Magnetic Fields in Starburst Galaxies and The Origin of the FIR-Radio Correlation

0.26 MB·English

by  Todd A. Thompson

SUBMITTEDTOAPJ:JANUARY26,2006
PreprinttypesetusingLATEXstyleemulateapjv.6/22/04
MAGNETICFIELDSINSTARBURSTGALAXIESANDTHEORIGINOFTHEFIR-RADIOCORRELATION
TODDA.THOMPSON,1,2 ELIOTQUATAERT,3ELIWAXMAN,4NORMANMURRAY,5,6 &CRYSTALL.MARTIN7
SUBMITTEDTOAPJ:JANUARY26,2006
ABSTRACT
We estimate minimum energy magnetic fields (B ) for a sample of galaxies with measured gas surface
min
densities. The sample spans more than four orders of magnitude in surface density from normal spirals to
luminousstarbursts.Weshowthattheratiooftheminimumenergymagneticpressuretothetotalpressureinthe
6
ISMdecreasessubstantiallywithincreasingsurfacedensity. Fortheultra-luminousinfraredgalaxyArp220,
0 thisratiois∼10- 4. Therefore,iftheminimumenergyestimateisapplicable,magneticfieldsinstarburstsare
0
dynamicallyweakcomparedtogravity,incontrasttoourGalaxyandothernormalstar-formingspiralgalaxies.
2
We argue, however, that rapid cooling of relativistic electrons in starbursts invalidates the minimum energy

n estimate. We criticallyassess anumberofindependentconstraintsonthemagneticfieldstrengthinstarburst
a galaxies. In particular, we argue that the existence of the FIR-radio correlation implies that the synchrotron
J
coolingtimescaleforcosmicrayelectronsismuchshorterthantheirescapetimefromthegalacticdisk;thisin
6 turnimpliesthatthetruemagneticfieldinstarburstsissignificantlylargerthanB . Thestrongestargument
min
2
against such large fields is that one might expect starbursts to have steep radio spectra indicative of strong

synchrotron cooling, which is not observed. We show, however, that ionization and bremsstrahlung losses
1
canflattenthenonthermalspectraofstarburstgalaxieseveninthepresenceofrapidcooling,providingmuch
v
betteragreementwithobservedspectra. Wefurtherdemonstratethationizationandbremsstrahlunglossesare
6
likely to be importantin shapingthe radiospectraof moststarburstsat GHz frequencies,therebypreserving
2
6 thelinearityoftheFIR-radiocorrelation. Wethusconcludethatmagneticfieldsinstarburstsaresignificantly
1 largerthanBmin. Wehighlightseveralobservationsthatcantestthisconclusion.
0
Subject headings: galaxies:general — galaxies:magnetic fields — galaxies:starburst — infrared:galaxies —
6
radiocontinuum:galaxies
0
/
h
p 1. INTRODUCTION implied magnetic field strength would be ∼0.03G on few-
- hundredparsecscales(Fig.1, Table 2). Thisissubstantially
o
The magneticenergydensity of the Galaxyis observedto largerthanthevalueof∼1mGtypicallyinferredinULIRGs
r
t beinroughequipartitionwiththecosmicrayenergydensity (e.g.,Condonetal. 1991)usingthe observedradioemission
s and the turbulent pressure. The sum of these yields a total andtheclassic“minimumenergy”argument(Burbidge1956;
a
: midplanepressureconsistentwiththatrequiredbyhydrostatic Longair1994).
v equilibrium,giventhemassdistributioninthesolarneighbor-
i hood (Boulares & Cox 1990). This equipartition magnetic Throughoutthispaperweusetheterm“minimumenergy”
X
to refer to the magnetic field strength inferred using the ob-
field is roughly 6 µG at the solar circle (e.g., Beck 2001).
r served radio flux and assuming comparable cosmic ray and
a Similar magnetic field strengths are found in most normal
magneticenergydensities(B ;eq.[1]).Wereservetheterm
star-forming spiral galaxies (Fitt & Alexander 1993; Niklas min
“equipartition”for magnetic field strengths that are dynami-
1995).
callyimportantwithrespecttogravity(B ;eq.[3]).
eq
Starbursts and ultra-luminous infrared galaxies (ULIRGs)
We find that the large discrepancy between the minimum
have much higher surface densities and turbulent velocities
energyandequipartitionmagneticfieldestimatesinArp220
than local spirals. As an example, Arp 220 has a gas sur-
is generic to starburst galaxies; this discrepancy motivates
face density ∼ 104 times higher than the Galaxy (Downes
theanalysisofmagneticfieldstrengthsinluminousstarbursts
& Solomon 1998). If Arp 220 has a magnetic energy den-
presentedin this paper. In §2we presenta sample of galax-
sity in rough equipartition with its hydrostatic pressure, the
ies with measuredgassurface densitiesand radiofluxes, for
whichbothminimumenergyandequipartitionfieldstrengths
1 DepartmentofAstrophysicalSciences,PeytonHall-IvyLane,Princeton
University,Princeton,NJ08544;[email protected] canbeestimated. In§3wesummarizeargumentsinfavorof
2LymanSpitzerJr.Fellow B∼Bmin and againstsignificantly largerfields. Rebuttals to
3 Astronomy Department & Theoretical Astrophysics Center, 601 a subset of these arguments are given in §4, where we also
Campbell Hall, The University of California, Berkeley, CA 94720; provide independentargumentswhy magnetic fields in star-
[email protected]
burstsarelikelytobe≫B . Theseargumentsdrawheavily
4PhysicsFaculty,WeizmannInstituteofScience,Rehovot76100,Israel; min
[email protected] ontheFIR-radiocorrelationforstar-forminggalaxiesandthe
5CanadaResearchChairinAstrophysics observedradio spectra of starbursts. Finally, in §5 we sum-
6 Canadian Institute for Theoretical Astrophysics, 60 St. George marizeourresultsanddiscusstheirimplications.
Street, University of Toronto, Toronto, ON M5S 3H8, Canada; mur-
[email protected]
7DepartmentofPhysics,TheUniversityofCalifornia,SantaBarbara,CA
93106;[email protected]
2
FIG. 1.—Minimumenergymagneticfield(Bmin;eq.[1])versusmeasuredgassurfacedensity(Σg). Normalstar-forminggalaxies(opensquares;Table1),
starburstgalaxies(filledsquares;Table2),andtheGalaxy(Bmin=6µGatthesolarcircle,e.g.,Beck2001;Σg≃2.5×10- 3gcm- 2,Boulares&Cox1990)are
shown. ToindicatetheuncertaintiesathighΣg,theopentrianglesshowBminforIC883(Arp193),Mrk273,andtheindividualnucleiofArp220,asinferred
usingΣgandtheradialsizefromDownes&Solomon(1998)(their“extreme”starbursts;seetheirTable12)andwiththeradiofluxfromCondonetal.(1991)at
8.44GHz(seeTable3). Thedottedlineistheequipartitionmagneticfield(eq.[3])andthedashedlineisthescalingBmin∝Σ2g/5(eq.[11]). Thelatterscaling
isexpectedifthecosmicrayelectroncoolingtimescaleismuchshorterthantheescapetimescalefromthegalacticdisk,inwhichcasethetruemagneticfieldis
alsosignificantlylargerthantheminimumenergyestimate(see§4;eq.[11]).ThesolidlinesshowthescalingsB∝Σa(witha=0.9,0.8,and0.7)usedin§4.3.
g
2. DATA&RESULTS facedensityΣg,hydrostaticequilibriumwiththeself-gravity
ofthediskimpliesamidplanepressure
Longair(1994)derivestheminimumenergymagneticfield
strengthas(Vol.II,eq.[19.30]) P≈πGΣ2g. (2)
B ≈7×10- 4δ2/7L2/7 V- 2/7ν1/7 G (1) WemeasureequipartitionviatheparameterηdefinedbyUB=
min 2 ν,23 6 GHz ηP. Theequipartition(η=1)fieldstrengthisthus
where L = L /1023 W Hz- 1, ν = ν/109 Hz, V =
ν,23 ν GHz 6 B ≈(8π2G)1/2Σ ≈2Σ mG, (3)
V/(100)3pc3isthevolumeoftheemittingregion,δ =δ/102 eq g g
2
is the ratio ofthe energyin cosmic rayionsto the energyin where Σ is in cgs units. Theoretically, there are significant
g
cosmicrayelectrons,andthederivationassumesthatthecos- uncertainties about the origin of magnetic fields in galactic
micrayelectronenergyspectrumisn(γ)∝γ- p with p=2.5. disks,andsoitisdifficulttoestimatetheexpectedvalueofη.
