What FIREs Up Star Formation: the Emergence of the Kennicutt-Schmidt Law from Feedback

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by  Matthew Orr

MNRAS000,1–14(2016) Preprint10January2017 CompiledusingMNRASLATEXstylefilev3.0
What FIREs Up Star Formation: the Emergence of the
Kennicutt-Schmidt Law from Feedback
M. E. Orr1(cid:63), C. C. Hayward2,3,1, P. F. Hopkins1, T. K. Chan4,
C.-A. Faucher-Gigu`ere5, R. Feldmann6, D. Kereˇs4, N. Murray7, and E. Quataert6
1TAPIR, Mailcode 350-17, California Institute of Technology, Pasadena, CA 91125, USA
2Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA
3Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA
7 4Department of Physics, Center for Astrophysics and Space Science, University of California at San Diego, 9500 Gilman Drive,
1 La Jolla, CA 92093, USA
0 5Department of Physics and Astronomy and CIERA, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA
2 6Department of Astronomy and Theoretical Astrophysics Center, University of California, Berkeley, CA 94720-3411, USA
7Canadian Institute for Theoretical Astrophysics, 60 St George Street, University of Toronto, ON M5S 3H8, Canada
n
a
J
Draftdate:10January2017
7

]
A ABSTRACT
We present an analysis of the global and spatially-resolved Kennicutt-Schmidt star
G
formation relation in the FIRE (Feedback In Realistic Environments) suite of cosmo-
h. logical simulations, including halos with z = 0 masses ranging from 1010 – 1013 M(cid:12).
p WeshowthattheKennicutt-Schmidt(KS)relationemergesrobustlyduetotheeffects
- of feedback on local scales, independent of the particular small-scale star formation
o
prescriptions employed. This is true for the KS relation measured using all of the gas
r
t and using only the dense (molecular) gas. We demonstrate that the time-averaged
s
KS relation is relatively independent of redshift and spatial averaging scale, and that
a
[ the star formation rate surface density is weakly dependent on metallicity ( Z1/4).
Finally, we show that on scales larger than individual giant molecular clo∝uds, the
1
primary condition that determines whether star formation occurs is whether a patch
v
of the galactic disk is thermally Toomre-unstable (not whether it is self-shielding):
8
onceapatchcannolongerbethermallystabilizedagainstfragmentation,itcollapses,
8
becomes self-shielding, cools, and forms stars.
7
1
Key words: galaxies: star formation – instabilities – opacity – galaxies: evolution –
0
methods: numerical.
.
1
0
7
1
1 INTRODUCTION 2010). Understanding what regulates the efficiency of star
:
v formation and results in the observed Kennicutt-Schmidt
Xi Understanding star formation and its effects on galactic relationisthereforekeytounderstandingtheformationand
scaleshasbeenintegraltoassemblingthestoryofthegrowth
dynamics of galaxies. Some authors (e.g. Thompson et al.
r and subsequent evolution of the baryonic components of
a 2005; Murray et al. 2010; Murray 2011; Ostriker & Shetty
galaxies.Observationally,therateatwhichgasisconverted
2011;Faucher-Gigu`ereetal.2013;Hayward&Hopkins2015;
into stars is characterized by the Kennicutt-Schmidt rela-
Semenovetal.2016)arguethatstarformationislocally ef-
tion, which is a power law correlation between the star for-
ficient, in the sense that tens of per cent of the mass of a
mation and gas surface densities in galaxies that holds over
gravitationallyboundgasclumpwithinaGMCcanbecon-
severalordersofmagnitude(Schmidt1959;Kennicutt1998;
verted into stars on the local free-fall time, and that local
see Kennicutt & Evans 2012 for a recent review).
stellar feedback processes – including supernovae, radiation
Numerous studies of the Kennicutt-Schmidt relation
pressure,photoheatingandstellarwinds–muststabilizegas
have shown that star formation is inefficient on galactic
discsagainstcatastrophicgravitationalcollapse,therebyre-
scales, with only a few per cent of a galaxy’s gas mass be-
sulting in the low global star formation efficiencies that are
ing converted to stars per galactic free-fall time (Kennicutt
observed. However, others claim that star formation is lo-
1998;Kennicuttetal.2007;Daddietal.2010;Genzeletal.
cally inefficient, with only of order a few per cent of the
massofclumpsbeingconvertedintostarsonafree-falltime
(cid:63) E-mail:[email protected]
(cid:13)c 2016TheAuthors
2 M. E. Orr et al.
independentoftheirdensity(Krumholz&Tan2007;Padoan 2 2
1995). 4
In either scenario, the Kennicutt-Schmidt law is con- 1 1
sidered to be an emergent relation that holds on galactic
0
scales and results from a complex interplay of the phys- 2 0
ical processes that trigger star formation and those that 1
−
regulate it. It has also been argued and observed that the c] −1
Kennicutt-Schmidtrelationbreaksdownbelowsomelength- kp 0 2
[ −
and time-scales (Onodera et al. 2010; Schruba et al. 2010; y 2
−
Feldmann & Gnedin 2011; Calzetti et al. 2012; Kruijssen 3
−
& Longmore 2014). Calzetti et al. (2012) found that the 2 3
− 4 −
Kennicutt-Schmidt relation breaks down due to incomplete −
sleanmgpthlinsgcaolfesstoafr-lfeosrsmtihnagnm∼ole1cuklpacr.cFloeulddms’amnanssetfuanlc.t(i2o0n1o2n) 4 −5 −4
−
claim that this breakdown on sub-kpc scales occurs due to 6 5
the stochastic nature of star formation itself. Furthermore, −4 −2 0 2 4logΣHI−+H2logΣ˙−(cid:63)
Kruijssen&Longmore(2014)arguethatthevarioustracers x[kpc]
ofgascolumndensityandstarformationratesurfacedensity
Figure 1. Example of one of our maps, made from a Milky-
requireaveragingoversomespatialandtemporalscales;con-
Waymasssimulatedgalaxyatz≈0(galaxym12ifromHopkins
sequently, when sufficiently small length scales are probed,
etal.2014),with100pcpixels.Neutralhydrogensurfacedensity,
a tight correlation between the star formation rate surface ΣHI+H2 [M(cid:12) pc−2] and instantaneous gas star formation rate
densityandthegascolumndensityshouldnotbeobserved. Σ˙(cid:63) [M(cid:12) yr−1 kpc−2]arecoloredinbluesandreds,respectively.
