An uncertainty principle for star formation. I. Why galactic star formation relations break down below a certain spatial scale

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by  J. M. Diederik Kruijssen

Mon.Not.R.Astron.Soc.000,1–14(2013) Printed20June2014 (MNLATEXstylefilev2.2)
An uncertainty principle for star formation. I. Why galactic star
formation relations break down below a certain spatial scale
4
1 2
1 J. M. Diederik Kruijssen and Steven N. Longmore
0 1Max-PlanckInstitutfu¨rAstrophysik,Karl-Schwarzschild-Straße1,85748Garching,Germany;[email protected]
2 2AstrophysicsResearchInstitute,LiverpoolJohnMooresUniversity,IC2,LiverpoolSciencePark,146BrownlowHill,LiverpoolL35RF,UnitedKingdom

n
a
J Accepted2014January15.Received2013November8;inoriginalform2013July9.

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1
ABSTRACT

] Galacticscalingrelationsbetweenthe(surfacedensitiesof)thegasmassandthestarforma-
A
tion(SF)rateareknowntodevelopsubstantialscatterorevenchangeformwhenconsidered
G belowacertainspatialscale.We quantifyhowthisbehaviourshouldbeexpectedduetothe
. incompletestatisticalsamplingofindependentstar-formingregions.Otherincludedlimiting
h
factorsaretheincompletesamplingofSFtracersfromthestellarinitialmassfunctionandthe
p
spatialdriftbetweengasandstars.WepresentasimpleuncertaintyprincipleforSF,whichcan
-
o beusedtopredictandinterpretthefailureofgalacticSFrelationsonsmallspatialscales.This
r uncertaintyprincipleexplainshowthescatterofSFrelationsdependsonthespatialscaleand
t
s predictsascale-dependentbiasofthegasdepletiontime-scalewhencenteringanapertureon
a gasorSFtracerpeaks.Weshowhowthescatterandbiasaresensitivetothephysicalsizeand
[
time-scalesinvolvedintheSFprocess(suchasitsdurationorthemolecularcloudlifetime),

1 andillustratehowourformalismprovidesapowerfultooltoconstraintheselargelyunknown
v quantities.Thankstoitsgeneralform,theuncertaintyprinciplecanalsobeappliedtootheras-
9 trophysicalsystems,e.g.addressingthetime-evolutionofstar-formingcores,protoplanetary
5 discs,orgalaxiesandtheirnuclei.
4
4 Keywords: galaxies:evolution—galaxies:ISM—galaxies:stellarcontent—ISM:evolu-
. tion—stars:formation
1
0
4
1
v: 1 INTRODUCTION numberofidealisedexamples.Weshowthatitaccuratelydescribes
i theobservedrangeofspatialscalesonwhichtheSFrelationscan
X Galactic star formation (SF) relations (e.g. Silk 1997; Kennicutt
beapplied,aswellastheirscatterandbiasonsmallerscales.Itis
1998; Elmegreen 2002; Bigieletal. 2011) break down below a
r alsoshown how theuncertainty principlecan beused toderivea
a certain spatial scale (Schrubaetal. 2010; Onoderaetal. 2010;
numberoffundamentalcharacteristicsoftheSFprocess.Wevali-
Liuetal. 2011). This failure of SF relations to describe small
datethemethodusingMonte-Carlomodelsofstar-formingregions
scalesmaypotentiallyprovideabetterunderstanding of galactic-
in galaxies. In a follow-up paper (Kruijssenetal. 2014, hereafter
scale SF than the relations themselves do (e.g. Blancetal. 2009;
K14),weapplytheuncertaintyprincipletoobservationaldata.
Schrubaetal.2010;Calzetti,Liu&Koda2012;Leroyetal.2012,
Athttp://www.mpa-garching.mpg.de/KL14principle,wehave
2013).
madeFortranandIDLmodulesforapplyingtheuncertaintyprin-
SFrelationsshouldbeexpectedtobreakdownatsomepoint–
ciplepubliclyavailable.Achecklistdetailingtherequiredstepsfor
theyrelatethegasandradiationpropertiesoftheSFprocess,which
theirobservationalapplicationissuppliedinAppendixA.
onsomescaleshouldnolongercorrelatebecausetheycoversubse-
quentphasesofSF.Forinstance,agas-free,youngstellarclusteris
detectableinSFtracerssuchasHαorUVemission,butnotinthe
gastracersthatwerelikelyvisibleatanearlierstage.Thisexample 2 ANUNCERTAINTYPRINCIPLEFORSTAR
canbeexpressedintermsofthestatisticalsamplingoftheSFpro- FORMATION
cess–galacticSFrelationsaverageoverallphasesandimplicitly
2.1 Thegeneraluncertaintyprinciple
assumethateachphaseisstatisticallywell-sampled.
Inthispaper,wepresentasimpleuncertaintyprinciplethatis WeadoptthehypothesisthatSFrelationsonlyapplyabovecertain
similarinformtothefamouscriterionofHeisenberg(1927).Itcan spatialscalesbecauseonsmallerscalesthedifferentphasesofthe
be used to identify the spatial scale on which SF relations break SFprocessarestatisticallynotwell-sampled.Thismayleadtothe
down due to the incomplete sampling of the SF process, as well perception of episodicity or a deviation from these SF relations:
as to quantify the resulting scatter around and bias of such rela- givenanensembleofstar-formingregions,certainpermutationsof
tions.Afterintroducingourframework,weillustrateitsusewitha theirstateswillleadtoSFRshigherorlowerthanpredictedbySF
2 J. M. D.Kruijssen& S.N. Longmore
relations.Forinstance,ifallregionsintheensemblehappentobe thatissampledwithinatwo-dimensionalapertureofdiameter∆x
onthevergeofinitiatingSF,theywillhaveverylowSFRsfortheir isN =(∆x/λ)2.Theconditionthatthisnumberexceedsthe
indep
gas masses in comparison to the galactic average. The appropri- numberofindependentregionsrequiredtoretrievethegalacticSF
ate sampling of the phases of the SF process can be achieved by relation(N >N )thusyields:
indep indep,req
increasingthesampleofstates,orbysomehowfollowingthedif-
ferentphasesintime.Ifawell-sampledSFrelationisachievedby ∆x 2 > τ . (3)
coveringalargeareaorvolume,thisimpliesthatthesize-scale∆x λ ∆t
(cid:16) (cid:17)
onwhichtheSFrelationisevaluatedshouldsatisfy: Thiscanberewritteninthefamiliarformofanuncertaintyprinci-
plethatneedstobesatisfiedforgalacticSFrelationstohold:
∆x>Aλ, (1)
where A and λ represent a dimensionless constant and a length- ∆x∆t1/2>λτ1/2. (4)
scale,respectively,bothofwhicharespecifiedbelow.
