Disruption of Molecular Clouds by Expansion of Dusty H II Regions

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by  Jeong-Gyu Kim

Accepted by ApJ
PreprinttypesetusingLATEXstyleemulateapjv.01/23/15
DISRUPTION OF MOLECULAR CLOUDS BY EXPANSION OF DUSTY HII REGIONS
Jeong-Gyu Kim1,2, Woong-Tae Kim1, & Eve C. Ostriker2
1CenterfortheExplorationoftheOriginoftheUniverse(CEOU),AstronomyProgram,DepartmentofPhysics&Astronomy,
SeoulNationalUniversity,Seoul08826, RepublicofKoreaand
2DepartmentofAstrophysicalSciences, PrincetonUniversity,Princeton,NJ08544, USA
Accepted by ApJ
6 ABSTRACT
1
DynamicalexpansionofHIIregionsaroundstarclustersplaysakeyroleindispersingthesurrounding
0
densegasandthereforeinlimitingtheefficiencyofstarformationinmolecularclouds. Weuseasemi-
2
analytic method and numerical simulations to explore expansion of spherical dusty H II regions and
n surrounding neutral shells and the resulting cloud disruption. Our model for shell expansion adopts
a the static solutions of Draine (2011) for dusty H II regions and considers the contact outward forces
J
on the shell due to radiation and thermal pressures as well as the inward gravity from the central
2 star and the shell itself. We show that the internal structure we adopt and the shell evolution from
1
the semi-analytic approach are in good agreement with the results of numerical simulations. Strong

radiationpressure in the interior controlsthe shell expansionindirectly by enhancing the density and
]
A pressure at the ionization front. We calculate the minimum star formation efficiency εmin required
for cloud disruption as a function of the cloud’s total mass and mean surface density. Within the
G
adopted spherical geometry, we find that typical giant molecular clouds in normal disk galaxies have
. ε . 10%, with comparable gas and radiation pressure effects on shell expansion. Massive cluster-
h min
formingclumpsrequireasignificantlyhigherefficiencyofε &50%fordisruption,producedmainly
p min
by radiation-driven expansion. The disruption time is typically of the order of a free-fall timescale,
-
o suggestingthatthe clouddisruptionoccurs rapidlyonce a sufficiently luminous HII regionis formed.
r We also discuss limitations of the spherical idealization.
t
s Keywords: galaxies: star clusters — galaxies: star formation — H II regions — ISM: clouds —
a
ISM:kinematics and dynamics — stars: formation
[

1
v 1. INTRODUCTION that the depletion time of molecular gas is more than
5 Giant molecular clouds (GMCs) are the sites of star an order of magnitude longer than the internal dynami-
3 formation in galaxies. They are highly structured, con- cal timescale (e.g., Krumholz & Tan 2007; Genzel et al.
0 sisting of hierarchy of clumps, filaments, and sheets re- 2010; Krumholz et al. 2012; Leroy et al. 2013).
3 Observations and theoretical arguments indicate that
sulting from shock interactions in supersonic turbulence
0 star formation efficiency tends to be higher within
(Elmegreen & Scalo 2004; Andr´e et al. 2014). Stars pre-
. high-density environments (Elmegreen & Efremov 1997;
1 dominantly form in groups (later becoming OB associ-
McKee & Ostriker 2007). For example, the estimated
0 ations or clusters) within dense, gravitationally-bound
star formation efficiencies of cluster-forming clumps for
6 clumpsinsideGMCs(Lada & Lada2003). Newbornstar
low-mass clusters in the solar neighborhood is ∼ 0.1–
1 clustershaveaprofoundinfluenceonthesurroundingin-
: terstellar medium (ISM) via protostellar outflows, stel- 0.3, higher than the few percent efficiency of entire
v GMCs (Lada & Lada 2003). The Orion Nebula Cluster,
lar winds, ionizing radiation, and supernova explosions,
Xi whicharecollectivelyreferredtoasstellarfeedback. The whichhasbeenformingstarsforseveraldynamicaltimes
(Tan et al.2006), appearstohavestellarfraction∼50%
r question of how eachfeedback process affects formation,
a evolution,anddispersaloftheir natalcloudsis anactive (Da Rio et al. 2014). Observations of massive, dense
cloudsindwarfstarburstgalaxiescontainingnascentsu-
and contentious area of research (see Dobbs et al. 2014
per starclusters indicate efficiencies >50%(Meier et al.
and Krumholz et al. 2014 for recent reviews).
2002;Turner et al.2015). Gasexpulsionfromprotoclus-
Animportantunsolvedprobleminstarformationthe-
tersisalsoimportantinthecontextofdisruptionorsur-
ory is what determines net star formation efficiency ε
vivalofastellarcluster(e.g.,Hills1980;Elmegreen1983;
of a cloud, defined as the fraction of the cloud’s mass
Goodwin 1997; Lada & Lada 2003; Banerjee & Kroupa
that is turned into stars over its lifetime. GMCs are
2015). If star formation efficiency is sufficiently high, a
knowntobeinefficientinconvertinggasintostars. Over
star cluster may remain gravitationally bound and thus
their lifetime, individual GMCs in the Milky Way ap-
become long-lived. If star formation efficiency varies
pear to turn only a few to several percent of their mass
with the mass of protocluster,it is likely to leave an im-
into stars (Myers et al. 1986; Williams & McKee 1997;
print on the shape of the cluster mass function distinct
Carpenter 2000; Evans et al. 2009; Kennicutt & Evans
fromthecloudmassfunction(e.g.,Ashman & Zepf2001;
2012; Garc´ıa et al. 2014). Furthermore, observations of
Kroupa & Boily 2002; Fall et al. 2010).
