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Magnetic Reconnection as a Mechanism for Energy Extraction
from Rotating Black Holes
Luca Comisso1, ∗and Felipe A. Asenjo2, †
1Department of Astronomy and Columbia Astrophysics Laboratory,
Columbia University, New York, New York 10027, USA.
2Facultad de Ingenier´ıa y Ciencias, Universidad Adolfo Ib´a˜nez, Santiago 7941169, Chile.
Spinning black holes store rotational energy that can be extracted. When a black hole is immersed
in an externally supplied magnetic ﬁeld, reconnection of magnetic ﬁeld lines within the ergosphere
can generate negative energy (relative to inﬁnity) particles that fall into the black hole event horizon
while the other accelerated particles escape stealing energy from the black hole. We show analyti
cally that energy extraction via magnetic reconnection is possible when the black hole spin is high
(dimensionless spin a∼1) and the plasma is strongly magnetized (plasma magnetization σ0>1/3).
The parameter space region where energy extraction is allowed depends on the plasma magnetization
and the orientation of the reconnecting magnetic ﬁeld lines. For σ01, the asymptotic negative
energy at inﬁnity per enthalpy of the decelerated plasma that is swallowed by a maximally rotating
black hole is found to be ∞
−' −pσ0/3. The accelerated plasma that escapes to inﬁnity and takes
away black hole energy asymptotes the energy at inﬁnity per enthalpy ∞
+'√3σ0. We show that
the maximum power extracted from the black hole by the escaping plasma is Pmax
extr ∼0.1M2√σ0w0
(here, Mis the black hole mass and w0is the plasma enthalpy density) for the collisionless plasma
regime and one order of magnitude lower for the collisional regime. Energy extraction causes a
signiﬁcant spindown of the black hole when a∼1. The maximum eﬃciency of the plasma ener
gization process via magnetic reconnection in the ergosphere is found to be ηmax '3/2. Since fast
magnetic reconnection in the ergosphere should occur intermittently in the scenario proposed here,
the associated emission within a few gravitational radii from the black hole is expected to display a
bursty nature.
PACS numbers: 52.27.Ny; 52.30.Cv; 95.30.Qd, 04.20.q
Keywords: Black holes; General relativity; Relativistic plasmas; Magnetic reconnection
I. INTRODUCTION
Black holes are believed to play a key role in a number
of highly energetic astrophysical phenomena, from active
galactic nuclei to gammaray bursts to ultraluminous X
ray binaries. The extraordinary amounts of energy re
leased during such events may have two diﬀerent origins.
It can be the gravitational potential energy of the matter
falling toward an existing or forming black hole during
accretion or a gravitational collapse. Or it can also be
the energy of the black hole itself. Indeed, a remarkable
prediction of general relativity is that a spinning black
hole has free energy available to be tapped. How this
occurs has fundamental implications for our understand
ing of high energy astrophysical phenomena powered by
black holes.
It was shown by Christodoulou [1] that for a spinning
(Kerr) black hole having mass Mand dimensionless spin
parameter a, a portion of the black hole mass is “irre
ducible”,
Mirr =Mr1
21 + p1−a2.(1)
The irreducible mass has a onetoone connection with
∗Electronic address: luca.comisso@columbia.edu
†Electronic address: felipe.asenjo@uai.cl
the surface area of the event horizon, AH= 4π(r2
H+a2) =
16πM 2
irr, which is proportional to the black hole enthropy
SBH = (kBc3/4G~)AH[2–5], where kB,G,~, and cde
note, respectively, the Boltzmann constant, the gravita
tional constant, the reduced Planck constant, and the
speed of light in vacuum. Thus, the maximum amount
of energy that can be extracted from a black hole with
out violating the second law of thermodynamics is the
rotational energy
Erot ="1−r1
21 + p1−a2#Mc2.(2)
For a maximally rotating black hole (a= 1), this gives
Erot = (1 −1/√2)M c2'0.29M c2. Therefore, a sub
stantial fraction of black hole energy can, in principle, be
extracted [6].
The possibility of extracting black hole rotational en
ergy was ﬁrst realized by Penrose [7], who envisioned a
thought experiment in which particle ﬁssion (0 →1 + 2)
occurs in the ergosphere surrounding a rotating black
hole. If the angular momentum of particle 1 is opposite
to that of the black hole and is suﬃciently high, then
the energy of particle 1, as viewed from inﬁnity, may be
negative. Hence, since the total energy at inﬁnity is con
served, the energy of particle 2 as measured from inﬁnity
will be larger than that of the initial particle 0. When
the particle with negative energy at inﬁnity (1) falls into
the black hole’s event horizon, the total energy of the
2
black hole decreases. Therefore, the energy of the escap
ing particle 2, which is higher than that of the original
particle 0, is increased at the expense of the rotational
energy of the black hole.
Although the Penrose process indicates that it is pos
sible to extract energy from a black hole, it is believed
to be impractical in astrophysical scenarios. Indeed, en
ergy extraction by means of the Penrose process requires
that the two newborn particles separate with a relative
velocity that is greater than half of the speed of light
[8, 9], and the expected rate of such events is too rare
to extract a sizable amount of black hole’s rotational en
ergy. On the other hand, Penrose’s suggestion sparked
the interest to ﬁnd alternative mechanisms for extract
ing black hole rotational energy, such as superradiant
scattering [10], the collisional Penrose process [11], the
BlandfordZnajek process [12] and the magnetohydrody
namic (MHD) Penrose process [13]. Among them, the
BlandfordZnajek process, in which energy is extracted
electromagnetically through the magnetic ﬁeld supported
by an accretion disk around the black hole, is thought
to be the leading mechanism for powering the relativis
tic jets of active galactic nuclei (AGNs) [e.g. 14–17] and
gammaray bursts (GRBs) [e.g. 18–20].
While diﬀerent mechanisms of energy extraction have
been carefully analyzed over the years, the possibility of
extracting black hole rotational energy as a result of rapid
reconnection of magnetic ﬁeld lines has been generally
overlooked. An exploratory study conducted by Koide
and Arai [21] analyzed the feasibility conditions for en
ergy extraction by means of the outﬂow jets produced in a
laminar reconnection conﬁguration with a purely toroidal
magnetic ﬁeld. In this simpliﬁed scenario, they suggested
that relativistic reconnection was required for energy ex
traction, but the extracted power and the eﬃciency of
the reconnection process were not evaluated. This is
necessary for determining whether magnetic reconnec
tion can play a signiﬁcant role in the extraction of black
hole energy. The recent advent of generalrelativistic ki
netic simulations of black hole magnetospheres [22] do in
deed suggest that particles accelerated during magnetic
reconnection may spread onto negative energyatinﬁnity
trajectories, and that the energy extraction via negative
energy particles could be comparable to the energy ex
tracted through the BlandfordZnajek process.
