DTIC ADA534483: A Physical Relationship Between Electron-Proton Temperature Equilibration and Mach Number in Fast Collisionless Shocks

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TheAstrophysicalJournal,654:L69–L72,2007January1
(cid:1)2007.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A.
A PHYSICAL RELATIONSHIP BETWEEN ELECTRON-PROTON TEMPERATURE EQUILIBRATION AND
MACH NUMBER IN FAST COLLISIONLESS SHOCKS
Parviz Ghavamian,1 J. Martin Laming,2 and Cara E. Rakowski2
Received2006September19;accepted2006November9;published2006December12
ABSTRACT
The analysis of Balmer-dominated optical spectra from nonradiative (adiabatic) SNRs has shown that the ratio
of the electron to proton temperature at the blast wave is close to unity at v (cid:1)400 km s(cid:1)1 but declines sharply
S
down to the minimum value of m /m dictated by the jump conditions at shock speeds exceeding 2000 km s(cid:1)1.
e p
We propose a physical model for the heating of electrons and ions in non–cosmic-ray–dominated, strong shocks
(v 1400 km s(cid:1)1) wherein the electrons are heated by lower hybrid waves immediately ahead of the shock front.
S
These waves arise naturally from the cosmic ray pressuregradientupstreamfromtheshock.Ourmodelpredictsa
nearly constant level of electron heating over a wide range of shock speeds, producing a relationship (T/T ) ∝
e p 0
v(cid:1)2 (∝M(cid:1)2) that is fully consistent with the observations.
S
Subject headings: cosmic rays—ISM: kinematics and dynamics—plasmas—shock waves—
supernova remnants
1. INTRODUCTION otherions.ThismakesthemeasuredwidthofthebroadBalmer
line directly proportional totheprotontemperaturesetbycol-
Thediscoveryofcollisionlessshockwavesinthesolarwind
lisionless heating at theshock front.Thebroad-to-narrowflux
in the 1960s ushered in a new era in the physics of space
ratio, I /I , on the other hand, is sensitive to both the degree
plasmas(seeTidman&Krall1971,Sagdeev1979,andKennel B N
of electron-proton temperature equilibrationattheshockfront
1985 for reviews and references). Due to the low density [i.e.,(T/T ) ]andtheshockvelocity,v (Chevalieretal.1980;
(n(cid:1)1 cm(cid:1)3) of the interplanetary medium, the jump in hy- e p 0 S
Smith et al. 1991). The ratio also depends (although lesssen-
drodynamicalquantitiesisproducednotbyCoulombcollisions
sitively)onthepreshockneutralfraction.Thebroadcomponent
but by collective plasma processes such as electromagnetic
width and I /I of an observed Balmer-dominated shock can
wavesandturbulence.However,despitetheavailabilityofex- B N
be modeled with numerical shock codesto simultaneouslyes-
tensive in situ observations of solar wind shocks and the ex- timate v and (T/T ) (Smith et al. 1991; Ghavamian 1999;
penditureofconsiderableeffortintheoreticalmodelingofthese S e p 0
Ghavamian et al. 2001, 2002).
structures, a detailed understanding of the processes at the
In this Letter, we draw together observed values of
shock transition responsible for partitioning the shock energy (T/T ) and v measured in five Balmer-dominated SNRs, in-
between different charged particle species has been slow to e p 0 S
cludingseveralpreviouslyunpublishedmeasurementsfromthe
emerge. The problem is far more acute for interstellarshocks,
SNR RCW 86. We then propose a physical model of electron
where in situ measurements of the shock structure are un-
heating by lower hybrid waves in a cosmic ray precursorthat
available and the very high Mach numbers (∼30–200) make obeys the observed relationship between (T/T ) and v. We
numericalsimulationsofthesestructuresextremelydifficultor e p 0 S
conclude by exploring some consequences of this interpreta-
impossible.
tion, the connection to other observables, and applicability to
The optical emission generated by nonradiative supernova
other collisionless shock situations.
