Table Of ContentDemagnetized Electron Heating at Collisionless Shocks
David Sundkvist1,∗ and F. S. Mozer1
1
Space Sciences Laboratory, University of California, Berkeley, CA. 94720 USA
(Dated: November14, 2012)
Seventy measurements of electron heating at the Earth’s quasi-perpendicular bow shock are an-
alyzed in terms of Maxwellian-temperatures obtained from fits to the core electrons that separate
thermalheatingfromsupra-thermalacceleration. Theperpendiculartemperaturesarebothgreater
andlesserthanexpectedforadiabaticcompression. Theaverageparallelandperpendicularheating
3 is the same. These results are explained because, over the electron gyroradius, δB/B ∼ 1 and
1 eδφ/Te ∼1, so electron trajectories are more random and chaotic than adiabatic. Because density
0 fluctuationsare also large, trappingand wavegrowth in density holes may be important.
2
PACSnumbers: 52.35.Tc
n
a
J
Collisionless plasma shocks play an important role in tant not only for determining the importance of disper-
3
astrophysicalandsolarsystemplasmasby redistributing sive effects [20] in addition to dissipation [14], but also
2
supersonic flow energy into thermal energy of plasma, fordetermining the adiabaticity andresultantheatingof
] fields, wavesandacceleratedparticles. Atthe veryheart theelectrons. Non-adiabaticityofelectronshasalsobeen
h
of the physics of collisionless shocks is the question of studiedin simulations[10, 12,13] wereitwas notedthat
p
dissipation, which is needed to limit the steepening of different sub-populations of electrons could be over- or
-
e the shock [9, 21]. A shock steady state is reached when sub-adiabatic depending on injection angle.
c
a the nonlinear steepening is balanced by dissipation with Inthisletterweusespaceplasmaobservationstoshow
p the addition of dispersion. thattheoverallthermalelectronheatingisrarelystrictly
s The shock heating mechanisms for the ions and elec- adiabatic but canbe super- as well as sub-adiabatic (de-
.
s tronsarebelievedtobeverydifferent. Inconstrasttothe pending on Mach number), and that this is due to the
c
i ions the problem of the electron heating is not fully un- fact that the ratio of perturbed to the average magnetic
s
derstood, and remains controversial after some 40 years field, and the ratio of electric potential to thermal tem-
y
h of research. Early theories focused on micro-instabilities perature(effectivelythe electricfieldgradient)areofthe
p such as ion-acoustic or lower-hybrid that should arise order of one, at all relevant length scales through the
[ fromthe presenceof unstable currentsin the shocklayer shockramptransition. Byintroducingtheconceptofthe
1 [11]. The current consensus however is focusing on the perpendicular Maxwellian-temperature and contrasting
v presence of the macroscopic (DC) magnetic and elec- it with both the conventional perpendicular moments-
1 tric fields and associated potential in the so called de temperature and the adiabatic-temperature, we also
7
Hoffman-Teller-frame. These fields inflate and also open show that previous studies have likely overestimatedthe
6
up a hole in the electron population when crossing the amountofelectronheating. We then discuss the proper-
5
. shock,iftheelectronsaremagnetizedandeffectivelyadi- ties of the perpendicular Maxwellian-temperature at 70
1
abatic [8, 16, 19]. This process is reversible and addi- quasi-perpendicular bow shock crossings.
0
3 tionalscatteringand/orinstabilities arerequiredtoinfill We use data from the THEMIS multi-spacecraft mis-
1 the distribution and thermalize the lower energy elec- sion[1],duringtimeswhenatleastthreeoftheTHEMIS
: trons. spacecraft were close together, and the time between in-
v
i Previousstudiessuggestedthatelectronsconservefluid dividual encounters of the Earth’s bow shock was short.
X
adiabaticity (Te,⊥/B ∼constant, where Te,⊥ is the elec- As an example, we will consider the THEMIS A
r
tron temperature perpendicular to the magnetic field) crossing on December 7, 2010, at 14:59:26 UT (Uni-
a
across the shock [17, 18], although a statistical study versal Time) near X = 8.26, Y = −7.53 and
GSE GSE
showed some heating above that expected from conser- Z = 2.84 (Geocentric Solar Ecliptic coordinates),
GSE
vation of the adiabatic moment [15]. Early observations withinabout30secondsofsimilarcrossingsbyTHEMIS
of super-adiabatic electron heating prompted the devel- D and THEMIS E. This was a quasi-perpendicular
◦
opment of alternative heating mechanisms [3–7] where (θ = 85 ) shock crossing with super-critical Alfv´enic
Bn
electrons are non-adiabatic on a particle level and the and fast Mach numbers (M = 13,M = 6). θ is
A F Bn
electrontrajectoriesarekinematicallydescribed. Thede- the angle between the upstream magnetic field and the
magnetizedandnon-adiabaticbehaviourinthesemodels normal to the shock and it was obtained from a mini-
isduetothepresenceofalarge-scale(macroscopic)elec- mum variance analysis of the magnetic field. The shock
tric field gradient, resulting in super-adiabatic heating normal direction in GSE was nˆ = (0.74,−0.67,−0.13).