In this paper we provide an analytical analysis of black
hole energy extraction via fast magnetic reconnection as
a function of the key parameters that regulate the pro
cess: black hole spin, reconnection location, orientation
of the reconnecting magnetic ﬁeld, and plasma magneti
zation. Our main objective is to evaluate the viability,
feasibility conditions, and eﬃciency of magnetic recon
nection as a black hole energy extraction mechanism. In
Section II we delineate how we envision the extraction
of black hole rotational energy by means of fast mag
netic reconnection, and we derive the conditions under
which such energy extraction occurs. In Section III we
show that magnetic reconnection is a viable mechanism
of energy extraction for a substantial region of the pa
rameter space. In Section IV we quantify the rate of en
ergy extraction and the reconnection eﬃciency in order
to evaluate whether magnetic reconnection is an eﬀective
energy extraction mechanism for astrophysical purposes.
We further compare the power extracted by fast mag
netic reconnection with the power that can be extracted
through the BlandfordZnajek mechanism. Finally, we
summarize our results in Section V.
II. ENERGY EXTRACTION BY MAGNETIC
RECONNECTION
The possibility of extracting black hole rotational en
ergy via negativeenergy particles requires magnetic re
connection to take place in the ergosphere of the spin
ning black hole since the static limit is the boundary of
the region containing negativeenergy orbits. Magnetic
reconnection inside the ergosphere is expected to occur
routinely for fast rotating black holes. Indeed, a con
ﬁguration with antiparallel magnetic ﬁeld lines that is
prone to magnetic reconnection is caused naturally by
the framedragging eﬀect of a rapidly spinning black hole.
In this paper, we envision the situation illustrated in Fig.
1, where the fast rotation of the black hole leads to an
tiparallel magnetic ﬁeld lines adjacent to the equatorial
plane. This scenario is also consistent with numerical
simulations of rapidly spinning black holes [e.g. 22–25].
The change in magnetic ﬁeld direction at the equa
torial plane produces an equatorial current sheet. This
current sheet forms dynamically and is destroyed by the
plasmoid instability (permitted by nonideal magnetohy
drodynamic eﬀects such as thermalinertial eﬀects, pres
sure agyrotropy, or electric resistivity) when the current
sheet exceeds a critical aspect ratio [26–28]. The forma
tion of plasmoids/ﬂux ropes (see circular sections in the
zoomedin region of Fig. 1) drives fast magnetic recon
nection [e.g. 31, 32], which rapidly converts the available
magnetic energy into plasma particle energy. Eventually,
the plasma is expelled out of the reconnection layer and
the magnetic tension that drives the plasma outﬂow re
laxes. The ﬁeld lines are then stretched again by the
framedragging eﬀect and a current layer prone to fast
plasmoidmediated reconnection forms again. This leads
to reconnecting current sheets that form rapidly and in
termittently.
Magnetic reconnection in the plasma that rotates
around the black hole has the eﬀect of accelerating part of
the plasma and decelerating another part. If the deceler
ated plasma has negative energy at inﬁnity and the accel
erated one has energy at inﬁnity larger than its rest mass
and thermal energies (see the example regions in orange
in Fig. 1(b)), then the plasma that escapes to inﬁnity ac
quires energy at the expense of the black hole rotational
energy when the negativeenergy particles are swallowed
by the black hole as in the standard Penrose process [7].
Therefore, we want to examine when magnetic reconnec
3
ϵ∞0
<
ϵ+
∞0
>
Δ
(b)
(a)
BH
BH
B
B
vr
out vr
out
βϕ>1
βϕ
<1
FIG. 1: Schematic illustration of the mechanism of energy
extraction from a rotating black hole by magnetic reconnec
tion in the black hole ergosphere. A conﬁguration with an
tiparallel magnetic ﬁeld lines adjacent to the equatorial plane
is favored by the framedragging eﬀect of the rapidly spin
ning black hole (panels (a) and (b) portray meridional and
equatorial views, respectively), and the resulting equatorial
current sheet is prone to fast plasmoidmediated magnetic re
connection (see circular structures in the zoomedin region
[29]). Magnetic reconnection in the plasma that rotates in
the equatorial plane extracts black hole energy if the deceler
ated plasma that is swallowed by the black hole has negative
energy as viewed from inﬁnity, while the accelerated plasma
with a component in the same direction of the black hole ro
tation escapes to inﬁnity. The outer boundary (static limit)
of the ergosphere is indicated by the shortdashed lines in
both panels. In panel (b), longdashed and solid lines indi
cate magnetic ﬁeld lines below and above of the equatorial
plane, respectively. Finally, the dashed lines in the zoomed
region indicate the two magnetic reconnection separatrices in
tersecting at the dominant magnetic reconnection Xpoint.
tion in the ergosphere of the black hole redistributes the
angular momentum of the plasma in such a way to sat
isfy these conditions. Furthermore, we want to evaluate
if the extraction of black hole rotational energy via fast
plasmoidmediated reconnection can constitute a major
energy release channel.
We describe the spacetime around the rotating black
hole by using the Kerr metric in BoyerLindquist coordi
nates xµ= (t, r, θ, φ), where ris the radial distance, θis
the polar angle, and φis the azimuthal angle. The Kerr
metric can be expressed in terms of the square of the line
element ds2=gµνdxµdxνas [e.g. 33]
ds2=gttdt2+ 2gtφ dtdφ +gφφdφ2+grr dr2+gθθdθ2,(3)
where the nonzero components of the metric are given
by
gtt =2Mr
Σ−1, gtφ =−2M2ar sin2θ
Σ,(4)
gφφ =A
Σsin2θ , gr r =Σ
∆, gθθ = Σ ,(5)
with
Σ = r2+ (aM)2cos2θ , (6)
∆ = r2−2Mr + (aM)2,(7)
A=r2+ (aM)22−(aM)2∆ sin2θ . (8)
The only two parameters that appear in the metric are
the black hole mass, M, and the black hole dimensionless
spin, 0 ≤a≤1. Here, and in all subsequent expressions,
we use geometrized units with G=c= 1.
