Comparing simulations of ionisation triggered star formation and observations in RCW 120
aa r X i v : . [ a s t r o - ph . GA ] M a y Mon. Not. R. Astron. Soc. , 1– ?? (2015) Printed 21. Juli 2018 (MN L A TEX style file v2.2)
Comparing simulations of ionisation triggered star formation andobservations in RCW 120
S. Walch ⋆ , A.P. Whitworth , T.G. Bisbas , D.A. Hubber , , R. W ¨unsch University of Cologne, Z¨ulpicher Str. 77, 50937 Cologne, Germany School of Physics & Astronomy, Cardi ff University, 5 The Parade, Cardi ff CF24 3AA, Wales, UK Department of Physics & Astronomy, University College London, Gower Place, London WC1E 6BT, UK Technical University Munich, Excellence Cluster Universe, Boltzmannstr. 2, 85748 Garching, Germany University Observatory Munich, Department of Physics, Ludwig-Maximilians-University Munich, Scheinerstr.1, 81679 Munich, Astronomical Institute, Academy of Sciences of the Czech Republic, Bocni II 1401, 141 31 Prague, Czech RepublicGermany.
Accepted . Received 2015, March 12; in original form
ABSTRACT
Massive clumps within the swept-up shells of bubbles, like that surrounding the galactic H ii region RCW 120, have been interpreted in terms of the Collect and Collapse (C&C) mecha-nism for triggered star formation. The cold, dusty clumps surrounding RCW 120 are arrangedin an almost spherical shell and harbour many young stellar objects. By performing high-resolution, three-dimensional SPH simulations of H ii regions expanding into fractal molecu-lar clouds, we investigate whether the formation of massive clumps in dense, swept-up shellsnecessarily requires the C&C mechanism. In a second step, we use RADMC-3D to computethe synthetic dust continuum emission from our simulations, in order to compare them withobservations of RCW 120 made with APEX-LABOCA at 870 µ m. We show that a distributi-on of clumps similar to the one seen in RCW 120 can readily be explained by a non-uniforminitial molecular cloud structure. Hence, a shell-like configuration of massive clumps does notimply that the C&C mechanism is at work. Rather, we find a hybrid form of triggering, whichcombines elements of C&C and Radiatively Driven Implosion (RDI). In addition, we inve-stigate the reliability of deriving clump masses from their 870 micron emission. We find thatfor clumps with more than 100 M ⊙ the observational estimates are accurate to within a factorof two and that, even at these long wavelengths, it is important to account for the radiativeheating from triggered, embedded protostars. Key words:
Galaxies: ISM - ISM: nebulae - H ii regions - bubbles - dust - Hydrodynamics -Stars: formation The possibility of star formation triggered by ionizing feedbackfrom young, massive stars has been explored for several decades.From a theoretical point of view, two main triggering mechanismshave been suggested: Collect and Collapse (C&C), and RadiationDriven Implosion (RDI).The C&C mechanism was first analyzed byElmegreen & Lada (1977). In this mechanism, an expandingH ii region sweeps up a layer of cold gas and dust beyond theionization front (e.g. Dale et al. 2007), and this shell eventuallybecomes gravitationally unstable due to the growth of perturbati-ons along its surface (Elmegreen 1994; Whitworth et al. 1994a;Dale et al. 2009; W¨unsch et al. 2010). One argument in favourof the C&C mechanism is that it is predicted to spawn massive ⋆ E-mail: [email protected] fragments (Whitworth et al. 1994b), and hence it a ff ords thepossibility of forming massive stars sequentially. The study byThompson et al. (2012) of massive, young stellar objects (YSOs)associated with bubbles suggests that triggering could be respon-sible for the formation of between 14% and 30% of all massivestars.In contrast, RDI (Sandford et al. 1982;Kessel-Deynet & Burkert 2003; Bisbas et al. 2011;Haworth & Harries 2012) triggers star formation by com-pressing pre-existing cold, but otherwise gravitationally stable,molecular cloud cores. Elmegreen et al. (1995) were the first toshow that bright rims are caused by ionization erosion around pre-existing turbulent clumps. Observational and theoreticalstudies (e.g. Gritschneder et al. 2009, 2010) suggest that RDI leadsto star formation in the tips of pillar-like structures, for exampleas seen in the Eagle nebula (White et al. 1999). Tremblin et al.(2012) have conducted a similar study but focus on the formationof cometary globules, which form if the turbulent pressure is high. c (cid:13) Walch et al.
