so-called “zero-angular-
momentum-observer”(ZAMO)frame[28],wheretheline X
i
element is ds2 =−dtˆ2+(cid:80)3 (dxˆi)2 =η dxˆµdxˆν, with 2L
i=1 µν
dtˆ= αdt and dxˆi = h dxi −αβidt (here and in the fol-
i
B
lowing, quantities observed in the ZAMO frame are de- r
noted with hats). For observers in the ZAMO frame, ϕ
the spacetime is locally Minkowskian. Using this frame,
o
any equation with the form of Eqs. (1) and (2) can be
cast into the general form ∇ Sµν = Hµ. For example,
ν
Sµν = hUµUν for the energy-momentum equation and
Sµν = 0 for the Ohm’s law, while the form of Hµ can
bededuced from Eqs.(1) and (2). This general equation
canthenbewrittenintheZAMOframeinvectorialform FIG. 1. Sketch of a reconnection layer in the azimuthal di-
as [29] rection. The rotating black hole is represented by the black
circle, while the magnetic diffusion region is marked by the
1 (cid:88)3 ∂ (cid:20)αh1h2h3 (cid:16)Sˆij +βjSˆi0(cid:17)(cid:21)+ Sˆ00 ∂α shaded area.
h h h ∂xj h h ∂xi
1 2 3 j i
j=1
3
−α(cid:88)(cid:16)G Sˆij −G Sˆjj +βjG Sˆ0i−βjG Sˆ0j(cid:17) Inouranalysis,weadoptaSweet-Parker-likeapproach
ij ji ij ji
[1], i.e., we look at magnetic reconnection under quasi-
j=1
stationary conditions (∂ ≈ 0) within narrow (δ (cid:28) L)
3 t
+(cid:88)σ Sˆ0j =αHˆi, (4) quasi-two-dimensionalcurrentsheets. Quasi-stationarity
ji
is generally satisfied not only in steady-state, but also at
j=1
the time of maximum reconnection rate. Current sheets
with i = 1,2,3, and where G = −(1/h h )∂h /∂xj,
ij i j i where magnetic diffusion takes place can form in differ-
and σ = −(1/h )∂(cid:0)αβi(cid:1)/∂xj. Vectors and tensors
ij j ent locations around black holes. Here, we consider two
observedbytheZAMOframearerelatedtothecovariant
paradigmatic cases with current sheets in the equatorial
vectors and tensors as Sˆ00 = α2S00, Sˆ0j = αh S0j −
j plane (θ = π/2) of the massive body. We assume that
βjSˆ00, and Sˆij = hihjSij − βiSˆ0j − βjSˆ0i − βiβjSˆ00. thesecurrentsheetsareinastableorbitaroundtheKerr
Analogously, Maxwellequationscanalsobeenwrittenin black hole [31].
the ZAMO frame (see Ref. [29]).
Reconnection layer in azimuthal direction.- We first
Since we are interested in analyzing magnetic recon-
consider a current sheet in the azimuthal direction, as
nection around black holes, we consider the spacetime
(cid:0)x0,x1,x2,x3(cid:1)=(t,r,θ,φ)givenbytheKerrmetric[30], showninFig.1. Themagneticfieldjustupstreamofthis
sheet is in the φ-direction and has magnitude Bˆ . In the
for which 0
diffusion region, vˆr vanishes at the neutral line (where
h =(1−2r r/Σ)1/2, h =(Σ/∆)1/2, Bˆφ = 0). Similarly, at the neutral line Jˆφ ≈ 0 ≈ Jˆr,
0 g 1
h2 =Σ1/2, h3 =(A/Σ)1/2sinθ, (5) implying that ρ(cid:48)e vanishes. We assume vˆθ ≈ 0, Bˆθ ≈ 0,
and that spatial variations of the fields with respect to
ω =ω =0, ω =2r2ar/Σ.
