Dust evolution with active galactic nucleus feedback in elliptical galaxies

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by  Hiroyuki Hirashita

Dust evolution with active galactic nucleus feedback in elliptical galaxies
HiroyukiHirashitaa,TakayaNozawab
aInstituteofAstronomyandAstrophysics,AcademiaSinica,P.O.Box23-141,Taipei10617,Taiwan
bNationalAstronomicalObservatoryofJapan,Mitaka,Tokyo181-8588,Japan
Abstract
7
Wehaverecentlysuggestedthatdustgrowthinthecoldgasphasedominatesthedustabundanceinellipticalgalaxieswhiledustis
1
efficientlydestroyedinthehotX-rayemittingplasma(hotgas). Inordertounderstandthedustevolutioninellipticalgalaxies,we
0
2constructasimplemodelthatincludesdustgrowthinthecoldgasanddustdestructioninthehotgas. We alsotakeintoaccount
the effect of mass exchangebetween these two gas componentsinducedby active galactic nucleus(AGN) feedback. We survey
n
areasonable rangesof the relevantparametersin the modeland find that AGN feedbackcycles actually producea variety in cold
Jgasmassanddust-to-gasratio.Bycomparingwithanobservationalsampleofnearbyellipticalgalaxies,wefindthat,althoughthe

5dust-to-gasratiovariesbyanorderofmagnitudeinourmodel,theentirerangeoftheobserveddust-to-gasratiosisdifficulttobe
2reproducedunderasingleparameterset. Variationofthedustgrowthefficiencyisthemostprobablesolutiontoexplainthelarge

varietyindust-to-gasratiooftheobservationalsample. Therefore,dustgrowthcanplayacentralroleincreatingthevariationin
]
Adust-to-gasratiothroughtheAGNfeedbackcycleandthroughthevariationindustgrowthefficiency.
G
Keywords: Activegalacticnuclei,Dust,Ellipticalgalaxies,Galaxyevolution,Interstellarmedium
.
h
p
-1. Introduction dustexistinginellipticalgalaxiesispossiblyinjectedfromout-
o
side via themergingoraccretionof externalgalaxies(Forbes,
r
t InthenearbyUniverse,ellipticalgalaxiesareknowntohave 1991;Temietal.,2004;Fujitaetal.,2013).Thelackofcorrela-
s
aless gas, dust, and ongoing star formation activity than spiral tionbetweendustFIRluminosityandstellarluminosityisalso
[galaxies. Yet,theystillhavesomeamountofinterstellargasin takenasevidenceofexternaloriginofdust(Temietal.,2007);

theformofhotX-ray-emittinghalogas(e.g.,O’Sullivanetal., however, this argument may not hold if dust is processed by
1
v2001) and cold gas (e.g., Wiklindetal., 1995). More- mechanismsunrelatedtostars.
0over, dust is detected for a significant fraction of elliptical
Recently, Hirashitaetal. (2015) have proposed that the ex-
0galaxies by optical extinction (e.g., Goudfrooijetal., 1994;
istence of dust in elliptical galaxies can be explained by dust
2
vanDokkumandFranx, 1995; Ferrarietal., 1999; Tranetal.,
7 growth by the accretion of gas phase metals in the cold in-
2001) or far-infrared(FIR) emission (e.g., Knappetal., 1989;
0 terstellar medium (ISM). They also suggest that the presence
.Smithetal.,2012;diSeregoAlighierietal.,2013). Dustmass of dust growth also explains the extinction curves observed
1
is estimated from the reddening in the optical or from FIR
0 in elliptical galaxies. The dust-to-gas ratio could become as
7dust emission, and ranges from ∼ 104 to ∼ 107 M⊙ (e.g., high as & 10−3 by accretion, explaining the high dust abun-
1Goudfrooijetal., 1994; Leeuwetal., 2004). Since the exis- danceinsomeellipticalgalaxies. However,dustgrowthinthe
:tence of dust could affect the cooling and chemical processes
v cold gas has not been considered as a major source of dust
(Dwek, 1987; Fabianetal., 1994; VoitandDonahue, 1995;
i in the context of dust evolution in elliptical galaxies. Dust
X
Seoketal., 2015; Hirashitaetal., 2015), the understandingof
growth has already been noted as a major mechanism of dust
rthe originandevolutionofdustinellipticalgalaxiesisimpor-
a mass increase in a wide context of galaxy evolution (e.g.,
tantinclarifyingtheirevolution.
Dwek, 1998; Hirashita, 1999; Inoue, 2003; Zhukovskaetal.,
Because the stellar population is dominated by old stars
2008; Valianteetal., 2011; MattssonandAndersen, 2012;
whose ages are comparable to the cosmic age, the dust in el-
Mancinietal., 2015; Poppingetal., 2016; Houetal., 2016;
lipticalgalaxiesispredominantlysuppliedbyasymptoticgiant
Zhukovskaetal., 2016), and some experimental studies have
branch(AGB)starsratherthan bysupernovae. However,dust
also shown that dust grains could grow by accreting the gas-
destructionbysputteringintheX-rayemittingplasmaissoef-
phase metals (Rouille´etal. 2014; but see Ferraraetal. 2016).
ficientthattheobserveddustmasscannotbe explainedbythe
Thus, constructing a dust evolution model that includes this
balance between the supply from AGB stars and the destruc-
dustgrowthmechanismwouldcontributetotheunderstanding
tion(e.g.,Patiletal.,2007). Thus,someauthorsarguethatthe
ofdustevolutioninellipticalgalaxies.
Theoverallevolutionofdustshouldbeconsideredinrelation
Emailaddress:[email protected](Hiroyuki tothegasevolution,especiallybecausegasanddustareusually
Hirashita) dynamically coupled on galactic scales. Recent studies have
PreprintsubmittedtoPlanetaryandSpaceScience January26,2017
proposedthat the gascoolingin galaxiesis stronglyregulated ofdustevolutioninanellipticalgalaxyinSection2. Weshow
by the energy input from active galactic nuclei (AGNs). The theresultsinSection3. Wediscussthemodelpredictions,and
energy input from AGN winds (SilkandRees, 1998; Fabian, comparethemwithobservationaldatainSection4. Finallywe
1999; King, 2005) or AGN jets (WagnerandBicknell, 2011; concludeinSection5.
