Radiative transfer of energetic photons: X-rays and helium ionization in C2-Ray

1.4 MB·English

by  Martina M. Friedrich

Mon.Not.R.Astron.Soc.001,1–19(2010) Printed4January2012 (MNLATEXstylefilev2.2)
Radiative transfer of energetic photons: X-rays and helium
2
ionization in C -RAY
2 Martina M. Friedrich1 ⋆, Garrelt Mellema1, Ilian T. Iliev2 and Paul R. Shapiro3
1
1DepartmentofAstronomy&OskarKleinCentre,AlbaNova,StockholmUniversity,SE-10691Stockholm,Sweden
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2AstronomyCentre,DepartmentofPhysics&Astronomy,PevenseyIIBuilding,UniversityofSussex,Falmer,BrightonBN19QH
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3DepartmentofAstronomyandtheTexasCosmologyCenter,TheUniversityofTexasatAustin,TX78712,USA

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4January2012

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ABSTRACT
C
We presentan extensionto the short-characteristicray-tracingand non-equilibriumphoton-
h. ionizationcodeC2-RAY.Thenewversionincludestheeffectsofheliumandimprovedmulti-
p frequency heating. The motivation for this work is to be able to deal with harder ionizing
- spectra,suchasforexamplefromquasar-likesourcesduringcosmicreionization.Wereview
o the basic algorithmic ingredients of C2-RAY before describing the changes implemented,
r
t which include a treatment of the full on the spot (OTS) approximation, secondary ioniza-
s tion,andmulti-frequencyphoto-ionizationandheating.Weperformedaseriesoftestsagainst
a
[ equilibriumsolutionsfromCLOUDYaswellascomparisonstothehydrogenonlysolutions
by C2-RAY in the extensive code comparison in Ilievetal. (2006). We show that the full,
1 coupled OTS approximation is more accurate than the simplified, uncoupled one. We find
v thatalso with heliumand a multi-frequencyset up, long timesteps(up to ∼ 10%of the re-
2
combinationtime) still give accurate results for the ionization fractions.On the other hand,
0
accurateresultsforthetemperaturesetstrongconstrainsonthetimestep.Thedetailsofthese
6
constraintsdependhoweverontheopticaldepthofthe cells. We use thenew versionofthe
0
. codetoconfirmthattheassumptionmadeinmanyreionizationsimulations,namelythathe-
1
lium is singly ionized everywherewere hydrogenis, is indeed valid when the sourceshave
0
stellar-likespectra.
2
1 Keywords: methods:numerical–radiativetransfer–galaxies:intergalacticmedium–HII
v: regions
i
X
r
a
1 INTRODUCTION warmandionized.Thisprocessisknownasreionizationandwas
theUniverse’slastglobalphasetransition(seeforexamplethere-
Photo-ionizationisoneofthemajorradiativefeedback processes
viewinBarkana2006).
inastrophysics. Theextremeultraviolet(EUV)photonsproduced
by massive stars in star formation regions are capable of heating Traditionallyphoto-ionizationcodesconcentratedonsolving
the gas to temperatures around 104 K and the heated ions pro- equilibrium situations, as for example CLOUDY (Ferlandetal.
ducecopiousamountsofcollisionallyexcitedlineradiation,lead- 1998), MAPPINGS (Sutherland&Dopita 1993) and the three-
ing tothewell-known and sometimes spectacular images of H II dimensional code MOCASSIN (Ercolanoetal. 2008) do. The
regions(suchasforexampletheOrionnebula,O’Dell2001).Ac- mainaimofthesecodesistoaccuratelycalculatelinestrengthsfor
creting black holes and neutron stars also produce ionizing radi- comparison tospectroscopic observations. However, sincethein-
ation and, depending on their mass, can ionize smaller or larger creaseinpressurecandrivepowerfulflowsinthegas,thereisalso
regionsaround themselves. Ongalacticscales, theemissionfrom aneedtocouple photo-ionization calculationstohydrodynamics.
supermassive central black holes are observed to produce ioniza- Thisnecessitatesasimpler versionof theradiativetransfer, since
tionconesstretchingintothegalaxy’simmediateenvironment(see ithastobecalculatedinstepwiththehydrodynamicsandfordy-
e.g.Pogge1989).Evenatthelargestscalesphoto-ionizationisim- namiccalculationstheindividuallinestrengthsarelessinteresting.
portant. Some time before redshift 6, ionizing radiation escaped Thehistoryofthesetypesofcalculationsgoesbackquitefar,see
from the first generations of galaxies and percolated through the forexampleYorke(1986)foranoverviewoftheworkdonebefore
intergalacticmedium(IGM),changingitfromcoldandneutralto 1986.
Forsomeoftheapplicationsmentionedabove,alowerdimen-
sional approach (one or two dimensional) is sufficient. However,
⋆ e–mail:[email protected] other applications, such ascosmic reionization, require thetrans-
2 Friedrichet al.
port tobeperformed inthefullthreedimensions. Because ofthe anoverviewofthebasicalgorithmicideasbehindC2-RAYtothen
higherdimensionalityhereaswellasimplerversionoftheradiation proceedinSection3withanoverviewofhowweextendeditscapa-
and photo-ionization physics is typically implemented, although bilitiestohandleharderphotons.Section4containsthedescription
thelevelofsophisticationvariesbetweenmethods(seee.g.thefirst ofaseriesofoneandthree-dimensionaltestsforthenewmethod,
generation methods of Razoumov&Scott 1999; Sokasianetal. alsoevaluatingtheeffectsofthevariousnewelementssuchassec-
2001;Nakamotoetal.2001;Ciardietal.2001).Insomecasesthe ondary ionizationsand thecoupled on-the-spot approximation. A
couplingtothedynamicswasalsodoneforcosmologicalapplica- seriesof appendices describe several important elements in more
tions(e.g.,Ricottietal.2002;Tracetal.2008;Wise&Abel2008). detail.
One of the simplifications which is often employed is to
consider hydrogen as the only element being photo-ionized.
Since close to 10% of the gas is helium, this approxima-
tion is crude. However, as shown for example in figure 2.4 of
Osterbrock&Ferland(2006),fortypicalO-starspectratheioniza-
tionofheliumfollowslargelythatofhydrogen.So,ifoneisonly
2 REMINDEROFBASICSTEPSOFTHEORIGINAL
interestedinthestructureoftheionizedregions,assumingthathe-
C2-RAYALGORITHM
liumfollowshydrogenisnotabadapproximation.