Here, ∆x isthe spatial scale on which the SFrelations are mea-
Thefactthatasinglestar-formingregionisobservedataspe-
suredand∆tisthedurationoftheshortestphaseoftheSFprocess
cifictimeimpliesthatthatregionbyitselfcannotsatisfygalacticSF
thatistracedbytheSFrelationinquestion.Ifequation(4)issat-
relations.However,ifitwerefollowedintimeforthefullduration
isfied,thentheshortestphaseoftheSFprocessisexpectedtobe
of theSFprocess τ such thatall ‘relevant’ phases(i.e.thosethat
well-sampled (modulo Poisson noise), and hence the galactic SF
aretracedingalacticSFrelations)oftheSFprocessarecovered,
relationisretrieved. Giventhevaluesof allt ,thisdefinesthe
thenthetime-averagedpropertiesoftheregionshouldbeconsistent ph,i
minimumsize-scale:
withthegalacticSFrelations–providedthatthephysicsofSFare
universal.Specifically,toretrievetheSFrelationforasinglestar- τ 1/2
formingregion,werequirethetime-scaleoverwhichtheproperties ∆x>∆xsamp ≡ ∆t λ, (5)
oftheregionareaveragedtocoveratleastthetimeτ coveringall (cid:16) (cid:17)
whichspecifiestheconstantAinequation(1).Hence,(1)themore
phasesoftheSFprocesstracedingalacticSFrelations.Observa-
similarthevarioust areor(2)thesmallerthesizeisofinde-
tionally,thisconditioncanneverbesatisfied–wecannotobserve ph,i
pendent regions, the smaller the minimum size-scaleis on which
astellarclusteranditsprogenitor cloudatthesametime.Thisis
galacticSFrelationsstillhold.
whygalacticSFrelationsmustconsiderspatialscaleslargeenough
Next to the statistical sampling in time and space of the
tocovermultiplestar-formingregions.
full SF process, additional scatter is introduced by the incom-
Observationally,eachphaseoftheSFprocessisprobedwitha
plete sampling of the SF tracer at low SFRs (Leeetal. 2011;
differenttracer(e.g.COformoleculargas,HαforSF).Theemis-
Fumagalli,daSilva&Krumholz2011).Observationalestimatesof
sion from these tracers is observable for some fraction of the to-
the SFR almost exclusively rely on emission from massive stars,
talSFprocess,althoughanindividualobservationonlyretrievesa
which statistically may not be produced at low SFRs, leading to
snapshotatadiscretemomentintime.ThedetectionofagasorSF
tracerdoesnotdistinguishatwhichtimealongthephaseprobedby anunderestimationoftheSFR.IfwedefineaminimumSFRmin
abovewhichacertainSFtracerarisesfromanadequatelysampled
thattraceritisobserved.Forinstance,thegasmassthatwilleven-
tuallyparticipateintheSFprocessisvisibleforacertainduration IMF(seeTable1),agivenSFRsurfacedensityΣSFR impliesthat
aspatialscaleof
andcan bedetectedthroughout. If wenow consider anensemble
opfrosctaers-sf,otrhmeiknegyrceogniosneqsuaetnrcaendisomthlaytadigsatrliabcutitcedStFimreelsatailoonncgatnheonSlFy ∆x>∆xIMF ≡ 4SFRmin 1/2 (6)
π ΣSFR
be retrieved if the shortest phase of theSF process issampled at (cid:16) (cid:17)
leastonce.Inother words,thephase withtheshortest durationis isrequiredtoretrieveareliableSFRestimate.Here,thefactorof
thelimitingfactorinwhetherornotthegalacticSFrelationisre- fourentersbecause∆xrepresentsadiameterratherthanaradius.
trieved.WedefinethedurationoftheSFprocessas Finally,theabovelimitsonthespatialscaleonwhichgalactic
SFrelationsholdonlyapplyiftherelativefluxofgasandSFtracers
N N−1
acrosstheboundaryofanapertureisnegligibleoverthedurationof
τ = tph,i− tover,i,i+1, (2) theSFprocessτ.Anynetdriftwillintroducescatterorpotentially
Xi=1 Xi=1 systematic deviations. Given a characteristic drift velocity σ, the
where tph,i is the duration of phase i and tover,i,i+1 represents size-scaleshouldthereforesatisfy
theduration of theoverlap between phasesiand i+1. Defining
1
∆t ≡ min(tph,i) asthe shortest phase of theSF process that is ∆x>∆xdrift≡ στ, (7)
2
tracedingalacticSFrelations,theratioτ/∆tthenreflectsthenum-
berofindependentstar-formingregionsNindep,reqthatneedtobe wherethefactor of 1/2 arisesfromtakingtheexpected timedif-
sampledinordertoretrievethestatisticallyconverged,galacticSF ferencebetweentworegionspositionedatrandomtimesduringthe
relation.Weillustratethisresultwithanexample. IftheSFwere gas and stellar phases. In practice, the condition of equation (5)
toconsistoftwophasessuchthattph,1/tph,2 =9(i.e.90percent impliesthatmultipleindependentstar-formingregionsarecovered
of the time is spent in the first, e.g. molecular gas, phase), then inasingleaperture.Statisticallyspeaking, thefluxofgasandSF
∆t/τ =0.1andhenceNindep,req =10independentstar-forming tracersacrosstheapertureboundaryisthereforemuchsmallerthan
regionsneedtobecoveredtoretrievethegalacticSFrelation. acrossindividualstar-formingregionsandequation(7)shouldtyp-
Thesizeofan‘independent’regionreferstothelargestspatial icallybesatisfied.Thiswillbeillustratedin§3below.
scaleonwhichSFeventswithinthatregionarecorrelated,e.g.by Together,theabovethreeconditionspostulatethatgalacticSF
theglobalgravitationalcollapseofamolecularcloud,triggeredSF, relationsholdonsize-scales
or a galaxy-scale perturbation such as a merger. The number of
independentstar-formingregionswithacharacteristicseparationλ ∆x>∆xmax≡max{∆xsamp,∆xIMF,∆xdrift}. (8)
An uncertaintyprincipleforstarformation 3
Table1.Examplestarformationtracerproperties
Tracer ∆t50%a ∆t95%a htiluma SFRmin
(Myr) (Myr) (Myr) (10−3M⊙yr−1)
Hα 1.7 4.7 2 1b
FUV 4.8 65 14 0.04c
aFromLeroyetal.(2012).bFromKennicutt&Evans(2012).
cAssumingaFUVfluxcontributionfromstarsM >3M⊙.
belowaregeneralisedanddonotdependonthedetailedintensity
Figure1.Schematicrepresentationoftgas,tstar,tover andτ.Depending evolutionofthegasandSFtracers.Theoverlapcanonlybeignored
ontheadoptedgasandSFtracers(andtheirdetectability), tover canrefer if(1)itsdurationismuchshorterthanthatoftheindividualgasand
tothedurationofSFitself,thedurationofgasremoval,oracombination
SF tracers and (2) multiple independent star-forming regions are
ofboth(seetext).
coveredintheaperture.Weillustratetheeffectofanon-zeroover-
lapin§3.
For two popular SF tracers, Table 1 shows times at which
2.2 Specificationofkeyvariables
50and95percentofthetotalfluxhasbeenemittedforatypical
Thegeneral uncertainty principleof equation(8)should besatis- stellarpopulation(seeLeroyetal.2012),aswellastheluminosity-
fiedwhenevaluatinggalacticSFrelations,irrespectiveofthepre- weighted durations hti of each tracer and the minimum SFR
lum
cisechoice ofthevariables itdepends on. However, thepractical requiredforadequately samplingthetracedstarsfromastandard
applicationoftheprinciplerequiresthespecificationofthe(tracer- Chabrier(2003)IMF.Fortheapplicationsbelow,wesettstarequal
dependent)durationsoftheseveralphasesoftheSFprocesst , to the luminosity-weighted duration, although in practice other
ph,i
theminimumSFRsrequiredfortheiruseSFRmin,thetotaldura- choicesmay bemoreappropriate depending on theobservational
tionoftheSFprocessτ,andthecharacteristicspatialseparationof sensitivitylimits.
independentstar-formingregionsλ.Notethatthenumbersinthis Thedurationofthegasphaseisnotwell-constrained,andde-
sectionarestrictlyadoptedforthepurposeofillustration.Inprac- pendsonthespecificgastraceraswellasthegalacticenvironment.
