The role of stellar feedback in the formation of galaxies

0.82 MB·English

by  Daniel Ceverino

The role of stellar feedback in the formation of galaxies.
Daniel Ceverino and Anatoly Klypin
Astronomy Department, New Mexico State University, Las Cruces, NM
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2 ABSTRACT

n Although supernova explosions and stellar winds happen at very small scales, they affect the
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interstellar medium (ISM) at galactic scales and regulate the formation of a whole galaxy. Pre-
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vious attempts of mimicking these effects in simulations of galaxy formation use very simplified
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assumptions. We develop a much more realistic prescription for modeling the feedback, which
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minimizes any ad hoc sub-grid physics. We start with developing high resolution models of the

] ISMandformulatetheconditionsrequiredforitsrealisticfunctionality: formationofmulti-phase
h
medium with hot chimneys, super-bubbles, cold molecular phase, and very slow consumption of
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- gas. We find that this can be achieved only by doing what the real Universe does: formation
o of dense (> 10 H atoms cm−3), cold (T ≈ 100 K) molecular phase, where the star formation
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t happens, and which is disrupted by young stars. Another important ingredient is the runaway
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stars: massivebinarystarsejectedfrommolecularcloudswhenoneofthecompanionsbecomesa
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[ supernova. Thosestarscanmoveto 10-100parsecsawayfrommolecularclouds beforeexploding

themselves as supernovae. This greatly facilitates the feedback. Once those effects are imple-
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v mentedintocosmologicalsimulations,galaxyformationproceedsmorerealistically. Forexample,
5 we do not have the overcooling problem. The angular momentum problem (resulting in a too
8 massive bulge) is also reduced substantially: the rotation curves are nearly flat. The galaxy
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formation also becomes more violent. Just as often observed in QSO absorption lines, there are
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substantial outflows from forming and active galaxies. At high redshifts we routinely find gas
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2 with few hundred km s−1 and occasionally 1000−2000 km s−1. The gas has high metallicity,
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whichmay exceedthe solarmetallicity. The temperature of the gas in the outflows and in chim-
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neys can be very high: T = 107−108 K. The density profile of dark matter is still consistent
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: with a cuspy profile. The simulations reproduce this picture only if the resolution is very high:
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better than 50 pc, which is 10 times better than the typical resolution in previous cosmological
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X simulations. Our simulations of galaxy formation reach the resolution of 35 pc.
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a Subjectheadings: hydrodynamics,methods: n-bodysimulations,ISM:general,stars: formation,galaxies:
formation, evolution
1. Introduction ΛCDM paradigm, because, after all, many obser-
vationalevidences ofcosmicstructures comefrom
Thecurrentcosmologicalparadigm,theΛCDM
the light emitted by galaxies.
Universe,hassuccessfullyexplainedtheoverallas-
Galaxy formation is driven by a complex set
sembly of cosmic structures (Blumenthal et al.
of physical processes with very different spatial
1984; Davis et al. 1985; Spergel et al. 2007). In
scales. Radiative cooling, star formation and su-
this picture ordinary matter (“baryons”), which
pernovaexplosionshappenatscaleslessthan1pc,
emits and absorbslight, passivelyfollows the evo-
but they affect the formation of a whole galaxy
lution of the dark matter. This should be cor-
(Dekel & Silk 1986). In addition, large-scale cos-
rected, if we want to make a realistic theory of
mologicalprocesses,suchasgasaccretionthrough
galaxy formation. It is necessary to include the
cosmic filaments, and galaxy mergers, control the
physics of the gas and galaxy formation into the
galaxy assembly. As a result, a complex interplay
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between verydifferent processesdrives the forma- Theproblemisthatothereffectscanbemissedat
tion of galaxies. Cosmological gasdynamical sim- the same time due to a lack of resolution and an
ulations have become useful tools to study galaxy inaccurate modeling of feedback.
formation. Anothermethodistointroduceasub-resolution
In early cosmological simulations “galaxies” model in which the energy from supernova ex-
formedwithtoosmalldisksandasignificantfrac- plosions is stored in an unresolved hot phase,
tionofangularmomentumwaslost(Navarro & Steinmetz which does not cool and looses energy through
2000). The situation has improved in the last theevaporationofcoldclouds(Yepes et al. 1997;
years. Sommer-Larsen et al. (2003);Governato et al. Springel & Hernquist 2003). In this model, the
(2004); Kaufmann et al. (2006) show that a sub- onlyeffectofstellarfeedbackistoregulatethestar
kpc resolution is necessary to prevent an arti- formation: the hot gas is coupled with the cold
ficial loss of angular momentum. Recent im- phase through cloud formation and evaporation.