However,asonemovestoharderspectra,thisassumptionbe- TheC2-RAYmethodwasdevelopedtobeatime-dependentphoto-
comeslessandlessvalid.Notonlydoesoneneedtotakeintoac- ionization algorithm that could be efficiently combined with a
count that helium can become doubly ionized, also the fact that hydrodynamics calculation, and not impose impractically short
thecrosssectionofthetwotypesofheliumstartcontributingsub- timesteps and small cell sizes on the latter. This is achieved by
stantially to the opacity of the gas becomes an issue. This prob- assuming that the ionization evolution of individual cells follows
lem becomes especially important when the ionizing spectrum is anexponential decay tothe equilibriumsolution and that a time-
powerlaw-like, as one expects from hot accretion disks around averagedvalueoftheopticaldepthcanbeusedtodescribetheef-
blackholes.Specificallyforthecaseofcosmicreionization,where fect of a cell on the radiative transfer during the entire timestep.
there is a possible contribution from powerlaw-like sources such Thisapproachisabletocorrectlytracktheprogressofionization
asquasarsandmini-quasars,anyproperstudyoftheircontribution frontsover many cellsduring onetimestep. Inaddition, optically
shouldconsiderbothhydrogenandhelium. thickcellsaredealtwithbydefiningthephoto-ionizationratesuch
This has motivated us to extend the capabilities of the code thatitisconsistent withthenumberofphotonsabsorbedinsidea
C2-RAY to include the effects of helium. C2-RAY is a photon- cell.C2-RAYwasdescribedandtestedindetailinM+06.Herewe
conserving radiative transfer code that uses short characteristic summarize some of the ideas in order to define our notation and
ray tracing and is described in detail in Mellemaetal. (2006b) provideanintroductiontotheextensionsdescribedinSect.3.
(hereafter M+06) . It has been used extensively for reionization The evolution of the ionized hydrogen fraction derives from
simulations(e.g.Mellemaetal.2006c;Ilievetal.2007,2008, to thesetofchemicalevolutionequations:
name a few). It was also combined with a hydrodynamics code
(CAPREOLE-C2, Mellemaetal.2006a),andanidealmagnetohy- d xHI = −ΓHI+CHIne αBHIIne xHI ,
drodynamicscode(PHAB-C2, deColle&Raga2006;Arthuretal. dt(cid:18) xHII (cid:19) (cid:18) ΓHI+CHIne −αBHIIne (cid:19) (cid:18) xHII (cid:19)
2011), to investigate galactic H II regions. Furthermore, it was
(1)
tested against other non-equilibrium radiative transfer codes in
Ilievetal.(2006)(hereafterI+06)andinconjunctionwithagrid- wherexHIistheneutralhydrogenfraction,xHIIistheionizedhy-
basedhydrodynamiccode(fordetails,seeMellemaetal.2006a)in drogen fraction, ne is the electron density, ΓHI is the hydrogen
a second comparison project, including gas dynamics (Ilievetal. photo-ionizationrate(seebelow),CHIisthecollisionalionization
2009). rateandαBHIIistherecombinationrate.SincexHI+xHII=1,we
Adding helium implies introducing another source of obtain
frequency-dependent opacity, thus making a multi-frequency ap-
d
proach inevitable. In addition the on-the-spot approximation be- dtxHII=−(ΓHI+CHIne+αBHIIne) xHII+(ΓHI+CHIne) .
comes morecomplicated asone hastotake intoaccount how re-
AH gH
combination photons fromheliumaffect thehydrogen ionization.
| {z } | {z }(2)
Asonemovestohigherphotonenergies,oneshouldalsotakeinto
accountthesecondaryionizationscausedbythesuperthermalelec- Here we introduce the notation, AH and gH, which allows us to
trons produced when high energy photons ionize the atoms and writetheequationinthegeneralvectorform,usefullateron,
ions. Including EUV and soft X-ray (SX) photons therefore is a
d
non-trivialextensionofthephoto-ionizationcalculationswhichwe x=Ax+g. (3)
dt
describeandtestinthispaper.
Asiswellknown,thegeneralsolutionx(t)toasetofequationsof
In terms of the physical processes included the method we
thistypeisthesumofthesolutiontothehomogeneouscase,x (t),
present here is similar to a number of others published in re- h
whereg=,andaparticularsolutionx :
centyears,butdiffersinthealgorithmsused.Thecodes CRASH p
(Masellietal.2009)andLICORICE(Baeketal.2010)useMonte n
Carlo techniques. The codes SPHRAY (Altayetal. 2008) and x(t)=xh(t)+xp with xh= cixietλi. (4)
TRAPHIC (Pawlik&Schaye 2011) implement ray tracing on Xi=1
particledatawhereasRADAMESH(Cantalupo&Porciani2011) Here,nistherankof A,i.e.thenumber of coupled equations (1
usesanadaptivemeshapproach. in the case of hydrogen), λi are the eigenvalues of A, xi are the
The lay-out of the paper is asfollows. In Section2 we give corresponding eigenvectors and ci are coefficients which can be
Radiativetransferofenergeticphotons:X-raysandheliumionizationin C2-RAY 3
calculatedfromtheboundaryconditionx(t=0)=x :
0
n
x0 = cixi+xp (5)
i=1
X
Insubsequent timesteps,x isthestateattheendoftheprevious
0
timestep.Inthecaseofaconstantg,theparticularsolutioncanbe
theequilibriumsolutiongivenby
Axp+g=.
Forthesimplest,hydrogenonlycase,thecoefficientsarethus
λH = −(ΓHI+CHIne+αBHIIne)
xH = 1
xH = ΓHI+CHIne
p ΓHI+CHIne+αBHIIne
cH = x −xH (6)
0 p
Here,weaddedthesuperscriptHforhydrogenandweskippedthe
subscript1sinceforhydrogen,n=1.
The photon-conserving photo-ionization rate Γ in each cell
usedinAiscalculatedusing
Γ = ∞ Lνe−hτνi1−e−h∆τνidν, (7)
HI Zνth hν hnHIiVshell iFnigMu+re061,.fiItgeurraetio2nschemeofC2-RAYforasinglecell,conceptionallyas
whereh∆τνiisthetimeaveraged opticaldepthover thecelland
V isthevolumeoftheshellthecellbelongsto.Thisquantity
shell
can be calculated from the time evolution of the neutral fraction cies higher than the ionization threshold of HeII, the ionization
(Eq.4).Bysolvingthesetwoequations(4and7)inalternatingor- cross-sectionsofHeIandHeIIareroughlyanorderofmagnitude
deroneiteratestoconvergenceasillustratedinFig.1.Thisiteration larger than the HI ionization cross-section. Therefore, neglecting
alsoinvolvestheelectrondensityne whichiscalculatedfromthe heliuminthecaseofsourceswithhardspectrawillunderestimate
timeaveragedionizedfraction.Thetimeaveragedopticaldepthto the optical depth substantially. We therefore have to add helium
the cell hτνi iscalculated by short characteristic ray-tracing over chemistrytoC2-RAY.