Galacticinfrareddarkclouds,nearbygalaxies,andhigh-
A number of theoretical studies have proposed that
redshift star-forming galaxies all point to a conclusion
H II regions may be the primary means of control-
ling the star formation efficiency within a cluster’s birth
[email protected],[email protected],[email protected]
2 Kim et al.
cloud (e.g., Whitworth 1979; Williams & McKee 1997; 2001; KM09; Fall et al. 2010; Murray et al. 2010). In
Matzner 2002; Krumholz et al. 2006; Goldbaum et al. particular, Fall et al. (2010) considered the potential for
2011;Dale et al.2012,2013). Anewbornclusterofstars disruptionofmolecularcloudsbyvariousfeedbackmech-
embedded in a cloud emits abundant ultraviolet (UV) anisms including protostellar outflows, photoionization,
photons, creating an ionization front that separates the and supernova explosions, and concluded that radiation
fully ionized gas close to the cluster from surrounding pressuremaydominatemomentuminjectionindenseand
neutral gas. Thermal balance between heating by pho- massiveprotoclusters. Utilizing the analytic solutions of
toionization and cooling by line emission keeps the tem- KM09 for shell expansion, they argued that the mini-
perature of the ionized gas roughly at ∼ 104 K (e.g., mum star formation efficiency, ε , required for cloud
min
Osterbrock 1989; Draine 2011). In the absence of radia- disruption primarily depends on the mean cloud sur-
tionpressure(seebelow)andconsideringconfinedrather face density. With ε independent of the mass, this
min
than blister-type HII regions, the density of ionized gas would explain the observed similarity between shapes of
isrelativelyuniform. Expansionoftheionizedgasdueto mass functions of molecular clouds and young star clus-
its high thermal pressure (initially ∼ 103 times ambient ters. Murray et al.(2010,hereafterMQT10)analytically
levels)drivesa shockwaveaheadof the ionizationfront. examined disruption of massive GMCs for sample sys-
An expanding shell of dense gas between the ionization tems representativeof various galactic environments,es-
and shock fronts is created, incorporating the ambient timatingeffectsofstellarfeedbackfromprotostellarjets,
neutral gas as the shock sweeps outward. Cloud disrup- shocked stellar winds, thermal pressure of photoionized
tionbyshellexpansionand/orassociatedphotoevapora- gas, radiation pressure, and the total gravity from stars
tion create hostile conditions for further star formation. and shell. They found that star formation efficiency for
Whitworth (1979) and Franco et al. (1994) found that massive GMCs in the Milky Way is only a few percent,
photoevaporation by massive stars born near the cloud while clouds instarburstgalaxiesandstar-forminggiant
boundary can limit ε to ∼ 5% in a typical molecular clumps in high-redshift galaxies require ∼20–40% of ef-
cloud. Williams & McKee (1997) and Matzner (2002) ficiency for disruption. In the part of parameter space
foundthatcloudsconvertonaverage∼10%oftheirmass they explored, direct and/or dust-reprocessed radiation
into stars before destruction by photoevaporation. Us- pressure dominates ionized-gas pressure in driving shell
ing the time-dependent virial theorem, Krumholz et al. expansion.
(2006) found that both mass ejection by photoevapora- While the previous analytic works mentioned above
tion and momentum injection by expanding HII regions are informative in understanding the effects of radiation
limit the net star formation efficiency of GMCs to ∼ 5– pressureontheshellexpansionandrelatedclouddisrup-
10% before disruption. tion, they are not without limitations. The minimum
The classical picture of an embedded H II region de- star formation efficiency derived by Fall et al. (2010)
scribedabovedoesnotaccountfortheeffectsofradiation does not allow for the effect of gas pressure and inward
pressure on dust grains, which are efficient at absorbing gravity that may be important in the late stage of the
UVphotons. Ifdustistightlycoupledtogasthroughmu- shell expansion. MQT10 applied their expansion model
tualcollisions,radiationforcesexertedonthe formerare to only a few representative cases, so that the general
readily transmitted to the latter. Not only does dust re- dependence of ε on the cloud mass and surface density
duce the size of the ionized zone (Petrosian et al. 1972), has yet to be explored. In addition, KM09 and MQT10
but it can also produce a central “hole” near the clus- set the radiation force on the shell equal to L/c, where
ter by the action of radiation pressure (Mathews 1967; L and c refer to the luminosity of the centralsource and
Arthur et al. 2004). Recently, Draine (2011, hereafter thespeedoflight,respectively,assumingthatalldustin-
Dr11)obtainedfamiliesofsimilaritysolutionsforthe in- side the HII regionis pushedoutto the shell. They also
ternal structure of dusty HII regions in static force bal- assumed the thermal pressure acting at the inner edge
ance. He found that radiation pressure is important for of the shell is equal to the mean thermal pressure in the
denseandluminousHIIregions,formingacentralcavity ionized region. Since the static solutions of Dr11 indi-
surroundedby anover-denseionizedshelljustinside the catethe ionizedgasinsideadustyHIIregionisstrongly
ionization front. Krumholz & Matzner (2009, hereafter stratified(athighluminosity)andabsorbsradiationfrom
KM09) showed that while expansion of H II regions ex- the central source, the assumption of the unattenuated
citedbyasmallnumberofmassivestarsiswelldescribed radiation up to the shell should be checked. All of the
bythegas-pressuredrivenclassicalmodel(Spitzer1978), above also adopt the assumption of spherical symmetry
the dynamics of H II regions around massive star clus- and a source luminosity that is constant in time.
ters is dominated by radiationpressure. They presented The numerical work of Dale et al. (2012, 2013) is not
an analytic formula for the shell evolution driven by the limitedbysymmetryidealizations,allowingforfullytur-
combination of gas and radiation pressures, finding that bulent gas dynamics and self-consistent collapse to cre-
radiation pressure is more important during the early ate sources of ionizing radiation. However, these works
phase of the expansion, while the late stage is governed do not include effects of radiation pressure, and con-
by a gas pressure force that increases with the shell ra- sider only a limited parameter space. Very recently,
dius. Raskutti et al. (2015) have conducted a set of numer-
SincegasexpulsionbythermallydrivenexpandingHII ical radiation hydrodynamic (RHD) simulations focus-
regions does not occur efficiently for clouds with high ing exclusively on the effects of (non-ionizing) UV in
escape velocities (& 10 kms−1) (e.g., Matzner 2002; turbulent clouds with surface densities in the range 10–
Krumholz et al. 2006; Dale et al. 2012, 2013), radiation 300 M⊙ pc−2. The Raskutti et al. simulations showed
pressurehasbeenconsideredthemostpromisingmecha- that turbulent compressions of gas can raise the value
nismfordisruptionofmassiveclouds(e.g.,Scoville et al. of ε by a factor of ∼5 for typical Milky Way GMC pa-
Cloud Disruption by Expanding HII Regions 3
rameters,becausestrongradiationforcesarerequiredfor internal structure of static, dusty H II regions, finding
dispersal of dense, shock-compressed filamentary struc- that radiationpressure acting on gas and dust gives rise
tures. Skinner & Ostriker (2015) used numerical RHD to a non-uniform density profile with density (and gas
simulationstoconsiderthe complementaryregimeofex- pressure) increasing outward. In this section, we revisit
tremely high surface density clouds, evaluating the abil- Dr11toobtaintheparametricdependenceofvariousHII
ityofradiationforcesfromdust-reprocessedinfrared(IR) region properties on the strength of ionizing radiation.
to disrupt clouds. This work showed that IR is effective This information will be used for our dynamical models
onlyifκ >15cm2 g−1,withκ beingthe gasopacity in Section 3.