UnderstandingthescaleswheretheKennicutt-Schmidtlaw Spiral arms and increasing density towards the galactic core are
holdsthereforeinformsourtheoriesofstarformationaswell. clearly visible, and the instantaneous star formation rate is seen
OnthelengthscaleswheretheKennicutt-Schmidtrela- tocloselytracethedensestgasstructures.
tiondoeshold,thecanonicalpowerlawofthetotalgasrela-
tionisΣ ∝Σ1.4 withΣ beingthestarformationrate
SFR gas SFR
surface density and Σ being the total gas surface density
gas
(Kennicutt1998).However,therehasbeenmuchdebatere-
gardingthepower-lawindexoftherelationanditsphysical becomessteeperowingtothemetallicitydependenceofthe
origin;thepreviousliteraturehasfoundKennicutt-Schmidt gas column needed to self-shield molecular gas.
relations ranging from highly sublinear to quadratic (Bigiel Large-volume cosmological simulations often use the
etal.2008;Daddietal.2010;Genzeletal.2010;Feldmann Kennicutt-Schmidt law as a sub-grid prescription for star
et al. 2011, 2012; Narayanan et al. 2012; Shetty et al. 2013, formation, both because of the prohibitive computational
2014a,b;Becerra&Escala2014).Someofthedisagreement complexity of including all of the physics relevant on the
owestotheparticularformulationoftheKennicutt-Schmidt scales of star-forming regions, and their inability to resolve
relation considered, such as whether Σ (total atomic even the most-massive giant molecular clouds ∼ 106 M
HI+H2 (cid:12)
+ molecular hydrogen column) or Σ (molecular column (e.g. Mihos & Hernquist 1994; Springel & Hernquist 2003).
H2
alone) is employed (e.g. Rownd & Young 1999; Wong & Even idealized disk simulations that have resolution on the
Blitz2002;Krumholz&Thompson2007),withtheΣ rela- order of 100 pc, but are unable to resolve a multiphase
H2
tiontypicallyhavingslope∼1.Therelationmayinprinciple ISM, employ star formation prescriptions that assume low
alsodependonthestarformationtracerused(e.g.Hα,far- starformationefficienciesa priori orimplementKennicutt-
infrared,orultraviolet).Furthermore,therearequestionsas Schmidt laws indirectly (Schaye & Dalla Vecchia 2008). It
to whether the index depends on spatial resolution – even has been shown that assuming a power-law star formation
on scales larger than the length scale below which the re- relation on the resolution scale can imprint a power-law re-
lation fails altogether – or if there are multiple tracks to lation of identical slope on the galactic scale (Gnedin et al.
the Kennicutt-Schmidt relation, each with different slopes 2014), demonstrating the importance of employing physi-
acrossseveraldecadesingassurfacedensity(Ostrikeretal. cally motivated sub-grid star formation prescriptions that
2010; Liu et al. 2011; Ostriker & Shetty 2011; Feldmann produce kpc-scale relations with the ‘correct’ slope if the
et al. 2011, 2012; Faucher-Gigu`ere et al. 2013). relevantphysicalprocessescannotbetreateddirectly.With
It has also been suggested that the Kennicutt-Schmidt advancesincomputingpower,andtheabilitytoexecutein-
relation may evolve with redshift or have a metallicity de- creasinglycomplexsimulationswithmorephysicsathigher
pendence (Bouch´e et al. 2007; Papadopoulos & Pelupessy mass resolution, simulations have only recently been able
2010; Gnedin & Kravtsov 2011). These are not entirely in- to predict the Kennicutt-Schmidt relation generically as a
dependent quantities, as metallicity generally increases as result of the physics incorporated in the simulations at the
galaxies process their gas through generations of stars over scales of molecular clouds (e.g. Hopkins et al. 2011, 2013a,
cosmictimescales.Becausethepresenceofmetalsresultsin 2014; Agertz & Kravtsov 2015). Simulations with realistic
moreefficientgascooling,andcanthusaidinthetransition feedbackphysicsthatresolveGMCscalesareabsolutelycrit-
from diffuse ionized and atomic species to dense molecular icaltounderstandingtheKennicutt-Schmidtrelationdueto
gas (Hollenbach & Tielens 1999), Krumholz et al. (2009b) itsemergentnatureandthemultitudeofcompetingphysical
havearguedthatthereisametallicity-dependentgassurface effects involved spanning a wide range of scales.
density cutoff below which the Kennicutt-Schmidt relation In this paper, we explore the properties and emer-
MNRAS000,1–14(2016)
Kennicutt-Schmidt on FIRE 3
genceoftheKennicutt-SchmidtrelationintheFIRE1(Feed- • The molecular fraction f is calculated as a function
H2
backInRealisticEnvironments)simulations(Hopkinsetal. of the local column density and metallicity using the pre-
2014).Specifically,byproducingmockobservationalmapsof scription of Krumholz & Gnedin (2011), and the molecular
thespatially-resolvedKennicutt-Schmidtlaw,weinvestigate gas density is used to calculate the instantaneous SFR (see
the form of the relation when considering several different below).
tracers of the star formation rate and gas surface densities, • Finally,weidentifyself-gravitatingregionsusingasink
and we characterize its dependence on redshift, metallic- particle criterion at the resolution scale, specifically requir-
ity, and pixel size. The FIRE simulations are well suited ing α ≡ δv2δr/Gm (< δr) < 1 on the smallest resolved
gas
for understanding the physical drivers of the Kennicutt- scale around each gas particle (δr being the force softening
Schmidt relation as they sample a variety of galactic envi- or smoothing length).
ronments and a large dynamic range in physical quantities
(chiefly, gas and star formation rate surface densities), and Regionsthatsatisfyalloftheabovecriteriaareassumedto
theydirectly(albeitapproximately)incorporatestellarfeed- have an instantaneous star formation rate of ρ˙ =ρ /t ,
∗ mol ff
backprocessesthatmaybecrucialfortheemergenceofthe i.e. 100 percent efficiency per free-fall time. As a large frac-
Kennicutt-Schmidt relation. tion of the dense (n > n ), molecular (f ∼ 1) gas is
crit H2
not gravitationally bound (α > 1) at any given time, the
global star formation efficiency (cid:15) is less than 100 per cent
((cid:15)<1)despitetheassumedlocal,instantaneous starforma-
2 SIMULATIONS & ANALYSIS METHODS
tion efficiency per free-fall time being 100 per cent. We will
In the present analysis, we investigate the star formation show below that the Kennicutt-Schmidt relation, with its
properties of the FIRE galaxy simulations originally pre- much lower global, time-averaged star formation efficiency
sented in Hopkins et al. (2014), Chan et al. (2015), and ((cid:15) (cid:46) 0.1), emerges as a result of stellar feedback prevent-
Feldmannetal.(2016),whichusedtheLagrangiangravity+ ing dense gas from quickly becoming self-bound and form-
hydrodynamicssolvergizmo(Hopkins2013)initspressure- ing stars and disrupting gravitationally bound star-forming
energy smoothed particle hydrodynamics (P-SPH) mode clumps on a timescale less than the local free-fall time. We
(Hopkins 2013). All of the simulations employ a standard stressherethattheemergentKennicutt-Schmidtrelationis
flat ΛCDM cosmology with h ≈ 0.7, Ω = 1−Ω ≈ 0.27, not a consequence of the star formation prescription em-
M Λ
and Ω ≈ 0.046. The galaxies in the simulations analyzed ployed in the simulations.
