Self-Consistent Dynamic Models of Steady Ionization Fronts I. Weak-D and Weak-R Fronts

0.83 MB·English

by  William J. Henney

DraftversionFebruary2,2008
PreprinttypesetusingLATEXstyleemulateapjv.6/22/04
SELF-CONSISTENTDYNAMICMODELSOFSTEADYIONIZATIONFRONTS:
I.WEAK-DANDWEAK-RFRONTS
W.J.Henney1andS.J.Arthur1
CentrodeRadioastronom´ıayAstrof´ısica,UNAMCampusMorelia,Apdo.Postal3-72,58090Morelia,Michoaca´n,Me´xico.
R.J.R.Williams
AtomicWeaponsEstablishment,Aldermaston,RG74PR,UK.
G.J.Ferland
5 DepartmentofPhysics&Astronomy,UniversityofKentucky,177Chem.-Phys.Building,600RoseStreetLexington,Kentucky40506-0055,USA.
0 DraftversionFebruary2,2008
0
2 ABSTRACT
We present a method for including steady-state gas flows in the plasma physics code Cloudy, which was
n
a previouslyrestricted to modelingstatic configurations. The numericalalgorithmsare describedin detail, to-
J getherwithanexampleapplicationtoplane-parallelionization-boundedHIIregions. Aswellasprovidingthe
4 foundationforfutureapplicationstomorecomplexflows,wefindthefollowingspecificresultsregardingthe
effectofadvectionuponionizationfrontsinHIIregions:

1 1. Significantdirecteffectsof advectiononthe globalemission propertiesoccuronly whenthe ionization
v parameter is lower than is typical for HII regions. For higher ionization parameters, advective effects are
4 indirectandlargelyconfinedtotheimmediatevicinityoftheionizationfront.
3 2Theoverheatingofpartiallyionizedgasinthefrontisnotlarge,evenforsupersonic(R-type)fronts. For
0 subsonic(D-type)frontswedonotfindthetemperaturespikethathasbeenpreviouslyclaimed.
1 3.Themostsignificantmorphologicalsignatureofadvectivefrontsisanelectrondensityspikethatoccursat
0
theionizationfrontwhenevertherelativevelocitybetweentheionizedgasandthefrontexceedsaboutonehalf
5
0 theionizedisothermalsoundspeed. Observationalevidenceforsuchaspikeisfoundin[NII]λ6584Åimages
oftheOrionbar.
/
h 4. Plane-parallel, weak-D fronts are found to show at best a shallow correlation between mean velocity
p andionizationpotentialforopticalemissionlinesevenwhentheflowvelocitycloselyapproachestheionized
- soundspeed.Steepgradientsinvelocityversusionization,suchasthoseobservedintheOrionnebula,seemto
o
requiretransonicflows,whichareonlypossibleinadiverginggeometrywhenradiationforcesareincluded.
r
st Subjectheadings:gasdynamics,HIIregions,numericalmethods
a
:
v
1. INTRODUCTION ized regions, ranging in scale from cometary knots in plan-
i
X The classic early work on the effects of dynamics on the etarynebula(Lo´pez-Mart´ınetal.2001)andphotoevaporated
r emissionstructureofHIIregionswascarriedoutbyHarring- circumstellar disks in HII regions (O’Dell 2001, and refer-
a ences therein) up to champagne flows in giant extragalactic
ton(1977),whostudiedweak-Dfrontsinwhichthegasmo-
tionsarealwayssubsonicwithrespecttothefront.Withinthe HII regions (Scowen et al. 1998) and the photoevaporation
ofcosmologicalminihalos(Shapiro& Raga2001). In such
fullyionizedinterioroftheHIIregionthegaswasfoundtobe
flows,non-equilibriumeffectswillbesomewhatmoreimpor-
closetothermalandionizationequilibrium. Significantnon-
tantthaninsubsonicweak-Dfrontsduetothehigherveloci-
equilibriumeffects induced by the dynamicsare confinedto
tiesinvolved.Sankrit&Hester(2000)madeafirstattemptat
theedgeoftheregion,neartheionizationfront,wherethere
detailedmodelingof theemissionstructureofthe flowfrom
existlargegradientsintheradiation-fieldintensityandinthe
theheadofthecolumnsinM16,usingstaticequilibriummod-
physicalconditionsofthegassuchastemperatureanddegree
elsfortheionizationstructure.
ofionization. Harringtonfoundthatthedynamicsonlyhada
Second, continuous improvement in the spatial resolution
small effect on the integratedforbiddenline spectrumof the
and wavelength coverage of observations, together with ad-
modelsheconsidered. However,therearevariousreasonsto
vances in theoretical and observationalatomic physics, now
revisitsuchcalculationsnow.
allowamuchmoredetailedcomparisonbetweenmodelpre-
First, ionization fronts are now studied in a diverse range
dictionsandspatiallyresolvedobservationsofamultitudeof
of astrophysical contexts in which the classical, spherically
emittingspecies. Inthiscontext,evenmoderateandlocalized
symmetric, subsonic expansion studied by Harrington may
changes in the predicted spectrum due to dynamical effects
be the exception rather than the rule. Transonic photoevap-
canbeimportant.
oration flows seem to be a ubiquitous feature of photoion-
Third,adynamicaltreatmentallowstheself-consistentcal-
Electronicaddress:w.henney,[email protected] culationof the velocity field, which allows comparisonwith
Electronicaddress:[email protected] high-resolutionspectrallineprofileobservationsthatprovide
Electronicaddress:[email protected]
further constraints on the models. Furthermore, it permits
1 Workcarriedoutinpartwhileonsabbatical atDepartmentofPhysics
a unified treatment of the entire flow from cold, molecular
andAstronomy,UniversityofLeeds,LS29JT,UK.
2
gas, through the photon-dominated or photodissociation re- Williams& Dyson1996). If the gasis magnetized,thenthe
gion (PDR), and into the ionized region. Previous studies classificationbecomesmorecomplicatedsincetherearenow
of non-equilibriummodelsof PDRs (London1978;Natta & three wave speeds to take into account (Alfve`n speed plus
Hollenbach1998;Stoerzer&Hollenbach1998)havetended fastandslowmagnetosonicspeeds)insteadofjustthesound
to treat the PDR in isolation withoutconsideringthe HII re- speed. Thus, one may have a slow-mode D-critical front, a
gioninanydetail.Richling&Yorke(2000)presentednumer- fast-mode weak-R front, etc (Redman et al. 1998; Williams
ical radiation-hydrodynamicsimulations of the photoevapo- etal.2000;Williams&Dyson2001).
rationofproplyddisksbutthephysicsofthePDRwascalcu- These classification schemes were developed for plane-
latedinasimplifiedmanner. parallelfrontsbutwillbeapproximatelyvalidsolongasthe
Incommonwithmostotherphotoionizationcodes,Cloudy radiusofcurvatureofthefrontgreatlyexceedsitsthickness.