We have scaled the results above for parametersappropriate Field strengths as large as equipartitionare, however, possi-
to compactstarbursts. For parameterstypicalof the Galaxy, bleifthemagneticenergydensityequilibrateswiththeturbu-
L ∼3×10- 2andV ∼7×104sothatB ≈10- 5G. lentenergydensity of the ISM. The scale-heighth of a self-
ν,23 6 min
gravitating galactic disk is given by h∼δv2/2πGΣ , where
The minimum energy magnetic field strength is often re- g
δvistheturbulentvelocity. Asaresult,theenergydensityin
ferred to as “equipartition” because the correspondingmag-
turbulentmotionsisρδv2∼πGΣ2,whereρisthemeanden-
neticenergydensity(U )isapproximatelyequaltothetotal g
Bmin sityoftheISM.Thus,ifthefieldisamplifiedbytheturbulent
cosmic ray (electron + ion) energy density if B actually
min motionssothatB2/8π∼ρδv2,equipartitionfollows.
obtains. In this paper, however, we use the term “equiparti-
tion” to denote magnetic energydensities comparableto the The gravitational potential of normal star-forming galax-
total hydrostatic pressure of the ISM. For a gas disk of sur- ies includes a significant contribution from stars (and per-
3
haps dark matter) on the scale of their effective radii. Ne- is physically motivated by setting the magnetic energy den-
glectingforsimplicitythedifferentscale-heightsofstarsand sityequaltothepressureintheISMproducedbystarforma-
gas,equation(2)shouldthenbereplacedbyP≈πGΣgΣtot≈ tion (P⋆). Because P⋆ ∝Σ˙⋆ (where Σ˙⋆ is the star formation
πGΣ2f- 1, where f is the gas fraction and Σ is the total rateperunitarea;see,e.g.,Chevalier&Clegg1985,Thomp-
g g g tot
surface density. Typically, f is in the range ∼0.1- 0.2 for son,Quataert,&Murray2005)andbecausetheSchmidtlaw
g
normalgalaxiesand fg ∼0.5 for starbursts. Thus, the mag- forstarformationisΣ˙⋆∝Σ1g.4 (Kennicutt1998),thescaling
neticfieldcouldinprinciplebelargerthanequation(3)bya B∝Σ0.7follows.
factorof f- 1/2 andstillbesub-dominantwithrespecttograv- g
g Our estimates of B in Figure 1 for normalstar-forming
ity. Inthispaper,thesedistinctionsarenotimportantbecause min
galaxiesareingoodagreementwiththoseestimatedbyother
we focuson gas-richstarburstsand use equation(3) primar-
authors (e.g., Fitt & Alexander 1993; Niklas 1995; Beck
ilyasausefulpointofcomparisonforfieldstrengthsinferred
2000). Whentheseminimumenergyfieldscanbecompared
usingtheminimumenergyargument(eq.[1]).
with other magnetic field estimates (e.g., from Faraday ro-
Figure 1 shows the minimum energy magnetic field Bmin tation) they are generally found to be consistent (e.g., Beck
inferredusingequation(1)asafunctionofthesurfacedensity 2000;Valleé1995).Inaddition,thecrucialassumptionofthe
Σg ofthegalacticdiskforasampleofgalaxiesrangingfrom minimumenergyargument,namelythattheenergydensityin
local spirals to ULIRGs. The sample is based on galaxies the magnetic field is roughly equal to the energy density in
with surface densitiesand sizes used byKennicutt(1998)to cosmic rays, is confirmed locally in the Galaxy using γ-ray
studytheglobalpropertiesofstarformationacrossarangeof observations(Strongetal.2000).
galaxies(the“SchmidtLaws”). Thepropertiesofthenormal
In keeping with previous studies, our results in Figure 1
star-formingandstarburstgalaxiesarelisted in Tables1and
showthatinnormalstar-forminggalaxies,B ∼B . How-
2,respectively. SystemsdescribedasstarburstsbyKennicutt min eq
ever, the striking result from Figure 1 is that the starburst
are denoted here by solid squares, whereas his normal star-
galaxies systematically have B < B . The ratio η =
forminggalaxiesarelabeledbyopensquares. min eq min
(B2 /8π)/P is a strongly decreasing function of Σ ; for a
In computing the radio emitting volume (eq. [1]), we as- typmiicnalstarburstsuchasM82,η ∼10- 2,whileforAgrp220,
sume that the scale height of the synchrotron emission is η ∼10- 4.9 min
h =500pcforthenormalgalaxies(Beuermannetal.1985; min
rad
Dumke & Krause 1998) and h =100pc for the starbursts If the minimumenergymagnetic field strengthestimate is
rad
(filled squares) (e.g., Klein et al. 1988; Seaquist & Odegard applicable, the direct implication of Figure 1 is that in star-
1991;Condonetal.1991).8 WeuseradialsizesfromKenni- bursts, magnetic fields are dynamically weak compared to
cutt(1998);fornormalgalaxies,thesearetakenfromtheRC2 gravity in the phase of the ISM in which the radio emitting
catalog (de Vaucouleurset al. 1976). Detailed studies show electronsreside. Because of the potentialimportanceof this
that for spiral galaxies the ratio of the optical to radio disk conclusion, it is worth critically examining the applicability
scale lengthis ∼0.9- 1.3(Hummel1980; Fitt & Alexander of the minimum energy estimate in the context of luminous
1993). For both the normal galaxies (open squares) and the starbursts(seeBeck&Krause2005foramoregeneraldiscus-
starbursts (filled squares), we have computed B using the sion).Theminimumenergyargumentrestsontheassumption
min
radioluminosityat1.4GHz. thatthemagneticenergydensityiscomparabletothecosmic
rayenergydensity. Giventhatthemagneticfieldandcosmic
Theopentrianglesillustratetheuncertaintiesintheresults
rayscouldhave rather differentenergygenerationand dissi-
of Figure 1 at high Σ . These points denote the “extreme”
g pation mechanisms(“sources and sinks”), the generic appli-
starburstsofDownes&Solomon(1998)(seetheirTable12):
cabilityofthisassumptionisunclear.
IC883(Arp193),Mrk273,andthetwoindividualnucleiof
Arp220. Forthesesystems,weusethegassurfacedensities In the context of local spirals, one argument in favor of
andradialsizesfromDownes&Solomon(insteadofKenni- the minimum energy estimate is precisely that it implies a
cutt1998), andwe combinethis data with radio fluxesfrom total cosmic ray + magnetic energy density in approximate
the 8.44GHz observations of Condon et al. (1991), which equipartitionwith gravity. If B6=Bmin, the net cosmic ray +
have sufficient resolution to probe the same spatial scales magneticenergydensitywouldexceedthegravitationalbind-
as the observations of Downes & Solomon (in contrast to ing energy of the gas, and the field and relativistic particles
the 1.4GHz observationsused for the rest of the systems in wouldpresumablyescapefromthegalacticdisk(e.g.,Parker
Fig.1). ThepropertiesofthesestarburstsarecollectedinTa- 1965,1966;Duric1990). Thisprovidesanaturalmechanism
ble3.AlthoughtheinferredB issomewhatlargeratagiven bywhichthemagneticfieldstrengthcanadjusttoa valueof
min
Σg thanfor the starburstsamplein Table 2 (filled squaresin ∼Bmin. However, such an argumentdoes not apply in star-
Fig. 1),thereasonablyclosecorrespondenceisencouraging. burstsgiventhattheminimumenergyargumentitselfimplies
η ≪1(Fig.1).