Mukherjeeetal., 2016) preventscoolingflowsfromoccurring
and/or makes cold gas evaporate(see also CiottiandOstriker,
2. Model
2001; Fabian, 2012). These kinds of energy input are called
AGN feedback, and are considered to play an important
Weconstructamodelthatdescribesthedustevolutioninan
role in galaxy formation and evolution (Crotonetal., 2006;
elliptical galaxy. For dust sources, in addition to AGB stars
BoothandSchaye,2009)
considered in previousstudies (MathewsandBrighenti, 2003;
Temietal. (2007) showed the existence of a dust emission
Patiletal., 2007), we also consider dust supply from the cold
componentextendedover5–10kpcinellipticalgalaxiesusing
gaswheredustgrowsbytheaccretionofgas-phasemetals.The
the Spitzer 70 µm band data. They suggestthatthis cold dust
dustinthecoldgasisinjectedintothehotgasbyAGNfeedback
componentwasoriginallycontainedinthecoldgas,whichwas
togetherwiththecoldgas. Thedustinjectedintothehotgasis
then heated by AGN feedback and eventually mixed with the
destroyed by sputtering. The hotcomponent(∼ 107 K) is the
hotgas. Intheirscenario,theheatedgasistransportedintothe
gaswhosetemperatureiscomparabletothevirialtemperature
hotgasbybuoyantforce,andthetime-scaleofthetransportis
determinedbytheglobalgravitationalpotentialoftheelliptical
around107yr. Kanedaetal.(2011)alsoshowedinanelliptical
galaxywhile the cold componentis cold and denseenoughto
galaxy (NGC 4125) a dust emission componentwhose exten-
host dust growth (. 100 K; Hirashitaetal., 2015). To make
sionissimilartothedistributionofthehotX-rayemittinghalo.
the modelassimple as possible, we treateach gascomponent
Themulti-phasestructuresandirregularmorphologiesofX-ray
as a single zone and consider the mass exchange between the
emittinghotplasmascouldbeexplainedbybubblescreatedby
twocomponents. Dustgrowthanddestructionaretreatedcon-
AGNfeedback(Buoteetal.,2003).
sistentlywiththeevolutionofeachgascomponentasexplained
Apartofthehotgasmaycooldowntoreformthecoldgas,
below.
which could contribute to the fueling of AGN (Werneretal.,
2014). AGNfeedbackmaycompressthesurroundinggasand
2.1. Basicequations
enhance gas cooling locally (ValentiniandBrighenti, 2015).
Because dust grains can grow in the cold gas as mentioned We consider the mass exchange between the hot and cold
above,thecoldgasinjectedintothehotgasbyAGNfeedback phases. The evolutionofthe coldgasmass Mg,c asa function
would supply the dust to the hot gas. This dust supply could oftimetiswrittenas
bedominantoverthedustproductionbyAGBstars. Ifso,for- dM
mationofcoldgasandthesubsequentAGNfuelingleadingto g,c = M˙in−M˙ret, (1)
dt
AGN feedback play a dominant role in the dust evolution in
ellipticalgalaxies. whereM˙inistheinfallrateofthecooledgasandM˙retisthegas
If the dustevolutionis affected byepisodic AGN activities, return rate from the cold to the hot phase by AGN feedback.
which appear as a result of a cycle of gas ejection and cool- Thesetwotermsaremodeledbelow.
ing, the statistical properties of the dust abundances in ellip- Thehotgasistreatedasaconstantreservoirofgasforsim-
tical galaxies are determined by the time-scales of dust pro- plicity;thatis,weassumethatthemassofhotgas,Mg,hiscon-
cessing relative to the period of an AGN cycle. Laueretal. stant(dMg,h/dt = 0). Thistreatmentneglectsthecomplication
(2005)inferredtheperiodofanAGNcycleinearly-typegalax- arisingfromthepossibilitythattherecouldbeasupply/lossof
ies based onthe dustlifetime againstsputteringin the hotgas hotgas from/tooutside. Because the hot gasbasically acts as
andthefractionofdustdetectionforasample,obtainingape- an efficient destroyer of dust regardlessof the choice of Mg,h,
riod of ∼ 108 yr. The dust contained in the cold gas may be thevalueofMg,hhasaminorinfluenceontheresultscompared
injected into the hot gas in an episodic way associated with withotherparameters. However,weshouldnotethatthedust-
theAGNcycle,ifAGNfeedbackefficientlyheatsthecoldgas to-gas ratio in the hot gas is directly affected by Mg,h since it
(Mathewsetal.,2013). directlyentersthe dust-to-gasratioin the hotgas. We discuss
Inthispaper,wemakeatheoreticalmodelofdustevolution thevalueofMg,hagaininSection2.3.
in AGN feedbackcycles by includingimportantphysicalpro- Thetimeevolutionofthedustmassinthecoldphase, Md,c,
cessessuchasthemassexchangebetweenthehotandcoldgas iswrittenas
componentsandthedustevolutioninthosegases.Fordustpro- dM M
cessing, we consider dust destruction by sputtering in the hot dtd,c =DhM˙in−DcM˙ret+ τ d,c, (2)
grow
gas and dust growth by the accretion of gas-phase metals in
the cold gas. This modelingenablesus to examinehowAGN whereD ≡ M /M andD ≡ M /M arethedust-to-gas
h d,h g,h c d,c g,c
feedback cycles affect the dust abundancesin elliptical galax- ratiosinthehotandcoldphases,respectively(M andM are
d,h d,c
ies. Wecanalsoexaminetheeffectofdustgrowthonthedust the dust mass in the hotand cold gas, respectively), and τ
grow
abundanceinellipticalgalaxiesforthefirsttime. is the time-scale of dust growth by the accretion of gas-phase
The paper is organizedas follows: we formulatethe model metalsinthecoldgas.Thedustgrowthtimescaleτ isgiven
grow
2
where [x] is the floor function, which indicates the largest in-
Table1:Parametersinthemodel.
teger that satisfies [x] ≤ x. This equation indicates that the
Parameter Unit Fiduciala Rangeb outflowisperiodicallyturnedoneveryτAGN withadurationof
M˙in,0 M⊙ yr−1 3 1–10 fretτAGN.
