Evolution of Prolate Molecular Clouds at Hii Boundaries: II. Formation of BRCs of asymmetrical morphology

6.3 MB·English

by  T. M. Kinnear

Mon.Not.R.Astron.Soc.000,1–??(20??) Printed20January2015 (MNLATEXstylefilev2.2)
ii
Evolution of Prolate Molecular Clouds at H Boundaries:
II. Formation of BRCs of asymmetrical morphology
T. M. Kinnear1?, J. Miao1, G. J. White2,3, K. Sugitani4 and S. Goodwin5
5 1 Centre for Astrophysics and Planetary Science, School of Physical Sciences, University of Kent, Canterbury, CT2 7NH, England
1 2 Department of Physics and Astronomy, The Open University, Milton Keynes MK7 6AA , England
0 3 Space Science and Technology Department,CCLRC Rutherford Appleton Laboratory, Oxfordshire OX11 0QX, England
2 4 Graduate School of Natural Sciences, Nagoya City University, Mizuho-ku, Nagoya 467-8501, Japan
5 Department of Physics and Astronomy, The University of Sheffield, Western Bank, Sheffield S10 2TN
n
a
J
7 Accepted20??Month??.Received20??Month??;inoriginalform20??Month??
1

] ABSTRACT
R A systematic investigation on the evolution of a prolate cloud at an Hii boundary is
S conductedusingSmoothedParticleHydrodynamics(SPH)inordertounderstandthe
. mechanismforavarietyofirregularmorphologicalstructuresfoundattheboundaries
h
of various Hii regions. The prolate molecular clouds in this investigation are set with
p
- their semi-major axes at inclinations between 0 and 90◦ to a plane parallel ionizing
o radiationflux.Asetof4parameters,thenumberdensityn,theratioofmajortominor
r axisγ,theinclinationangleϕandtheincidentfluxF ,areusedtodefinetheinitial
t EUV
s stateofthesimulatedclouds.Thedependenceoftheevolutionofaprolatecloudunder
a
RadiationDrivenImplosion(RDI)oneachofthefourparametersisinvestigated.Itis
[
foundthat:i)inadditiontothewellstudiedstandardtypeA,BorCBrightRimmed

1 Clouds(BRCs),manyothertypessuchasasymmetricalBRCs,filamentarystructures
v and irregular horse-head structures could also be developed at Hii boundaries with
6
only simple initial conditions; ii) the final morphological structures are very sensitive
9
to the 4 initial parameters, especially to the initial density and the inclination; iii)
1
The previously defined ionizing radiation penetration depth can still be used as a
4
good indicator of the final morphology.
0
Basedonthesimulationresults,theefficiencyoftheRDItriggeredstarformation
.
1 fromcloudsofdifferentinitialconditionsisalsoestimated.Finallyaunifiedmechanism
0 for the various morphological structures found in many different Hii boundaries is
5
suggested.
1
v: Key words: Hii regions - hydrodynamics - stars: formation - ISM: evolution - ISM:
i kinematics and dynamics - radiative transfer
X
r
a
1 INTRODUCTION (RDI) mechanism for EUV radiation triggered star forma-
tion in BRCs.
As the intensive EUV radiation from newly formed star(s)
ionizesthestar-facingsurfacelayerofamolecularcloud,the Multi-wavelength observations have revealed various
ionization heating induces a shock which propagates into physical and morphological properties of BRCs at Hii re-
the cloud, compressing material to form highly condensed gions. BRCs observed so far tend to be classified as type
core(s). These cores are the potential sites for EUV radia- A, B and C in increasing order of the curvature of their
tion triggered new star formation. Simultaneously the Hα bright rim (Sugitani et al. 1991; Sugitani & Ogura 1994),
emission produced by the recombination of ions with elec- though some of BRCs are found in M shaped morphology
tronscreatesabrightrimaroundthestarfacingsideofthe (Osterbrock 1957; Urquhart et al. 2006; Karr et al. 2005).
cloud. The structures formed are termed Bright Rimmed Starting with a uniform and spherical molecular cloud
Clouds (BRCs), the best candidate for studying the feed- illuminated by a plane-parallel ionizing radiation from one
back of massive stars to surrounding and parental molecu- side,theoreticalmodellingbasedontheRDImechanismsuc-
larcloud(s).ThisprocessistheRadiationDrivenImplosion cessfullyrevealedtheformationprocessofstandardtypeA,
B or C BRCs. These simulated BRCs are all symmetric to
their structure axis, and the latter aligns with the radia-
? E-mail:[email protected] tionfluxdirection,i.e.,thelineconnectingthetipofaBRC
(cid:13)c 20??RAS
2 T. M. Kinnear, J. Miao, G. J. White, K. Sugitani and S. Goodwin
Figure 1. The distribution of the inclination angle ϕ, which is
defined as the angle between the major axis and the x axis as
shown in Figure 2, of the BRCs in about 10 emission nebulæ
(Osterbrock1957).
and the centre of the exciting star (Bertoldi 1989; Lefloch
&Lazareff1994;Kessel-Deynet&Burkert2003;Miaoetal.
2006,2009;Gritschnederetal.2009;Miaoetal.2010;Bisbas
et al. 2011; Haworth & Harries 2012; Tremblin et al. 2012).
However, further investigation on the observed struc-
turesatHiiboundariesfindsthat:i)someBRCsdonotshow
a comet-like morphology, but fragment-core linear struc-
turesperpendiculartotheEUVradiationfluxdirection(e.g.