b
in this paper range in z ≈ 0 halo masses from 9.5×109 In Appendix A we demonstrate this explicitly. We ran
to 1.4×1013 M , and minimum baryonic particles masses severaltestsrestartingoneofthestandardFIREsimulations
(cid:12)
m of 2.6×102 to 3.7×105 M . For all of the simulations, with varying physics and star formation prescriptions. For
b (cid:12)
the mass resolution is scaled with the total mass such that anyreasonablesetofphysics,onlyvariationinthestrength
the characteristic turbulent Jeans mass is resolved. Conse- of the feedback affected the galactic star formation rates,
quently, the simulations are able to resolve a multiphase because the simulated galaxies self-regulate their star for-
ISM, allowing for meaningful ISM feedback physics. This is mation rates via feedback. A number of independent stud-
crucial because the vast majority of star formation occurs ies have also shown that once feedback is treated explic-
in the most massive GMCs due to the shape of the GMC itly,thepredictedKennicutt-Schmidtlawbecomesindepen-
mass function (Williams & McKee 1997). dent of the resolution-scale star formation criterion (Saitoh
Thestellarfeedbackphysicsimplementedinthesesimu- etal.2008;Federrath&Klessen2012;Hopkinsetal.2012b,
lationsincludeapproximatetreatmentsofmultiplechannels 2013c,a,b, 2016; Agertz et al. 2013).
of stellar feedback: radiation pressure on dust grains, su- To quantify the spatially resolved Kennicutt-Schmidt
pernovae(SNe),stellarwinds,andphotoheating;adetailed relationinthesimulations,weanalyzedatafromsnapshots
description of the stellar feedback model can be found in spanning redshifts z = 0 − 6. The standard FIRE snap-
Hopkinsetal.(2014).Starparticlesinthesimulationseach shots from Hopkins et al. 2014 and the dwarf runs in Chan
represent individual stellar populations, with known ages, et al. (2015) are roughly equipartitioned amongst redshift
metallicities, and masses. Their spectral energy distribu- bins z ∼3−6, 2.5−1.5, 1.5−0.5, and <0.5, whereas the
tions, supernovae rates, stellar wind mechanical luminosi- snapshots of halos from Feldmann et al. (2016) have red-
ties, metal yields, etc. are calculated directly as a function shifts evenly spaced between 2 < z < 6 (these were run to
of time using the starburst99 (Leitherer et al. 1999) stel- only z ∼ 2). To compare the snapshots with observational
lar population synthesis models, assuming a Kroupa (2002) constraintsoftheKennicutt-Schmidtrelation,wemadestar
IMF. formation rate and gas surface density maps of each snap-
Inthesesimulations,thegalaxy-andkpc-scalestarfor- shot’s central galaxy. We summed the angular momentum
mation efficiency is not set by hand. Star formation is re- vectors of the star particles in the main halo of each snap-
strictedtodense,molecular,self-gravitatingregionsaccord- shot to determine the galaxy rotation axis and projected
ing to several criteria: along this axis to generate face-on galaxy maps. The pro-
jected maps were then binned into square pixels of varying
• The gas density must be above a critical threshold, size, ranging from 100 pc to 5 kpc on a side. Only parti-
ncrit ∼ 50 cm−3 in most runs (and 5 cm−3 in those from cles within 20 kpc above or below the galaxy along the line
Feldmann et al. 2016). of sight were included in the maps (this captures all of the
star-forminggas,butexcludesdistantgalaxiesprojectedby
chancealongthesameline-of-sightinthecosmologicalbox).
1 http://fire.northwestern.edu An example of the resulting maps can be found in Figure
MNRAS000,1–14(2016)
4 M. E. Orr et al.
1, which shows maps of the gas surface density and the in- each pixel, including the gas metallicity4 Z, the Keplerian
stantaneousstarformationratesurfacedensityinthem12i velocity v , and the dynamical angular velocity Ω, defined
c
simulationfromHopkinsetal.(2014),atredshiftz≈0with here as
100 pc pixels.
v (GM(<R))1/2
Using the star particle ages, we calculated star forma- Ω= c = , (1)
R R3/2
tion rates averaged over the previous 10 and 100 Myr, cor-
rectingformasslossfromstellarwindsandotherevolution- where R is the galactocentric radius of the pixel and M(<
ary effects as predicted by starburst99 (Leitherer et al. R) is the total mass enclosed within that radius (and G
1999). We also considered the instantaneous star formation is the gravitational constant). These quantities allow us to
rate of the gas particles (defined above). A time-averaging investigate the dependence of star formation on gas phase
interval of 10 Myr was chosen because this approximately metallicity, approximate the optical depth of star-forming
corresponds to the timescale traced by recombination lines regions,andrelategalacticdynamicaltimestostar-forming
such as Hα, whereas UV and FIR emission traces star for- regions.
mation over roughly the past 100 Myr (e.g. Kennicutt & Inouranalysiswetreatpixelsfromallsimulationsand
Evans 2012).2 The instantaneous star formation rate of the all times equally, unless otherwise stated. However we wish
gas particles covers a larger range of star formation rates to ensure that only well-resolved pixels contribute. Recall-
because it is not constrained at the low end directly by the ing that each of our simulations has a fixed baryonic par-
mass resolution of our simulations; it is a continuous quan- ticle mass, m , we discard pixels which contain <3 gas
b
tity intrinsic to the gas particles themselves. This quantity particles; for a pixel size l, this means only gas surface
best demonstrates the direct consequences of feedback on densitiesΣ >3×10−3M pc−2(m /1000M )(l/kpc)−2
gas (cid:12) b (cid:12)
the gas in situ by locally tracing the star formation rate, will be considered. However, in the example above (m ∼
b
whereas the other two SFR tracers are more analogous to 1000M , l ∼kpc), the observed Kennicutt (1998) rela-
(cid:12)
observables. tion gives a typical star formation surface density Σ ∼
SFR
Thegassurfacedensitytracerswerealsochosenonthe 10−7M yr−1kpc−2 at this minimum Σ , so in∼10Myr,
(cid:12) gas
basis of observable analogues, including all gas, neutral hy- theexpectednumberofm ∼1000M starparticlesformed
b (cid:12)
drogen gas (total HI + H column, accounting for metal- willbejust0.001.Obviously,thedistributionofstarforma-
2
licity and He corrected), and “Cold & Dense” gas which tionrateswillnot,then,beresolved(evenifthesimulations
wespecificallydefinehereandthroughoutthispaperasgas capturethemeanstarformationratecorrectly,thediscrete
with T < 300 K and n > 10 cm−3. These roughly corre- natureofstarformationmeansonly1in1000pixelswillhave
H
spond to the total gas, atomic + molecular gas (HI+H ), a star particle, while 999 have none). We therefore adopt a
2
and cold molecular gas reservoirs observed in galaxies. We more appropriate criterion: (1) we first calculate the mean
have checked that the approximation for the molecular gas Σ perpixelfromeachsimulation,foralltheirpixelswith
SFR
component(T <300K,n >10cm−3)yieldsmoleculargas agivennumberofgasparticles(fixedΣ );(2)weestimate
H gas
fractions similar to themolecular fractionf calculatedin the average number (cid:104)N (∆t)(cid:105) of star particles this would
H2 (cid:63)
gizmo itself. We have opted to use the aforementioned ap- produceinthetime∆t(10or100Myr,asappropriate);(3)
proximation rather than reconstruct the f predicted by if this is <1(=N ), we discard all pixels which contain
H2 min,(cid:63)
the Krumholz & Gnedin (2011) model (which not output this number or fewer gas particles. For the example above,
inthesnapshots)asthef modelassumesasimplifiedge- for ∆t = 10Myr (100Myr), this requires > 500 (> 100)
H2
ometryattheresolutionscale,thatcangetthelocaloptical gas particles per pixel for a “resolved” star formation rate.