τ yr 107 (0.3–3)×107 Theinflowofcooledhotgasisassumedtooccuronlywhen
tr
τAGN yr 108 fixed thereisnooutflowactivity(i.e.,whenMret =0):
f — 0.3 0.1–0.5
ret 0 if0≤t/τ −[t/τ ]≤ f ,
τ yr 107 106–108 M˙ (t)= AGN AGN ret (7)
τacc yr 107 106–108 in ( M˙in,0 otherwise,
sput
Z — 0.02 fixed wherewetreatM˙ asaconstantfreeparameter.
in,0
DAGB — 0.01 fixed Based on the inferred dust lifetime and dynamical time in
α yr−1 1.6×10−12 fixed thecenterofanellipticalgalaxyandtherateofdustdetection
M∗ M⊙ 1010.5 1010–1011 forellipticalgalaxies,Laueretal.(2005)suggestedthatthecy-
cle ofdustappearanceanddisappearanceis∼ 108 yr. We can
aFiducialvalue.
regard this time as τ if that dust cycle is induced by the
bRangeofvaluesinvestigatedunlesstheparametervalue AGNfeedbackcycle.ATGhNus,weadoptτ =108yr. Sincethe
AGN
isfixed.
timeevolutionofgasanddustisdeterminedbythetime-scale
relative to τ , we fix τ and change other time-scales as
AGN AGN
in Section2.3(equation9), andistreatedasafunctionofD . describedbelow. Themaximumcoldgasmassisroughlyesti-
c
Thetimeevolutionofthedustmassinthehotphase, Mh,c, on matedasM˙in,0τAGN(1−fret),whichshouldbecomparabletothe
theotherhand,iswrittenas observedgasmass∼afew×108M⊙(e.g.,Wiklindetal.,1995).
dMdtd,h =−DhM˙in+DcM˙ret− Mτ d,h +DAGBαM∗, (3) TinhviessmtigeaatnesathraatnMg˙eino,0f∼∼a1f–e1w0×M(⊙1−yr−fr1e.t)−S1inMce⊙cyor−ld1.gTahsuiss,dwee-
sput tectedinasignificantfractionofellipticalgalaxies, f cannot
ret
beverynearto1. Thus,wefocusonarangeof0< f ≤0.5.
where τ is the dust destruction time-scale by sputtering in ret
sput
thehotgas,D isthedust-to-gasratioinAGBstarwinds,α For τtr, it may be reasonable to adopt the dynamical time.
AGB
AsmentionedintheIntroduction,Temietal.(2007)suggested
isthemasslossratesofAGBstarsperstellarmass,and M∗ is
dusttransportbybuoyancyactingontheheatedcoldgasinor-
the total stellar mass (αM∗ is the total mass loss rate of AGB
stars). UsingM =D M , M =D M ,andequation(1), der to explain the origin of extended dust FIR emission com-
d,c c g,c d,h h g,h
ponent. Theyestimatedthetime-scaleoftransportas∼ 107 yr
we rewite equations(2)and(3)as thefollowingequationsfor
based on the dynamicaltime-scale. Thus, we adoptτ ∼ 107
thedust-to-gasratios: tr
yr,andalsoexamineanorder-of-magnituderangeforτ .
tr
dD D M˙
c = c +(D −D ) in , (4)
dt τ h c M 2.3. Parametersfordustevolution
grow g,c
We have introduced two time-scale parameters that govern
dDh =−D M˙in − Dh +D M˙ret +D αM∗. (5) thedustevolution: dustgrowthtime-scale(τgrow)anddustde-
dt hM τ cM AGBM struction(sputtering)time-scale(τsput).
g,h sput g,h g,h
We adopt the following estimate of the time-scale of grain
We solve equations (1), (4) and (5). Below we formulate growth(Asanoetal.,2013):
some undeterminedtermsand estimate a reasonablerangefor
eachparameter(Table1). τ = 2×106 a nH,c −1 Tgas −1/2
acc 0.1µm!(cid:18)103cm−3(cid:19) 50K!