Chauhan et al. (2011)); ii) most type A, B and C BRCs Figure2.Theinitialgeometryofaprolatecloudinsimulations.
are not symmetric about their structural axes, which also xp,xaandxmaxarethexprojectionsoftheapex,thesemi-major
do not necessarily align with the radiation flux direction of axisaandthefarthestpointofthecloudfromzaxisrespectively.
their exciting stars (Thompson et al. 2004; Morgan et al.
2004). As shown in Figure 1, there is a distribution in the
anglebetweentheaxisofaBRCandthedirectionfromthe of which is illustrated in Figure 2. In this way we can ex-
tip of the BRC to the centre of the exciting star, based on pectthatthevarietyofthemorphologicalstructuresderived
theobservationaldataongroupsofBRCsinabout10emis- from RDI simulations must be greatly increased.
sion nebulæ (Osterbrock 1957); iii) There are more objects A set of RDI simulations were conducted using an ini-
with structures which can not be categorized by type A, B tially prolate cloud with ϕ = 0◦, which we call ‘perpen-
or C than those which can be; iv) In those asymmetrical dicular’ to the EUV radiation flux. It is found that a per-
BRC structures, RDI triggered multi-star or sequential for- pendicular prolate cloud would evolve into a perpendicular
mation are often found not only at the head of BRCs, but fragment-core linear structure to the ionizing flux, covered
also in the more compressed side layer (Fukuda et al. 2013; byabrighttoplayeronthestarside.Boththeinitialphys-
Ma¨kela¨ & Haikala 2013; Sicilia-Aguilar et al. 2014; Panwar ical/geometrical conditions of the cloud and the strength
etal.2014).Itisobviousthatitisdifficulttoaccommodate of radiation flux affect the distribution of condensed cores
thevarietyofstructuralandphysicalfeaturesobservedwith over the structure, the total condensed core mass and the
RDI modelling starting with a spherical cloud. triggered star formation time. The sporadic fragment-core
Observations on the physical properties of molecular structures at Hii boundaries can be interpreted by the ef-
cloudshaverevealedthatsphericalcloudsareveryrarecases fectoftheinteractionbetweenaperpendicularprolatecloud
(Gammieetal.2003;Dotyetal.2005;Rathborneetal.2009) and a parallel-plane EUV radiation field. More details are
and there are many physical processes which lead to for- presented in Kinnear et al. (2014).
mation of prolate molecular clouds (Gong & Ostriker 2011; In this paper, we expand the investigation to the evo-
Gholipour&Nejad-Asghar2013).Toinvestigatethemecha- lutionofaprolatecloudatHiiboundaries,withthecloud’s
nismforthedevelopmentofstructuresofvariousmorpholo- major axis inclined to the EUV radiation flux by an angle
gies other than standard (i.e., symmetrical) BRCs at Hii 0◦ (cid:54)(90◦−ϕ)(cid:54)90◦. This is performed in order to under-
boundaries, using an initially non-spherical cloud, such as stand the mechanism for the formation of a broad range of
a prolate cloud, in RDI modelling seems the most feasible asymmetrical structures and the consequent triggered star
choice.Definitionintermsoftwoadditional,geometric,pa- formation inside them. In the rest of the paper, we first
rametersareintroduced:theratioofthemajor(a)tominor presentabriefdescriptionofthecodesused,theinitialcon-
(b)axisγ = a andtheinclinationangleϕ,theconfiguration ditionsofthecloudsforsimulations,andaderivedquantity
b
(cid:13)c 20??RAS,MNRAS000,1–??
Evolution of Prolate Clouds at Hii Boundaries 3
which gives an indication of the dynamic evolution of pro- In Figure 2, x is the x coordinate of the apex of the
p
late clouds at different Hii boundaries. In the results and cloud.lettingrbethedistanceofanyapointonthesurface
discussion section we present simulation results and discuss of a prolate cloud in the xz plane, and θ the angle between
thephysicalmechanismresponsiblefortheformationofdif- r and the x axis, we have,
ferent mophological structures at Hii boundaries. Finally, a
r= . (1)
conclusions are drawn. p
cos2(θ−ϕ)+γ2sin2(θ−ϕ)
Themaximumxcoordinateandthexprojectionofthesemi-
major axis
2 THE CODES AND INITIAL CONDITIONS
r
2.1 The codes sin2ϕ
x = a cos2ϕ+ , (2)
max γ2
An updated Smoothed Particles Hydrodynamics (SPH)
x = acosϕ. (3)
Code ii is used for all of the simulations investigated in a
this paper, which is an extended version of Code i (Nelson As an important indicator to the dynamical evolution
& Langer 1999). Code i is an SPH code to investigate the of a prolate cloud with an inclination angle ϕ, the effective
effectofanisotropicFUVinterstellarbackgroundradiation illuminated area by the EUV radiation is,
on the evolution of a molecular cloud, by solving the full r
setofhydrodynamicequations,includingself-gravityofthe A= πaxmax = πa2 cos2ϕ+ sin2ϕ, (4)
cloud and a chain of chemical differential equations. Based γ γ γ2
on Code i, a numerical solver with a ray-tracing algorithm
which is a decreasing function with ϕ, has the maximum
(Kessel-Deynet & Burkert 2000) for a plane-parallel EUV πab at ϕ = 0◦ and the minimum πb2 at ϕ = 90◦. For the
radiationtransferringequationwasimplemented,aswellas
convenienceofthefollowingdiscussion,wecallthehalfcloud
the relevant heating and cooling processes. We refer to this withϕ(cid:54)θ(cid:54)π+ϕasthe‘front’semi-ellipsoid,andπ+ϕ<
versionasCode ii,whichcanbeusedtoinvestigatethein- θ(cid:54)2π+ϕ the ‘back’ semi-ellipsoid.
teractionofaplane-parallelionizingradiationfrommassive
star(s)withamolecularcloudofarbitraryinitialgeometry.