thesolutionsfoundforcellslyingnearertothesource.Thismakes Inordertoincludehelium,boththechemicalevolutionequa-
thealgorithmcausal. tion,Eq.(2),andthecalculationoftheionizationrate,Eq.(7),have
Inthecaseofmultiplesources,theiterationasshowninFig.1 tobechanged.Additionally,asshownbelow,theiterationscheme
is split up in two parts: The first part, including step three (find- fromFig.1hastobemodified.Wedescribeeachofthesechanges
ingtheionizationand heatingratesineach cell)isdonefor each here.
source, looping through theentirecomputational gridusing short
characteristicraytracing.Foreachcell,theratesfromallsources
areadded.Thesetotalratesareusedintheremainingtwostepsin 3.1 Chemicalevolutionequation
theiteration.SeealsoM+06foraflowchart andadescriptionof
Theprocedurefor addingheliumphoto-ionization toour calcula-
theimplementationhowtoloopthroughthesourcelist.
tionsisbyitselfrelativelystraightforwardasthebasicalgorithmic
NotehoweverthattheflowchartinM+06forthesinglesource
idea described in Sect. 2 provides the framework for this. How-
loop(figure4)incorrectlyincludesthelasttwostepsofthesingle
ever,acomplicatingfactoristhepresenceofionizingrecombina-
cell loop. The electron density and the time averaged ionization
tionphotonssincetheycoupletherateequationsofhydrogenand
fractionsareinfactnotupdatedinthesourceloopbutareupdated
helium. Here we present two approaches for dealing with these,
firstafterthephoto-ionizationratesfromallsourcesaresummedto
wherethefirstoneislessaccurate,butsimplerandmoresimilarto
aglobalphoto-ionizationrate.
whatotherauthorshaveused.Wecomparetheresultsofthesetwo
approachesinSect.4.
3 EXTENDINGC2-RAY
3.1.1 Simplerecombination:nocouplingofspecies
The original C2-RAY methodology works well for soft, stellar
spectra. In this case, the IGM can be considered to be hydrogen Whenionsrecombine,photonsareemitted.Incaseofrecombina-
only since there are not many photons capable of ionizing He II tionofhydrogenions,onlyrecombinationstothegroundstatere-
andheliumcanbeassumedtobesinglyionizedeverywherewhere sultinphotonsenergeticenoughtoionizehydrogen.Ifoneassumes
hydrogen isionized. InSect. 4.3.1weshow that themorphology thesephotonstoionizeimmediatelyanotherhydrogenatomclose
oftheionizationfractionfieldinacosmologicalreionizationsim- by, thisiscalled theon the spot (OTS)approximation for hydro-
ulationwithstellarsources(only),isindeedhardlyaffectedbythe gen (e.g. Osterbrock&Ferland 2006). In this approximation, the
inclusionofhelium.However,whenthespectrumhasasignificant recombinationcoefficienttoallstatesofhydrogen,αAisreplaced
amountofSXphotons,heliumcontributessignificantlytotheop- bytherecombinationcoefficienttoallstatesbutthegroundstate,
ticaldepthandamulti-frequencyapproachisrequired:atfrequen- αB.Foramixofhydrogenandhelium,theOTSapproximationis
4 Friedrichet al.
morecomplicatedasheliumrecombinationphotonscanionizeboth HeII,HeIandHI,dependingontherelativeopticaldepthoverthe
hydrogenandhelium.However,asafirststep,weassumethatpho- cellinquestionatνHeII.Ofthoserecombinationphotons,afraction
th
tons fromrecombinations to the ground statecan only ionize the yagoesintoHeIIionization,afractionybgoesintoHeIionization
2 2
samespeciesfromwhichtheyoriginateandthatinrecombinations andafraction1−ya−ybgoesintoHIionization.Here,thefrac-
2 2
tootherstatesthanthegroundstatenoionizingphotonsareemit- tionsy,ya,andybaredependentontherelativeopticaldepthofthe
2 2
ted.Thatmeans,weusetheαB recombination-coefficientsforall speciesatthethresholdfrequenciesofHeIandHeII,asdescribed
species.Inthefollowingwerefertothisasthe“uncoupledon-the- below.
spotapproximation”(U-OTS).Inthisapproximationhydrogenand RecombinationsofHeIIItootherstatesthanthegroundstate
helium can be treated separately. For helium, theset of chemical contributetoionizationofHIandHeI.Thoserecombinationslead
evolutionequations(inanalogytoEq.1forhydrogen)is: eithertotwophotonemission(inafractionv ofthecases,where
vistemperaturedependent,Hummer&Seaton1964),ofwhichon
x
d HeI average a fraction l is energetic enough to ionize HI and a frac-
x =
dt HeII  tion m is energetic enough to ionize HeI; therefore, a fraction
x
HeIII vw = v(l −m+ my) goes into HI ionization and a fraction
 
−UHeI neαHeII 0 xHeI v(m(1−y)) goes into HeI ionization. The remaining fraction,
UHeI −neαHeII−UHeII neαHeIII xHeII 1−v,leadstoemissionofaHeLymanαphoton.Thosephotons
   
0 UHeII −neαHeIII xHeIII areabsorbedbyanyspeciestoafractionf.Bylettingescapesome
   (8) of the helium Ly α photons (f 6= 1) we lose them since we do
not include those at larger distances from the source. Of the ab-
wherex istheneutralheliumfraction,x isthesinglyion-
HeI HeII sorbedHeLymanαphotons, afractionz goesintoHIionization
izedheliumfractionandx isthedoublyionizedheliumfrac-
HeIII andtheremainingfraction1−zintoHeI.Additionally,theBalmer
tion.Wegroupedtheionizing‘uprates’intooneterm
continuumemissionphotons(α2 )canionizehydrogen.Table1
HeIII
Un ≡Γn+neCn. (9) summarizesthephotonemittingrecombinationprocessesincluded
in the on the spot treatment. The numerical parameters used are
ThesubscriptsonC,αandΓindicateonwhichspeciestheyact,so
listedinTable2.