remnants (SNRs) in partially neutral gas provides a valuable
diagnostictoolforprobingtheheatingprocessesincollisionless
2. RELATIONSHIPBETWEENEQUILIBRATIONANDSHOCKSPEED
shocks. The optical spectra of these SNRs (which lose a neg-
ligible fraction of their energy to radiation) are dominated by In Figure 2, we plot 11 measurements of (T/T ) and v,
e p 0 S
Balmerlineemission,producedbycollisionalexcitationwhen obtained from long-slit spectra of four Galactic remnants: the
neutral hydrogen is overrun by the blast wave (Chevalier & Cygnus Loop (one position from the northeast), RCW 86 (po-
Raymond 1978; Bychkov & Lebedev 1979). Each emission sitionsfromthesouthwestern,northern,andeasternlimbs),Ty-
line consists of two components: (1) a narrow velocity com- cho’s SNR (“knot g” from the eastern rim), SN 1006 (one po-
ponentproducedwhencold,ambientHioverrunbytheshock sitionfromthenorthwesternrim),andoneremnantintheLarge
is excited by electron and proton collisions and (2) a broad Magellanic Cloud, DEM L71 (five positions from the full cir-
velocity component produced when fast neutrals created by cumference of the shell). In the first four remnants, the shock
postshockchargeexchangearecollisionallyexcited(Chevalier parametershavebeenestimatedvialong-slitspectroscopy.First,
et al. 1980). An example of a Balmer-dominated shock spec- we used the broad component Ha width to constrain therange
trum from the Galactic SNR RCW 86 is shown in Figure 1. ofshockspeedsforeachBalmerfilamentinthelimitsofminimal
Theopticalemissionarisesinaverythin((cid:1)1016cm)ionization [(T/T ) pm /m ] and full [(T/T ) p1]equilibration.Next,
e p 0 e p e p 0
zone, thin enough so that the protons transformed into hot we fine-tuned (T/T ) and v by using shock models to match
e p 0 S
neutrals have had little time to equilibrate with electrons and theobservedbroad-to-narrowfluxratio(Ghavamianetal.2001,
2002).
InthecaseofRCW86,wehaveaddedtwomoredatapoints,
1DepartmentofPhysicsandAstronomy,JohnsHopkinsUniversity,Balti-
indicated by dashes in Figure 2, derived from previously un-
more,MD;[email protected].
publishedopticalspectraofGhavamian(1999).Thesewereob-
2Naval Research Laboratory, Washington, DC; [email protected],
[email protected]. tained from the eastern rim [v (Ha)p640(cid:2)35 km s(cid:1)1;
FWHM
L69
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4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
A Physical Relationship Between Electron-Proton Temperature
5b. GRANT NUMBER
Equilibration and Mach Number in Fast Collisionless Shocks
5c. PROGRAM ELEMENT NUMBER
6. AUTHOR(S) 5d. PROJECT NUMBER
5e. TASK NUMBER
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Naval Research Laboratory,Washington,DC,20375 REPORT NUMBER
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13. SUPPLEMENTARY NOTES
14. ABSTRACT
The analysis of Balmer-dominated optical spectra from nonradiative (adiabatic) SNRs has shown that the
ratio of the electron to proton temperature at the blast wave is close to unity at v 400 km s1 but declines
sharply S down to the minimum value of m /m dictated by the jump conditions at shock speeds exceeding
2000 km s1. e p We propose a physical model for the heating of electrons and ions in
non?cosmic-ray?dominated, strong shocks (v 1 400 km s1) wherein the electrons are heated by lower
hybrid waves immediately ahead of the shock front. S These waves arise naturally from the cosmic ray
pressure gradient upstream from the shock. Our model predicts a nearly constant level of electron heating
over a wide range of shock speeds, producing a relationship (T /T ) &#8733 e p 0 v2 (∝M2) that is
fully consistent with the observations.