The inner boundary of the ergosphere of the Kerr black
hole, which coincides with the outer event horizon, is
given by the radial distance
rH=M+M(1 −a2)1/2,(9)
while the outer boundary (static limit) is given by
rE=M+M(1 −a2cos2θ)1/2,(10)
which yields rE= 2Mat the equatorial plane θ=π/2.
In this paper we make the simplifying assumption that
magnetic reconnection happens in the bulk plasma that
rotates circularly around the black hole at the equatorial
plane. This corresponds to a Keplerian angular velocity
ΩK=±M1/2
r3/2±aM3/2,(11)
as seen by an observer at inﬁnity. The upper sign refers to
corotating orbits, while the lower sign applies to counter
rotating orbits. Circular orbits can exist from r→ ∞
4
down to the limiting circular photon orbit, whose radius
is given by
rph = 2M1 + cos 2
3arccos(∓a).(12)
For a maximally rotating black hole (a= 1), one has
rph =M(corotating orbit) or rph = 4M(counter
rotating orbit). However, for r > rph not all circular
orbits are stable. Nonspinning test particles can stably
orbit the black hole if they are at distances larger than
or equal to the innermost stable circular orbit [8]
risco =M3 + Z2∓(3 −Z1)(3 + Z1+ 2Z2)1/2,
(13)
where
Z1= 1 + (1 −a2)1/3[(1 + a)1/3+ (1 −a)1/3],(14)
Z2= (3a2+Z2
1)1/2.(15)
For a maximally rotating black hole risco =M(corotat
ing orbit) or risco = 9M(counterrotating orbit). Here
we focus on corotating orbits since we are interested in
magnetic reconnection occurring inside the ergosphere.
We also assume that the plasma acceleration through
magnetic reconnection is localized in a small region (close
to the dominant reconnection Xpoint) compared to the
size of the black hole ergosphere.
In what follows, it is convenient to analyze the
plasma energy density in a locally nonrotating frame, the
so called “zeroangularmomentumobserver” (ZAMO)
frame [8]. In the ZAMO frame, the square of the line ele
ment is given by ds2=−dˆ
t2+P3
i=1 (dˆxi)2=ηµνdˆxµdˆxν,
where
dˆ
t=α dt , dˆxi=√gii dxi−αβidt (16)
(no implicit summation is assumed over i), with αindi
cating the lapse function
α= −gtt +g2
φt
gφφ !1/2
=∆Σ
A1/2
(17)
and βiindicating the shift vector (0,0, βφ), with
βφ=√gφφ ωφ
α=ωφ
αA
Σ1/2
sin θ(18)
and ωφ=−gφt/gφφ = 2M2ar/A being the angular veloc
ity of the frame dragging. An advantage of this reference
frame is that equations become intuitive since the space
time is locally Minkowskian for observers in this frame.
Hereinafter, quantities observed in the ZAMO frame are
denoted with hats. Vectors in the ZAMO frame are re
lated to the vectors in the BoyerLindquist coordinates
as ˆ
b0=αb0and ˆ
bi=√gii bi−αβib0for the contravari
ant components, while ˆ
b0=b0/α+P3
i=1 (βi/√gii)biand
ˆ
bi=bi/√gii for the covariant components.
We evaluate the capability of magnetic reconnection to
extract black hole energy by examining the conditions for
the formation of negative energy at inﬁnity and escaping
to inﬁnity of the plasma accelerated/decelerated by the
reconnection process in the ergosphere (in this work we
do not address the origin of the plasma properties but
rather assume a plasma with a given particle density and
pressure). From the energymomentum tensor in the one
ﬂuid approximation,
Tµν =pgµν +wU µUν+FµδFνδ −1
4gµν FρδFρδ ,(19)
where, p,w,Uµ, and Fµν are the proper plasma pressure,
enthalpy density, fourvelocity, and electromagnetic ﬁeld
tensor, respectively, one has the “energyatinﬁnity” den
sity e∞=−αgµ0Tµ0. Therefore, the energyatinﬁnity
density is given by
e∞=αˆe+αβφˆ
Pφ,(20)
where
ˆe=wˆγ2−p+ˆ
B2+ˆ
E2
2(21)
is the total energy density and
ˆ
Pφ=wˆγ2ˆvφ+ˆ
B×
ˆ
Eφ(22)
is the azimuthal component of the momentum density,
with ˆγ=ˆ
U0=1−P3
i=1 (dˆvi)2−1/2,ˆ
Bi=ijk ˆ
Fjk /2,
and ˆ
Ei=ηij ˆ
Fj0=ˆ
Fi0.
The energyatinﬁnity density can be conveniently sep
arated into hydrodynamic and electromagnetic compo
nents as e∞=e∞
hyd +e∞
em, where
e∞
hyd =αˆehyd +αβφwˆγ2ˆvφ(23)
is the hydrodynamic energyatinﬁnity density and
e∞
em =αˆeem +αβφˆ
B×
ˆ
Eφ(24)
is the electromagnetic energyatinﬁnity density, with
ˆehyd =wˆγ2−pand ˆeem = ( ˆ
B2+ˆ
E2)/2 indicating the
hydrodynamic and electromagnetic energy densities ob
served in the ZAMO frame. In this paper we assume
an eﬃcient magnetic reconnection process that converts
most of the magnetic energy into kinetic energy, so that
the electromagnetic energy at inﬁnity is negligible with
respect to the hydrodynamic energy at inﬁnity. Then,
from Eq. (23), we can evaluate the energyatinﬁnity
density of the expelled plasma using the approximation
that the plasma element is incompressible and adiabatic,
which leads to [21]
e∞
hyd =αh(ˆγ+βφˆγˆvφ)w−p
ˆγi.(25)
5
To analyze the localized reconnection process, we in
troduce the local rest frame xµ0= (x00, x10, x20, x30) of
the bulk plasma that rotates with Keplerian angular ve
locity ΩKin the equatorial plane. We choose the frame
xµ0in such a way that the direction of x10is parallel
to the radial direction x1=rand the direction of x30
is parallel to the azimuthal direction x3=φ. The ori
entation of the reconnecting magnetic ﬁeld lines is kept
arbitrary as it ultimately depends on the large scale mag
netic ﬁeld conﬁguration, the black hole spin, and is also
time dependent. Indeed, the complex nonlinear dynamics
around the spinning black hole induces magnetic ﬁeld line
stretching, with magnetic reconnection causing a topo
logical change of the macroscopic magnetic ﬁeld conﬁgu
ration on short time scales. Therefore, here we introduce
the orientation angle
ξ= arctan v10
out/v30
out,(26)
where v10
out and v30
out are the radial and azimuthal compo
nents of the outwarddirected plasma in the frame xµ0.