Inadditiontotheinferredminimumenergymagneticfield, min
Figure1also includesseveraltheoreticalcurves. Thedotted Amoresignificantworryconcerningtheapplicabilityofthe
lineshowstheequipartitionmagneticfield(eq.[3])whilethe minimum energy argument in starbursts is that strong syn-
dashedlineshowsthescalingB ∝Σ2/5derivedin§4.1. Fi-
nally,thesolidlinesshowthesmcianlingsgB∝Σa (witha=0.9, 9Free-freeabsorptionmaysuppresstheobservedradiofluxdensityatGHz
g frequenciesindensestarbursts(e.g.,Condonetal. 1991). However,evenin
0.8, and 0.7) used in our discussion of the FIR-radio corre- ourmostextreme case, theULIRG Arp220, this correction amounts toa
lationin§4.3. Althoughthesealternative(non-equipartition) smallincrease(∼10%)inourinferredBmin.Anotheruncertaintyisthecon-
scalingsaresomewhatarbitrary,theparticularchoiceB∝Σ0.7 tributionoffree-freeemissiontotheobservedradioflux.AtGHzfrequencies
g thethermalfractionisoforder10%fornormalstar-forminggalaxies(e.g.,
Niklas1995). Forstarburststhethermalfractionislesscertain. Inanycase,
8Bmin∝h-ra2d/7souncertaintiesinhradintroduceonlymodestuncertainties subtractingthethermalcontributiontotheradioluminositywoulddecrease
inBmin. theinferredBmin,strengtheningourconclusionthatBmin≪Beqinstarbursts.
4
chrotronandinverseCompton(IC)lossescouldleadtorapid whereh=100h pc is the scale heightof the galacticdisk
100
electron cooling. If the steady state electron energy density andv =500v kms- 1isthespeedofthegalacticwind(see,
w 500
is significantly lower as a result, then the minimum energy e.g.,Martin1999forestimatesofthelatter).
magneticfieldisanunderestimate.
Theradiospectraofbothnormalspiralsandluminousstar-
In the following sections we critically review indepen- bursts are similar with hαi≈0.75 from ≈1- 10 GHz (e.g.,
dentobservationalandtheoreticalconstraintsonthemagnetic Condonetal. 1991;Condon1992;Niklasetal. 1997).There
fields in starbursts. We begin by summarizing argumentsin appears to be no strong variation of α with the luminosity
favor of B∼B and against the hypothesis that magnetic of the galaxy (though Niklas et al. 1997 note a weak trend
min
fieldsinstarburstsaresignificantlylarger.Severalofthesear- forstarburststohavesomewhatflatternonthermalspectralin-
gumentsdrawheavilyonexistingresultsintheliterature,but dices).Thelackofasystematicvariationofthespectralindex
areworthreviewinginthepresentcontext. Rebuttalstothese withluminosity,togetherwiththefactthattheobservedradio
argumentsaregivenin§4,wherewealsoprovideindependent spectra are significantly flatter than expected for a “cooled”
argumentsinfavorofB≫Bmin. electrondistribution, suggeststhatτcool&τesc, evenin lumi-
nousstarbursts. Usingequation(7)toestimateτ ,equation
esc
3. ARGUMENTSFORB∼BminINSTARBURSTGALAXIES (4)thenimpliesfieldstrengths.2×10- 4G,reasonablycon-
sistentwith the minimumenergyestimatesforthe starbursts
3.1. ZeemanMeasurements inFigure1andinconsistentwithB≫B .
min
To our knowledge, there are no extragalactic detections
3.3. SynchrotronHalos
of Zeeman splitting. In four ULIRGs, however, Killeen et
al. (1996) derive upper limits of B . 3- 5 mG using OH The spatial variation of the radio spectrum can provide a
masers. Theselimitsarelargerthantheminimumenergyes- morestringentconstraintontheimportanceofelectroncool-
timateof∼1mGforULIRGs,butareafactorof∼5smaller ingthan the integratedradiospectrumalone. To take a con-
than B for Arp 220 in Figure 1. In our own Galaxy OH crete example, observations of M82 by Klein et al. (1988)
eq
masers trace regions of higher than average surface density andSeaquist&Odegard(1991)revealanextendedradiohalo
andmagneticfieldstrength,withB≈2- 10mG(e.g.,Fishet along the minor axis of M82, coincident with the observed
al.2003);itisunlikelythattheoppositeistrueinULIRGs,in galactic wind. The radio spectrum at GHz frequencies is
whichcasetheKilleenetal.(1996)upperlimitsargueagainst observed to be roughly constant in the disk of M82, but to
B∼B . However,alargersampleofZeemanmeasurements steepen significantly in the halo a distance h ∼ 300 pc
eq break
is needed, particularly for galaxies which also have reliable above the midplane. The usual interpretation of this halo is
gassurfacedensitymeasurements. that relativistic electrons generated in the starburst are ad-
vectedoutwiththegalacticwind. Thesteepeningath ∼
break
3.2. CoolingBreaks 300 pc can then be modeled as a consequence of electron
cooling(e.g.,Seaquist&Odegard1991). Theradioemitting
Thesynchrotroncoolingtimescaleforcosmicrayelectrons electronsinthehalomustthushaveh ∼τ v . Thisap-
break cool w
emittingatfrequencyν is proximateequalitycanbeusedtoputalimitonthemagnetic
τ ≈106B- 3/2ν- 1/2yr, (4) fieldinthediskofM82. Forexample,theequipartitionfield
syn 100 GHz of B ≈1.6 mG in M82 (see Fig. 1) implies a synchrotron
eq
whereB100=B/100µG. IClossescanbeappreciableaswell coolingtimeofτsyn∼104yrs,which,inturn,requiresanun-
andleadtocoolingonatimescale physicallylargev ∼30,000kms- 1 toexplaintheobserved
w
τ ≈5×105B1/2ν- 1/2U- 1 yr, (5) scale of the spectral break in M82. Thus a magnetic field
IC 100 GHz ph,- 9 strengthof∼B isstronglydisfavoredundertheinterpreta-
eq
whereUph=10- 9Uph,- 9ergscm- 3istheenergydensityof(pri- tionthatthe synchrotronhaloin M82is dueto electronsac-
marilyinfrared)photonsinthestarburst. celeratedinthediskandadvectedoutinagalacticwind. By
contrast,afieldstrengthofB∼B impliesv ∼500kms- 1
The coolingtimescale for electrons can be indirectly con- min w
to explaina spectral break at h ∼300 pc — much more
strainedusingtheradiospectrum.Forcontinuousinjectionof break
reasonableinthecontextofgalacticwinds. Thus,thespatial
relativisticelectrons,ifthecoolingtime
variationoftheradiospectruminM82appearstoprovidein-
τc-o1ol=τs-y1n+τI-C1 (6) dependent evidence that τcool &τesc in the galactic disk and
of the electrons emitting at frequency ν is shorter than the hencethatB∼Bmin≪Beq.