HeIIα⇌HeIIIHeIII.UsingthefactthatxHeI+xHeII+xHeIII =1, We can now introduce these terms in the general equation,
ΓHeII
theequivalentequationtoEq.(2)canbewrittenas Eq.(3):
d xHeII =AHe· xHeII +gHe, (10) xHII UHI
dt(cid:18) xHeIII (cid:19) (cid:18) xHeIII (cid:19) x= xxHeII  g= UH0eI  (12)
HeIII
wherethevectorgHeandthematrixAHehavethefollowingforms    
and
U
gHe= HeI
0
(cid:18) (cid:19) nHene×
AHe =(cid:18) −neαHeII−UHeUIIH−eIIUHeI ne−αnHeeIαIIH−eIIUI HeI (cid:19)(11) +−α(UBHHnIe+) (yα1HneHInIeH+npeα×BHeII) α2H((efIIzI(+1(−1v−)ny+H2av−wy)2bα)BHαe1HIIeII+II)) 
 
A= −UHeI+ 
Thesolution of this set of linear differential equations for the U-  −UHeII−UHeI− ne(α1HeIIIy2b+ 
OTScasecanbefoundinAltayetal.(2008)orwiththehereintro-  0 (αAHeII−(1−y)α1HeII)ne (αAHeIII−α1HeIIIy2a)+ 
ducednotationinAppendixA.  αBHeIII(f(1−z)(1−v)+(l−w)v))
 
 0 UHeII (−ne(αAHeIII−y2aα1HeIII)) 
 
3.1.2 Onthespotapproximation:couplingofspecies (13)
Inreality,recombinationphotonsfromheliumcanionizeeitherhy- ThedensityfractionsnHe/nH inA12 andA13 areduetothe
drogenorhelium,introducingtheneedtocoupletherateequations factthattheequationsarewrittenintermsoffractions,notinterms
for the two elements. Thisis the proper OTS approximation. Ta- ofdensities.Thecompletesolutionforthissetofequationsispre-
ble 1 gives an overview of the recombination processes affecting sentedinAppendixB.
hydrogenandheliumfractionsintheOTSapproximation.
To implement the OTS approximation we follow
3.2 Extendingtheiterationmechanism
Osterbrock&Ferland (2006) for dealing with the recombina-
tions of HeII to HeI and Flower&Perinotto (1980) for those of In Sect. 3.1.2, we treat the set of chemical evolution equations
HeIII to HeII (except for recombinations to the ground-state). dx/dt = f(x) as a set of linear differential equations, Eq. (3),
Mostlywealsousetheirnotation. anduseiterationstoobtainthecorrectelectrondensityandioniza-
Forhydrogen,wetaketheαB-recombinationcoefficientαB . tionrates.However,furtherdependencesonthedifferentionization
HII
ForHeII,thephotonsfromrecombinationstothegroundstateare fractionsarehiddenintheparametersy,z,w,yaandyb.Wefound
2 2
distributed between helium and hydrogen depending on the frac- thatthishiddenextranon-linearitycancomplicatetheconvergence
tionofopticaldepthattheheliumionizationthresholdfrequency, oftheiteration.Wethereforeextendedouriterationschemetomake
νHeI:afractiony goesintohydrogenionization,afraction1−y itmorerobust.Theextensionconsistsofupdating theparameters
th
goesintoheliumionization.Thephotonsfromstatesotherthanthe y,z,w,ya andyb afterthechemicalevolutionequationhasbeen
2 2
ground state contribute with a fraction p to hydrogen ionization. solvedandthensolvingthechemicalevolutionequationforasec-
Similarly,for HeIIItherecombinations totheground stateionize ondtimewiththisnew setofparameters. Wethentakethemean
Radiativetransferofenergeticphotons:X-raysandheliumionizationin C2-RAY 5
Table1.SummaryofrecombinationprocessesincludedintheOTStreatment
HII α−1H→IIHI groundstaterecomb ⇒ HI→HIIionization
HeII α−BH→eIIHeI deexciationsfrom ⇒p HI→HIIionization
recombinationston≥2
HeII α−1H→eIIHeI groundstaterecomb ⇒y HI→HIIionization
1−y
⇒ HeI→HeIIionization
HeIII α−1H→eIIIHeII groundstaterecomb ⇒y2a HeII→HeIIIionization
⇒y2b HeI→HeIIionization
1−y⇒2a−y2b HI→HIIionization
HeIII α2H→eIIIHeII recombton=2 ⇒ HeIIBalmercontinuum ⇒ HI→HIIionization
HeIII αBH→eIIIHeII deexcitationsfrom ⇒v 2photondecay ⇒w HI→HIIionization
m(1−y)
recombinationston≥2 ⇒ HeI→HeIIionization
1−v f(z−1)
⇒ HeIILyαphoton ⇒ HeI→HeIIionization
fz
⇒ HI→HIIionization
Table2.OverviewofthenumericalparametersusedintheOTSapproximation
p = 0.96or0.66dependingonne(Osterbrock&Ferland2006),inourcasesofinterest(cosmologicalsimulations)always0.96
y = τH/(τH+τHeI)atνtHheI(ionizationthresholdofHeI)
y2a = τHeII/(τH+τHeI+τHeII)atνtHheII(ionizationthresholdofHeII)
y2b = τHeI/(τH+τHeI+τHeII)atνtHheII(ionizationthresholdofHeII)
z = τH/(τH+τHeI)athν =40.8eV(HeILyα)
f = 1to0.1(”escape”fractionofLyαphotons)dependingontheneutralfraction
v = temperaturedependentcoefficient(Hummer&Seaton1964)
w = (l−m)+my
l = 1.425,fractionofphotonsfrom2-photondecay,energeticenoughtoionizehydrogen(Flower&Perinotto1980)
m = 0.737,fractionofphotonsfrom2-photondecay,energeticenoughtoionizeneutralhelium(Flower&Perinotto1980)
ofthesefirstandsecondsolutionsforthetime-averagedionization inthebins,itisusefultofurthersubdividebin2andbin3,hence-
fractionsandusethismeantocalculatethephoto-ionizationrates. forth referred to as (frequency) sub-bins. To refer to a particular
These extrasteps mean that the iterationscheme now consists of sub-bin,weusethefollowingnotation:n.mreferstosub-binmof
seveninsteadoffoursteps,seeFig.2. binn(n=2,3).InAppendixCweshowourpower-lawfitstothe
cross-sectiondatafromVerneretal.(1996).
The general treatment in every such sub-bin follows M+06
3.3 Calculatingthephotonrates
andisschematicallyillustratedinFig.3.Toavoidexpensiveinte-
For a pure hydrogen medium Eq. (7) gives the photo-ionization grationsforeachphoto-ionizationcalculation,wetabulatethetotal
rate,i.e.Γ=Γ andτ =τ .Inthecaseofamediumconsisting ionizationrateΓ asafunctionoftotalopticaldepthτ atthe
HI HI tot tot
ofhydrogenandhelium,thissimpletreatmentcanonlybeusedfor minimumfrequencyofthesub-bininquestion,assumingforallrel-
photons with frequencies below the ionization threshold for neu- evantspeciesthesamefrequencydependenceforthecross-section,
tralhelium,νHeI.Abovetheionizationthresholdfrequencyofion- followingtheconceptoutlinedinTenorio-Tagleetal.(1985).The
th
izedhelium,νHeIIallthreespecies,HI,HeIandHeIIIcontribute assumption of having the same frequency dependence allows the
th
totheopticaldepth.Inthefrequencybinbetweenthosethreshold summationoftheopticaldepthofeachspeciesandthetabulation
frequencies, HI atoms and HeI atoms contribute to the total op- ofthephoto-ionizationrateasafunctionofthissingletotaloptical
ticaldepth. Therefore,theminimumnumber offrequencybinsto depth.Sincethefrequencydependencesofthecross-sectionsofthe
considerseparatelywhencalculatingthephotoionizationratesfor differentspeciesareinfactnotidentical(seeforexampleFig.C1),
eachspeciesisthree,henceforthreferredtoas(frequency)bins.As thisisanapproximation. Themoresub-binsweusethemoreac-
we explain below, we use a single frequency dependence for all curatelywefollowtheactualshapeofthe(frequencydependence)
speciesineachbin.Sincethethedifferentfrequencydependences curves of the different cross-sections. The frequency dependence
oftheionizationcrosssectionsσ ofthespeciesareverydifferent imposedonallspeciesisthatofσ forallsub-binsofbin2and