(Ferland et al. 1998; Ferland 2000) has traditionally calcu- This is usually the case since ionization fronts are in gen-
lated static equilibrium models in which time-dependentef- eral very thin comparedwith the sizes of HII regionsunless
fects are neglected, such as isochoric (constant density) or the ionization parameter is small (see below). What type of
isobaric(constantpressure)configurations. Thetaskofcom- ionizationfrontactuallyobtainsin a givensituationdepends
bininghydrodynamicswithdetailedsimulationoftheplasma ontheupstreamanddownstreamboundaryconditionsofthe
microphysics can be approached from one of two angles. front, in particular the upstream gas density and the down-
One method would be to add the atomic physics and radia- streamgaspressureandionizingradiationfield,togetherwith
tive transfer processes to an existing time-dependenthydro- thelarge-scalegeometryoftheflow,whichneednotbeplane-
dynamiccode.Theother,whichisthemethodpursuedinthis parallel.
work, is to add steady-state dynamics to an existing plasma Since the gas velocity through the front is high for an R-
physicscode. typefront,soisthefluxofneutralparticlesthatmustbeion-
The current paper is the first in a series of three that will ized for a given upstream density, which in turn requires a
present detailed results from our program to include a self- high ionizing flux at the downstream boundary. As a result,
consistent treatment of steady-state advection in a realistic R-typefrontsareusuallytransientphenomenaaccompanying
plasmaphysicscode. Thisfirstpaperintroducesthemethod- temporalincreases in the ionizing flux, such as the “turning
ology employed in the series as a whole and then concen- on”ofanionizingsource.Inthemostcommoncase,thefront
trates on the restricted problem of “weak” ionization fronts will be propagating rapidly through slowly moving gas. In
in a plane-parallelgeometry. The secondpaperof the series thelimit ofan extremeweak-Rfront,the gasvelocityin the
includesthemolecularreactionnetworksnecessaryformod- ionizationfrontframeanddensityareconstantthroughoutthe
eling the neutral/molecular PDR, while the third paper con- flow.
siders“strongD”ionizationfronts,wherethegasaccelerates D-type fronts are more common and the limit of an ex-
throughasonicpoint,asfoundindivergentgeometriessuch tremeweak-Dfrontcorrespondstoastatic constantpressure
asthephotoevaporationofglobules. frontinionizationequilibrium. Weak-Dfrontsrequireahigh
We first discuss the general problem of advection in ion- downstream pressure and therefore are likely to be found in
izationfronts(section2). Wethendescribethemodifications caseswheretheionizationfrontenvelopstheionizingsource.
that have been made to the Cloudy photoionization code in Strong-D and D-critical fronts, on the other hand, are con-
order to treat steady-state flow (section 3). Results from a sistentwiththefreeescapeofthedownstreamgasandhence
small sample of representative models are presented in sec- applytodivergentphotoevaporationflows,forexample,from
tion4andtheapplicationofourresultstoobservationsofthe globules.
Orionnebulaisdiscussedin section5. Furthertechnicalde- Advectionof materialthroughthe ionizationfrontmaybe
tails of the physicalprocessesand computationalalgorithms expectedtohavevariouseffectsontheemissionpropertiesof
arepresentedinaseriesofappendices. anebula.Inordertosimplifythediscussion,wewillconsider
aplane-parallelnebula,illuminatedatonefacebyagivenra-
2. ADVECTIONINIONIZATIONFRONTS diationfieldandinaframeofreferenceinwhichtheioniza-
The classification of ionization fronts depends on the be- tionfrontisatrest. ThisisillustratedinFigure1a.Thegasis
haviorofthe gasvelocityinthe frameofreferenceinwhich supposedtoenterthefrontfromtheneutralsidewithvelocity
theionizationfrontisfixed(Kahn1954;Goldsworthy1961). vn and to leave on the ionized side with velocity vi. Results
Inthisframe,thegasflowsfromtheneutralsideofthefront fromthissimplifiedmodelaredescribedin 2.2butwefirst
§
(denotedupstream)towardstheionizedside (denoteddown- discusstherelationbetweenthismodelandrealHIIregions.
stream). Iftheupstreamgasvelocityonthefarneutralsideis
2.1. PhysicalContextofAdvectiveFronts
subsonicwith respecttothefrontthenthefrontissaidto be
D-type,whileifitissupersonicthefrontissaidtobeR-type. Inthissection,weconsidertwotypicalscenariosinwhich
A further distinction is made between those fronts that con- advective ionization fronts may be encountered and investi-
tain an internalsonic point, whichare said to be strong, and gatetowhatextenttheymayapproximatedassteadyflowsin
those that do not, which are said to be weak. For example, theframeofreferenceoftheionizationfront. Inthisdiscus-
aweak-Rfrontwillhavesupersonicvelocitiesthroughoutthe sion we follow Shu (1992) in denoting gas velocities in the
front,whereasinastrong-Dfrontthegasstartsatsubsonicve- frameofreferenceoftheionizingstarbyu,gasvelocitiesin
locities on the neutralside, accelerating throughthe frontto theframeofreferenceoftheionizationfrontbyv,andpattern
reachasupersonicexhaustvelocityontheionizedside.When speedsofionizationandshockfrontsbyU.
thedownstreamgasvelocityisexactlysonicthefrontissaid
2.1.1. ClassicalStro¨mgrenSphere
tobecritical. Thereisalsothepossibilityofarecombination
front,inwhichthesenseofthegasvelocityisreversedandthe TheevolutionofaclassicalStro¨mgren-typeHIIregionina
flowisfromtheionizedsidetowardstheneutralside,witha constantdensitymediumhasbeendescribedbymanyauthors
similarrangeofpossiblestructures(Newman&Axford1968; (e.g., Goldsworthy 1958; Spitzer 1978; Shu 1992; Dyson &
DynamicModelsofIonizationFronts 3
(a) (b) (c)
5 Rif/rinit
MachNumber,M
Uif 4
ionized neutral
Us 3
u
v u 2
vn z un ci
ui(rif)
vi rs r r 1
rif
Fig. 1.—Advective fronts: (a)Plane-parallel idealization; (b)Classical 0
Stro¨mgrensphere;(c)Photoevaporationflow. 0 2 4 6 8 10
Time,t/t0
Williams 1997). If the ionizing source turns on instanta- Fig. 2.—Evolutionoftheionizationfrontradiusandthe“exhaust”Mach
neously,thentheionizationfrontisinitiallyR-typeandprop- numberofnewlyionizedgasintheframeofreferenceoftheadvancingion-
agatessupersonicallythroughthe surroundinggaswith little izationfrontinaclassicalStro¨mgrensphere.Thesequantitiesareplottedasa
accompanyinggas motion. By the time the ionization front functionoftimesincethefrontreachedtheinitialStro¨mgrenradius,inunits
ofthesoundcrossingtimeatthatradius.