depthquitewrong,3 andamoredetailedanalysisofthetrue We have repeated this exercise using instead the observed
molecular fraction of the gas would require a careful radia- Kennicutt-Schmidt relation (instead of the predicted one),
tivetransferpost-processing,whichweleavetoalaterwork. to estimate the resolved thresholds, and find it gives nearly
We acknowledge that because of the rather strict density identical results. We have also verified that changing the
and temperature criteria, and the lack of any consideration threshold N by an order of magnitude in either direc-
min,(cid:63)
to the local UV field or the geometry of the gas, we may tion does not change any of our conclusions here; we note
underestimate the molecular gas column by up to a factor toothattheaveragestarformationratesfromlow-resolution
of a few. simulationscontinuetoagreewellwithourhigher-resolution
Other quantities are calculated as the mass-average in simulations down to (cid:104)N (∆t)(cid:105) as low as ∼0.001.
(cid:63)
Comparingwithobservations,wecompiledtheresolved
Kennicutt-Schmidtobservationsfromalargenumberofpa-
2 Directly computing SFR indicators from the simulations (e.g. pers at various resolution scales commensurate with our
mockobservationalmaps(Kennicutt1998;Kennicuttetal.
Hayward et al. 2014; Sparre et al. 2015) rather than computing
the SFR averaged over the past 10 or 100 Myr would provide a 2007; Bigiel et al. 2008; Genzel et al. 2010; Bothwell et al.
moreaccuratecomparisonwiththeobservedKennicutt-Schmidt 2010; Onodera et al. 2010; Verley et al. 2010; Shapiro et al.
relation,butdoingsowouldconsiderablyexpandthescopeofthis 2010; Wei et al. 2010; Daddi et al. 2010; Shi et al. 2011;
work,soweleaveittoafuturestudy. Lisenfeld et al. 2011; Boquien et al. 2011; Tacconi et al.
3 Forthepurposesofourstarformationcriteria,however,thisis 2013; Freundlich et al. 2013). We then represent the range
for the vast majority of cases not an issue. Due to the steep-
ofthecompileddatawithshadedcontours,includingtheex-
ness of the exponential attenuation of the local UV field, we
care only whether, strictly speaking, the gas is optically thin or
thick. Not necessarily how thin or thick, as any optical depth
τ (cid:29)1effectivelyyieldsexp(−τ)(cid:28)1,andτ (cid:28)1similarlyyields 4 Inthispaper,wetakesolarmetallicitytobeZ(cid:12)≈0.0142when
exp(−τ)≈1. scalingmetallicities(Asplundetal.2009).
MNRAS000,1–14(2016)
Kennicutt-Schmidt on FIRE 5
AllGas NeutralHydrogen Cold&DenseGas
2]) 1
−
R pc 0
F k
S 1 1
vg. yr− −2
A (cid:12)−
M
r 3
y [ −
M R
F 4
10 (ΣS −5
og − ObservationalDataRange ObservationalDataRange
l 6
−
) 1
2]
c− 0
R p
F k 1
S 1− −
t. yr 2
s −
n (cid:12)
I M 3
Gas ˙Σ[(cid:63) −−4
(
g 5
o −
l
6
− 1 0 1 2 3 4 1 0 1 2 3 4 1 0 1 2 3 4
− − −
log(Σ [M pc 2])
gas −
(cid:12)
Figure 2.Kennicutt-SchmidtrelationintheFIRErunsin1kpc2 pixels,binnedbyΣgas,forseveralgasandstarformation“tracers”.
Neutral(andmolecular)hydrogenis∼ΣHI+H2,andCold&DensegasincludesparticleswithT<300KandnH>10cm−3 (∼ΣH2).
The gas instantaneous star formation rate is calculated directly from the gas particles in each pixel, whereas the 10 Myr average star
formationrateiscalculatedfromtheyoungstarparticlesineachpixel.Linearaveragevaluesofthepixeldistributionareplottedinbins
of gas surface density, with error bars denoting the 5-95% range of resolved star formation in the bins. The yellow and green shaded
regionsdenotetheobservationaldatarangefromthestudiesenumeratedin§2.Thestarformationrelationsderivedin§4.1and§4.2are
plotted with dashed black lines, using the fiducial values assumed there and Σ(cid:63) = 102 M(cid:12) pc−2. The simulations robustly reproduce
a resolved Kennicutt-Schmidt relation in broad agreement with observations, regardless of the particular star formation or gas tracer
considered.
tentofthestarformationdistributions(determinedbyeye) are taking pixels at a given star formation rate and moving
for two tracers of gas surface density (Σ and Σ ). themtolowergassurfacedensities(totheleft)byreducing
HI+H2 H2
Nodistinctionismadebetweenthemanyestimatorsofstar whatgascontributestotheoverallgascolumndensity.The
formation rate used in the aforementioned papers; they are restrictionsarenon-linear:athighsurfacedensities,thegas
taken at face value. is predominately molecular, and added restrictions do lit-
tle to change the participating gas column, whereas at low
surface densities, relatively little of the gas column may re-
main after making these additional cuts. Little difference is
3 KENNICUTT-SCHMIDT RELATION IN THE seenbetweenthestarformationdistributionsinΣ −Σ
SFR gas
SIMULATIONS spacewhenconsideringthesurfacedensitiesofallgasversus
neutral gas (first and second columns of Figure 2) because
3.1 Dependence of the Kennicutt-Schmidt
the contribution of ionized gas to the total gas column is
Relation on Star Formation and Gas Tracers
small in regions where significant star formation is occur-
Figure 2 demonstrates the broad emergence of the ring. In contrast, there is a marked change in slope of the
Kennicutt-Schmidt power law relation, irrespective of spe- Kennicutt-Schmidt relation when moving from the surface
cific choice of star formation or gas tracer, in the FIRE density of neutral hydrogen gas to that of Cold & Dense
simulations. Two of our star formation rate tracers, the 10 gas (T < 300 K, nH > 10 cm−3), with the slope shifting
Myr-averaged and gas instantaneous star formation rates, from ∼ 1.7 to ∼ 1.2 for the gas instantaneous star forma-
yield very similar Kennicutt-Schmidt relations. The points tion rate. This is due to the fact that significant amounts
denote the linear average of the star formation rate distri- of star formation can occur in “small” pockets of molecular
bution. The error bars in Figure 2 denote the 5% to 95% gas, relative to the overall gas column, yielding a shallower
inclusion interval in the distribution of the star formation slope than when considering neutral gas.