Accordingly, the plasma escaping from the reconnection
layer has velocities v0
±=vout(±cos ξe0
3∓sin ξe0
1), with
vout indicating the magnitude of the outﬂow velocity ob
served in the frame xµ0and the subscripts + and −in
dicating the corotating and counterrotating outﬂow di
rection, respectively. In the plasmoidmediated recon
nection regime, a large fraction of the plasma is evac
uated through plasmoidlike structures [34], which can
also contain a signiﬁcant component of nonthermal par
ticles. Such particles gain most of their energy from the
motional electric ﬁeld [e.g. 35] and are carried out by
the plasmoids (where most of them are trapped) in the
outﬂow direction [e.g. 36].
The outﬂow Lorentz factor ˆγand the outﬂow veloc
ity component ˆvφobserved by the ZAMO can be con
veniently expressed in terms of the Keplerian velocity in
the ZAMO frame and the outﬂow velocities in the local
frame xµ0. From Eq. (11), we can express the corotating
Keplerian velocity observed in the ZAMO frame as
ˆvK=A
∆1/2(M/r)1/2−a(M/r)2
r3−a2M3−βφ.(27)
Then, using ˆvφ
±= (ˆvK±vout cos ξ)/(1 ±ˆvKvout cos ξ) for
the azimuthal components of the two outﬂow velocities
and introducing the Lorentz factors ˆγK= (1 −ˆv2
K)−1/2
and γout = (1 −v2
out)−1/2, we can write the energyat
inﬁnity density of the reconnection outﬂows as
e∞
hyd,±=αˆγK"1+ˆvKβφγoutw
±cos ξˆvK+βφγoutvout w
−p
(1±cos ξˆvKvout )γoutˆγ2
K#,(28)
where the subscripts + and −indicate the energy
atinﬁnity density associated with corotating (v0
+) and
counterrotating (v0
−) outﬂow directions as observed in
the local frame xµ0.
The outﬂow velocity vout can be evaluated by assum
ing that the local current sheet at the dominant Xpoint
has a small inverse aspect ratio δX/LX1, where δX
and LXare the halfthickness and halflength of this local
current sheet. If we consider that the rest frame rotat
ing with Keplerian velocity is in a gravityfree state and
neglect general relativistic corrections [37–39], then, the
conservation of momentum along the reconnection neu
tral line gives
wγ2
outv2
out/LX+B2
upδ2
X/L3
X'(Bup/δX)(Bup δX/LX),
(29)
where Bup is the local magnetic ﬁeld strength immedi
ately upstream of the local current sheet. Here we have
used Maxwell’s equations to estimate the current density
at the neutral line in addition to the outﬂow magnetic
ﬁeld strength [40, 41]. We also assumed that the ther
mal pressure gradient force in the outﬂow direction is
small compared to the magnetic tension force, as veriﬁed
by numerical simulations of relativistic reconnection with
antiparallel magnetic ﬁelds [43]. Then, from Eq. (29) one
gets
vout '"1−δ2
X/L2
Xσup
1 + (1 −δ2
X/L2
X)σup #1/2
,(30)
where σup =B2
up/w0is the plasma magnetization imme
diately upstream of the local current sheet at the dom
inant Xpoint. Consequently, for δX/LX1, the out
ﬂow velocity reduces to vout '[σup/(1 + σup)]1/2.
The local magnetic ﬁeld Bup can be connected to the
asymptotic macroscale magnetic ﬁeld B0by considering
force balance along the inﬂow direction. In the magnet
ically dominated regime, thermal pressure is negligible,
and the inwarddirected magnetic pressure gradient force
must be balanced by the outwarddirected magnetic ten
sion (the inertia of the inﬂowing plasma is negligible if
δX/LX1). Then, from geometrical considerations one
gets [43]
Bup =1−(tan ϕ)2
1 + (tan ϕ)2B0,(31)
where ϕis the opening angle of the magnetic reconnec
tion separatrix. Estimating tan ϕ'δX/LX, we have
simply
vout 'σ0
1 + σ01/2
, γout '(1 + σ0)1/2,(32)
where we have deﬁned σ0=B2
0/w0as the plasma magne
tization upstream of the reconnection layer. Accordingly,
in the magnetically dominated regime σ01, the recon
nection outﬂow velocity approaches the speed of light.
We ﬁnally note that in the presence of signiﬁcant embed
ding of the local current sheet, the scaling of the outﬂow
6
velocity could be weakened with respect to B0, while Eq.
(30) remains accurate [36, 43].
We must point out that in the plasmoidmediated re
connection regime considered here, the continuous forma
tion of plasmoids/ﬂux ropes prevents the formation of ex
tremely elongated “laminar” reconnection layers, thereby
permitting a high reconnection rate [e.g. 31, 32]. De
pending on the plasma collisionality regime, plasmoid
mediated reconnection yields an inﬂow velocity (as ob
served in the frame xµ0)
vin =(O(10−2) for δX> `k[44−47]
O(10−1) for δX.`k[42,43] ,(33)
where `kis the relevant kinetic scale that determines
the transition between the collisional and collisionless
regimes. The collisional regime is characterized by δX>
`k, while the collisionless regime occurs if δX.`k. For
a pair (e−e+) dominated plasma, we have [41] `k=
√γth,e λe, where λeis the nonrelativistic plasma skin
depth and γth,e is the electron/positron thermal Lorentz
factor. If there is also a signiﬁcant ion component, then
[31] `k=√γth,i λi, where λiis the nonrelativistic ion in
ertial length and γth,i is the ion thermal Lorentz factor.
We emphasize that the reconnection rate is independent
of the microscopic plasma parameters when magnetic re
connection proceeds in the plasmoidmediated regime. In
particular, plasmoidmediated reconnection in the colli
sionless regime has an inﬂow velocity vin that is a sig
niﬁcant fraction of the speed of light, which potentially
allows for a high energy extraction rate from the black
hole (see Sec. IV).