Bisbas et al. (2011) have published a detailed numerical study ofRDI in initially stable Bonnor-Ebert spheres.The nearby H ii region RCW 120 is one of the best studiedH ii regions in the Galactic plane. As observed with Spitzer at 8 µ m(Churchwell et al. 2006), it appears to be an almost perfectly roundbubble with a well defined ionization front. Zavagno et al. (2007)and Deharveng et al. (2009) have combined Spitzer and 2MASSdata with observations at 870 µ m and 1.2 mm, to analyze the starformation associated with RCW 120. They conclude that star for-mation in the shell of RCW 120 has probably been triggered by acombination of triggering mechanisms. Zavagno et al. (2010) stu-dy the YSO properties with Herschel
PACS and SPIRE. They con-firm the existence of a YSO with mass M ⋆ = ⊙ in oneof the condensations, and identify a number of lower mass (0.8 to4 M ⊙ ) Class 0 sources within the shell. Massive clumps have al-so been found within the swept-up shells of other bubbles, e.g. Sh104 (Deharveng et al. 2003) and RCW 79 (Zavagno et al. 2006),and it has been suggested that the C&C mechanism, which invokesthe formation of massive clumps via shell fragmentation, could beat work. For all of these bubbles, the mass of the swept-up shellis in agreement with the expected swept-up mass (Anderson et al.2012), as estimated using a simple model of an H ii region expan-ding into a uniform-density ambient medium (e.g. Spitzer 1978;Whitworth et al. 1994b).Walch et al. (2012, 2013) show that clumpy, shell-like struc-tures like that seen in RCW 120 are probably attributable to pre-existing density structures in the natal molecular cloud. During theexpansion of the H ii region and the collection of the dense shell,the pre-existing density structures are enhanced and lead to a clum-py distribution within the shell. The masses and locations of theswept-up clumps depend on the fractal density structure of the mo-lecular cloud, through the parameters n and ρ (see Section 2 andWalch et al. 2012, 2013). Subsequently, the clumps grow in mass,and at the same time they are overrun and compressed by the H ii region, until they become gravitationally unstable and collapse toform new stars. Due to the formation of massive clumps, it is pos-sible that there is a second generation of massive star formation.This is a hybrid triggering scenario, which combines elements ofboth the C&C and RDI mechanisms. In this paper we show that thedistribution of massive clumps formed around an H ii region, whichexpands into a structured molecular cloud, is in good agreementwith observations of massive clumps around e.g. the Galactic H ii region RCW 120.The plan of the paper is the following. In section 2 we describethe algorithm used to generate initial fractal molecular clouds, andthe numerical method used to evolve them, including the treatmentof ionizing radiation. In section 3 we describe the resulting H ii re-gions and the modeled synthetic 870 µ m observations. We discussthe shell and clump masses inferred from the synthetic observati-ons in section 4, and compare them with the true masses. Our mainconclusions are summarized in section 5. The initial three-dimensional fractal density structure is construc-ted using an algorithm based on Fourier Transformation. The algo-rithm has three main input parameters, (i) the 3D power spectralindex n , where P ( k ) ∝ k − n , (ii) the random seed R used to generatea particular cloud realisation, and (iii) the density scaling constant ρ (see below). We populate the integer modes k = x , y , z ), where k = D , of afractal structure embedded in three-dimensional space is related tothe power spectrum by D = − ( n − D is equivalent todefining the power spectral index n . Here, we choose setups with D = .
4, in agreement with observations of molecular clouds inthe Milky Way (Falgarone et al. 1991; Vogelaar & Wakker 1994;Stutzki et al. 1998; Lee 2004; S´anchez et al. 2005). This corre-sponds to n = . ρ FFT ( x , y , z ) on a128 grid, the resulting density field is scaled exponentially, usinga scaling constant ρ : ρ ( x , y , z ) = exp ρ FFT ρ ! . (2) ρ ( x , y , z ) has a log-normal density probability density function(PDF) and a clump mass distribution in agreement with observa-tions (as described in Shadmehri & Elmegreen 2011). In particular,for a given spectrum of density fluctuations, changing ρ allows usto adjust the width of the density PDF, i.e. the variance σ of the log-normal distribution, whilst leaving the underlying topology of thedensity field unchanged. In Figure 1 we show the resulting densityPDFs for the two setups we discuss in this paper. Both clouds haveequal mass (see section 2.2), the same fractal dimension D = . R , but two di ff erent values of ρ .Before populating the computational box with SPH particles,we shift the point of maximum density to the center of the compu-tational domain, and position the ionizing star there. Then we par-tition the computational box with a 128 grid, compute the massin each cell of the grid, and apportion each cell the correspondingnumber of SPH particles, distributed randomly within the cell. Fi-nally we cut out a sphere with radius R MC , centered on the ionizingstar. The caveat of the fractal density setup is that the particles allhave no initial velocities. This assumption neglects possible flowstowards / away from the source of ionising radiation, which mightchange the dynamics of the expanding HII region. We choose a cloud with total mass of M MC = M ⊙ , and ra-dius of R MC = . ρ = . × − g cm − , or equivalently ¯ n = − for molecu-lar gas having mean molecular weight µ = .