reachestheinitialStro¨mgrenradius(wheretherateofrecom-
binations in the ionized region approximately balances the
ionizingluminosityof the source), the frontpropagationve-
locityhasslowedtotheorderofthesoundspeedintheionized 2.1.2. PhotoevaporationFlow
gasand the frontbecomesD-type, precededby a shock that
Photoevaporation flows are very common in ionized re-
acceleratesandcompressestheneutralgas.TheinitialR-type
gions,occurringwhenevertheionizationfrontisconvexfrom
phase is very short (of order the recombination timescale),
the point of view of the ionizing source, thus allowing the
and consequently is of little observational importance. The
ionized gas to freely stream away from the front. Examples
structureoftheregionduringitssubsequentD-typeevolution
isshowninFigure1b. include bright-rimmed globules and proplydsin HII regions
and cometary knots in planetarynebulae. On a larger scale,
Thepropagationspeedoftheshockisroughlyequaltothat
oftheionizationfrontandalsotothevelocityofthegasinthe blister-typeHIIregionscanalsobeconsideredphotoevapora-
tionflows(Bertoldi&Draine1996).
neutralshell:
Figure1cshowthestructureofanidealizedphotoevapora-
U U u , (1)
s ≃ if ≃ n tionflow,roughlycorrespondingtotheequilibriumcometary
whereastheionizedgasexpandshomologouslywithaveloc- globulesofBertoldi&McKee(1990). Theneutralgasisas-
itythatincreaseslinearlywithradius,reachinghalfthespeed sumed to be approximately at rest with respect to the ion-
oftheionizationfrontatthefrontitself: izing source and the ionization front to be D-critical with
1 r v u c. The ionization front eats slowly into the neu-
ui = Uif. (2) trial≃gasiw≃ithiU = v =c /(2c2) c,andtheMachnumber
2 Rif! i − n n i ≪ i
reachedbytheionizedgasatthefrontwillbe 1.Outsidethe
Hence, the velocity of the ionized gas immediately down- ≃
front,theionizedgasacceleratesasanapproximatelyisother-
stream of the ionization front in the ionization front frame
malwind,asobservedaroundtheOrionproplyds(Henney&
is
Arthur1998).
v =u(R ) U = 0.5U . (3)
i i if if if
− −
The ionization front propagates very slightly faster than the 2.2. DirectandIndirectEffectsofAdvection
gas in the neutral shell, giving a small upstream neutral gas
Inordertoprovidesomephysicalinsightintoadvectiveion-
velocityintheionizationfrontframeof
izationfrontsandtoguidetheinterpretationofthenumerical
v =u U < 2c2/c 0.1kms 1, (4) simulations,wehavedevelopedasimpleanalyticmodelfora
n n− if − n i ∼ − plane-parallelweak-Dionizationfront,inwhichthegastem-
wherec ,c aretheisothermalsoundspeedsintheneutraland
n i peratureisassumedtobeaprescribedmonotonicfunctionof
ionizedgas,respectively.Theevolutionoftheionizationfront
thehydrogenionizationfraction,x. Withthisassumption,and
propagationspeedcanbedescribedintermsofitsradiusas
consideringmassandmomentumconservation,itispossible
U = 2ci , (5) to find algebraic solutions for the density, gas velocity, and
if 4(R /R )3/2 1 1/2 soundspeedasfunctionsof x(seeAppendixA). Thesesolu-
if init − tionsformaone-parameterfamilycharacterizedbythemaxi-
h i
where R is the initial Stro¨mgren radius. The Mach num- mumMachnumberofthegasintherest-frameoftheioniza-
init
berreachedbytheionizedgasjustinsidetheionizationfront, tionfront,reachedasymptoticallyas x 1. Ifwenowtake
→
measuredin theframeofreferenceinwhichthefrontissta- intoconsiderationtheionizationbalanceandradiativetrans-
tionary,isgivenbyM =0.5U /c. ThisisplottedinFigure2 fer, one can find a solution for x(z), the ionization fraction
if i
as a function of ionization front radius. During the lifetime asafunctionofphysicaldepth,bysolvingapairofordinary
of a typical O star (and assuming an ambient density of or- differentialequations.
der 1 cm−3), the radiusof a classical HII regionwill expand ApartfromthemaximumMachnumber, m,thesolutions
M
by roughly a factor of 4, so, as can be seen from Figure 2, are found to depend on two dimensionless parameters, ξ
ad
M = 0.3–0.5 is typical of the majority of the evolutionary and τ . The first of these, ξ (defined in equation A18), is
ad
lifetime. rough∗ly the ratio of recombination length to mean-free-path
4
(a) Υ model, the direct global effects of advection are expected
1.2 tobesignificantandindeedthethicknessoftheionizedslab
ne is reduced by 50% as approaches unity. In the high-Υ
1 v model(morerepresentaMtivmeoftypicalHIIregions),thedirect
globaleffectsofadvectionareexpectedtobenegligiblesince
0.8 λ 3 10 3 . However,we find thatevenin thiscase
ad − m
≃ × M
the thickness of the ionized slab varies by about5% as
m
0.6 isvariedbetween0and1. ThisisduetoapronouncedpMeak
that develops in the electron density distribution at the ion-
0.4 ization front for & 0.5 (for a static model the electron
m
M
density declines monotonically through the front). We also
0.2 findthattheionizationfrontbecomessubstantiallysharperas
is increased, which is due to a decrease in the ioniza-
m
0 Mtionfractionforagivenvalueoftheopticaldepthtoionizing
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
radiation. Both these indirect effects of advection have sig-
z
nificanteffectson the emissivity profilesof opticalemission
(b)
lines(seeFigureA19ofAppendixA),especiallythosesuch
1.4
ne as [OI]λ6300 Å and [SII]λ6716 + 6731 Å that form close
1.2 v to the ionization front. One can also calculate the spectral
profilesof emissionlines fromthemodels, asshownin Fig-
1 ures A20 and A21 of the Appendix. Again, it is lines that
form in the partially ionized zone that show the most inter-
0.8
esting behavior. These may show RMS line widths roughly
equaltothesoundspeedandsignificantvelocityoffsets,both
0.6
due to the gas acceleration in the ionization front. The de-
0.4 rivedwidthsareroughlyfourtimesthethermalwidthoflines
emittedbylightmetals.
0.2 Although this analytic model has provided insight into
some of the effects of advection, it is obviously deficient in
0
many respects. Many physical processes have been ignored
0 0.2 0.4 0.6 0.8 1
and in particular the use of a fixed temperature profile T(x)
z
doesnotallowforthefactthatT(x)itselfmaybeaffectedby
Fig. 3.— Dynamic slab solutions from the simplified model developed
theadvectedflow.
in Appendix A. (a) τ = 30, low ionization parameter, fat IF. (b) τ =
3000, highionization p∗arameter, thinIF. Electron density (solidline)∗and
gasvelocity(dashedline)areshownineachcasefor6modelswith m = 3. ADDINGDYNAMICSTOCLOUDY
0.0,0.2,0.5,0.7,0.9,0.99(correspondingrespectivelytosuccessiveMcurves
from right to left). Electron density is normalized by the fiducial density Inordertostudythestructureofadvectiveionizationfronts
nm,velocityisnormalizedbythemaximumsoundspeedcm,anddistanceis ingreaterdetail,weneedtoincludeawiderangeofadditional
normalizedbythestaticStro¨mgrendepthz0—seeAppendixAfordetails. physics.Thiscouldbedoneinavarietyofways,forexample
byintegratingthroughthesteadystateequationsorbyinclud-
ing source termsin a time-dependenthydrodynamicsimula-
tion. In each case, an implicit treatmentof the source terms
of ionizing photons in a D-critical front and is not expected
isnecessary,asthe manyphysicalprocesseswith timescales
tovarygreatlybetweenHIIregions,havingatypicalvalueof shorterthanthedynamicaltimescaleleadtotheproblembe-
ξ 10.Thesecondparameter,τ (definedinequationA17),
isadro≃ughlytheratioofthethicknes∗softhefullyionizedslabto ingverystiff.