tice,theycanbemeasuredfromtheobservationaldata(see§3and IfamoleculargastracerisusedinaMilkyWay-likegalaxy,aplau-
§4.3). sibledurationofitsdetectabilityisthedynamicaltime-scaleofthe
Intherestofthispaper,weillustratetheuseofouruncertainty host galaxy, whichsetsthetimeinterval betweenexternal pertur-
principlebyassumingthattheSFprocessastracedbygalacticscal- bationssuch ascloud-cloud collisionsor spiralarmpassages. By
ingrelationsconsistsoftwosteps,asisshownschematicallyinFig- contrast, if a tracer of atomic gas is used, then the condensation
ure1.Duringthefirstphase,thegastracercanbedetectedwhileno time-scaletothemolecularformenters.Inthefollowingexamples,
SFisseen,whereasduringthesecondphase,ayoungstellarpopu- weassumetheuseofamoleculargastracer,andsettgas = Ω−1,
lationisinplaceandthegashasbeenexpelledduetostellarfeed- where Ω ≡ V/R is the angular velocity at circular velocity V
back(oranyothermechanismthatleadsthegasphasetobecome and galactocentric radius R. In galaxy discs in hydrostatic equi-
undetectableintheadoptedgastracer).Thedurationoftheentire librium, this time-scale is similar to the free-fall time of GMCs
SFprocessisthereforegivenbyτ =tgas+tstar−tover.Here,tgas (e.g. Krumholz&McKee 2005). However, equilibrium is not al-
denotesthedurationofthefirstphase, i.e.thetimeforwhichthe wayssatisfied,andthereforeanalternativedefinitionwouldbeto
gasisvisibleintheadoptedtracerbeforeenteringtheSFprocess, usethetypical observed free-fall timeof GMCs. Asshown in§3
includingpossibleinterruptions,tstarrepresentsthedurationofthe and§4.3,ouruncertaintyprinciplecanactuallybeusedtoempir-
secondphase,andtover isthedurationoftheoverlapbetweenthe ically determine the time-scales during which gas tracers are de-
gasandstellarphases.Asaresult,∆t = min{tgas,tstar},andin tectable.
thefirstexamples weshall set tover = 0.Inreality, morephases In galactic discs, a good proxy for the typical separation of
mayexistandinevitablytover 6=0(see§3.3). independent star-forming regions λ is the Toomre length, which
ThemeaningoftoverdependsontheadoptedgasandSFtrac- foraflatrotationcurveisgivenby
ers.IfadiagnosticisusedthattracesSFonlyintheunembedded
πGΣ
state(suchasHα),thentovermaybedominatedbythetimeittakes lT= Ω2 , (9)
thestellarfeedbacktoremovethegasfromtheaperture(eitherbya
whereΣisthegassurfacedensity.Substitutingtheaboveexpres-
phasetransitionorbymotion).IfadiagnosticisusedthattracesSF
sionsforτ andλinequation(5),wecanspecifythespatialscales
fromtheonsetofSF,evenwhenitisstilldeeplyembedded(suchas
abovewhichgalacticSFrelationsapplyduetothestatisticalsam-
cmcontinuumoryoungstellarobjectcounts),thentoverreflectsthe plingofindependentstar-formingregions
durationofSFitselfplusthetime-scaleforgasremoval.Inevitably,
tthheeSpFretcriasceeornasnedttohfetthimeeo-veevrolaluptitohnenofdtehpeesntdasrfoonrmthaetisoennesiftfiivciietynctoy ∆xsamp = Ωm−1in+{Ωts−ta1r,t−sttaorv}er 1/2 πΩG2Σ, (10)
(orfeedback). Forseveraloftheexamplesbelow,weassumethat (cid:18) (cid:19)
andduetodrift
duringtheoverlapthegasandSFtracersarestatisticallydetected
at50percentoftheirnormalintensity,consistentwithalinearde- ∆xdrift= 1σ(Ω−1+tstar−tover), (11)
crease(gas)orincrease(stars).Higher-orderfunctionalforms(e.g. 2
Burkert&Hartmann2012)arelikelymoreaccurate,butthesetypi- whereσisassumedtobethegasvelocitydispersionmeasuredon
callyintroducecorrectionsofonlyafew10percent,whichjustifies the largest scale young stars are assumed to inherit the velocity
theuseofalinearform.However,notethattheequationsprovided dispersion of their natal cloud). Equation (6) does not require to
4 J. M. D.Kruijssen& S.N. Longmore
Table2.Adoptedpropertiesofidealisedgalaxiesandregions.
Galaxy/ Σ Ω−2 σ ΣSFR tgas lT ∆xsamp ∆xIMF ∆xdrift ∆xmax
region (M⊙pc−2) (100Myr)−1 (kms−1) (M⊙yr−1kpc−2) (Myr) (kpc) (kpc) (kpc) (kpc) (kpc)
SN 10 2.6 7 0.0063 38 0.21 0.94 0.45 0.14 0.94
CMZ 120 85 35 0.20 1.2 0.0023 0.0039 0.080 0.057 0.08
Disc 15 4.0 10 0.011 25 0.13 0.49 0.34 0.14 0.49
Dwarf 10 3.0 10 0.0063 33 0.16 0.66 0.45 0.18 0.66
SMG 103 30 50 20 3.3 0.16 0.26 0.0080 0.14 0.26
Thesolarneighbourhood(SN)valuesarebasedonaflatVc = 220kms−1rotationcurveandtheWolfireetal.(2003)gasmodel;theCentral
MolecularZone(CMZ)valuesaretakenfromKruijssenetal.(2013,230pc-averaged);thediscanddwarfgalaxyvaluesarebasedonLeroyetal.
(2008);thesub-mmgalaxy(SMG)valuesarebasedonGenzeletal.(2010).ExceptfortheCMZandtheSMG,theSFRsurfacedensitiesassume
aKennicutt(1998)SFrelation.Boldface∆xvaluesindicatethemaximaofequation(8).Inallcases,tstar =2Myrisadopted(seetext).
be specified further. The above equations show that the gas sur-
facedensityΣ,theSFRsurfacedensityΣSFR,therotationcurve,
andthegasvelocitydispersionσ mustbeknowntoapplyourun-
certaintyprincipleinthewayspecifiedhere.Galacticenvironments
otherthandiscs(suchasgalaxymergers)mayrequiredifferentdef-
initionsfortgas,τ andλ.
3 APPLICATIONTOIDEALISEDEXAMPLES
3.1 ThefailureofSFrelationsonsmallspatialscales
We now illustrate the use of the uncertainty principle quantita-
tivelywithanumberofarchetypicalgalacticenvironments,which
are listed in Table 2 together with their characteristic properties
andtheresulting∆x .Weconsiderthesolarneighbourhood(SN),
i
the Central Molecular Zone (CMZ) of the Milky Way, a ‘disc
galaxy’, a ‘dwarf galaxy’, and a starbursting ‘sub-mm galaxy’
(SMG). Throughout this section, we will assume the use of Hα Figure2.Exampleoftheminimumsize-scales ∆xi abovewhichgalac-
totraceSF.Adoptingtheluminosity-weightedduration,wedefine ticSFrelationsholdasafunctionofgalactocentric radiusinanidealised
tstar =2Myr(seeTable1). discgalaxy(seetext).Thereddashedlineshows∆xsamp,whichaccounts
forthestatistical samplingofthedifferentphasesoftheSFprocess.The
InallcasesotherthantheCMZ,thegalacticSFrelationbreaks
down first due to the incomplete sampling of independent star- blue dotted line shows ∆xIMF, which accounts for the sampling of the
high-mass(SF-tracing)endoftheIMF.Thegreendash-dottedlineshows
forming regions (i.e. below ∆xsamp), whereas in the CMZ the ∆xdrift,whichaccountsforthedriftofgasandyoungstarsacrosstheaper-
SFRistoolow toadequately sampletheSFtracer fromtheIMF tureboundary.Ateachradius,theblacksolidlineshowsthemaximumof
on scales lap < 80 pc. This is due to the low SFR measured in thesethreelimits,∆xmax.Theverticaldottedlinesindicatethescalera-
theCMZ(Longmoreetal.2013),buteveniftheCMZwereform- diusRsandtheopticalradiusR25 ∼ 5Rs(e.g.Leroyetal.2008),while
ing stars at a higher rate, the effect of drift would still be more thegreyboxindicatestheradiusintervalwheremostoftheSFoccursinan
important than the incomplete sampling of star-forming regions exponentialdisc(R=0.5–3Rs).