35. The gas is initiallyisothermal at T MIN =
30 K. In this paper, we discuss two simulati-ons, both of which result in a shell-like structure very similar to theone observed in RCW 120. Apart from the scaling parameter ρ , theinitial clouds are identical, i.e. their density fields have the same to-pological structure. The simulation with ρ = . Run 1 , and the simulation with ρ = . Run 2 . By fitting their densityPDFs using a χ -squared minimisation method, we can estimate thestandard deviation of the logarithmic density, viz. σ = .
88 (
Run1 ), and σ = .
31 (
Run 2 ).We can relate σ to the conditions produced in turbulent gasby introducing the scaling relation between σ and the turbulent c (cid:13) , 1– ?? omparing simulations and observations in RCW 120 Figure 1.
Volume-weighted density PDFs for fractal clouds having the samefractal dimension D = . R , but di ff erent density scalingparameters: ρ = . Run 1 ) and ρ = . Run 2 ). Both distributions have been fitted with a log-normal (red lines).
Mach number derived by Padoan et al. (1997), Padoan & Nordlund(2002) and Federrath et al. (2008), σ = ln (cid:16) + b M (cid:17) , (3)with b ≃ .
5. According to this relation, the density field in
Run1 corresponds to a molecular cloud with Mach 2.2 turbulence; and
Run 2 corresponds to Mach 4.3 turbulence.In both
Run 1 and
Run 2 , a star emitting hydrogen-ionisingphotons at rate ˙ N LyC = s − is placed at the center of thecloud. This corresponds approximately to an O7.5 ZAMS star witha mass of ∼
25 M ⊙ and a surface temperature of ∼ ,
000 K(Osterbrock & Ferland 2006, their Table 2.3). Thus, the ionizingsource is more powerful than the central star within RCW 120,which is estimated to be an O8.5 to O9 star emitting ˙ N LyC = . ± . s − (Zavagno et al. 2007). We use the SPH code S eren (Hubber et al. 2011), which isextensively tested and has already been applied to many pro-blems in star formation (e.g. Walch et al. 2011; Bisbas et al. 2011;Stamatellos et al. 2011). The ionizing radiation is treated with aHEALPix-based adaptive ray-splitting algorithm, which allows foroptimal resolution of the ionization front in high resolution simu-lations (see Bisbas et al. 2009). The implementation is based onthe On-The-Spot approximation (Zanstra 1951; Osterbrock 1974;Spitzer 1978), which is valid if the hydrogen number density is suf-ficiently high; this is usually the case, in particular in the vicinityof the ionisation front. The ionisation front is located along eachray by assuming radiative equilibrium. We employ the standardSPH algorithm (Monaghan 1992). The SPH equations of motionare solved with a second-order Leapfrog integrator, in conjunctionwith a block time-stepping scheme. Gravitational forces are cal-culated using an octal spatial decomposition tree (Barnes & Hut1986), with monopole and quadrupole terms and a Gadget-styleopening-angle criterion (Springel et al. 2001). We use the standardartificial viscosity prescription (Monaghan & Gingold 1983), mo-derated with a Balsara switch (Balsara 1995). In both simulationswe use 2 . × SPH particles. Thus, each particle has a mass m SPH = . × − M ⊙ and the minimum resolvable mass is ∼ . ⊙ (Bate & Burkert 1997).The temperature of ionized gas particles well inside the io-nisation front is set to 10 ,
000 K. The temperature of neutral gasparticles well outside the ionisation front is given by a barotropicequation of state that mimics the gross thermal behaviour of proto-stellar gas (Masunaga et al. 1998, their Fig. 4), T ( ρ ) = T MIN + ρρ CRIT ! ( γ − , (4)with T MIN =
30 K, ρ CRIT = − g cm − , and γ = /
3. Thus, fordensities below ρ CRIT , the gas is approximately isothermal at 30 K,and for densities above ρ CRIT , the gas is approximately adiabaticwith adiabatic index γ = / γ has the monatomic value, because,although the gas is primarily molecular hydrogen, it is too cold forthe internal degrees of freedom of the hydrogen molecules, eventhe rotational ones, to be excited). The choice of T MIN =