1 2 3 g θ (latitude) are negligible, i.e., ∂ ≈ 0. The same con-
θ
Here,r =GM isthegravitationalradius(Gisthegrav- figuration has been adopted by Koide and Arai [32] to
g
itational constant and M is the mass of the compact ob- examine the possibility of energy extraction from a ro-
ject) and a = J/J ≤ 1 is the rotation parameter (J tating black hole. As in their work, we assume that the
max
is the angular momentum and J =GM2). Moreover, plasma in this configuration rotates in a circular orbit
max
Σ = r2+(ar )2cos2θ, ∆ = r2−2r r+(ar )2, and A = with Keplerian velocity.
g g g
(cid:2)r2+(ar )2(cid:3)2 −∆(ar )2sin2θ. Finally, α = (∆Σ/A)1/2 Wecancalculatetheoutflowvelocityfromthediffusion
g g
andβj =βφδjφ,whereβφ =h ω /αisameasurementof region, vˆ , from the spatial φ-component of the energy-
3 3 o
the rotation of this spacetime. Note that in Kerr space- momentum equation (1). This equation can be written
time we have βjG ≡0. in the ZAMO frame by using Eq. (4). Then, evaluating
ij
3
along the neutral line we get Inside the current sheet, the θ-component of Eq. (2)
written in the ZAMO frame reduces to
1 ∂ (cid:2)αh h hγˆ2vˆφ(cid:0)vˆφ+βφ(cid:1)(cid:3)=−h JˆθBˆ − ∂p ,
αh1h2∂φ 1 2 3 r ∂φ γˆEˆθ+γˆvˆφBˆr =ηJˆθ. (10)
(6)
When evaluated at the X-point, this equation simply is
where γˆ = (1−vˆ2)−1/2 is the Lorentz factor. The out-
Eˆ | = ηJˆθ. On the other hand, outside the current
flowvelocityvˆ canbefoundbyintegratingEq. (6)from θ X
o
sheet the plasma response to the electromagnetic field is
φ (X-point) to φ (outflow-point). For this calcula-
X o
well described by the ideal Ohm’s law. Therefore, the
tion, we assume that the rotation of the black hole is
rhs of Eq. (2) can be neglected in this outer region. This
slow compared to the outflow velocity, vˆ (cid:29) βφ. This
o
implies that the θ-component of the electric field at the
assumption is justified a posteriori. Note that pressure
balanceacrossthelayergivesp ≈Bˆ2/2,whilemagnetic inflow-point becomes just Eˆθ|i ≈vˆiBˆ0.
X 0 The final step requires the estimation of δ, which
flux conservation yields
can be achieved from flux conservation. The in-
(cid:18) (cid:19)(cid:12)
Bˆr|o ≈ rhh31 (cid:12)(cid:12)(cid:12)o LδBˆ0, (7) flboawlanflcuext∂hre(αohu2thfl3oγˆwiρˆvˆfliu)x/(h∂1φh(2αhh31)h≈2γˆαoρˆρˆvˆγˆoi)vˆ/i(/h(h1h1δ2)h3m)us≈t
αrρˆγˆ vˆ /(h L). This leads us to find
o o 3
where the symbol | indicates that the relevant quanti-
o (cid:18) (cid:19)(cid:12)
tiesareevaluatedattheoutflow-point. Strictlyspeaking, δ ≈ h3 (cid:12)(cid:12) γˆivˆi L, (11)
the magnetic flux in the diffusion region is not fully con- h r (cid:12) γˆ vˆ
1 o o o
served, but Eq. (7) is an estimated approximation for
and, finally, the inflow velocity (reconnection rate)
the reconnected magnetic field. In principle, one can de-
rthivies ctohnedSitwioene.t-PFaurrktehrerrmecoorne,nefcrotimonMraatxewewlli’tshoeuqutautsioinngs vˆi ≈S−1/2(cid:18)hr (cid:12)(cid:12)(cid:12)(cid:12) (cid:19)1/2 . (12)
we can evaluate 3 o
Jˆθ|o ≈− h1 (cid:12)(cid:12)(cid:12)(cid:12) Bˆδ0 . (8) H(ce=re,1Sin o≡urLu/nηitsi)s, wthheichrelcaotmivpisatriecsLthuenddqyuniasmt incualmabnedr
1 o resistivediffusiontimescalesoftheplasma. Wehavecon-
Thereby, by using the above assumptions and Eqs. (7) sideredγˆi oftheorderofunityasS (cid:29)1forastrophysical