15. SUBJECT TERMS
16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF
ABSTRACT OF PAGES RESPONSIBLE PERSON
a. REPORT b. ABSTRACT c. THIS PAGE Same as 4
unclassified unclassified unclassified Report (SAR)
Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
L70 GHAVAMIAN, LAMING, & RAKOWSKI Vol. 654
Fig.2.—Electrontoprotontemperatureratioattheshockfrontasafunction
ofshockvelocityforfiveBalmer-dominatedSNRs.MagnetosonicMachnumbers
(M)appropriatefortypicalISMconditionsareindicatedalongthetopaxis.The
Fig. 1.—Example of the optical spectrum of a Balmer-dominated shock, S
datashownhereweremeasuredfromBalmer-dominatedshocksintheCygnus
showingthebroadandnarrowHalinescharacteristicofnonradiativeshocks
Loop,RCW86,Tycho’sSNR(Ghavamianetal.2001),SN1006(Ghavamian
in partially neutralgas. This spectrum, originally presentedby Sollermanet
etal.2002),andDEML71(Rakowskietal.2003;Rakowski2005).Thedashed
al.(2003),wasobtainedfromthesouthwesternrimoftheGalacticSNRRCW
86, with high enough spectral resolution (∼10 km s(cid:1)1) to resolve the broad errorbarsforRCW86markpreviouslyunpublishedresults.Below400kms(cid:1)1
(∼500kms(cid:1)1FWHM)andnarrow(∼30kms(cid:1)1FWHM)Halines.Thenight- (MS≈30),thedataareconsistentwith(Te/Tp)0p1.Thepredictionofthepro-
posed lower hybrid wave-heating mechanism in the cosmic ray precursor,
skyOHlines(indicatedbythecircledplussigns)havebeenleftintodem-
(T /T) ∝V(cid:1)2(∝M(cid:1)2),isshownforv 1400kms(cid:1)1.
onstratetheirrelativelynarrowerwidthscomparedtotheHalines.Thebroad e p0 S S S
HawidthandratioofthebroadtonarrowHafluxforthesetypesofshocks
wereusedtoproducetherelationshipshowninFig.2. perpendicular)shockischaracterizednotbytheshockspeedbut
rather by the magnetosonic Mach numberM ({v /v , where
S S MS
v {(c2(cid:2)v2)1/2 is the magnetosonic speed, c is the sound
3IB2/5IN(cid:2)p110.0km(cid:2)s0(cid:1).12;]IB/aINndpt1h.e06n(cid:2)or0th.1e]rnofrRimCW[v8F6W.HInM(tHheafi)fpth ssppMeeSeeddo(fptSh[(e5p/3rAe)(sPh/orc)k]1/g2)asa).ndThveAp[r{eshBo/c(4kptreim)1p/2e]rSaitsurteh,eioAnldfvee´nn-
SNR of our sample, DEM L71 (Ghavamian et al. 2003), we sity, and magnetic field strength arenot stronglyconstrainedin
usedthebroadcomponentHawidthonlytoconstraintherange theobservedBalmer-dominatedshocks.Inparticular,M ismost
ofshockspeedsandtheprotontemperatureTp,duetotheanom- sensitive to the choice of preshock magnetic field duSe to the
alouslylowIB/INvaluesinthisSNR(seebelow).Wethencom- dependenceof theAlfve´nspeedonB2.However,ifweassume
bined this information with Te measured from Chandra X-ray standard values for the warm neutral interstellar medium
spectra of the blast wave to obtain (Te/Tp)0 (Rakowski et al. (ISM)—preshock temperature of 10,000 K, density of 1 cm(cid:1)3,
2003; Rakowski 2005). magneticfieldstrengthof3mG,and50%preshockionization—
AlthoughtheequilibrationswereobtainedfromdifferentSNRs the magnetosonic speed is then approximately 13 km s(cid:1)1. The
inarangeofenvironmentsanddistances,theyshowacleartrend corresponding values of M are marked at the top of Figure 2.