30 K is so-mewhat arbitrary, and was initially chosen simply because it agreeswith the estimate in Deharveng et al. (2009). However, it agreeswell with the mean dust -temperature calculated a posteriori usingRADMC-3D (see Section 4.3). The temperatures of SPH particleswithin one local SPH smoothing length of the ionisation front areforced to vary smoothly between the temperature of the ionised gason one side and the temperature of the neutral gas on the other; thisis required to avoid a numerical instability.We use a new sink particle algorithm to capture forming pro-tostars (Hubber et al. 2013). Sinks are introduced at density peaksabove ρ SINK = − g cm − , provided that the density peak in que-stion is at the bottom of its local gravitational potential well. Sin-ce ρ SINK ≫ ρ CRIT , a peak that is converted into a sink is alwayswell into its Kelvin-Helmholtz contraction phase. Once created, asink has a radius of twice the SPH smoothing length at ρ SINK , i.e. R SINK =
20 au. A sink then accretes mass smoothly from the sur-rounding SPH particles within R SINK , over many dynamical times-cales, but transfers their angular momentum to SPH particles justoutside R SINK (see Hubber et al. 2013, for more details). This pro-cedure ensures that the masses and locations of sink particles areessentially independent of the arbitrary parameters of sink creati-on and evolution, ρ SINK and R SINK . Sinks are identified as protostars,and their luminosity can be included in the radiative transfer mo-dels produced with R admc -3D in the post-processing step (Section3.3).
In this subsection we discuss the results of the SPH simulations. InFigure 2, we show the initial and final column density distributionsfor the simulations. Note that
Run 2 has the broader density PDF,and hence the more pronounced density contrasts in the initial con-ditions. Both setups develop an H ii region with a diameter of ∼ ∼
500 M ⊙ has been converted into stars, which is t END = .
98 Myrfor
Run 1 and t END = .
68 Myr for
Run 2 . The black dots in theevolved H ii regions mark sinks, i.e. protostars. At t END there are 79 Both runs could be followed further, but become extremely time-consuming and slow once the dense shell is collapsing in many places.c (cid:13) , 1– ?? Walch et al.
Run 1 Run 2 C M SINK , tot N SINK C M SINK , tot N SINK [M ⊙ ] [M ⊙ ]Shell 400. 31 Shell 483. 36C1 0.0 0 C1 34.6 3C2 0.0 0 C2 54.2 6C3 140. 11 C3 171. 9 Table 1.
Properties of the protostars embedded in the shell, and in the mainclumps. For
Run 1 ( Run 2 ), Column 1 (4) gives the ID of the shell or clump,Column 2 (5) the number of protostars it contains, and Column 3 (6) thetotal mass of the protostars it contains, at t END . Note that the sink masses donot add up to the total mass in protostars at t END , because some protostarsare located outside the main clumps. sinks in
Run 1 and 38 sinks in
Run 2 . We stress that the simula-tions are highly dynamical. New protostars are constantly formedand existing protostars continue to accrete at di ff erent rates.In Figure 3, we plot the mass accretion rate onto each protostaras a function of its mass. The masses range from ∼ M ⊙ to ∼
40 M ⊙ .The accretion rates of protostars that have essentially stopped ac-creting are arbitrarily set to ˙M MIN = − M ⊙ yr − ; most of theseare from Run 1 . For all other sinks there is no clear correlation bet-ween sink mass and mass accretion rate, and the mean accretion ra-te is ∼ − M ⊙ yr − in both Runs . Run 2 forms more massive starsthan
Run 1 , and has a clutch of eight massive and rapidly accretingprotostars at t END . As with RCW 120, we find that the shells formed in our simula-tions are not perfectly spherical, but elongated and perforated. Forexample, in the simulations illustrated in Fig. 2 the initial cloud hasa region of reduced column density in the northwest corner, wherethe ionized gas breaks through and streams out, thereby relievingthe pressure of the H ii inside the bubble.The SPH mass distribution does show some small pillars andEvaporating Gaseous Globules (EGGs) close to the northwest hole,but in general pillars are not a prominent feature of H ii regions ex-panding into clouds with fractal dimension D = .