IR IR
to dust-reprocessed IR radiation, and even in this case Let us consider a central point source with luminosity
the predictedefficiency is∼50%,whichmayexplainob- L=L +L embeddedina cloud,whereL andL refer
i n i n
servations of nascent super star clusters (Turner et al. totheluminositiesofhydrogenionizingandnon-ionizing
2015). photons, respectively. The number of ionizing photons
In this paper, we use a simple semi-analytic model per unit time emitted from the source is Q = L/hhνi,
i i i
as well as numerical simulations to investigate expan- where hhνi is the mean photon energy above the Ly-
i
sion of dusty H II regions and its effect on disruption of man limit. For simplicity, we ignore the effect of He and
star-forming clouds across a variety of length and mass assume that the ionized gas has a constant temperature
scales. Thisworkimprovesuponpreviousanalyticworks T = 104 K under photoionization equilibrium. Dust
i
in several ways. First, shells in our model expand due grains absorb both ionizing and non-ionizing radiation,
to both radiation and thermal pressures explicitly, ex- with a constant absorption cross-section per hydrogen
tending Fall et al. (2010), which considered solely radi- nucleus of σ = 10−21 cm2H−1 (Dr11). The outward
d
ation pressure. We also include the inward force due photon momentum absorbed by the dust is transferred
to gravity of the stars and shell, which extends the non- to the gas via thermal and Coulomb collisions, resulting
gravitatingmodelsofKM09. Second,weadoptthestatic inanon-uniformgasdensityprofilen(r). Letf denote
ion
solutions of Dr11 for non-uniform internal structure of the fraction of photons absorbed by the gas. Assuming
HII regions. This allows us not only to accurately eval- “Case B” recombination, the radius of the Stro¨mgren
uate the contactforces on the shell arising from thermal sphere centered at the source is then given by
and radiation pressures, but also to compare them with
the effective forces adopted in the previous studies (e.g., 3fionQi 1/3
r ≡ , (1)
KM09; MQT10). Third, we have also run direct nu- IF 4πα n2
(cid:18) B rms(cid:19)
merical simulation to check the validity of the solutions
of Dr11 for representing the interior structure in an ex- where α = 2.59×10−13 T/104 K −0.7 cm3 s−1 is the
B
pandingHIIregionandalsotoconfirmoursemi-analytic effectiverecombinationcoefficient(Osterbrock1989)and
shell expansion solutions. Fourth, we conduct a system- (cid:0) (cid:1)
atic parameter survey in the full space of cloud mass 3 rIF 1/2
n ≡ n2(r)r2dr (2)
and surface density, evaluating at each (Mcl, Σcl) the rms r3
minimum efficiency ε requiredfor disruption, the rel- (cid:18) IF Z0 (cid:19)
min
ative importance of radiation and ionized-gas pressures, is the root-mean-square number density within the ion-
and the timescale of cloud disruption. We also consider ized region.
theeffectsofthemass-dependentlight-to-massratio,the The luminosity L(r) at radius r inside the Stro¨mgren
density distribution of the backgroundmedium, and the sphereisgivenby L(r)=Lφ(r)+L e−τd,whereφ(r) is
i n
dust-reprocessedradiation. Ourmodeldoesnotallowfor the dimensionless quantity that describes attenuation of
r
potentially important effects of stellar winds, which we ionizing photons and τ = nσ dr is the dust optical
d 0 d
briefly discuss in Section 5.2. The chief idealization of depth of non-ionizing photons. The functions φ(r) and
our study is the adoption of spherical symmetry. While τ (r) ought to satisfy R
d
realcloudsarenotsymmetric,ourresultsprovideaguide
dφ
and baseline for future work that will relax this restric- =−nσ φ−4πα n2r2/Q , (3)
d B i
tion. dr
The rest of this paper is organized as follows. In Sec- dτ
d
tion 2, we briefly summarize the solutions of Dr11 for =nσd. (4)
dr
dusty H II regions in static equilibrium. In Section 3,
we describe our semi-analytic model for shell expansion, The gas/dust mixture (assumed to be collisionally cou-
and evaluate the contact forces on the shell in compar- pled) in the H II region is subject to both thermal
ison with the effective forces adopted by other authors. pressure Pthm(r) = 2n(r)kBTi and radiation pressure
We also run numerical simulations for expanding H II P (r) = L(r)/(4πr2c). The condition of static force
rad
regions, and compare the results with those of the semi- balance can thus be written as
analytic model. In Section 4, we calculate the minimum
1 d dn
efficiencyofstarformationrequiredforclouddisruption. Lφ+L e−τd +2k T =0. (5)
4πr2cdr i n B idr
Wealsopresentanalyticexpressionsforε intheradia-
min
tionorgaspressuredrivenlimits. Finally,wesummarize Equations (3)–(cid:0)(5) can be so(cid:1)lved simultaneously for
our results and discuss their astrophysical implications n(r), φ(r), andτ (r) subjectto the boundaryconditions
d
in Section 5. τ (0) = 0, φ(0) = 1, and φ(r ) = 0. Dr11 showed
d IF
that there exists a family of similarity solutions that are
2. INTERNALSTRUCTUREOFDUSTYHIIREGIONS completely specified by three parameters: β ≡ L /L,
n i
Dr11 studied the effect of radiation pressure on the γ ≡2ck Tσ /(α hhνi),andQn ,characterizingthe
B i d B i i rms
4 Kim et al.
Figure 2. Dependence on Qi,49nrms of the ratios of the edge-
to-rms densities nedge/nrms (red), the rms-to-mean densities
nrms/nmean (green),thedustopticaldepthtotheionizationfront
τd,IF (purple),andthefractionfion (blue)ofionizingphotons at-
tenuatedbyphotoionizationpriortoreachingtheshell. Theblack
dottedlineshowstherelationshipbetweenQi,49nrmsandtheratio
rIF/rch of the ionization front radius to the characteristic radius
(seeEquations (16)and(17)).