(i.e. ∆xdrift > ∆xsamp). The typical size-scales above which
galacticSFrelationsholdvaryfrom∼ 100pcintheCMZtoal-
mostakpcintheSN.Itcanbeinferredfromequation(10)thatthis σ = 10kms−1.Inthecentralkpc,thevelocitydispersionisas-
lattervalueisrepresentativeformuchofthepopulationofnearby, sumedtoriselinearlytoacentralvalue of σ = 50kms−1,and
star-forminggalaxies.ThesegalaxiessatisfyabroadtrendofΩ∝ therotationcurveisassumedtobesolid-body(i.e.Ωisconstant).
Σ0.5 with about 0.3–0.5 dex scatter (Krumholz&McKee 2005; TheKennicutt (1998) relationisused to translatethe gassurface
Kruijssen2012)becausetheysimultaneouslysatisfytheSchmidt- densitytoaSFRsurfacedensity.
Kennicutt and Silk-Elmegreen SF relations (Schmidt 1959; Silk Inthecentral500pc,thedriftofgasandyoung starsacross
1997;Elmegreen1997;Kennicutt1998).Substitutionofthistrend theapertureboundarysettheminimumscaleonwhichgalacticSF
intoequation(10)resultsinaroughlyconstant∆x∼1kpcforall relations hold (∆x ∼ 150 pc), which is consistent with the
drift
surfacedensities,withavariationof0.6–1.0dexasinTable2.Be- knownoffsetbetweenthedensegasand24µmsourcesintheCMZ
causethesize-scalesofGMCsandstellarclustersaremuchsmaller oftheMilkyWay(Yusef-Zadehetal.2009;Longmoreetal.2013).
(Longmoreetal.2014),itisclearthattheycannotsatisfygalactic However,theincompletesamplingofindependentstar-formingre-
SFrelations. gions sets ∆xsamp = 100–900 pc over the largest part of the
InFigure2,weillustratetheradialprofilesofthethreediffer- galaxy, in the radius range R = 1–9 kpc. At larger radii, the
ent∆xifromequations(5)–(7)aswellas∆xmaxforadiscgalaxy incomplete sampling of SF tracers from the IMF kicks in and
withalogarithmicpotential(V =200kms−1),anexponentialgas ∆xIMF>1kpc.AdirectimplicationisthatwhenusingSFtracers
surfacedensity profilewithcentral value Σ(0) = 200 M⊙ pc−2 that are visible over a longer age range (e.g. FUV), the IMF re-
and scale radius Rs = 2.5 kpc, and agas velocity dispersion of mainsproperlysampledouttolargerradii.Thisiswhydiscsdonot
An uncertaintyprincipleforstarformation 5
showthesametruncationintheUV(Thilkeretal.2007)aswhen scatter.Inthesimplecaseof tgas = tstar,thetransitionbetween
observed in Hα (Martin&Kennicutt 2001; also see Bigieletal. bothregimesoccursatlap =2–3λ.
2010).ConsideringthatthemajorityoftheSFinexponentialdisc The other terms of equation (12) will vary substantially be-
galaxiesoccursbetween0.5and3gasscaleradii,Figure2shows tween different tracers, galaxies, and specific observations. The
that∆xsamp moststronglyrestrictstheapplicationofgalacticSF scatter due tothe luminosity evolution of gaseous regions during
relationsonsmallscales. tgasandstellarregionsduringtstar dependsonitsparticularfunc-
tionalform.Likewise,thescatterduetothemassspectrumofthe
independentregionsalsodependsonitsdetailedcharacteristics.In
3.2 ThescatterofSFrelations
bothcases,thescatterdoesdecreasewiththeaperturesize,simply
Galactic SF relations are often characterised by defining the gas because thescatterof themeandecreases withthenumber of re-
depletiontime-scaletdepl ≡ Mgas/SFR,whichallowsthescatter gionssampled,i.e.σmean =σ1N−1/2,whereσ1isthescatterfora
oftheSFrelationtobequantifiedastheroot-mean-square(RMS) singleregion.Asstatedpreviously,thenumberofsampledregions
scatter of the depletion time-scale σlogt (e.g. Bigieletal. 2008; N ∝ (lap/λ)2 forlap ≫ λ,butforsmallaperturestheselection
Schrubaetal. 2011; Leroyetal. 2013). In the framework of the biasofrequiringthepresenceofbothtracerspreventsN < 1(if
uncertaintyprinciplepresentedin§2.1,thescatteroftheSFrela- tover 6=0)orN <2(iftover =0).Detailedexpressionsareagain
tionforanaperturewithdiameterlapisdeterminedbythePoisson providedinAppendixB,andareincludedinthepubliclyavailable
statisticsofthenumberoftimestheSFphasesaresampledwithin routines (see Appendix A). Weleave the scatter due tothe lumi-
theaperture,whichwillbedominatedbytheshortestphase.Addi- nosityevolutionandthemassspectrumforsingleregionsasafree
tionalsourcesofscatteraretheluminosityevolutionofthegasand parameter. Reasonable values are σevo,1g = σevo,1s = 0.3 dex
starsduringtgas andtstar,respectively, themassspectrumofthe fortheluminosityevolutionofasinglegaseousorstellarpeak,re-
independentregions,andtheintrinsicobservationalerror: spectively(cf.Leroyetal.2012),andσMF,1 ∼ 0.8dex,whichis
roughlyappropriateforapowerlawmassspectrumwithaslopeof
σl2ogt =σs2amp+σe2vo+σM2F+σo2bs, (12) −2over afactor of 40 in mass. Finally, the observational scatter
actsasaconstantlowerlimitoverallsize-scales,whichinpractice
whereσsampindicatesthePoissonerror,σevorepresentsthescatter
causedbytheluminosityevolutionofindependentregions,σMFis flattenstheσlogt–laprelationatlargeaperturesizes.
Inadditiontotheanalyticexpressionprovidedabove,wees-
thescatterduetothemassspectrum,andσ denotestheintrinsic
obs
observational error. The full derivation of these four components timate the scatter in logtdepl with a simple Monte-Carlo experi-
ment,inwhichwerandomlydistribute50,000pointsoveranarea
is presented in Appendix B and we describe their qualitative be-
haviourhere. suchthattheirmeanseparationisλ=130pcandpositioneachre-
gionrandomlyonthetimesequenceofFigure1,usingtime-scales
Forlap ≫ λ,thescatterduetothePoissonstatisticsofsam-
plingindependentregionsdecreaseswithaperturesizeasσsamp ∝ tgas = tstar andtover = 0.Wedonotincludeanypossibleevolu-
l−1, because the relative Poisson error of the number of regions tionofeachregion’sluminosityineithertracer,noraregionmass
ap
coveredinanapertureisσlnN =σN/N =N−1/2 =λ/lap.The functionoranintrinsicobservationalerror.Wethenrandomlyplace
50,000 apertures to measure the scatter in logt , only includ-
distributionof theseN regionsover gasand stellarregions isset depl
ingthoseaperturesthatincludenon-zerofluxforboththegasand
bythefractionstgas/τ andtstar/τ,respectively,implyingthatfor
SFtracers.Theresultingrelationbetweenthescatterandtheaper-
largelapwehave:
turesizeisshowninthetoppanelofFigure3(blackline),which
σlogt =α 1+min ttsgtaasr,ttsgtaasr 1/2 ∆xlampax, (13) ainlsgooinllluys1tr0a0teasptehratutrtehseivsamriiantoiorn(gorfeythaereoab)t.aAinteldarrgeelaatpioenrtuwrheesnizuess-,
(cid:20) (cid:18) (cid:19)(cid:21) thescatterdecreasesasσ ∝N−1/2 ∝l−1,whichisexpected
logt ap
whereα ≡ 1/ln10 ≈ 0.43convertsthelogarithmicscatterfrom forPoissonstatistics.Asexplainedabove,thescatteralsodoesnot
baseetobase10,theterminbracketsaccountsfortheconversion increaseindefinitelytowardssmallapertures,becauseaperturesnot
ofthescatterofasingletracertothecombinedscatterofbothtrac- includingbothtracersarediscarded.Fortheadoptedparameters(in
ersontdepl,andtheratio∆x/lapcountsthenumberofshortestSF particulartover =0),thisrequiresatleasttwoindependentregions
phases withinthe aperture.1 Ifthe apertures weretrulyrandomly tobepresentintheaperture.Theprobabilityoffindingmorethan
positioned,thisexpressionwoulddescribethescatterforallspatial thatdecreasesrapidlywhenlap ≪λ,implyingthatmostapertures
scales. However, the scatter does not keep increasing indefinitely that are not discarded have the same content of one gaseous and
towardssmallaperturesizes(lap <λ),becauseonlyaperturesthat onestellarregion.Asaresult,thescattergoestozeroforlap ↓0.