rates of pixels in that bin of gas surface density. Broadly speaking, all of our various formulations of
More restrictive gas tracers (e.g. taking gas with T < the Kennicutt-Schmidt relation in the FIRE simulations
300 K and n > 10 cm−3 instead of all neutral hydrogen agree with the corresponding spatially resolved observa-
H
gas) yield shallower power-law slopes. This is intuitive be- tional data, as represented by the shaded regions in the
cause by placing more restrictions on the gas column, we panels of Figure 2. There is some offset between the sim-
MNRAS000,1–14(2016)
6 M. E. Orr et al.
AllGas NeutralHydrogen Cold&DenseGas
2]) 1
−
R pc 0
F k
S 1 1
vg. yr− −2 (cid:15) = 1
MyrA [MFR(cid:12)−−43 (cid:15)(cid:15) == 00..10 1
10 (ΣS −5
og − ObservationalDataRange ObservationalDataRange
l 6
−
) 1
2]
c− 0
R p
F k 1
S 1− −
t. yr 2
s −
n (cid:12)
I M 3
Gas ˙Σ[(cid:63) −−4
(
g 5
o −
l
6
− 2 1 0 1 2 2 1 0 1 2 2 1 0 1 2
− − − − − −
log(ΣgasΩ[M yr−1 kpc−2])
(cid:12)
Figure3.Elmegreen-SilkrelationintheFIRErunsin1kpc2pixels,inthestyleofFigure2.Linesofconstantstarformingefficiencyare
plotted,with(cid:15)=(0.01−1).Thesimulatedgalaxieshavekpc-scalestarformationefficienciesincreasingfrom∼1%(allgas)to∼10%
(Cold&Densegas)asdensergastracersareselected.
ulations and observations for our Cold & Dense gas surface 3.1.2 100 Myr Averaged Star Formation Rate
density, which may indicate that our approximation under-
InFigure4,weseeaclearflatteningofthe100Myr-averaged
estimates the true molecular gas column, but they agree in
starformationratesurfacedensityrelativetothe10Myrav-
form with the observed range of molecular gas data. More-
erage,forbothneutralhydrogenandmoleculargascolumns
over, the distribution of star formation rates in the FIRE
at low gas surface densities, Σ (cid:46) 1 M pc−2. This is
simulationsoverlapwiththeDampedLyαSystems(DLAs) gas (cid:12)
ascribable to effects discussed in §3.2 and §3.4, where indi-
observed at high redshift by Rafelski et al. (2016). Though
vidualorsmallnumbersofyoungstarparticlesarescattered
weappeartoseeanaloguestothesesystemsat1kpc2 pixel
into regions of very low gas surface density that are not ac-
sizes,weleaveittoafutureworktoinvestigatethedetailed
tually forming stars. Moreover, dynamical changes in star-
physical properties of these systems.
forming regions over the averaging period (100 Myr) cause
gas complexes to dissipate and produce small numbers of
star particles left in now-diffuse galactic environments. At
high gas columns, Σ > 10 M pc−2, the 100 Myr aver-
gas (cid:12)
agestarformationratesurfacedensitiesagreewellwiththe
3.1.1 Elmegreen-Silk Relation (Alternative KS Law) shorter time-scale estimators.
Alternatively, in Figure 3, we probe the global efficiency of
gas turning into stars in a dynamical time according to
3.2 Pixel Size Dependence
Σ˙(cid:63) =(cid:15)ΣgasΩ, (2) The Kennicutt-Schmidt relation that we find in the FIRE
simulationsdoesnotappeartohaveasignificantdependence
where (cid:15) represents the “star formation efficiency” on kpc onpixelsize(i.e.mapresolution)forpixelswithsufficiently
scales; this relation is known as the Elmegreen-Silk rela- resolvedgasandstarformationratetracers((cid:38)fewgas/star
tion (Elmegreen 1997; Silk 1997). For both the neutral hy- particlesperpixel),asshowninFigure5.Overtherangeof
drogen and Cold & Dense gas surface densities for which pixel sizes we investigate, 100 pc - 5 kpc (0.01 - 25 kpc2),
we have observations to compare with (where (cid:15) ranges be- the slope of the power law varies only weakly between ∼ 1
tween10−3−1),weseesystematicagreement.Wefindkpc- and∼4/3.AtthelowendoftherelationinΣ ,weexpect
gas
averaged star formation efficiencies of (cid:15) ∼ 0.01−0.1 con- the scatter to grow as Poisson statistics become important
sistently for our entire range of star formation rate surface when only a few star particles are present in the pixels on
density.Dashedblacklinesindicateconstantefficienciesbe- average. However, because we exclude poorly-sampled pix-
tween 0.01 and 1. Without feedback, one would expect to els,thissimplymanifestsasalowerlimittotheplottedΣ
gas
see (cid:15)∼1. for smaller pixels sizes (c.f. §2).
MNRAS000,1–14(2016)
Kennicutt-Schmidt on FIRE 7
100MyrAverageSFR by the stochastic nature of the star formation in the simu-
lations.
ObservationalDataRange
1
)
2]− 0 3.3 Redshift Independence
c
p
k
1 1 WefindnosignificantredshiftdependenceoftheKennicutt-
− −
r SchmidtrelationoftheFIREgalaxies.Therelation’sinsen-
y
(cid:12) 2 sitivity to redshift in the simulations can be seen in the
M −
[ top panel of Figure 6, where the snapshots are colored by
FR 3 redshift bin (z < 0.5, 0.5−1.5, 1.5−2.5, 3−6), and the
S −
Σ
g( 4 10 Myr-averaged ΣSFR and neutral gas surface density are
lo − considered. Some scatter is seen in the average values be-
tweenredshiftbins,butanydependenceonredshiftismuch
5 NeutralHydrogen
− smaller than the range of the data itself. The absence of
ObservationalDataRange anyredshiftdependencepersistsforallmeasuresofstarfor-
1
mationrate.Thoughtheabsoluteamountofstarformation
2c])− 0 vstaarriefsowrmitahtiroendsrhaitfet,stuhrefaccoerrdeelantsiiotnybreemtwaeinens cgoanssciostluenmtn. and
p
k
1 1
− −
r
y 3.4 Metallicity Dependence
(cid:12) 2
M −
[ We see in the bottom panel of Figure 6 evidence of a weak
SFR −3 dependence on metallicity for the Kennicutt-Schmidt rela-
Σ
( tion in the FIRE runs. For all gas surface densities, more
g 4
lo − metal-rich gas exhibits elevated star formation rates, with
anadmittedlylargescatter.Interestingly,noneofourforms
5 Cold&DenseGas
− ofthe Kennicutt-Schmidtlawexhibit anotablemetallicity-
2 1 0 1 2 3 dependent cutoff in star formation, as some models pre-
− −
log(Σgas[M pc−2]) dict (Krumholz et al. 2009b). For our 10 and 100 Myr-
(cid:12)
averaged star formation rates, this may be an issue of ade-
Figure 4.Kennicutt-Schmidtrelationforthe100Myr-averaged
quatelysamplingstarformationratesinthe“cutoff”regime
starformationratein1kpc2 pixels,asFigure2.The“observed” of Σ˙ ∼ 10−(3−4) M yr−1 kpc−2. However, for the well-
contours are those measured with ∼10 Myr tracers in Figure 2. (cid:63) (cid:12)
resolved instantaneous star formation rate, the form of the
At high Σgas, the ∼ 10 and ∼ 100 Myr estimators agree well.
starformationrelationdoesnotchangeatallforanyofthe
AtlowΣgas,theΣSFR fromthe∼100Myrtracerflattens.This
appearstostemfromabreakdowninthecorrelationbetween100 metallicitybinsinthe“cutoff”regimethatKrumholzetal.