nated by gas pressure,except in the very centralregions
where P > P , resulting in an almost flat density
rad thm
distribution with n ≈ n . In this case, the dust op-
rms
tical depth up to r = r is less than unity, indicating
IF
that a large fraction of non-ionizing radiation survives
absorption by dust. For Q n & 104 cm−3, on the
i,49 rms
other hand, radiation pressure is crucial in controlling
the density structure. Gas close to the center is pushed
out to large radii, forming a central cavity and an outer
ionized shell, making the density at the ionization front
n ≡ n(r ) larger than n .1 Note that although
edge IF rms
P ≫ P in the cavity, the gas density there is too
rad thm
small to attenuate radiation significantly. Consequently,
Figure 1. Radial distributions of (a) the normalized density the radiation force exerted on the gas is almost negli-
n/nrms (proportional to gas pressure, solid lines) and the nor- gible in the central cavity and rises only in the outer
malized radiation pressure P˜rad = Prad/(2nrmskBTi) (dashed parts of the H II region, as Figure 1(b) illustrates. In
lines), and (b) the normalized radiation force per unit volume
fact,Equation(5)showsthatf achievesitsmaximum
f˜rad=frad/(2nrmskBTi/rIF)(equaltogaspressureforceperunit at the position of the steepest draednsity gradient (see also
volume in magnitude and opposite in direction) for Qi,49nrms =
1,102,104,106 cm−3. Lopez et al. 2011, 2014). Thus, radiation strongly mod-
ifies the interior of the H II regions such that both gas
(inverseof)importanceofionizingphotons,thedustcon- and radiation forces are applied primarily in the outer
tent, and the radiation pressure, respectively. Through- portion.
out this paper, we take β = 1.5, hhνi = 18eV, and Figure 2 plots the dependence on Q n of τ ≡
i i,49 rms d,IF
γ =11.1as standardvalues,appropriateforHII regions τd(rIF),fion,nedge/nrms,andnrms/nmean,wherenmean =
formed around massive star clusters (see Appendix A). 3 rIFn(r)r2dr/r3 is the mean density. Note that τ
0 IF d,IF
Figure 1 plots the radial profiles of the normal- and f saturate to finite values (τ = 1.97 and
ion d,IF
ized gas density and radiation pressure in the upper fR = 0.26) as Q n increases. This is consistent
ion i,49 rms
panel, and the radiation force per unit volume frad = with the results of Yeh & Matzner (2012) who showed
−r−2d(r2P )/dr in the lower panel, for Q n =
rad i,49 rms
1,102,104,106 cm−3, where Qi,49 ≡ Qi/(1049 s−1). For 1 Dust-deficient HIIregionswithγ<1donotpossessacavity
Q n . 104 cm−3, the internal structure is domi- (Dr11,Yeh&Matzner2012).
i,49 rms
Cloud Disruption by Expanding HII Regions 5
that Equations (3)–(5) yield born instantly at the cloud center. Copious energetic
photons emitted by the cluster begin to ionize the sur-
γ(β+1)e−γ−γ1τd,IF =β(γ−1)e−τd,IF +β+1, (6) rounding medium, causing the ionization front to ad-
vance to the initial Stro¨mgrenradius, r , within a few
IF,0
fion = β+1 τd,IF− γ 1−e−γ−γ1τd,IF , (7) recombination timescales (∼ 103 yr for n ∼ 102 cm−3).
γ−1 γ−1 Theoverpressured,ionizedgascreatesashockfrontthat
(cid:20) (cid:16) (cid:17)(cid:21) moves radially outward, sweeping up the ambient neu-
nedge = 1+β(1−e−τd,IF)n r σ ∝(n Q)1/3, tral medium into a dense shell. We assume that the
nrms 3fionγ rms IF d rms i H II region remains in internal quasi-static equilibrium
(8) throughoutitsdynamicalexpansion,whichisreasonable
in the limit of Qinrms →∞. The saturation of τd,IF and since the sound-crossing time over the ionized region is
fion occurs because the radiation-induced ionized shell sufficiently small compared to the expansion timescale.
near r = rIF has such large density that neutral hy- We further assume that all of the swept-up gas resides
drogen produced by recombinations can compete with in a thin shell of mass M located at r . Since we are
sh sh
dust in absorbing ionizing photons. This is in contrast interested in the evolution of the shell well after the for-
totheclassicalcaseofuniform-densityHIIregionswith- mationphase,wemay taker ≪r indescribingshell
c IF,0
out radiation pressure, for which τd,IF → ∞, fion → 0 expansion in a power-law envelope.
as Qinrms → ∞ (Petrosianet al. 1972). Since the ion- The momentum equation for the shell is
ized shell has a saturated column rIFndr = τ /σ ,
the gas mass Mion within a dusty 0H II regiondi,InFthdis d(Mshvsh) =Ftot =Fout−Fin, (10)
limit depends on r as M ∝ r2 R, as opposed to clas- dt
IF ion IF
sical H II regions for which nmean ≈ nrms and Mion ∝ wherevsh =drsh/dtistheexpansionvelocityoftheshell,
nmeanrI3F ∝rI3F/2 for given Qi. and Fout and Fin denote the outward and inward forces
actingonthe shell, respectively. The inwardforceis due
3. EXPANSIONOFDUSTYHIIREGIONS to the cluster gravity and shell self-gravity,given by
The high pressure in the interior of an H II region GMsh(M∗+Msh/2)
compared to the ambient levels leads to radial expan- Fin = r2 , (11)
sion. Spitzer (1978) provides approximate solutions for sh
this expansion in the case where just gas pressure is in- so that the cluster gravity and gaseous self-gravity de-
cluded, under the assumption that ambient gas is swept pend on r as r1−kρ and r4−2kρ, respectively. The radi-
sh sh sh
up into a thin shell surrounding the ionized interior (see ationandthermalpressuresgiverisetothetotaloutward
also Franco et al. 1990; Shu 1992). In this section, we force
explore expansion of spherically-symmetric, dusty H II L
F = ne−τd,IF +8πk Tn r2 . (12)
regions driven by both thermal and radiation pressures, out c B i edge sh
making a thin-shell approximation. Here we ignore
Note that F is the contact force acting on the sur-
pressure resulting from diffuse recombination radiation, out
faceimmediately outsidethe ionizationfront. Onlynon-
which is negligible in comparison to gas pressure (e.g.,
ionizing photons reach the neutral shell and exert the
Henney & Arthur 1998; KM09; Dr11). We also assume
outward force, if the shell is optically thick to them.3
that the shell is optically thin to IR photons emitted
In Equation (10), we neglect the reaction force on the
by dust grains, the effect of which will be examined in
shell exerted by evaporation flows away from the ion-
Section 4.3. Expansion models by KM09 and MQT10
ization front because it is not significant for embedded
adopted the assumption that all of the radiation is ab-
H II regions. The thrust term would be important in
sorbed only by the shell, while ionizing radiation pro-
driving expansion of classical, blister-type H II regions
ducesauniformpressure,uniformdensityinterior. Here,
for which ionization fronts are kept D-critical (Matzner
we relax these assumptions and calculate the direct out-
2002; Krumholz et al. 2006; KM09). We also ignore the
ward forces based on the Dr11 solutions.