MoredetaileddescriptionsofthecodecanbefoundinMiao 2.2.2 Initial mass distribution
et al. (2006) and Kinnear et al. (2014).
As we have done in the investigation of the evolution of a
Byvirtueoftheasymmetricalmorphologyofthesimu-
perpendicular prolate cloud at an Hii boundary (Kinnear
latedclouds,thecondensedcorestriggeredbyRDIinduced
et al. 2014), all of the molecular clouds in our simulations
shocks are not usually aligned with the structural axis and
start with a uniform density, which is rendered by a glass-
multi-starformationisalsofrequentlyobserved.Toidentity
like distribution of SPH particles created using Gadget-2
the positions of these triggered condensed cores and anal-
(Springel 2005). A glass-like distribution is taken as a good
ysetheirphysicalproperties,acodecalled‘CoreFinder’is
approximation to a uniform mass distribution. We choose
used,forwhichadetaileddescriptionisincludedinKinnear
theinitialmassofthecloudtobe30M ,thesameasused
etal.(2014).Forthesimulationspresentedinthispaper,all (cid:12)
for Kinnear et al. (2014). The number of SPH particles for
‘cores’ are defined as a region of at least 0.03 M (> 100
(cid:12) each molecular cloud is determined by the mass resolution,
SPH particles) and the density of each SPH particle sam-
3.0×10−4M perSPHparticle,for100,000particles,higher
pled must be n (cid:62) 106 cm−3. As in Kinnear et al. (2014), (cid:12)
thanrequiredbytheconvergencetestoftheCodeii.Azero
the cores formed from the simulations are the first gener-
initial velocity field is set for all of the molecular clouds in
ation of RDI triggered objects. Simulations for the subse-
the simulations.
quent generations of RDI triggered star formation becomes
Sincetheobjectiveofourinvestigationistoexplorethe
excessivelyslowerafterinitialextremelyhighdensitycore(s)
evolution of the RDI triggered collapse of an initially in-
form,whichleadstoadecreasetowardsinfinitesimallysmall
clinedprolatecloud,anyinitiallyunstablecloudagainstthe
timesteps,andtheeffectivehaltofthesimulation.Thiscan
self-gravity before interacting with the ionization radiation
beavoidedbytheuseofthe‘sink’particlesofBate&Burk-
flux should not be chosen. Stability is assured by applying
ert (1997), but is not currently implemented in Code ii.
the Jeans criteria for an isolated prolate cloud, in terms of
Allofthecolumndensityimagesareproducedusingthe
Jeans number J, the ratio of the initial gravitational en-
softwareSPLASH,whichisspeciallydesignedforprocessing
ergy of a prolate cloud to the thermal energy and can be
SPH numerical data (Price 2007).
expressed as (Bastien 1983),
πGρµb2 (cid:16)1+e(cid:17)
2.2 Initial conditions J = ln (cid:54)1.0, (5)
15eR T 1−e
g
2.2.1 Initial geometry
where ρ, T and µ are the mass density, the initial temper-
Thegeometricalshapeofaprolatecloudinthesimulations ature and the mean molecular mass of the prolate cloud
is described by a pair of initial parameters ( a, γ ) and its respectively, G and Rg are the standard physical constants
orientation to the plane-parallel ionization radiation flux is and the eccentricity,
0◦ (cid:54) ϕ (cid:54) 90◦, as shown in Figure 2. The ionization flux r p
b2 γ2−1
onto the surface of the cloud on the star side is treated as e= 1− = . (6)
a2 γ
plane-parallel to the z axis, for an assumed large ratio of
distance to the illuminating star against cloud size. Using the relation between density, mass M and volume of
(cid:13)c 20??RAS,MNRAS000,1–??
4 T. M. Kinnear, J. Miao, G. J. White, K. Sugitani and S. Goodwin
a cloud, we can derive the condition for the major axis a of n γ ϕ F0
an initially stable prolate cloud, Seriesname cm−3 ◦ ×109 cm−2 s−1
G1200 1200 1-10 0-90 1
p !
M∗ γ γ+ γ2−1 density&flux 100-1200 2 60 1&2
a(cid:62)acrit =0.052 p ln p , (7) irregular 400 2-3.5 60-85 2
T γ2−1 γ− γ2−1
Table 1.Summaryofthetestseriesconducted.nisinitialden-
where M∗ is the mass of the prolate cloud in units of solar sity,γtheaxialratio,ϕtheinclinationangleandF0 theincident
masses, and a and a have units of Parsecs. For a given flux.
crit
molecularcloudofmassM∗,andinitialtemperatureT and
γ,aminimumvalueacanbeestimated,themajoraxisofan
by‘fociconvergence’mode,i.e.,twohighdensitycoresform
initiallygravitationallystablecloudshouldsatisfya>acrit. atthetwoendsofafilament;whend (cid:54)1,thecloudcan
EUV
stillbeRDItriggeredtocollapsebutthrough‘linearconver-
gence’, i.e., a few high density cores form along the whole
2.3 Boundary condition
filament; when d (cid:62) 1, the cloud is in photo-ionization
EUV
The clouds in our simulations are subject to an isotropic dominant region, and would be photo-evaporated. As d
EUV
interstellar background FUV radiation of one Habing unit canpossessasimilarrangeofvaluesforallϕinEquation8,
(Habing1968)andanionizingEUVradiationwithafluxof we would expect it to play a similar role for ϕ6=0.