2.2. Massexchangebetweenthephases Z −1
× yr, (8)
The mass exchangebetween the cold and hot phases is de- (cid:18)0.02(cid:19)
scribed by M˙ (inflow of cooled hot gas to the cold gas) and
in where a is the grain radius, n is the hydrogennumber den-
M˙ (returnofcoldgastothehotphasebyAGNfeedback).For H,c
ret sity in the cold gas, T is the gas temperature, and Z is the
simplicity, we assume that AGN feedbackoccurs periodically gas
metallicity (e.g.,Inoue, 2011; Asanoetal., 2013). Because of
with a period of τ . During an episode of AGN feedback,
AGN somepoorlyconstrainedparameterssuchasaandn ,wetreat
the cold gas is transported to the hot gas, and this transporta- H,c
τ as a given constant parameter for simplicity, but investi-
tion time-scale is parameterized by τ . This transport (or the acc
tr gateawiderangeinτ . Moreover,sinceaccretionisefficient
return of cold gas to the hot gas) lasts for a fraction of τ , acc
AGN only in the dense medium, it depends on the mass fraction of
andthisfractionisparameterizedby f . Thatis, theduration
ret densegasinthecoldgas,whichisdenotedas f . Thedust
of an AGN feedbackepisode is f τ . We assume that the dense
ret AGN growthtime-scaleiseffectivelyestimatedasτ /f . Based
outflow(M˙ )ofthecoldgasisonduringtheAGNfeedbackas acc dense
ret on the aboveestimate, we examineτ = 106–108 yr (107 yr
acc
M˙ (t)= Mg,c/τtr if0≤t/τAGN−[t/τAGN]≤ fret, (6) for the fiducial case), allowing for a small value for fdense or
ret ( 0 otherwise, variationinaandnH,c(Kuoetal.,2013;Schneideretal.,2016;
3
Aoyamaetal.,2017). Inreality,theaccretiontime-scaleisalso sizedistributionisconsistentwiththeopticalextinctioncurves
affected by grain size distribution (KuoandHirashita, 2012); observedbyPatiletal.(2007),althoughtheydidnotconsidera
thus,theabovearepresentstheappropriateaverageofthegrain fullcycleofAGNfeedback.Atthesametime,theyalsoshowed
size.1 Considering the uncertainties in grain size distribution, thattheresultinggrainsizedistributiondependsonthesizedis-
τ maybeoutoftherangeabove(106–108 yr). Ifτ . 106 tributionofdustgrainsproducedbyAGBstars. Toavoidsuch
acc acc
yr, the grain growth is so efficientthat the dust-to-gasratio in acomplication,wetreatτ andτ asgivenconstantparam-
acc sput
the cold gasstays at a maximumvaluegiven below(D ) for eters and concentrate on the total dust mass. The grain size
sat
most of the time regardless of the value of τ (. 106 yr). If distributionrealizedafterAGNfeedbackcycleswillbeinvesti-
acc
τ &108yr,incontrast,dustgrowthisnotefficientenoughto gatedinthefuturework.
acc
raise the dust-to-gasratio above10−4. Therefore,the rangeof We adopt DAGB = 0.01, α = 1.6× 10−12 yr−1, and M∗ =
τacc = 106–108 yr coversall the resulting behavior of interest 1010–1011 M⊙(TsaiandMathews,1995). However,theprecise
for D , i.e., coversallthe interestingrangeof D aswe show valuesofthesequantitiesdonotaffectthedustabundanceinthe
c c
below. coldgasaslongasthe dustformedin AGBstarsis efficiently
Using a given value of τ , the dust growth time-scale is destroyedby thehotgas. We fix D andα, andonlyfocus
acc AGB
givenby onthevariationofM∗.
τgrow = 1−Dτac/cD , (9) forAssimepxlpiclaitiyn.edThinehSoetcgtiaosnm2a.1ss,iwnethfiexcetnhetrahloatfgeawskmpacs,swMhicg,hh
c sat
coulddirectlyexchangethegasmasswiththecentralcoldcom-
where the denominator on the right-hand side means that ac- ponent,isestimatedas108–109 M⊙ (TsaiandMathews,1995).
cretionis saturatedas Dc approachesDsat, which isthe abun- Thus, we adopt Mg,h = 3×108 M⊙. The dust-to-gasratio in
dance of the metals available for dust formation. We assume thehotgas(D )isinverselyproportionalto M bydefinition;
h g,h
Dsat = 0.01 in this paper, since we are considering solar however,becausedustinthehotgasisefficientlydestroyedby
metallicity environment (i.e., Dsat ∼ Z⊙). The abundance sputtering,asshownbelow,Dh ≪Dc issatisfiedwhenthehot
of available metals for dust formationmay be correlated with gasis actively accretedon to the cold gas, exceptfor the case
the stellar mass because of the stellar mass–metallicity rela- where the sputtering time-scale is as long as the AGN feed-
tion (Gallazzietal., 2005). However, this relation has a large backcycle(τ &τ ). Therefore,theinflowispracticallya
sput AGN
scatter, anditisalsosuggestedthatthevelocitydispersion(or “dust-free”gasforthecoldgas,sothatthevalueofD doesnot
h
the virial mass) is more tightly related to the stellar metallic- influenceD . Thismeansthatthechoiceof M ,whichcould
c g,h
ity (Smithetal., 2009). Moreover, it is not clear how tightly affectD ,doesnothaveasignificantinfluenceonD .
h c
the stellar metallicity is related to the gas metallicity because We assume the following initial conditions: D = D =
c h
of accretion of external gas (SuandIrwin, 2013). Thus, we 10−6, and Mg,c = 107 M⊙. As seen later, the initial condition
simply adopt a constant Dsat, but we also note that increas- isnotimportant,sincethesystem“forgets”theinitialcondition
ing/decreasing Dsat has broadly the same effect of short/long in∼τAGN.
τ .
acc
The time-scale of dust destruction by sputtering is es-
timated as (TsaiandMathews, 1995; Nozawaetal., 2006; 3. Results
Hirashitaetal.,2015)
3.1. Fiducialcase
a n −1
τsput =7.1×106 0.1µm!(cid:18)10−2Hc,hm−3(cid:19) yr, (10) Firstofall,weshowthetimeevolutionofrelevantquantities
underthefiducialparametervalueslistedinTable1. InFig.1,
wheren isthenumberdensityofhydrogennucleiinthehot weshowtheevolutionsofthedust-to-gasratiosinthehotand
H,h
gas. Aswedidforτ above,wetreatτ asagivenconstant coldgas,thecoldgasmass,andthedustmassesinthecoldand
acc sput
parameter. Based on this estimation, we adopt τ = 107 yr hotgas. Herewedescribehoweachofthesequantitiesevolves
sput
forthefiducialcaseandexaminearangeof106–108yrconsid- asafunctionoftime.
