The infrared jet in Centaurus A: multiwavelength constraints on emission mechanisms and particle acceleration

0.37 MB·English

by  M. J. Hardcastle

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed4February2008 (MNLATEXstylefilev2.2)
The infrared jet in Centaurus A: multiwavelength constraints on
emission mechanisms and particle acceleration
1 2 3
M.J. Hardcastle , R.P. Kraft and D.M. Worrall
1SchoolofPhysics,AstronomyandMathematics,UniversityofHertfordshire,CollegeLane,Hatfield,HertfordshireAL109AB
6 2Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,Cambridge,MA02138,USA
0 3DepartmentofPhysics,UniversityofBristol,TyndallAvenue,BristolBS81TL
0
2

n 4February2008
a
J

4 ABSTRACT
2 WereportonSpitzerandGeminiobservationsofthejetofCentaurusAintheinfrared,which
wecombinewithradio,ultravioletandX-raydata.Spitzerdetectsjetemissionfromabout2

2 arcminfromthenucleus,becomingparticularlybrightafterthejetflarepointat∼3.4arcmin.
v
WhereX-rayandinfraredemissionareseentogetherthebroad-banddatastronglysupporta
1
synchrotronoriginfortheX-rays.Thejetflarepointismarkedbyabroad,diffuseregionof
2
X-rayswhichmaybeassociatedwithashock:wediscusspossiblephysicalmechanismsfor
4
this.TheinfraredjetpersistsaftertheflarepointregionalthoughX-rayemissionisabsent;itis
1
0 plausiblethathereweareseeingtheeffectsofparticleaccelerationfollowedbydownstream
6 advection with synchrotron losses. Gemini data probe the inner regions of the jet, putting
0 limitsonthemid-infraredfluxofjetknots.
/
h Keywords: galaxies:active–galaxies:individual:CentaurusA–galaxies:jets
p
-
o
r
t
s 1 INTRODUCTION trastswiththesituationinotherwell-studiedlow-powerjets,such
a
asM87(e.g.Perlmanetal.2001)or3C66B(Hardcastleetal.2001)
:
v CentaurusAistheclosestradiogalaxy(weadoptD = 3.4Mpc, inwhichdataintheinfrared,opticalandultravioletsupportasyn-
Xi Israel 1998) aswell asthe closest activegalaxy and largeellipti- chrotronoriginandprovideconstraintsonparticleaccelerationand
cal. Its proximity makes it a vital laboratory for AGN studies of energy-productionprocesses.
r
a allkinds.Itsnucleusshowsbothheavilyobscuredandunobscured However,recentSpitzerobservationshavedetectedthejetof
X-ray components (Evans et al. 2004), making it one of the few CenAinthemid-infrared(Brookesetal.2006),whileGALEXde-
low-power radiogalaxiestoshow X-rayevidence for theobscur- tectsitintheultraviolet(Neffetal.2003).Inthisletterwecombine
ing torus of canonical unification models (Evans et al. 2006). Its thesedatawithnewandarchivalradioandX-rayobservationsand
X-ray jet, one of the first to be discovered (Schreier et al. 1979) discusstheirimplicationsforparticleaccelerationprocesses.Inad-
has been studied in great detail with Chandra (Kraft et al. 2000, dition,weuseobservationsofthenuclearregionswithGeminiat
2002;Hardcastleetal.2003,hereafterH03;Kataokaetal.2006). 10µmtoplaceconstraintsonthepropertiesofthenucleusandthe
Becauseofthehighspatialresolution(1arcsec=17pc)compared innerjet.
tothatavailableformoretypicalX-rayjetsinmorepowerfulFRI Exceptwhereotherwisestated,spectralindicesαaretheen-
sources,observationsofthejetprovideevidenceforbothlocalized ergyindices,definedinthesensethatflux∝ν−α.