109 cm−2 s−1 (typicaloftheboundaryofanHii region)di-
rectedparalleltothez-axis(alongthenegativez direction)
as illustrated in Figure 2. 3 RESULTS AND DISCUSSIONS
Aconstantpressureboundaryconditionisappliedwith
thevalueoftheexternalpressurebeingsetequivalenttoan Three sets of simulations were conducted using prolate
external medium of atomic hydrogen of n(Hi) = 10 cm−3 clouds of different geometrical shapes (γ), inclination an-
andtemperatureof100K.Anoutflowconditionisimposed gles (ϕ), initial densities (n) and ionising fluxes (FEUV), to
at a fixed boundary which is a few times of the initial size investigate the effect of their variations on the evolution of
of the cloud (Nelson & Langer 1999). a prolate cloud. Table 1 contains a summary of the ranges
of properties for each set. All of the clouds have an initial
temperatureof60Kandareilluminatedbya50,000Kstar
2.4 The ionization penetration depth parameter providing a flux at the cloud, F0 = 109 cm−2 s−1 (or 2.0
×109 cm−2 s−1,whenspecified),thesameasthatforKinn-
In order to classify the initial dynamic regime of a perpen-
earetal.(2014).Inthissection,wefirstdescribetheevolu-
dicular prolate cloud (ϕ = 0), a dimensionless parameter,
tionofaprolatecloudofaninclinationangleof45◦indetail,
d ,wasdefined,beingtheratiobetweenthephysicalion-
EUV then investigate the influence of changing initial inclination
ising radiation penetration depth to the semi-minor axis of
angle ϕ on the evolution of the same cloud. Next we dis-
the cloud. For this scenario, d = FEUVγ (Kinnear et al.
EUV αBan2 cuss how the initial geometry of a prolate cloud affects the
2014).Whenϕ6=0◦,wemodifyitsdefinitiontotheratioof
morphological evolution of a cloud and RDI triggered star
the physical ionization penetration depth to the character-
formation inside it. Finally we propose a formation mecha-
isticdepthofthecloud.Thischaracteristicdepthisdefined
nismforthevarietyinmorphologicalstructuresfoundatH
ashalfofthelongestpaththroughthecloudinthedirection ii boundaries.
of the radiation, which is always the depth of the cloud at
x=0, y=0.
3.1 Evolution of cloud G1200(5) with ϕ=45◦
FEUV
dEUV = r(θα=Bn920◦) Table 2 lists all of the relevant parameters of clouds with
initialdensity1200cm−3 andinclinationangleϕ=45◦,for
r
= FEUVγ sin2ϕ +cos2ϕ variations of the initial ratio γ . The cloud is the fifth one
αBan2 γ2 in the G1200 set, notated G1200(5). It has an initial shape
F∗ γrsin2ϕ definedbyγ =2.0,itsmajoraxis(0.784pc)is≈10timesof
= 1.6×103 EUV +cos2ϕ (pc) (8) itscriticalmajoraxis(0.079pc)derivedbyusingEquation7;
n2a∗ γ2
indicating its initial stability against gravitational collapse.
where the major axis a∗ is in the unit of pc, and F∗ is Figure3showsanevolutionarysequenceofcolumnden-
EUV
theEUVionisingradiationfluxinunitsof109 cm−2s−1,α sityofthecloudoveraperiodof0.13Myr.Wealsoplotthe
B
is the recombination coefficient of hydrogen ion - election corresponding mean axial density distribution along the x
under the ‘on-the-spot’ approximation (Dyson & Williams axis in Figure 4, with the established binning method for
1997) and has the value of 2.0×10−13 cm3 s−1 at a tem- describing axial profiles of SPH particle properties (Kinn-
perature of about 104 K (Dyson & Williams 1997). This is earetal.2014;Nelson&Langer1999;Nelson&Papaloizou
thentakenasaconstant,astheequilibriumtemperaturefor 1993). This view provides a qualitative picture of the loca-
ionized material is ≈ 104 K and the dependence of α on tion of condensed cores formed in the shocked layer of gas
B
temperatureisnotstrongintheregionaroundthattemper- in the cloud by the RDI effect.
ature. It is seen from the above two figures that 28 Kyr after
As already discussed in Kinnear et al. (2014), when theradiationfluxwasswitchedon,ashockisestablishedby
d <<1 for ϕ=0, the prolate cloud is in the RDI trig- the ionization heating. This shock starts compressing the
EUV
geredshockdominantregion,andthecollapseofthecloudis cloud surface on the star-facing side. A slightly compressed
(cid:13)c 20??RAS,MNRAS000,1–??
