Estimating gas accretion in disc galaxies using the Kennicutt-Schmidt law

0.77 MB·English

by  Filippo Fraternali

Mon.Not.R.Astron.Soc.000,1–??(2011) Printed3July2012 (MNLATEXstylefilev2.2)
Estimating gas accretion in disc galaxies using the
Kennicutt-Schmidt law
Filippo Fraternali1,2(cid:63) and Matteo Tomassetti1,3,4†
2 1Astronomy Department, University of Bologna, via Ranzani 1, 40127, Bologna (IT)
1 2Kapteyn Astronomical Institute, Postbus 800, 9700 AV, Groningen (NL)
0 3Argelander Institut fu¨r Astronomie, University of Bonn, Auf dem Hu¨gel 71, 53121, Bonn (D)
2 4Max Planck Institut fu¨r Radioastronomie, Auf dem Hu¨gel 69, 53121, Bonn (D)

n
u
J
Accepted2012Monthday.Received2012Monthday
0
3

] ABSTRACT
O
C We show how the existence of a relation between the star formation rate and the
gas density, i.e. the Kennicutt-Schmidt law, implies a continuous accretion of fresh
.
h gas from the environment into the discs of spiral galaxies. We present a method to
p derive the gas infall rate in a galaxy disc as a function of time and radius, and we
-
apply it to the disc of the Milky Way and 21 galaxies from the THINGS sample. For
o
r the Milky Way, we found that the ratio between the past and current star formation
t rates is about 2−3, averaged over the disc, but it varies substantially with radius.
s
a In the other disc galaxies there is a clear dependency of this ratio with galaxy stellar
[ mass and Hubble type, with more constant star formation histories for small galaxies

oflatertype.ThegasaccretionratefollowsverycloselytheSFRforeverygalaxyand
1
v it dominates the evolution of these systems. The Milky Way has formed two thirds
3 of its stars after z = 1, whilst the mass of cold gas in the disc has remained fairly
9 constant with time. In general, all discs have accreted a significant fraction of their
0 gas after z =1. Accretion moves from the inner regions of the disc to the outer parts,
0 and as a consequence star formation moves inside-out as well. At z = 0 the peak of
. gas accretion in the Galaxy is at about 6−7kpc from the centre.
7
0
Key words: Galaxy: evolution – galaxies: star formation – galaxies: ISM – galaxies:
2
evolution
1
:
v
i
X
1 INTRODUCTION the need for continuous accretion of cold gas onto galaxy
r
a discs (e.g. Sancisi et al. 2008).
Starformationisthefundamentalprocessthatshapesgalax-
ies into different classes. Although the majority of stars in
Severalpiecesofevidenceshowthatdiscgalaxiesshould
the local Universe are found in spheroidal systems, most of
collect fresh gas from the environment in order to sustain
thestarformationiscontributedbydiscgalaxiesofthelater
their star formation. In the Milky Way, the star formation
types (beyond Sb). The key ingredient for star formation is
rate (SFR) in the solar neighbourhood appears to have re-
cold (star-forming) gas, which is present almost exclusively
mained rather constant in the last ∼ 10Gyr (e.g. Twarog
in disc galaxies. However, the amount of cold gas currently
1980;Rocha-Pintoetal.2000;Binneyetal.2000),suggest-
available in galaxy discs appears rather scant. Kennicutt
ing a continuous replenishment of the gas supply. Simple
(1998b) estimated that disc galaxies have current star for-
(closed-box) models of chemical evolution for our Galaxy
mation rates ranging from a few to about (cid:39) 10M yr−1.
(cid:12) predicttoofewmetaldeficientG-dwarf starsthanobserved
Thus, considering a typical gaseous mass of a few 109M ,
(cid:12) (Searle & Sargent 1972; Pagel & Patchett 1975; Haywood
the gas consumption time scale (i.e. the time needed to ex-
2001).Theseobservationsareeasilyexplainedbyaccounting
haust the gas fuel with a constant star formation rate) is
forinfalloffreshunpollutedgas(e.g.Chiappinietal.1997,
always of the order of a few Gigayears. This result, known
2001).ObservationsofDampedLymanAlphasystemsshow
asthegas-consumptiondilemma(Kennicutt1983),suggests
almostnoevolutionintheneutralgascontentofstructures
intheUniverse(Zwaanetal.2005;Lahetal.2007;Jorgen-
(cid:63) E-mail:fi[email protected] sonetal.2009).Hopkinsetal.(2008)pointedoutthatthis
† E-mail:[email protected] constancy of gas density can be explained assuming a rate
(cid:13)c 2011RAS
2 Filippo Fraternali & Matteo Tomassetti
ofgasreplenishmentproportionaltotheuniversalSFRden- evolution to the gas reservoir in the disc. We treat the stel-
sity.Bauermeisteretal.(2010)convergedtoasimilarresult larfeedbackwiththeinstantaneousrecyclingapproximation
whencomparingtheevolutionofthemoleculargasdepletion (I.R.A.),meaningthatthefractionalmassthatstarsreturn
rate with that of the cosmic star formation history (SFH). totheISMateachtimeisassumedtobeconstantandequal
Finally, the derivations of SFHs for galaxies with different tothereturnfactorR.Equations(1)-(2)modifyasfollows:
stellar masses consistently show that late type systems do
have a rather constant SFR throughout the whole Hubble − dΣ∗ =(1−R)SFRD (3)
time(e.g.Panteretal.2007).Thesefindings,otherthanbe- dt
ing the signature of downsizing in cosmic structures, point − dΣgas =−(1−R)SFRD+Σ˙ (4)
at a continuous infall of gas onto galaxies of late Hubble dt ext
types.
In Section 4.1 we discuss the effect of a delayed return.
The way gas accretion into galaxies takes place is still
The parameter R in eqs. (3)-(4) depends on the IMF
a matter of debate. The classical picture states that ther-
andthefractionη(M)ofmassthatastarofmassM returns
malinstabilitiesshouldcausethecoolingofthehotcoronae
to the ISM, called the stellar initial versus final mass rela-
that surround disc galaxies and be the source of cold gas
tion. Following the prescription of Kennicutt et al. (1994)
infall (e.g. Maller & Bullock 2004; Kaufmann et al. 2006).
weassumethathigh-massstars(M >8M )allleavea1.4
However,recentstudieshaveshownthat,duetoacombina- (cid:12)
M remnant.Low-massstarsareassumedtoleaveremnants
tionofbuoyancyandthermalconduction,hotcoronaeturn (cid:12)
withmassesbetween∼0.5and∼1.3M ,withalinearde-
outtoberemarkablystableandthermalinstabilitydoesnot (cid:12)
pendencyfromtheinitialmass(seeChapter2ofMatteucci,
appear to be a viable mechanism for gas accretion (Binney
F.2001,fordetails).AssumingaSalpeterIMF,ourfiducial
etal.2009;Joungetal.2011).Ontheotherhand,accretion
value for the recycling parameter is R (cid:39) 0.30. Assuming
maytakeplaceintheformofcoldflowsbuttheimportance
Kroupa (Kroupa et al. 1993) or Chabrier (Chabrier 2003)
of this process is expected to drop significantly for redshift
IMFs would make little difference, giving R = 0.31 and
z < 2 (Dekel & Birnboim 2006; van de Voort et al. 2011).
