In Situ Discovery of an Electrostatic Potential, Trapping Electrons
and Mediating Fast Reconnection in the Earth’s Magnetotail
J. Egedal,1M. Øieroset,2W. Fox,1and R.P. Lin2,3
1Massachusetts Institute of Technology, Plasma Science Fusion Center, Cambridge, Massachusetts 02139, USA
2Space Sciences Laboratory, University of California, Berkeley, California 94720, USA
3Physics Department, University of California, Berkeley, California 94720, USA
(Received 11 February 2004; revised manuscript received 10 August 2004; published 19 January 2005)
Anisotropic electron phase space distributions, f, measured by the Wind spacecraft in a rare crossing of
a diffusion region in Earth’s far magnetotail (60 Earth radii), are analyzed. We use the measured f to probe
the electrostatic and magnetic geometry of the diffusion region. For the first time, the presence of a strong
electrostatic potential (1 kV) within the ion diffusion region is revealed. This potential has far reaching
implications for the reconnection process; it accounts for the observed acceleration of the unmagnetized
ions out of the reconnection region and it causes all thermal electrons be trapped electrostatically. The
trapped electron motion implies that the thermal part of the electron distributions are symmetric around
vk? 0: f?vk;v?? ’ f??vk;v??. It follows that the field aligned currents in the diffusion region are
limited and fast magnetic reconnection is mediated.
DOI: 10.1103/PhysRevLett.94.025006 PACS numbers: 94.30.–d, 52.20.–j, 52.35.Vd
Magnetic reconnection, the process in a plasma during
which magnetic field lines rearrange and change topology
[1,2], plays a fundamental role in a variety of plasma
phenomena. It controls the evolution of solar flares , it
allows the solar Wind to enter Earth’s magnetosphere ,
and it is an integral part of magnetic substorms and the
aurora phenomena [5,6]. Mounting evidence suggests that
the fast rate at which reconnection proceeds is strongly
influenced by the ion-scale Hall effect . However, for
fast reconnection to be maintained, a hitherto unknown
electron scale process must limit the electron current in the
reconnection region, allowing electrons to diffuse through
the magnetic field at the same or at even a higher rate than
the ions .
Recently, the first in situ measurements of the electron
distribution function have been reported for a reconnection
event encountered by the Wind spacecraft in Earth’s mag-
netotail [9,10]. This event, observed in the deep magneto-
tail (60 Earth radii), is characterized by a kinetic regime in
which the electron temperature is high, Te? 400 eV, and
the magnetic field is weak, B ? 6–12 nT, implying an
electron thermal speed, vth, much larger than the Alfve ´n
speed, vA(for the observed density n ? 0:1 cm?3), and
thus larger than the outflow speed, vout, of plasma from the
Because of the computational requirements posed by
this kinetic regime (vth? 40vout), so far it has not been
accessible to self-consistent particle simulations. Mean-
while, in Ref.  the regime was investigated through
the use of Liouville’s theorem applied to the electron
trajectories in prescribed fields. This technique of follow-
ing particle trajectories for understanding particle distribu-
tion functions measured in space was pioneered by Speiser
 and has since been applied in numerous investigations
[13–16]. More recently, prompted by experimental obser-
vation in the Versatile Toroidal Facility (VTF) magnetic
reconnection experiment, the kinetic regime was also
studied in Ref. . Applying an approach similar to those
in Refs. [11,17], the Wind electron distributions are used to
‘‘probe’’the electrostatic and the magnetic geometry of the
reconnection region; the presence of a strong electrostatic
potential (?1 kV) is revealed. This potential causes all
thermal electrons to follow trapped trajectories inside the
reconnection region, significantly limiting the number of
‘‘free’’ current carriers. As has been proven experimentally
at the VTF [18,19], the trapped orbit dynamics effectively
mediates fast reconnection.
Figure 1 illustrates the ion flow, magnetic field compo-
nents, and plasma density measured by Wind during its
fortunate encounter with the reconnection region. The
gradual ion flow reversal and the measured magnetic field
components, including Hall-current-generated out-of-
plane (By) magnetic fields, indicate that Wind passed
through the ion diffusion region of a quasisteady collision-
less reconnection region [9,10]. Based on these measure-
ments and assuming a magnetic X-line structure as
characteristic for reconnection, the approximate path of
Wind through the diffusion region can be reconstructed
as shown in the schematic of Fig. 2(a).