HeI
6 Friedrichet al.
eachsub-bin.Doingsooverestimatesthecontributionofhydrogen
totheopticaldepthinfrequencybin2sincetheHIionizationcross
section drops faster withfrequency than theionization cross sec-
tionofHeI.ThisleadsthereforetoanoverestimateofΓ /Γ .
HI HeI
Infrequencybin3,theslopesofthecross-sectioncurvesareingen-
eralmoresimilar(seeAppendixC)butstillsteeperforHIwhich
resultsinanoverestimateofΓ /Γ .Themoresub-binsused,
HI HeII
thesmaller theoverestimateswill be. Wetest theconvergence of
thisinAppendixE.Wefindthatincreasingthenumberofsub-bins
inbin2improvestheslopeoftheionizationfrontwhileincreasing
thenumber ofsub-binsinbin3improvestheionizationfractions
insideandoutsidethefront.
3.4 Heatingandsecondaryionizations
OneofthemotivationsforextendingC2-RAY,istoinvestigatethe
effectsthatheliumandhardspectrahaveonthetemperatureevo-
lutionoftheIGMduringtheEoR.Inthissectionwefirstdescribe
theimplementationofthetemperaturecalculation.Thisisfollowed
byremarksontheinclusionofsecondaryionizationsandconsider-
ationsabouttheadditionaltimesteprestrictionscausedbythetem-
peraturecalculation(inadditiontotheionizationcalculation).
WeusetheidealgaslawPV = NkBT,wherekB isBoltz-
mann constant, to calculate the pressure P from the temperature
T and the gasparticlenumber density N/V = nH +nHe +ne
in each cell of volume V. From the pressure we then calculate
the internal energy per volume V (in each cell), using the di-
mensionless heat capacity (per particle) at constant volume, cV:
uint = Uint/V = cVP Foramonatomicgas,cV = 3/2.Sothe
equationtoconverttemperatureintointernalenergydensityreads:
3
uint = 2kBT(nH+nHe+ne). (15)
Thisenergydensityisaffectedbytheheating(H)andcooling(C)
Figure 2.Extended iteration scheme for C2-RAY including helium with ofthegasper(cell-)volumeandperunittime.Inordertofollow
couplingofthespecies. thetemperatureevolutionofthegas,wethereforehavetosolvethe
equation
∂uint
thatofσHeIIforallsub-binsofbin3.Thetableentriesarethein- ∂t =H−C. (16)
tegralsofionizationrateoverthefrequencyrangeofthesub-binin
As the only contribution to the heating rate H, we consider
question,thusthetotalionizationrateduetophotonswithfrequen-
photo-ionizationheating.Ineveryphoto-ionizationofspeciesi,the
ciesinthatfrequencysub-bin.
excessenergy,(ν−ν (i))istransferredtotheelectronreleased.If
Inordertousethispre-calculatedtableanddeterminethetotal th
oneassumesthatthecellsareopticallythick,(almost)allphotons
photo-ionizationrateinagivencell,weneedtodeterminetheion-
areabsorbedandtheaverageenergyperphotoionizationissimply
izationcross-sectionsforeveryspeciesattheminimumfrequency
theaverageexcessenergyover thewholespectrum. Thiswasthe
of every sub-bin. We use the fitting formula from Verneretal.
approachusedbytheoriginalC2-RAYinthetestswithtemperature
(1996) for the cross-sections and compute the total optical depth
evolutioninI+06.1
τtotasthesumoftheopticaldepthsofallspeciesattheminimum To properly take into account the effects of different optical
frequencyofeachsub-bin.Thisopticaldepthisusedtodetermine
depthontheheatingwecancalculateHinanalogytotheioniza-
thetotalphotoionizationrateinthetable.
tionrateΓ(Eq.7).Itshouldberememberedthat,althoughwewrite
Next, this total photo-ionization rate has to be split up into
thisasone equation, for thephoto-ionization rate itisin fact the
ionizationratesforHI(Γ ),HeI(Γ )andincaseofbin3,HeII
HI HeI differencebetweentheingoingphoto-ionizationrate(calculatedon
(Γ ).Thisisdoneaccordingtotherelativefractionsofoptical
HeII thebasisofhτi)andtheoutgoingphoto-ionizationrate(calculated
depthineachfrequencyinterval.
onthebasisofthehτi+h∆τi).Fortheheating,thesequantities
Γi =Γtotττtoit, τtot = τi, i=HI,HeIand HeII (14) athreefroatrhmeroafbpshtroatcotnssinwcheentheeyntesyrimngbothliezecethlleaenxdctehsesaemneorugnytsotifllexin-
i
X
cessenergystillintheformofphotonsleavingthecell.Neverthe-
InAppendix D weevaluate other approaches for thisdistribution
thathavebeenproposedintheliterature.
Wecalculatethesefractionsofopticaldepthattheminimum 1 The H-only C2RAY used in Mellemaetal. (2006a) and Arthuretal.
frequencyofeachsub-bin.Thisimpliesthatwealsohereassumea (2011)actuallyusedaonespecies,onefrequencybinversionoftheheating
singlepowerlawfitforthefrequencydependenceofallspeciesin methoddescribedbelow.