R = 0.32 (or R = 0.46 for a flatter high-mass slope) re-
Observations of local gas accretion at 21-cm emission seem
spectively. With this value the gas consumption timescales
to show too little gas around galaxies in the form of HI
are extended by a factor (1−R)−1 = 1.4 (see Kennicutt
(high-velocity) clouds to justify an efficient feeding of the
et al. 1994).
disc star formation (Sancisi et al. 2008; Thom et al. 2008;
The third term on the RHS of eq. (2) is the in-
Fraternali 2009). However, UV absorption towards quasars
flow/outflowterm.Thecombinationofeqs.(1)and(2)leads
and halo stars point to the possibility that ionised gas at
temperatures between a few 104 and a few 105K could fill to an expression for this term:
the gap between expectations and data (Bland-Hawthorn
dΣ dΣ
2009; Collins et al. 2009; Lehner & Howk 2011). Σ˙ext =− dt∗ − dgtas (5)
Inthispaper,weestimategasaccretioningalacticdiscs
indirectlybycomparingbasicphysicalpropertiesofgalaxies In the following we often refer to this term as the accretion
today. Our model relies on the existence of a law relating (or infall) rate.
thestarformationratedensity(SFRD)andthegassurface Equations (1) and (2) can be applied to a galactic disc
density (Σgas) holding at every redshift, i.e. the Kennicutt- assuming that they are valid at each radius, i.e. that the
Schmidt (K-S) law (Schmidt 1959; Kennicutt 1998b). This disc can be divided into annuli that evolve independently.
approachhassimilaritieswiththemodelofNaab&Ostriker Thisassumptionisequivalenttosaythatstarsthatformat
(2006) that we describe in detail in Section 4. The paper acertainradiusremainatthatradiusforthelifetimeofthe
is organized as follows. Section 2 describes our method. In galactic disc. In Section 4.3 we discuss the effect of stellar
Section 3 we apply our model to the Milky Way disc and migration and show that our main results do not change
to a sample of external discs. In Section 4 we discuss our significantly. As a consequence of this assumption we can
results. Section 5 sums up. integrate eq. (1) from t (the lookback time at the disc
form
formation)tot=0togetthecurrentstellarsurfacedensity.
Clearly, in order to solve the above system one needs
2 DESCRIPTION OF THE MODEL theSFRDasafunctionoflookbacktimeandradius,i.e.the
SFH as a function of radius. In the Milky Way, the SFH is
Theevolutionofgasandstellardensities(Σ andΣ )ina
gas ∗ known with fairly good precision only in the Solar Neigh-
galacticdiscasafunctionofthelookbacktimetisdescribed
borhood(e.g.Rocha-Pintoetal.2000;Cignonietal.2006).
by the following equations (Tinsley 1980):
Inexternalgalaxiestherehavebeenpioneeringattemptsto
dΣ
− ∗ =+SFRD−Σ˙ (1) estimatetheSFHasafunctionofgalacticradius(e.g.Gog-
dt fb arten et al. 2009; Weisz et al. 2011). In general, however,
dΣ
− gas =−SFRD+Σ˙ +Σ˙ (2) aprecisedeterminationofSFH(R)isbeyondthecapability
dt fb ext of the current data. Therefore, for the time being, we must
where SFRD is the star formation rate density, Σ˙ is contentourselveswithusingsimpleparametrisationsforthe
fb
the contribution from stellar feedback and Σ˙ is the in- shape of the SFH. In the following section we describe how
ext
flow/outflow rate from/to the external environment. SFRD weestimatethetrendoftheSFHateachradiusinagalactic
gives the rate at which stars are formed or, with a sign in- disc.Weconsideronlythedisc,leavingasidethebulge/bar,
version, the rate at which gas is consumed, whilst Σ˙ rep- which may have formed and evolved differently as it shows
fb
resents the gas density per unit time returned by stellar a markedly different SFH (e.g. Hopkins et al. 2001).
(cid:13)c 2011RAS,MNRAS000,1–??
Gas accretion using the Kennicutt-Schmidt law 3
z Universeasawholeisdescribedbyavalueofγ muchlarger
0 0.1 0.2 0.3 0.5 0.7 1 1.8 4 than unity, as expected given that the SFR density in the
1 Hopkins and Beacom (2006) past was much higher than now. The shape of the univer-
best fit with f(t) sal SFR density is produced by the combined contribution
of galaxies with very different SFHs. Massive red-sequence
3)
-c galaxies dominate the overall SFR at high-z, whilst late-
p
M typegalaxiesbecomerelativelymoreandmoreimportantas
1
-yr 0.1 time passes (e.g. Panter et al. 2007; Vincoletto et al. 2012).
! Although our parametrisation is only suitable for galaxies
M
D ( with non-negligible current SFR, taking γ as the ratio of
R SFR(t )/SFR(0),clearlyredsequencegalaxiesmusthave
F form
S γ (cid:29)1.
0.01 The above assumptions for the SFRD (eqs. 6 and 7)
allow us to integrate eq. (3) from t to 0 to obtain:
form
0 2 4 6 8 10 12
2 Σ (R,0)
Lookback time (Gyr) γ(R)= ∗ −1 (10)
(1−R) t SFRD(R,0)
form
Figure1. EvolutionoftheaverageSFRdensityoftheUniverse, whereΣ (R,0)isthesurfacedensityprofileofthestellardisc
∗
the points are taken from Hopkins & Beacom (2006). The thick
that we derive from the observed surface brightness profile.
black curve shows a fit with our function f(t) described in the
Eq. 10 reveals that our parameter γ(R) is ultimately
text.
a ratio between the stellar density and the current SFRD,
given a certain formation time for the disc, which is of the
2.1 Reconstructing the SFH of a galaxy disc orderoftheHubbletimeandcanbeassumedroughlyequal
for all discs. A γ(R) close to 1 means that the current SFR
We assume that the SFH in a galaxy disc can be described
densityextendedtoanHubbletimeproducesasmanystars
by a dimensionless and positive function f = f(R,t). This
as observed at that location in the disc. Integrating γ(R)
allows us to write the star formation rate density as:
over the disc and making used of eq. (8) we find an expres-
SFRD(R,t)=SFRD(R,0)×f(R,t) (6) sion for the global γ as:
where SFRD(R,0) is the radial distribution of the current 2 M (0)
γ = ∗ −1 (11)
star formation rate density, which we derive from the data. (1−R) t SFR(0)
form
Thesimplestexpressionforf onecanassumeisafirstorder
whereM (0)andSFR(0)arerespectivelythecurrentstellar
polynomial with time: ∗
mass and star formation rate of the entire disc.