Myr-oldstarsandtheobservedgastracers,eitherfrommigration (2009b) find. The instantaneous star formation rate tracer,
orotherdynamicaleffects(e.g.mergersorstrongoutflowevents). as with the averaged star formation rate tracers, presents
higher star formation rates for metal-enriched gas even at
these low gas surface densities, but the change is smooth,
rather than a “threshold” effect.
Figure 7 illustrates the strength of the metallicity de-
Broadly,ourresultsareinagreementwithobservations
pendence. Binning pixels by gas surface density Σ and
at various pixel sizes, in spite of our likely systematic un- gas
normalizingbytheaveragestarformationrateineachΣ
derestimation of column density in our “molecular” (Cold gas
bin, we find that star formation rate surface density in-
& Dense) gas. However, given our mass resolution, at our
creases weakly with metallicity, across all Σ bins. This
smallestpixelsize(100pc),noneofoursimulationsareable gas
presentationofthedatanormalizesouttheΣ dependence
toadequatelysamplestarformationatgassurfacedensities gas
(cid:46)10 M pc−2, where observations exhibit the largest scat- tohighlightthemuchweakerZ dependence.Aby-eyefitof
(cid:12)
a power law with slope 1/4 is plotted as a dashed black
ter.Atleastthreeprocessescausethecorrelationofthestar
line. This slope is much shallower than the slope derived
forminggasandyoungstarparticlestobreakdownonscales
in §4.2 but on the order of the predicted metallicity depen-
lessthanl∼500pc.(1)Therelativevelocitiesbetweenstar
denceofSNefeedback’smomentuminjection(rangingfrom
forminggasandtheyoungstarstheyproducecausethemto
∼ 1/10−3/14; Cioffi et al. 1988; Martizzi et al. 2015)5.
wanderintodifferentpixels,thustheybecomeuncorrelated
onapixel-by-pixelbasis,whenv ∼l/∆t.For100pcpixels
p
and100Myrtimebins,thisisarelativevelocityofonly∼1
km s−1 (10 km s−1 for ∆t ∼ 107 yr), so we would expect 5 As described in Hopkins et al. (2014), when SNe explode in
regionssuchthattheircoolingradiiwillbeunresolved(common
significant scatter to arise from this effect at the smallest
in the lower-resolution simulations here with particle masses (cid:38)
pixel sizes. (2) Dynamical processes affecting gas and star
104 M(cid:12)),theejectaareassignedaterminalmomentumbasedon
particles, like dispersion of GMCs, major mergers, or SNe,
thedetailedindividualexplosionmodelsfromCioffietal.(1988),
over the time bin (i.e. 10 –100 Myr) cause greater fluctua- which scale as pt ∝Z3/14. However, the metallicity dependence
tionsfromthepowerlawaverageaspixelsizedecreases.(3) in Figure 7 persists if we restrict only to our highest-resolution
When considering small (< 1 kpc) pixels, scatter is caused simulations.
MNRAS000,1–14(2016)
8 M. E. Orr et al.
) 1
2]− 0.1kpc 0.5kpc 1kpc Observations
c 0
p 5kpc
k
1
1
− −
yr 2
(cid:12)−
M 3
[ −
R 4
F −
S
Σ 5
( −
g
o 6
l − 2 1 0 1 2 3 2 1 0 1 2 3 2 1 0 1 2 3 2 1 0 1 2 3
− − − − − − − −
log(Σ [M pc 2])
mol −
(cid:12)
Figure5.PixelsizedependenceoftheKennicutt-SchmidtrelationintheFIREsimulations.Σ isthesurfacedensityofCold&Dense
mol
gas(T <300K,nH>10cm−3);ΣSFR isthe10Myraveragedstarformationratesurfacedensity.Plottedpointsanderrorbarsareas
inFigure2.Greenregionsdenotesobservationsatappropriateresolutionforcomparisonfrompapersenumeratedin§2,withthe1kpc
observations used for both 500 pc and 1 kpc, as a number of the source observations lie between those two resolutions. Star formation
relations derived in §4.1 are plotted with dashed black lines using the fiducial values assumed there and Σ(cid:63) =102 M(cid:12) pc−2. For each
pixelsize,thereisathresholdinΣ belowwhichstarformationispoorlyresolvedgivenevenourhighestmassresolution,asdescribed
mol
in§2.Abovethisthreshold,thesimulatedgalaxies’KSrelationexhibitsnosystematictrendwithpixelsizedespitedynamicalprocesses
thatmightbeexpectedtobreakdownthecorrelationsbetweenyoungstarsandgasonsmallscales(prominentatsmallpixelssizes).
Though the scatter within bins is quite large, this weak de- with σ being the turbulent velocity and Ω being the lo-
T
pendence is rather robust. calorbitaldynamicalfrequency.Weareconcernedwiththe
largest eddies, which contain most of the turbulent energy.
Hence, the timescales of turbulent energy dissipation scale
ast ≈t ∼H/σ ∼Ω−1.Equatingtheseratesoftur-
4 PHYSICAL INTERPRETATION eddy diss T
bulent momentum injection and momentum dissipation in
On the scales of tens or hundreds of millions of years, it gas, we find,
is possible to understand star formation as an equilibrium
Σ˙ (P /m )≈σ Σ /t , (3)
process(ongalacticscales)inwhichtheinputsofeithermo- (cid:63) (cid:63) (cid:63) T gas diss
mentuminjectionfromstellarfeedback(athighgassurface substitutinginourrelations,thisyieldsastarformationrate
density) or energy from photoheating (at low gas surface of,
density) balance gravitational collapse.