provements in both the resolution and model- As a result, this high entropy gas is artificially
ing of the feedback have resulted in simulations trapped within the galactic disk. Thus, galactic
with extended galactic disks (Governato et al. winds are introduced in a simplified way in or-
2004; Robertson et al. 2004; Brook et al. 2004; der to reproduce other natural effects of stellar
Okamoto et al. 2005; D’Onghia et al. 2006; feedback, such as galactic outflows.
Governato et al. 2007). However, simulated An alternative approach assumes kinetic feed-
galaxies are still too concentrated, and more real- backinsteadofthermalfeedback(Navarro & White
istic simulationswith better resolutionand better 1993). In that case, the energy from supernova
physics are needed to reproduce the shape of the explosions or stellar winds is transfered to the ki-
rotation curves of observed galaxies. netic energy of the surrounding medium. This
Current simulations lack the necessary resolu- energyis notdissipateddirectlybyradiativecool-
tion to follow correctlythe effect of supernova ex- ing. However, in order to resolve this effect accu-
plosions in the ISM. Because of this lack of reso- rately, simulations should be able to resolve the
lution, the modeling of stellar feedback has relied expansion of individual supernova explosions or
on ad-hoc assumptions about the effect of stel- the stellar winds fromindividualstars. Currently,
lar feedback at scales unresolved by simulations this is not possible. At larger scales, the picture
(0.2-1 Kpc). Early attempts to introduce stellar is more complicated. Different blastwaves from
feedback into simulations found the obstacle of a different supernova explosions can collide, dissi-
strongradiativecooling. The energydepositedby pating their kinetic energy. The same dissipation
supernova explosions was quickly radiated away of energy happens in collisions of stellar winds in
without any effect in the ISM (Katz 1992). Sev- stellar clusters. So, it is commonly assumed that
eral shortcuts have been proposed to get around most of the kinetic energy from stellar feedback
this over-coolingproblem. is dissipated into thermal energy at the small-
The most common method is to artificially est scales resolved by simulations. Nevertheless,
stop cooling when the stellar energy is deposited this feedback-heated gas can expand. As a result,
(Gerritsen & Icke 1997; Thacker & Couchman thermalenergycanbetransferedtokineticenergy.
2000; Sommer-Larsenet al. 2003; Kereˇs et al. The net results are flows at large scales powered
2005; Governato et al. 2007) This approach pro- bythethermalfeedback. However,feedbackheat-
longs the adiabatic phase of supernova explosion ing should dominates over radiative cooling: only
(the Sedov solution) to about 30 Myr. The moti- in this case those flows are produced.
vation behind this ad-hoc assumption is that the To summarize, the main problems of current
combination of blastwaves from different super- simulationsofgalaxyformationarethelackofthe
nova explosions and turbulent motions produces necessary resolution and too simplified models of
hot bubbles much larger than individual super- the complex hydrodynamic processes in the mul-
novaremnants andlastlonger. All this effects are tiphase ISM.
not resolved with the current resolution. They ThegalacticISMhasaverywiderangeofdensi-
do not develop in a self-consistent way. Instead, ties and temperatures (for review see Cox (2005)
the delay in the cooling is introduced by hand. and Ferri`ere (2001)). Three distinct phases are
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distinguished: the dense cold gas (giant molecu- 30 km s−1, much higher than the velocity disper-
lar clouds (GMC), cold HI gas or diffuse clouds) sion of the population of massive stars in clusters
with densities above 10 cm−3 and temperatures (10 km s−1) (Stone 1991). Some of these stars
bellow 100 K, the warm component with densi- have large peculiar velocities, up to 200 km s−1
ties between 0.1 and 1 cm−3 and temperatures (Hoogerwerf et al. 2000). This is why they are
of several thousands degrees, and the hot phase called runaway stars. The current scenario of the
with temperatures above 105 K and densities bel- origin of runaway stars is the ejection of these
low 10−2 cm−3. This multiphase medium is set massive stars from stellar clusters. There are two
by the competition of cooling and heating mech- possible mechanisms of this ejection. One possi-
anism and the onset of thermal instabilities. The bility is the ejection due to a supernova explosion
hot ISM component (T > 105 K) is usually asso- in a close binary system (Zwicky 1957; Blaauw
ciated with gas heated by shocks. They can be 1961). The second mechanism is the ejection due
produced by turbulent motions driven by gravi- to dynamical encounters in the crowded regions
tational and thermal instabilities. However, these of stellar clusters (Poveda et al. 1967). In spite
turbulentdrivenshockscanonlyheatthegasupto of the fact that a significantfraction of the stellar
106 K (Wada & Norman 2001). Only supernova feedback occurs far from star forming regions, no
explosions and stellar winds can produce larger attention has been paid to its effect on the galaxy
gastemperatures(McCray & Snow 1979;Spitzer formation.