Evolution of Prolate Clouds at Hii Boundaries 5
G1200
No. γ acrit (pc) a1200 (pc) dEUV (%)
1 1.00 0.052 0.494 0.225
2 1.25 0.060 0.573 0.218
3 1.50 0.067 0.648 0.219
4 1.75 0.073 0.718 0.221
5 2.00 0.079 0.784 0.224
6 2.25 0.084 0.849 0.228
7 2.50 0.039 0.910 0.232
8 2.75 0.093 0.970 0.237
9 3.00 0.097 1.028 0.241
10 3.25 0.101 1.084 0.246
11 3.50 0.104 1.139 0.250
12 3.75 0.107 1.193 0.256
13 4.00 0.110 1.245 0.260
14 4.50 0.116 1.347 0.268
15 5.00 0.121 1.445 0.277
16 5.50 0.126 1.540 0.285
17 6.00 0.130 1.632 0.292
18 7.00 0.138 1.808 0.307
19 8.00 0.145 1.977 0.320
Figure 5.Magnificationofthecolumndensityprofilestimet=
Table 2. Parameters of molecular clouds of G1200 series with
0.11(left)and0.13Myr(right)attheheadofthecloudG1200(5)
aninclinationangleϕ=45◦.Fromlefttoright,columns1-3are
withϕ=45◦
the number identity, axial ratio and the critical semi major axis
definedbyEquation7.Columns4-5arethemajoraxisanddEUV
defined by Equation 8. All of the semi-major axes and critical ϕ(◦) 0 15 30 45 60 75 90
sinem%i-.major axes are in units of pc and the penetration depth is d(%EU)V 0.283 0.276 0.255 0.224 0.187 0.155 0.141
Table 3.VariationofdEUV overdifferentϕforcloudG1200(5).
thinlayerofmeandensity≈3.0×103 cm−3 appearsatthe
EUV exposed surface, as shown in the upper-middle panels the apex of the cloud moves down from z = 0.61 pc to
in Figures 3 and 4. The density at x = 0.49 pc is slightly z = 0.035 pc, a net recoil velocity of hv i = 4.3 km s−1,
z
higher than the axial mean density, which reflects the very due to the rocket effect causedby the evaporating gas from
early stage of a condensed core formation around the apex the star-facing surface irradiated by the ionizing radiation
of the cloud. (Oort&Spitzer1955).Nextweinvestigatetheeffectofthe
Fromt=0.028to0.11Myr,theshockcontinuespropa- inclination of a prolate cloud on its dynamical evolution.
gatingintothecloud.Thedensityinthethinshockedlayer
continues increasing and the structure of the head becomes
more distinctive. The non-head mean axial density and the 3.2 Evolution of G1200(5) with 0◦ (cid:54)ϕ(cid:54)90◦
central density of the head increase to 104 and 105 cm−3
respectively at 0.054 Myr, and to 2.0×104 and 106 cm−3 To investigate the dependence on the inclination, the cloud
G1200(5) is simulated for a range of inclinations 0◦ (cid:54) ϕ (cid:54)
respectivelyat0.08Myr.Att=0.11Myr,thedensityinthe
90◦. The sequence of inclination angles are chosen as 0, 15,
shockedlayerincreasesdramaticallytowardstheapexofthe
cloud(atx=0.32pc).Thisdensityjumpsfrom≈8×104to 30, 45, 60, 75 and 90 ◦. The values of dEUV for these seven
8×105 cm−3 atx=0.24pc,thentothemaximum4×106 clouds are listed in Table 3, from which we can see that
cm−3attheapex.Itisinterestingtofindfromthemagnified theionizingpenetrationdepthdecreaseswithϕ,andisvery
shallow,withvalues<<1.Asaresult,allofthesecloudsare
imageintheleftpanelofFigure5thatafilamentstructure
in the RDI induced shock-dominated regime and triggered
forms at the head, aligning with the EUV radiation direc-
collapse is expected.
tion. The structure has a length of roughly 0.03 pc and the
top end of it coincides with the apex of the cloud.
Thegascontainedinboththeshockedsurfacelayerand
3.2.1 Morphological variation with ϕ
inthefilamentbecomesfurthercompressedbyt=0.13Myr.
At this stage the maximum mean density in the shocked It is found that the evolutionary sequences of the 7 clouds
layer and in the filament are 105 −106 and > 109 cm−3 are qualitatively similar to that with ϕ=45◦, i.e., the for-
respectively. The right panel of Figure 5 further reveals mationofacondensedlayeronthestarfacingsurfaceofthe
the formation of two high density cores in close proximity. cloudandhighlycondensedcore(s)neartheapex,although
These dual cores may be the result of fragmentation of the thefinalmorphologiesthemselvesdependontheinitialincli-
filament-like structure observed at t=0.11 Myr. nationangleofthecloud.Asthegeneralevolutionissosim-
Overall, the cloud G1200(5) with an initial inclination ilar, they are not discussed individually, but with focus on
angle of 45◦ evolves to an asymmetrical type B BRC, with comparisonofthedifferencesintheirfinalmorphologiesand
double RDI triggered high density cores embedded at its thephysicalpropertiesofanyhighdensitycore(s).InFigure
head, very close to the apex of the BRC. The cores are the 6, the final snapshots of column density of cloud G1200(5)
potential sites for new star formation. At the same time from 6 different simulations with ϕ = 0,15,30,60,75, and
(cid:13)c 20??RAS,MNRAS000,1–??
6 T. M. Kinnear, J. Miao, G. J. White, K. Sugitani and S. Goodwin
Figure 3.SequentialevolutionofthecolumnnumberdensityforCloudG1200(5)withtheinclineangleϕ=45◦.Timesequenceisleft
toright,thentoptobottomthenlefttoright.
Figure 4. Axial mean density distribution along the x axis of the cloud G1200(5) corresponding to the column density evolutionary
snapshotsinFigure3
(cid:13)c 20??RAS,MNRAS000,1–??
Evolution of Prolate Clouds at Hii Boundaries 7
90◦,arepresented.Theequivalentforϕ=45◦ canbefound
in Figure 3.