Wind passed through the ion diffusion region are dis-
played in Figs. 3(a)–3(c). The measurements provide
the phase-space density, f, of the electrons as a func-
tion of pitch angle, ? ? ??v;B?, and kinetic energy,
Ek. The anisotropy is most pronounced in the distribu-
tion shown in Fig. 3(a), which was obtained near the center
of the diffusion region. The anisotropy gradually decreases
as Wind exits the X-line region [Figs. 3(b) and 3(c)].
Because the Coulomb collision frequency of the elec-
trons, ? ? 10?9s?1, is very small compared to any other
frequency in the magnetotail, f is governed by the colli-
sionless Vlasov equation df=dt ? 0. Here d=dt represents
PRL 94, 025006 (2005)
21 JANUARY 2005
2005 The American Physical Society
the total time derivative along particle orbits. The Vlasov
equation states that the distribution is constant along par-
ticle orbits through phase-space ?x;v?. Hence, we can
equate f for a point ?x0;v0? in the reconnection region to
the isotropic distribution, f1, in the ambient plasma sheet
outside the ion diffusion region by following particle orbits
back in time until they reach points ?x1;v1? in the sheet. It
then follows that
f?x0;v0? ? f1?v1?;v1? jv1j:
⋅ ⋅ ⋅
~ 100 km
FIG. 2 (color).
the trajectory of Wind. The green arrows indicate the flow of
magnetic flux towards and away from the X line. At locations
away from the X line the in-plane magnetic field approaches
values of about 10 nT. Besides the in-plane magnetic fields, an
out-of-plane guide magnetic field, By? 6 nT, was also ob-
served. The red arrows illustrate the electrostatic electric field
trapping the thermal electrons within the ion diffusion region
(the yellow area). (b) An example of a passing electron guiding
center trajectory reaching the location of the Wind spacecraft.
(c) An example of a trapped electron trajectory. The electron
bounces back and forth along a field line, while slowly drifting
with the magnetic field towards the X line.
(a) Illustration of the magnetic geometry and
Hall magnetic fields
vx [km s−1]
vy [km s−1]
vz [km s−1]
components, and density during the encounter with an active
reconnection region. Further away from the X line vyand vz
remain small, while the ion outflow speed, jvxj, approaches the
Alfve ´n speed of 400 km=s.
Wind measurements of the ion flow, the magnetic field
0 90 1800 90180
0 90 180
090 1800 90180
0 901800 90180
Pitch Angle (degrees)
08:03:38 UT 08:06:53 UT
(d) (e) (f)
08:00:22 UT: Φ0 = 0 V
Φ0 = 300 V
Φ0 = 900 V
Φ0 = 1150 V: 08:00:22 UT
FIG. 3 (color).
sured by the 3D plasma and energetic particle instrument  on
the Wind spacecraft at locations indicated in Fig. 2. Each line
represents the phase-space density for a given energy; the re-
spective energies are given on the right of the figure. In (a) the
blue parts correspond to trapped electrons, and the red parts
correspond to passing electrons. At sufficiently high energies the
electron orbits become unmagnetized, leading to pitch angle
scattering; this effect reduces the level of anisotropy observed
above 6 keV [the green parts in (a)]. (d)–(i) Theoretical distri-
butions calculated for locations [inferred from the measured
values of ?Bx;Bz?] corresponding to the data in (a)–(c). The
distributions in (d)–(g) correspond to the center of the diffusion
region (t ? 08:00:22UT) calculated with increasing values of
the trapping potential (?02 f0;300;900;1150g V). The best
match to the data in Fig. 3(a) is obtained with ?0? 1150 V.
The distributions in (h) and (i), also obtained with ?0? 1150 V,
correspond to the Wind observations in the outflow region [see
(b) and (c)].
(a)–(c) Electron pitch angle distributions mea-
PRL 94, 025006 (2005)
21 JANUARY 2005
The measured distributions are therefore closely related to
changes in kinetic energy that the particles undergo along
their trajectories into the X-line region.
The electrons can reach the X-line region via two dis-
tinct types of trajectories: passing and trapped. Anexample
of a passing trajectory is shown in Fig. 2(b). The passing
electron reaches the location of the measurement along a
field line. Electrons of this type (if not intercepted by the
spacecraft) continue to follow the magnetic field line out of
the region, without experiencing any significant changes in
An example of a trapped electron is given in Fig. 2(c). In
their rapid bounce motion back and forth along field lines,
the total energy is conserved. Consequently, the variations
in the kinetic energy during the bounce motion are linked
to thevariations in the electrostatic potential along the field
lines. Thus, the trapping of electrons is a consequence of
the electric field structure in the reconnection region as
well as the local minimum in the magnetic field strength.