anddiffuseparticleaccelerationprocesses.Finally,CenA’sSWra-
diolobe,inexpanding throughtheISMofthehostgalaxy,drives
whatiscurrentlytheonlyclearexampleofahighMach number,
2 OBSERVATIONS
attachedbowshocktobeobservedinX-raysaroundaradiogalaxy
(Kraftetal.2003). 2.1 Spitzerdata
CenA’smaindisadvantageasasubjectforbroad-bandstud- TheSpitzerdataweused,describedinmoredetailbyBrookeset
iesisthestrongdustlane,whichobscurestheinnerregionsofthe al. (2006), are taken from the public archive, and consist of two
source in the optical to ultraviolet. Partly as a result of this, and datasets,aset of IRACobservations taken on2004 Feb10 anda
partlybecausethejetisrelativelyweakcomparedtotheemission set of MIPS observations taken on 2004 Aug 06. Several sets of
fromstars,therehaveuntilrecentlybeennocleardetectionsofsyn- MIPSobservations areavailableinthearchive:theonewechose
chrotronemissionfromthejetatfrequenciesbetweenradioandX- to use (AOR 4940288) covers a wide area around the centre of
ray, although several claims of optical and infrared emission that CenA.ThedatausedwerethePost-BasicCalibratedData(PBCD)
mayberelatedtothejetormaterialaroundithavebeenmade(e.g. filesavailablefromthearchive.Theseincludeanautomated,flux-
Brodieetal.1983, Joyetal.1991,Leeuwetal.2002). Thiscon- calibratedmosaic(‘MAIC’file)ofthenumerous individual maps
2 M.J. Hardcastleetal.
thatgotomakeupanobservation.ThePBCDfilesarestatedinthe
instrumentdatahandbooks(http://ssc.spitzer.caltech.edu/irac/dh/;
http://ssc.spitzer.caltech.edu/mips/dh/)tobesuitableatthetimeof -42 56
writingforbasicscientificanalysisforallIRACchannelsandfor
the24-µmchannel(channel1)oftheMIPSdata.Thelowangular
resolution and calibration issues of the longer-wavelength MIPS
57
channelsmeantthatthesewerenotsuitableforouranalysisinany
case.IRACchannel1(3.6µm)wastoodominatedbystarlightfrom
the host galaxy and from foreground objects to be useful in our
aIRnaAlyCsicsh.aAncnceolsr,diantg4ly.5,,th5e.8daatnadw8e.0uµsema,raenfdrofmromtheMthIrPeSeraetm2a4inµimng; N (J2000) 58
O
Brookes et al. (2006) show a selection of images in these bands. TI
A
eWxeclcuadrirnigedpoouinttapsoerutrucreespfhrootmomseoturryc,eusainndgbaalcokcgarlobuancdkgrreoguionndsa,ntdo ECLIN 59
D
measurefluxdensitiesfromcomponentsofthejet.Ourphotometry
isconsistentwiththeindependentanalysisofBrookesetal.
-43 00
2.2 Geminidata
ThenuclearregionsofCenAwereobservedwiththeT-ReCSin-
01
strumentonGeminiSouthatN-band(10µm)on2004Mar06and
2004Mar11-12.Weobtainedtheseobservationsinanattemptto
detectthebrightradioandX-raycomponentsoftheinnerjet,and 13 25 50 45 40 35 30
sotheT-ReCSfieldofview(28.8×21.6arcsec)wasalignedalong RIGHT ASCENSION (J2000)
the jet, with the active nucleus in one corner. The standard nod
Figure1.TheCenAjetat24µm,usingalogarithmictransferfunction,
andchopmodewasusedforbackgroundsubtraction,andthebase-
withoverlaidcontoursfroma6-arcsecresolution4.9-GHzradiomap.Black
linecalibrationwasusedforphotometryandpoint-spreadfunction isthebackgroundlevelof32.8MJysr−1 andthepeakis724MJysr−1.
(PSF)determination,usingobservationsofstandardstars.Intotal Thelowestcontourisat0.01Jybeam−1;contoursincreasebyafactor2.
theon-sourceexposuretimewasaround2.1h. Theflarepoint(peaksurfacebrightness33.4MJysr−1)ismarkedwithan
arrowandtheregionsusedforfluxmeasurementareshownasredboxes.