containboththegasandSFtracerareincluded–otherwisethegas The top panel of Figure 3 also shows that the analytic ex-
depletiontime-scalewouldbezeroorinfinity.Iftover =0,thisse- pressionofequation 12andAppendixB(reddashedline)agrees
lectionbiasimpliesthatverysmallapertureseachtypicallycontain verywellwiththeMonte-Carloexperiment.2 Thedecreaseofthe
thebareminimumofonegaseousregionandonestellarregion(the scatteratsmallaperturesizesvanisheswhenincludingσevo,1g =
probabilityofcatchingmoreisnegligible),whichcausesthescat- σevo,1s = 0.3 dex scatter (blue dashed line). Even when all
tertovanish. Thismeansthatatsmallaperturesizes,thePoisson apertures have the same content of (at least) one stellar and one
scatteractuallyincreaseswithaperturesizeratherthanthedecrease gaseous region, the evolution of the gas-to-stellar fluxratio leads
thatisseenatlap ≫ λ.Inthispartofthesize-scalerange,thelu-
minosityevolutionandthemassspectrumthereforedominatethe
2 Thesmalldiscrepancy atlap ∼ 2λisnotduetostatistical noiseand
1 Wehavetestedtheinfluenceofanon-homogeneousenvironment(e.g.an arisesbecause thederivation inAppendixBdoesnotinclude thecovari-
exponential discgalaxywithascaleradius Rs <∼ lap)andtheresulting ancebetweenthenumberofgaseousandstellarregions.Thecompleteex-
variationof∆xmaxwithintheaperture.Wefindthatequation(13)remains pressionisconsiderablymorecomplex,whichisundesirableinviewofthe
accuratetowithinafewpercentunderallphysicalcircumstances. satisfactoryaccuracyofthepresentedform.
6 J. M. D.Kruijssen& S.N. Longmore
massfunctionaffectsthegasandstellarfluxinthesamewayand
hencedoesnotintroduceadditionalscatter.Finally,includinganin-
trinsicobservationalerrormarginofσ =0.15dex(greydashed
obs
line)causesthescattertosaturateatσlogt =σobsforlap →∞.
For reference, Figure 3 includes the relation be-
tween the scatter and the aperture size that was found by
Feldmann,Gnedin&Kravtsov(2011)ingrid-basedhydrodynami-
calsimulationsof(disc)galaxies.Althoughadetailedcomparison
isobstructed by the somewhat arbitraryposition on the x-axis of
eachmodel (our models assumeλ = 130 pc), bothresultsagree
forlap > 300pc.Atsmalleraperturesizes,thecomparisonisnot
representative because in that regime the size-scale dependence
ofthescatter depends onthedetailsoftheunderlying luminosity
evolution and the adopted mass spectrum. Nevertheless, the
prediction that some flattening of the relation must occur below
size-scalesofafew100pcseemstoberobust.
Usingequation(12),wecannowpredictthescatteroftheob-
served gas depletion time-scale as a function of aperture size for
severalofourexamplesystemsfromTable2.Thisisshowninthe
bottompanelofFigure3forthediscgalaxy,dwarfgalaxyandSMG
parametersets.Thegalaxiesfollowroughlythesametrendofde-
creasingscatterwithaperturesize,butthereareseveralrelevantdif-
ferences.Forinstance,thebumpsandslightlywave-likebehaviour
iscausedbythedissimilarvaluesoftgasandtstarandtheresulting
increaseof∆x–thebumpvisibleatlap ∼500pcforthedwarfand
discgalaxiescoincideswith∆x = τ/∆tλ ∼ 3.9λ ∼ 500pc.
Fortgas ∼ tstar,thisbumpwould havemoved tolap ∼ λ,asis
p
thecasefortheSMGparameterset.Thedotted,dashedanddash-
dottedlinesshowhowthesize-scaledependenceofthescatterde-
Figure 3. Scatter in the gas depletion time-scale as a function of aper- pends on the luminosityevolution of individual regions and their
turesizelap.Toppanel:Theblacksolidlinereflectstheresultofasim- underlyingmassspectrum.Whilethevariationisnon-negligible,it
pleMonte-Carloexperimentusing50,000randomlyplacedapertures.The
clearlyrepresentsasecondaryeffect.Theoveralltrendisthatthe
grey area indicates the RMS scatter of the scatter when using only 100
apertures. The red dashed line represents the analytic model from equa- scattervariesfromσlogt ∼ 0.9atlap = 50pctoσlogt ∼ 0.2at
tion(12)andAppendix Bforσevo = σMF = σobs = 0,andthered lap =1kpc,indicatingaroughpower-lawrelationof:
dottedlineindicatesitspower-lawbehaviourforlap ≫λasexpressedin −0.5
equation(13).The{blue,green,grey}dashedlinessubsequentlyaddscat- σ ∼0.2 lap , (14)
ters of{σevo,1,σMF,1,σobs} = {0.42,0.8,0.15}dextoillustrate the logt (cid:18)kpc(cid:19)
effects ofthe luminosity evolution ofindependent regions, acloud mass
spectrum,andanintrinsicobservationalerror,respectively.Theblackdot- forlap = 0.05–1kpc.Notethatthedetailsofthisrelationareby
no means universal and should vary substantially between galax-
tedlineindicatestheresultfromFeldmann,Gnedin&Kravtsov(2011)for
aparticularnumericalsetup(seetext).Bottompanel:The{blue,green,red} iesduetovariationsinλ,tgas,tover,σevoandσMF.Whileaslope
solidlinesshowthe{dwarf,disc,sub-mm}galaxiesfromTable2.Wehave of −1 is expected for pure Poisson statistics, we see that a shal-
setσevo,1g =σevo,1s =0.3dex,σMF,1=0.8dexandσobs=0.15dex lowerslopecanemergeduetothecombinedeffectoftheflattening
asinthetoppanel.The{dotted,dashed,dash-dotted}linesrefertothedisc atsmallaperturesizesthatwasexplainedabove, theintrinsicob-
galaxymodelwithσMF,1 = 0dex,σMF,1 = 1.6dex,andσevo,1g = servationalerror,andthedissimilarityoftgas andtstar.Although
σevo,1s = 0 dex, respectively. The grey-shaded area indicates the part a detailed comparison with observations is deferred to K14, Fig-
of parameter space covered by nine galaxies from the HERACLES sur- ure3doesshow whichpart ofparameter spaceiscovered bythe
vey(Leroyetal.2013).ThefilleddiamondsrepresentM33(Schrubaetal.
observed nearby galaxies of Leroyetal. (2013), as well as M33
2010)andtheopendiamondindicatesM51(Blancetal.2009).