and (8), we can finally integrate Eq. (6) to obtain plasmas. The spacetime curvature leads to h3/r|o > 1,
implying a lower reconnection rate with respect to the
1
γˆ vˆ ≈ . (9) flat spacetime limit. This can be seen more explicitly in
o o 2
the scenario where the diffusion region is sufficiently far
Here, we have considered a relativistically hot plasma, from the black hole, where the reconnection rate can be
i.e. h≈4p [33]. From Eq. (9) we have that the Lorentz approximated as
factor of the plasma outflow is γˆ ≈O(1), which implies
o (cid:32) (cid:33)
a mildly relativistic outflow velocity vˆ ≈O(1), as in the a2r2
o vˆ ≈S−1/2 1− g , (13)
flat spacetime limit [3, 4]. Note that vˆo (cid:29)βφ ≈2arg2/r2 i 4ro2
when the current sheet is far from the black hole.
We proceed further by seeking the solution for the in- with r = r| . We clearly see that the spacetime cur-
o o
flow velocity vˆ, which is a measure of the reconnection vature effects lead to a decrease of the reconnection rate
i
rate, and the current sheet width δ, which, by means of in this configuration. The responsible factor is the rota-
Eq. (7),alsoleadstothereconnectedmagneticfieldBˆ | . tionoftheblackhole,whilethecurvaturecreatedbythe
r o
For this purpose, it is useful to consider separately the black hole mass itself does not play an important role if
inner region, where magnetic diffusion occurs, from the the current sheet is in the azimuthal direction far from
outer region, where the plasma moves with a transport the black hole.
velocitythatpreservesthemagneticconnectionsbetween Reconnection layer in radial direction.- We now con-
plasma elements [34–37]. In the flat spacetime limit, sider a magnetic reconnection configuration in which a
∂ ≈ 0 and ∂ ≈ 0 imply that Eˆ is uniform and can narrowcurrentsheetislocatedintheradialdirection, as
t φ θ
be used as a matching condition for these two regions. shown in Fig. 2. The magnetic field just upstream of the
In a more general case, Eˆ can change in the equatorial currentsheetisinther-directionandhasmagnitudeBˆ .
θ 0
plane, but for our purpose we only need to consider Eˆ In the diffusion region, the θ-component of the velocity
θ
at the X-point and the inflow point. From Maxwell’s vanishes at the neutral line (where Bˆr =0). We assume
equations we have ∂ (αh Eˆ ) = 0 along the inflow line vˆφ ≈0≈Bˆφ and ∂ ≈0. This implies that ρ(cid:48) vanishes.
r 2 θ φ e
passing through the X-point, implying that Eˆ | ≈Eˆ | This configuration is particularly relevant for the split-
θ i θ X
if the current sheet width δ is small, as can be seen a monopole magnetic field geometry, where reconnection
posteriori. layers form in the radial direction [23].
4
In this case, the outflow velocity vˆ can be calculated
o
by considering the spatial r-component of the energy- B
momentum equation (1). Writing it in the ZAMO frame X
2δ o
(4), and evaluating it along the neutral line, we get i
∂ (cid:2)αh h hγˆ2(vˆr)2(cid:3)+h h ∂αhγ2 = 2L
∂r 2 3 2 3∂r
(cid:18) (cid:19) Symmetry
∂p axis
−αh2h3 ∂r +h1JˆφBˆθ . (14) ϕ
Intheintegrationofthisequationfromr tor (inorder θ
X o
toobtaintheoutflowvelocity),weassumethatthegrav-
itational tidal field is negligible. This implies that the r
gravitational effects on different zones of that region, i.e.