When the inclination angle ϕ is small ((cid:54) 30◦), x ≈
a
x , the EUV radiation mainly illuminates the surface of
max
the front semi-ellipsoid. The effect of the RDI is to drive a
pseudo-plane-parallelshockpropagatingintothecloud,and
the prolate cloud evolves to a filamentary structure with
a condensed surface layer on the star side and highly con-
densed core(s) at the apex (when ϕ 6= 0◦) or at either the
apex or the two ends (when ϕ = 0◦). For the latter case,
the evolution has been analysed in detail by Kinnear et al.
(2014).
When 45◦ (cid:54) ϕ < 90◦, the EUV radiation illuminates
notonlythesurfaceofthefrontsemi-ellipsoid,butalsothe
toppartofbacksemi-ellipticalsurface,sothattheshockin-
ducedbyRDIcreatesacondensedandcurvedsurfacelayer
with the apex as a convergence point of two compressed
curving surface layers, those surfaces on opposing sides of
theapex.Theregionaroundtheapexisalsothesiteforthe Figure 7. The x displacement of the cores formed in G1200(5)
formation of high density core(s). These clouds evolve to with different initial inclination angles. The short ‘-’ indicates
asymmetricalBRCsofdifferentmorphologies.Themorpho- xmax, ‘+’ xp, ‘◦’ for the cores with 106 (cid:54) npeak (cid:54) 1012 cm−3,
logicalsequenceissimilartothatofthecloudwithϕ=45◦, and ‘•’ for the extremely high density cores with npeak > 1012
cm−3.
as shown in Figure 3. Comparing the three final morpholo-
gies of the simulations at 45, 60 and 75◦ in Figures 3 and
6, we can see that as the inclination angle increases, the
final structure of the asymmetrical BRC becomes increas-
inglyextendedbothintheheadandtail.Thisisbecause,as
the effective EUV illumination area decreases as described
in Equation 4, the total volume of the cloud affected by
the RDI induced shock becomes smaller. Consequently the
original prolate shape of a cloud is less distorted.
Whenϕ=90◦,thecloudG1200(5)developsintoasym-
metrical type B/C BRC but having a wide tail structure,
with the widest part defined by the initial semi-minor axis.
Themorphologydevelopedisdifferentfromthatofstandard
typeB/CBRCstructure,whichhasamuchnarrowerwidth
in its tail, if one assumes that the progenitor cloud was ini-
tially spherical (Lefloch & Lazareff 1994; Kessel-Deynet &
Burkert 2003; Miao et al. 2006, 2009). Observations have
also reported several type B/C BRCs with wide tail mor-
phological structures, e.g. SFO74 (Thompson et al. 2004; Figure 8. The evolution of maximum density of the core(s) in
Kusune et al. 2014). thecloudG1200(5)withdifferentinitialinclinationangles.
In summary, a variety of non-standard morphological
structures can be obtained by changing the inclination of
3.2.3 RDI triggered core formation efficiency
cloud G1200(5). At low inclination, linear fragment-core
structures form, at middle inclinations, asymmetric type B Inordertoexaminetheeffectofvaryinginclinationangleon
andCstructures,andapproaching90◦inclinationsthesere- the efficiency of RDI triggered high density core formation,
turn to symmetrical type B and C structures. we investigate the angle dependence of the core formation
timeandtheaccumulatedcoremass.Figure8showstheevo-
lutionofthemaximumdensityinthe7simulationswithdif-
ferentinclinationangles.Ingeneral,theevolutionofn in
3.2.2 The location of high density cores max
eachsimulationpassesthroughthreedifferentphases:quasi-
Figure 7 illustrates the distribution of the x-displacements linear, quasi-stable and steeply rising, although the time of
of condensed cores formed in the 7 simulations of different eachphasevarieswithinclinationangleϕ.Thesephasesmay
initial inclination angle ϕ. A point to note is that, where correspondtothedifferentstagesofRDItriggeredhighden-
coresforminverycloseproximity,theirpositionscannotbe sity core formation. Beginning with initial compression by
distinguishedonthescaleused,regardlessofthesizechoice an RDI induced shock, mass accumulates to form the con-
of the marker. It can be seen that when ϕ > 0◦, core(s) densed region, which finally collapses to form high density
alwaysformatthesideofpositivexaxisandthechangeof core(s).Thetimeforasimulationtoreachthehighestden-
the x displacement with ϕ follows the trend of x with ϕ. sity(highdensitycoreformation)decreaseswithinclination
p
Thisillustratesanextremelyhighpreferenceforhighdensity angle, from 0.19 down to 0.086 Myr for simulations with
core(s) to form around the apex x in each simulation. ϕ = 0 and 90◦ respectively. This is because the curvature
p
(cid:13)c 20??RAS,MNRAS000,1–??
8 T. M. Kinnear, J. Miao, G. J. White, K. Sugitani and S. Goodwin
Figure6.ThefinalsnapshotofthecolumndensityfromsixsimulationsofG1200(5)withdifferentinitialinclinationangles;lefttoright
thenuptobottomareper:0,15,30,60,7590◦.
ϕ,becoming0.4M atϕ=90◦.Thisisconsistentwiththe
(cid:12)
aboveexpectation.Howeverthetotalmassoftheextremely
high density core(s) (n > 108 cm−3) doesn’t show an ob-
vious similar relation with ϕ to the mass of high density
core(s).Thismaybeforseveralreasons.Firstly,asthesim-
ulationiscurtailedbeforetheendofallofthedynamicpro-
cesses, extremely high density cores may be on the cusp of
formation, but have not yet had the opportunity. Secondly,
a large degree of random variation is expected with such
gravitational instability-based collapse. Accumulating mass
in compact high density regions can be fairly consistent,
butthecollapseoftheseregionsorpartsoftheseregionsto
extremely high densities through gravitational instability is
chaotic and varies strongly with the exact configuration of
the dynamics.