The expression for the energy at inﬁnity associated
with the accelerated/decelerated plasma as a function
of the critical parameters (a,r/M,σ0,ξ) can be ﬁnally
obtained by substituting the magnetization dependence
of the outﬂow velocity into Eq. (28). Then, the hydro
dynamic energy at inﬁnity per enthalpy ∞
±=e∞
hyd,±/w
becomes
∞
±=αˆγK"1+βφˆvK(1+σ0)1/2±cos ξβφ+ ˆvKσ1/2
0
−1
4
(1+σ0)1/2∓cos ξˆvKσ1/2
0
ˆγ2
K(1 + σ0−cos2ξˆv2
Kσ0)#,(34)
where we have assumed a relativistically hot plasma with
polytropic index Γ = 4/3. Similarly to the original Pen
rose process [7], energy extraction from the black hole
through magnetic reconnection occurs when
∞
−<0 and ∆∞
+>0,(35)
where
∆∞
+=∞
+−1−Γ
Γ−1
p
w=∞
+(36)
for a relativistically hot plasma. Therefore, black hole
rotational energy is extracted if the decelerated plasma
ϵ+
∞�σ�
ϵ
∞σ�/�

σ�
ϵ+
∞�ϵ
∞
FIG. 2: Energy at inﬁnity per enthalpy ∞
+(gray line) and
∞
−(orange line) for maximal energy extraction conditions
(a, r/M →1 and ξ→0). Energy extraction requires σ0>
1/3. For σ01, ∞
+'√3σ0(dashdotted black line) and
∞
−' −pσ0/3 (dashed black line).
acquires negative energy as measured at inﬁnity, while
the plasma that is accelerated acquires energy at inﬁnity
larger than its rest mass and thermal energies.
The energy at inﬁnity per enthalpy ∞
±given by Eq.
(34) depends on the black hole spin aand the Xpoint
distance r/M, as well as the plasma magnetization σ0and
the orientation angle ξ, which encodes the information
of the magnetic ﬁeld conﬁguration surrounding the black
hole. Equations (34)(36) indicate that energy extraction
is favored by lower values of the orientation angle ξand
higher values of the magnetization σ0. It is instructive to
consider the limit a→1, ξ→0, and r→M(the metric
(3) has a coordinate singularity at the event horizon that
can be removed by a coordinate transformation). In this
case, from Eq. (34) we obtain ∞
+>0 and ∞
−<0 when
σ0>1/3.(37)
Therefore, in principle, it is possible to extract rotational
energy via magnetic reconnection for values of σ0below
unity. However, higher σ0values are required to extract
sizable amounts of energy. If, in addition to a, r/M →1
and ξ→0, we also consider σ01, from Eq. (34) we
obtain
∞
+'p3gφφ ωφγoutvout '√3σ0,(38)
∞
−' −rgφφ
3ωφγoutvout ' −rσ0
3.(39)
These relations give us the energy at inﬁnity per enthalpy
of the accelerated (+) and decelerated (−) plasma in the
maximal energy extraction regime (as can be seen from
Fig. 2, they provide a fairly accurate estimate already at
values of σ0moderately larger then unity).
In the next sections, we will show that magnetic re
connection is a viable mechanism for extracting energy
7
ξ=π12/
0.80 0.85 0.90 0.95 1.00
1.0
1.2
1.4
1.6
1.8
2.0
r/M
a
ϵ∞00 0
1σ)( <
=
0
ϵ∞0 0
3σ)( <
=
0
ϵ∞0 0
1σ)( <
=
0
ϵ∞0
3)( <
=
0
σ
ϵ∞0
1
0
σ)( <
=
FIG. 3: Regions of the phasespace {a, r /M}where the en
ergies at inﬁnity per enthalpy from Eq. (34) are such that
∆∞
+>0 (gray area) and ∞
−<0 (orange to red areas),
for a reconnecting magnetic ﬁeld having orientation angle
ξ=π/12 and diﬀerent values of the magnetization param
eter σ0∈ {1,3,10,30,100}. The area with ∞
−<0 increases
monotonically as σ0increases. The solid black line indicates
the limit of the outer event horizon, Eq. (9), the dashed
black line represents the limiting corotating circular photon
orbit, Eq. (12), while the dashdotted black line corresponds
to the innermost stable circular orbit, Eq. (13). The limit
r/M = 2 corresponds to the outer boundary of the ergosphere
at θ=π/2.
from rotating black holes for a signiﬁcant region of the
parameter space, we will evaluate the rate of black hole
energy extraction, and we will determine the eﬃciency of
the reconnection process.
III. ENERGY EXTRACTION ASSESSMENT IN
PHASE SPACE
We analyze the viability of energy extraction via mag
netic reconnection by considering solutions of Eq. (34).
In particular, in Figs. 3 and 4 we display the regions of
the phasespace {a, r/M}where ∞
−<0 and ∆∞
+>0,
which correspond to the conditions for energy extraction.
This is done for a reconnecting magnetic ﬁeld with orien
tation angle ξ=π/12 and diﬀerent values of the magne
tization parameter σ0∈ {1,3,10,30,100}(Fig. 3), and
for a plasma magnetization σ0= 100 and diﬀerent values
of the orientation angle ξ∈ {π/20, π/12, π/6, π/4}(Fig.
4).
As the magnetization of the plasma increases, the re
gion of the phasespace {a, r/M}where magnetic recon
0.80 0.85 0.90 0.95 1.00
1.0
1.2
1.4
1.6
1.8
2.0
r/M
a
=100
0
σ
ϵ∞6 0
ξ)( <
=�/
ϵ∞2 0
1ξ)( <
=�/
ϵ∞4 0
ξ)( <
=�/
ϵ∞2 0
0ξ)( <
=�/
FIG. 4: Regions of the phasespace {a, r /M}where the en
ergies at inﬁnity per enthalpy from Eq. (34) are such that
∆∞
+>0 (gray area) and ∞
−<0 (green areas), for plasma
magnetization σ0= 100 and diﬀerent values of the orientation
angle ξ∈ {π/20, π/12, π /6, π/4}. Other lines are the same as
in Figure 3. The area with ∞
−<0 increases monotonically
as ξdecreases.
nection extracts black hole rotational energy extends to
larger r/M values and lower values of the dimensionless
spin a(Fig. 3). From Eq. (34) we can see that ∞
−is a
monotonically decreasing function of σ0, while ∞
+mono
tonically increases with σ0.∞
+>0 is easily satisﬁed
for rph < r < rE,a > 0, and ξ < π/2. On the other
hand, ∞
−<0 requires σ01 in order for reconnec
tion to extract black hole energy in an extended region
of the phasespace {a, r/M}. High values of the plasma
magnetization can extend the energy extraction region
up to the outer boundary of the ergosphere, while energy
extraction for moderate values of the spin parameter a
is subject to the occurrence of particle orbits inside the
ergosphere.