at r and r , are essentially the same. This also allows
X o
us to assume that the current sheet does not fall toward
the black hole, and it can be considered in a rotational FIG.2. Sketchofareconnectionlayerintheradialdirection.
equilibrium. The above assumptions are no longer valid The rotating black hole is represented by the black circle,
if the length of the current sheet [38] is comparable with while the magnetic diffusion region is marked by the shaded
theradialdistancetotheblackholeorifthecurrentsheet area.
is too close to it. In that case, a more detailed approach
mustbeused,whichisleftforfutureworks. Here,were-
match the electric field considering the Ohm’s law inside
strictourselvestothissimplifiedmodelinordertoobtain
and outside the current sheet at the inflow point. Inside
the first approximated contribution of gravity to mag-
the current sheet, the φ-component of Eq. (2) written in
netic reconnection. Thus, at the integration level, the
the ZAMO frame formalism (4) becomes
contributionofthespacetimecurvaturecanbeevaluated
at the outflow-point. The φ-component of the current γˆEˆ +γˆvˆrBˆ =ηJˆφ. (19)
φ θ
densityandtheθ-componentofthemagneticfieldatthe
outflow-pointcanbeestimatedfromMaxwell’sequations At the X-point, this equation gives Eˆφ|X = ηJˆφ. The
and magnetic flux conservation. From them we obtain φ-component of the electric field of this equation has to
be matched with that coming from the ideal Ohm’s law
(cid:34) (cid:35)(cid:12)
∂ (αh Bˆ ) (cid:12) Bˆ evaluated at the inflow-point, i.e., Eˆ | ≈ vˆBˆ . This
Jˆφ| ≈− θ 1 r (cid:12) ≈− 0 (15) φ i i 0
o αh h (cid:12) δ leads us to find the inflow velocity once that the current
1 2 (cid:12)
o sheet width is evaluated. From the balance between the
and energy inflow and outflow, we obtain
(cid:12)
Bˆθ|o ≈ h11(cid:12)(cid:12)(cid:12)oLδBˆ0, (16) δ ≈ h1|oγˆγˆoivvˆˆioL. (20)
where Bˆ0 is in the radial direction. Then, considering Therefore, the inflow velocity (reconnection rate) turns
a relativistically hot plasma and pressure balance across out to be
the layer, Bˆ2/2 ≈ p , from the integration of Eq. (14)
along r we fi0nd (at tXhe same order) vˆ ≈ 1 (cid:20)1−L∂rlnα|o(cid:21)1/4 , (21)
γˆ2vˆ2+L ∂lnα(cid:12)(cid:12)(cid:12) γˆ2 ≈1, (17) i (cid:112)Sh1|o 1+L∂rlnα|o
o o ∂r (cid:12) o
o where again γˆi is of unity order because S (cid:29)1. Eq. (21)
where Eqs. (15) and (16) have been used. Hence, the contains the contribution of the gravitational field, from
plasma outflow is characterized by a Lorentz factor and where we can clearly see that the spacetime curvature
an outflow velocity given by of the Kerr black hole modifies the reconnection rate. If
we consider the ordering r (cid:28) r , the reconnection rate
g o
(cid:18) ∂lnα(cid:12)(cid:12) (cid:19)−1/2 (cid:18)1 L ∂lnα(cid:12)(cid:12) (cid:19)1/2 from Eq. (21) can be approximated as
γˆo ≈ 1+L ∂r (cid:12)(cid:12) , vˆo ≈ 2 − 2 ∂r (cid:12)(cid:12) .
o o (cid:34) (cid:35)
(18) r (2a2−3)r2
vˆ ≈S−1/2 1− g + g . (22)
In the flat spacetime limit (α → 1), we recover again a i 2r 8r2
o o
mildly relativistic outflow velocity [3, 4].
FromMaxwell’sequations[29]wehave∂ (αh Eˆ )=0 This expression shows that the curvature created by the
θ 3 φ
alongtheinflowlinepassingthroughtheX-point,imply- black hole mass leads to a decrease of the reconnection
ing that Eˆ | ≈ Eˆ | for a small δ. Therefore, we can rate. On the contrary, the black hole rotation produces
φ i φ X
5
the opposite effect of increasing the reconnection rate. ton University Press, 2005).