Inthepresentwork,wehavechosentoadaptthephotoion-
thethicknessoftheionizationfrontandisproportionaltothe
izationcode,Cloudy,whichalreadyincludesacomprehensive
ionization parameter at the ionized face: τ 106Υ, where
Υ F /(n c). The global importance of∗ad∼vection for the treatment of the physical source terms. Cloudy, in common
≡ 0 0 withothertraditionalplasmacodes,searchesforequilibrium
systemasawholecanbecharacterizedbytheparameterλ ,
ad solutionsoftheionizationequations. Inordertotreatsteady
definedastheratioofthefluxofhydrogenatomsthroughthe
advectiveflows, we haveincludedadvectivesourceandsink
ionizationfrontto the fluxof ionizingphotonsat the illumi-
termsintheequilibriumbalanceequations. Theeffectofthis
natedfaceoftheslab(equationsA10andA19).Forsmallval-
is that the equilibrium search phase now in fact determines
uesofλ ,itsvaluecanbeapproximatedasλ ξ /τ .
This canadbe understood as follows: the locaalde≃ffecadtsMomf ad∗- theimplicitsolutionoftheadvectiveequations,andsotreats
short-timescaleprocessesinastablemanner.
vectionattheionizationfrontitselfarealwayssubstantial(so
In this section, we present the basic equationsand outline
longas isnottoosmall),beingoforderξ . Onthe
Mm adMm themethodswhichweusetosolvethem.
other hand, the partially-ionizedzone occupies only a small
volume compared with the fully ionized gas unless the ion-
ization parameteris small, so the globaleffects of advection 3.1. Equations
arereducedbyafactorofτ . Cloudy takes into account the conservation equations for
Detailedresultsarecalcul∗atedinAppendixAforbothalow each speciesand also the heating andcoolingbalance under
ionization parameter model (τ = 30) and a high ionization the simplifying assumptions of constant density or constant
parametermodel(τ = 3000).∗Thestructureofthesemodels pressure. This procedure can be summarized as balancing
fordifferentvalues∗of isshowninFigure3. Inthelow- sourceandsinktermsfortheionizationandenergyequations.
m
M
DynamicModelsofIonizationFronts 5
For the ionization equation in the static case this can be ex- the iterative scheme. Instead, we take advantage of the cur-
pressedas rent algorithm used by Cloudy and difference the equations
implicitly.Suchanimplicitschemehastheadvantagethatthe
dni =G + R n n S + R =0, (6) timestepisnotlimitedtotheshortestionizationorrecombina-
where ddnt/dt isi theXj,riatej→oif cjh−angie oif thXje,ivoil→umje density of tslioyonsntgeetmirmthiena,nwwthhhiiecchhioitnshieczlaedtaiyornlnyamournicrseaacltoitsimfmabceitsnocaratylieofsnoratiramenegasescntarelorepashl.ymsuiccahl
i
aparticularionizationstate, whichinequilibriumisequalto At iteration m, the advection terms may be approximated
zero.TheR aretheratesforionization(where jisahigher basedonthevalueinthepresentzoneandanupstreamvalue
i j
statethani)→andrecombination(where jisalowerstatethan inthepreviousfulliterationas
i). G andS caterforprocessesnotincludedwithintheion-
izatioin laddeir, and are respectively the source of ions from d x xmi (z)−xim−1(z−∆z), (12)
suchprocessesandthesinkrateintothem. Adetaileddiscus- dz i → ∆z
sionofthesolutionmethodfortheionizationnetworksinthe where xm(z)isthevalueof x = n/natpositionzatthemth
i i i
equilibriumcaseisgiveninAppendixB. iteration of the scheme, and ∆z is an adjustable advection
The general Cloudy solution method works by a series of length. For the first iteration no upstream values are avail-
nested iterations. The innermost loop is the ionization net- ablesonoadvectiontermsareincludedintheequations. Itis
work, externalto this isthe electrondensityiteration, which usefultodefinethelook-backoperator
enforceschargeneutrality, then the temperatureloop, which
L [xm(z)]= xm 1(z ∆z) (13)
enforces thermal balance. Finally, an optional outermost it- ∆z i i − −
eration loop varies the density to achieve pressure (or more forvalues, such as x, givenperunitmaterial. Values speci-
i
generallymomentumflux)balance.Thewholesystemisiter- fiedperunitvolumeneedtobescaledtoaconservedvariable
ateduntilconvergencewithinagiventolerance. beforethelookbackisapplied,sothat
Oncedynamicsisincluded,thecontinuityandmomentum
L [nm(x)]=nmxm 1(z ∆z). (14)
equationsmustbeaddedtothesetofequationstobesolved, ∆z i i − −
kineticandinternalenergytransportandpressureworkmust Thismaybethoughtofasafirst-orderLagrange-remapsolu-
betakenintoaccount,andadvectiontermsmustbeaddedto tionfortheadvectionequation.
the ionization balance equations. For example, for a plane- FortheschemediscussedinAppendixBfortheionization
parallel steady-state flow (the simplest case, but one that is ladder,theadvectivetermsmaythenbeincludedsimplyasan
applicable to blister HII regions), the equations to solve in additionalsourceterm,
fluxconservativeformare L [xm(z)]
G =nv ∆z i , (15)
∂ i ∆z
(ρu)=0, (7)
∂x andsinkrate
v
∂ S = (16)
(p+ρu2)=ρa, (8) i ∆z
∂x
inthelinearizedformoftheequations,anditeratedtofindthe
∂ ρu w+ 1u2 =H C, (9) non-linearsolutionasinthetime-steadycase.
∂x" 2 !# − The energy balance equation may be treated in a similar
manner. Using the mass conservation equation (eq. 7), we
∂ havefromequation(9)
(nu)=G + R n n S + R . (10)
Here,∂axisianacceileraXjt,iionje→.gi.,jg−raviityiorrXaj,diiatii→onjdriving,w ρv.∇ w+ 21v2!= H−C, (17)
is the specific enthalpyw = ε+ p/ρ, whereε is the specific whichmaybedifferencedas
internal energy, and H C is heating minus cooling. Here
tohfetrsapnescliafiticonin,tseornεal=e3n/e2−rg(yp/iρn)c,luasdetrsaonnsfleyrsthferotmheromthaelrepnheyrsgiy- ρv(cid:16)w+ 12v2(cid:17)−∆Lz∆z(cid:16)w+ 12v2(cid:17) = H−C. (18)
calenergycomponents(ionizationenergy,bindingenergy,vi-
Thetermsontheleftsideofthisequationmaythenbetreated
brationandrotationenergyofmolecules,etc.) aretreatedas
as additional heating and cooling terms in the temperature
explicitheating and cooling termsin the underlyingthermal
solver.
balancescheme.