(cid:18)P (cid:19)−1
Σ˙ ≈σ ΩΣ (cid:63) . (4)
(cid:63) T gas m
(cid:63)
4.1 High Gas Surface Density Regime
Relating σ Ω back to the disk surface density with a
T
In our analysis, there is a marked transition in the star modified Toomre-Q parameter (Toomre 1964),
formation rate distribution at gas column densities above (cid:112)
κ c2+σ2
Σ ∼ 100 M pc−2. Above this threshold, almost all the Q= s T , (5)
gas (cid:12) πGΣ
gasformsstarsonorveryneartheKennicutt-Schmidtpower disk √
law. Here, supernova feedback becomes an increasingly im- where κ is the epicyclic frequency ∼ 2Ω for galactic po-
portant mechanism for injecting momentum into the ISM, tentials, cs is the sound speed, Σdisk ≈ Σgas +Σ(cid:63) is the
asthemassiveyoungstarsproducedareembeddedindense disksurfacedensity.Hereweincludetheself-gravitycontri-
molecularenvironmentstowhichtheycaneffectivelycouple. bution from the collisionless stellar component of the disk,
Astarformationrelationcanbederivedconsideringthe whichiscorrectuptosomeorderunityprefactor.Assuming
limit wherein the ISM is supported against collapse by tur- that we are turbulently rather than thermally supported,
(cid:112)
bulent pressure (Ostriker & Shetty 2011; Faucher-Gigu`ere c2s+σT2 ≈σT. Substituting this in we find,
et al. 2013; Hayward & Hopkins 2015; Torrey et al. 2016).
π (cid:18)P (cid:19)−1
Here, stellar feedback injects momentum into the ISM at a Σ˙ ≈ √ GQ (cid:63) Σ (Σ +Σ ). (6)
rate per area proportional to Σ˙ (P /m ), where (P /m ) (cid:63) 2 m(cid:63) gas gas (cid:63)
(cid:63) (cid:63) (cid:63) (cid:63) (cid:63)
is the characteristic momentum injected per mass of young Adoptingafiducialvaluefor(P /m )of∼3000kms−1
(cid:63) (cid:63)
stars formed, and is dissipated in the mass of nearby gas (e.g. Ostriker & Shetty 2011; Faucher-Gigu`ere et al. 2013;
per area Σgas on some characteristic timescale related to Kim & Ostriker 2015; Martizzi et al. 2015), this yields
the coherence time of the turbulent eddies t , where
eddy
t ∼l /σ ,l beingthespatialscaleoftheeddy (cid:18) P /m (cid:19)−1(cid:18)Σ (Σ +Σ )(cid:19)
eddy eddy eddy eddy Σ˙ ≈3.3×10−2 (cid:63) (cid:63) gas gas (cid:63) Q
and σeddy the turbulent velocity σT. As we are consider- (cid:63) 3000km/s 104M2(cid:12)pc−4
ing an approximately disk-like environment for star forma-
M yr−1kpc−2. (7)
tion in the high gas surface density regime, the largest ed- (cid:12)
dies will likely have length scales on the order of the disk For the gas-dominated regime, where Σ (cid:29) Σ , we
gas (cid:63)
scale height H (Martizzi et al. 2016), so l ∼H∼σ /Ω, recover a quadratic relation for star formation. Similarly,
eddy T
MNRAS000,1–14(2016)
Kennicutt-Schmidt on FIRE 9
1
z 3 6
∼ −
z 2 1
2])− 0 z∼1
kpc z<∼0.5 )R(cid:105)
1− 1 ΣSF 0
yr − /(cid:104)
(cid:12) R
M SF
[ 2 Σ 1
R − g( −
F o
ΣS l −1<logΣHI+H2≤0
log( −3 −2 01<<llooggΣΣHHII++HH22<<12
Redshift 2<logΣHI+H2<3
4
− 3 2 1 0 1
1 − − −
0<Z(cid:48)<0.1 log(Z/Z )
(cid:12)
0.1<Z <0.5
2kpc])− 0 02.5<<Z(cid:48)Z(cid:48)(cid:48)<2 Fb10yigΣMuHryeIr+-a7Hv.2eSratatagre1foΣkrpSmcF2aRtpinoixnoerrlmasatiezliezd.eedPpoebinnydtestnhacereeaovanevremargaeegtaeΣllviSacFiltuRye,sibnoinfenathechde
1yr− −1 ΣfoHrmI+aHti2onbiinn.eEarcrhorbbina.rsAdweneoatkedtehpee5n-d9e5n%ceraonngmefeotralrleicsiotlyveisdsseteanr
(cid:12) forallgassurfacedensities,asdemonstratedbythedashedblack
M
[ 2 lineofslopeΣSFR∝Z1/4.
R −
F
S
Σ
g( 3 Schmidt relation at moderately high surface density in the
o −
l FIREsimulations.Remarkably–giventhesimplicityofthe
Metallicity derivation–whenEquation7isusedonapixel-by-pixelba-
4
− 0 1 2 3 sistopredictstarformationratesfromΣHI+H2 andΣ(cid:63),the
predicted rates are nearly identical to the 10 Myr-averaged
log(Σ [M pc 2])
HI+H2 (cid:12) − and instantaneous star formation rates, extending down to
Σ ≈10−1 M pc−2.
Figure 6. Kennicutt-Schmidt relation binned in redshift and HI+H2 (cid:12)
metallicity at 1 kpc2 pixel size. In both panels, within their re-
spective redshift and metallicity bins, average values of the 10
4.2 Low Gas Surface Density Regime
Myr average ΣSFR are plotted in bins of ΣHI+H2, in the style
of Figure 2. Yellow region denotes appropriate observations and At the other extreme of galactic environments, we consider
dottedlinesderivedstarformationrelations,asinFigure2.Top
thelowgassurfacedensityregimeinwhichgasissupported
Panel: Simulation snapshots binned by redshift, with markers
by thermal – rather than turbulent – pressure. We expect
denoting different epochs. No significant dependence on redshift
thistransitiontooccurforΣ (cid:46)10M pc−2;seeOstriker
is seen – the range of data in each bin is greater than any sys- gas (cid:12)
et al. (2010) and Hayward & Hopkins (2015) for details. In
tematic difference between them. Bottom Panel: Pixels from
snapshots binned by gas metallicity, with markers indicating in- thisregime,astarformationequilibriumratecanbederived
tervalsinZ(cid:48)=Zgas/Z(cid:12).Thoughweakcomparedtodatarangein by balancing photoheating from young stars with gas cool-
eachbin,ametallicitydependenceofΣSFRisseenatallΣHI+H2. ing.Atextremelylowgassurfacedensities,whereΣgas (cid:28)1
Nometallicity-dependentcutoffisevident. M pc−2, the metagalactic UV background itself may be-
(cid:12)
come the predominant source of heating, requiring no star
formationatalltomaintainathermalpressureequilibrium
(Ostriker et al. 2010).
should the stellar component dominate, as may be the case
As derived in Ostriker et al. (2010) as an “outer-disk”
instellarsystemswitholderpopulations,alawlinearinΣ
gas
law,wecanbalancephotoheatingwithradiativegascooling.
isfound;thisappearstobeingoodagreementwiththeslope
For ionizing and photo-electric photons dominating the gas
of the Kennicutt-Schmidt relation seen in the FIRE runs.
heating, the heating rate per area is
TheobservedweakmetallicitydependenceseeninFig-
ures 6 & 7, combined with the result shown in the lower E˙ f βL
heat = abs (cid:63) =f β(cid:15)c2Σ˙ , (8)
leftpanelofFigureA1,whichshowsexplicitlythatthestar l2 l2 abs (cid:63)
formation rate varies with the strength of feedback, can be
where f ((cid:46)1) is the fraction of the emitted photoheating
partlyexplainedbyaweakdependenceofthefinalmomen- abs
photonsabsorbedbysurroundinggas,β ∼0.1isthefraction
tum injection from SNe feedback on the metallicity of the
ofionizingradiationemittedbyyoungstars(Leithereretal.
surrounding gas, e.g. (P /m )−1 ∼ Z0.114 (Martizzi et al.