From the above result, it is understood that as a cloud
is rotated from ϕ = 0 to 90◦, the morphology of the cloud
Figure 9. The variation of the total core mass of the cloud varies from fragment-core filamentary structures to varied
G1200(5) over different initial inclination angles. The symbols asymmetricalBRCs,thentostandardsymmetricalBRCsas
andtheirrepresentedpropertiesofcoresareshowninthelegend ϕ approaches 90◦. At high ϕ, RDI triggered star formation
box. occursearlierwithlowermassesathighdensities.However,
for the sampling performed, such pattern is not obvious for
theextremelyhighdensitycoreformation,whichappearsto
of the EUV radiation illuminated surface around the apex
bechaotic.Nextwelookattheeffectoftheinitialshapeof
of the prolate cloud increases with ϕ, this is beneficial for
aninclinedprolatecloudonitsfinalmorphologicalstructure
mass accumulating towards a geometric focus underneath
and RDI triggered star formation efficiency.
the curved surface. This decreases the path length to the
focus, and so also the time for accumulating mass around
it and triggering gravitational instability of the condensed
region; albeit with less material having been ‘swept up’ by
3.3 Evolution of G1200 series with 1.0(cid:54)γ (cid:54)8.0
thefront.Duetothisshorteningpathlength,thetotalmass
ofthehighdensitycore(s)isthereforeexpectedtodecrease Theinitialshapeofthecloudsinvestigatedhereisdefinedby
with ϕ. theratioofthemajortominoraxiswhichareintherangeof
The relation between the total high density core mass 1.0 (cid:54) γ (cid:54) 8.0. When the initial density and total mass are
and inclination angle ϕ is shown in Figure 9, which reveals constants, the major axis changes with γ. The inclination
that the total high density (n (cid:62) 106 cm−3) core mass is angle is kept at 45◦. The ionization radiation penetration
about 4.1 M , in simulation of ϕ = 0, then decreases with depth ranges from 0.225 to 0.320% as listed in Table 2.
(cid:12)
(cid:13)c 20??RAS,MNRAS000,1–??
Evolution of Prolate Clouds at Hii Boundaries 9
Figure11.ThexdisplacementofcondensedcoreinG1200series Figure 12.TotalmassofcondensedcoresincloudG1200series
withaninclinationofϕ=45%forvariedγ.Theshort‘-’indicates ataninclinationof45%forvariedaxialratioγ.
xmax, ‘+’ xp, ‘◦’ for the cores with 106 (cid:54) npeak (cid:54) 1012 cm−3,
and ‘•’ for the extremely high density cores with npeak > 1012
cm−3. Thispatternisdisplayedinthesameformforthecondensed
cores. A very clear trend of the core formation occurring at
or around the apex for all ratios is exhibited.
3.3.1 Morphology of final structures
Thedynamicalevolutionofthesecloudsisalsoqualitatively
3.3.3 The total core mass
similar to that discussed in Section 3.1. We focus on the
analysis of the effect of the initial shape of a cloud on the The trend of total mass of both the high density core(s)
final morphology, the location(s) of the condensed core(s) (emptycircles)andextremelyhighdensitycores(solidblack
andthetotalcoremasses.Figure10presentssixsimulation circles)isadecreasewithγ.ThisisillustratedinFigure12.
result of cloud G1200 with different initial major to minor The total mass of high density cores has an approximate
axisratioγ =1.0,1.5,2.0,2.5,3.0,5.0,selectedfromsimula- pattern of maximum mass of 2.0 M at γ ≈ 2.25 (with an
(cid:12)
tionresultsfor19differentinitialclouds,describedinTable outlierashighasalmost3.0M atγ =2.75)andaminimum
(cid:12)
2. With increasing γ, the final morphology changes from a of 0.6 M at γ = 8 (with an unusually low value of only
(cid:12)
standard(symmetrical)typeA(γ =1.0),toanasymmetric around 0.2 M for γ = 6.0). Clouds with lower ellipticity
(cid:12)
typeABRC(γ =1.5),thentoanasymmetrictypeBBRC appeartohaveahigherprobabilityofcollectinggreatermass
(γ = 2.0), next to evolve to an asymmetrical type C BRC inthefinalcores.Thiscanbeexpectedasthemassperunit
(γ =2.5).Withfurtherincrease(γ (cid:62)3.0)onlyafilamentary length along the major axis decreases with γ. As a result,
structure can form. theamountofmassabletocollapsetowardtheapexregion
In these six simulations, high density cores form at the decreases,whichissimilartothecasewhenϕ=0◦(Kinnear
headofthestructureexceptinthecaseofγ =2.5,wherethe etal.2014).Whenϕ6=0◦,duetothewiderangeofvariation
highlycondensedcoreisembeddedinthesmall‘nose’struc- inthefinalmorphologiesoftheclouds,theamplitudeofthe
turejustoutsidethehead.Itisinterestingtonoticethatthe fluctuation is similarly increased.
morphologicalstructureofSFO46(CG1)fromobservations Fromthesamefigure,wecanseethemassofextremely
(Harju et al. 1990; Haikala et al. 2010; Ma¨kela¨ & Haikala condensedcores(solidblackcircles)inacloudalsodecreases
2013)bearsasimilarfeaturetothatofγ =2.5.Weleavethe with γ in a similar way as the high density core. The mass
discussion for the formation mechanism of this morphology of these cores corresponds to the precursor mass for possi-
to the Section 3.4, where we present and analyse more re- ble proto-stars forming later in the evolution of the cloud,
sults of various kinds of morphological structures. It is also having a maximum of ≈1.6 and a minimum of ≈0.2 M .