1990). We first study the effect of stellar feedback in
2D and 3D hydrodynamical simulations of the ISM, using simulations ofa Kpc-scalepiece of
the ISM have enough resolution (parsecs) to re- the ISM with few parsecs resolution. Then, we
solve the multi-phase nature of the ISM and check if this picture holds when the resolution is
to explore complicated effects of stellar feed- degraded to the resolution that our cosmological
back ondifferent scales(Rosen & Bregman 1995; simulations can achieve at high redshift. Finally,
Scalo et al. 1998; Korpi et al. 1999; de Avillez we study the effect of stellar feedback in galaxy
2000; de Avillez & Breitschwerdt 2004, 2007; formation at high redshift using cosmological hy-
Wada & Norman 2001, 2007; Slyz et al. 2005). drodynamical simulations.
There is much to learn from these simulations. This paper is organized as follows. Section 2
However,they typicallyfocus onconditionsinthe describesthe necessaryconditionsinwhichstellar
solarneighborhood,whicharedifferentfromwhat feedbackdominatesoverradiativecooling. Section
one may expect during early stages of galaxy for- 3 describes all details of the modeling of stellar
mation. Not always they follow the whole gas feedback. Section 4 shows Kpc-scale simulations
cycle: cooling, star formation, and stellar feed- of the ISM. Section 5 describes the cosmological
back. For example, de Avillez & Breitschwerdt simulations of galaxy formation. Finally, section
(2004) include star formation but artificially re- 6 is the discussion and conclusion. Throughout
strict the rate of supernova explosions around a the paper we give quantities in physical units.
fixed value. However, this rate could be much
higher in large star forming regions. As a result, 2. Physical conditions for the heating
the effect of stellar feedback is underestimated in regime
these regions. Nevertheless, the effect of the stel-
lar feedback in the ISM, such as the formation of The thermodynamicalstate of the gas depends
hot bubbles and super-bubbles is resolved. on two competing processes: heating from stel-
lar feedback and cooling from radiative processes.
It is crucial to understand where and how
They appear as source and sink terms of internal
the energy from massive stars is released back
energy in the equation of the first law of thermo-
to the ISM. While a large fraction of massive
dynamics:
stars are found in stellar clusters and OB as-
sociations, 10-30% are found in the field, away du
+p∇·v=Γ−Λ (1)
from any molecular cloud or stellar cluster (Gies dt
1987; Stone 1991). This population have pecu-
where u is the internal energy per unit volume,
liar kinematics. Their velocity dispersionis about
p is the pressure of the gas, and v is its velocity.
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ParameterΓistheheatingrateduetostellarfeed- feedback is not able to heat the gas beyond 104
back, and Λ is the net cooling rate from radiative K for densities higher than 0.1 cm−3 and typical
processes. values of Γ′. This is the well known overcooling
The heating rate from stellar feedback can be problem for simulations, which allow cooling only
expressedastherateofenergylossesfromayoung to a temperature of 104 K at which the star for-
and active single stellar population with a given mation is assumed to happen. The energy from
density, ρ∗,young : stellar feedback is radiated away very efficiently
andthethermalfeedbackcannotplayanyrole. In
Γ=ρ∗,youngΓ′ (2) this case one needs to invoke “subgrid physics” –
aguesshowthe systemshouldreacttotheenergy
where Γ′ is the specific rate of energy losses of
released by the stars.