escape time from the galactic disk, the radio spectrum will In fact, more careful estimates account for both IC cool-
steepenby∆α=1/2athigherfrequencies,whereFν ∝ν- α. ingand adiabatic coolingof relativistic electronsas theyare
For canonical electron power-law indices of p=2 expected advectedoutinto the halo. Thisincreasesthe requiredwind
theoretically for strong shocks (e.g., Blandford & Eichler speedby a factor of few to ∼2000km s- 1 (e.g., Seaquist&
1987)andobservedinsituinsomesupernovaremnants(e.g., Odegard 1991). Although this is substantially smaller than
Aharonianetal.2005;Broganetal.2005),α=1/2intheab- thespeedof∼30,000kms- 1requiredforB∼B ,itislarger
eq
senceofcoolingandsteepenstoα=1inthepresenceofcool- thantheinferredspeedsofgalacticwinds(e.g.,Martin1999)
ing.Theescapetimeofrelativisticparticlesfromthegalactic andiscomparabletothe initialvelocityofsupernovaejecta.
diskisuncertain.Inlocalspiralstheescapeisduetodiffusion Onecannot, however,ruleouta verylow densityphaseof a
acrossthemagneticfield,whileinstarburstsitmaybemuch galacticwindwiththerequiredvelocity∼2000kms- 1.
morerapidduetoadvectionoutofthegalaxywithagalactic
WenotethatNGC253(Carillietal.1992),NGC1569(Is-
wind.Theescapetimeinthelattercaseisroughly
rael&deBruyn1988;Lisenfeldetal.2004),NGC4631(Ek-
τ ∼h/v ∼3×105h v- 1 yr, (7) ers & Sancisi 1977), NGC 891 (Allen, Sancisi, & Baldwin
esc w 100 500
5
1978),andNGC4666(Dahlemetal. 1997)allexhibitradio 4.2. TheFIR-RadioCorrelation
halos qualitatively similar to those in M82. Unfortunately,
with the exception of NGC 253, these systems have lower Equation (9) implies a linear correlation between the FIR
gassurfacedensitiesthanM82andlieinthepartofFigure1 luminosity(LFIR)andtheradioluminosity(Lrad)ofgalaxies.
where B ∼B . Clearly, a more extensive survey for low Suchacorrelationisinfactobservedovermorethanfouror-
min eq
surfacebrightnessextendedradioemissionaroundstarbursts dersof magnitudein LFIR (vander Kruit1971, 1973; Helou
withΣ ∼1gcm- 2wouldbeveryusefulindiscriminatingbe- et al. 1985; Condon 1992; Yun, Reddy, & Condon 2001).
g
tween the minimum energy and equipartition magnetic field Equation(9) is a consequenceof the fact that if τcool .τesc,
strengths(seeTable2). thentheradiofluxisessentiallyindependentofthemagnetic
field strength and depends only on the rate at which energy
4. ARGUMENTSFORB≫BminINSTARBURSTGALAXIES is supplied to the relativistic electrons. The linearity of the
FIR-radiocorrelationthenfollowsnaturallyfromthefactthat
In this section we rebut the arguments for B∼B pre-
min the supernova rate is proportional to the star formation rate
sentedinSections3.2and3.3andprovideadditionalevidence
(e.g., the “calorimeter” theory; Völk 1989; see also Bressan
that B≫B . Mostof this section centers on showing that
min etal.2002). Thusaslongasτ .τ onecanaccountfor
cool esc
τcool.τescisconsistentwithradioobservationsofstarbursts. theobservedlinearFIR-radiocorrelationwithnofinetuning.
4.1. UnderstandingBminasaFunctionofΣg By contrast, if τcool &τesc (as argued in §3.2 & 3.3) then
significantfinetuningisrequiredtoexplaintheFIR-radiocor-
A magneticfield strengthof B≫B in starburstswould relation. Becausethesynchrotroncoolingtimescale(eq.[4])
min
imply that the cooling time for relativistic electrons is very variesbyafactorof∼103forthesystemsinFigure1(afactor
short compared to the escape time (τ ≪τ ). To show of∼60forthestarburstsalone),theescapetimescalewould
cool esc
this, we assume that B=η1/2B , that τ =106τ yr, and havetovarybyexactlythesamefactorinordertopreservea
eq esc esc,6
wecombineequations(3)and(4)toestimatethat linearFIR-radiocorrelation. Webelievethatthisleveloffine
τsyn ∼0.01Σ- 3/2η- 3/4ν- 1/2τ- 1 . (8) tuningarguesstronglyagainstτcool&τesc(andthusindirectly
τ g GHz esc,6 againstB∼Bmin).
esc
For Σ &0.05τ- 2/3η- 1/2 g cm- 2, equation (8) implies that The magnitude of the radio flux from star-forming galax-
g esc,6 ies is also fully consistent with the hypothesis that τ .
τ <τ forelectronsemittingatGHzfrequencies. Includ- cool
syn esc τ . To demonstrate this, we note that if each supernova
ingICcoolingdecreasesthiscriticalsurfacedensitybyafac- esc
supplies an energy of ξ1048 ergs to cosmic ray electrons
toroforderunity.Inaddition,forlowersurfacedensitygalax-
with a spectral index p=2, and if τ ≪τ , then the ra-
ies,theelectronescapetimebecomesincreasinglydetermined cool esc
dio luminosity should be related to the FIR luminosity via
bydiffusionacrossthemagneticfieldratherthanescapeina
νL ≈10- 5ξL /2ln(γ )≈2.5×10- 7ξL , where γ is
galacticwind.Thislikelyincreasesτ substantiallyandthus ν IR max IR max
esc themaximumLorentzfactoroftheacceleratedelectrons.For
decreasesthecriticalsurfacedensitywhereτ ∼τ .
syn esc comparison, the 1.4 GHz FIR-radio correlation from Yun et
TheaboveconsiderationsshowthatifB≫Bminthennearly al. (2001) is νLν ≈ 2×10- 6LIR. This implies that ξ ≈ 8
allofthestarburstsinFigure1areinthelimitwhereτcool< is needed to account for the FIR-radio correlation. That is,
τesc. Our observationally inferred results for Bmin(Σg) can ≈0.8%ofeach supernova’senergyis radiatedawayas syn-
be understood quantitatively in this limit as follows. In the chrotronradiation.Thisissimilartostandardestimatesforthe
rapidly cooling limit, the electrons radiate all of the energy fraction of SN energy supplied to relativistic electrons (see,
suppliedtothembysupernovashocksandthustheradioflux e.g.,§2.2.2ofKeshetetal.2003forasummary),whichsup-
perunitvolumeisdeterminedbythesupernovarateperunit portsthehypothesisthatτ .τ . Inparticular,thisargu-
cool esc
volume: ment rules out the possibility that τ ≫τ , since in this
νLν/V ∝Σ˙⋆/h, (9) casetheamountofsupernovaenergycosoulpplieedsctocosmicrays
whereΣ˙ isthestarformationrateperunitareaandV isthe wouldhavetobeprohibitivelylarge(ξ≫1).
⋆
radio emitting volume. Using the Schmidt Law for star for- Anoft-citedobjectiontothecalorimetermodelfortheFIR-
mation(Σ˙ ∝Σ7/5;Kennicutt1998)andequation(1),wefind radiocorrelationisthatthenonthermalradiospectraofgalax-
⋆ g
B ∝Σ2/5h- 2/7. (10) iesshouldthenbesignificantlysteeperthanareobservedbe-
min g cause of strong synchrotroncooling (see §3.2; e.g., Condon
Toestimatetheproportionalityconstantinequation(10),we 1992). In §4.3 we show that ionization and bremsstrahlung
assumethat10- 3ξofeachSN’senergyissuppliedtorelativis- lossescansystematicallyflattentheradiospectraofstarbursts
tic electrons (and is radiated away), in which case equation evenwhenτ ≪τ ,providingmuchbetteragreementwith
cool esc
(10)becomes observations.