The continuity and momentum equations are more easily
Theadvectiontermshavethegeneralformof (nv)(for
i dealt with. The continuity equation is taken into considera-
∇·
steady state), where v is the advection velocity. This can be
tion simply by eliminatingthe velocity v in termsof ρ in all
writtenas
equationsusingthesubstitution
(nv)=nv (n/n). (11)
i i
∇· ·∇ ρvrd =constant, (19)
3.2. DifferencingScheme
whichcomesfromintegratingthegeneralformofequation7
Althoughitispossibletosolvetheionizationequationsin whered =0,1,2indicatesplaneparallel,cylindricalorspher-
an explicitlytime-dependentway, thisis notthe bestway to ical geometry, respectively. The initial condition is given at
proceed. Thephotoionizationtermsin the steady-state solu- theilluminatedface.
tion will often have very short timescales, so stability con- The dynamicalpressure, which appearsin the momentum
straintswouldlimitthetimesteptothisshortphotoionization equation, is taken into account by adding the ram pressure
timescale,andhencecauseanextremelyslowconvergenceof termρv2tothetotalpressure.
6
1
low enough value, then all physical processes that occur on
∆zad/1015cǫm1 timescaleslongerthan∆t = ∆z/vhaveconverged,sothead-
0.1 ǫ2 vectionlength can be reduced. This has the effect of lower-
ingǫ butalso temporarilyincreasesǫ due to the release of
2 1
shorter-timescale processes from strict local equilibrium, so
0.01 severaliterationsmustbecarriedoutatthenewvalueof∆z.
Thisprocedureiscontinueduntilǫ hasfallentoasufficiently
2
lowvalue.
0.001
4. RESULTS
10−4 In this section we present results for selected advective
ionization fronts calculated using Cloudy, all using a plane-
parallelslabgeometry.Theprincipalinputparametersforthe
10−5
0 10 20 30 40 50 60 70 80 90 modelsarethehydrogennumberdensity,n0,gasvelocity,u0,
IterationNumber hydrogen-ionizingphotonflux,F0,allspecifiedattheillumi-
Fig. 4.—Convergencebehaviorofanexamplemodelasafunctionofiter- natedface. Thespectraldistributionoftheincidentradiation
ationnumber,m. Solidline: advectionlength∆z(unitsof1015cm);dashed fieldwasassumedtobeablackbodywitheffectivetempera-
line:convergenceerror,ǫ1;dottedline:discretizationerror,ǫ2. ture, Teff. All these parametersare shown in Table 1 for the
threemodelspresentedhere. Thegas-phaseelementalabun-
dances for all the models were set at the standard ISM val-
Varying the Advection Length— The advection length, ∆z, in ues(Baldwinetal.1996)andOrion-typesilicateandgraphite
this scheme determines the manner in which different pro- grainswereincluded(Baldwinetal.1991). Sincethispaper
cesses are treated. The differencingwe have chosen has the is concerned with the effects of advection on the ionization
effect that processes far more rapid than ∆z/v are treated as front, all molecules were turned off and the integration was
instaticequilibrium,whileslowerprocessesarefollowedex- stopped when the electron fraction fell below 10 3. The in-
−
actly. The correct steady-state solution is found in the limit clusionofmolecularprocessesin thePDRwillbe described
∆z 0, but the smaller the advection length chosen, the in a following paper. The models also include an approxi-
→
longerthesystemwilltaketoreachanequilibriumstate. The matetreatmentofatangledmagneticfield(seeAppendixC),
natural procedure is then to use a first iteration solution by characterizedbythefieldstrengthattheilluminatedface,B .
0
ignoringtheadvectionterms,andthengraduallydecrease∆z One further physical process was disabled in these mod-
until∆z ∆z , where∆z isthesizeofaspatialzonein els: theradiativeforceduetotheabsorptionofstellarcontin-
grid grid
∼
thesimulation,atwhichstagethetreatmentofadvectionwill uumradiation(principallybydustgrains). Thiswasdonefor
beasaccurateasthatofphotoionization. purelypragmaticreasons, since the inclusionof this process
Inordertotracktheconvergenceofthemodelsandtodeter- forhighionizationparametermodelsmakesitverydifficultto
minewhentoreducetheadvectivetimestep, we monitorthe settheapproximatedesiredconditionsattheionizationfront
behavior of two error norms, ǫ and ǫ . Both are calculated byvaryingconditionsattheilluminatedface. Modelsthatdo
1 2
as the squared norm over all zones, z, and ionic/molecular includethisprocessarediscussedfurtherinsection5.2.
species, i. Thefirstofthese normsistheconvergenceerror, Thefirsttwomodels,ZL009andZH007,areweak-Dfronts
definedas withlowandhighionizationparameter,respectively,withpa-
ǫ = nmi −nim−1 , (20) rameterssimilartothetoymodelsdiscussedinAppendixA.
1 (cid:13)(cid:13)(cid:13) ∆z/v (cid:13)(cid:13)(cid:13)z,i Tthheeivsoetlhoecritmyaaltstohuenidonsipzeeeddfaicneo,rud0e,rwtaosasveotisdomtheewphoastsbibeilloitwy
(cid:13) (cid:13)
which measures the diffe(cid:13)(cid:13)rence betw(cid:13)(cid:13)een the model solutions of gas passing through a sonic point during an intermediate
for the last two iterations. The second is the discretization iteration (transonic fronts will be considered in a following
error,definedas paper). Thethirdmodel,ZH050,isaweak-Rfrontinwhich
ǫ = ni−L∆z(ni) ni−L(∆z/2)(ni) , (21) tShuecghaRs-vteylpoeciftryonretslaatirveelitkoetlhyetforobnettirsasnuspieenrtsoannidctthhurosuogfhloimut-.
2 (cid:13)(cid:13) ∆z/v − ∆z/2v (cid:13)(cid:13)z,i ited observational significance. Nevertheless, this model is
(cid:13) (cid:13)
which measures(cid:13)(cid:13)the accuracy of the present e(cid:13)(cid:13)stimate of the usefulsince it providesa stringenttest of our simulationsin
advective gradients in the solution compared to an estimate thelimitofhighadvectivevelocities.
withhalftheadvectionlength. As well as the advective models, we also calculate equiv-
Ifǫ ǫ , thenthesolutionisconvergedwiththepresent alent static models for each of the three cases considered,
1 2
∆z,whi≪leiftheerrorsintheLagrangianestimateofthegradi- whichhaveconstantpressureforcomparisonwiththeweak-
entofthevaluearestillsignificant,thenthetimestepshould Dfrontsorconstantdensityforcomparisonwiththeweak-R
bedecreased. Cutting∆zwhenǫ2 < 0.1ǫ2 producessubstan- front.