(cid:12)
emphasised that we don’t intend to compare the detailed
physical structures between simulation and observation on
SFO46inthispaper,butonlytopointoutthelinkbetween 3.4 Morphological evolution of clouds of different
theinterestingfeaturesfromthesesimulationstosuchobser- initial density and under different ionizing
vations.Detailedcomparisonofphysicalpropertiesrequires flux
further simulations, and will be addressed in a subsequent
After the investigation of the effects of both initial shape
paper.
andinclinationangleofaprolatecloudonitsRDItriggered
dynamical evolution, we now explore the consequences of
changing the initial density of a cloud and the ionizing ra-
3.3.2 Displacement of the high density cores
diation flux in the morphological development of a cloud of
Figure 11 displays the x-displacement of condensed core(s) mass 30 M . This is essentially to examine the consistency
(cid:12)
with changing geometry, based on the simulation results of of the d parameter in characterising a wider variety of
EUV
19 clouds with different γ. cloud parameters.
Withincreasingγ,x movesfurtherinthe+xdirection. A sequence of 8 different initial densities are chosen as
p
(cid:13)c 20??RAS,MNRAS000,1–??
10 T. M. Kinnear, J. Miao, G. J. White, K. Sugitani and S. Goodwin
Figure 10. The final snapshots of the column density from six simulations of G1200 with γ equal to the values shown in the top-left
cornerofeachpanel.Theinclinationangleis45◦ foralloftheinitialclouds.
Index n a dEUV(F1) dEUV(F2) 3.4.1 An overview of the final morphological structures
cm−3 pc % %
G100 100 1.797 11.779 23.558 Figure13displaysthefinalsnapshotsofthecolumndensity
G200 200 1.427 3.708 7.416 of the 8 simulations with ionization flux of F . The mor-
1
G400 400 1.132 1.17 2.34 phology of the final structure of a cloud changes from an
G600 600 0.989 0.594 1.188 asymmetrical type C BRC to a filamentary structure and
G700 700 0.940 0.460 0.920 then to an irregular structure, as the initial density is de-
G800 800 0.899 0.368 0.736 creased from 1200 to 100 cm−3, which leads to an increase
G1000 1000 0.834 0.254 0.508
of d from 0.187 to 11.779%.
G1200 1200 0.784 0.187 0.374 EUV
As shown in the top row panels, an asymmetrical type
Table 4. Parameters of molecular clouds of mass 30 M(cid:12), an C BRC is developed in G1200, G1000, G800, and G700,
inclination angle ϕ = 60◦, γ = 2, and acrit = 0.079 (pc). From with d of 0.187, 0.254, 0.368 and 0.460% respectively.
EUV
lefttoright,columnsarethecloudindex,initialnumberdensity,
Although they all bear an asymmetric type C BRC mor-
the major axis and dEUV defined by Equation 8, with different phology, the minor axis becomes narrower with increasing
ionizationfluxF1andF2being1.0×109and2.0×109cm−2s−1 d , which results in more cloud material being photo-
respectively. EUV
evaporatedfromtheirstarfacingsurfaces.Whentheinitial
density n = 600 cm−3 (d = 0.594%), the prolate cloud
EUV
evolves into a filamentary structure, as shown in the first
panelinthebottomrowofFigure13.Cloudsofinitialden-
100,200,400,600,700,800,1000,and1200cm−3.Twodif- sities of 400, 200 and 100 cm−3 (dEUV = 1.17, 3.708 and
ferentionizationfluxesofF =1.0×109 andF =2.0×109 11.779 % respectively) are all seen to form irregular struc-
1 2
cm−2 s−1 areappliedforatotalof16simulations.Anincli- tures.
nation angle of 60◦ and γ = 2.0 is used for all cases. This Thefinalmorphologicalstructuresfromthesimulations
is as a result of the simulation of G1200(5) exhibiting the usingdoubledionizationfluxF2 arepresentedinFigure14.
formation of the ‘nose’ structure in the vicinity of these ge- Similar morphological structure sequences to those seen in
ometries.Assuch,wecanalsoexpecttogetsomeclueofhow Figure 13 are observed. An asymmetrical type C BRC is
the‘nose’structureisdevelopedandchangeswiththeinitial formed from G1200, G1000 and G800, a filamentary struc-
properties of a cloud in this set of simulations. The corre- ture from G700, and irregular structures in G600, G400,
sponding parameters of the clouds are listed in Table 4. It G200andG100.Theonlydifferenceisthatthemorphologi-
isseenthatthechangeinEUVradiationpenetrationdepth caltransitionpointshiftstoacloudofhigherinitialdensity
overtheinitialdensity(100(cid:54)n(cid:54)1200cm−3)isalmosttwo compared to the F1 simulations.
orders of magnitude, a much more dramatic variance than The first transition from type C BRC to filamentary
from changing γ or ϕ. morphologyoccursaroundG600(F )andG700(F ),andthe
1 2
(cid:13)c 20??RAS,MNRAS000,1–??