Bmin≈7×10- 5δ22/7ξ2/7Σg2/5h1- 020/7νG- 1H/z7 G, (11) Equation(8) impliesthatτsyn &τesc for Σg.0.005(0.05)
Takingh∼constantand ξ =10 (see §4.2), we plotequation g cm- 2, if τ ≈3×107(106) yr. The former value of τ
esc esc
(11) in Figure 1 (dashed line) and find that it is in reason- is comparable to that estimated for the Milky Way (Garcia-
able agreementwith the resultsforstarbursts. Thisprovides Munoz et al. 1977; Connell 1998). According to the argu-
strongsupportforthehypothesisthatτ .τ inthesesys- mentspresentedinthissection,thecalorimetertheoryforthe
cool esc
tems. Furthermore,theagreementbetweenthestarburstdata FIR-radiocorrelationthusdoesnotapplytolowsurfaceden-
andequation(11)emphasizesthattheinferredminimumen- sity normal star-forming galaxies. One might then expect
ergy field in starbursts is not a reliable magnetic field mea- a change in the FIR-radio correlation at low Σ because a
g
surement,butrathersimplyreflectsthefactthattheradioflux smallerfractionoftheelectronenergyisradiatedaway. Ob-
isproportionaltothestarformationrate(eq.[9]). servationally,thereisonlyaverymildsuppressionintheratio
6
FIG. 2.—Theratiooftheminimumenergymagneticenergydensitytothephotonenergydensity(UBmin/Uph)asafunctionofthegassurfacedensity(Σg).
ThesymbolsarethesameasthoseinFig.1.Uphiscalculatedusingeq.(12)(seealsoeq.[13])forallsystemsexcepttheGalaxy,forwhichwetakeF=L/πR2
withL=1.5×1010L⊙andR=8.5kpc(asappropriatefortheestimateofBminatthesolarcircle). UB<Uphisdifficulttoreconcilewiththelinearityofthe
FIR-radiocorrelationandsuggeststhantheminimumenergyfieldstrengthisanunderestimateforluminousstarbursts.
of the radio to FIR luminosities for low-luminosity galaxies where F is the radiative flux, Σ˙ is the star formation rate
⋆
(Yunetal. 2001). Bell (2003)hasshown, however,thatthis per unit area, and we take ǫ = 3.8×10- 4 for consistency
isinpartbecauseneithertheradionortheFIRemissionfrom with Kennicutt (1998). Above the dotted line in Figure 2,
lowluminositygalaxiesaregoodproxiesforthetotalstarfor- U /U >1, τ <τ , and synchrotron dominates IC as
Bmin ph syn IC
mation rate. Because the focus of this paper is on starburst the primary coolant for the relativistic electrons. Below the
galaxies,wedeferamoredetailedanalysisoftheconnection dottedlineτ >τ andICdominates.
syn IC
betweentheFIR-radiocorrelationinnormalspiralsandstar-
Figure2 shows thatforhigh surfacedensity starburststhe
burststoafuturepaper.
minimumenergyargumentimpliesmagneticenergydensities
smallerthanthephotonenergydensity. Forthemostextreme
4.2.1. SynchrotronversusInverseComptonCooling
caseofArp220(andULIRGsingeneral),theminimumen-
Regardless of the value of τ /τ , the linearity of the ergy estimate impliesUB/Uph ∼0.1. For most starbursts in
FIR-radio correlation suggests cthooalt tehsec synchrotron cooling Figure2,amodestincreaseofBbyafactorof∼2- 3would
timescale τ (eq. [4]) must be shorter than the IC cooling leadtoUB&Uph. Thisisplausiblywithintheuncertaintiesof
syn
theminimumenergyestimate—andisaratherminorcorrec-
timescaleτ (eq.[5]). ThisinturnimpliesthatU &U . If
IC B ph
thisconstraintisnotsatisfied,anyvariationinU /U would tiononthescaleofBeq/Bmin inFigure1—butdoessuggest
B ph
thatB isanunderestimate.
implylargechangesinthefractionofcosmicrayelectronen- min
ergy radiated via synchrotron radiation. A linear FIR-radio Equation (12) may significantly underestimate U for
ph
correlationwouldthenrequiresignificantfinetuning. galaxieswith Σ &0.5 g cm- 2 because the ISM is optically
g
InFigure2weplottheratio(B2 /8π)/U versusgassur- thicktoitsownFIRradiationinthesesystems. Inthiscase,
min ph
facedensityforthesampleofgalaxiesfromFigure1(Tables
1-3). Thephotonenergydensityiscomputedusing U = τIRF ≈τ ǫΣ˙ c, (13)
ph c IR ⋆
Uph= Fc =ǫΣ˙⋆c, (12) whereτIR≈κIRΣg istheopticaldepthofthegalacticdiskto
7
theIRradiationandκ ≈3- 10cm2 g- 1 isthetemperature-
IR
dependent dust opacity in the FIR (Thompson, Quataert, &
Murray2005). In principle, this factor of τ couldincrease
IR
U byafactorof10- 100forULIRGs. Ifonerequiresthat 1.2
ph
U >U in order to satisfy the FIR-radio correlation, the
B ph
necessary field strength would then be ≫B , and perhaps
min
aslargeas ∼Beq. The applicabilityof equation(13) hinges, 1
however,onthe assumptionthatthe radioemittingelectrons
are co-spatial with optically thick gas at the average density
oftheISM.Iftheradioemittingelectronsareinalower-than-
0.8
average density (optically thin) environment, then equation
(12)isstillvalid.
4.3. CoolingBreaks 0.6
In §3.2 we noted that because starburst radio spectra do
not show significant evidence for synchrotron cooling, the
0.4
straightforward interpretation is that τ &τ and, hence,
cool esc
B∼B . Onedifficultywiththisconclusionisprovidedby
min
ULIRGssuchasArp220,wheretheICcoolingtimealoneis
∼104 yrs. Unless h.10 pc, the cooling time in ULIRGs 0.2
0.1 1 10 100
is shorter than the advection time across the galactic disk,
for reasonable values of the advection velocity. Given that
the size of the radio emitting region resolved by the VLA FIG. 3.—Predictedradiospectralindexasafunctionoffrequency. Elec-
in ULIRGs is typically ∼ 100pc (Condon et al. 1991), one tronsareinjectedwithn(γ)∝γ- p,withp=2(solidlines)andp=2.5(dashed
lines), andloseenergythroughsynchrotronradiation, bremsstrahlung, and
might thus expect the nonthermalspectrum to be systemati- ionization losses. Escape is neglected, which corresponds to the limit
callysteeperinULIRGsthaninotherstarbursts.10 Thisdoes τcool≪τesc. Thecalculations assumeΣg=1gcm- 2,η=1,h100=1,and
not however, appearto be the case (Condonet al. 1991; see f =1,0.5,0.25(frombottomtotop).Ionizationlossesflattentheradiospec-
alsoFig.19ofDownes&Solomon1998forthecaseofArp trumatlowfrequencies. Therefore, eveninthelimitτcool≪τesc,spectral
indices atGHzfrequencies arelessthanthecanonical valueofα=1(for
220). The lack of a systematic differencebetween the radio
p=2)appropriateforstrongsynchrotroncooling.Thisprovidesmuchbetter
spectralindices of ULIRGs and other systems raises signifi- agreementwiththeobservedradiospectraofstarbursts.
cantquestionsabouttheextenttowhichtheradiospectracan
beeasilyusedtoconstraintheelectroncoolingtimeandthus
themagneticfieldstrength.
gas at the mean density of the ISM (f ∼1) ionization and
We suggest that the radio spectra of starbursts are in fact
bremsstrahlung losses can significantly modify the distribu-
compatible with τ ≪τ because of the effect of ioniza-
cool esc tionfunctionforelectronsemittingat∼GHzfrequencies.