t
f (R,t)=1+[γ(R)−1] (t (cid:62)t(cid:62)0) (7) Clearly from eq. (11), one can also describe γ as pro-
1 tform form portionaltotheratiobetweentheaveragepastSFRandthe
wheretisthelookbacktimeandt istheageofthestellar currentSFR.Thislatteristheinverseoftheso-calledScalo
form
disc. With this definition, γ(R) is the value of the function b-parameter (Scalo 1986) to which γ is closely related (see
f at t = t and sets the steepness of the SFH. A value Section 3.2). We also note that any modification of eq. (7)
form
of γ(R) equal to unity represents a constant SFR over the with power law terms of t would produce the same de-
tform
disc lifetime, γ(R) > (<)1 describes a SFR that increases pendenciesforγ aseq.(10).Wediscussthesepointsfurther
(decreases) with increasing lookback time. The dependence in Section 4.2.
of γ on the radius R allows us to describe different shapes
of SFH within the disc. In Section 4.2 we discuss the effect
2.2 The star formation law
of taking different forms for f, we anticipate that our main
results remain unchanged. Themainingredientofourmodelistherelationbetweenthe
We define the global γ for a galactic disc as: rateofstarformationandthegasdensity.Theempiricalstar
2π(cid:82)RmRγ(R)SFRD(R,0)dR formation law is broadly accepted to be a power-law rela-
γ = 0 (8) tion between the “cold” gas density and the star formation
2π(cid:82)RmRSFRD(R,0)dR
0 ratedensity(Schmidt1959;Kennicutt1998b).Thisrelation
wheretheintegrationisperformedouttothemaximumra- has been tested on global (i.e. averaging gas and star for-
dius R ; the denominator is just the current SFR of the mationonthewholedisc)andlocalscales(Kennicuttetal.
m
disc.Thisdefinitionallowsustowriteanexpressionforthe 2007) considering the molecular gas or the sum of atomic
global SFR analogous to eqs. (6) and (7), and molecular gas. Remarkably, it appears to hold across
(cid:20) (cid:21) various orders of magnitudes, from quiescent discs to star-
t
SFR(t)=SFR(0)× 1+(γ−1) (9) busting galaxies (Kennicutt 1998a; Krumholz et al. 2011).
t
form Thedependenceongasmetallicityisdifficulttoinvestigate
Asanillustrationofthemeaningofourparameterγ in due to the related dependency of the X factor (Boissier
CO
Fig. 1 we fit the average SFR density of the Universe with etal.2003).However,thecompilationofazimuthalaverages
eq. (9). The data points are taken from Hopkins & Beacom of SFRD and Σ presented by Leroy et al. (2008) for the
gas
(2006).Herewetaket =12Gyr.Fortheotherparame- THINGS sample, which includes galaxies with very differ-
form
tersweinferSFR(0)=0.012M yr−1Mpc−3andγ =13.4as ent masses and presumably different metallicities, seems to
(cid:12)
thebest-fitvalues.ThisfitshowsthattheSFRdensityofthe show that the effect is limited (see Wyder et al. 2009, and
(cid:13)c 2011RAS,MNRAS000,1–??
4 Filippo Fraternali & Matteo Tomassetti
Section4.4).Thus,inthefollowingweassumethefollowing mustbeaddedtothediscatacertainradiusandasafunc-
star formation law: tion of time to produce the star formation density given by
SFRD(R,t).Asshownabove,theshapeoftheSFHisregu-
SFRD=AΣN (12)
gas latedbythevalueofγ(R).Forγ(R)=1,SFRandaccretion
with N = 1.4 and A = 1.6×10−4 if Σ is measured in are constant in time. More in general, when γ is close to 1
gas
M pc−2 and SFRD in M yr−1kpc−2 (Kennicutt 1998b). the second term on the RHS of eq. (14) is small leading to
(cid:12) (cid:12)
Note that the above value of A takes into account the cor- Σ˙ext(R,t) (cid:39) (1−R)SFRD(R,t). Therefore the gas needed
rection for the presence of Helium (a factor 1.36 in mass), is directly proportional to the gas consumed by the star
not applied in Kennicutt (1998b). formation and this proportionality is simply (1−R). For
Deviationsfromtheaboverelationareobservedatcol- γ(R) > 1 the second term in eqs. (14) and (15) becomes
umn densities lower than Σ (cid:39)10M pc−2, referred to gradually more negative and less accretion is needed. High
gas,th (cid:12)
asthedensitythreshold.Belowthesedensitiesthelawsteep- values of γ(R) may result in a negative Σ˙ext(R,t), hence at
ensandthescatterincreasesmakingitmoredifficulttode- thattimeandradius,theamountofgasavailableinthedisc
scribe (Schaye 2004; Bigiel et al. 2008). In the context of exceedswhatisneededbystarformation.Thiswouldbethe
this paper, it may have an effect in the outer parts of the signature of an outflow of gas.
discs (see Sections 3.1 and 4.4). We define the global accretion rate M˙ext(t) as the gas
Krumholz et al. (2012) have recently pointed out that massaccretedperunittimeoverthewholegalacticdisc,ob-
thestarformationlawexpressedintermsofvolumedensity tained integrating eqs. (14) or (15) over the disc out to our
is more general than the one expressed in terms of surface maximumradiusRm.Itisevidentthattheglobalaccretion
density.Thisisveryrelevanttounderstandthephysicalori- ratewillhavedependenciessimilartothoseofthelocalone
gin of the law and to perform numerical simulations. How- andtheaboveconsiderationsapply.Inparticular,forgalax-
ever, for our investigation a surface-density law is far more ieswithγ oforderunity,theglobalinfallratecloselyfollows
practical.Clearly,thechoicesareequivalentifintheregion the SFR at every time M˙ext(t)(cid:39)(1−R)SFR(t).
ofinterest,i.e.withinthestarformingdisc,thescale-height ItisworthnotingthatΣ˙ext(R,t)representsthegassur-
of the gaseous disc does not change much as a function of face density that has to be added to (or removed from) the
radius. The validity of this condition (within a factor 2) is discperunittimetoproducethereconstructedSFRD(R,t).
wellestablishedinourGalaxy,asthediscappearstobeflar- Ifweareconsideringasingleannulus,Σ˙ext(R,t)refersgener-
ingsignificantlyonlybeyondR(cid:39)16kpc(Kalberla&Dedes icallytotheexternalenvironmentincludingtheadjacentan-
2008). nuli. Therefore, Σ˙ext(R,t) coincides with the accretion (in-
fall) rate at that radius only if there are no radial flows in
the gaseous disc. This is because, with this method we do
2.3 Inferring the gas accretion rate nottracethelocationofthegasinfallbutrathertheradius
at which such gas turns into stars. We discuss this issue
The above assumptions on the SFH and the star formation a bit further later on. On the other hand, M˙ (t) repre-
ext
law allow us to infer the local gas accretion rate using eq.
sents the total mass needed to be accreted by the galaxy
(5). If the star formation law is described by eq. (12), the
disc from outside, at the time t, to sustain the SFR at the
accretion rate density can be written as:
reconstructed rate. Note however that also in this case the
Σ˙ (R,t) = (1−R)SFRD(R,0)f(R,t) definitionofoutside thediscismerelybeyond themaximum
ext
radiusR assumedforthedisc,whichwetooktobe5times
SFRD1/N(R,0) 1−N ∂f(R,t) m
− f(R,t) N (13) the scale-length of the stellar disc.