slowdownoftheshellcausedbyturbulentrampressureof
theexternal,neutralmedium(e.g., Tremblin et al.2014;
3.1. Model
Geen et al. 2015), which is difficult to model within our
Weconsiderasphericalneutralcloudwithdensitydis- spherically-symmetric,one-dimensional model.
tribution Figure 3 plots F as well as the respective contribu-
out
tionsofradiativeandthermalpressuresagainstthe shell
n , for r/r <1,
n(r)= c c (9) radiusr ,takentoequalr ,astheblack,blue,andred
(cid:26)nc(r/rc)−kρ, for r/rc ≥1, solidlinsehs,respectively,utilIiFzing the Dr11solutionswith
β = 1.5 and γ = 11.1. Apparently, the thermal term
where n and r are the density and radius of a flat
c c
dominates the radiative term by non-ionizing photons
core, respectively, and k is an index of a surrounding
ρ overthe whole rangeshowninFigure3. This shouldnot
power-law envelope.2 A star cluster with total mass
be interpreted as an indication that radiation is unim-
M∗, total luminosity L, and total ionizing power Qi is portantfor the shell expansion. Rather, the effect of the
2 While GMCs do not appear to possess density stratifica- 3 The condition that the dusty shell should be opaque to
tion on a global scale (McKee&Ostriker 2007), the density dis-
tribution of clumps within them is well described by a power- UV photons can be expressed as rsh . (κUVMsh/4π)1/2 =
law profile including a central core (see e.g., Francoetal. 1990; 8.4 pc(Msh/104 M⊙)1/2, where κUV = σd/µH = 4.3 ×
Hillenbrand&Hartmann1998). 102cm2g−1 (Thompsonetal.2015).
6 Kim et al.
sufficientlysmallradius,thetotaloutwardforceisinfact
log r /r
10 IF ch equal to F .4
-1 0 1 2 rad,eff
Since F ∝ n r2 ∝ f1/2r1/2, while F re-
Dr11 thm,eff rms sh ion sh rad,eff
mains constant, one can write
KM09
10.0 L f r 1/2
ion sh
F = 1+ , (15)
out,eff
c f r
" (cid:18) ion,ch ch(cid:19) #
with the characteristic radius r at which F =
ch rad,eff
F defined by (KM09)
) therm,eff
c
/
L L/c 2
F/( 1.0 rch ≡ (1+β)σd Qi
γ 12πα f (16)
8πk Tr2n (cid:18) (cid:19) B ion,ch
B i IF edge →5.3×10−2 pcQ ,
i,49
8πkBTirI2Fnrms where fion,ch ≡ fion(rch) is 0.32 for our fiducial parame-
Lne−τd,IF/c tersβ =1.5andγ =11.1.5 Expansionisdrivenpredom-
0.1 inantlybyradiationintheregimewithrsh/rch <1,while
ionized-gas pressure is more important for r /r >
sh ch
1. Thus, the relative importance of radiation pressure
3 2 1 0 1 within an HII region can be assessed by the ratio
log r /Q [pc]
10 IF i,49
r γ(36πα )1/3 2 f 1/3
IF B ion
=f .
Figure 3. Outward forces on the shell surrounding the H II re- r ion,ch (1+β)σ Q2n2
gion, shown as a function of the shell radius rsh = rIF, scaled ch (cid:20) d (cid:21) (cid:18) i rms(cid:19) (17)
as rIF/Qi,49 (lower x-axis) or rIF/rch (upper x-axis). The solid f 1/3 Q n −2/3
linesdenotetheforcesbasedonthequasi-staticequilibriummodel → ion i,49 rms ,
of Dr11, while the dotted lines are the effective forces assuming 0.32 2.55×104 cm−3
noattenuation of radiation insidethe shell (e.g., KM09). The to- (cid:18) (cid:19) (cid:18) (cid:19)
tal forces and the radiative and thermal contributions are drawn shownas a black dotted line in Figure 2 as a function of
as black, blue, and red curves, respectively. An H II region with Q n . Thus, shell expansion is dominated by radia-
gfoivrceenpQuisehveoslvtehsefrgoams wleiftthtionrtihgehtH. WIIhernegrioIFn≪outrwcha,rdth,eerffaedcitaivtieolny tioi,n49prremsssure when Qi,49nrms &104 cm−3.