(Schrubaetal.2010)andM51(Blancetal.2009),indicatingthat
ourmodelagreesverywellwiththetrendoftheseobservations.
to residual variance and hence the scatter approaches σ2 =
logt
σe2vo,1g + σe2vo,1s for lap ↓ 0. Similarly, including a scatter of
σMF = 0.8 dex due to an underlying mass function (green 3.3 HowtheuncertaintyprincipleconstrainsSFphysics
dashed line) causes the scatter at small aperture sizes to saturate
Thusfar,wehaveassumedthataperturesarerandomlypositioned
at σ = σ2 +σ2 +σ2 /2. It also increases the
logt evo,1g evo,1s MF,1 on a galaxy. However, it is also possible to estimate the relative
scatteratlargeaperturesizes.Notethatthepresenceorabsenceof
amassfunctpion doesnot affectthescatterat smallaperturesizes change of themeasured gasdepletion time-scale asafunction of
aperturesizewhencenteringitonaconcentrationofgas(increasing
whentover 6= 0–thescatteratsmallaperturesizesisthendomi- t withrespecttothegalacticaverage)oryoungstars(decreas-
natedbysingleregionsresidingintheoverlapphase3,forwhicha depl
ing t with respect to the galactic average, see Schrubaetal.
depl
2010).Inparticular,weshowinthissectionthatthisrelativechange
3 Fortypicalparameters,thisismorelikelythancatchingonegaseousand (orbias)isaveryusefulquantitytoconstrainthetime-scalesgov-
onestellarregionindependentlyinasingleaperture. erningtheevolutionofgasandSFingalaxies.
An uncertaintyprincipleforstarformation 7
Therelativechangeofthegasdepletiontime-scalewhencen-
tering apertures on gas or stellar peaks can be estimated with a
simplestatisticalmodel.Thedepletiontimeisdefinedast ≡
depl
Mgas/SFR ∝ Fgas/FSF,whereFgas and FSF indicatetheflux
emittedbygasandSFtracers,respectively.Theexpectedfluxfrom
both tracers in apertures focussed on gasor stellar peaks follows
fromthe Poissonstatisticsof independent regions inapertures of
varyingsize.Theresultingfluxratiocanthenbecomparedtothe
galaxy-wide flux ratio to obtain the relative change of the mea-
suredgasdepletiontime-scale.Thederivationispresentedindetail
inAppendixCandonlytheresultisprovidedhere.
Bycenteringanapertureonagaspeak,thegasfluxisguar-
anteedtobenon-zeroandmayincreaseduetoadditionalgas-rich
regionsresidingintheaperturebychance–theexpectednumberof
these‘contaminants’increaseswiththeaperturesize.Bycontrast,
thestellarfluxcouldpotentiallybezero,becauseitisconstitutedby
thesumofthefluxemittedbystellarregionsresidingintheaperture
bychanceandthefluxemittedbythecentralgaspeakifithappens
tobeintheoverlapphase(whichcanonlyoccuriftover 6=0).The
relativechangeofthegasdepletiontime-scalethenbecomes
Figure 4. Expected relative change of the measured gas depletion time-
2 scale as a function of aperture size when centering on gas peaks (top
[tdepl]gas = 1+ tgτas laλp ,(15) curves) or stellar peaks (bottom curves) for several combinations of
[tdepl]gal βsttostvaerr 1+(βs−1)ttos(cid:16)tvaerr −(cid:17)1+ tgτas laλp 2 m{togvaasl,dtsutearto,tfoeveedrb}ac(kseoeccleugrseninds).taEntxacneepotuswlyhearnednhoetnecdeoththeeorwveisrlea,pgeaqsuarels-
(cid:2) (cid:3) (cid:16) (cid:17) the duration ofSF (tover = tSF). The thick black curves represent the
Analogously,foranaperturecenteredonastellarpeakwefind referencevariableset,withtgas = tstar andtover = 0.Theorangeand
redcurvesillustratetheeffectofanon-zerooverlaptime,thegreencurve
−1 2 showstheeffect oftheratio betweenthedurations ofthegasandstellar
[tdepl]star = βgttogvaesr 1+(βg−1)ttogvaesr + tsτtar laλp .(16) padhdassetsh,eanefdfetchteocfynanonc-uinrvsteanintadniceaotuessgthaescreommobvinaleddueeffetoctf.eTedhbeabclku,ewciuthrvea
[t ] h i 2 (cid:16) (cid:17)
depl gal 1+ tsτtar laλp SFtime-scaleoftSF=3Myrandacharacteristicgasremovalvelocityof
(cid:16) (cid:17) vej = 100kms−1.Theverticaldottedlinedenotestheadoptedtypical
separationbetweenregionsλ = 0.13kpc(cf.Table2).The(asymptotic)
Ithnethmeeseaneqguaastfliounxs,oβfgpe≡akFsgin,otvheer/oFveg,rilsaopinFdgi,coavteersatnhdetrhaetiombeeatnwfleuenx fivagluureesirneagcrheeyd. Afosrlcaapn↓be0vaerreifiineddicbaytesdubtoswtitaurtdinsgtheeqruigathiot-nha(n5d)sinidtoeoefquthae-
of those in isolation Fg,over. Likewise, βs ≡ Fs,over/Fs,iso in- tions(15)and(16),thebiasoftdeplneverexceedsafactoroftwoaslong
dicates the same ratio for stellar fluxes. These flux ratios can be aslap>∆xsamp.
directlymeasuredfromobservationsifthespatialresolutionallows
thesmallestaperturestocontainonlyasingleregion(i.e.lap <λ).
Byonlyconsideringthesmallestapertures,onecanthenobtainβg thatonlydetectsunembeddedstars,thentSFmustbeclosetozero
andβsbydividingthemeanfluxinaperturescontainingbothtrac- andhencethedurationoftheoverlapismainlysetbygasremoval.
ersbythemeanfluxinthosecontainingonlyasingletracer.Ifthe InFigure4,weshowtherelativechangeofthegasdepletion
spatial resolution isinsufficient, some parametrization of theflux time-scaleresultingfromequations(15)and(16)fordifferentcom-
evolutionneedstobeassumed.Forinstance,ifthegas(stellar)flux binations of tgas, tstar and tover. WhileFigure3 already showed
decreasestozero(increasesfromzero)linearlyduringtheoverlap substantialscatteronscalessmallerthanafew100pc,wenowsee
and is constant otherwise, then βg = βs = 0.5. The advantage that centering an aperture on an overdensity of gas or stars sys-
ofmeasuringthefluxratioratherthanadoptingsomeparametriza- tematicallybiasesthegasdepletiontime-scalebyuptoanorderof
tion of the flux evolution is that equations (15) and (16) become magnitudeormore.Thisiscausedbecausefocussingonacertain
independentofanypriorassumptions. tracerguaranteesittobepresentintheaperture,whichleadstoa
Inequations(15)and(16),thenumber1inthenumeratoror different depletion time-scalethan measured on average through-
denominator indicates the guaranteed gas or stellar peak, respec- out the galaxy. If tgas = tstar, this occurs symmetrically around
tively. The terms containing (lap/λ)2 represent the gas and stel- thegalacticvalueofthedepletiontime,butinallothercasesthere
lar peaks residing in the aperture by chance. If lap ≫ λ, then existsanasymmetrybetweenthecurvesfocussingongasorstars,
both equations approach unity and the bias of the gas depletion which depends on the ratio tgas/tstar. This is easily understood,
time-scalevanishes. Finally,theterminthedenominator (numer- becausethebiasiscausedbytheguaranteeofhavingagaseousor
ator) containing βs (βg) reflectsthe non-zero probability of find- stellarpeak–ifthevisibilitytime-scaleofeithertracerexceedsthat
ing stars (gas) in the central gas (stellar) peak in case the gas oftheother,itwillalsobemorenumerousatagiveninstant.There-
and stellar phases overlap (i.e. tover 6= 0). As explained in §2.2, fore, the guarantee of including at least one region bright in that
tover encompassesthedurationofSFtSFaswellasthetime-scale tracer will change the depletion time-scale by an amount smaller
for the removal of gas from the aperture or region by feedback thanwhenfocussingontheshorter-livedtracer.