tionandbremsstrahlunglossesonthelowenergypartofthe
electron distribution function. The timescale for a relativis- Undertheactionofionizationlossesalone,lowenergycos-
ticelectronthatemitssynchrotronradiationatfrequencyν to mic rays lose energy more rapidly than high energy cosmic
loseitsenergyionizingneutralhydrogenofdensityn=10n rays(thebremsstrahlunglosstimeisindependentofelectron
10
cm- 3is energy). The opposite is true for synchrotron losses. For
τ ≈107n- 1ν1/2 B- 1/2yr. (14) thisreason,ionizationlossescansystematicallydecreasethe
ion 10 GHz 100
spectralindexfromwhatonewouldexpectforasynchrotron-
Relativistic electrons also lose energy to bremsstrahlung cooled electron distribution (α≈1- 1.25). To demonstrate
emissiononambientneutralnuclei,onatimescale
this explicitly, we have calculated the steady state electron
τ ≈3×106n- 1yr. (15) distribution function for a model in which relativistic elec-
brem 10
tronsareinjectedwithaspectralindex pandloseenergyvia
Setting B=η1/2Beq and n= fhni= fΣg/2hmp we find that synchrotronradiation, bremsstrahlunglosses, and ionization
theratiooftheionizationlosstimetothesynchrotroncooling losses. Escape is neglected, which corresponds to the limit
timeis τ ≪τ . Figure 3 shows the resulting synchrotronspec-
τ cool esc
ion ≈1.6ν h f- 1η1/2 (16) tralslopeαasafunctionoffrequency.Thesolidlinesarefor
GHz 100
τ
syn p=2 and the dashed lines are for p=2.5. The calculations
andthattheratioofthebremsstrahlunglosstimetothesyn- showninFigure3assumeΣg=1gcm- 2,η=1,h100=1,and
chrotroncoolingtimeis f =1,0.5,and0.25(frombottomtotop).
τbrem ≈2.4ν1/2 h f- 1η3/4Σ1/2, (17) The spectral indices in Figure 3 at GHz frequencies are
τ GHz 100 g compatiblewith the observedvaluesevenin the presenceof
syn
strong cooling. We find similar results for reasonablevaria-
whereh100=h/100pcisthegasscaleheight. Equations(16) tionsofourmodelparameters.Forexample,if p=2,η=0.1,
& (17) show that if the cosmic ray electrons interact with and f =0.25, we find α≈0.6 at 1 GHz from Σ ≈0.1- 10
g
g cm- 2 and α≈0.7- 0.8 at 5 GHz over the same range of
10Althoughtheradiosizesare∼100pc,asmallscaleheightforULIRGs Σ . Our calculations predict that the non-thermal spectral
isnotruledout:forarandomvelocityδv,thescaleheightofagalacticdiskis g
h≈δv2/2πGΣ,whichimpliesh≈10pcfortheobservedvaluesofδv≈100 index should steepen to α∼1 at sufficiently high frequen-
kms- 1andΣ≈10gcm- 2inArp220(e.g.,Scovilleetal.1997). cies, but this may be difficult to detect because of free-free
8
emission. A second prediction of this model is that the ra-
diospectralindexshouldbelargerathighfrequenciesthanit
isatlowfrequencies,withatransitionat∼GHzfrequencies
where the ionization, bremsstrahlung, and synchrotron loss
0.2
times are comparable(see eqns. [16] & [17]). For example,
inthemodelfromFigure3with p=2and f =0.5(themid-
dle solid line), the effective spectral index from 1- 10 GHz
is α=0.8, whereas from 0.1- 1 GHz it is α=0.55. Using
theradiodatainTable2,aswellas57.5,151,408,750,and
0
4850MHzdataavailableintheNASAExtragalacticDatabase
(from Hales et al. 1993; Ficarra et al. 1985; Israel & Ma-
honey1990;Beckeretal.1991;McCutcheon1973;Waldram
etal.1996;Heeschen&Wade1964;Largeetal.1981;Grif-
fith etal. 1995; Kuehretal. 1981),and additionaldata from
-0.2
Oly & Israel(1993)(forNGC3504),Irwin& Saikia (2003)
(for NGC 3079), and Hummel et al. (1991)(for NGC 891),
wefindthatforthesampleofstarburstgalaxiesconsideredin
thispaperthespectralindexbelow1GHzisindeedtypically
flatterthanthatabove1GHzby∆α≈0.3(althoughthereare
-0.4
someexceptionsandscatter).
Although encouraging, a larger and more homogeneous 0.01 0.1 1 10
sample of data is clearly needed to test our predictions of
spectralindexvariationsasa functionoffrequency. Inaddi-
FIG. 4.—PredictedratiooftheFIRto1GHzsynchrotronluminosityas
tion,lowfrequencyradioemissionmaybestronglymodified afunction ofgas surface density Σg. Electrons are injected with a p=2
by free-free absorption in many starbursts. Moreover, elec- spectrumandloseenergythroughsynchrotronradiationandionizationand
tronsemittingat∼100MHzina∼mGfieldhaveenergiesof bremsstrahlunglosses;escapeisneglected. TheFIRluminosityisassumed
∼100MeV.Attheseenergiessecondaryelectronsproduced tobeproportionaltothetotalrateatwhichenergyissuppliedtorelativistic
electrons.Thecalculationsassumethat f =0.5(n=0.5hni)andthatB∝Σa
by charged pion decay are important and may significantly with a= 1, 0.9, 0.8, and 0.7(bottom to top) and B=Beq at Σg =0.01 gg
modifytheelectronspectrumaswell. cm- 2(seeFig.1forthesescalings). Theseresultsdemonstratethatalthough
ionizationandbremsstrahlunglossesmodifytheradiospectraofstarbursts
Because the synchrotron, ionization, and bremsstrahlung
(Fig.3),theynonethelessmaintainanearlylinearFIR-radiocorrelation, in
loss rates all scale differentlywith the gas density and mag- agreementwithobservations.
neticfieldstrength,thereisnoguaranteethatthelinearityof
the FIR-radiocorrelationis preservedunderthe influenceof
ionization and bremsstrahlunglosses. To assess this, Figure
4showsmodelcalculationsofL /L asa functionofΣ . assumingelectronsareinjectedwitha p=2spectrum.Ifion-
FIR rad g
The synchrotron flux at 1 GHz is calculated assuming that izationandbremsstrahlunglossesareimportantinstarbursts,
electronsareinjectedwitha p=2spectrum,andloseenergy as argued in this section, then the required energy injection
via synchrotron, ionization, and bremmstrahlung losses (as rateincreasesto≈1.6%ofthesupernovaenergy.Inaddition,
before,escapeisneglected). TheFIRluminosityisassumed iftheelectroninjectionspectrumis p=2.2,thentherequired
to be proportionalto the rate at which energy is supplied to energyinjectionincreasesbyanadditionalfactor≈2.
relativistic electrons. The modelsshownin Figure4 assume
In contrast to the results of Figure 4, in a model in which
that f =0.5 (n=0.5hni), h100=1, and thatB∝Σag with a = f =0.5andB∝Σ0.4,whichcorrespondstotheobservedscal-
1,0.9,0.8, and 0.7 (bottom to top; see Fig. 1 for the B(Σg) ing of B (Σ ) ingFigure 1, the ratio of the FIR to radio lu-
curves). AlloftheassumedmodelsforB(Σg)haveB≫Bmin minositiemsinincgreases by a factor of ≈ 10 from Σ = 0.01 g
g
forluminousstarbursts. cm- 2 to Σ =10 g cm- 2 because of the increasing suppres-
g
Figure4demonstratesthatthepredictedFIR-radiocorrela- sion of the radio flux by bremsstrahlung losses. Thus field
tionislineartobetterthanafactorof2over3ordersofmag- strengths≈B are only consistentwith the observedradio
min
nitudein Σ . Themodelswith a=0.7, 0.8, and0.9 provide emissionfromstarburstsifcosmic-rayelectronsinteractwith
g
particularlygoodagreementwiththeobservedlinearityofthe gasatmuchlessthanthemeandensityoftheISM(f ≪1).