NA1/Nt ∂t
form
whereSFRD(R,0)isthecurrenttotalgasdensity,AandN
are the parameters of the star formation law and f is the
functional form of the star formation history. The RHS of 3 RESULTS
eq. (13) can also be expressed in terms of Σ (R,0) (using
gas InthisSectionweapplythemethoddescribedabovetoreal
theK-Slaw)insteadofSFRD(R,0).Thisequationisgeneral
galaxy discs. We impose the following boundary conditions:
and it could be used if the exact shape of the SFH(R) were
i)thecurrentstellardensityprofileΣ (R,0),ii)thecurrent
known.Iff isgivenbyeq.(7),thegasaccretionratedensity ∗
SFRdensityprofileSFRD(R,0).Thelattercouldbeinprin-
becomes:
ciple derived from the current gas density profile using the
(cid:20) (cid:21)
Σ˙ (R,t)=(1−R)SFRD(R,0) 1+(γ(R)−1) t star formation law (Section 2.2). However here we restrict
ext tform ouranalysistocaseswherethecurrentSFRDisknownfrom
SFRD1/N(R,0) (cid:20) t (cid:21)1−NN independentestimates.WefirstconsidertheMilkyWayand
− (γ(R)−1) 1+(γ(R)−1) (14) then a sample of spiral galaxies with known SFRD profiles.
NA1/Nt t
form form
or, equivalently in terms of the gas density:
(cid:20) (cid:21) 3.1 Application to the Milky Way disc
t
Σ˙ (R,t)=A(1−R)ΣN (R,0) 1+(γ(R)−1)
ext gas tform We assume that the stellar disc of the Milky Way is ex-
−Σgas(R,0)(γ(R)−1)(cid:20)1+(γ(R)−1) t (cid:21)1−NN(15) pTorenmenatiinael 2w0i0t8h)aansdcaalemlaexnigmthumhRext=ens3io.2nkopfcR(Bin=ne5yh&.
Nt t m R
form form The normalization of this profile is chosen so that the to-
The above equations give the rate at which fresh gas tal stellar mass of the disc is 4 × 1010 M (Dehnen &
(cid:12)
(cid:13)c 2011RAS,MNRAS000,1–??
Gas accretion using the Kennicutt-Schmidt law 5
0.025 100
From HI+H2 using the K-S law
Lyne et al (1985)
Case and Bhattacharya (1998)
0.02 Fit to the data (see text)
2)
-c
p
1 k 0.015
-M yr! (R)(cid:97) 10
D ( 0.01
R Global (cid:97) = 2.8
F
S
0.005
1
0
0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16
Radius (kpc) Radius (kpc)
Figure2. CurrentstarformationratedensityoftheMilkyWay Figure 3. Steepness of the SFH as a function of radius for the
discestimatedusingpulsars(opensquares),andsupernovarem- Milky Way disc, parametrised by γ(R) (see Section 2.1). The
nants (filled squares). These data, originally normalized to the horizontallineshowstheglobalvalueobtainedforthewholedisc.
Solar neighbourhood value, have been fitted with a function de-
scribed in the text (solid line) and rescaled so that the integral
overthediscgivesSFR(0)=3M(cid:12)yr−1.Theshadedregionshows oftheSFRD(R,0)fromthegassurfacedensitybyinverting
theSFRD(R,0)onewouldgetfromthetotal(neutral+molecu- theK-Slaw,soweprefertoadoptthisvaluethroughoutthe
lar)gasdensityusingtheK-Slaw.Thelowerandupperbound- paper. In the following we further fix the age of the stellar
aries are due to the different determinations of the HI surface disc t =10Gyr or, equivalently, z =1.8.1 Note that
form form
densitiesfromBinney&Merrifield(1998)andKalberla&Dedes
takingalargervaluefort ,suggestedforinstancebythe
(2008)respectively. form
investigation of Aumer & Binney (2009) would strengthen
our results.
Binney 1998). The present distribution of the star forma- We reconstruct the SFH of the Milky Way disc as de-
tion rate density with respect to the Solar neighborhood, scribedinSection2.1.Wefindasteepnessofγ =2.8forthe
SFRD(R,0)/SFRD(R ,0),hasbeenestimatedusingseveral wholedisc.Thebehaviorasafunctionofradiusisshownin
(cid:12)
methods. In Fig. 2 we show the estimates coming from the Fig.3.Theparameterγ(R)appearstobeadecreasingfunc-
distribution of pulsars (open squares, Lyne et al. 1985) and tionoftheradiusR.Recallingthatγ(R)istheratiobetween
supernova remnants (filled squares, Case & Bhattacharya the initial and the current SFRDs, this trend shows, as ex-
1998).Wefitthesepointswithafunction(solidlineinFig. pected,thatstarformationinthecentralregionsmusthave
2) of the form: been faster than in the outer disc, and it has spread from
SFRD(R,0)=SFRD(0,0)(cid:18)1+ R (cid:19)αe−RR∗, (16) ianpspidroeaocuhtis(sfeuellyalasoxiN-syaambm&etrOicstarnikderit2d0o0e6s).nNototaecctohuanttofuorr
R∗ the presence of a bar, thus our results for R<∼3kpc should
whereα=3.10isanexponentdefiningthetaperingtowards be taken with some caution.
thecentreandR∗ =1.96kpcisascaleradiussuchthatthe The SFR(t) of the Galaxy reconstructed using the lin-
peak of the distribution is located at R = R∗(α−1). The ear temporal dependency of eq. (7) is shown in Fig. 4 (top
centralSFRdensitySFRD(0,0)=2.3×10−3M(cid:12)yr−1kpc−2 panel) (γ = 2.8). We remind that the key parameters to
is normalized in order to have a global current star forma- obtain this curve are the current SFR and the current
tion rate of the Milky Way SFR(0)=3M(cid:12)yr−1 (see Table stellar mass of the disc (eq. 11). Changing these parame-
1 in Diehl et al. 2006). The same normalization is then ap- ters and the formation time of the disc (t ) would pro-
form
pliedtothedatapoints.TheshadedregioninFig.2shows, duce rather intuitive variations in the value of γ and the
as a consistency check, the SFRD inferred from the current steepness of the SFH. Extreme values can be obtained by
total gas density using the K-S law. The upper and lower taking SFR(0) = 2M yr−1, M (0) = 5 × 1010M and
(cid:12) ∗ (cid:12)
boundariesrefertotwodifferentdeterminationsoftheneu- t = 8Gyr on the one hand and SFR(0) = 4M yr−1,
form (cid:12)
tralgasdensitybyBinney&Merrifield(1998)andKalberla M (0)=3×1010M and t =12Gyr on the other, cor-
∗ (cid:12) form
&Dedes(2008).ThelatterisnotdeterminedforR<4kpc. responding to γ =7.9 and γ =0.8 respectively.
max min
The molecular gas density is taken from Nakanishi & So- Having derived γ(R) we can proceed to estimate the
fue (2006). On the whole the agreement between the direct gas infall rate at each radius and its global value by in-
determinations of SFRD and that inferred by the K-S law tegrating over the whole disc surface using eq. 14. Fig. 4
is very good. In the outer parts the K-S predicts a larger shows the global gas accretion rate necessary to maintain
SFR than observed, likely due to the low densities and the the star formation in the Milky Way disc. The accretion
steepening of the law. We quantify this effect in Section rate is positive at every redshift starting from 5.4M yr−1
(cid:12)
4.4. Some recent determinations of the current SFR of the
Milky tend to give values around 1−2M yr−1 (see e.g.