Radiativetransferofenergeticphotons:X-raysandheliumionizationin C2-RAY 7
Figure3.Sketchshowinghowtocalculatetheionizationratecontributionforbin3.Instep1,thetotalopticaldepthineachsub-biniscalculated.Instep
2,thepre-calculatedphotoionizationratetableisusedtocalculatethetotalphotoionizationrateineachsub-bin.Onthebasisoftheopticaldepthofeach
speciesattheminimumfrequencyνminineachsub-bin,thefractionsfigoingintohydrogen-,helium-andionizedhelium-ionizationarecalculatedinstep
3.Finally,theproductsofthefractionsandtotalphotoionizationratesaresummedoverallsub-binstoresultintheionizationrateforeachspecies,ΓH,ΓHeI
andΓHeII.
less, what is tabulated as function of optical depth is this excess ferred energy is greater than the binding energy of this electron,
energyasafunctionofopticaldepth.Thecorrespondingequation this electron is released. This process is called secondary ioniza-
toEq.(7)isthen tion. The probability of such an event depends on the energy of
the primary electron and on the ionization state of the gas. The
H(i)= h(ν−νth(i))Lνe−hτνi1−e−h∆τνidν fullprocessrequirescarefulmodelling,butShull&vanSteenberg
Zν(sub−bin) hν Vshell (1985),Ricottietal.(2002)andValde´s&Ferrara(2008),toname
(17) a few, published separable functional relations dependent on pri-
maryelectronenergyandhydrogenionizationfractiontodescribe
forspeciesi.Sincetheexcessenergyisdifferentforeachspecies, what fraction of the energy deposited in primary electrons goes
instead of having one table of heating rates in each sub-bin, we into secondary ionizations of either HI or HeI, fHI/HeI, and what
ion
need three tables, where the excess energy is with respect to the intoheat,f .Intheserelationsitisassumedthatx = x
heat HII HeII
thresholdfrequencyofνth(HI),νth(HeI)andνth(HeII).Theto- andsecondaryionizationsofHeIIareneglected.Weimplementthe
talheatingineachfrequencysub-binisnaturallyanalogoustothe separable functional relationship as in Ricottietal. (2002) which
photo-ionizationgivenby converges for high electron energies to the functional form of
H= τi H(i). (18) Shull&vanSteenberg(1985).
τ
i tot Furlanetto&Stoever(2010)pointedout thatthereisingen-
X
eralnosimpleseparablefunctionalformfordecidingthefractions
Someoftheelectronsthatreceivedexcessenergy(ν−ν (i)) thatgointosecondary ionizationsandheating. Ratherthedepen-
th
after being released from the bound state of atom i, will collide dence of the relativefractionsvaries withionization fractionina
withboundelectronsofotheratoms(orions)ratherthanotherfree complex way, see their figure 5. However, the complex relation
electrons.Thistransfersenergytotheboundelectron.Ifthetrans- givenbyFurlanetto&Stoever(2010)stillassumesx = x
HII HeII
8 Friedrichet al.
andno secondary ionizations of HeII.Giventheselimitationswe
Table3.Parametersfortestingthecode.
decidedthatatthispointthereisnoclearbenefitinimplementing
thesemorecomputationallyexpensiverelations. nH/cm−3 constanthydrogennumberdensity
Forthecoolingrate,weincludefree-freeandrecombination nHe/cm−3 constantheliumnumberdensity
coolingforHII,HeIIandHeIIIandcollisionalexcitationcooling N˙γ/s rateofphotonsintheenergyinterval
for H I, He I and He II. A full overview of the radiative cooling [13.6,5441.6]eV
ratesusedisgiveninAppendixG.Incaseofcosmologicalsimu- Teff/K effectivetemperatureoftheblackbodysourcein
lations,cosmologicalcoolingduetotheexpansionoftheuniverse caseofBBsource
and Compton cooling against the cosmic microwave background β powerlawindexincaseofPLsource
photonsareincludedaswell. Tini,x initialtemperatureandinitialionizationstates,HII,
HeIIandHeIII
In order to numerically solve Eq. (16) we use forward Eu-
∆r thecellsize
ler integration. However, the cooling rate depends sensitively on
∆t thetimestep
the gas temperature which is changing because f the heating. To
(n2/n3) numberofsub-binsinfrequencybins2and3
accuratelyfollowthisbehaviourweusetheforwardEulermethod OTS/U-OTS/αA assumingthe(U-)OTS-approximation,orusing
withsub-timestepsdeterminedbyalimitonthetemperaturechange αArecombinationrates
(typically10%).WekeeptheheatingtermH(whichrepresentsthe
averageheatingrateoverthetimestep)constantbutusethetemper-
aturefromtheprevioussub-timesteptocalculatethecoolingrate
C. time,anaccuratevalueforthetemperaturerequirestimestepsofthe
orderoftheionizationtime.Thisconstraintbecomesmorestrictif
Above,wedescribedhowtocalculatetheheatingrateinanal-
thecellsareveryopticallythick.
ogy to the photo-ionization rate. Specifically, we use the time-
averagedopticaldepth(tothecellandinthecell)bothinEq.(7) Given these conclusions it is difficult to provide a simple
andEq.(17).Thereishoweveranimportantdifferencebetweenthe recipeforchoosingthetimestep.Wethereforerecommendtesting
dependence ofthephoto-ionization ratesandtheheatingrateson fornumericalconvergenceifaccuratetemperaturesarerequired.
opticaldepth.AlthoughEq.(7)givesthecorrectrateofabsorbed
photons,fortheheatingthefrequencyofthephotonsmatters.Let
usfirstconsiderthesourcecell,i.e.,theopticaldepthtothecell,τ,
4 TESTINGTHECODE
isconstantwithtimeandequal0.Whileforthephotoionization,
Totestthevalidityoftheapproximationswemadeforthesakeof
∆tΓ(τ(t′))dt′
Γ(hτi )= 0 , (19) codeefficiency,weperformedaseriesoftests.Firstofallwevali-
∆t R ∆t datedthenewversionagainsttheoldversionofC2-RAYbysetting
fortheheatingingeneral the helium abundance to a very low value. This test showed that
thenewversionhasthesamephoton-conserving propertiesasthe
∆tH(τ(t′))dt′
H(hτi )≤ 0 , (20) methodpresentedinM+06.Fortestingthetimedependentsolution
∆t ∆t withhelium,includingtheOTSapproximation,secondaryioniza-
R
becauseofthe(ν−ν (i))termintheintegralforH.Qualitatively tion and temperature evolution, we would need a fully validated
th
onecouldsaythatinthebeginningofthetimesteptheheatingper timedependentphoto-ionizationcode,whichasfarasweknowis
photo-ionizationistheopticallythickvalue(averagephotonenergy notpubliclyavailable.Wewerethereforeforced,justasforexam-
overthewholespectrum),forlatertimesitshiftstolowervalues, pleBaeketal.(2010)andPawlik&Schaye(2011)tocompareour
thelimitofwhichisgivenbytheopticallythinvalue(averagepho- results against the one-dimensional photo-ionization equilibrium
ton energy over the spectrum weighted withthe photo-ionization code CLOUDY (version 08.00, last described in Ferlandetal.
cross-section) 1998). Below we present the results of two sets of such one-
Forthisfirstcellwheretheonlydependenceisontheoptical dimensional tests, one set in which the source has a black body
depthinsidethecell,itwouldbepossibletotabulateacorrection (BB)spectrumwithaneffectivetemperatureofTeff,theotherfor
factorbasedonthechangeoftheopticaldepthduringatimestep. a source with power-law (PL) like spectrum, where the energy-
However, inthemoregeneral casewheretheopticaldepthtothe distributioncanbedescribedasL(ν) ∝ ν−β.FortheBBsource
cell depends on the time varying optical depth of the other cells wetestvariousaspectsofthecalculationoftheionizationfractions
along the ray, a local fix (such as using an energy-average based whilekeeping the temperature constant. For the PL case we also
on thechange of optical depth inthecell or sub-timestepping on consider thetemperatureevolution. Wechosethisapproach since
a cell-by-cell basis) to the heating rate is no longer possible. We thelattercasehavingawideenergyrangeofhardphotons,consti-
exploreddifferentapproachesforcalculatingaccurateheatingrates tutes a more difficult test for the photo-heating. Table 3 gives an
usinglargetimesteps,butwewereunabletofindageneralsolution. overviewoftheparametersusedforalltheone-dimensionaltests.