Energy extraction via magnetic reconnection is also fa
vored by reconnection outﬂows whose orientation is close
to the azimuthal direction. The region of the phase
space {a, r/M}where energy extraction occurs increases
to larger r/M values and lower avalues as the orien
tation angle ξdecreases. Notwithstanding, even an an
gle as large as ξ=π/4 admits a feasible region of the
phasespace where magnetic reconnection extracts rota
tional energy. The increase of the energy extraction re
gion for decreasing angle ξis due to the fact that only
the azimuthal component of the outﬂow velocity con
tributes to the extraction of rotational energy. For an
angle ξ=π/20, the extraction of black hole energy hap
pens for Xpoints up to r/M ≈1.96 (for σ0= 100), while
8
ξ→0 can extend this margin up to the outer boundary
of the ergosphere.
The ergosphere of spinning black holes (rH< r < rE)
can reach very high plasma magnetizations (e.g, σ0
100 close to the event horizon of the black hole M87*
[48]). Furthermore, for rapidly spinning (aclose to unity)
black holes, we expect a reconnecting magnetic ﬁeld with
small orientation angle, ξ.π/6, as the strong frame
dragging eﬀect inside the ergosphere stretches the mag
netic ﬁeld lines along the azimuthal direction [e.g. 49, 50].
Therefore, the plots shown in Figs. 3 and 4 indicate that
magnetic reconnection is a viable mechanism for extract
ing energy from rotating black holes with dimensionless
spin aclose to unity. On the other hand, energy ex
traction via magnetic reconnection becomes negligible
for spin values a.0.8. The availability of reconnec
tion regions inside the ergosphere decreases as the spin
parameter decreases, with no circular orbits inside the
ergosphere for spin a≤1/√2. Magnetic reconnection
could still be capable of extracting energy in such cases
if a circular orbit is sustained thanks to the help of the
magnetic ﬁeld or if one considers noncircular orbits.
IV. ENERGY EXTRACTION RATE AND
RECONNECTION EFFICIENCY
We now evaluate the rate of black hole energy extrac
tion. This depends on the amount of plasma with nega
tive energy at inﬁnity that is swallowed by the black hole
in the unit time. Therefore, a high reconnection rate can
potentially induce a high energy extraction rate. The
power Pextr extracted from the black hole by the escap
ing plasma can be estimated as
Pextr =−∞
−w0AinUin ,(40)
where Uin =O(10−1) for the collisionless regime, while
Uin =O(10−2) for the collisional one. Ain is the cross
sectional area of the inﬂowing plasma, which can be es
timated as Ain ∼(r2
E−r2
ph) for rapidly spinning black
holes. In particular, for a→1 one has (r2
E−r2
ph) =
(r2
E−r2
H)=3M2.
We show in Fig. 5 the ratio Pextr/w0as a function of
the dominant Xpoint location r/M for a rapidly spin
ning black hole with a= 0.99 and magnetic reconnec
tion in the collisionless regime. This is done for a typ
ical reconnecting magnetic ﬁeld with orientation angle
ξ=π/12 and diﬀerent values of the magnetization pa
rameter σ0∈10,102,103,104,105(top panel), and for
a typical magnetization σ0= 104and diﬀerent values of
the orientation angle ξ∈ {0, π/20, π/12, π/8, π/6}(bot
tom panel). The power extracted from the black hole in
creases monotonically for increasing values of the plasma
magnetization and for lower values of the orientation an
gle. It reaches a peak for Xpoint locations that are close
to the limiting circular orbit until it drops oﬀ. The peak
of the extracted power can continue to raise up to a max
imum value that is achieved for r/M →1 if a→1. The
/
/
�= ξ=π/
σ=
σ=
σ=
σ=
σ=
ξ=π/ξ=π/ξ=π/
ξ=π/ ξ=
/
/
�= σ=
FIG. 5: Pextr/w0=−∞
−AinUin as a function of the dominant
Xpoint location r/M for a rapidly spinning black hole with
a= 0.99 and reconnection inﬂow fourvelocity Uin = 0.1 (i.e.,
collisionless reconnection regime). ∞
−is evaluated using Eq.
(34), while Ain = (r2
E−r2
ph). We have also set M= 1.
Diﬀerent colors (from indigo to red) refer to diﬀerent plasma
magnetizations (from σ0= 10 to σ0= 105) and ξ=π/12
(top panel) or diﬀerent orientation angles (from ξ=π/6 to
ξ= 0) and σ0= 104(bottom panel). The vertical dashed line
indicates the limiting circular orbit rph(a= 0.99).
theoretical limit of the maximum power is given by
Pmax
extr 'pσ0/3w0AinUin ∼0.1M2√σ0w0,(41)
which follows directly from Eqs. (39) and (40). We can
see from Fig. 5 that the peak of the extracted power
is already close to the maximum theoretical limit when
ξ.π/12.
The proposed mechanism of energy extraction via mag
netic reconnection generates energetic plasma outﬂows
that steal energy from the black hole, but it also necessi
tates magnetic ﬁeld energy to operate. Magnetic energy
is indeed needed in order to redistribute the angular mo
mentum of the particles in such a way to generate parti
cles with negative energy at inﬁnity and particles escap
ing to inﬁnity. Therefore, it is convenient to deﬁne the
9
�=
�=
�=
�=
�=
/
η
σ�= ξ=π/
FIG. 6: Eﬃciency ηof the reconnection process as a func
tion of the dominant Xpoint location r/M for a reconnection
layer with upstream plasma magnetization σ0= 100 and re
connecting magnetic ﬁeld having orientation angle ξ=π/20.