1 2
tialimprovementsintherateofconvergenceoftheadvective
4.1. LowIonizationParameter,Weak-D
solutions. However, it can still take a substantial number of
iterationstoreachequilibriumforlargeadvectionvelocities. This model, ZL009, has physical parameters that are in-
Anexampleofthewayinwhich∆z,ǫ ,andǫ varyduring spired by those of cometary globules in planetary nebulae
1 2
a modelcalculationis givenin Figure 4 (which corresponds such as the Helix, although the geometry is plane-parallel
to model ZL009 discussed in the following section). It can ratherthansphericallydivergent.Theionizationparameterof
beseenthatwhile∆z remainsconstant,theconvergenceer- themodelis verylow(Υ = 3.33 10 5), whichaccentuates
ad −
ror,ǫ ,decreaseswitheachiteration,whileǫ hardlychanges theglobaleffectsofadvectionand×alsoleadstotheionization
1 2
after the first iteration for a given ∆z. When ǫ /ǫ falls to a frontthicknessbeingcomparabletotheStro¨mgrenthickness
1 2
DynamicModelsofIonizationFronts 7
3·10−18
TABLE1 Hαλ6563˚A(advective)
ModelParameters 2·10−18 Hαλ6563˚A(static)
Parameter ZL009 ZH007 ZH050 1·10−18
log(n0/cm−3) 3.5 4.0 4.0 a
ulloo0gg/((kTFme0ff/s/c−Km1)−2) −949...065 −1753...000 −51034...006 34··1100−−11088 [[NNIIII]]λλ66558844˚A˚A((astdavteicc)tive)
log(B0/Gauss) none −4.0 −4.0 2·10−18
logΥ0 4.5 1.5 1.5 1·10−18
τ −9.9 6−800 6−800 b
Mξa∗dm 08.8.41 01.37.38 41.30.18 −−13cm) 3·10−108 [OI]λ6300˚A(advective)
λad 2.2 0.0015 0.008 (ergs 2·10−18 [OI]λ6300˚A(static)
Emissivity 1·10−18 c
Line 4·10−108
3·10−18 [[SSIIII]]λλ66771166++66773311˚A˚A((astdavteicc)tive)
2·10−18
1·10−18
d
0
2·10−19
1.5·10−19 [[OOIIIIII]]λλ55000077˚A˚A((astdavteicc)tive)
1·10−19
5·10−20
e
0
0 1·1015 2·1015 3·1015 4·1015
Depth,z(cm)
Fig. 6.—EmissivitystructureofmodelZL009. AsFigure5butshowing
volumeemissivityof(a)Hαλ6563Å,(b)[NII]λ6584Å,(c)[OI]λ6300Å,
(d)[SII]λ6716+6731Å,and(e)[OIII]λ5007Å.
1
Hαλ6563˚A
[NII]λ6584˚A
0.8 [OI]λ6300˚A
[SII]λ6731˚A
intensity 0.6 [OIII]λ5007˚A
line
Relative 0.4
0.2
Fig. 5.—StructureofmodelZL009asafunctionofdepthfromtheillu- 0
minatedface.(a)Velocityandisothermalsoundspeed.(b)Gastemperature. −40 −30 −20 −10 0 10 20 30
(c)Numberdensityofhydrogennucleons.(d)Hydrogenionizationfraction. Dopplershift(kms−1)
(e)Heliumionizationfractions.Panelsb–eshowtheadvectivemodelresults Fig. 7.—Face-onemissionlineprofilesofmodelZL009including ther-
(solidline)andtheresultsfromanequivalentstaticconstantpressuremodel malbroadening: Hαλ6563 Å (thick solid line), [NII]λ6584 Å (thin solid
(dashedline). line),[OI]λ6300Åthickdashedline),[SII]λ6731Å(thindashedline),and
[OIII]λ5007Å(thickdottedline). Alllinesarenormalizedtotheirpeakin-
tensities.
oftheionizedlayer.
As can be seen from Figure 5, the advectionhas large ef-
fects on the modelstructure as would be expected given the the intrinsic Balmer decrementis increased in the advective
largevalueofλ . Thedepthoftheionizedregionisreduced model, which gives Hα/Hβ = 3.29 (reddening by internal
ad
bymorethanafactoroftwowithrespecttothestaticmodel, dustisnotincludedinthelineratiosgiveninTable2).Thisis
with concomitant reductions in the brightness of the hydro- becausethetemperatureatlow ionizationfractionsissignif-
gen recombinationlines (see Table 2). The collisional lines icantly higherin the advectivemodel, which leads to a non-
of [OI], [SII], and [NII] are also reduced in intensity, albeit negligiblecollisionalcontributiontotheBalmeremissionthat
to a lesser degree ([OIII] emission from this model is neg- preferentiallyexcitesHα. Thetemperatureinthemorehighly
ligible due to the low ionization parameter). Interestingly, ionizedzoneisalsoincreasedsomewhatbytheeffectsofad-
8
vection. −1)
s
Also listed in Table 2 are the mean velocities and full- (km 15 v
alwisnisdeutmphr-iohnfiagllfet-shmaaatrxethiimelluamsotorfdatteheledidsinioffFbeisrgeeurnvrteeed7m.fiaTscsheieooncnol.inntTerishb,euctaeiolmcnuisflsariotoemnd soundspeed 150 c
eeancehdcuosminpguttahteiolnoaclazlotneemtpoetrhaetulirneeapnrdofitlheeisatthoemrmicawllyeibgrhotaodf- velocity, 0 a
Gas
bldtihruneoeeaetdimoseritbthttleihunaeagncscshtpehilefeetcericadeotsil.wolinAistihosofnargaerlasyepsseetuchxltrtc,otituothegde[hSHmtIhIαe]etaaliiolnnndleiinzi[seOastsi.Ioi]g,nTnwhfifirehocin[acNtnh,taIliIyss] mperature(K) 51··110034 TTee((astdavteicc)tive)
canbeseeninFigure5a.TheHαlinehastwocomponents:a Te b
0
broadblue-shiftedcomponentdue to emission fromthe ion-
ibizoyendaizsflautoibowsn,idfairanardcytiapoennasakrwrionhwetrheceotehmleepctoetrnmoenpnetdreaanttuszrieteyrocthhvaaentlgooeccscitusyhr,sacraaptullyso.ewd −3Density(cm) 110045 nn((astdavteicc)tive)
Forthe[SII]line,thethermalbroadeningandthegasaccel- Gas c
erationbothcontributeinequalamountstothepredictedline
width.Forthelightermetallines,theaccelerationbroadening 1 x(H+)(advective)
rseommaeiwnsharto.uFgohrlyHtαh,ethsaemtheerbmutalthberothaderemnianlgwdiodmthinisatiensc.reased Fraction00.7.55 x(H+)(static)
In thismodel, the gasvelocityandMach numberincrease Ion0.25
monotonicallyasthegasflowsfromtheneutraltotheionized d
0
side,reachingamaximumMachnumberof =0.84atthe
illuminatedface.Asaresult,emissionlinesMfrommmorehighly 1 x(He+)(advective)
btihoeentwizeeeffedencstp[OeiscIi]sealsingadhret[OmwIoiItIrh]e.obnlulye-s2h.5iftkemd(ss−e1eTvaebloleci2ty).dHiffoewreevnecre, IonFraction000..27.555 xxx(((HHHeee+++++)))(s((tasatdtavitcei)cc)tive)
e
0
4.2. HighIonizationParameter,Weak-D 0 2.5·1016 5·1016 7.5·1016 1·1017 1.25·1017 1.5·1017 1.75·1017
Depth,z(cm)
Thismodel,ZH007,hasphysicalparameters(seeTable1)
Fig.8.—StructureofmodelZH007asafunctionofdepthfromtheillumi-
inspired by the central region of the Orion nebula (Baldwin natedface.AllpanelsasinFigure5.