ering the variety in a and n . Note that fixing τ and τ Allthequantitieshavealmostperiodicbehaviorswithape-
H,h acc sput
implicitlyassumesafixedgrainsize(orgrainsizedistribution). riodgivenbyτ . InassociationwiththeonsetofAGNfeed-
AGN
In reality, the grain size distribution itself is determined as a back,thedustgrowninthecoldgasisinjectedintothehotgas,
result of the cycle of dust destruction and growth. Therefore, sothatthedust-to-gasratiointhehotgasincreases. However,
a consistent treatment of those time-scales and the grain size the dust supplied to the hot gas is quickly destroyed by sput-
distributionwouldbenecessary. Hirashitaetal.(2015)consid- tering, and the dust-to-gasin the hot gas tends to convergeto
eredtheevolutionofgrainsizedistributionbydustdestruction thevaluedeterminedbythebalancebetweendustsupplyfrom
andsubsequentdustgrowthandshowedthattheresultinggrain stars and dustdestructionby sputtering. After AGN feedback
stops,thecoldgasincreasesitsmassbygasinfall.Inthisphase,
becausethedust-to-gasratiointheinfallinggas,whichisequal
1More precisely, a is the ratio of the mean a3 to the mean a2
to the dust-to-gas ratio in the hot gas, is smaller than that in
(HirashitaandKuo,2011). As shownin HirashitaandKuo(2011), the time
thecoldgas,thedust-to-gasratiointhecoldgasdecreasesasa
variationofacouldbeeffectivelyincorporatedinthedefinitionofτaccsothat
τacccouldstillbetreatedasconstant. resultofdilution.
4
Figure1: Timeevolutions ofrelevant quantities. (a)Dust-to-gas ratiointhe Figure2:Timeevolutionsofrelevantquantities.(a)Dust-to-gasratiointhehot
cold(Dc)andhotgas(Dh)(solidanddottedlines,respectively). (b)Coldgas gas(Dh).(b)Dust-to-gasratiointhecoldgas(Dc).(c)Coldgasmass(Mg,c).
mass(Mg,c)(solidline),anddustmassesinthecoldgas(Md,c)(dashedline) Thesolid,dotted,anddashedlinesshowtheresultsforM˙in,0 =3(fiducial),1,
andinthehotgas(Mh,c)(dottedline). and10M⊙,respectively,inallthepanels.
The above general behavior does not change by the choice expected, the cold gas mass is larger if M˙ is larger, but the
in,0
of parameter values. Thus, for the purpose of simplification, logarithmicamplitudeofthecoldgasmassdoesnotdependon
we hereafter concentrate on the dust-to-gas ratios in the cold M˙ , sincewefixthetime-scalesofAGNfeedbackandmass
in,0
and hotgas (D and D ), and the cold gas mass (M ), since transport.
c h g,c
thedustmassesinthehotandcoldgasjusttracetheevolution
of the corresponding dust-to-gas ratios. Below we show the 3.3. Masstransporttime-scaleinAGNfeedback
dependenceoneachparameterlistedinTable1.
The mass transport time-scale of AGN feedback, τ , regu-
tr
lates the mass loss rate of the cold gas when AGN feedback
3.2. Massinflowrate
ison(equation6). InFig.3,weshowthetimeevolutionofthe
WeconsiderthemassaccretionontothecoldgaswhenAGN quantitiesofinterestforτ =107(fiducial),3×106,and3×107
tr
feedbackisoff.Themassinflowrateisregulatedbytheparam- yr.
eter M˙ . In Fig. 2, we show the time evolution of D , D ,
in,0 c h Theincreaseanddecreaseofthedust-to-gasratiointhehot
and Mg,c forM˙in,0 =3(fiducial),1,and10 M⊙ yr−1. Belowwe gas is sharper if τ is shorter just because the effect of mass
describehowtheresultsdependonM˙ . tr
in,0 transportfromthecoldtohotgasoccursonashortertime-scale.
The dust-to-gas ratio in the hot gas has a larger amplitude
Theamplitudesofallthe quantitiesshowninFig. 3 arelarger
foralarger M˙ . Thisisbecausemoredustisaccumulatedin
in,0 for shorter τ because larger mass is exchanged between the
tr
the coldgas. Whenthisdustycoldgasisinjectedintothehot
coldandhotphases.Inparticular,theamplitudeofthecoldgas
gas by AGN feedback, the hot gasis enrichedwith dust. The
massissensitivetoτ . Accordingly,thedust-to-gasratiointhe
tr
dustinjectedintothehotgasisreadilydestroyedbysputtering; coldgasdropsmoreforshorterτ becausetheeffectofdilution
tr
thus, the hot gas is dust-poorfor most of the time. The dust-
by the inflow is more significant. In contrast, the dust-to-gas
to-gasratio in the hotgas dropsslightly morein the case of a
ratiointhehotgasdropstothevalueinsensitivetoτ whenthe
tr
higherMin,0becausetheinflowtransportsthedustfromthehot AGNfeedbackisoff(i.e.,whenthereisnosupplyofdustfrom
gastothecoldgasinourmodel. However,thelowestlevelof thecoldgas)becauseofsputtering. ThelevelofthelowestD
h
thedust-to-gasratiointhehotgasisnotverysensitiveto M
in,0 is determinedbythe balancebetweendustformationbyAGB
sinceitisdeterminedbythebalancebetweendustsupplyfrom starsanddustdestructionbysputtering.
AGB stars and dust destructionby sputtering. In terms of the
dust-to-gas ratio in the cold gas, the periodic decrease is due
3.4. DurationofAGNfeedback
to the inflow of dust-poorgas. However, since dust growthis
efficientenough,theoriginalhighdust-to-gasratioisrecovered ThedurationofAGNfeedbackrelativetotheAGNdutycy-
as soonasthe coldgasattainsenoughmassbythe inflow. As cleisparametrizedby f ,sothatanepisodeofAGNfeedback
ret
5
Figure5:SameasFig.2butforvarioustime-scalesofdustgrowthbyaccretion
(τacc). The solid, dotted, dashed, and dot-dashed lines show the results for
Figure3: SameasFig.2butforvariousmasstransporttimes(τtr). Thesolid, τacc=107(fiducial),106,3×107and108yr,respectively,inallthepanels.
dotted,anddashedlinesshowtheresultsforτtr =107(fiducial),3×106,and
3×107M⊙,respectively,inallthepanels.
lastsfor f τ . InFig.4,weshowthetimeevolutionofthe
ret AGN
quantitiesofinterestfor f =0.3(fiducial),0.1,and0.5.
ret
ComparingFigs. 3and4, we findthatthedurationofAGN
feedback has an effect similar to the mass transport time in
terms of the amplitudes: a larger f makes the amplitudesof
ret
thecoldgasmassandthedust-to-gasratiointhecoldgaslarger.