the stellar population according to its age. The
The situation is completely different if the gas
cooling rate can be expressed as:
is allowed to cool to 100 K. The cooling is very
Λ=n2 Λ′, (3) inefficient at that temperature: Λ′ = 10−25 erg
H s−1 cm3. So, stellar feedback can produce the net
where nH is the hydrogen number density. gas heating even if the density is large: nH ≈100
cm−3 for ρ∗,young ≈ ρgas. Our conclusion is that
2.1. Heating versus radiative cooling simulations should include cooling process bellow
104 K.The coldphase shouldbe resolvedinorder
Now, we can ask ourselves under which con-
to get a high efficiency of stellar feedback.
ditions the feedback heating dominates over the
However, heating to high temperatures is still
radiative looses. Using the expression, n =
H
problematic because as the gas is heated, the
ρ /(µ m ), where ρ is the gas density, µ
gas H H gas H
cooling rate increases. So, the peak of the cool-
is the molecular weight per hydrogen atom and
ing rate at 104 K is a bottle-neck for heating
m is the hydrogen mass, the condition for heat-
H
gas to higher temperatures. Nevertheless, tem-
ing (Λ≤Γ) can be expressed as:
peratures of diffuse gas as high as 106 − 107 K
n Λ′ ≤ ρ∗,youngµ m Γ′ (4) have been observed around star-forming regions
H H H
ρgas such as the Rosette nebulae (Townsley et al.
2003; Wang et al. 2007), M17 (Townsley et al.
Using typicalvalues, we can rewrite the condition
2003), and the Orion nebula (Feigelson et al.
for the heating regime in the following way:
2005; Guedel et al. 2007). The main question
n Λ′ ishowyoungandmassivestarscanheattheirsur-
H ≤ (5) rounding medium to these high temperatures, if
0.1cm−3 10−22ergs−1cm−3
(cid:16) (cid:17)(cid:18) (cid:19) theoriginalmedium, inwhichtheywerebornhad
high densities.
ρ∗,young Γ′ Theanswerlikelydependsonthedistancefrom
ρ 1034ergs−1M−1 those young stellar clusters. At small 1-2 pc dis-
(cid:18) gas (cid:19) ⊙ !
tances it is likely to have the collisions of stel-
The cooling rate, Λ′, is a strong function of gas lar winds (Townsley et al. 2003; Feigelson et al.
temperature. So,thetemperatureandthedensity 2005) . At larger distances the heating is related
of the gas are two key properties in establishing with the formation of superbubbles: the cumula-
the cooling or the heating regimes. The following tiveeffectofwinds andshocksgeneratedbymany
two examples illustrate common situations. young stars. One way or another, the density of
Attemperaturesaround104K,thecoolingrate gasaroundthe youngstellarpopulation decreases
is close to its maximumvalue. We use Λ′ =10−22 and the ratio ρ∗,young/ρgas increases as the over-
erg s−1 cm3 as a fiducial value. In this case pressuredbubbleofgasexpands. Oncethedensity
eq.(5) shows that the heating overcomes the cool- goes below 0.1 cm−3, eq.(5) can be fulfilled even
ing only at very low densities n ≤0.1 cm−3, op- at 104 K. The net result is a heating regime, in
H
timistically assuming that the ratio of densities, which the surrounding gas can be heated to very
ρ∗,young/ρgas is about unity. As a result, stellar high temperatures. In other words, the process
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starts with the expanding bubbles at low temper- 2.2. Localgravityversuspressuregradient
aturesandthenproceedstoarunawayoverheating
Aswesawintheprevioussection,lowdensities
regime.
are required in order to heat the gas beyond the
Asanexample,weconsideratypicalGMCwith
peak ofthe coolingcurve. Stellar feedback should
a mass of 105 M⊙ and a size of 50 pc. These evacuate the gas by creating an expanding bub-
are the typical values found in recent catalogs
ble around young stellar clusters. However, the
of GMCs in M33 (Rosolowsky et al. 2007), M31
over-pressuredbubbleexpandsonlyifthepressure
andtheMilkyway(Sheth et al.2008). Therefore,
gradient overcomes self-gravity.
the mean density is n = 50 cm−3. This value
H If we consider an over-pressured bubble of ra-
seems low compared with typical observed den-
dius R in a homogeneous medium of density ρ,
sities of molecular clouds. However, GMCs are
we can derive a Jeans-instability type of condi-
highly clumpy. High-density clumps are usually
tion. As a result, the bubble expands only if the
embedded in a low density inter-clump medium.