Diﬀerent colors (from indigo to red) refer to diﬀerent black
hole spin values (from a= 0.9 to a= 1).
eﬃciency of the plasma energization process via magnetic
reconnection as
η=∞
+
∞
++∞
−
.(42)
Extraction of energy from the black hole takes place when
η > 1. Figure 6 shows the eﬃciency ηas a function of
the dominant Xpoint location r/M for a reconnection
layer with magnetization parameter σ0= 100, orienta
tion angle ξ=π/20, and diﬀerent black hole spin val
ues a∈ {0.90,0.96,0.99,0.999,1}. The eﬃciency ηsig
niﬁcantly increases for reconnection Xpoints that are
closer to the black hole event horizon and falls oﬀ below
unity when the inner radius reaches rph. The maximum
eﬃciency can be evaluated by considering the optimal
energy extraction conditions (a, r/M →1, ξ→0) and
σ01. In this case, Eq. (42) gives
ηmax '√3σ0
√3σ0−pσ0/3= 3/2.(43)
Therefore, the additional energy extracted from the black
hole, while nonnegligible, does not extensively modify
the energetics of the escaping plasma.
We can also compare the power extracted from the
black hole by fast magnetic reconnection with the one
that can be extracted via the BlandfordZnajek mecha
nism, in which the rotational energy is extracted electro
magnetically through a magnetic ﬁeld that threads the
black hole event horizon. For maximum eﬃciency condi
tions [51–53], the rate of black hole energy extraction via
the BlandfordZnajek mechanism is given by [12, 54]
PBZ 'κΦ2
BH Ω2
H+χΩ4
H+ζΩ6
H,(44)
=
=
=
=
=
σ�
�/�
�= ξ=π/
FIG. 7: Power ratio Pextr/PBZ as a function of the plasma
magnetization σ0for a black hole with dimensionless spin
a= 0.99 and a reconnecting magnetic ﬁeld having orien
tation angle ξ=π/12. Diﬀerent colors (from indigo to
red) refer to diﬀerent dominant Xpoint locations r/M ∈
{1.7,1.6,1.5,1.4,1.3}. We considered Uin = 0.1 (i.e., colli
sionless reconnection regime), Ain = (r2
E−r2
ph), and κ= 0.05.
where ΦBH =1
2RθRφBrdAθφ is the magnetic ﬂux
threading one hemisphere of the black hole horizon (with
dAθφ =√−g dθdφ indicating the area element in the θ
φplane), ΩH=a/2rHis the angular frequency of the
black hole horizon, while κ,χ, and ζare numerical con
stants. The numerical prefactor κdepends on the mag
netic ﬁeld geometry near the black hole (κ≈0.053 for a
split monopole geometry and κ≈0.044 for a parabolic
geometry), while χ≈1.38 and ζ≈ −9.2 [54]. Equation
(44) is a generalization of the original BlandfordZnajek
scaling [12] PBZ 'κΦ2
BH(a/4M)2, which is recovered in
the small spin limit a1.
In order to provide a rough order of magnitude es
timate of the power extracted during the occurrence of
fast magnetic reconnection with respect to the Blandford
Znajek process, we assume ΦBH ∼ Brr2
H∼B0sin ξ r2
H
(we point out that a precise evaluation of ΦBH requires
direct numerical simulations that reproduce the detailed
magnetic ﬁeld conﬁguration at all latitudes, while the
angle ξis a good estimate for the magnetic ﬁeld conﬁgu
ration only at low latitudes [e.g. 49, 50]). Then, we can
evaluate the ratio Pextr/PBZ as
Pextr
PBZ ∼−∞
−AinUin
κΩ2
Hr4
Hσ0sin2ξ(1 + χΩ2
H+ζΩ4
H).(45)
Figure 7 shows the ratio Pextr/PBZ given by the right
hand side of Eq. (45) as a function of the plasma magne
tization σ0for the fast collisionless reconnection regime.
Pextr/PBZ 1 for an extended range of plasma mag
netizations. For σ0∼1, the forcefree electrodynamics
approximation (the inertia of the plasma is ignored, i.e.
w0→0) that is used to derive the extracted power in
the BlandfordZnajek process becomes invalid. In this
10
case, magnetic reconnection is an eﬀective mechanism
of energy extraction provided that the plasma magneti
zation is suﬃcient to satisfy the condition ∞
−<0 (as
well as ∆∞
+>0). On the other hand, for σ0→ ∞,
energy extraction via fast magnetic reconnection is al
ways subdominant to the BlandfordZnajek process since
Pextr/PBZ →0 in this limit. If we neglect higher order
corrections with respect to Ω2
H(which leads to an over
prediction of PBZ by about 25% as a→1 [54]), and
recalling that ΩH= 1/2Mfor a→1, we can estimate
the ratio Pextr/PBZ for a rapidly spinning black hole as
Pextr
PBZ ∼−∞
−
κ σ0sin2ξ,(46)
where we considered plasmoidmediated reconnection in
the collisionless regime. Therefore, the power extracted
via fast collisionless magnetic reconnection can exceed
the one extracted through the BlandfordZnajek process
for an extended range of plasma magnetizations if there is
a signiﬁcant toroidal component of the magnetic ﬁeld in
the black hole ergosphere. Note that this energy extrac
tion mechanism is expected to be bursty in nature, with
a continuous buildup of the magnetic ﬁeld conﬁguration
storing the magnetic energy that is eventually dissipated
via fast magnetic reconnection.
V. CONCLUSIONS
In this paper, we envisioned the possibility of extract
ing black hole rotational energy via fast magnetic recon
nection in the black hole ergosphere. We considered a
conﬁguration with antiparallel magnetic ﬁeld lines near
the equatorial plane, which is induced by the frame drag
ging of the spinning black hole. The change in magnetic
ﬁeld direction at the equatorial plane produces an equa
torial current sheet that is disrupted by the plasmoid in
stability when its aspect ratio reaches a critical value (for
a collisionless relativistic pair plasma, the critical aspect
ratio condition is derived in Ref. [55]). The formation of
plasmoids/ﬂux ropes drives fast magnetic reconnection,
which rapidly converts the available magnetic energy into
plasma particle energy. When the plasma is expelled
out of the reconnection layer, the magnetic tension that
drives the plasma outﬂow relaxes. The ﬁeld lines are then
stretched again as a consequence of the frame dragging
and a current layer prone to fast plasmoidmediated re
connection forms again. This process leads to reconnect
ing current sheets that form rapidly and intermittently.