tfb =min(lap,λ)/vej,wherevejisthecharacteristicremovalve- If the gas and SF tracers never overlap (tover = 0), the de-
locityofthegas,beitbyaphasetransitionorbymotion.Together, pletiontime-scaleinthelimitlap ↓ 0goestoinfinity(zero)when
thisyieldstover = tSF +min(lap,lT)/vej.Whenusing atracer focussingongaseous(stellar)peaks.Bycontrast,anon-zeroover-
8 J. M. D.Kruijssen& S.N. Longmore
lapbetweenthegasandstellarphases(i.e.thedurationofSFand
gasremoval) introduces aflatteningof thecurves inFigure4for
aperture sizes smaller than the typical separation of independent
regions. When both phases overlap for some duration, then there
isalwaysanon-zeroprobabilitythatbothtracersarepresentinthe
aperture,evenwhenonlyasingleregioniscovered.Thisprevents
thedepletiontime-scalefromapproachingzeroorinfinityforsmall
aperturesizes. Inparticular, whenfocussing ongaspeakswesee
thatthebiasoft saturatesatavalueof
depl
lim [tdepl]gas =1+β−1 tstar −1 , (17)
lap↓0 [tdepl]gal star tover
(cid:16) (cid:17)
whereaswhenfocussingonstellarpeaksitapproaches
lim [tdepl]star = 1+β−1 tgas −1 −1. (18)
lap↓0 [tdepl]gal gas tover
h (cid:16) (cid:17)i
Becausetstarisknownfromstellarpopulationmodelling,thisim-
pliesthattherelevanttime-scalesoftheSFprocesscansimplybe
readofffigureslikeFigure4.Thisisapotentiallyverypowerfulap-
plicationofourframework,whichisdiscussedinmoredetailbelow
andinAppendixA.Figure4alsoshowsthattheinclusionofafinite
gasejectionvelocitycausestheflatteningforsmallaperturestobe-
come more gradual, which occurs because the non-instantaneous
Figure 5. Comparison of the gas depletion time-scales seen in a simple
removal of thegasincreases theduration of theoverlapfor large Monte-Carloexperiment(seetext)tothoseoftheanalyticexpressionsof
aperturesizes(seeabove). equations(15)and(16),centeringtheaperturesongaspeaks(topcurves)
Notethatthetime-scalesobtainedthroughthismethodshould orstellarpeaks(bottomcurves).Topleft:Thereddashedlineindicatesthe
beinterpretedcarefully.ThetimesequenceofFigure1followsthe analyticexpression.Thegreyareasindicatetheuncertaintyrangeforsmall
massflowofthegastowardsandthroughSF,i.e.itis‘Lagrangian’. numbersofapertures (seelegend) asobtained fromtheMonte-Carlo ex-
BecausenotallofthegasisconsumedintheSFprocess, amass periment. Topright:Asintheprevious panel, butassigningapower-law
massspectrumtotheregions,coveringtwodecadesinmasswithaslope
unitlikelycompletesthesequenceofFigure1multipletimes,either
of−1.7.Bottom left: Effect of the spatial randomisation, comparing the
byonlyperipherallyparticipatingintheSFprocessorbyactually
analyticexpression(reddashedline)totheMonte-Carloexperimentfora
formingastarandsubsequentlybeingejectedbyfeedback.Inthis
hexagonalequidistantgrid(green),foranadditionalrandomscatterupto
context,tgasreflectsthetotaltimeinbetweensubsequentSFevents 0.5λ(blue),andforafullyrandomdistribution(black).Bottomright:Ef-
during which the mass unit is visible in the gas tracer. This also fectofincompleteness,comparingtheanalyticexpression(reddashedline)
meansthattgasmayspaninterruptionsduetophasetransitionsun- totheMonte-Carloexperiment(black),usingthesamemassspectrumasin
relatedtoSF.Whilethedurationoftheseinterruptionsthemselves thetoprightpanel.Inthecaseof‘coupledlimits’,thegasandstellarmass
doesnotcontributetotgas,anypriorvisibilityofamassunitinthe spectra are undetected below twice their minimum mass, whereas in the
adoptedgastraceriscontainedintgas. caseof‘independent limits’thegas(stellar)massspectrumisundetected
As a first test of equations (15)–(18), we have performed a belowfive(three)timestheminimummass.Dashedlinesindicatetheana-
set of Monte-Carlo experiments very similar to those discussed lyticmodelwhensettingtgas,tstar andtover tothemeantime-scalesfor
whichthetracersaredetectable(ordetectablyoverlapping).
in §3.2. We randomly distribute 20,000 points over an area such
that their mean separation is λ = 130 pc and position each re-
gionrandomlyonthetimesequenceofFigure1,usingtime-scales
{tgas,tstar,tover}={10,5,3}Myr.Thegas(stellar)luminosityis the model to interpret observational data – for larger numbers of
takentodecrease(increase)linearlyduringtheoverlapphasewhile apertures, theory and simulation converge to high accuracy. The
remainingconstant whenonlyasingletracerispresent,implying scatter increases by about afactor of twowhen including amass
βg = βs = 0.5.Wethenplaceaperturesofdifferentsizes,which spectrum(top-rightpanel),to∼ 0.2dex,suggestingthattheanal-
are focussed on each of these regions. The entire gas and stellar ysisproposedhererequiresaminimumofabout 100aperturesto
fluxforthesubsetsofaperturescenteredongasandstellarpeaks, yield statisticallyuseful results. Note that the applicability of the
respectively,arethenaddeduptoobtainthebiasofthegasdeple- analyticexpressionsisunaffectedbythepresenceofamassspec-
tion time-scale as a function of aperture size. In addition to this trum.Thethirdpanel inFigure5addresses theassumptionmade
standard model, insome cases weassign a mass spectrum tothe thusfarthatindependent regionsarerandomlydistributed.Ifthey
regions, account for detection limits, or consider different spatial are distributed on an equidistant, hexagonal grid with inter-point
distributionsofthepoints. separationλ,thefamiliarsaturationofthedepletiontime-scaleat
Figure 5 shows the comparison between the Monte-Carlo smallaperturesizesisalreadyattainedatλ–atsmallersize-scales
modelandtheanalyticexpressionsofequations(15)and(16).The there is never more than a single region residing in the aperture.
valuesofβg andβs usedintheanalyticexpressionsaremeasured Theaddition of random perturbations tothisfixed distributionof
fromtheMonte-Carlomodelastheywouldbedeterminedfromob- pointsshiftsthesaturationpointto0.5λ.Ineithercase,thesatura-
servedgalaxies(seetheearlierdiscussion)–theothervariablesare tionvalueisstillareliablemeasureoftgasandtover.
simply set according to the initial conditions of the Monte-Carlo Thebottom-rightpanelofFigure5considerstheeffectofin-
model. The top-left panel shows that when using only 100 aper- completeness–thepartialdetectionofthemassspectraofgaseous
tures,anuncertaintyof∼ 0.1dexshouldbeexpectedwhenusing and stellar regions. Unsurprisingly, the bias of the gas depletion
An uncertaintyprincipleforstarformation 9
time-scale changes when only part of the mass spectrum is de- the inference of SF physics from galactic SF relations (see e.g.