2.3 VLAdata
2.5 Chandradata
The 8.4-GHz VLA data that we have described in earlier papers
(Kraft et al. 2002, H03) were not ideal for comparison with the Forthehigh-energyconstraintsonthespectrumofthejetweused
large-scale Spitzer jet because of the VLA’s small primary beam twoChandradatasetstakenusingtheACIS-Sinstrument:theob-
atthiswavelength.Wethereforere-reducedthedatadescribedby servationtakenon2002 Sep03(obsid2978) whichwastakenas
Clarkeetal.(1992)at1.5and4.9GHz.Thesearewellmatchedto partoftheHRCguaranteedtimeprogramme,andtheobservation
theangular scalesandresolutionof theSpitzerdata.Forsmaller- takenon2003Sep14(obsid3965)whichwastakenbyusinguest
scale mapping we used our existing 8.4-GHz data. VLA data observer time. These two observations are well matched in posi-
fromdifferentconfigurationswerecalibratedandcombinedwithin tion on the instrument and roll angle. The data were reprocessed
AIPS,andaprimarybeamcorrectionwasappliedtoallimages. and filtered using CIAO 3.2.2 and CALDB 3.1 (applying new bad
pixel files, removing afterglow detection, and removing the 0.5-
arcsecpixelrandomization)andwerebothalignedtotheradiocore
2.4 GALEXdata position. After filtering they had livetimes of 44592 and 49518
s respectively, giving a total effective on-source time of 94.1 ks.
The GALEX data we use were taken from the archive
Spectrawereextractedfromregionsmatchedtothoseusedatother
(http://galex.stsci.edu/GR1/) and were derived from observa-
wavelengths,withlocalbackgroundsubtraction,usingtheacisspec
tions made on 2003 Jun 07 as part of the Nearby Galaxies
toolwithinCIAOandappropriateresponsematricesweregenerated
Survey, as reported by Neff et al. (2003); Brookes et al. (2006)
withmkacisrmf.SpectralfittingwasdonewithinXSPEC11.3.
show an image. Two broad bandpasses are available, with
mean wavelengths of 153 and 231 nm. We use the background-
subtracted intensity map, with units of (corrected) counts s−1,
for our measurements. Photometry was carried out in the 3 RESULTS
same way as for the Spitzer data, using ground-based calibration
3.1 Thelarge-scalejet
(http://galexgi.gsfc.nasa.gov/Documents/ERO data description 2.htm),
correcting for a Galactic E(B −V) of 0.114 mag using the ex- Inthe24-µmdataextendedemissionfromthejetisclearlydetected
tinction curves of Cardelli, Clayton & Mathis (1989), which fromabout2arcmin(Fig.1)andextendsatleastuntiltheendofthe
give correction factors of 0.94 mag at both mean wavelengths. clearlydefinedradiojet.(Twostronginfraredpointsourcesinthe
Since the photometric zero point is not yet well defined and the lobetotheNWofthejetareprobablyunrelatedtoit.)Thebrightest
extinctioncorrectionvariessignificantlyoverthebandpasses,there regionoftheinfraredjet,andthepartmostclearlydetectedagainst
are potentially large systematic errors in the conversion between thehigherbackgroundintheshorter-wavelengthIRACimages,oc-
GALEXcountrateandfluxdensity. cursataregionwheretheradiojetbecomesabruptlybrighter,and
Infraredjetin Cen A 3
Table1.Fluxdensitiesfromregionsofthelarge-scale jet:seeFig.1forregions.Errorsarenominal3percentcalibration errors(radio),errorsbasedon
calibrationandbackgrounduncertainties(infraredandultraviolet)orstatisticalerrors(X-ray).