ofdecreasing(T/T ) withv.TheplotinFigure2suggeststhat S
e p 0 S
atransitionalshockspeedexistsaround400kms(cid:1)1,belowwhich
3. LOWERHYBRIDWAVEHEATINGMODEL
collisionlessprocessespromptlyequilibratetheelectronsandpro-
tonsattheshockfront.Abovethatspeed,(T/T ) rapidlydeclines, The constant electron heating with shock velocity [giving
e p 0
eventuallyreachingvaluesconsistentwithmass-proportionalheat- (T/T ) ∝v(cid:1)2] suggests a process occurring in the preshock
ing.Thisdeclineappearstofollow(T/T ) ∝v(cid:1)2.Weshowthat meedipum0 rathSer than the shock front itself. Waves in the pre-
e p 0 S
thisbehaviorispredictedbyaphysicalmodelofelectronheating, shock medium can be excited by shock-reflected ions, which
where the electrons are heated in the cosmic ray precursor to a gyratearoundthefieldlinesbeforereturningtotheshockitself,
level that is nearly independent of shock speed beforeacquiring or by a cosmic ray precursor. Cargill & Papadopoulos (1988)
the mass-proportional increment attheshockfront.Theprotons, performed hybrid simulations of the first possibility, wherein
on the other hand, receive only mass-proportional heating, re- shock-reflectedionsgenerateLangmuirandion-acousticwaves
sultingintheinversesquareddependenceofequilibrationonshock ahead of the shock, which then heat electrons as they are
speed. Note that while our proposed model relies on the accel- damped. They predicted a temperature ratio (T/T ) ∼0.2
e p 0
erationofcosmicraysandtheexistenceofacosmicrayprecursor nearly independent of shock velocity, a result that clearlydis-
toheattheelectrons,itdoesnotrequiretheenergeticsoftheshock agreeswiththeobservationaldatainFigure2.Laming(2001a,
to be dominatedby thecosmicrays. 2001b)modeledtheexcitationoflowerhybridwavesbyshock-
In physical models, the strength of the (assumed quasi- reflected ions, following the suggestion of McClements et al.
No. 1, 2007 FAST COLLISIONLESS SHOCKS L71
(1997) that lower hybrid waves could stream away from the would be t∼d/v ∼1/Q , where the precursor depth d is ap-
S p
shockwithagroupvelocityequaltotheshockvelocity.Inthis proximatelythegyroradiusofacosmicrayproton.Substituting
way, waves stay in contact with the shock for arbitrarilylong this relation into equation (2) gives (T/T ) , independent of
e p 0
periodsoftimeandhencecangrowtolargeintensitiesdespite v,similartothebehaviorpredictedbyCargill&Papadopoulos
S
low intrinsic growth rates. The electron heating predicted by (1988),althoughtheseauthorsconsiderdifferentwaves.Inthis
such a model (Vink & Laming 2003) depends on the product case, the shallowness of the reflected ion precursor prevents
of the maximum electron energy and the fraction of electrons the collisional redistribution of energy among the electrons.
that are resonant with the waves that can be accelerated.This Ifaconstantkineticenergyisimpartedtoelectronsbylower
gives (T/T ) ∝v(cid:1)1exp((cid:1)M2), where M is the shock Mach hybrid waves, then for shock speedsof400kms(cid:1)1andabove
e p 0 S
number. (cf. Fig. 2) the observations require m v2/2∼4#10(cid:1)10 ergs
e e
Here we explore another approach. Cosmic rays have been (∼0.3 keV) immediately behind the shock front. In that case,
long known to generate waves upstream of shocks; the gen- weinferQ k(cid:1)7#1021cm2s(cid:1)2,andk∼4#1014/Bcm2s(cid:1)1.
e
eration of Alfve´n waves is an intrinsic part of cosmic ray ac- For typical magnetic fields B∼3 mG, k is several orders of
celeration models. Drury & Falle(1986)showedthattheneg- magnitude below that inferred for the undisturbed ISM but
ative gradient of cosmic pressure with distance ahead of the entirely consistent with values estimated for the solar windor
shock can generate sound waves via the “Drury instability.” interplanetary medium. We emphasize that this estimate is a
This is the mechanism that smooths out the hydrodynamical lower limit on k. This estimate neglects the electron-electron
jump in cosmic ray modified shocks. Mikhailovskii (1992) collisionalequilibrationnecessarytobringmoreelectronsinto
gives a corresponding expression for the excitation of mag- resonance with the turbulence. Furthermore, lower hybrid