of electrons depending on the gradient scale. The shock Multi-spacecraft timing analysis gave a shock speed of
transition scale size and other gradients are thus impor- 5.5 km/s, consistent with the above shock normal. The
2
Electron temperature [eV]12345000000 (a) TTTT⊥|a⊥| d ,MmMiaobaamaxxtewiwcn..t ffiitt
−2−1−1−1mssreV]109 (b) 21 eV electrons (75°−105°)
def [c
108
−2−1−1−1mssreV]110078 (c) 435 eV electrons (75°−105°)
def [c106
14:59:00 14:59:10 14:59:20 14:59:30
FIG. 2: Shock electrons sampled by the THEMIS A space-
craft. (a) Electron perpendicular and parallel temperatures.
Fitting a Maxwellian to the thermal part of the electron
FIG. 1: Snapshots of the electron distribution function sam- distribution function gives a perpendicular (red solid) and
pledover0.25s, forpitchanglesof75◦−105◦. Thesnapshots parallel (blue dashed) heating of electrons that is less than
are taken at distinctive regions across the shock. Arrows in that expected from adiabatic compression (black dashdot).
Fig. 2 show where the snapshots were taken. (a) Upstream Also plotted is the perpendicular moment-temperature that
region. (b)Behindtheramp. (c)Downstream. (d)Fardown- includes supra-thermal electrons, calculated using high reso-
stream. lution plasma data. Arrows indicatelocations of distribution
functions in Fig. 1. (b) Differential energy flux (DEF) for 22
eV (thermal) perpendicular electrons (c) DEF 435 eV (non-
magnetic field increased from 5 nT to 30 nT across the thermal) perpendicularelectrons.
shock and the plasma density increased from 30 cm-3
to more than 300 cm-3. Quarter-second snapshots of
typical 75◦−105◦ (hereafter called perpendicular) elec- Time-domain data across the shock of Figure 1 are
tron distributions across this bow shock are illustrated illustrated in Figure 2, in which the upstream direc-
in Figure 1. These distributions consist of low energy tion is at the right of the figure and the shock ramp
thermal cores and supra-thermal power law distribu- is indicated by the yellow rectangle. The black dash-
tions of halo electrons, originating in the upstream solar dotted curve in panel (a) is the perpendicular adiabatic-
wind. Maxwellian-temperatures are defined as the fits temperature expected for adiabatic compression of the
ofMaxwell-Boltzmanndistributionstothethermalcores plasma. It is computed from the magnetic field data
of the plasma and are illustrated as the red curves in by making the first invariant, Te,⊥/B = Te,⊥,sw/Bsw,
Figure 1. These and all distributions in this paper are a constant whose value was determined in the solar
analyzed in the spacecraft frame because their transfor- wind. The magnetic field is here sampled with 128 vec-
mation to the more appropriate frame of the plasma has tors/s by the fluxgate magnetometer. The red solid
been shown to affect the Maxwellian-temperature esti- curve in panel (a) is the perpendicular Maxwellian-
matebynomorethanaboutoneeV,becausetheplasma temperature and the green curve with diamonds is the
flow speed is small compared to the thermal electron perpendicular moment-temperature obtained from mo-
speed. It is also noted that the spectra have been ad- ments of the spin period distribution function. The blue
justed for the spacecraft potential. The four snapshots dashedcurveistheparallelMaxwellian-temperature. Be-
of the perpendicular electron distributions in Figure 1 cause the moment-temperature includes contributions
are taken at select positions that highlight distinctive from the supra-thermal electrons, it is greater than the
features of electron shock physics. The positions for Maxwellian-temperature, as shown in panel (a). In the
these snapshots are indicated with arrows in Fig. 2. In current data set, the moment-temperature exceeds the
the upstream region, panel (a), a mixture of 9 eV per- Maxwellian-temperature by factors from 1.2 to 2.0.
pendicular core electrons co-exist with halo electrons of Panels (b) and (c) of Figure 2 give the fluxes of 22 eV
higherenergyand,perhaps,someshockacceleratedelec- thermal electrons and 435 eV supra-thermal electrons.