Region Fluxdensity
1.4GHz 4.9GHz 8.4GHz 24µm 8.0µm 5.4µm 4.5µm 231nm 153nm 1keV
(Jy) (Jy) (Jy) (mJy) (mJy) (mJy) (mJy) (µJy) (µJy) (nJy)
Inner 3.25±0.10 1.67±0.05 1.0±0.03 9.0±1.8 – 2.4±0.7 – 80±12 – 27.2±0.3
Middle 9.73±0.30 5.21±0.16 3.79±0.11 15.9±3.1 6.0±1.8 4.9±1.4 3.8±1.1 180±25 70±18 16±0.5
Outer 20.2±0.6 10.1±0.3 – 24.3±4.9 4.2±1.3 5.2±1.6 4.0±1.2 – – <3
startstobendnorthwards,at∼ 3.4arcminfromthenucleus.Here directaperturephotometrywithbackgroundsubtraction,excluding
werefertothisasthe‘flarepoint’(nottobeconfusedwiththein- pointsources.Tofirstorder,thetworegionswithdetectionsinall
nerjetflarepointataround14arcsecfromthenucleus:seeH03). wavebandsareroughlyconsistentwiththetypeofmodelwehave
UltravioletemissionisdetectedintheGALEXdatabothfromthis fittedelsewhere (e.g. Hardcastle et al. 2001) inwhich the energy
flarepointandfromregionsofthejetclosertothenucleus. spectrum of all theelectrons intheregion isabroken power law
withanon-standardbreakconnectingtheradioandX-rayandre-
Fig.2showsthattheflarepointismarkedbyaregionofrela-
producing the X-ray photon indices; in this case the break must
tivelystrongX-rayemissionwhichbeginsabout0.2arcmincloser
occuratenergieslowerthanthosecorrespondingtotheinfraredre-
to the nucleus than the flare point and continues for about 1 ar-
gion(Fig. 4).However, indetail, thebest-fittinginfrared spectral
cmin.ThereafteralmostnoX-rayemissionisseenfromthejet,but
indices(0.89±0.25and0.84±0.18fortheinnerandmiddlere-
theinfraredemissioncontinues.TheX-rayregionaroundtheflare
gionsrespectively)aresomewhatflatterthanwouldbeexpectedin
point(whichwedenoteregionG,followingthenotationofFeigel-
this model, and the ultraviolet data points lie significantly above
sonetal.1981)isresolvedintoseveralcompactknotsandextended
it,closertoalinearextrapolationfromtheinfrared.Inthemiddle
emission.Thecompactknotsareallclosertothenucleusthanthe
region, in particular, where the GALEX fluxes are probably most
flarepoint,butthebrightestdiffuseemissioniscoincidentwiththe
reliable,thedataappeartorequirea‘bump’abovethebest-fitting
flarepoint,althoughthepeakradioandinfraredsurfacebrightness
lineinwhichthespectrumsteepensandthenflattens,whichwould
(atabout3.5arcmin)isoffsetfromthepeakX-raysurfacebright-
implyamorecomplexelectronpopulationthanourmodelallows
ness.ThisisillustratedbyFig.3,whichshowsaprofilealongthe
for.Theouter region, inwhichnosignificantultravioletor X-ray
jetinradio, infrared andX-ray. Thefact that theinfraredsurface
emission is detected, can also be fitted with a simple model, but
brightnessbeginstorisearoundknotG1mightimplythatthatthe
in this case either the change of the electron spectral index, ∆p,
threeknotsareinvolvedinhigh-energyparticleacceleration,rather
mustbegreaterortheremustbeacutoffintheelectronspectrum
thanbeingunrelatedtothejet.Therearenodetectedradiocoun-
betweentheinfra-redandX-rayregions;theX-rayupperlimitpre-
terpartstotheseknots,butwedonothavesensitivehigh-resolution
cludesfittingthisregionwithamodelidenticaltothatusedinthe
radiodataatthisdistancefromthenucleus.TheX-rayspectraof
othertworegions(Fig.4).Thebest-fittingpowerlawspectralindex
theknotsareallwellfittedwithpowerlawswithGalacticabsorp-
tion(N = 7×1020 cm−2)andhavesteepspectra(withphoton totheinfrareddataaloneis1.13±0.19fortheouterregion,which
H
indicesof1.84±0.16,2.01±0.16,and2.02±0.23respectively). wouldbeconsistentwithalarger∆p,thoughtheerrorsarelarge.
However, theirfluxdensitiesarelow(∼ 2nJyeach) andsothey Insuchamodeltheremaybelow-levelX-rayemissionfromthis
regionthatwouldbedetectableindeeperobservations.
wouldnotcontributesignificantlytothefluxintheinfraredifthese
Noobviousjet-relatedinfraredemissionisseenonthecoun-
spectrawereextrapolatedbacktothosefrequencies.
terjetsideofthesource.ThecounterjetradioandX-rayknotsdis-
Intheabsenceofcounterpartstotheknotsatotherwavebands,
cussedbyH03areatsmalldistancesfromthenucleus,wherethe
weexcludetheminwhatfollows,andaskthequestion:istheover- infraredbackgroundfromthehostgalaxyishigh.