netoacoustic waves. We argue that the high-frequency exten- waves can only be excited in the cosmic ray precursor if the
sion of such processes by the cosmic ray pressure gradient cosmicraypressuregradientissufficientlystrong.Therequired
wouldbetheexcitationoflowerhybridwaves.Thefrequencies gradientscalelengthsare1/L18c /3kforsoundwaves(Drury
S
of these waves would lie between the gyrofrequencies of the & Falle 1986) and 1/L1v /cr ∼v /k for magnetoacoustic
A g A
electrons and the protons, with electron heating occurring as waves (Mikhailovskii 1992), where r is the gyroradius of a
g
thewavesdampalongmagneticfieldlines.Withoutcalculating cosmic ray proton. An upper limit on k will come from the
the growth rate of the lower hybrid waves explicitly, we can requirement that the neutral H surviveagainstelectronimpact
estimate the magnitude of the electron heating. The parallel ionization in the precursor to reach the shock front. Numeri-
diffusion coefficient for electrons in lower hybrid wave tur- cally, k(cid:1)1024(v /1000 km s(cid:1)1)2/n cm2 s(cid:1)1. Further con-
S e
bulence is (derived from eqs. [10.83] and [10.93] of Melrose straints on the cosmic ray diffusion coefficient and pressure
1986; see also Begelman & Chiueh 1988) will be obtained from calculation of the lower hybrid wave
growthraterequiredtosustaintheelectronheating.Thisgrowth
1(qdE)2 q2 1(qdE)2k2 1 rate must be significantly largerthan thechargeexchangefre-
Dkkp4 m k2v2k v p4 m kk2q, (1) quency of the partially neutral gas. Under typical ISM con-
e ⊥ k k k e ⊥ ditions, (Q Q )1/2∼1 Hz and n Aj vS∼10(cid:1)8 Hz, so the con-
e p H0 cx
dition for electron heating is nearly always satisfied.
where qp(Q Q )1/2 is the lower hybrid wave frequency, the
e p The electron heating by lower hybrid waves is inherently
geometric mean of the electron and proton cyclotronfrequen-
anisotropic. If some of this anisotropy survives the electron-
cies (Q {eB/m c), and k and k are wavevectorsparallel
e,p e,p k ⊥ electron collision equilibration, a polarization signal may be
and perpendicular to the magnetic field. Taking dEp
B(Q /q)1/3(q/4k c) (Karney 1978), k2/k2 pm /m , and the present in the narrow component of Ha (see, e.g., Laming
p ⊥ k ⊥ e p 1990),whichmightprovidemoreinsightintotheelectronheat-
time spent by an electron inside the cosmic ray precursor t∼
ing process.
k/v2, where k is the cosmic ray diffusion coefficient, we find
S
1 1 m (qB)2(Q )2/3 q2 m 1 k 4. THEWIDTHOFTHEHaNARROWCOMPONENT
mv2p mD tp e p e
2 e 2 kk 128 me q k⊥2c2mpqv2S The same cosmic ray precursor should also generate lower
frequency waves, which for a quasi-perpendicular shock will
m (Q )5/3 q2
p e Q p k. (2) include magnetoacoustic waves. Below the ion-neutral charge
128 e q k2v2 exchange frequency, these waves are not effectively damped
⊥ S
(Drury et al. 1996) and may reveal themselves via broadening
Ifthelowerhybridwavegroupvelocity(cid:2)q/(cid:2)k pv ,thenthe ofthenarrowHacomponent(Smithetal.1994).Thoseauthors
⊥ S
phase velocity q/k p2v when k2/k2 pm /m (Laming suggested that waves in the cosmic ray precursor actuallyheat
⊥ S k ⊥ e p
2001a), and the electron heating depends on a few constants thepreshockgasto30,000–40,000Kandthattheobservedline
times the product Q k. Thus, for a Bohm-like cosmic ray dif- width is thermal in nature. An upper limit to the cosmic ray
e
fusion coefficient k∝1/B and independent of v , we have an diffusion coefficient of ∼1024/n cm2 s(cid:1)1 then results from the
S e
electronheatingprocessindependentofboththeshockvelocity constraint that sufficient neutral H survive passage throughthe
andtheupstreammagneticfield,and(T/T ) ∝v(cid:1)2asrequired. precursor to encounter the shock. However, no precise heating
e p 0 S
Thisestimateneglectsthefactthatonlyelectronswithvelocity mechanism was specified, and we now suggest that the large
greater than v ∼q/k may be heated by the waves. However, width of the narrow component Ha line in the observedSNRs
e k
forthelikelydepthofacosmicrayprecursor(estimatedbelow), is not of thermal origin but ratherdueto themotionofprotons
the electrons will have sufficient time to at least partiallycol- inthelowestfrequencywavesofthemagnetoacousticspectrum.