4, because – forthis fractal dimension – large-scale density fluctuations dominatethe cloud structure (Walch et al. 2012).
In a post-processing step we map the SPH density distributions at t END onto a three-dimensional grid of 128 (0 .
12 pc) cells, usingkernel-weighted interpolation. Hence the spatial resolution of thegrid at the distance of RCW120, D = .
34 kpc, corresponds to theangular resolution of 19 . ′′ achieved when observing RCW 120with APEX-LABOCA at 870 µ m (Deharveng et al. 2009). All themain clumps and condensations are well resolved with & SPHparticles. Assuming the standard, uniform gas-to-dust ratio of 100,the distribution of SPH particles translates directly into the distri-bution of dust. The only modification to this rule is that we assumedust to be destroyed at gas temperatures T > T cut = ff set value in cells abo-ve this temperature, which are exclusively those in the H ii region;the o ff set value is 10 − times the minimum dust density in the rest Figure 3. U pper P anel : Cummulative sink mass functions at t END , for
Run1 (black line) and
Run 2 (red line), showing that
Run 2 forms a higher pro-portion of massive stars than
Run 1 . The vertical dotted line marks 8 M ⊙ ,which is usually taken as the lower limit for a massive star, so at this stage ∼
17% of the stars in
Run 1 are massive, and ∼
34% of the stars in
Run 2 are massive L ower P anel : Mass accretion rate, ˙ M , plotted against currentmass, M , at t END , for the protostars formed in
Run 1 (black open diamonds)and
Run 2 (red stars). For protostars that have essentially stopped accreting,we set ˙ M = − M ⊙ yr − , so they appear in a line along the bottom ofthe plot; these stars are predominantly formed in Run 1 . For the remainingprotostars, there is no strong correlation between ˙ M and M , and no obviousdi ff erence between the stars formed in Run 1 and
Run 2 , apart from a clutchof eight massive, rapidly accreting stars in
Run 2 . of the computational domain. We only take into account silicatedust grains, on the assumption that these dominate the opacity at870 µ m. We use the standard opacity table for this species given byDraine & Lee (1984).Using the gridded density distribution, we model the transport c (cid:13) , 1– ?? omparing simulations and observations in RCW 120 Figure 2. U pper P anels : Initial column-density distributions in M ⊙ pc − for the two runs. In both cases the initial conditions have been generated with thesame fractal dimension, D = .
4, and the same random seed, R , but di ff erent scaling densities: ρ = . Run 1 ; left column) and ρ = . Run 2 ; rightcolumn). L ower P anels : The column density distributions at t END , after ∼
500 M ⊙ of the cloud ( ∼ ×
12 pc. of continuum radiation against dust opacity using R admc -3D (ver-sion 0.25, written by C. Dullemond; see also Peters et al. 2010). Inthe first step R admc -3D performs a thermal Monte Carlo (MC) ra-diative transfer simulation to determine the equilibrium dust tempe-rature distribution. The MC implementation is based on the methodof Bjorkman & Wood (2001), but includes various improvements,for instance the continuous absorption method of Lucy (1999). Thetotal luminosity of all sources within the computational domain isdistributed amongst N PHOT = × photon packages, where halfof them, i.e. N SCAT = × photon packages, are used to com-pute scattering events. In one set of radiative transfer calculationswe only invoke radiation from the central ionizing star; henceforthwe refer to these calculations as ionizing source only . In addition,we perform radiative transfer calculations in which radiation fromthe newly-formed protostars is also included; henceforth we referto these calculations as secondary sources included , and distinguish http: // / dullemond / software / radmc-3d quantities derived from these calculations with a superscript ⋆ . Theluminosities of the protostars are estimated on the assumption thatthey are ZAMS main-sequence stars. This probably results in asignificant underestimate of their time-averaged luminosities, butsince the luminosities of young protostars are expected to be high-ly variable, due to infrequent episodic accretion bursts, this seemsthe safest way to proceed.In the second step, R admc -3D computes isophotal maps at870 µ m, using ray tracing. Figure 4 shows the resulting isophotalmaps of our simulations, as seen in their ( x , y )-projection. The ou-ter white contour marks the 0 . / beam cuto ff value, which weuse to define the total mass of the shell. The inner white contourmarks the 0 . / beam value, which we use to define the massesof the clumps. With radiation from the ionizing source only (toprow of Fig. 4), the total flux an observer at 1 .