The cold gas naturally decreases more if AGN feedback lasts
longer, and the subsequent dilution of dust abundance by in-
flowbecomesmoresignificant. Thedust-to-gasratiointhehot
gashasa largeramplitudefor a smaller f since thecold gas
ret
massislarger(i.e.,thedustmasssuppliedfromthecoldgasis
larger).
3.5. Dustgrowthtime-scale
Theaccretiontime-scaleτ regulatesthedustgrowthinthe
acc
coldgas.InFig.5,weshowthetimeevolutionofthequantities
ofinterestforτ =107(fiducial),106,3×107,and108yr.
acc
Naturally,thecoldgasmass,whichisnotrelatedtodust,is
not affected by τ . As expected, the dust-to-gasratio in the
acc
cold gas becomes higher for shorter τ . Moreover, if τ is
acc acc
as long as τ , the overall level of dust-to-gas ratio is sup-
AGN
pressedbecausedustdoesnotgrowsufficientlywithinanAGN
feedbackcycle. Thedust-to-gasratioin thehotgasisalsoaf-
fected,sincethemajorsourceofdustinthehotgasisthedust
growninthecoldgas. WeobserveasecularincreaseofD for
c
τ = 3× 107 yr. This indicates that, if τ is smaller than
acc acc
Figure4: SameasFig.2butforvariousratiosoftheAGNfeedbackduration τ ,thedust-to-gasratiointhecoldgastendstoincreaseun-
totheAGNfeedbackdutycycletime(fret).Thesolid,dotted,anddashedlines leAsGsNitreachesthesaturationlimit(D ).Therefore,weobserve
showtheresultsfor fret = 0.3(fiducial), 0.1,and0.5,respectively, inallthe sat
panels. abifurcationforthevalueofDc:ifτacc <τAGN,thedust-to-gas
ratioincreaseswhileifτ >τ ,thedust-to-gasratioissup-
acc AGN
pressedandD ∼ D . Inthelattercase,dustgrowthhaslittle
c h
6
Figure 6: Same as Fig. 2 but for various time-scales of dust destruction by Figure7:SameasFig.2butforvariousstellarmasses(M∗).Thesolid,dotted,
sputtering (τsput). The solid, dotted, and dashed lines show the results for anddashedlinesshowtheresultsforM∗=1010.5(fiducial),1010,and1011M⊙,
τsput=107(fiducial),106,and108yr,respectively,inallthepanels. respectively,inallthepanels.
influenceonthedustbudgetandthedust-to-gasratioissimply isdeterminedbythebalancebetweendustsupplyanddustde-
determinedbythebalancebetweendustsupplyfromAGBstars structionisproportionaltothestellarmass.Themaximumlevel
anddustdestructioninthehotgas. ofthedust-to-gasratiointhehotgasisinsensitivetothestellar
mass, because it is determined by the dust injection from the
coldgas.Thedust-to-gasratiointhecoldgasisnotsensitiveto
3.6. Dustdestructiontime-scale
thestellarmassaslongasdustgrowthisefficientenough.
Dustinthe hotgasisdestroyedbysputtering. Thedestruc-
tiontime-scaleisregulatedbyτ . InFig.6,weshowthetime
sput
evolution of the relevant quantities for τ = 107 (fiducial), 4. Discussion
sput
106,and108yr.
In this section, we mainly compare the theoretical calcula-
Naturally, the cold gas mass, which is not related to dust,
tionswithobservationaldata.Hereweutilizetheobservational
is not affected by τ . As expected, the dust-to-gas ratio in
sput datafordustandgasinellipticalgalaxies. Becauseofthegen-
thehotgasislargelyaffectedbythesputteringtime-scale,with
eraldeficiencyofgasanddustinellipticalgalaxies,dataislim-
longer τ predictinghigherD . The dust-to-gasratio in the
sput h ited, while the stellar mass is relativelyeasy to be determined
hot gas is roughly proportionalto τ (i.e., inversely propor-
sput inthesesystemsusingopticalphotometricobservations. Thus,
tionalto thedustdestructionefficiency). Thedust-to-gasratio
wealsoadoptthestellarmassandutilizeit(mainlyfornormal-
inthecoldgasisalsokepthighforτ =108yr:ifD remains
sput h ization).
ashighasD ,thedust-to-gasratiointhecoldgasiskepthigh
c
becausethedust-to-gasratiointheinfallinggasishigh(i.e.,the
4.1. Sample
dustabundanceinthecoldgasisnotdilutedbytheinfall).This
Oneofthenewestsystematicdustobservationsinnearbyel-
effectissignificantonlyifτ &τ .
sput AGN
lipticalgalaxiesiscarriedoutbyHerschel.Weadoptthesample
from Smithetal. (2012) and diSeregoAlighierietal. (2013).