difference in pressure with its surroundings, ∆P,
As a result, the volume-averaged density inside
satisfies the following relationship:
clouds is much smaller than the typical observed
mass-weighted density (McKee 1999). 4π
∆P/k ≥ G(ρR)2 =10−1(n R )2 (6)
Now, we consider an Orion-like stellar cluster 3k H pc
formedatthecenterofthecloud. Themassofthe
wherekistheBoltzmannconstant,Gisthegravi-
stellarclusteris5×103M (Hillenbrand & Hartmann
⊙ tationalconstant,andR istheradiusinpc. The
1998). In a region of mass 104 M⊙, the stellar pc
above equation sets the conditions for the bubble
cluster has the ratio ρ∗,young/ρgas equal to 0.5,
expansion. For the Galactic plane the pressure
and the condition for heating, eq.(5), is fulfilled.
is P/k ∼ 2 ×104 cm−3K (Cox 2005). For ex-
This heating produces an over-pressuredhot bub-
ample, a region of 50 pc in radius and a density
ble with a pressure 100 times higher than the
of 100 cm−3 will only expand, if the difference in
surrounding unperturbed medium. As a result,
pressureisbiggerthan2×106cm−3K.Thiscanbe
the bubble expands, the density decreases, and
achieved, if the bubble is over-pressured by more
the ratio ρ∗,young/ρgas increases. Then, we get a
than 100 times. Stellar feedback can produce this
runaway bubble, which proceeds to blowing away
overpressurejust by raising the temperature from
all gas (Kroupa et al. 2001).
100 K to 104 K. The resulted over-pressured re-
Simulations should resolve the expansion of
gion will expand, and the density as well as the
bubbles over-pressured by stellar feedback. The
cooling rate will decrease. So, the efficiency of
density of young (and active) stars and the den-
stellarfeedbackincreases,raisingthetemperature
sity of gas should be comparable at the smallest
and pressure further.
scales resolved by the simulations. The minimum
Eq. 6 also sets a upper limit on the resolution.
value of the ratio ρ∗,young/ρgas depends on the
Using the equation of state of the ideal gas P =
gas density (eq. 5). For moderate gas densities,
n = 10−100 cm−3, the above ratio should be nkT, where n is the mean number density and T
H
is the temperature of the gas, the over-pressured
around 0.1-1.
bubbleshouldberesolvedwithaspatialresolution
Theaboveconditioncanbe achievedifthe star
X =R /2, such that the expansionis resolved:
pc pc
formation efficiency, the fraction of the progeni-
tor cloud consumed in stars is 10%-50% at the X 2 T n −1
resolution scale. This high efficiency is consis- pc ≤ H (7)
75pc 104K 10cm−3
tent with the observedvalue of 10%-40%found in (cid:18) (cid:19) (cid:18) (cid:19)(cid:16) (cid:17)
Galactic stellar clusters, (Greene & Young 1992; As a result, for typical values of these over-
Elmegreen et al. 2000; Kroupa et al. 2001). Due pressured regions, the resolution should be bet-
tothefactthat80%oftheGalacticstarformation ter than ∼ 70 pc. Otherwise, the bubble cannot
occursinstellarclusters(Lada & Lada2003),this overcome its self-gravity and cannot expand.
highefficiency ofstar formationshouldbe consid-
eredinanystarformationmodelwhichcanresolve
the sites where star formation occurs.
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3. Stellar feedback model
Weassumeamodelofthermalfeedbackforthe
injection of energy from stellar winds and super-
nova explosions. The kinetic energy from these
processes is efficiently dissipated into thermal en-
ergyduetoshocksatscalesbellowthespatialres-
olution.
The net thermalrate (Γ−Λ) is used to update
the internalenergy in each step of the simulation.
This approach is rather different than the deposi-
tion of energy. Instead, the energy injection from
stellar feedback is treated in a self-consistent way
along with the radiative looses.