(cid:12)
Murray & Rahman 2010). However, Fig. 2 shows that only 1 We assume a standard cosmology with Ωb = 0.27 and ΩΛ =
a SFR(0)(cid:39)3M(cid:12)yr−1 is self-consistent with the derivation 0.73.
(cid:13)c 2011RAS,MNRAS000,1–??
6 Filippo Fraternali & Matteo Tomassetti
z
0.025
0 0.1 0.2 0.3 0.5 0.7 1 1.8
9 MILKY WAY
8 MILKY WAY -2c ) 0.02
p
7 1 k
-1M yr)! 456 SFR -n rate (M yr! 0 .00.1051 3 Gyrs ago
( o 1.5 Gyrs ago
3 eti
Gas accretion rate cr 0.005
2 Ac
NOW
1
0
0
0 2 4 6 8 10 12 14 16

Radius (kpc)
1
)(cid:161)
ncy ( 0.8 F3iGgyurr,e16.5.GGyars,aacncdrentoiown.pNrootfielethaetvtherryeereepceoncthsd:elvoeolkobpamceknttimofea=n
e
ci inner depression and of a peak in the gas accretion distribution,
effi 0.6 nowataboutRpeak(cid:39)6−7kpcfromtheGalacticcentre.
n
o
eti 0.4
cr
Ac 0.2 initial conditions. A closed system is only able to produce
(cid:39) 8.4×109M of stars from t = t to t = 0, virtually
(cid:12) form
0 leaving no remnant gas mass at present. On the contrary,
0 2 4 6 8 10
our model with gas infall predicts a present total gas mass
Lookback time (Gyr) of7.1×109M ,comparabletowhatobserved.Thebottom
(cid:12)
panel of Fig. 4 shows the gas accretion efficiency defined as
Figure 4. Toppanel:SFRandglobalgasaccretionrate,versus
theratiooftheaccretionrateoverthenetgas-consumption
timeandredshiftforthediscoftheMilkyWay.Thedashedline
including feedback, ε(t) = M˙ (t)/(1−R)SFR(t). For the
showstheevolutionofaclosedbox.Bottompanel:Gasaccretion ext
efficiencyasafunctionoftime,ε=1correspondstoacomplete whole disc of the Milky Way we obtain εMW(t) ∼ 0.6 at
replenishmentandthusaconstantSFR. t = 0. εMW(t) has a mild dependence on time, reaching a
valueofabout0.9atz=1.Thisshowsthattheefficiencyof
gasaccretionisratherhighbutnotenoughtokeeptheSFR
z constant(ε=1).Thus,gasaccretiondoesnotfeedthediscs
0 0.1 0.2 0.3 0.5 0.7 1 1.8 indefinitely,howeveritdelaystherunning-outoffuelsignif-
4.0
0<R<6 kpc icantlybeyondthe(closed-box)gas-consumptiontimescale.
6<R<10 kpc The value of the current accretion rate and efficiencies are
1) 10<R<16 kpc
- yr!3.0 rzaertho-eprouinntcse.rFtaoirninasstathnecye,slteraovnignlgyadlleptehnedotohnerthpearaasmsuemteerds
M
e ( unchanged and taking SFR(0) = 2M(cid:12)yr−1 would give a
n rat 2.0 currentaccretionofonly0.14M(cid:12)yr−1.Thislargedifference
o istotallylocalizedtothecurrenttime,indeedtheefficiency
creti of accretion remains always above 0.5 from the disc forma-
as ac 1.0 ctiaosnetoof NzG=C05.105p5luinngSiencgtitohnen3.2to. 0.1 at z = 0, see also the
G
We now turn to the local infall rate Σ˙ (R,t). For vi-
ext
0.0 sualization purposes, we split the galaxy in three represen-
0 2 4 6 8 10 tativezones:thecentralpart(between0−6kpc),thesolar
Lookback time (Gyr) circle(6−10kpc)andtheouterdisc(10−16kpc).Foreach
region we calculate the local gas accretion rate integrating
Figure 5. Gas accretion into the Milky Way’s disc integrated
over the annulus surface and we plot the results in Fig. 5.
over three annuli centered at 3 kpc (short dashed line), 8 kpc
We find three different regimes: i) in the central region the
(solid line) and at 13 kpc (long dashed line) versus time and
gasaccretiondeclinesverysteeply,ii)inthesolarcirclethe
redshift.
gas accretion declines by a factor 1.7, iii) in the outer disc
it increases slightly reaching its maximum at z = 0. The
at t and decreasing to 1.3M yr−1 at the current time. behaviour visible in Fig. 5 reflects the simple fact that the
form (cid:12)
It follows very closely the shape of the SFR. These results ratio between the stellar density and gas density decreases
areinagreementwithearlierdeterminationsbasedonchem- stronglywithradiusintheGalaxy.Thesamekindoftrends
icalevolutionmodels(e.g.Tinsley1980;Tosi1988a).Inthe are visible in the SFH reconstructed for the same annuli,
same figure, we also plot the SFR that our Galaxy would given that as seen above, Σ˙ (R,t)∼(1−R)SFRD(R,t).
acc
haveifitevolvedlikeaclosedsystemstartingfromthesame Fig.6showsthegasaccretionprofileasafunctionofR
(cid:13)c 2011RAS,MNRAS000,1–??
Gas accretion using the Kennicutt-Schmidt law 7
intheMilkyWay’sdiscnowandattwoepochsintherecent
past.ThecurrentprofileisnegativeforR<3kpc,peaksat
about R = 6kpc and then falls further out. The negative 20
value in the central regions shows that there the amount
of gas is sufficient to sustain the star formation and the
15
model would predict outflow (of about 0.2M yr−1). This
(cid:12)
inner feature is of recent formation as 3 Gyrs ago the peak
(cid:97)
ofaccretionwasrightinthecentre.Soweconcludethatthe 10
N5055
bulk of gas accretion is moving away from the inner disc.
As a consequence star formation also moves out although
5 N628
with a delay with respect to the accretion. Note that the
shape of the accretion profile at t = 1.5Gyr is remarkably
N4449
similar to the current SFRD(R,0) (see Fig. 2), hinting at 0
a delay between accretion and star formation of a Gyr or 7 7.5 8 8.5 9 9.5 10 10.5 11
so. Interestingly, this is of the order of the gas depletion Log(M*/M )
!
timescale.