InAppendixFweinvestigatehowtheheatingdependsonthe To demonstrate the multi-dimensional and multi-source capabili-
choiceofthetimestep.Fromthesetestsweconcludethefollowing: tiesoftheextendedC2-RAYwealsopresenttheresultsoftwotest
problemsusingthree-dimensionalcosmologicaldensityfields.
• Ifweareinterestedintimescaleslargerthantherecombina-
tiontime,wecanstilluselargetimesteps,astheinitialheatingis
4.1 Test-suite1:Blackbodysource
nolongerdominating.
• Ifthecoolingtimet <∆twecanalsouselargetimesteps, Thefirstsetoftestsconsiders theexpansion ofanionizedregion
cool
as the temperature is set by the equilibrium heating and cooling produced byasinglesourceintoaconstant densitymediumwith
rates.Thisisthecasefortypicalinterstellarmediumconditions. constanttemperature.Undertheassumptionofsphericalsymmetry
• If we are interested in time scales below the recombination thiscanbecalculatedinonedimensionfortheradialdistancerto
Radiativetransferofenergeticphotons:X-raysandheliumionizationin C2-RAY 9
thesource.Theparametersthatarenotvariedinthissetoftestsare: also plot the ionization profiles after 1010 yr for the latter ap-
nH/cm−3 =1.0×10−3,nHe/cm−3 =8.70×10−5,N˙γ/s=5.0× proach (thin black lines). It can be seen that the differences be-
1048,T /K=105.T =10000K,x=(10−40,10−40,10−40) tweentheOTSandU-OTSresultsarelargerthanbetweentheOTS
eff ini
(i.e.completelyneutral),∆r/pc=150,∆t/yr=107and(n ,n )= approximationandthefullradiativetransferresults,whencompar-
2 3
(10,14). ingthe(almost)equilibriumsolutions.Fromthisweconcludethat
Wechoosetheseparameterssothattheheliumfractionf = althoughmorecomplicatedtoimplement,thecoupledOTSapprox-
He
n /(n +n ) = 0.08andtheremainingphysicalparameters imationshouldbethepreferredapproach.
He H He
arethesameasinTest 2ofI+06,exceptthatwedonotfollow SinceC2-RAYcannot(now)dealwiththediffusephotonsin
thetemperatureevolutionhere.Thetimestep∆tis∼ 10%ofthe another way than using the OTS approximation, we are not able
recombinationtimeinafullyionizedmedium,n αA,B,and∼104 to investigate the effects of using this approximation during the
H
timestheionizationtimeforthefirstcell. growth of the H II region. Cantalupo&Porciani (2011) compare
resultsfromwhat wecallthe U-OTSapproximation withfullra-
diativetransferofdiffusephotonsusingtheircodeRADAMESH.
4.1.1 Test1A Theyfindratherlargeeffectsattimeswhentheequilibriumsolu-
tionhasnotyetbeenreached.Thiswarrantsfurtherinvestigation,
Inthefirsttestofthissuiteweassumethatallionizingphotonsfrom
probablybestpursuedinmannersimilartoI+06,involvingresults
recombinationscanescape,whichmeansthatweareusingtheαA
frommultiplecodes.
rates.Thistestismeanttoshowhowwellourbasicapproachcan
matchtheequilibriumsolutionfromCLOUDY.InFig.4weshow
the resultstogether withthat equilibrium solution. It can be seen 4.2 Test-suite2:Apowerlawsource
thataftert = 1010 yrthegeneralagreement withthe CLOUDY
Inasecondset oftests,weuseUV-photon emittingsourceswith
equilibriumsolutionisexcellentfordistancesbelow12kpc,butnot
apower-lawspectrum,L(ν) ∝ ν−β,implyingthatthenumberof
forlargerdistances.However,sincetherecombinationtimescaleis
given by (n α)−1, it follows that for ionization fractions below photonsgoesasf(ν)∝ν−(β+1).Fortheseweconsideredthetwo
e
∼ 1%, the recombination timescale is larger than ∼ 1010 yr. In casesβ =1andβ=2.Wepresenttheresultsforsimulationswith
temperature evolution using T = 102 K. We only use the full
otherwords,theseouterregionshavenotyetreachedequilibrium. ini
OTS approximation, apply the secondary ionizations and choose
AnothernoticeablefeatureinFig.4isthattheHeIIIfractionshows
thenumberofsub-binsinbin2and3tobe(n ,n ) = (26,20).
atransientbumparoundr = 4kpc.Thisbumpdisappearsasthe 2 3
TheremainingparametersareasinTEST 1.
equilibrium solution is approached, but both, our results and the
Wefoundatimestepof∆t=105yrtobesufficienttoobtain
equilibriumsolutionstillshow aslopechangeintheHeIIIcurve
convergenceinthetemperatureevolution.Choosingatimestepof
around thatsamedistance. Fromallthisweconclude that there-
the order of the ionization time of the first cell (∆t = 103 yr)
sultsofthistestshowthatourbasicmulti-frequencyradiativetrans-
gavetemperature results whichfor cellsclose tothe source were
ferisconsistentwithCLOUDY’s.
only different by at most 7%. For theparameters of this test, the
opticaldepthattheionizationthresholdforhydrogenforonecell
isτ ∼ 3.2,inbetweenthetwocasestestedinAppendixFandso
4.1.2 Test1B
thisresultforthetimestepisconsistentwiththose.