trons. Just downstream of the ramp in panel (b), the The thermal electrons were heated predominately in the
Maxwellian-temperatureincreasedbyafactoroftwoand shock ramp and the more energetic electrons were accel-
flat-topdistributionswereseen. Theseelectronswereob- erated about three seconds downstream of the ramp (as
served with relatively little change in the thermal com- can also be seen in panel (c) of Figure 1). Thus, ther-
ponent further downstream, in panels (c) and (d). malheatingandsupra-thermalelectronaccelerationarise
3
3
2.5
2
Adiabaticity1.5
1
0.5
0 5 10 15 20 25
MA
FIG. 3: Adiabicity TMaxwellian/Tadiabatic as a function of the
Alfv´en Mach number. The dashed curve is proportional to
1/MA. Circled pointscorrespondtothetwoeventspresented
in thisLetter.
from different physical processes, so separating them, as FIG. 4: Dimensionless perturbation ratios of magnetic field,
is done by considering the Maxwellian-temperature in- electric field and density for time scales τ around the con-
stead of the moments-temperature, is necessary for a vected electron gyrodiameter. Quantities are measured with
more careful analysis of electron heating at the bow 128 samples/s. (a) Total magnetic field. (b-c) δB/B. (d-e)
shock. An important result in panel (a) of Figure 2 δE·2ρe/Te,⊥ ∼δE/20. (f-g) δn/n.
is that the thermal electron heating described by the
Maxwellian-temperature is less than half that expected
for adiabatic electron heating in the compressed mag- angles between 0◦ −30◦ and 150◦−180◦. The average
netic field, as described by the adiabatic-temperature. [(parallelheating)/(perpendicularheating)]is0.95±0.14,
Thisisthefirstexperimentalevidencethatelectronsmay which means that the parallel and perpendicular elec-
be heated less than adiabatically at the bow shock. We trons are generally heated by the same amount when
note that in this event also the moment-temperature is crossing the bow shock.
less than that expected from adiabatic heating. The conservation of the first adiabatic invariant,
To determine statistically how thermal electrons Te,⊥/B, is derived under the assumption that the mag-
are heated relative to adiabatic heating, 70 quasi- netic field does not change over the gyration orbit of
perpendicular bow shock crossings with θ between an electron or the time for it to complete one gyra-
Bn
46 and 89.5◦ have been analyzed analogously to the tion. Also assumed is that the electric potential across
above example. The ratio of the observed Maxwellian- the electron gyrodiameter is small comparedto the elec-
temperature to the adiabatic-temperature (called ”adia- tron thermal energy, i.e. that the electric field gradi-
baticity”) is plotted in Figure 3 versus the Alfv´en Mach ent is small. To test the validity of these conditions,
number. It is seen that, at individual crossings, thermal we compute the fluctuations of the magnetic field, B,
electrons may be heated more or less than adiabatically and electric field, E. In this case the magnetic field re-
by factors as great as three, although the average adia- quirementfor conservationofthe firstinvariantbecomes
baticity is 1.06±0.52. The adiabaticity depends on the δB/B(t,τ)=[B(t+τ)−B(t)]/[B(t+τ)+B(t)]/2≪1
Alfv´en Mach number, with the adiabaticity decreasing where τ is a time scale, here corresponding to the con-
withincreasingMachnumber. Inthisdataset,thereare vected gyrodiameter.
no examples of sub-adiabatic heating for Mach numbers In Figure 4, δB/B is calculated for a randomly se-
less than five and the few measurements above a Mach lected bow shock crossing by THEMIS E, for which δB
number of 10 all have adiabaticity less than one. The was determined from search coil data and B came from
dashed curve in Figure 3 is proportional to 1/MA and fluxgate data. The local time of the crossing was 12:40
it describes the average behavior of the adiabaticity as and the latitude was 11◦, with the characteristic shock
a function of MA, as is expected because Te,⊥ ∼ B and parameters MA = 8.8, θBn = 64◦, and plasma β = 3.5.
MA ∼1/B. Panel (a) of this figure gives the magnetic field strength
Parallel heating has also been studied by computing andpanels(b-c)givethetimedomainvaluesofδB/B for
the Maxwellian-temperatures for electrons having pitch the time scales, τ, associated with each panel. For Te,⊥
4
= 20 eV, B = 20 nT, and a shock velocity of 5 km/sec,
theelectrongyrodiameterisabout1kmandthetimefor
theshocktomoveonegyrodiameterisabout0.2seconds.
∗
Electronic address: [email protected]
Overatypicalshockrampthe gyrodiametermaychange
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This work was supported by NASA contract
sionlessShocks.PhysicalReviewLetters,107(21):215002,
NAS5-02099-09/12 and NASA grants NNX13AE24G,
Nov. 2011.
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5
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