FIR-radio correlation. Because galaxies of a given Σ can
g To summarize this subsection, if B≫B and if cosmic-
havearangeofFIRluminosities(dependingongalaxysize), min
rayelectronsinteractwithgasatapproximatelythemeanden-
thesystematictrendsinFigure4mayshowupinpartasscat-
sityoftheISM,ionizationandbremsstrahlunglosseswillsys-
terinthecanonicalplotsofL /L vs. L ;the.factorof
FIR rad FIR tematicallyflattenthenonthermalspectraofstarburstgalaxies
2variationinFigure4isconsistentwiththeobservedscatter
(Fig. 3)andyetmaintainthelinearityoftheFIR-radiocorre-
in the FIR-radio correlation. The approximate constancy of
lation(Fig.4).Thiseliminatesoneofthestrongestarguments
L /L in Figure 4 arises because if f ∼1 and B≫B ,
FIR rad min againstτ ≪τ andarguesforB≫B instarbursts.
thenτ /(τ- 1+τ- 1 )- 1isrelativelyconstantovermostofthe cool esc min
syn ion brem
parameterspacerelevanttoobservedstarbursts.
4.4. PionProduction
In§4.2weestimatedthat≈0.8%oftheenergyofeachsu-
pernovamust be supplied to relativistic electronsto account An independent test of the density of the ISM in which
forthemagnitudeofthe≈1.4GHzradiofluxfromstarburst the cosmicraysreside can be providedby gamma-rayemis-
galaxies,takingintoaccountsynchrotronradiationaloneand sion from neutral pion decay. The timescale for cosmic ray
9
protons to lose their energy to pion production is t ∼ As an alternative to the standard interpretation (§3.3) that
pion
108(n/1cm- 3)- 1 yr. If n∼hni, this is significantly shorter extended synchrotron halos are due to relativistic electrons
thanτ instarbursts(seeeq.[7])andsopionproductionwill generatedinthediskandadvectedintothehalo(beforecool-
esc
leadtoasubstantialγ-rayluminosity. Assumingthat10%of ing),wesuggestthatthehalosareinsteadduetoelectronsac-
the energy of each supernova is supplied to relativistic pro- celeratedinsituinthegalacticwind,byeitherdirectelectron
tons with a spectral index p=2, and that ≈1/3 of this en- acceleration in shocks or via charged pion production. We
ergygoesintoγ-raysratherthan neutrinosorpairs, we esti- notethatinseveralsimulationsofgalacticwinds(e.g.,Strick-
mateaγ-rayluminosityfromneutralpiondecayinstarbursts land&Stevens2000),mostoftheX-rayemissionisduetoin
ofνL ≈3×10- 4L /ln(γ )≈2×10- 5L , whereγ is situshocks,whicharealsolikelytobesitesofparticleaccel-
ν IR max IR max
the maximum Lorentz factor of the accelerated protons and eration. Sincethetotalkineticenergyfluxinagalacticwind
the estimate is only applicable above ≈100 MeV (see Tor- is comparable to the energy produced by supernovae in the
res2004formoredetailedcalculations). Thispredictedflux starburst, only a small fraction of the wind energy must be
is just below the EGRET upper limits for systems such as convertedinto relativistic particles in situ to account for the
M82andArp220(Cillisetal.2005),butshouldbedetectable extended synchrotron emission observed. In this interpreta-
with GLAST. Because the timescales for electron ionization tion,whichclearlyrequiresadditionalwork,B≫Bminwould
and bremsstrahlung losses and proton pion losses are quite befullyconsistentwithextendedsynchrotronemission. Ob-
similar,detectionofgamma-rayemissionatthislevelwould servedspectraldifferencesbetweenthegalacticdiskandhalo
alsoconfirmtheimportanceofionizationandbremsstrahlung inM82andothersystemsmayarisefromspatialvariationsin
lossesfortheobservedradiospectraofstarbursts. τsyn/τesc,τIC/τesc,τbrem/τsyn,andτion/τsyn.
Relativistic protonslose ≈2/3 of their energyto the pro-
5. DISCUSSION&CONCLUSIONS
ductionofchargedpions,whichsubsequentlydecayintohigh
energyneutrinosandelectron/positronpairs;thepairsreceive
Basedontheanalysisoftheprecedingsections,itisworth-
≈1/4ofthepionenergy.Thusif∼10%oftheenergyofeach
while contrasting the following two scenarios for the radio
supernovaissuppliedtorelativisticprotons,andift ≪t ,
pion esc emissionfromstarbursts:
then ∼1.5% of the energyof each supernovagoes into sec-
ondary electrons and positrons. This is in good agreement B∼Bmin:Fortheminimumenergymagneticfieldestimateto
with the energy injection rate estimated in §4.2 and §4.3 to beapplicable,theelectronsmustescaperapidlyinagalactic
accountfortheradiofluxfromstarbursts. Thissuggeststhat windwithτcool&τesc. Therapidescapeimpliesthatthereis
secondaryelectronsmaydominatetheradioemissioninstar- nocoolingbreakatGHzfrequencies,whichisconsistentwith
bursts (see also Rengarajan 2005). If correct, it remains to the observednonthermalspectra of α≈0.75. Rapid escape
be understood why normal spirals and starbursts lie on ap- is plausible given that hot outflowing galactic winds are ob-
proximatelythesameFIR-radiocorrelationeventhoughsec- servedfromstarburstsandthatthecosmicraysarelikelygen-
ondariesdonotappeartobeimportantforelectronsemitting erated in the same supernovashocksthat generate the wind.
at∼ GHz frequenciesin normalspirals(Strongetal. 2004). Thisinterpretationofmagneticfieldsinstarbursts—thestan-
Asdiscussedin§4.2variationsinτ /τ mightalsobeex- dard one — has two major drawbacks: (1) it is difficult to
syn esc
pectedtomodifytheFIR-radiocorrelationfornormalspirals see how τcool &τesc is applicable in ULIRGs where cooling
relativetostarbursts. times from IC alone are ∼104 yr and (2) it is very difficult
to accountfor the FIR-radio correlationwith τ &τ be-
cool esc
causevariationsinτ orτ shouldleadtovariationsinthe
4.5. InverseComptonEmission cool esc
fractionoftheelectronenergyradiatedawayviasynchrotron
radiation. Eitherseveralcoincidencesoracomplexfeedback
IC upscattering of infrared photons by cosmic-ray elec-
looparethenrequiredtoaccountforthelinearityoftheFIR-
trons can provide an independent constraint on U /U be-
B ph
radio correlation. The latter would be somewhat surprising
causetheratiooftheICpowertotheradiopowerisgivenby
giventhatB itselfimpliesthatmagneticfieldsandcosmic
L /L ≈U /U . Ifwe useequation(12) toestimateU , min
IC rad ph B ph
raysaredynamicallyweakcomparedtogravity(Fig.1).
thenB∼B instarburstswouldleadtonegligibleICpower
eq
becauseitwouldimplyUB/Uph≫1. However,iftheinfrared B≫Bmin: An alternative possibility is that magnetic fields
optical depth is larger than unity, equation (13) may instead instarburstsaremuchstrongerthanissuggestedbythemin-
beapplicable,inwhichcaseB∼B impliesU /U ∼1. imumenergyestimate. Rapidcoolingofrelativisticelectrons
eq B ph
in starbursts will invalidate the minimum energy estimate if
ICemissionhaslongbeenarguedtocontributesignificantly
τ ≪τ . Magneticfieldsinstarburstscouldtheninprin-
totheX-rayemissioninavarietyofsystems,includingM82 cool esc
ciple be as large as ∼B , which is ∼10 times larger than
(e.g.,Hargrave1974;Moran&Lehnert1997)andNGC3256 eq
B for typicalstarbursts (Fig. 1). This modelnaturallyac-
(e.g.,Moran,Lehnert,&Helfand1999). Inbothoftheseex- min
counts for the linearity of the FIR-radio correlation because
amples,however,muchofthehardX-rayemissionhasbeen
in the limit τ .τ the radio flux is nearly independent
resolvedintopointsourceswithChandra,althoughICemis- cool esc
ofthemagneticfield strength(e.g.,Völk1989). Inaddition,
sionmaystillcontributetothehardX-raysfromtheverycen-
theobservedradioluminositiesfromstar-forminggalaxiesare
tralnucleusofthestarburst(e.g.,Liraetal.2002;Strickland
comparabletotheestimatedrateatwhichenergyissupplied
etal.2002;Strickland&Heckman,privatecommunication).