tected.Thisgivesincorrectresultswhenusingthemodeltoderive Kennicutt&Evans 2012), because it relies exclusively on tracer
tgas and tover from observations. It makes no difference whether fluxratios.
thedetectionlimitsofthegasandSFtracersarecoupled(i.e.the
Fortran and IDL modules for applying the uncertainty principle
gas limit turns into the stellar limit and the number of regions
are available at http://www.mpa-garching.mpg.de/KL14principle.
is conserved between both tracers) or independent. However, we
Achecklistdetailingtherequiredstepsfortheirobservationalap-
alsoshowtheanalyticmodelwhenadoptingthemeandetectability
plicationissuppliedinAppendixA.
time-scalesof the gasand SFtracersrather than their underlying
lifetimes,aswellasthemeandurationoftheirdetectableoverlap
(dashedlines).TheseagreewellwiththeMonte-Carloexperiment,
4.2 Assumptions,observationalcaveatsandbiases
indicating that the time-scales that are obtained from incomplete
observations refer to the time-scales during which thetracers are ThestatisticalargumentsusedtoderivetheexpectedscatterinSF
detectable. When the observations are incomplete, this naturally relations rely on several implicit assumptions. If this theoretical
differsfromtheunderlying,truelifetimesofthegasandSFtracers framework is applied in regimes where these assumptions break
andtheiroverlap. down,thederivedscatterwillvaryfromthatpredicted.Forexam-
We conclude that measurements of the gas depletion time- ple,wehaveassumedthatthegalacticSFRisroughlyconstantover
scaleinsmallaperturescenteredongasorSFtracersdirectlyprobe τ,which willbreak down ina localised‘starburst’ event.4 Alter-
thedurationoftheseveralphasesoftheSFprocess.Thismayopen natively, if the physical properties of a galaxy vary substantially
upanewavenuetoinferthephysicsofSFandfeedbackasafunc- within a given observational aperture, the characteristic size and
tionofthegalacticenvironment. mass of independent regions may also change, potentially giving
risetoadditionalscatter.Also,theaboveframeworkhasbeende-
finedunderthesimplestassumptionthatthegalaxyisface-on.As
inclination will directlyaffect several key variables (e.g. the pro-
4 DISCUSSION
jectedaperturearea,gas/starsurfacedensity, rotationcurves), the
4.1 Summary deprojectedvaluesshouldbeused.
Ourframeworkalsoassumesthatobservationsaccuratelyre-
Wehavepresentedasimpleuncertaintyprincipleforspatiallyre-
cover the full underlying gas and young star distributions. In the
solvedgalacticSFrelations.Thisexplainsthefailureoftheserela-
idealisedcasewhereallindependentstar-formingregionshavethe
tionsonsmallspatialscalesastheresultoftheincompletestatis-
samecharacteristicmass(e.g.theToomremassMT =σ4/G2Σ),
ticalsamplingofindependentstar-formingregions.Themaincon-
the observations need to be sufficiently sensitive to both (1) de-
clusionsofthisworkareasfollows.
tect thischaracteristic gas mass and (2) detect the stellar popula-
(i) Throughoutmostoftheknownstar-formingsystems,thein- tioneventuallyresultingfromthisgas(e.g.theToomremasstimes
completesamplingofindependentstar-formingregionsdetermines someSFefficiency).However,inpracticethegasmassdistribution
thespatialscale∆xbelowwhichgalacticSFrelationsbreakdown. willbecontinuousandhencesensitivitylimitationsmeanthatob-
Itprovidesamorestringentcriterionthantheincompletesampling servationswillonlydetectemissionfromafractionofthegasand
ofSFtracersfromtheIMF(whichdominatesintheouterregionsof youngstars.Wehaveshownin§3.3thatthisaffectsthetime-scales
galaxydiscs)orspatialdrift(whichdominatesingalaxycentres).If that are retrieved when considering the bias of the gas depletion
theToomrelengthsetstheseparationofindependentstar-forming time-scaleinsmallapertures.Ideally,thegasandSFtracerobser-
regions,wepredictthatthereshouldbelittlevariationof∆xasa vationsarecomplete,butatleasttheyshouldencompassthesame
functionofthegalacticgassurfacedensity. totalfractionofgascloudsandresultingyoungstellarpopulations
(ii) ThePoissonstatisticsofsamplingindependentstar-forming theyproduce. Arobust applicationofourframeworkrequiresthe
regions cause the scatter of the gas depletion time-scale of the recovered fraction to be large. The requirement that observations
spatially resolved SF relation to depend on the aperture size as besensitivetoasimilar,substantialfractionofthegasandyoung
σlogt ∝ la−pγ withγ = 0.5–1 forlap = 0.1–1 kpc.Theincrease starsbeingproducedeffectivelyplacesadistancelimitontheap-
ofthescatterwithdecreasingsize-scaleflattensforsmallaperture plicabilityofthisanalysis.Theseandotherobservationalpointsof
sizes, where it isdominated by the cloud mass spectrum and the cautionarediscussedinmoredetailinAppendixA2.
detailsoftheluminosityevolutionduringtheSFprocess.Wefind Onefinal thingtobear inmind, isthat our uncertainty prin-
goodagreementwiththeobserveddependenceofthescatteronthe ciple mainly considers the statistics of observational tracers of a
spatialscale. physical system rather than directly describing that system itself.
(iii) Whenfocussingaperturesongasorstellarpeaks,themea- As described in §3.3, the physics of SF can be characterised by
sured gas depletion time-scale is biased to larger or smaller val- consideringtheireffectsonthesestatistics.
ues,respectively.Thisbiasdirectlyprobesthetime-scalesgovern-
ing the SF process, such as the duration of the gas phase and its
timeoverlap with thestellar phase. Thesetime-scales can be ob- 4.3 Implicationsandfutureapplications
tainedfromgalaxy-wideobservationswithouttheneedtospatially
The uncertainty principle presented in this paper shows that care
resolve independent star-forming regions – resolving their mean
must be taken when comparing galactic SF relations to small-
separationsuffices.SimpleMonte-Carlomodelsoflargenumbers
scale SF relations and provides quantitative limits on their appli-
ofstar-formingregionsshow thatthemethod isinsensitivetothe
cloudmassspectrumandcanbeappliedreliablywhenatleast100
gas or stellar peaks are used. Another important strength of this 4 Intheframeworkofthispaper,thecloudsandregionsinagalaxy-wide
methodisthatitisnothamperedbytheuncertainconversionfac- starburstarenolongerindependent, whichimplies that theentire galaxy
torsbetweengastracerfluxandgasmassthattraditionallyplague actuallyconstitutesasingleindependentregion.
10 J. M.D. Kruijssen&S. N. Longmore
cability. For instance, SF relations measured in the solar neigh- tailed comments on the manuscript, and to Frank Bigiel, Simon
bourhood (Heidermanetal.2010;Lada,Lombardi&Alves2010; Glover,AdamLeroy,AndreasBurkertandRalfKlessenforstimu-
Gutermuthetal.2011;Ladaetal.2013)arefundamentallydiffer- latingdiscussionsand/orcommentsonanearlydraftofthiswork.
ent from their galactic counterparts. Small-scale SF relations de- We acknowledge the Aspen Center for Physics for their hospi-
scribetheconversionofdense(andlikelyself-gravitating)gasinto tality and to the National Science Foundation for support, Grant
stars, whereas large-scaleSF relationsadditionally cover galactic No.1066293.
physics such as feedback, cooling and inflow dynamics. Despite
this added complexity, the clear advantage is a better statistical
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