The evolution of the kinetic energy in the slower motion
where the trapped electrons drift with the magnetic field
towards the X line is governed by adiabatic invariance. For
thermal electrons the magnetic moment, ? ? mv2
and the second adiabatic invariant of the bounce motion,
J ?Hvkdl, are conserved . The conservation of ? and
the decrease in B as the X line is approached causes a
decrease in v2
served in vk, which is due to the conservation of J and
the decreasing orbit bounce length (Hdl) in the inflow
region. The combination gives particles with vk* v?
(vk& v?) a net gain (loss) in Ekin the inflow region. By
is large enough that ? is conserved for thermal electrons at
all locations, and no pitch angle diffusion is observed at the
X line. In the outflow region the changes in Ekare reversed.
The trapped orbit motion also implies that f is nearly
f?v;180?? ?? .
For sufficiently high energies (Ek> 1 keV) only the
magnetic forces are important in determining if an electron
is trapped or passing. The flat parts of the distribution in
Fig. 3(a), indicated by the red lines, correspond to passing
orbits, whose energies do not change on the path from the
ambient plasma. The blue parts are associated with trapped
electrons; the dip in phase-space density as a function of ?
is caused by the cooling of these electrons in the inflow
region. This interval of trapped electrons is determined by
the magnetic field strength, By? 6 nT, along the X line
and the asymptotic value, B1? 10:5 nT, away from the X
line. From the finite ratio By=B1, standard magnetic mirror
trapping arguments predict a phase-space boundary be-
tween passing and trapped particles (a loss-cone boundary)
at the pitch angles, ?b? 49?and ?b? 131?. These
boundaries agree with the interval covered by the dips in f.
For the bulk portion of the distribution (Ek? 639 eV)
the dips in phase-space density extend from ? ? 0?to ? ?
?[11,13]. Meanwhile, an increase is ob-
180?. This is evidence that the bulk electrons, at all pitch
angles, enter the X-line region along trapped trajectories.
The broadening of the interval for trapping implies the
presence of an electrostatic potential, ?trap, that traps all
In the remaining part of this Letter, we describe our
theoretical modeling of the measured distributions. The
magnetic geometry close to the X line is approximated
by B ? b0??z=??^ x ? ?x^ z ? l0^ y?. Consistent with the
Wind magnetic field measurements (see Fig. 1) we apply
By? b0l0? 6 nT, and l0? 65 km. The case with ? ? 1
corresponds to a currentless cusp for which the angle of the
separatrix is 90?(the equation of the separatrix is jzj ?
j?2xj). For other values of ? the separatrices meet at some
other angle. Several values of ? have been applied in our
modeling scheme; the results prove to be independent of
this parameter. For simplicity, here we consider a configu-
ration with ? ? 1. The electric fields are approximated by
E ? ?r??x;z? ? Ey^ y. Here ? ? ?rec? ?trap, where
?trap? ?0exp???x2? z2?=l2
rated to trap the thermal electrons. The reconnection rate
is estimated by assuming that vinflow? 0:1vA, from which
Ey? 0:1 ? 400 km=s ? 6 nT ? 0:24 mV=m
(note that our results are insensitive to Ey). The role of
E ? B ? 0 (required from E ? v ? B ? 0) outside the
electron diffusion region . We apply a recent experi-
mental result that the size ? of the diffusion region scales
with the drift orbit width of the trapped electrons, ? ?
also been found applicable to configurations without a
guide magnetic field ). The potential ?rec? l0Ey?
15 V is small compared to the electron temperature
(400 eV), and the theoretical distributions are therefore
nearly independent of ?recand ?.
We assume that the electrons have reached a point x1
outside the X-line region when B?x1? ? B1. Hence, orbits
are followed back in time from their initial locations
? B1, where x1? ?x1;y1;z1?. The am-
bient plasma sheet is approximated by an isotropic distri-
bution f1?jv1j?, with a temperature of Te? 400 eV for the
bulk electrons and a power law dependence, f / E?4:5, for
the high energy tail, Ek> 2 keV, consistent with Wind
measurements away from the diffusion region. Additional
simulations (not included here) show that the fluctuation
level in the magnetic field observed in Fig. 1 does not
influence the results of our analysis.