3.1. Heating rate from stellar feedback
The heating rate from stellar feedback in a
givenvolumeelementismodeledastherateofen-
ergy losses from a set of single stellar populations
Fig. 1.—Rateofenergylossesperunitmassfrom
present in that volume. This is just a generaliza-
a single stellar population. Top panel shows the
tion of eq.(2):
resultsfromtheSTARBURST99code,assuminga
1
Γ= M Γ′(t ), (8) Miller-ScaloIMFforamassrange(0.1−100)M⊙.
i i
V The dotted line shows the contribution of super-
i
X
nova explosions and the full line shows the total
where M and t are the mass and the age of each
i i rate. Althoughsupernovaexplosionsdominatethe
single stellar population.
overall energy release, stellar winds are the only
The modeling of the specific release of energy
mechanism of energy release during the first few
overtime,Γ′,ismotivatedbytheresultsfrompop-
Myr. Middle and bottom panels show two dif-
ulation synthesis codes, such as STARBURST99
ferent models: a constant feedback model and a
(Leitherer et al. 1999). Figure 1 shows differ-
modelofstellarwindpluscore-collapsesupernova.
ent models of Γ′ and the results of a STAR-
Although the total energy released is the same
BURST99 computation with a Miller-Scalo IMF
in both models, the SN model is more elemen-
from 0.1 M⊙ to 100 M⊙. Parameter Γ′ is dom- tary and takes into account the explosive nature
inated by stellar winds from massive OB main-
of core-collapse supernova.
sequence stars and WR stars during the first
few Myr. Later the energy is produced by core-
collapse supernovaefromstars more massivethan injectionof2×1051ergofenergyfromstellarwinds
8 M⊙. After 40 Myr, the release of energy comes and supernova explosions per each massive star
from stellar winds of AGB stars and other less withM >8M⊙ duringitslifetime. We assumea
powerful sources, and the injection rate drops 6 Miller-ScaloIMFinthemassrange(0.1−100)M⊙.
ordersofmagnitudes. SupernovaeIadominatethe Note that this constant heating rate is the sum of
feedback at much longer time-scales. We assume thecontributionsfromallmassivestarsinasingle
a peak of the SNIa rate at 1 Gyr. However, this stellar population. We also consider a more sim-
peak is 3 orders of magnitude lower than the con- ple model, which we calla SN model. In this case
tribution from core-collapse supernovae. This is 1051 ergsisinjectedatconstantrateduetostellar
because the energy from a population of SNIa is winds over10 or40 Myr. Then it follows a strong
dilutedoveramuchlongertimescalethantheen- peakofenergyreleaseduetothesupernovaexplo-
ergy from core-collapse supernovae. sion,in which1051 erg arereleasedduring105 yrs
WemodelΓ′ withaconstantrateof1.18×1034 – the typical age of young supernova remnants.
ergs−1M−1 over40Myr. Thisisequivalenttothe Although the total energy released is the same in
⊙
6
bothmodels,the SNmodeltakesintoaccountthe metallicity-dependentcoolingandUVheatingdue
explosive nature of core-collapse supernovae. to a cosmologicalionizing background.
3.2. A model of runaway stars 4. Results of ISM runs
The effect of runaway stars is implemented by
Our first step in the understanding of stellar
adding a random velocity to a fraction of stellar
feedback in galaxies is to understand its effect in
particles (10%-30%). This extra velocity has a
the ISMatgalacticscales. Therefore,we runsim-
random orientation and the value is taken from ulations of a 4 ×4 ×4 Kpc3 piece of a galactic
an exponential distribution with a characteristic
disk with 8-16 pc resolution. These simulations
scale of 17 km s−1. This choice is motivated
fully resolve the effect of massive stars at galactic
by Hipparcos data (Hoogerwerf et al. 2000) and
scales. So, resolution is not longer an issue.
Monte-Carlo simulations (Dray et al. 2005). For
We can use this ISM-scale simulation as a
comparison, a Gaussian distribution is also used
benchmark for the effect of stellar feedback at
(Stone 1991). However, the effect of runaway
galactic scales. Then, we can degrade the reso-
stars in the ISM is not very sensitive to the de-
lution to see which model of feedback reproduces
tails of this velocity distribution.
thesameoverallpictureatlowerresolution. These
simulations can then be used as testing grounds
3.3. Radiative cooling
forthesemodelsatdifferentresolutions. Theytell
Radiative cooling counterbalances feedback uswhatarethenecessaryingredientstoreproduce
heating. So it is very important to have an accu- thetrulyeffectofstellarfeedbackattheresolution
rate model of radiative cooling in order to study that we can afford in cosmological simulations of
the net effect of stellar feedback in the ISM. galaxy formation.