Magnetic reconnection accelerates part of the plasma
in the direction of the black hole rotation, while another
part of the plasma is accelerated in the opposite direc
tion and falls into the black hole. Black hole energy ex
traction occurs if the plasma that is swallowed by the
black hole has negative energy as viewed from inﬁnity,
while the accelerated plasma that gains energy from the
black hole escapes to inﬁnity. Therefore, diﬀerently from
the BlandfordZnajek process, in which the extraction of
rotational energy is obtained through a purely electro
magnetic mechanism, the energy extraction mechanism
described here requires nonzero particle inertia. This
mechanism is also diﬀerent from the original Penrose pro
cess, since dissipation of magnetic energy is required to
produce the negativeenergy particles. Clearly, all mech
anisms extract black hole rotational energy by feeding the
black hole with negative energy and angular momentum.
We showed analytically that energy extraction via
magnetic reconnection is possible when the black hole
spin is high (dimensionless spin a∼1) and the plasma is
strongly magnetized (plasma magnetization σ0>1/3).
Magnetic reconnection is assumed to occur in a circu
larly rotating plasma with a reconnecting ﬁeld having
both azimuthal and radial components. The region of
the phasespace {a, r/M}where magnetic reconnection
is capable of extracting black hole energy depends on
the plasma magnetization σ0and the orientation ξof the
reconnecting magnetic ﬁeld. We showed that high val
ues of the plasma magnetization and mostly azimuthal
reconnecting ﬁelds can expand the energy extraction re
gion up to the outer boundary of the ergosphere. For
a dimensionless spin parameter that approaches unity,
the extraction of black hole energy is maximal when the
dominant reconnection Xpoint (where the two magnetic
reconnection separatrices intersect) is close to the event
horizon. For σ01, we showed that the asymptotic neg
ative energy at inﬁnity per enthalpy of the plasma that
is swallowed by the black hole is ∞
−' −γoutvout /√3'
−pσ0/3. On the other hand, the plasma that escapes
to inﬁnity and takes away black hole energy asymptotes
the energy at inﬁnity per enthalpy ∞
+'√3γoutvout '
√3σ0.
We calculated the power extracted from the black hole
by the escaping plasma and evaluated its maximum when
the dominant reconnection Xpoint is close to the event
horizon. This corresponds to Pmax
extr ∼0.1M2√σ0w0for
the collisionless plasma regime and one order of mag
nitude lower for the collisional regime. The overall eﬃ
ciency of the plasma energization process via magnetic re
connection can reach a maximum of ηmax '3/2. There
fore, the additional energy extracted from the black hole,
while important, does not extensively modify the energet
ics of the escaping plasma. On the other hand, the power
extracted via fast magnetic reconnection can induce a
signiﬁcant reduction of the rotational energy of the black
hole, dErot/dt =∞
−w0AinUin . This is eﬀective when ais
close to unity. Therefore, if we consider a black hole with
dimensionless spin parameter close to unity and deﬁne
$= 1 −a1, we have dErot/dt ' −(M/4√$)d$/dt
and the spindown time can be obtained as
tsd =O(10)
2√σ0w0M(√$f−√$i),(47)
where the subscripts f and i are used to label ﬁnal and
initial values, respectively. This indicates that mag
netic reconnection can cause a signiﬁcant spindown of
11
the black hole when a∼1. For example, fast mag
netic reconnection in the ergosphere can reduce the black
hole dimensionless spin from a= 0.999 to a= 0.99 in
tsd ∼1/(√σ0w0M). On the other hand, at lower spin
values, especially for a < 0.9, magnetic reconnection loses
its eﬃcacy as the plasma available in the ergosphere di
minishes.
Various systems hosting a black hole are expected to
have magnetization σ0&1 in the ergosphere. For the
typical conditions around supermassive black holes in ac
tive galactic nuclei (AGNs), the energy density of the
electromagnetic ﬁeld far exceeds the enthalpy density
of the plasma and σ0∼104or larger [48, 56, 57] is
foreseeable. Likewise, long and short gammaray bursts
(GRBs) may have σ0∼1 or larger [58–61] in the ergo
sphere (a central black hole is assumed). Under these
magnetization conditions (in addition to a∼1), mag
netic reconnection is capable of extracting energy from
the black hole. For σ0∼1−104, we have shown that
the bursty energy extraction rate occurring during fast
magnetic reconnection can exceed the more steady en
ergy extraction rate expected from the BlandfordZnajek
mechanism. On the other hand, as the plasma magneti
zation increases, energy extraction via fast magnetic re
connection becomes always subdominant since it requires
nonvanishing plasma inertia.
In the scenario proposed here, fast magnetic reconnec
tion occurs rapidly and intermittently, so that the as
sociated emission within a few gravitational radii from
the black hole is expected to be bursty in nature. This
bursty behavior of fast magnetic reconnection might be
responsible for triggering ﬂares in the vicinity of rotat
ing black holes. Indeed, frequent Xray and nearinfrared
ﬂares are detected on a regular basis from the Galactic
Center black hole Sgr A* [e.g. 62–65], and magnetic re
connection close to the black hole is often conjectured to
induce these ﬂares [e.g. 25, 56, 66]. Recent observations
by the GRAVITY collaboration [67] have been able to pin
down the motion of nearinfrared ﬂares originating near
the last stable circular orbit of Sgr A*. Reconnection
layers originate naturally in the ergosphere of rotating
black holes and produce plasmoids/ﬂux ropes that are
ﬁlled with energized plasma with an energy budget that
can exceed the energy originally stored in the magnetic
ﬁeld.
In this paper we have assumed that the plasma rotates
circularly around the black hole. This assumption may be
relaxed in order to treat more complex scenarios in which
reconnection occurs in noncircular orbits. In this case,
the plasma could approach the event horizon even when
the black hole spin is not particularly high, expanding
the parameter space region where magnetic reconnection
can extract black hole energy. Another situation that
could increase the eﬃcacy of magnetic reconnection is the
simultaneous presence of equatorial and nonequatorial
current sheets [25], which may result in an increase of the
extracted power to some degree. Finally, for reconnecting
magnetic ﬁelds that have a signiﬁcant radial component,
particle acceleration owing to the reconnection electric
ﬁeld can increase the rate of energy extraction and the
overall eﬃciency of the reconnection process.
Acknowledgments
We gratefully acknowledge discussions with Lorenzo
Sironi, Daniel Groˇselj, Russell Kulsrud, Manasvi Lingam,
YiHsin Liu, Joonas N¨attil¨a, Kyle Parfrey, Bart Rip
perda, Daniel Siegel, and Yajie Yuan. L.C. acknowl
edges support by the NASA ATP NNX17AG21G and
NSF PHY1903412 grants. F.A.A. acknowledges support
by the FondecytChile Grant No. 1180139.
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