(cid:63) (cid:63) 1999), and (cid:15)∼4×10−4 is the fraction of rest-mass energy
2015).
radiated by stars in their lifetimes. On the other hand, the
The Kennicutt-Schmidt relation predicted by a
cooling rate per area is
turbulence-supported model assuming that the stellar sur-
face density Σ(cid:63) (cid:29) Σgas, with its approximately linear E˙cool = ΛneniV ≈ ΛZngΣgas , (9)
power-lawslope,agreesremarkablywellwiththeKennicutt- l2 l2 µ
MNRAS000,1–14(2016)
10 M. E. Orr et al.
4 0.8
DefaultPhysics
Unstable Stable 102 3
3 0.4
102
2c])− 2 Self-shielding 2 0.0 2pc]−
p 1 k
M(cid:12) 1 Non-self-shielding c] −0.41yr−
[(cid:12) 0 nts [kp 0 0.8 M(cid:12)
/Zgas −1 101 101 Cou y −1 −1.2 Σ[SFR
Z − g
as 2 2 lo
Σg − − 1.6
log( −3 −3 −2.0
−4 Star-formingAnnuli No3Self-S2hieldi1ng 0 1 2 3 −1.2
5 Non-star-formingAnnuli 100 100 3 − − − x[kpc]
− 2 1 0 1 2 3 0.8
− − log(Q˜uthne−rmshailelded) 2 0.4 2]−
c
p
1 0.0 k
Finisgtaubrielit8y.dCeotmerpmairniseosnthoefwohnesethteorfseelffif-csiheinetldsintagrofrorgmraavtiitoantio(nseael kpc] 0 0.41yr−(cid:12)
§4.3). For each radial annulus in each galaxy (500 pc wide an- y[ − [M
n(Euqli., a1t3)eaacnhdtΣimgeasZfrgoams/Zz(cid:12)=. W0.5e−pl0o)t,awehemateamsuarpe Qo˜futhnte−hrmeshainleuldmed- −1 −0.8 ΣSFR
bcoelroro-fcopdixeedlssowsitthar-efaocrhmvinagluaenonfuQl˜iutahnre−ermsrheaidleld(emdeaanndΣ˙Σ(cid:63)g>as1Z0g−as3/ZM(cid:12)(cid:12), −2 −11..62 log
kpc−2 yr−1)andnon-star-formingannuliareblue.Q˜un−shielded 3 −
istheToomre-Qparameterifthegaswerepurelythertmhearlmlyalsup- − 2.0
ported with T =104 K; this indicates whether the gas could be Onl3ySelf2-Shie1lded0GasC1anCo2llapse3 −1.2
thermally stabilized against gravitational instabilities if it were 3 − − − x[kpc]
notself-shielding.ΣgasZgas/Z(cid:12) isaproxyforopticaldepth,ap- 0.8
proximating whether or not the gas is self-shielding to ionizing 2
radiation. Vertical and horizontal dotted red lines indicate the 0.4 2]−
c
aQitmnhnaen=Ktutiror1luianmscatkcrhaleeoboanflizlrovilttayeyrtsoyetaclihfnlc-.rguse(hsr2Σhsi0egoal0adlr9dsiobnaua)gtnn,–fiddreix.tsQee˜hp.duteeghncZers−trae(cid:48)imsvlvhfaiea-itnslleayhdldt.iieeΩBodlnd.la∼aiTnclkhg1ine,dtshoeotvnrateetsbseneihdtliottolhlyifdnoisuendtgaiesthrrhiiaofvttoweherdess- y[kpc] 01 −0.00.41[Myrkp−(cid:12)
1 0.8 R
collapse,whichthenproducesdenseself-shieldingclumps. − − SF
Σ
1.2 g
−2 − lo
with Λ ∼ 10−22 erg s−1 cm−3 being the net cooling rate 1.6
(Robertson&Kravtsov2008);n ,n ,andn beingtheelec- 3 −
e i g −
tron, ion, and gas number densities; and V ∼l2h being the 2.0
−
volumeofgasconsidered.Equatingtheheatingandcooling −3 −2 −1 0 1 2 3
rates, we find x[kpc]
Σ˙ ≈ ΛZngΣgas . (10) Figure 9. Demonstration of the importance of gravitational in-
(cid:63) f µβ(cid:15)c2 stabilityversusself-shieldinginstarformation.OneMilkyWay-
abs
mass simulation, m12i presented in Hopkins et al. (2014), was
Furthermore, we have n = ρ /µ ≈ Σ /2hµ and
g gas g(cid:112)as restarted from z ≈ 0.07 and rerun with three sets of physics,
h ≈ c /Ωinthethermallysupportedlimit,as c2+σ2 ≈
s s T the default physics implemented in the FIRE runs (top), one
cs. Thus, we have ng ≈ΣgasΩ/2csµ, and Σ˙(cid:63) becomes withbothself-shieldingandcoolingbelow104 Kdisabled(mid-
ΛZΩΣ2 Σ ZΩ(cid:18)Σ (cid:19)2 dle),andonewhereshieldedgashadnormalpropertiesbutnon-
Σ˙ ≈ gas = 0 gas , (11) shielded gas had a large artificial pressure floor, effectively dis-
(cid:63) 2f µ2β(cid:15)c2c f Σ
abs s abs 0 ablinggravitationalfragmentationuntilthegasfirstbecameself-
where Σ ≈ 2µ2β(cid:15)c2c /Λ ≈ 4 M pc−2, assuming T = shielding(bottom).Coloredpixelsindicatetheextentandinten-
0 s (cid:12)
104 K, for which c ≈ 12 km s−1. Scaling this to approx- sity of star formation since the runs were restarted in the form
s
imately Milky-Way values (Ω ≈ v /R ≈ 220 km s−1/R, ofthe500Myraveragedstarformationratesurfacedensity.The
c star formation largely has the same structure with or without
Z ≈Z ), we have
(cid:12) self-shielding, as long as fragmentation is allowed (gas can still
(cid:18) Z (cid:19)(cid:18)10kpc(cid:19)(cid:18)Σ (cid:19)2 1 isothermally collapse and meet the star formation criteria). But
Σ˙ ≈1.3×10−3 gas whenonlyannuliinagalaxywhichareentirelyself-shieldingcan
(cid:63) Z R Σ f
(cid:12) 0 abs fragmentorcollapse,onlythecentral∼kpcofthegalaxy(where
M yr−1kpc−2. (12) thegasisentirelymolecular)efficientlyformsstars.
(cid:12)
MNRAS000,1–14(2016)