Inthesecond test,TEST1 B,weusetheonthespotapproxima- WeshowtheprofilesforHI/HIIandHeI/HeII/HeIIIandT
tionasdescribed inSection3.1.2and otherwiseset the samepa- forthreetimesandthetwodifferent power lawspectrainFig.6.
rameters as in the previous test. We show the ionization profiles WealsoplottheequilibriumsolutionsfromCLOUDYwithOTS
inFig.5togetherwithtworesultsfromCLOUDY,oneusingthe forthosetwospectra.Wefindthatatt = 109 yrthe C2-RAYre-
OTS approximation (green dashed line) and the other using full sults are close to the CLOUDY ones for those distances where
radiativetransferofrecombinationphotons(bluedashedline).As the equilibrium solution has been reached, although not as close
isapparent inFig.5,theagreement betweenthe C2-RAYandthe asfor theresultsof TEST 1.Thesesomewhat larger differences
OTSsolutionof CLOUDYisgoodforthosedistanceswherethe withtheCLOUDYresultscannotbeexplainedbyeffectsfromthe
equilibriumsolutionhasbeenreached(r<12kpcatt=1010 yr, temperaturecalculationsincewefoundcomparablediscrepancies
justasinTEST1 A).Thegeneralpatternofevolutionoftheion- whenimposingaconstanttemperature.However,overallthematch
izationfractionsisquitesimilartotheoneinTEST1 A,although isstillreasonable.
theOTSapproximationdoesofcoursechangethedetailedvalues The time evolution shows that the two values of β produce
ofthefractions. similarresults,withinitiallyquitesteepprofilesfortheionization
TheCLOUDYsolutionusingfullradiativetransferofrecom- fractionsforalltimesandarelativelysteepfrontinthetemperature
bination photons can be used to evaluate the validity of the OTS evolutionevenforthequitehardβ = 1spectrumattherelatively
approximationforthisparticulartestproblem.InspectionofFig.5 latetimeoft = 107 yr.Comparingtheβ = 1andβ = 2results
shows that the error introduced by using on the spot approxima- itcanbenoticedthattheformergivesahigherdegreeofhydrogen
tion is barely noticeable, but larger for hydrogen and largest for andheliumionizationoutsidethefrontthanthelatter.Furtheritcan
theHIfractioninsidetheHIIregion.Closerinspectionshowsthat beseenthattheresidualneutralhydrogenfractioninsidethefront
theneutralhydrogenprofileinsidetheHIIregion,uptoaneutral ishigherfortheresultswithβ = 1.JustasinTEST1,thebumps
fractionofabout1%,followsthecurvefromtheαArecombination inthe HeIII fractionare transient phenomena, although even the
coefficientfromTEST1 A.Thisisnotsurprisingsincemostofthe equilibrium solution shows a slope change around a distance of
recombinationphotonsfromthishighlyionizedregionwillnotbe ∼6kpc.
absorbedonthespot,butescapetolargerdistances. Tittley&Meiksin(2007)reportedtheHIIfronttobetrailing
TogaugetheimportanceofapplyingthecoupledOTSrather behindtheHeIIIfrontforspectrawithβ <1.8.However,evenin
thanthepopular U-OTS(i.e.usingαB rates,seeSect.3.1.1),we ourβ =1resultswefindtheHIIfractiontobealwayslargerthan
10 Friedrichet al.
1
t/yr=107
H I H II
t/yr=108
t/yr=109
t/yr=1010
0.1 CLOUDY
ni
o
cti
a
fr
0.01
1
He I He II He III
0.1
ni
o
cti
a
fr
0.01
1 5 10 15 1 5 10 15 1 5 10 15
Distance /kpc Distance /kpc Distance /kpc
Figure4.ResultsfromTEST1 A.FractionsofHI(upperleftpanel),HII(uppermiddlepanel),HeI(lowerleftpanel),HeII(lowermiddlepanel)andHeIII
(lowerrightpanel)attimest/yr=[1×107,1×108,1×109,1×1010]asindicatedbylinethicknessinthelegend.Wealsoshowtheequilibriumsolutionof
CLOUDY(greendashed).Notethatduetothelowelectrondensityintheouterregions,theequilibriumsolutionhasnotyetbeenreachedbeyondadistance
ofabout12kpcatt=1010yr.
theHeIIIfraction.Wewereabletoobtain(double)frontcrossings 4.3.1 TEST3:Effectofheliumonthemorphologyofthe
when using the αA recombination rates (no OTS) and disabling hydrogenionizationfractionfieldduringEoRwithout
thesecondaryionizations.However,thelocationswheretheHeIII temperatureevolutionforstellartypesources
fractionwasfoundtobelargerthantheHIIfractionwereinalim-
ited area in the ionization fraction-time space, roughly in the in- Many cosmological reionization simulations only include hydro-
terval [0.1, 108 yr] to [0.01, 109 yr]. We therefore conclude that gen and implicitly assume that helium is singly ionized every-
trailingHIIfrontsareatbestamarginaleffect. wherewherehydrogenisionized.Theusedhydrogennumberden-
In order to evaluate the effects of secondary ionizations we sity is therefore equal to the total number density. In this sec-
alsoranTEST 2withoutthem.Wefoundthatthisledtochanges tion, we test if the morphology of H II regions in a reioniza-
larger than the differences we found between our results and the tionsimulationwithstellarsourceschangesifheliumisincluded.
equilibriumsolutionofCLOUDY.Wethereforeconcludethatitis For this comparison, we use simulation 53Mpc g8.7 130S from
worthtoincludesecondaryionizationsinthecalculation. Friedrichetal. (2011) and Ilievetal. (2011). The electron scat-
teringopticaldepth produced bythissimulation, τ = 0.083, is
es
consistent withthe 1–σ range allowed by theseven year WMAP
results,τ = 0.088±0.015(Komatsuetal.2011).Inthissimu-
es
lation,thenumberofionizingphotonsproducedbyadarkmatter
haloofmassM isdefinedthrough
MΩ
4.3 Cosmologicaltests N˙γ =gγ10Ωmmb p , (21)
Inthissection,wetestthe3D-versionoftheextendedC2-RAYon where N˙γ is the number of ionizing photons emitted per Myr,
twocosmologicaldensityfields.Thefirsttestisalargerscalecos- Ωb = 0.044, Ωm = 0.27andmp istheprotonmass(Thisequa-
mologicaltestwithouttemperatureevolutiontoevaluatetheeffect tionincludedincorrectlyaµinFriedrichetal.2011, equation1).
ofheliumatafixedtemperatureinasimulationwithsourceswith Massivehalosareassignedanefficiencygγ =8.7whilelowmass
asoftspectrum.Thesecondtestisarathersmallcosmologicalvol- sourceshaveanefficiencyofgγ = 130andaresuppressedinre-
umebutincludestemperatureevolution:weredothecosmological gions where the ionization fraction (of hydrogen) is higher than
test-problem with multiple sources from I+06 (their test4) to test 10%.Toevaluate theeffectofheliumonthemorphology of HII
theeffectofheliumontheheatingandonthehydrogenionization regionsweusethedimensionlesspowerspectrumoftheHIIfrac-
fractionfield. tion∆2 InFig.7weshowthepowerspectraforthissimulation
xx