34 kpc would recei-ve at 870 µ m is F tot =
312 Jy for
Run 1 and F tot =
535 Jy for
Run 2 . If radiation from the secondary sources is included (bot-tom row of Fig. 4) the resulting total fluxes are significantly higher: F tot =
500 Jy for
Run 1 and F tot =
760 Jy for
Run 2 . The fluxes c (cid:13) , 1– ?? Walch et al. are higher in the second case, because most of the newly formedprotostars are located in the shell, and in or near the clumps.Qualitatively, the synthetic isophotal maps of our simulationsare very similar to the 870 µ m observations of RCW 120. In thefollowing, we compare synthetic and observed images in greaterdetail, in order to assess the SPH simulation, but also to evaluatethe uncertainty of the mass estimates for clumps in the shell ofRCW 120.We use the 0 . / beam contour to define clumps within theshell. In particular, we identify the 3 most massive clumps, whichwe label C1, C2, C3, and focus our analysis on their properties.The total mass and number of sinks located within each clump arelisted in Table 1. The mass in gas and dust in the shells and in theindividual clumps are listed in Table 2, which we describe morethoroughly in the next section. In order to compare the simulations with observations of RCW 120,we adopt the same procedure as in Deharveng et al. (2009) to cal-culate the masses of the shells, and of the individual clumps, fromthe synthetic emission maps, M = F D κ B ( T DUST ) . (5)Here D is the distance of the source ( D = .
34 kpc for RCW 120), κ = . g − (Ossenkopf & Henning 1994) is the opacity perunit mass of dust at 870 µ m, and B ( T DUST ) is the Planck functionat 870 micron for dust temperature T DUST . We adopt T DUST =
30 K,since we do not allow the gas to cool below this temperature in thesimulations; Deharveng et al. (2009) also invoke this temperatureto analyse their results. A constant gas-to-dust mass ratio of 100has been assumed.In Figure 5 we show the mass distribution derived fromthe isophotal maps using Eq. 5. For reference we overplot the0 . / beam and 0 . / beam contours from Fig. 4, which define– respectively – the shell and the most massive clumps. For Run 1 we calculate a total shell mass of 1818 M ⊙ with radiation from theionizing source only, and 2928 M ⊙ with radiation from secondarysources included. Both of these estimates are similar to the shellmass of 1100 M ⊙ found in RCW 120 with T DUST =
30 K. For
Run 2 the corresponding shell masses are 3013 M ⊙ and 4420 M ⊙ . We divide the shells up into clumps using the 0 . / beam contouron the synthetic isophotal maps. These clumps typically have mas-ses between a few and a few hundred M ⊙ . In Fig. 5 we identify thethree main clumps formed in each Run , and their properties are li-sted in Table 2. All these clumps are su ffi ciently massive that theymay spawn massive protostars, leading to sequential propagationof massive star formation (see Koenig et al. 2012, for observatio-nal evidence of sequential triggering).The individual clumps are well aligned with the overall struc-ture of the ionisation front and the swept-up dense shell. However,our simulations clearly show that the formation of separate clumpsin these simulations is not really due to C&C – in the sense thatat no time do we observe the formation of a coherent shell, which grows to subsequently become gravitationally unstable and under-go fragmentation. The seeds for the clumps are already present inthe initial fractal density structure of the cloud, and as the H ii regionexpands it sweeps additional low-density material into the clumps,and the clumps themselves are pushed outwards and collect addi-tional material that way. At the same time, the ionizing radiationpenetrates low-density regions much more easily than high-densityregions, and as a result the hot, high-pressure ionised gas tendsto envelop the dense clumps and compress them, as in Radiative-ly Driven Implosion. It is the combination of collecting additionalmass and being enveloped by the H ii region that renders the clumpsunstable and drives them into collapse. This is therefore a hybridprocess, combining elements of both C&C and RDI. Based on 2Dradiation-hydrodynamic simulations, Elmegreen et al. (1995) dis-cuss a similar scenario for massive core formation by shock focu-sing of turbulent clumps inside a moving post-shock layer.We are unable to comment on whether individual proto-stars are regularly spaced within individual clumps (as notedby Deharveng et al. (2009) in the unsharp-masked 24 µ m Spitzerimage of their Condensation 1) because properly modelling the24 µ m emission is beyond the scope of this paper (see, for exam-ple, Koepferl et al. 2015). However, we note that regularly spacedprotostars could also arise in the scenario we have described above;they are not necessarily a product of C&C.The masses estimated for the three main clumps are strongly de-pendent on whether the radiative transfer modeling includes theradiation from secondary sources or not (see Table 2 and section4.3). The variations can be as high as a factor of 8, as is the ca-se for clump C3 in Run 1 , for which the estimated mass increasesfrom 61 M ⊙ to 493 M ⊙ when the radiation from secondary sourcesis included. This is because the extra heating from newly-formedprotostars makes the dust in their vicinity hotter, and therefore mo-re material falls within the 0 . / beam threshold. µ m fluxes? In this section we compare how well the actual mass distributionin the simulations is recovered by applying Eq. 5 to the synthe-tic isophotal maps at 870 µ m. This comparison can provide usefulinsights into the reliability of clump mass estimates from observa-tional data.We define four masses, and list their values in Table 2. M is the mass obtained using Eq. 5 on synthetic isophotal maps cal-culated with radiation from the ionizing source only, and M ⋆ isthe mass obtained in the same way, but with secondary sources in-cluded. Likewise, M TRUE is the actual mass falling within a shell orclump on synthetic 870 µ m isophotal maps calculated with radiati-on from the ionizing source only, whilst M ⋆ TRUE is the correspondingquantity obtained when radiation from secondary sources is inclu-ded in the radiation transfer modelling. M TRUE and M ⋆ TRUE are obtai-ned by integrating the surface density of SPH particles over the areacovered by the shell or clump (i.e. the area inside the 0 . / beamor 0 . / beam contours, respectively).The resulting fractional errors, ǫ = (cid:12)(cid:12)(cid:12) M − M TRUE (cid:12)(cid:12)(cid:12) M TRUE , (6)and ǫ ⋆ = (cid:12)(cid:12)(cid:12) M ⋆ − M ⋆ TRUE (cid:12)(cid:12)(cid:12) M ⋆ TRUE , (7) c (cid:13) , 1– ?? omparing simulations and observations in RCW 120 Figure 4.
870 micron emission calculated using RADMC-3D. The frames on the left refer to
Run 1 , and those on the right to
Run 2 ; the frames on the toprow have been calculated with radiation from the ionizing source only, and those on the bottom row with radiation from secondary sources included.The leftcolumn shows the images for
Run 1 , and the right column for
Run 2 . The white contours are set to fluxes of 0 . / beam and 0 . / beam in all images. Eachframe is 12 pc ×
12 pc. are plotted against the true masses ( M TRUE , M ⋆ TRUE ) on Fig. 6. The shell masses obtained using Eq. 5 on synthetic 870 µ m isophotalmaps are always lower than the true masses, typically by a factor .
2, irrespective of whether the radiation from secondary sourcesis included or not. In contrast, the masses of individual clumps ob-tained using Eq. 5 on synthetic 870 µ m isophotal maps are someti-mes higher and sometimes lower than the true masses; for massiveclumps ( >
100 M ⊙ ) the fractional error is always less than two, butfor lower-mass clumps it can be very large, and such masses shouldprobably not be trusted.In general, masses obtained using Eq. 5 on synthetic 870 µ mmaps calculated including radiation from secondary sources aremore accurate than those obtained with radiation from the ioni-sing source only, as they should be. In order to quantify the ef-fect of including secondary sources, we have generated the mass-weighted distribution of temperature, both with and without se-condary sources; in addition, we have generated the correspondingdistributions for radiation transport calculations without the gas-temperature cut-o ff at T cut = Run 1 are shown in Fig. 7; similar results are obtained for
Run 2 .We note two features of this plot. First, when there is no cut-o ff , themean dust temperature is lower. This is because the dust in the H ii region absorbs much of the short wavelength radiation from the io-nising source and the newly-formed protostars, and then radiates itat long wavelengths, with the result that it escapes without heatingthe neutral gas further out. Second, when the secondary sources areincluded, the mean dust-temperature is almost exactly 30 K, as wehave assumed following the observational interpretation of RCW120. We have performed high resolution SPH simulations of H ii regi-ons expanding into fractal molecular clouds, and compared syn-thetic 870 µ m isophotal maps of these simulations (obtained usingR admc -3D) to 870 µ m observations of the well-studied, galactic H ii c (cid:13) , 1– ?? Walch et al.