Weusethemodelofradiativecoolingdescribed We want to see the effect of stellar feedback in
in Kravtsov (2003). It is a metallicity-dependent thetypicalconditionsofnormaldiskgalaxieswith
cooling plus a UV heating due to a cosmologi- moderate gas surface densities. So, we are not
calionizingbackground(Haardt & Madau 1996). modelingstarburstgalaxieswithlargeamountsof
ThemodelincludesComptonheating/coolingand gas and high star formation rates. This type of
molecular cooling. The temperature range of the study will be done in the future.
model is between 102 K < T < 109 K. Thus, A 4 Kpc box of ISM represents a significant
this model includes cooling below 104 K and the piece of a galactic disk. The simulation resolves
gas can reach the thermodynamical conditions of the dense galactic plane, where molecular clouds
molecular clouds. As we saw in section 2, this is are formed. This is important to follow star for-
crucial for the efficiency of the stellar feedback. mation correctly. At the same time, the simula-
The cooling and heating rates from radiative tion follows the gas at few Kpc above the galactic
processes are tabulated using the CLOUDY code plane. This height is similar to the scale-heightof
(version 96b4; Ferland et al. 1998). As a result, the diffuse warm phase of the ISM (Cox 2005).
thenetcoolingratefromradiativeprocesses,Λ′,is The simulation includes radiative cooling and
available for a given density, temperature, metal- UV heating from a uniform UV field at redshift 0
licity and redshift. as described in section 3. Star formation happens
inthehighestdensitypeakswithadensitythresh-
3.4. Description of the code old of 100 cm−3. In each star formation event,
The numerical simulations were performed us- 5 % of the mass in gas inside a volume element
ing the Eulerian gasdynamics + N-body Adap- is converted into a stellar particle with a mass
tive Refinement Tree code (Kravtsov et al. 1997; of 88 M⊙within a time-step set by the Courant
Kravtsov 1999, 2003). The physical processes condition (∼ 2 ×103 yr). The supernova model
of the gas include star formation, stellar feed- was used for stellar feedback and SNIa was not
back, metal enrichment, self-consistent advec- included. The metallicity was assumed solar and
tion of metals, cooling and heating rates from constant throughout the simulation.
7
4.1. Initial conditions
Theinitialdistributionofgasdensityisuniform
in the x and y directions of the box. In the z-
direction,thedensityprofiledeclinesatbothsides
of the middle plane, z = z = 2 Kpc. This plane
0
defines the galactic plane for this ISM model:
z−z
n = n cosh−2 0 (9)
H 0
z
(cid:18) d (cid:19)
wheren is the gasdensity inthatplaneandz is
0 d
the scale-height.
The choice of parameterssets the conditions of
a quiescent normal galactic disk, n = 1 cm−3
0
and z = 250 pc. Thus, the surface density is
d
,Σ = 16 M pc−2. The system is originally
gas ⊙
in hydrostatic equilibrium with a temperature of
104 K. No stars are present at the beginning of
the simulation. The box has open boundaries in
the z-direction. So, all material that cross these
boundaries escapes the system.
The initial velocity field consists of a sum of
plane-parallel velocity waves:
z−z 2
u = A (i,j,k)sin(~k·~r)exp− 0 (10)
x x
z
i,j,k (cid:18) d (cid:19)
X
z−z 2
u = A (i,j,k)sin(~k·~r)exp− 0 (11)
y y
z
i,j,k (cid:18) d (cid:19)
X
z−z 2
u = A (i,j,k)sin(~k·~r)exp− 0 (12)
z z
z
i,j,k (cid:18) d (cid:19)
X
The amplitudes are taken from a Gaussian field
with a tilted power spectrum, P ∝ k−3, where k
k
is the wavenumber, k = 2π i2+j2+k2. i,j and
Fig. 2.— Formation of a galactic chimney. Edge- L
k are integers running from -20 to 20 (excluding
on slices through the simulation show density, 0)andu =20kms−1.Thispisatypicalspectrum
0
temperatureandvelocityinthe verticaldirection,
of a compressible turbulent medium (Kraichnan
perpendicular to the galactic plane. The bottom
1967; Va´zquez-Semadeni et al. 1995).
panelshowsgascolumndensity. Thechimneyout-
flowisnotahomogeneous,coherentflow: itistur- R
Gauss
A (i,j,k)=u (13)
bulent and has dense and cold clumps embedded x 0(i2+j2+k2)3/2
into the flow. The core of the chimney reaches
R
107-108 K. Outflow velocities exceed 103 km s−1. Ay(i,j,k)=u0(i2+jG2a+usks2)3/2 (14)
This hot material is able to escape the disk and
R
generate a galactic wind. A (i,j,k)=u Gauss (15)
z 0(i2+j2+k2)3/2
R isarandomnumbertakenfromaGaussian
Gauss
distribution.