allspectrumoftheextendedemissionconsistentwithasynchrotron
model?Toinvestigatethis,weextractedfluxdensitiesfromthree
matched regions of the jet around the flare point at all available
frequencies,excludingpointsourcesand,inthecaseoftheX-ray,
the compact knots labelled on Fig.2, and measuring background
3.2 Thesmall-scalejet
fromadjacentoff-sourcebackgroundregions.TheX-rayemission
intheseregionsisdominatedbytheextendedemissionandsothe The Gemini observations do not detect any component of the jet
exclusionoftheknotsmakeslittledifferencetoourresults.Fig.1 within 24 arcsec of the nucleus. Using the baseline photometric
showstheextractionregions,whichwecalltheinner,middleand calibrationweestimatethatanupperlimitonanycompactjet-like
outer regions, and the results are tabulated in Table 1. The high componentis1mJyat10µm.Theobservationsarelesssensitive
background and low signal intheinner jet means that wecannot thanwouldhavebeenpredicted,presumablybecauseofthebright
measure reliablefluxes for theinner jetat 8 and 4.5µm. For the emissionfromthedustlane.Theupperlimitallowsustosetalower
X-ray data, we fitted power laws with Galactic absorption to the limit on the spectral index between radio and infrared, given the
tworegionsinwhichsignificantcountsweredetectedtodetermine knotradiofluxesmeasuredpreviously(H03),ofα > 0.5,consis-
a 1-keV flux density (finding photon indices of 2.29±0.05 and tentwiththeαRI measuredfor thelarge-scalejet,∼ 0.75. Since
2.44 ±0.07 for the inner and middle regions respectively), and the knots in the inner region generally have steep X-ray spectral
the upper limit to the flux density in the outer region was deter- indices, it seems likely that their spectrum, like that of the outer
mined assuming a spectrum similar to that of the middle jet re- jet,turnsoverbeforetheinfraredregion.Sub-mmobservationsare
gion. Atotherwavelengths thefluxdensitiesweredetermined by requiredtotestthis.
4 M.J. Hardcastleetal.
Figure2.UnsmoothedChandragreyscaleimageofthejetofCenAinthe0.5-5.0keVbands.OverlaidarethecontoursfromFig.1.Pixelsarethestandard
Chandrapixels,0.492arcseconaside.Greenboxesshow(fromrighttoleft)theboundariesofinner,middleandouterextractionregions.
Figure4.Thespectraoftheinner,middleandouterregionsofthejet.Tri-
Figure3.Theprofileofthepartofthejetneartheflarepoint.Thex-axis
anglesdenotetheinnerregion,filledcirclesthemiddleregion,andstarsthe
showslinear distance fromthenucleus alongthejet. Redindicates radio
emission (from a 4.9-GHz map with 6.2×2.0 arcsec resolution, beam outerregion.Thesolidlineisthebest-fittingreferencesynchrotronmodel
tothemiddleregion,asdescribedinthetext.Whereerrorbarsarenotvisi-
elongatedN-S),greenindicates24-µminfrared,andblueindicates0.5-5.0
bletheyaresmallerthansymbols.TheX-rayupperlimitismarkedwithan
keVX-rays.ThepositionsoftheX-rayknotsF2,G1,G2,G3andoftheflare
arrow.
pointaremarkedwithvertical lines.Theextraction regionfortheprofile
wasarectangle42arcsecinthetransversedirection:eachpointrepresentsa
0.9-arcsecslice.InfraredandX-raydatawerebackground-subtractedusing
FWHM of 0.27 arcsec. Some of the apparent extension with re-
adjacentidenticalregions.RepresentativePoissonerrorsareplottedonthe
specttothePSFmaybetheresultofthelongerobservation,with
X-raypoints.
morechop/nodcycles,usedforthenucleus,butwecanconserva-
tivelysaythatthesizeofthenucleusat10µmis <∼0.27arcsec,or
3.3 Thenucleus <∼4.5pc,consistentwiththemeasurement of0.17±0.02arcsec
byKarovskaetal.ThetorusinCenAmustthusbecompact.