We stress again that with our approach we are able to 12
trace the locations in the disc where the new gas begins to
formstars.Theinfallitselfcouldhavehappenedsomewhere 10
else,realisticallymorefurtheroutandtheinfallinggascould
havequiescentlyflowntoinnerradiibeforestartingforming 8 N5055
stars and becoming detectable by our method.
6
(cid:97)
3.2 External galaxies 4 N628
In this Section we extend the application of our method to
2
asampleofexternalgalaxies.Weconsidergalaxiesfromthe
THINGSsurvey(Walteretal.2008)forwhichradialprofiles N4449
0
of gas density and SFRD are available (Leroy et al. 2008).
As for the Milky Way, we assume a common return factor Irr Sd Scd Sc Sbc Sb Sab
R=0.3,anageofthediscsoft =10Gyrandthelinear Hubble type
form
f(t) described in Section 2.1. We derive the global values
Figure 7. The (disc-integrated) parameter γ (steepness of the
of γ both using eq. (11) and by integrating γ(R) (eq. 10)
SFH)forthesampleof21discgalaxiesofLeroyetal.(2008)asa
overthedisc.Intheformercasethevaluesofstellarmasses functionofthestellarmass(top panel)andgalaxytype(bottom
and SFRs were taken from Table 4 of Leroy et al. (2008), panel). The star indicates the Milky Way. A common time of
in the latter we used their SFR densities available online. disc formation of 10 Gyr and the SFH-function f(t) have been
The two methods led, in some cases, to significantly differ- assumedforallgalaxies.Theerror-barsshowtheintervalallowed
ent values for γ. This is partially due to the different range bytwodifferentmethodsofintegrationdescribedinthetext.The
of integration, Leroy et al. (2008) integrate out to 1.5R dashedcurveinthetoppanelshowsthevalueofγ derivedfrom
25
theScalob-parameterfortheSLOANsurvey(Brinchmannetal.
while we integrate out to 5 scale-lengths (when the profiles
2004).
extend this far, i.e. for roughly for half of the galaxies) for
consistency with the Milky Way analysis.
Fig.7showsthederivedvaluesfortheglobalγasafunc-
tionofthestellarmass(top)andHubbletype(bottom),the pletely different ways. Our values tend to be slightly above
black star shows the Milky Way. The bars show the ranges the curve for masses lower than 10.5 in the log and below
allowed by the two methods described above. Note that for for larger masses. This is likely due to the shape adopted
somegalaxiesalsothestellarmassesresultingfromourinte- fortheSFH,seediscussioninSection4.2.Notethat,inthis
grations differ from those of Leroy et al. (2008). The values respect, Fig. 7 is analogous to Fig. 3 in Kennicutt (1998a).
of γ inferred for these galaxies are all included between 0 Fortheabovediscgalaxieswehaveperformedthesame
and 11, with one outlier at γ = 23, NGC2841, a massive analysisasfortheMilkyWay.Wederivedtheradialprofiles
Sb galaxy with rather low current SFR (shown only in the of γ(R) and Σ (R,t) and integrated the latter to derive
ext
topplot).ThereisacleartrendwithstellarmassandHub- theglobalaccretionrates.Fig.8showstheresultsforthree
ble type, showing that big galaxies had much larger SFRs galaxiesrepresentativeofthreeclassesofstellarmassesand
in the past. On the other hand, low values of γ, typical for galaxytypes(seelabelsinFig.7).Ingeneral,smalllate-type
smallergalaxiesbetrayadisccontinuouslyformingthrough galaxies tend to have a flatter SFH and an efficiency of gas
the acquisition of fresh gas. accretion close to 1, see the case of NGC4449 (left panels).
As mentioned, we can relate our γ to the Scalo b- Galaxies of intermediate types have a shallower SFR and
parameter.Weusetherecentdeterminationofbversusstel- efficiency gradually decreasing with time. Finally, massive
lar mass relation from the SDSS (Brinchmann et al. 2004) discswithlowcurrentstarformationratesseemtohaveex-
andconvertitintoγ.Weplotthisrelationasadashedcurve hausted their capability of acquiring significant amount of
in the top panel of Fig. 7. The two parametrisations agree gasfromtheenvironment.Theynowappearto beconsum-
remarkably well, considering that they are derived in com- ing their remnant gas before moving to the red sequence
(cid:13)c 2011RAS,MNRAS000,1–??
8 Filippo Fraternali & Matteo Tomassetti
z z z
0 0.1 0.3 0.7 1 2.0 3.0 0 0.1 0.3 0.7 1 2.0 3.0 0 0.1 0.3 0.7 1 2.0 3.0
0.5
25
NGC 4449 4 NGC 628 NGC 5055
0.4
20
SFR 3 SFR
-1yr) 0.3 SFR 15
(M ! 0.2 Gas accretion rate 2 10
0.1 1 Gas accretion rate 5
Gas accretion rate
0 0 0

1 1 1
cy ()(cid:161) 0.8 0.8 0.8
n
e
ci
effi 0.6 0.6 0.6
n
o
eti 0.4 0.4 0.4
cr
c
A 0.2 0.2 0.2
0 0 0
0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12
Lookback time (Gyr) Lookback time (Gyr) Lookback time (Gyr)
Figure 8. Top panels: SFRs and gas accretion rates as a function of time for three representative galaxies from the sample of Leroy
etal.(2008).Theshadedareasshowtheuncertaintiesinthedeterminationoftheglobalγ describedinthetext.Theopendotsarethe
average SFHs estimated by Panter et al. (2007) for galaxies of the same stellar masses. Bottom panels: Gas accretion efficiencies as a
functionoftime,ε=1correspondstoacompletereplenishmentandthusaconstantSFRwithtime.
(e.g.Cappellarietal.2011).InthesameFig.8wealsocom- of the disc or decrease the return factor the value of the
pare our results with the average SFHs derived using the current accretion rate may increase substantially. What re-
SLOAN survey by Panter et al. (2007), shown as open cir- mainsindisputableisthefundamentalrelationbetweenstar
cles.Panteretal.dividetheirgalaxiesinbinsofcurrentstel- formation and accretion and the fact that the latter must
larmasses,thoserelevanthereare3×1010 <M <1×1011, beamajorplayerintheevolutionofgalaxiesfromtheirfor-
∗
1×1010 <M <3×1010, M <1×1010, respectively ap- mation to the present. There is no way a disc galaxy could
∗ ∗
plicableforNGC5055,NGC628,andNGC4449.Theagree- become as we see it today without substantial gas infall at
mentbetweenourSFRsandtheiraverageSFRsisverygood, z<1.
exceptforthegalaxiesintheintermediatemassclassasour
method tends to find steeper SFHs. Note however that this
may also depend on the way the stellar masses are esti-
mated. We used the estimate of Leroy et al. (2008) who 4 DISCUSSION
simplymultiplythe3.6-µmsurfacebrightnessbyaconstant
In the previous Sections we have shown how a simple com-
mass-to-light ratio of 0.5.