communicating most of the radiation force L/c to the surround-
ingshellthrough a dense, ionizedshell (nedge/nrms ≫1). As the 3.2. Non-gravitating Similarity solutions
H II region expands to rsh > rch, direct radiation forces become
unimportantandnedge≈nrms. Intheabsenceoftheinwardgravitationalforces(Fin =
0)andinthelimitofnegligiblecoreradius,itisstraight-
radiation force is indirectly communicated to the shell forward to show that Equations (10) and (15) yield the
by increasing the ionized gas density n above n , similarity solutions
edge rms
with the limiting ratio n /n given by the factor in
edge rms 4−k L 3
EqTuhaetiobneh(8a)viworheonf Qthinermoust→wa∞rd. forces for varying rIF nmeanrs4h = 2 ρ c 4πµHt2, (18)
can be readily understood in comparison with the
in the limit of r /r ≪1, and
usual approximation adopted by previous authors (e.g., sh ch
Harper-Clark & Murray 2009; KM09; MQT10). Assum- 3 k T 3f Q 1/2 (7−2k )2
ingthat allthe photons fromthe sourceareabsorbedby n r7/2 = B i ion i ρ t2, (19)
the shell, that the H II region has uniform interior den- mean sh 2 µH (cid:18) 4παB (cid:19) 9−2kρ
sity n , and that infrared photons re-radiated by dust
rms in the limit of r /r ≫ 1, where f is regarded as a
sh ch ion
freely escape the system, the effective outward forces on constant and µ = 2.34×10−24 g is the mean atomic
tahseshellfromdirectradiationandgaspressurearetaken mass per hydroHgen and nmean = 3nc(rsh/rc)−kρ/(3 −
k ) is the mean number density interior to r (e.g.,
F =L/c, (13) ρ sh
rad,eff Krumholz et al. 2006; KM09). Therefore, the shell ra-
and dius and velocity vary with time as rsh ∝ t2/(4−kρ)
Fthm,eff =8πkBTinrmsrs2h, (14) and vsh ∝t(kρ−2)/(4−kρ) in the radiation-pressure driven
respectively, where n is given in terms of Q, f , limit, and rsh ∝ t4/(7−2kρ) and vsh ∝ t(2kρ−3)/(7−2kρ) in
rms i ion the gas-pressure driven limit. KM09 presented an an-
and r = r using Equation (1). These together with
sh IF alytic approximation valid in both limits by combining
the total effective force F = F +F are
out,eff rad,eff thm,eff Equations (18) and (19). Note that for k = 3/2, the
plotted as dotted lines in Figure 3. Evidently, F ρ
out,eff velocity approaches a constant at sufficiently late time.
agrees well (within 20%) with F from Equation (12).
out
Comparison of the thermal term in Equation (12) with
4 Combining Equations (1), (8), and (12), one can show that
Equation (14) shows that the enhanced thermal force Fout→L/cinthelimitofQi,49nrms→∞.
for small radius is due to increased nedge/nrms, which is 5KM09adoptedaconstantvalueof0.73forbothfionandfion,ch
caused by radiation pressure in the Dr11 solutions. At fortheirapproximatetreatment.
Cloud Disruption by Expanding HII Regions 7
3.3. Valid Range of k solutions discussed in Section 3.2 (Equations (18) and
ρ
(19)), although strong stellar gravity in the case with
When a cloud has too steep a density gradient, the
shell mass becomes smaller than the ionized gas mass Qi,49 = 103 makes the shell expansion deviate from the
within the ionization front. In this case, the thin shell non-gravitating solutions early time. Self-gravity of the
approximation we adopt would no longer be valid. For swept-upshellbecomesimportantinthelatestage,even-
instance,Franco et al.(1990)foundthatforathermally- tually halting the expansion. The maximum shell veloc-
driven H II region formed in a cloud with k = 3/2, ity is only mildly supersonic with respect to the ionized
ρ
the shock front moves twice as fast as than the ionized gas owing to the rapid increase in the shell mass. Note
sound speed, without significant mass accumulation in that when Qi,49 = 10, the driving force changes from
the shocked shell. For 3/2 < kρ < 3, an H II region radiation to gas pressure at small rsh, while shell ex-
becomes “density bounded” and develops “champagne” pansionis alwaysdominated by radiationpressure when
flows rather than forming a shell (see also self-similar Qi,49 =103.
solutions by Shu et al. 2002). To be consistent with our
3.5. Comparison with Numerical Simulations
thin-shell approximation, therefore, we consider clouds
only with k < 1.5 in the following analysis for shell So far we have used a very simple model to study dy-
ρ
expansion. namical expansion of a spherical shell surrounding an
H II region. In order to check how reliable our results
3.4. Shell Expansion
are,werundirectnumericalsimulationsusingtheAthena
When stellar gravity is included, we can relate M∗ code in spherical geometry (Stone et al. 2008), as de-
to Qi through M∗ = Qi/Ξ, where Ξ is the conversion scribed in Appendix B. As an initial state, we consider
factor representing the ionizing photon output per unit a sourceof radiationatthe center ofa radiallystratified
stellar mass. The corresponding light-to-mass ratio is cloud with k = 1. To handle the transfer of radia-
ρ
Ψ=L/M∗ =(1+β)hhνiiΞ. Obviously, these quantities tion from the source, we adopt a ray-tracing technique
depend on M∗ and vary from cluster to cluster, espe- explained in Mellema et al. (2006) and Krumholz et al.
cially for low-mass ones due to stochastic fluctuations (2007). While gas inside the H II region is evolved self-
in the stellar population. In Appendix A, we perform consistently,weensurethattheouterenvelopeunaffected
Monte-Carlo simulations for the spectra of coeval popu- byradiationmaintainsitsinitialhydrostaticequilibrium.
lations using the SLUG code (Krumholz et al. 2015) to Figure 5 plots as solid lines the radial density distri-
find spectral properties as functions of M∗. Equations butions at a few epochs for Model A (with Qi,49 = 1
(A1) and (A2) provide the fits to the resulting median and n(r )=103 cm−3; top), Model B (with Q = 102
0 i,49
values of Ψ and Ξ. While Ψ and Ξ saturate to constant and n(r ) = 104 cm−3; middle), and Model C (with
values for M∗ & 104 M⊙, they decrease sharply as M∗ Q =0103 and n(r ) = 104 cm−3; bottom), all with
decreases below ∼ 103 M⊙, due to a rapid decrease in ki,4=9 1 and r = 10 pc. In Model A, the effect of
the number of O-type stars in the realizations of low- ρ 0
radiation on shell expansion is almost negligible since
mass clusters. In this work, we use Equation (A1) for Q n < 104 cm−3. Its overall expansion is in good
conversion between M∗ and Qi, while fixing to β = 1.5 agir,4e9emrmenst with the classical picture of thermally driven
and hhνi = 18eV: we have checked that varying β and
i expansion(e.g., Spitzer 1978). As soonas the ionization
hhνi does not affect our results much.
i front reaches the initial Stro¨mgren radius (∼0.3pc), an
To integrate Equation (10), we first choose a set of
isothermal shock wave forms and propagates outward.
values for β, γ, Q, n , and k . We then spec-
i rms,0 ρ Atthesametime,rarefactionwavesexcitedattheioniza-
ify f from the Dr11 solutions and r from Equa-
ion IF,0 tion front propagate radially inward (e.g., Arthur et al.
tion (1). At t = 0, a shell with a zero velocity is
2011), gradually turning into acoustic disturbances that
located at r (0) = r . The outside density pro-
sh IF,0 travel back and forth between the ionization front and
file is taken as n(r) = nrms,0(r/rIF,0)−kρ for r ≥ rsh the center. In this model, the density profile in the ion-
and the shell mass at any radius is given by Msh = ized region is nearly flat.