Cen A’s nucleus has been observed at wavelengths around N-
band by several other groups (Krabbe, Bo¨ker & Maiolino 2001;
Karovskaetal.2003; Siebenmorgen, Kru¨gel&Spoon2004). We
4 DISCUSSION:THEORIGINOFTHEFLAREPOINT
estimatethebackground-subtracted 10-µmfluxdensityofthenu-
cleusinourGeminiobservationsas1.1Jy,witha10percentpho- TheSpitzerobservationsconfirmthatthebroad-bandspectrumof
tometriccalibrationerror.Thisissignificantlyhigherthanseenin the large-scale Cen A jet can be described with a synchrotron
someearlierobservations,thoughconsistentwiththe1.5±0.4Jy model,asinotherFRIsources,althoughthedetailedspectralshape
reportedbyKarovskaetal.fromdatatakenin2002May,andmay almost certainly requires a multi-component model for the syn-
indicate variability on timescales of years. Variability has earlier chrotron emission. However, the infrared detection points up the
beenclaimedatshorterinfraredwavelengths(Turneretal.1992). importance of the region wehavecalled theflarepoint. Boththe
Comparing825-sindividualobservationsfrom2004Mar06with radio and infrared brighten by a factor ∼ 3 here (Fig. 3) while
the short, 43-s observation of the standard star HD 110458, we thereislittleor no X-rayemission after the extended component
find that the nucleus appears slightly resolved, with a Gaussian ofregionG.TheshortsynchrotronlifetimeofX-ray-emittingelec-
Infraredjetin Cen A 5
trons means that X-ray emission must always be associated with Qualitatively, the bulk deceleration at the flare point in ei-
high-energyparticleacceleration:butcouldthelackofX-raysafter ther of these scenarios is also consistent with the observed onset
regionGimplythatthisregion,whichisroughlycoincident with of bending of the jet there. However, if the jet really has a high
theentryofthejetintothelobe,representsthe‘lastgasp’ofpar- bulkspeed, as required toavoidin situparticleacceleration after
ticle acceleration in Cen A’s jet? In a field strength of 3 nT, the theflarepoint,itsdensitymustbelow.Forexample,ifweassume
equipartitionvalueforthispartofthejet,theelectronenergyloss thatthejetisbentbytherampressureofhotexternalmaterialmov-
timescale(E/(dE/dt))is2×104 yearsforelectronsemittingat ingatthesoundspeed,then,applyingEuler’sequation(e.g.Eilek
4.5µm,and4×104yearsforelectronsemittingat24µm.Thepro- etal.1984)andusingtheparametersofKraftetal.(2003),thejet
jecteddistancefromtheendoftheX-rayemissionattheflarepoint densitymustbe10−4timestheexternaldensityifthespeedis0.2c.
totheendofthejet,whereonlyradioand24-µmemissionisde- Thisisstillafactor5abovetheminimumpossibleeffectivejetden-
tected,isroughly2.5kpc,implyingalighttraveltimeof∼8×103 sity(fromtheminimum-energycondition,assumingthatthejetisa
years.Thus,forparticleaccelerationtobeabsentinthisregion,we purelepton/magneticfieldplasma),butifanyentrainmentofbary-
requirejetspeedsof(0.2/sinθ)c,whereθistheangletotheline onicmaterialtakesplaceintheinnerjet,orifthereisasignificant
ofsight.Asspeedsintheinnerjetare >∼0.5c,andtheangletothe departure fromequipartion, thejet willbe heavier, inwhich case
lineofsightmayberelativelylarge(seediscussioninH03)thisis relativisticbulkspeedsatthebendwouldbeunrealistic.Inthatsit-
notimpossible,sothatitcouldindeedbethecasethatsignificant uation,thepost-shockjetspeedwouldhavetobeslower,andsome
high-energyparticleaccelerationceasesattheflarepoint. continuinginsituparticleaccelerationwouldberequiredtoexplain
This motivates us to ask a further question: what is the na- theextendedinfraredjet.
tureoftheextendedX-rayemissionattheflarepoint?Mostofthe
X-rays(Fig.3)come fromaregion only10arcsec, or 170 pc,in
size. This is still larger than the expected travel distance for 1- ACKNOWLEDGEMENTS
keV-emitting electrons (at most 100 pc) and in fact extended X-
WearegratefultoCharlesLawrenceandMairiBrookesfordiscus-
ray emission isseen on scales up to around 30 arcsec, making it
sionoftheirresultsonCenApriortopublication.MJHthanksthe
difficulttosustainamodelinwhichtheparticleaccelerationhere
RoyalSocietyforaresearchfellowship.
takes place at a single point if the magnetic fieldstrength has its
equipartition value, although only modest decreases in the mag-
netic field strength, by a factor of a few, would be necessary to
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