lisionally equilibrate among themselves before crossing the Thesewavesliebelowthechargeexchangefrequency,allowing
shock, allowing a much larger fraction of preshock electrons acoherentoscillationofpreshockneutralsandpreshockprotons.
tointeractwiththewaves.Thetimespentbyanelectroninside Inthiscase,dv(cid:1)v dB/BforAlfve´nwaves(v shouldbereplaced
A A
areflectedionprecursor,asopposedtoacosmicrayprecursor, byv formagnetoacousticwaves).TheAlfve´nspeedupstream
MS
L72 GHAVAMIAN, LAMING, & RAKOWSKI Vol. 654
from the shock is in the range 1–10 km s(cid:1)1, so an observed dominated shocks is limited by ion-neutral damping to ∼0.1–
broadening dv/v 11 also implies dB/B11. Such a magnetic 1 TeV (Drury et al. 1996). It is possible that the shocks with
A
field amplification has already been proposed in the context of stronger cosmic ray acceleration generate sufficient electron
cosmic ray acceleration in shocks (Lucek & Bell 2000; Bell& heatingintheirprecursorsthatnoneutralssurvivetocrossthe
Lucek2001).Asuccessfulmodelofprecursorionswillneedto shock.TheanomalouslylowI /I ratioobservedinDEML71
B N
reproduce the existing measurementsofnarrow-componentHa (Ghavamian et al. 2003) may indicate that this is an interme-
line widths (Smith et al. 1994; Hester et al.1994;Sollermanet diate case. Electron heating in a shock precursor willproduce
al. 2003), which indicate that they remain relatively constant added narrow-component Ha. However, since the broad com-
(30–50 km s(cid:1)1) over a wide range in shock speed (180–3000 ponent can only arise by charge exchange with shocked pro-
km s(cid:1)1). We defer this topic to future work. tons, the neutral H must penetrate either the shock or the re-
flected ion precursor. We emphasize that of the points plotted
in Figure 2 from Ha I /I measurements, the electron tem-
5. DISCUSSIONANDSUMMARY B N
peratures from RCW 86 (Vink et al. 2006), the Cygnus Loop
The plot of (T/T ) in Figure 2 is qualitatively similar to (Levenson et al. 2002), SN1006 (Laming etal.1996;Vinket
e p 0
results of a survey of electron heating at solar wind shocks al. 2003), and Tycho (Hwang et al. 2002; Warren et al. 2005)
(Schwartz et al. 1988). The main difference is that T ∼T is have been independently corroborated by measurements in
e i
obtained for Alfve´nic Mach numbers ≤5 in the solar wind, other wave bands.
whereasinFigure2thisoccursformagnetosonicMachnumbers Several lingering questions remain from Figure 2: What
around 20–30, for assumed preshock magnetic fields of 3 mG. mechanismcausesthepromptelectron-protonequilibrationbe-
A preshock magnetic field amplification in SNRs, of a similar low 400 km s(cid:1)1, and why does the decline in (T/T ) begin
e p 0
amountrequiredtoreproducethewidthoftheHanarrowcom- at that shock speed? Even more importantly, does the inverse
ponent (dB/B∼5–10), would shift the (T/T ) ratios in Fig- square relationship between equilibration and shock speed/
e p 0
ure 2 onto Mach numbers similar to those in the solar wind. Machnumberalsoholdforcollisionlessshocksinfullyionized
Thismaysupportourinterpretation.Collisionlessshocksplaya gas? These issues will be addressed in future work.
dominantroleinheatingtheintergalacticmedium(IGM)during
large-scale structure formation (Ryu et al. 2003; Yoshida et al.
2005; Kang et al. 2005). If the preshock magnetic field is am- P.G.wouldliketothankR.CenandK.Sembachforhelpful
plified as described above, then the relationship (T/T ) ∝ discussions and acknowledges support from NASA contract
e p 0
M(cid:1)2mayalsobeapplicabletomodelingtheionizationandtem- NAS8-03060.J.M.L.andC.E.R.acknowledgesupportfrom
S
perature structure of the IGM. NASA contract NNH06AD66I (LTSA Program) and basicre-
The maximum energy attainable by cosmic rays inBalmer- search funds of the Office of Naval Research.
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