3.7. Stellarmass
ThesamplegalaxiesadoptedarelistedinTable2. Weselected
Thetotalstellarmassaffectsthedustevolutionthroughdust galaxieswithamorphologicaltypeofellipticalgalaxies(E)and
productionbyAGBstars. InFig.7,weshowthetimeevolution excludedobjectswithstellarmasswith<109.5 M⊙. NGC4374
of the relevantquantitiesfor M∗ = 1010.5 (fiducial), 1010, and is common for both samples. The stellar mass is taken from
1011 M⊙. Corteseetal.(2012). ThegasmassesaretracedbyHiandH2
The stellar mass does not affect the cold gas mass in our (CO);therefore,theobservedgasmassesarecomparedwiththe
modelsinceitonlycontributestothedustproduction.Because cold gas mass in our models. We only compare the observed
the dust supply from AGB stars to the hot gas is proportional dust mass with the dust mass in the cold gas in our models,
to the stellar mass, the lowest levelof dust-to-gasratio which because the observationalsensitivity to the diffuse component
7
in the hot gas depends on the spatial extension, which is not which affect the relative abundance of dust to gas (i.e., dust-
known.Inanycase,sincethedustmassinthecoldgasismuch to-gasratio). Thoseparametersareτ , f , τ andτ . The
tr ret acc sput
larger than that in the hot gas in most of the time, adding the samerangesfortheparametersasabove(i.e.,therangesinTa-
dustmassinthehotgasdoesnotsignificantlychangethecom- ble1)areinvestigated.Weonlyplottherelationatt>2×108yr,
parisonbelow. whenthetrajectoryalmostconvergestoalimitcycle(by“for-
getting the initial condition”) except for the case with τ =
acc
4.2. Coldgasmass 3 × 107 yr in Panel (c). In this case, there is a secular drift
towardhighdustabundanceasshowninFig.5.
ThevariationofthecoldgasmassinourmodelsinSection3
Overall,weobserveinFig.8thatthevariationisdominated
is broadly in the range of the observed gas masses shown in
Table 2 (∼ 109 M⊙ down to ∼ 107 M⊙ or less). Therefore, bythechangeofthegasmassbecausethevariationismostlyin
thediagonaldirectiononthediagram.Ifthegasmasschanges,
we confirm that the choices of parameter values in Section 2
are reasonable as far as the cold gas mass is concerned. The both Md,c/M∗ and Mg,c/M∗ moves on the diagonal line under
a constantD . Inotherwords, thevariationofthedust-to-gas
largestcoldgasmassisrealizedifwe adoptthelargestinflow c
rate (M˙in,0 = 10 M⊙ yr−1; Fig. 2). Therefore,we confirmthat ratio can be seen in off-diagonal behavior. Indeed, such off-
diagonalbehavioris seen clearly for a short τ in Fig. 8a and
themaximuminflowratebytheaccretionofthecooledgasis. tr
10M⊙yr−1. Theamplitudeofthecoldgasmassis,ontheother a large fret in Fig. 8b. In bothcases, the cold gasmass is lost
drastically in the epochof AGN feedback,so thatthe dilution
hand, determined by the time-scale of gas transport in AGN
of dust abundance by the inflow of dust-poor gas has a large
feedback (Fig. 3) and the duration of AGN feedback(Fig. 4).
impactonthedust-to-gasratiointhecoldgas(Sections3.2and
If the variationof the gasmass in the sample is interpretedas
due to time evolution, τ & 3 × 107 yr or f . 0.1 is not 3.4).
tr ret
We observe in Fig. 8 that a single model doesnot cover an
favoredbecausethevariationofthecoldgasissmallerthanan
order of magnitude. The non-detection of Hi and H gas for area wide enoughto explain all the data points. In particular,
2
a large part of the sample galaxies is consistent with even the thedatapointsathighMg,c/M∗andlowMd,c/M∗aredifficultto
mostextremecaseofτ =3×106yror f =0.5. Thismeans bereproduced.Asshownabove,thedilutioneffectappearsjust
tr ret
after the epoch of AGN feedback, when the cold gas mass is
that it is difficult to constrain the lower bound for τ and the
tr
low. Therefore,itisdifficulttoreproducethelowdustcontent
upperboundfor f .
ret
at the high gas mass. In our scenario, the only way to repro-
ducesucha dust-poorandgas-richdata pointis to changethe
4.3. Dustmass
efficiencyofaccretionasshowninFig.8c. Thesameeffectis
The dust mass in the cold gas varies between ∼ 105 and ∼ alsorealizedbyloweringD (equation9).Therefore,thelarge
106M⊙inthefiducialmodel(Fig.1).Themaximumofthedust dispersionofdustmassathsaitghgasmasscanbeinterpretedas
mass is achieved when the cold gas mass becomes the largest
variousdustgrowthefficiencies. Avarietyindustgrowtheffi-
(i.e., just before AGN feedback). In this phase, the dust-to-
ciencycouldbecausedbyavarietyindensegasfraction,cold
gasratioisalmostsaturatedifτ issignificantlyshorterthan
acc gasdensity,metallicity,orgrainsizedistribution(Section2.3).
τ . Ifwe adoptthemaximumvalueofthedust-to-gasratio
AGN Somestudiesarguethatthedust-to-stellarmassratiochanges
(D =0.01),weobtainthemaximumdustmassinthecaseof
sat as the system is enriched with dust (e.g., Re´my-Ruyeretal.,
thelargestcoldgasmass(109M⊙;seeabove)as∼107M⊙.The 2015). Dustenrichmentisindeedthemostimportantfactorof
observeddustmassisindeedsmallerthanthisvalue. Because
changingthedust-to-stellarmassratioingas-richstar-forming
thelogarithmicvariationofthedust-to-gasratioissmallerthan
galaxies,sincedustproductionbystarsanddustgrowthinthe
thatofthecoldgasmass,thetimeevolutionofthedustmassis
cold denseISM are activelyoccurring. In thispaper, we have
dominatedbythechangeofthecoldgasmassinourmodels.
shownthat, inellipticalgalaxies, thedust-to-stellarmassratio
is still varied by dust growth, although it is driven by a cycle
4.4. Dust-to-stellarmassratio
ofAGNfeedback,notbychemicalenrichment. Therefore,our
Tocanceloutthesizeeffectsofellipticalgalaxies,itisuseful results imply that, if AGN feedback affects the galaxy evolu-
to normalizethe dustmass to an indicatorof totalmass scale. tion, we also needto considerthevariationofdustabundance
Herewe usethestellarmassforthenormalization;thatis, we byAGNfeedback.