Figures 3(d)–3(g) represent theoretical distributions
corresponding to the Wind measurements at time
08:00:22 where Wind was near the center of the diffu-
sion region. These theoretical distributions are obtained
with increasing values of the trapping potential (?02
f0;300;900;1150g V). As seen, the measured distributions
0? is the potential incorpo-
4l0Eylog??x2? ?2?=?z2? ?2?? is to ensure that
? 10 km  (the length scale ?chas
PRL 94, 025006 (2005)
21 JANUARY 2005
are best matched for ?0? 1150 V. The value ?0?
1150 V was also applied in calculating the distribu-
tions in Figs. 3(h) and 3(i) corresponding to the location
of Wind for times 08:03:38 and 08:06:53. The agree-
ment between observations [Figs. 3(a)–3(c)] and model
[Figs. 3(g)–3(i)] covers several points which support the
(i) For Ek> 1 keV,thewidth and the depth of the dipsin
the theoretical phase-space density are highly sensitive to
the ratio of By=B1; agreement between the measured f and
the theoretical f is obtained only when the measured
values of Byand B1are used in the simulation.
(ii) The theoretical anisotropy agrees with the measure-
ments not only near the center of the diffusion region, but
also at locations in the outflow region; compare Figs. 3(b)
and 3(c) with Figs. 3(h) and 3(i).
(iii) For Ek< 1 keV, agreement with the measured dis-
tributions are obtained by including an electrostatic poten-
tial, ?trap, whose strength is uniquely determined by the
anisotropy in the electrons. Furthermore, the energy gained
by the unmagnetized ions in their acceleration in the out-
flow region agree with ?trap. For the applied electrostatic
geometry, the potential drop from the center to the edge of
the simulation region is about 1000 V. The reported out-
flow speed of the ions, vion;out? 400 km=s , corre-
sponds to acceleration in a potential drop of ?835 V.
This suggests that ?trapis responsible (and adequate) for
accelerating the unmagnetized ions out of the X-line re-
gion. Such electrostatic structures have also been observed
in self-consistent particle simulations  (but were not
associated with trapping because vth? vout).
(iv) For electrons with energies above 6 keV, the Larmor
radii of the electrons are larger than the characteristic
length scale of the magnetic field curvature, l0. Thus, these
electrons see significant changes in B over a single Larmor
period which causes pitch angle diffusion . Such non-
adiabatic effects are evident in both the measured and the
theoretical distributions: the levels of anisotropy decrease
rapidly above 4 keV.
(v) Inside the ion diffusion region the trapped electrons
bounce back and forth along field lines while drifting with
the magnetic field. They remain ‘‘frozen in’’ to the mag-
netic field to within the length scale ?cfrom the X line.
Thus, the electron dynamics are consistent with the Hall
magnetic field evident in Fig. 1 and discussed in Ref. .
Simulations with vth? vout[7,21,23,24] do not capture
the electron dynamics evident in the Wind measurements:
the electrons do not have time to bounce back and forth
along the field lines, so theoretical distributions, in general,
will not display the observed symmetry (typically in these
simulations jk? qnvth). However, the regime vth? voutis
likely to be relevant for reconnection sites closer to Earth
(characterized by higher values of B and lower ninflow
producing a higher vA).
In summary, we have applied the electron phase-space
distributions f, measured by Wind within a reconnection
region in Earth’s magnetotail, to infer properties of the
electrostatic geometry inside the ion diffusion region. For
the first time, the presence of a strong electrostatic poten-
tial, trapping all thermal electrons, is revealed. The sym-
metric properties of f associated with trapping can be
expressed as f?vk;v?? ? f??vk;v??. It follows that jk?
qRvkf?vk;v??dv3? qnvth, which allows the plasma to
support an electric field along the X line, mediating fast
reconnection [17,19]. Additional investigations in the re-
gime vth? vout, numerical as well as experimental, are
needed to determine how the trapped orbit dynamics trans-
late into dissipation free r ? P and v ? rj terms in the
generalized Ohm’s law.
This work is partly funded by NSF/DOE Grant No. DE-
FG02-03ER54712 at MIT and NASA Grants No. NAG5-
12941 and No. NAG5-10428 at U.C. Berkeley.
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