parison between the stellar density in galactic discs and
In general, the analysis of these 21 THINGS galaxies
their current SFRD (or gas density) implies the need for
gives results similar to those found for the Milky Way’s
alargeamountofgasinfallduringtheirlifetimes.Thisneed
disc.GasaccretionproceedsalmostparalleltotheSFRand
isrootedintheexistenceoftheKennicutt-Schmidtlawand
it remains significantly important throughout a large frac-
intheassumptionthatthisdoesnotsignificantlyevolvewith
tion of the lifetime of a typical disc galaxy. The efficiency
time. Under this assumption we were able to derive the ac-
of accretion seems to decrease fast for large galaxies, while
cretionratesasafunctionofgalactocentricradiusandtime
smalllate-typediscsremainquiteefficientincollectingcold
foranumberofdiscgalaxiesincludingtheMilkyWay.Here
star-forminggasfromtheenvironment.Thecurrent(z=0)
we discuss the limitations of our approach and the implica-
values of gas infall that we derive are quite low for most
tions of our results.
galaxies in the sample and a few also have negative val-
ues showing that the available gas is more than sufficient
for star formation to proceed. Thus, it would appear that
4.1 Delayed stellar feedback
thesediscgalaxiesdonotneedgasaccretiontosustaintheir
current star formation (see e.g. NGC5055). This is poten- Treating stellar feedback with the I.R.A. saves computing
tially a very interesting result but unfortunately, as shown time and it allows us to write simple analytic expressions
for the Milky Way, these values are very uncertain because for the equations of the model, however the delayed return
they come from an extrapolation. Moreover, they depend from stars with mass M< 8M may be an important ef-
(cid:12)
onparameterslikethestellarmassesthathaveintrinsicun- fect that needs to be quantified. We added this ingredient
certainties and other parameters that are simply assumed using the approximation that a star with mass M returns
(seeeq.14).Forinstanceifoneincreasestheformationtime a fraction η(M) of its mass to the interstellar medium at a
(cid:13)c 2011RAS,MNRAS000,1–??
Gas accretion using the Kennicutt-Schmidt law 9
timet (M)(seeKennicuttetal.1994).Thestellarlifetime z
MS
(t ) are taken from Maeder & Meynet (1989): 0 0.1 0.2 0.3 0.5 0.7 1 1.8
MS
9
t (Gyr) = 12 (cid:18) M −2.78(cid:19) M (cid:54)10 M 8 MILKY WAY
MS M (cid:12)
(cid:12) 7
(cid:18) M −0.75(cid:19)
0.11 M >10 M (17) 6
M(cid:12) (cid:12) -1yr) 5 SFR
given that tMS(1 M(cid:12)) > 10 Gyr, stars below ∼ 1 M(cid:12) do ! 4
M
not contribute to the feedback. The difference between the ( Gas accretion rate
3
I.R.A. and the delayed return in eqs. (1) and (2) is in the
term Σ˙ that becomes: 2
fb
(cid:90) 100 1 with delayed feedback
Σ˙fb = SFRD(t−tMS)η(M)φ(M)dM (18) 0
1
where φ(M) is the IMF. Given the assumed form of the 1
)(cid:161)
tSoFeRqD.3(.RC,to)n(seeqq.u7e)n,ttlyh,etdheeleaxypedrefseseiodnbaocfkγa(Rdd)staakneesxtthrea-ftoerrmm: ncy ( 0.8
e
ci
γDF(R)= −(11−−RR+−22tτ2ftoDτfroDFΣmrFm∗)(tRfo,r0m)SFRD(R,0) (19) ccretion effi 00..46
1−R+2 τDF A 0.2
tform
with delayed feedback
where: 0
0 2 4 6 8 10
(cid:90) 100
τ = t (M)η(M)φ(M)dM (20) Lookback time (Gyr)
DF MS
1
Figure 9. Toppanel: SFRandglobalgasaccretionrate,versus
is a characteristic delayed-feedback time that, with our as-
timeandredshiftforthediscoftheMilkyWaywithinstantaneous
sumptions, turns out to be τ = 3.9 × 108yr (6.0 and
DF recycling approximation (thin solid lines) and delayed feedback
6.3×108 for Kroupa and Chabrier IMFs respectively). Eq.
(thick dashed lines). Bottom panel: Gas accretion efficiency as
(19) clearly reverts to eq. (10) when τ = 0. Finally, the
DF a function of time with instantaneous recycling approximation
following extra-term: SFRD(R,0)(γ(R)−1)tτfoDrFm needs to (thinsolidline)anddelayedfeedback(thickdashedline).
beaddedtotheaccretionrateΣ (R)(eq.14)totakeinto
ext
account the delay.
Fig. 9 shows the comparison between the global SFR
and accretion rate for the Milky Way with I.R.A. and with
delayedstellarfeedback.ThevaluesofcurrentSFRandstel-
lar mass are kept fixed in the two models. As expected the at higher lookback times; ii) a triangular function of time
globalgammawithdelayedfeedbackislower,γDF =2.6(2.5 f3(t) that linearly increases to a maximum SFR at t = tc
for a Chabrier IMF), than that with I.R.A. This because if andthendecreasestoSFR(0)att=t ;iii)anexponential
form
themassreturntotheISMisdelayedthenlessstarsneedto functionofthisform:f (t)=et/τ whereτ istheexponential
4
forminordertohavethesamestellarmassattheend.Asa e-foldingtimeoftheSFR.Withthesechoicesweencompass
consequencealsothegasaccretionisslightlylowerformost both monotonic and non-monotonic functions. In ii) γ is
of the life of the galaxy, reaching however a slightly higher defined as the ratio between SFR(t )/SFR(0).
c
value of 1.4M(cid:12)yr−1 at the current time. Given that the In Table 1, we show the values of the global γ, and the
extra-termintheaccretionformulaispositive(forγ(R)>1) present accretion rate M˙ (0) and accretion efficiency ε(0)
ext
theefficiencywithdelayedfeedbackisalwaysslightlyabove obtained with the four types of f(t) for the Milky Way. We
theI.R.A.value.Inconclusion,includingadelayedfeedback took t = 7.7Gyr (z = 1). For the exponential function,
c c
in our calculations makes little difference and it goes in the often used for early type galaxies (e.g. Bell et al. 2003), we
direction of increasing the efficiency of gas accretion. found a rather long scale-time of τ = 8.5Gyr and a shape
of the SFH very similar to that obtained with f (t). Even
1
thoughthesethreefunctionsdepictverydifferenttrendsfor
4.2 Parametrisation of the SFH
the SFH, the disc-integrated γ varies only by ∼ 17% and
The reconstruction of the star formation rate density as a the values of the present accretion rate and ε(0) change by
functionoftimerequirestheparametrisationoftheSFH.In no more than 13 %. Note that the inverse of the Scalo b-
Section 2.1 we have chosen a simple first order polynomial. parameterwouldcorrespondtoapiece-wiselinearfunctional
Wealsopointoutthatamoregeneralpower-lawwouldgive form f (t) for t →0. All the above values are obtained us-
2 c
similar dependencies for γ(R). Here we consider three dif- ingtheI.R.A.Usingthedelayedreturnwithdifferentshapes
ferent parametrisations and show that our results do not of the SFH can produce different effects. For instance an
changesignificantly.Wetakethefollowingfunctionalforms: exponential SFH of the type f (t) maximizes the gas con-
4
i) a piecewise-linear function of time f (t) that linearly in- sumptiontimescales,whichcanincreaseuptoafactor2−3
2
creases to a maximum SFR at t=t and remains constant at the earliest times (see Kennicutt et al. 1994).
c
(cid:13)c 2011RAS,MNRAS000,1–??