8
Thishotgascannotstayintheplaneofthedisk,as
aresult,thebubbleexpandsfasterinthedirection
perpendicular to the disk, because the density de-
clines in that direction. The bubble develops into
agalacticchimney(Norman & Ikeuchi1989). The
chimney outflow does not look as a homogeneous,
coherent flow. Instead, the chimney is turbulent
andhasdenseandcoldclumpsembeddedintothe
flow. Eventually, the gas expands in the halo and
cools (Figure 2).
Another interesting feature seen in this model
is a population of isolated bubbles in the warm
medium. These are the results of individual su-
pernova explosions of runaway stars
4.3. Star formation rate
After an initial burst of star formation, the
star formation rate is nearly constant for the rest
of the evolution (Figure 3). We found a low
Fig. 3.— Top panel: Star formation rate surface
star formation rate surface density, Σ = 3×
density of the whole simulation. The value shown SFR
is also averaged over a period of ∼ 2 ×105 yr 10−3M⊙ yr−1Kpc−2, temporally averagedover a
period of 2×105 yr (100 time-steps). This value
(100 time-steps). After an initial burst, the star
is consistent with the expected value from the
formation rate surface density is consistent with
correlation between the star formation rate sur-
the Kennicutt et al. (2007) empirical fit (horizon-
face density and the gas surface density found in
tal line). Bottom panel: Fraction of volume filled
nearbygalaxies(Kennicutt 1998;Kennicutt et al.
with each gas phase over time. The volume oc-
2007). For a gas surface density of Σ = 12
cupied by the warm and the hot phase oscillates.
M pc−2 at t=90 Myr, the expected value from
The hotphase dominates after a burst ofstar for- ⊙
the Kennicutt et al. (2007) fit is Σ = 2 ×
mation and the warm phase dominates when the SFR
10−3M yr−1Kpc−2. This is veryclose to ourre-
gasis cooleddown. The coldphase coversa small ⊙
sults.
volume, which remains constant after an initial
collapse. As observers usually do, we also calculate the
gas consumption time-scale, τ = M /SFR, in
GMC
thesimulatedmolecularclouds,assumingthatgas
4.2. Galactic Chimney formation with a density higher than 30 cm−3 is mainly
within GMCs. In our simulations, the amount of
At the beginning of the simulation, the gas
gasinmolecularcloudsisM =8×106 M at
starts to move according to the turbulent veloc- GMC ⊙
t=90Myr. Thestarformationrateatthattimeis
ity field. As a result, the gas accumulates where
different flows converge and molecular clouds 1 SFR = 4.8×10−2M⊙ yr−1. As a result, the gas
consumption time-scale in the simulated clouds is
naturally appear in form of filaments and shells.
τ ≈170Myr. Thisisquitelongcomparedwiththe
However, around 90% of the volume is filled with
typicalfree-falltime-scaleinsidemolecularclouds,
warm and diffuse gas heated by UV background.
t =(3π/32Gρ)1/2 ≈4 Myr for n = 100 cm−3.
Star formation occurs in the cores of the cold ff H
phase. Newly formed massive stars inject energy Inoursimulations,thestarformationefficiency
and a cavityfilled with hotand verydiffuse gasis over a free-fall time-scale, the fraction of gas con-
formed. Thisover-pressuredmaterialexpandsand sumedinstarsduringafree-falltime-scale,isonly
the net result is the formation of super-bubbles. 2.5%. This value is consistent with observations
(Zuckerman & Evans 1974). Krumholz & Tan
1coldanddensephasewithnH ≥30cm−3 andT ≤300K (2007) report a range of 0.6%-2.6% for the whole
9
Fig. 4.— Snapshot of the model after 113 Myr, showing the density in cm−3 (first row), temperature in
Kelvin (second row), gas velocity in the z-direction (third row), and surface density in cm−2 (forth row).
Left panels show a face-on view of the galactic plane (z = z ) and right panels show an edge-on view
0
perpendicular to that plane. The three phases of the ISM are clearly visible: cold and dense clouds, warm
and diffuse medium and hot bubbles with very low densities. Velocities exceeding 300 km s−1 can be seen
in hot outflows at both sides of the galactic plane.
10