4πµ n r3−kρrkρ /(3−k ). Ontheotherhand,ModelCisdominatedbyradiation.
H rms,0 sh IF,0 ρ
As illustrative examples, we fix β =1.5, γ =11.1,and Gas at small r is pushed out supersonically to create a
n = 104 cm−3, and vary k and Q. Figure 4 plots central cavity together with an ionized shell within t ∼
rms,0 ρ i
as solid lines the temporal behavior of the shell radius 103 yr.6 After the shell formation, the gas in the ionized
(upper panels)andvelocity(lowerpanels)forQ =10 regionrelaxesintoaquasi-staticequilibriumstatewithin
i,49
(left) and Q = 103 (right). The solutions without asoundcrossingtimeacrosstheHIIregion. InModelB,
i,49
the shell expands mostly due to radiation pressure until
gravity are compared as dotted lines. The cases with
it reaches r = 6.3 pc at t = 0.82 Myr, after which the
k = 0, 1, and 1.4, plotted in red, black, and green, sh
ρ
driving force switches to ionized-gas pressure.
respectively, show that the shell expansion is faster in
Figure 5 alsoplots as reddashedlines the correspond-
an environment with a steeper density profile, because
ingDr11quasi-staticsolutionsinsidetheionizationfront,
the shell mass increases more slowly for larger k . The
ρ
whichareingoodagreementwiththeresultsofthetime-
horizontaldashedlines markthe the characteristicradii,
r = 0.53 and 53 pc for Q = 10 and 103, respec-
ch i,49 6 Assuming that the dust absorption dominates over the pho-
tively, inside (outside) of which shell expansionis driven
toionization, the acceleration on dusty gas at r is given by
primarily by radiation (gas) pressure. The star symbols L(r)σd/(4πr2cµH). Then, the timescale for a gas parcel to
marktheradiuswhereF =0,beyondwhichexpansion travel over a distance d in the ionized region is roughly t ∼
tot
is slowed down by the shell self-gravity significantly. 103(r2d/rI3F,0)1/2trec, wheretrec =(αBn)−1 isthe recombination
The solutions converge to the asymptotic power-law timescale(Arthuretal.2004).
8 Kim et al.
Figure 4. Temporalevolutionoftheshellradius(upper)andtheshellvelocity(lower)forQi=1050s−1(left)andQi=1052 s−1(right).
The initial rms density in the ionized region is fixed to nrms,0 =104 cm−3. The cases with kρ =0, 1, and 1.4 are plotted in red, black,
and green. The solid and dotted lines correspond to the models with and without gravity, respectively. The thin horizontal lines in the
upperpanelsmarkthecharacteristicradiusrch(Equation(16)),atwhichthedominantdrivingforceswitchesfromradiationtoionized-gas
pressure. ThestarsymbolsindicatetheradiiwhereFtot=0.
dependent simulations. This not only confirms that the 2011; MQT10). If it is luminous enough, the cluster is
staticsolutionsofDr11areapplicableevenforexpanding able to disrupt the entire parent cloud. For a low-mass
HIIregions,butalsovalidatesourray-tracingtreatment cluster, onthe other hand, only a small volume near the
of radiation in the numerical simulations. center is affected by the shell expansion, with the major
Figure6plotsthetemporalchanges(solidlines)ofthe portion of the cloud remaining intact. Therefore, there
shellradiusfromthe simulationsin comparisonwith the existsaminimumstarformationefficiencyε suchthat
min
solutions(dashedlines)ofEquation(10). Thecaseswith a cloud with ε < ε is not disrupted and will un-
min
and without gravity are plotted in blue and red, respec- dergofurtherstarformation,whileacloudwithε>ε
min
tively. Smalldifferencesatearlytimebetweentheresults willbe disruptedcompletely,withnofurther starforma-
from the different approaches are caused mainly by the tion. In real turbulent clouds, both star formation and
assumption of a vanishing shell velocity at t=0 in solv- gas mass loss occur continuously (e.g., Dale et al. 2012,
ing Equation (10), while the shell in simulations has a Skinner & Ostriker 2015, Raskutti et al. 2015), and one
non-zerovelocitywhenitfirstforms. Nevertheless,semi- can expect ε increases with time until ε≈ε at which
min
analytic and numerical solution agree with each other point feedback is able to destroy or evacuate all the re-
within ∼ 5% after 0.05 Myr, suggesting that our semi- maining gas (see Section 5.2 for more discussion).
analytic model of shell expansion is quite reliable. Al- Inthissection,weusethesimplemodelofshellexpan-
thoughtheinclusionofgravitydoesnotmakesignificant sion described in Section 3 to evaluate ε . Fall et al.
min
changesinthe internalstructureof the ionizedregion,it (2010) studied how gas removal due to various feedback
is self-gravity that makes the swept-up shell decelerate processes regulates the efficiency of star formation, and
and eventually stall at r = 51 pc when t = 23 Myr estimated ε for massive, compact protoclusters by
sh min
in Model A. Gravity becomes more significant for larger considering only the radiation pressure. Our model ex-
Qn , stopping the shell expansion at t ≈ 3–4 Myr in tends their work by including both ionized gas pressure
i rms
Models B and C, despite correspondingly stronger out- and gravity. MQT10 integrated equations for shell ex-
ward forces. pansion due to radiation pressure, gas pressure, stellar
4. MINIMUMEFFICIENCYOFSTARFORMATIONFOR winds, and protostellar outflows, etc., as applied to sev-
CLOUDDISRUPTION eral specific cases of massive GMCs. Our aim here is to
find the systematic dependence of ε on cloud param-
Consider a star cluster comprising a mass fraction ε min
eters such as mass, surface density, and density profile,
of its birth cloud, and located at the cloud center. The
etc.,whichwillbeusefultoassesstheeffectivenessofHII
H II region produced by the luminous cluster launches
region feedback in a wider range of physical conditions.
an expanding shell that sweeps up the bulk of the gas
in the cloud as it grows. (see also, e.g., Harper-Clark
Cloud Disruption by Expanding HII Regions 9
Figure 5. Evolution of the radial density distributions in nu- Figure 6. Temporal evolution of the shell radius in the simu-
mericalsimulationsofexpanding HIIregionsinapower-lawden- lations (solid lines) compared to the solutions of Equation (10)
sity background with kρ = 1 and r0 = 1 pc for (a) Model A (dashed lines). Thecases withandwithout gravity areplotted in
with Qi = 1049 s−1 and n(r0) = 103 cm−3, (b) Model B with blue and red, respectively. Other than small differences at early
QQii ==11005521s−s−11anadndn(nr(0r)0)==10140c4mc−m3−.3T,haendred(cd)oMtteoddellinCeswpiltoht ttihmosee,tohfethseemsiim-aunlaaltyitoincsr.esultsareoverall ingood agreementwith
the corresponding static equilibriumsolutions of Dr11, describing virial parameter of order unity (e.g., Bertoldi & McKee
theinterioroftheHIIregion.