usethedust-to-stellarmassratiofortheindicatoroftherichness
of dust. As an indicator of AGN feedback, we also show the 4.5. Dust-to-gasratio
ratioofthecoldgasmasstothestellar mass, andexaminethe As shown above, the behavioron the Md,c/M∗–Mg,c/M∗ di-
relation between Md,c/M∗ and Mg,c/M∗. For the data points, agram is dominated by the overall oscillation of the cold gas
weexcludetheobjectswithoutanyconstraintontheHimass, mass. Toexaminetheeffectofdustabundancevariationmore
sinceitisimpossibletoconstrainthegasmassinsuchacase.If clearly,itisusefultoconsideraquantitynormalizedtothegas
Hiisdetected,wesumtheHimassandtheH masstoobtain mass. In this way, we can cancel the effect of oscillatory be-
2
thetotalcoldgasmass(ifH isnotdetected,weaddtheupper havior of the cold gas mass, and concentrate on the variation
2
limitofH gasmasstotheHigasmass). of dust abundance. We still use the gas-to-stellar mass ratio,
2
In Fig. 8, we show the relation between Md,c/M∗ and Mg,c/M∗, for the horizontal axis to show the cycle of AGN
Mg,c/M∗. We only show the dependence on the parameters feedback. Thus, we consider the Dc–Mg,c/M∗ diagram here.
8
Table2:EllipticalgalaxysampleadoptedfromSmithetal.(2012)anddiSeregoAlighierietal.(2013).
Name Othername logM∗ logMdust logMHI logMH2 logD Ref.a
(M⊙) (M⊙) (M⊙) (M⊙)
VCC763 NGC4374,M84 11.18 5.05b 8.96 <7.23 −3.91c 1,2,3
HRS150 NGC4406,M86 11.22 6.63 7.95 <7.4 −1.43c 1,3
HRS186 NGC4494 10.88 5.08 8.26 <7.35 −3.23c 1,3
HRS241 NGC4636 10.98 5.06 9.0 <7.02 −3.94c 1,3
VCC345 NGC4261 11.32 5.81 <8.45 <7.70 >−2.71d 2,3
VCC881C NGC4406 11.22 5.47 7.95 — −2.48e 2,3
VCC1226 NGC4472,M49 11.39 5.49 <7.90 <7.26 >−2.50d 2,3
VCC1619 NGC4550 10.03 5.41 <7.90 7.20 >−2.57d 2,3
HRS3 NGC3226 10.21 5.96 — <7.13 — 1,3
HRS43 NGC3608 10.27 <4.88 — <7.28 — 1,3
HRS49 NGC3640 10.70 <5.20 — <7.25 — 1,3
HRS125 NGC4339 10.30 <5.17 <7.84 <7.2 — 1,3
HRS135 NGC4365 11.48 <6.17 <8.18 <7.62 — 1,3
HRS179 NGC4473 10.71 <5.19 <7.92 <7.16 — 1,3
HRS181 NGC4478 10.09 <4.85 — <7.31 — 1,3
HRS211 NGC4552,M89 10.80 <5.67 <7.92 <7.36 — 1,3
HRS214 NGC4564 10.25 <5.30 <7.79 <7.33 — 1,3
HRS218 NGC4570 10.48 <5.67 <7.31 <7.47 — 1,3
HRS236 NGC4621,M59 10.99 <5.76 <7.92 <7.24 — 1,3
HRS245 NGC4649,M60 11.34 <5.40 <7.92 <7.59 — 1,3
HRS248 NGC4660 10.05 <4.85 <7.92 <7.30 — 1,3
HRS258 NGC4697 11.10 5.46 — <7.25 — 1,3
HRS312 NGC5576 10.60 <5.51 — <7.46 — 1,3
HRS316 NGC5638 10.52 <5.23 — <7.54 — 1,3
VCC1316 NGC4486,M87 11.24 5.34 — <7.18 — 2,3
VCC1327 NGC4486A 10.05 4.92 — — — 2,3
aReferences:1)Smithetal.(2012);2)diSeregoAlighierietal.(2013);3)Corteseetal.(2012).
bAdoptedfromRef.1. Ref.2gives5.30withthesamemassabsorptioncoefficient.
cTheupperlimitofH massisused,butthisdoesnotaffecttheestimateddust-to-gasratiosincethegasmassisdominatedby
2
Higasmass.
dWeusetheupperlimitsofgasmasstoderivethelowerlimitsofdust-to-gasratio.
eWeneglecttheH massforthegasmass.
2
9
Figure8:Relationbetweendust-to-stellarmassratio(Md,c/M∗)andgas-to-stellarmassratio(Mg,c/M∗).(a)Dependenceonτtr.Thethick(green),medium(blue)
andthin(red)linesrepresentthetrajectoryforτtr=107(fiducial),3×106,and3×107yr,respectively. (b)Dependenceon fret.Thethick(green),medium(blue),
andthin(red)linesshowthetrajectoryfor fret=0.3(fiducial),0.5,and0.1,respectively. (c)Dependenceonτacc.Thethick(green),mediumthick(blue),medium
thin(red),andthin(brown)linesrepresentthetrajectoryforτacc=107(fiducial)106,3×107,and108yr,respectively.(d)Dependenceonτsput.Thethick(green),
medium(blue),andthin(red)linesshowthetrajectoryforτsput =107(fiducial),106,and108yr,respectively. Thesethreelinesarealmostoverlapped. Wealso
showobservationaldatapointsinallthepanelslistedinTable2: Thediamondsshowthegalaxiesforwhichbothdustmassandgasmassaredetected,whilethe
crosseswitharrowspresentthedatapointsforwhicheitherdustmassorgasmassisnotdetected(onlyanupperlimitisobtained). Thecrossontherightlower
cornershowsthetypicalerrorfortheobservationaldatapoints.
10