10 Filippo Fraternali & Matteo Tomassetti
f(t) γ M˙ext(0) ε(0) vt[kpc] Σ(cid:48)∗ h(cid:48)R (γ(cid:48)/γ)MW
linear(1) 2.8 1.31 0.62 -3 1226.96 2.31 1.25
piece-wiselinear(2) 2.5 1.27 0.61 -2 980.63 2.59 1.08
triangular(3) 2.4 1.21 0.58 -1 792.97 2.88 1.02
exponential(4) 3.2 1.49 0.71 0 648.22 3.2 1.00
1 534.77 3.54 1.04
2 444.79 3.89 1.12
Table 1. The effect of using different parametrisations for the
3 372.61 4.27 1.21
SFHdescribedinthetext.γisalwaysdefinedastheratiobetween
theinitialoverthecurrentSFRexceptforthetriangularfunction
whereitistheratiobetweentheSFRatzc=1overthecurrent Table2.Theeffectofradialmigrationofstarsonanexponential
value. stellarprofileandonthedisc-integratedγ forourGalaxy
4.3 Effect of stellar migration in a manner that preserves the overall angular momentum
distribution.Starsmovebothinwardsandoutwardswithout
In the above we have assumed that radial motions of stars
any significant radial spreading of the disc or increase in
inthediscarenotsignificant,herewerelaxthisassumption
non-circular motions. In their Fig. 13 they plotted the final
and estimate the effect of a global migration (e.g. Roˇskar
distributionofhomeradiiforsixdifferentradialbinsforour
et al. 2008; Minchev et al. 2011). For simplicity we take
Galaxy. The distributions are roughly Gaussian for every
anexponentialstellarprofilewithcentralsurfacebrightness
radial bins, with the largest dispersion σ (cid:39) 3kpc at about
Σ (0), a scale length h , and a maximum extent at radius
∗ R R . So as to give an upper limit on this effect we infer the
(cid:12)
Rm =5hR. The stellar mass of the disc is: new stellar profile Σ (R)(cid:48) as the average of that modified
∗
(cid:20) (cid:18) (cid:19)(cid:21)
M(Σ∗(0),hR)=2πΣ∗(0)h2R 1−e−RhRm 1+ Rhm (21) bthyatinowfaardn oraudtwiaalrmdomtiootniowniwthitphapraamraemteertevrtv=t=−33 kkppcc.aWnde
R
foundthatthechangesinγ(R)aresignificantforsomeradii
Consider now a radial motion with a constant velocity v at
but there is basically no effect on the global value of γ. We
R that lasts a time t directed inward or outward. Since
m conclude that both migration and random redistribution of
the initial stellar profile is truncated beyond R , all stars
m stars over the disc do not affect our global results.
atthisradiuswillmovetothenewpositionR +vt,which
m
represents a new maximum radius and is greater or smaller
than Rm depending on the sign of v. This condition trans- 4.4 Comparison to other studies
lates to the following relation involving the new (i.e. after
migration) stellar profile Σ(cid:48): Itisamatterofdebatewhethergalaxieshadtheirreservoir
∗
of gas in place from the beginning or they have been har-
Σ(cid:48)∗(Rm+vt)=Σ∗(Rm) (22) vesting the gas they needed throughout their lives. We find
that the second must be the case for all late type galaxies.
that,togetherwiththeassumptionthattheprofileremains
A concentration of gas as large as the current disc stellar
exponential and that the mass is conserved:
masseswouldimplyanextremelyhighSFRatearliertimes,
M(cid:48)(Σ(cid:48)∗(0),h(cid:48)R)=M(Σ∗(0),hR), (23) asuddenformationofmostofthestarsandasteeplydeclin-
allowsustoestimateΣ(cid:48)(0)andh(cid:48) ,thenewcentralsurface ingstarformationatlatertimes.Thisishowaredsequence
∗ R
brightness and scale length. galaxy forms and evolves. Galaxies like the Milky Way or
In Table 2 we report the variation of the exponen- spirals of later types clearly did not have this kind of star
tial profile of the Milky Way (Σ (0) = 648.2M pc−2 and formationhistory.Theevidencecomesfromthereconstruc-
∗ (cid:12)
h = 3.2 kpc) in the presence of coherent radial motions. tionofthestarformationhistoriesfromthecolormagnitude
R
Whentheflowofstarsisdirectedinward,obviouslythenor- diagrams(e.g.Harris&Zaritsky2009),fromchemicalevolu-
malizationofthestellarprofileincreasesandthescalelength tionmodels(e.g.Chiappinietal.2001),andfromcosmology
decreases. Eq. (10) tells us that γ(R)+1 ∝ Σ (R). There- (e.g. Hopkins et al. 2008). Our study further supports this
∗
fore, the profile of γ(R) will change to γ(cid:48)(R) according to point making use of the Kennicutt-Schmidt law.
this relation: Apossiblewayoutwouldbetoinvokeavariationofthe
K-Slawwithredshiftorwithmetallicityasproposedbye.g.
Σ(cid:48)(R)
γ(cid:48)(R)= ∗ (γ(R)+1)−1 (24) Krumholz & Dekel (2011). However, to reconcile the obser-
Σ (R)
∗ vations with negligible accretion at z < 1 the variations in
For radii R where Σ(cid:48)(R)/Σ (R) > 1, eq. (24) predicts the K-S law should be very large. If one required the initial
∗ ∗
an increase in the parameter γ(R) and vice versa for gas mass of the Milky Way disc to be as large as the stel-
Σ(cid:48)(R)/Σ (R)<1.Thiscanbeexplainedconsideringthata lar mass today, the K-S law would imply an initial SFR of
∗ ∗
radialmotionthatbringsmorestarsatacertainradius(i.e. ∼50M yr−1.Atthatrate,thestellardiscwouldformina
(cid:12)
Σ (R)(cid:48)/Σ (R) > 1) makes the latter as it had been more Gyrandthesubsequentevolutionwouldbepassive(similar
∗ ∗
star-forming. Despite the changes in γ(R) it is remarkable shapetotheclosed-boxshowninFig.5).Thereforetokeep
that the new global γ changes very little, see the rightmost the current SFR at the observed value the transformation
column of Table 2, showing that our global results are only of gas into stars should have been an order of magnitude
slightly affected by radial migration. lessefficientthantheK-Slawwouldpredict.Moreover,this
Sellwood & Binney (2002) showed that the dominant efficiencyshouldhaveevolvedwiththegalaxyitselfinorder
effect of spiral waves in galaxy discs is to churn the stars tokeepthestarformationroughlyconstantintime.Thisis
(cid:13)c 2011RAS,MNRAS000,1–??