etal.1991).ThedensityisonlyslightlyhigherthaninZL009
buttheionizingfluxismuchhigher,resultinginafarhigher
ionizationparameter(Υ=3.3 10 2).Theionizationfrontis
−
×
muchthinnerthanthedepthofthefullyionizedslab. Thelo- profiles,itcanbeseenthattheinitialheatingofthegasasit
caladvectionparameter,ξ (seeSection2andAppendixA), isionizedismoregradualintheadvectivemodelthaninthe
ad
is somewhat higher than in the previous model due to the staticmodelduetothephoto-heatingtimescalebeinglonger
hotter gas temperature and softer ionizing spectrum but the than the dynamic timescale. However, once the ionization
muchhighervalueofτ thataccompaniesthehigherioniza- fraction exceeds about 20% the photo-heating rate exceeds
tionparametermeansth∗attheglobaladvectionparameter,λ , the static equilibrium model because the neutral fraction at
ad
isverysmall. a givenvalue of the ionizing flux is higherthan in the static
Thestructureoftheadvectivemodelasafunctionofdepth case. Thisproducesacharacteristicoverheating,whichisof-
intotheslabfromtheilluminatedfaceisshownbysolidlines tenseeninadvectivefronts.Inthecurrentcaseitisrelatively
inFigure8.Forsomepanels,anequivalentconstant-pressure, modest, producingan extended104 K plateau for x = 0.3–
e
staticmodelisalsoshown(dashedlines). Theindirecteffects 0.9. For x > 0.9 the gas approaches static thermal equi-
e
of advection are much greater than the direct loss of 0.1% librium again and the two curves coincide with temperature
oftheincidentionizingfluxduetotheionizationoffreshgas. variationsdeterminedbyradiationhardeningandthevarying
Thelargesteffectisaroughly3%increaseinthemeandensity importanceofthediffusefield. Figure9cshowsthevariation
intheionizedzone(thedensitiesattheilluminatedfacesare in the gas density and it can be seen that the density on the
setequalinthestaticandadvectivemodels)duetothevarying farneutralsideissignificantlyhigherintheadvectivemodel.
importanceoframpressureasthevelocityvariesthroughthe Thisisaresultofthelackoframpressureontheneutralside
slab. Due to the differencein density dependenceof recom- (see Figure 9e), which means the thermal pressure must in-
binationanddustabsorption,thisleadstoaslightdecreasein crease to compensate. The magnetic field at the illuminated
the ionized column density together with an increase in the faceinthismodelwassettobe0.1mG,implyinganegligible
emission measure, caused by a reduction in the fraction of contributiontothetotalpressureintheionizedgas. However,
ionizingphotonsthatareabsorbedbydustgrains. onthefarneutralside,wherethedensityishigherthefieldis
Figure9showssomeofthesamequantitiesasinFigure8 muchlarger(assumedtogrowwithcompressionasB ρ2/3,
butthistimeplottedagainsttheelectronfraction, x = n /n. seeAppendixC), so thatthemagneticpressurebecom∝esap-
e e
This effectively ‘zoomsin’ on the ionization fronttransition preciable,limitingthedensityinthecoldgas. TheinferredB
itself, allowingoneto appreciatedetailsof the structure that intheneutralgasissimilar toobservedvaluesin theneutral
arenotapparentintheplotsagainstdepth.Notethatthemore veilofOrion(Trolandetal.1989).
neutral gas is on the left in Figure 9, whereas it was on the Theline emissivitiesasafunctionofdepthforthe ZH007
rightinFigure8. InFigure9b,whichshowsthetemperature modelareshowninFigure10. Themainchangebetweenthe
DynamicModelsofIonizationFronts 9
TABLE2
EmissionLinePropertiesfromWeak-DPhotoionizationModels
ModelZL009 ModelZH007
LineIntensity LineIntensity
Line λ Static Advect v¯ ∆v Static Advect v¯ ∆v
Hα 6563 2.95 3.29 5.89 21.6 2.92 2.92 6.46 20.0
− −
[OI] 6300 2.40 4.91 5.04 6.3 3.5( 3) 2.8( 3) 6.99 6.4
− − − −
[NII] 6584 3.97 6.23 6.07 6.9 0.42 0.43 7.56 6.0
− −
[SII] 6731 3.52 5.33 5.17 5.3 0.04 0.03 7.29 4.3
− −
[OIII] 5007 0.05 0.03 7.50 6.1 5.21 5.21 6.32 5.0
− −
Hβ 4861 2.868 3.382 0.435 0.440
− −
Note. —Lineintensities areall calculated foraface-on orientation andare
given relative to Hβ except for Hβ itself, which is log(intensity) in units of
ergcm 2s 1.
− −
−1) 6·10−17
s
(km 15 v 4·10−17
soundspeed 105 c 2·10−17 a HHααλλ66556633˚A˚A((astdavteicc)tive)
Temperature(K)Gasvelocity,51··1100034 ba TTee((astdavteicc)tive) −3m) 1.12345·····1111100000−−−−−111110077777 b [[NNIIII]]λλ66558844˚A˚A((astdavteicc)tive)
−3GasDensity(cm) 1100045 c nn((astdavteicc)tive) −1LineEmissivity(ergsc 251···111000−−−1110787 c [[OOII]]λλ66330000˚A˚A((astdavteicc)tive)
1 1.5·10−17 [[SSIIII]]λλ66771166++66773311˚A˚A((astdavteicc)tive)
x(H+) 1·10−17
Fraction0.07.55 xx((HHee+++)) 5·10−18 d
Ion0.25 1.5·10−106
d
0
−2) 1·10−16
Pressures(dynecm 123···111000−−−888 PPPgmraaamsg 5·10−107 0e [[OOIIIIII]]λλ55000077˚A˚A5·((1as0td1av6teicc)tive) 1·1017 1.5·1017 2·1017
Partial 00e 0.2 0.4 0.6 0.8 1 1.2 Fig. 10.— Emissivity structureDeoptfh,zm(comd)el ZH007 in units of 2
ElectronFraction,ne/n 10−16ergcm−3s−1).PanelsasinFigure6. ×
Fig.9.—StructureofmodelZH007asafunctionofelectronfraction,ne/n.