We observe, however, that these effects are small in the [2] M. Hoshino and Y. Lyubarsky, Space Sci Rev 173, 521
regime of validity of Eq. (22). (2012).
[3] Y. E. Lyubarsky, Mon. Not. R. Astron. Soc. 358, 113
Conclusions.- The presented analysis has allowed us
(2005).
to calculate the reconnection rate and other important [4] L. Comisso and F. A. Asenjo, Phys. Rev. Lett. 113,
properties of the reconnection layer for two configura- 045001 (2014).
tionsintheequatorialplaneofarotatingblackhole. We [5] S.Zenitani,M.HesseandA.Klimas,Astrophys.J.696,
have shown how the spacetime curvature modifies the 1385 (2009).
[6] G. Kowal, A. Lazarian, E.T. Vishniac, and K.
magnetic reconnection process in comparison to the flat
Otmianowska-Mazur, Astrophys. J. 700, 63 (2009).
spacetimelimit. Therotationoftheblackholeleadstoa
[7] S.Zenitani,M.HesseandA.Klimas,Astrophys.J.Lett.
decreaseofthereconnectionrateifthereconnectionlayer
716, L214 (2010).
is in the azimuthal configuration. On the other hand, it [8] H.R. Takahashi, T. Kudoh, Y. Masada and J. Mat-
is the spacetime curvature due to the black hole mass sumoto, Astrophys. J. Lett. 739, L53 (2011).
that acts to decrease the reconnection rate if the current [9] N.BesshoandA.Bhattacharjee,Astrophys.J.750,129
sheet is in the radial direction. (2012).
[10] M. Takamoto, Astrophys. J. Lett. 775, 50 (2013).
Theseresultshavebeenobtainedforaplasmanotclose
[11] Y.-H. Liu, F. Guo, W. Daughton, H. Li, and M. Hesse,
totheeventhorizonoftheblackhole, andinastableor-
Phys. Rev. Lett. 114, 095002 (2015).
bit [31]. We have also assumed a small rotation rate of [12] M. Takamoto, T. Inoue and A. Lazarian, Astrophys. J.
the black hole. In future studies these assumptions may 815, 16 (2015).
be relaxed. We also observe that collisionless effects are [13] S. Zenitani and M. Hoshino, Astrophys. J. 670, 702
important in plasmas around black holes, and thereby, (2007)
[14] G. Kowal, E. M. de Gouveia Dal Pino and A. Lazarian,
it is expected that they can also couple to gravity and
Astrophys. J. 735, 102, (2011).
affect the reconnection rate. A poloidal configuration
[15] G. Kowal, E. M. de Gouveia Dal Pino and A. Lazarian,
could also be examined, and the black hole rotation is
Phys. Rev. Lett. 108, 241102 (2012).
expected to modify the reconnection rate in a similar [16] B. Cerutti, D.A. Uzdensky, and M.C. Begelman, Astro-
fashion to Eq. (13). This is relevant for exploring recon- phys. J. 746, 148 (2012)
nection events in the corona and jet launching regions of [17] L.SironiandA.Spitkovsky,Astrophys.J.Lett.783,L21
the black hole [21, 39–43]. (2014)
[18] F. Guo, H. Li, W. Daughton, and Y.-H. Liu, Phys. Rev.
We finally observe that the potential formation of ex-
Lett. 113, 155005 (2014)
tremely elongated current sheets would result in slow re- [19] F. Guo, Y.-H. Liu, W. Daughton, and H. Li, Astrophys.
connection rates. However, this situation is prevented J. 806, 167 (2015)
by the occurrence of the plasmoid instability [44], which [20] G.R. Werner, D.A. Uzdensky, B. Cerutti, K. Nalewajko,
leads to the breakup of the current sheet in a fractal-like andM.C.Begelman,Astrophys.J.Lett.816,L8(2016).