torelativisticelectronsinsupernovashocks(§4.2). Thissup-
More detailed tests of this hypothesis would be very worth-
portsthehypothesisthatτ .τ . Independentsupportfor
whilegiventhepossibilityof providingan independentcon- cool esc
thishypothesisisprovidedbythefactthattheobservedscal-
straintonthemagneticfieldstrengthinstarbursts.
ing B ∝Σ2/5 for the minimum energy field in starbursts
min g
(Fig. 1) followsdirectly from the SchmidtLaw for star for-
4.6. SynchrotronHalos
mationwhenτ .τ (eq.[10]).
cool esc
10
A standardobjectionto τ .τ is thatthe typicalnon- ing gas to smaller radii to fuel a central active galactic nu-
cool esc
thermalspectralindicesofstarbursts(α≈0.75)donotshow cleus. Towards this end, we briefly summarize several ob-
the expected steepening due to strong cooling (α≈1). We servationsthatshouldhelptestthepredictionsofourmodel:
haveargued,however,thatifB≫B andifcosmicrayelec- (1)Ifionizationandbremsstrahlunglossesareimportant,the
min
tronsinteractwith gas atapproximatelythe mean density of radio spectra of starbursts should be flatter below ∼1 GHz
theISM,thenionizationandbremsstrahlunglossesflattenthe than they are above ∼1 GHz (§4.3 and Fig. 3). (2) If cos-
radiospectraofstarburstsat∼GHzfrequenciesandreconcile mic rays interact with gas at about the mean density of the
τ .τ with the observedspectra (Fig.3). An important ISM, as is requiredfor ionizationand bremsstrahlunglosses
cool esc
partofthisargumentistherealizationthattheionizationand to be important, there should be an appreciable gamma-ray
bremsstrahlunglosstimesaresimilartothesynchrotroncool- fluxfromstarburstsduetoneutralpionproduction(§4.4).We
ingtimeinallstarburstgalaxiesforcosmicrayelectronsemit- estimate νL ≈2×10- 5L above≈100 MeV. This predic-
ν IR
tingatGHzfrequencies.Thus,ionizationandbremsstrahlung tion is testable with GLAST. (3) Zeeman measurements of
losses can modifythe nonthermalspectra of starburstgalax- starburstscandirectlyprobethemagneticfieldstrengthinthe
iesandyetmaintainthelinearityoftheFIR-radiocorrelation densephaseoftheISM,anddistinguishbetweenB∼B and
min
(Fig. 4). B≫B . Existing upper limits in 4 ULIRGs from Killeen
min
al. (1996) are above B , but below the equipartition field
This interpretation of magnetic fields in starbursts — that min
(§3.1). (4)IfB≫B ,ICemissionisunlikelytocontribute
B ≫ B — has two potential drawbacks: (1) Given that min
min significantlytotheX-rayemissioninstarbursts(§4.5).
galacticwindsefficientlyremovemassandmetalsfromgalax-
ies, it is unclear whether the cosmic rays actually interact Itisinterestingtocompareourresultsonmagneticfieldsin
with the bulk of the ISM in starbursts, which is requiredfor starburstswiththeinferredcorrelationbetweenmagneticfield
ionization losses to be significant. (2) The nonthermal syn- strength,columndensity,andmassdensityinGalacticmolec-
chrotron halos observed in several systems are typically in- ular clouds (e.g., Troland & Heiles 1986; Crutcher 1999).
terpretedascosmic-rayelectronsadvectedoutwithagalactic Using Zeeman splitting to determine the magnetic field
wind, with synchrotroncooling only becoming importantin strength directly, Crutcher (1999) finds that over the range
the halo (e.g., Seaquist & Odegard1991). If this interpreta- 0.01gcm- 2.Σ .2gcm- 2.and2.7.log [n (cm- 3)].
tioniscorrect,itrequiresB∼Bmin(§3.3).Wehavesuggested, 6.8, that B∝n1H/22 and B∝Σ . These find1in0gsHi2mply both
hpoarwtiecvleesr,atchcaetleerxatteenddiendssiytuncinhrgoatlraocntiecmwiisnsdiosn(§m4a.6y),biendwuheictoh thatthattheAlfHv2énspeedvA∼H2constant∼10kms- 1andthat
equipartition obtains. Observations of polarization in H O
case the observed halos cannot be readily used to constrain 2
masers in dense star-forming regions provide evidence that
themagneticfieldstrengthingalacticdisks.
these trends extend to yet higher densities (108- 1010 cm- 3;
We believe that the arguments presented in this paper Sarmaetal.2002;Vlemmingsetal.2005).Theseresultssug-
stronglyfavorτcool.τesc fortherelativisticelectronsinstar- gestthatthemagneticfieldisalwayscomparabletothetotal
bursts. In this limit, the minimum energy argument under- pressureinGalacticmolecularclouds,withB≈B .
eq
estimates the true magnetic field strength. The observed ra-
diofluxis,however,nearlyindependentofthemagneticfield The sample of Galactic molecular clouds reviewed by
anddependsprimarilyontherateatwhichenergyissupplied Crutcher (1999) covers precisely the same surface densities
to relativistic electrons. Thus direct constraints on the true representedbystarburstsinourFigure1.Itisthusparticularly
field strength are difficult to come by. By themselves, these strikingthattheminimumenergyestimateimpliesB≪Beqin
arguments do not imply that B∼B , only that τ .τ starbursts,whiledirectZeemanmeasurementsoflocalmolec-
eq cool esc
and B&Bmin. Our strongest argument that magnetic fields ularcloudswiththesameΣg implyB∼Beq. Oursuggestion
are likely to be ≫B derivesfrom the ratio of the ioniza- isthatthisdifferencearisesinpartfromtheinapplicabilityof
min
tionandbremsstrahlunglosstimestothesynchrotroncooling theminimumenergyestimateinstarbursts.
time:ifcosmicrayelectronsinteractwithgashavingroughly
themeandensityoftheISM,magneticfieldsmustbe≫B
min
instarburstsinorderforionizationandbremsstrahlunglosses
We are grateful to Rainer Beck, Tim Heckman, and Jim
to not completely dominate over synchrotron losses (which
Condon for a close reading of the text and for a number
would be inconsistent with the FIR-radio correlation and
of very useful comments. We also thank Alberto Bolatto,
would require that an unreasonablylarge fraction of the su-
Bruce Partridge, and Carl Heiles for valuable conversations.
pernovaenergybe supplied to cosmic ray electronsin order
E.Q.issupportedinpartbyNASAgrantATP05-54,anAlfred
toaccountfortheobservedradiofluxesfromstarbursts;§4.3).
P.SloanFellowship,andtheDavidandLucilePackardFoun-
Our conjecture is thus that the magnetic field is signifi- dation. E.W.thankstheMillerFoundationforsupportinghis
cantly larger than ∼ B (the canonical estimate) in many visit to UC Berkeley and the Institute for Advanced Study.
min
starbursts. Confirmingthispredictionwouldhavesignificant N.M. is supported in part by a Canadian Research Chair in
implications for understandingthe physics of star formation Astrophysics.ThisresearchmadeextensiveuseoftheNASA
in starbursts, the effects of magnetic fields on the dynamics ExtragalacticDatabase.C.L.M.issupportedbytheDavidand
of galactic winds, and the role of magnetic stresses in driv- LucilePackardfoundationandAlfredP.SloanFellowship.
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