Panels(a)–(c)asinFigure8.(d)IonizationfractionsofH+(mediumweight
line),He+ (thickline),andHe2+ (thinline). (e)Partialcontributionstothe
totalpressure:thermalgaspressure(mediumweightline),magneticpressure
(thinline),andrampressure(thickline). ionization front itself, such as [SII] and [OI], are somewhat
reduced in intensity in the advective model due to the nar-
rowingoftheionizationfrontbyadvection(seediscussionin
AppendixA).
staticandadvectivemodelsisthesharppeakattheionization Unlike in the the low ionization parameter model of the
frontthatisseenintheemissivityoflinesfromsingly-ionized previoussection, in this model the velocity and Mach num-
speciessuchas[NII]. Thispeakisduetotheelectrondensity berdonotincreasemonotonicallyfromtheneutraltotheion-
peakseeninFigure3anddiscussedinAppendixA. izedside. Instead,themaximumMachnumber( =0.73),
m
M
The integrated emergentemission line spectrum (Table 2) which also corresponds to the maximum velocity, occurs at
isbarelyaffectedbytheadvection.Asaresultoftheincrease theHe0/He+ front. Althoughthisdoesnotcorrespondtothe
inemissionmeasurediscussedabove,althoughtheionization maximum temperature, it is a maximum in the sound speed
frontmovesappreciablyinward(Figure8d),theBalmerline due to the decrease in the mean mass per particle when He
fluxisactuallyslightlyhigherintheadvectivemodelthanin is ionized. There is a second, slightly lower, maximum in
the static model. On the other hand, lines that form in the thesoundspeed,Machnumber,andvelocityjustinsidetheH
10
1 −1)
H[NαIIλ]6λ5665384˚A˚A (kms 75 v
Relativelineintensity 0000....2468 [[[OSOIIII]I]Iλλ]6λ63750300107˚A˚A˚A mperature(K) 1Gasvelocity,soundspeed.515···11125000050344 a TTcee((astdavteicc)tive)
Te b
0
2·104
0−40 −30 −20 Do−pp10lershift(km0s−1) 10 20 30 −3(cm) nn((astdavteicc)tive)
Fig. 11.—Face-onemissionlineprofilesofmodelZH007includingther- Density 1·104
mal broadening. Hαλ6563 Å (solid lines), [NII]λ6584 Å (dashed lines), Gas c
[OI]λ6300Å(dottedlines),and[SII]λ6731Å(dot-dashedlines). Vertical 5·103
axisunitsarearbitrary. 1
x(H+)(advective)
ionizationfrontwhereradiationhardeningcausesatempera- Fraction 00.7.55 x(H+)(static)
turemaximum.Thevelocityalsostartstoincreaseagainvery Ion 0.25
d
closetotheilluminatedface. 0
Due to this complex velocity structure, the trends of 1
x(He+)(advective)
bmshluooedwseshl,isfaitmscwualinathtbeedioseenmeiznisafstriiooonmnlTpinaaerbalpmero2efitaelnreds.afrreomleFssigucrleea1r1,inwhtihcihs IonFraction 000..27.555 xxx(((HHHeee+++++)))(s((tasatdtavitcei)cc)tive)
e
0
4.3. HighIonizationParameter,Weak-R 0 5·1016 1·1017 1.5·1017 2·1017
This model, ZH050, is identical to ZH007 exceptthat the Depth,z(cm)
gasvelocityattheilluminatedfaceissetto50kms 1,produc- Fig. 12.—StructureofmodelZH050asafunctionofdepthfromtheillu-
− minatedface.AllpanelsasinFigure5.
ingaweak-Rfront.Themodelstructureasafunctionofdepth
isshowninFigure12andasafunctionofelectronfractionin
Figure 13. In both cases, the advective model is now com-
paredwith a constant-densitystatic modelas opposedto the beduetotheseauthorsincludingfewercoolingprocessesthan
constant-pressure model that was used for comparison with areincludedinCloudy(forfurtherdiscussion,seeWilliams&
the weak-D models. As can be seen from Figure 12a and c Dyson2001, andreferencestherein).Thegreaterwidthofthe
thevelocityanddensityareroughlyconstantacrossthefront, temperaturespike in the Rodriguez-Gaspar& Tenorio-Tagle
asisexpectedintheweak-Rcase. TheextremalMachnum- modelsisduetothefactthattheywereconsideringlowerden-
ber in the front (which for R-type fonts is a minimum, see sities,sothegasmovesfurtherfromtheionizationfrontinthe
AppendixA)is m =4.01,whichoccursontheionizedside timeittakestocooltotheequilibriumtemperature.
M
ofthefront(seeFig.13c)atx 0.93.
e
≃ 5. DISCUSSION
4.4. TemperatureStructureoftheIonizationFronts In this section, we look for evidence of advective effects
Figure12bshowsthatourweak-Rmodelhasapronounced in one of the closest and best-studied HII regions, the Orion
temperature spike at the ionization front, together with a nebula.Weconcentrateontheclearestsignaturesofadvection
smaller spike at the He ionization front. Figure 13b shows to emerge from our simulations: the electron density spike
that this is a more extreme manifestation of the overheating andtheionization-resolvedkinematics.
in the ionization front that was seen in the weak-D model
(Fig.9b).However,eveninthiscaseofarapidlypropagating 5.1. ElectronDensityStructureoftheOrionBar
frontwe do notfind the overheatingto be verygreat, reach- Onefirmpredictionoftheadvectivemodelthatdiffersfrom
ingamaximumtemperatureofonly13,000K. Otherauthors thestaticcaseistheexistenceofasharpelectrondensitypeak
havefoundmorepronouncedover-heatingeffectsindynamic at the position of the ionization front. This peak manifests
ionization fronts (Rodriguez-Gaspar & Tenorio-Tagle 1998;
itself most clearly in the emission of the [NII]λ6584 Å line
Marten&Szczerba1997)butfordifferentvaluesofthephys-
(see Figure 10), producing a narrow emissivity peak at the
ical parameters. Marten & Szczerba found temperatures as
edgeofthebroaderpeakthatcomesfromtheneutralHelium
highas20,000K behindR-typefrontsmovingatasubstan-
zone. InFigure14weshowan[NII]imageofthebrightbar
tial fraction of the speed of light, which would be difficult
regionintheOrionnebulathatshowsevidenceofjustsucha
to model using our steady-state code. Rodriguez-Gaspar & structure.2 Thebarisbelievedtobeasectionoftheprincipal
Tenorio-Taglestudiedthetime-dependentevolutionofanHII ionizationfrontin Orion that is seen almost edge-on. It can
regionaftertheturning-onofanOstar,findingtemperatures
of order 15,000 K behind the ionization front soon after its 2ThisimageisbasedondataobtainedwiththeWFPC2instrumentonthe
transition fromR to D-type. Their highertemperaturesmay HubbleSpaceTelescope,providedbyC.R.O’Dell.