1992). At t = 0, the cloud forms a star cluster with
4.1. Fiducial Case mass M∗ = εMcl instantaneously at its center. The re-
maining gas mass (1−ε)M is distributed according to
cl
Let us consider an isolated spherical cloud with mass Equation (9) with r = 0.1R within the cloud radius
c cl
Mcl, radius Rcl, and mean surface density Σcl ≡ Rcl; we have checked that the choice of the core radius
Mcl/(πRc2l). We assume the cloud is gravitationally has little influence on the resulting εmin as long as it is
bound with the one-dimensional turbulent velocity dis- sufficiently small. The cluster emits ionizing photons at
persion σ =(αvirGMcl/5Rcl)1/2, where αvir is the usual a rate Qi =ΞM∗, producing a shell atrsh =rIF,0, which
10 Kim et al.
we determine by substituting n (< r ) for n in
rms IF,0 rms
Equation (1).
To determine ε , we first take a trial value of 0 <
min
ε < 1 and integrate Equation (10) over time until the
shellreachesthe cloudboundary. Atr =R ,wecheck
sh cl
iftheshellmeetsoneofthe followingfourdisruptioncri-
teria: (1)v (R )=v ≡[(1+ε)GM /R ]1/2, corre-
sh cl bind cl cl
spondingtoavanishingtotal(kineticplusgravitational)
energy of the shell (cf. Matzner 2002; Krumholz et al.
2006)7; (2) v (R ) = σ, corresponding to a stalling of
sh cl
theshellexpansionbyturbulentpressureinsidethecloud
(e.g.,Matzner2002;KM09;Fall et al.2010);(3)v (R )
sh cl
equal to a large-scale turbulent ISM velocity dispersion,
v ∼ 7 kms−1 (Heiles & Troland 2003), correspond-
turb
ing to merging of the shell with the diffuse ISM; and
(4) F (R ) = 0, corresponding to a balance between
tot cl
gravityandoutwardforces(e.g.,Ostriker & Shetty2011;
MQT10). If none of these conditions are satisfied or if
r isunabletoexpandtoR ,wereturntothefirststep
sh cl
and repeat the calculations by changing ε.
Wetakeβ =1.5,γ =11.1,α =1,andk =1forour
vir ρ
fiducialparameters. Figure7plots assolidlines ε (a)
min
asafunctionofΣclforcloudswithMcl =105 M⊙and(b)
as a function of Mcl for clouds with Σcl =103 M⊙ pc−2,
for the four disruption criteria given above. As ex-
pected,cloudswithsmallersurfacedensityaremoreeas-
ily destroyed by feedback, while more compact clouds
need to have ε increasingly closer to unity for dis-
min
ruption. The minimum efficiency depends more sensi-
tively on Σ than M . In the case of the condition
cl cl
v (R ) = v , for instance, ε rises rapidly from
sh cl bind min
0.02 to 0.66 as Σcl increases from 102 to 103 M⊙ pc−2
for fixed Mcl = 105 M⊙, whereas it changes less than a
factor of two over 102 M⊙ ≤ Mcl ≤ 106 M⊙ for fixed
Σcl = 103 M⊙ pc−2. Furthermore, a criterion based on
a higher shell velocity at the cloud boundary results in
higher ε .
min
Figure 8 plots the contours of ε on the M –Σ
min cl cl
plane for the various criteria. The solid curves draw
theresultsforthevariablelight-to-massratio(seeEqua-
tions (A1) and (A2)), while the case of constant Ψ =
943L⊙ M⊙−1 andΞ=5.05×1046s−1 M⊙−1 areshownas
dottedcurves. The differencebetweenthe solidanddot-
ted contours is caused by an incomplete sampling of the
initialmass function (IMF) which lowersΨ for low-mass
clusters and thus yields higher εmin for M∗ . 103 M⊙.
For the criterion v (R ) = v , typical GMCs with
sh cl bind
104 M⊙ .Mcl .106 M⊙ andΣcl ∼102 M⊙ pc−2 in the Fruipgtuiorneb7y.HMIIinriemgiuomnesxtapranfosiromnaatsioanfeuffincctiieonncyofε(ma)inthfoertocltoauldmdeaisn-
Milky Way could be destroyed by a handful of O stars
(Qi,49 . 10) with efficiency of . 10%. On the other sfourrffiaxceeddΣencslit=y1fo0r3 fiMx⊙edpMc−c2l,=ba1s0ed5 oMn⊙t,heanfodu(rbd)iffthereentottcarlitmeraisas.
hand, denser and more compact cluster-forming clouds DottedlinesshowtheanalyticapproximationusingEquation(24).
or clumps with Σcl & 103 M⊙ pc−2 require the star for-
mation efficiency larger than ∼50% for disruption.
Inourmodel,the gasdensityisproportionalto(1−ε)
and the ionizing rate Q is proportional to ε, so that
i
the size of the H II region becomes increasingly larger
for higher ε. It is thus possible to have r ≥ R ,
IF,0 cl
indicating that the whole cloud is completely photoion- izedfromthe beginning. Theupper rightpartabovethe
thickbluelineinFigure8(a)correspondstotheseclouds
7Asadisruptioncriterion,Matzner(2002)andKrumholzetal. destroyedby photoionizationwithout involving shell ex-
pansion. This can be seen more quantitatively by con-
(2006) took vsh(Rcl) = vesc = p2GMcl/Rcl, which is the escape
speedofanunboundtestparticle,ratherthantheminimumspeed sidering the critical efficiency εS that makes the initial
forunbindingathinshellatr=Rcl. Stro¨mgren radius equal to Rcl, for which Equation (1)