[21] KoideS.,KudohT.,ShibataK.,Phys.Rev.D74,044005
fashion [45] that ends when the smallest elementary cur-
(2006).
rent sheets are sufficiently thick to avoid the plasmoid
[22] R.F. Penna, J.C. McKinney, R. Narayan, A.
instability. Our results can then be applied to these el-
Tchekhovskoy, R. Shafee and J.E. McClintock, Mon.
ementary current sheets, which are the actual locations Not. R. Astron. Soc. 408, 752 (2010).
where the magnetic energy is converted into plasma par- [23] M. Lyutikov and J.C. McKinney, Phys. Rev. D 84,
ticleenergy. Furthermore,ouranalysiscanalsobeuseful 084019 (2011).
to better understand fast reconnection driven by turbu- [24] J.C.McKinney,A.Tchekhovskoy,R.D.Blandford,Mon.
Not. R. Astron. Soc. 423, 3083 (2012).
lence [6, 12, 15], where the Sweet-Parker configuration
[25] A.Lichnerowicz,RelativisticHydrodynamicsandMagne-
occurs in several current sheets simultaneously due to
tohydrodynamics (New York: Benjamin, 1967).
the wandering of the magnetic field lines.
[26] A.M.Anile,Relativistic fluids and magneto-fluids(Cam-
Itisapleasuretoacknowledgefruitfuldiscussionswith bridge University Press, 1989).
Gabriele Brambilla, Luis Lehner, Russell Kulsrud, and [27] K.S.ThorneandD.A.Macdonald,Mon.Not.R.Astron.
Alexander Tchekhovskoy. FAA thanks Fondecyt-Chile Soc. 198, 339 (1982).
[28] J.M. Bardeen, W.H. Press and S.A. Teukolsky, Astro-
Grant No. 11140025. L.C. is grateful for the hospitality
phys. J. 178, 347 (1972).
oftheUniversidadAdolfoIb´an˜ez,wherepartofthiswork
[29] S. Koide, Astrophys. J. 708, 1459 (2010).
was done.
[30] S.Weinberg,GravitationandCosmology: Principlesand
Applications of the General Theory of Relativity (John
Wiley and Sons, 1972).
[31] M. A. Abramowicz, A. Lanza and M. J. Percival, Astro-
phys. J. 479, 179 (1997).
∗ [email protected] [32] S. Koide and K. Arai, Astrophys. J. 682, 1124 (2008).
† [email protected] [33] S.Chandrasekhar,AnIntroductiontotheStudyofStellar
[1] R.M.Kulsrud,Plasma Physics for Astrophysics (Prince- Structure (New York: Dover, 1938).
6
[34] W. A. Newcomb, Ann. Phys. 3, 347 (1958). (2014).
[35] F. Pegoraro, Europhys. Lett. 99, 35001 (2012). [41] L. H. S. Kadowaki, E. M. de Gouveia Dal Pino and C.
[36] F. A. Asenjo and L. Comisso, Phys. Rev. Lett. 114, B. Singh, Astrophys. J. 802, 113 (2015).
115003 (2015). [42] C.B. Singh et al. Astrophys. J. Lett. 799, L20 (2015).
[37] F.A.Asenjo,L.Comisso,S.M.Mahajan,Phys.Plasmas [43] B. Khiali and E. M. de Gouveia Dal Pino, Mon. Not. R.
22, 122109 (2015). Astron. Soc. 445, 838 (2015).
[38] Note that too elongated current sheets cannot form be- [44] L. Comisso, M. Lingam, Y.-M. Huang, and A. Bhat-
cause of the occurrence of the plasmoid instability [44]. tacharjee, Phys. Plasmas 23, 100702 (2016).
[39] E.M.deGouveiaDalPinoandA.Lazarian,A&A,441, [45] K.ShibataandS.Tanuma,EarthPlanetsSpace53,473
845 (2005). (2001).
[40] J. Dexter et al., Mon. Not. R. Astron. Soc. 440, 2185