Figure 5.
Mass distribution derived from the thermal dust emission at 870 µ m shown in Fig. 4. For comparison we overlay contours marking 0 . / beam and0 . / beam from Fig. 4. As in Fig. 4, the frames on the left refer to Run 1 , and those on the right to
Run 2 ; the frames on the top row have been calculatedwith radiation from the ionizing source only, and those on the bottom row with radiation from secondary sources included. The three largest clumps, whichare discussed in the text, are marked with C1, C2, and C3. Each frame is 12 pc ×
12 pc. region RCW 120. Our model reproduces a swept-up shell with amass between 3000 and 7000 M ⊙ . The shell contains several star-forming clumps, with masses between 30 and 700 M ⊙ . These coresare not formed by shell fragmentation.We suggest that finding massive clumps and sites of high massstar formation within an expanding shell around an H ii region neednot, and probably should not, be taken as evidence for the Col-lect & Collapse mechanism at work. The clumpiness of the denseshell can be attributed to density structures in the fractal, turbulent,molecular cloud into which the H ii regon expands. Two proces-ses combine to render the larger clumps gravitationally unstable.First the expanding H ii region sweeps lower-density gas into theclumps, and pushes the clumps outwards so they also sweep uplower-density gas further out. Second, the H ii region advances morerapidly through the lower-density gas around a clump, thereby en-veloping and squeezing it. In other words, a hybrid mechanism is atwork, which combines elements of C&C and RDI; it is not standardC&C because at no stage does a coherent shell form and then be-come gravitationally unstable and fragment, but it relates to C&C because the clumps do collect additional material, due to the ex-pansion of the H ii region. A detailed study of the growth of clumpsin the shells of expanding H ii regions is presented in Walch et al.(2013).Overall, we find good agreement (to within a factor of 2) betweenthe actual mass distribution and the mass distribution inferred fromthermal dust emission, for shells and clumps having masses grea-ter than ∼
100 M ⊙ . In particular, masses estimated from synthetic870 µ m isophotal maps always under-predict the mass of the thelarge-scale shell structure, whereas for individual clumps both over-and under-estimates are possible, and usually the result is accurateto within a factor of two. To obtain a realistic estimate of the shelland clump masses from synthetic 870 µ m isophotal maps, it is im-portant to take into account the heating from embedded protostars. c (cid:13) , 1– ?? omparing simulations and observations in RCW 120 Figure 6.
Fractional errors in the masses of shells and clumps obtainedusing Eq. 5 on synthetic isophotal maps, with and without radiation fromnewly-formed protostars (see Eqs. 6 and 7), plotted against their true mas-ses; a value of 1 is a perfect match. Open diamonds represent results from
Run 1 , and stars represent results from
Run 2 . Values pertaining to the shellare contained within the red ring; all the other values represent clumps.
Figure 7.
Mass-weighted dust temperature distribution for di ff erent R admc -3D models of Run 1 . All dust temperature distributions were derived using10 photon packages. The black line shows the fiducial case, where weinclude the emission from triggered protostars and cut the dust abundanceat gas temperatures above T cut = T cut ’)shows the case where also dust within the H ii region is included. Most of themass is found at T dust =
30 K. In the latter case the distribution shifts to aslightly colder mean temperature since less photons are available to heat thedust within the shell due to the increased absorption within the H ii region.The blue and pink lines show the resulting dust temperature distribution inthe case that only the emission from the central O-star is taken into account. ID M M ⋆ M TRUE M ⋆ TRUE M ⊙ M ⊙ M ⊙ M ⊙ Run 1
Shell 1818 2928 3031 4450C1 33 91 5.0 23.C2 75 146 114 262C3 61 493 32. 336
Run 2
Shell 3013 4420 6291 6853C1 193 302 129 228C2 74 184 114 270C3 160 783 89 655
Table 2.
Column 1 identifies the di ff erent structures analysed. Columns 2and 3 give mass estimates derived by applying Eq. 5 to the synthetic 870 µ mimages which were calculated with radiation from the ionizing source only, M , and with radiation from secondary sources included, M ⋆ (see Fig.5). Columns 4 and 5 give the true masses, obtained by integrating the SPHcolumn density over the area inside the 0 . / beam contour for shells, andthe 0 . / beam contour for the clumps. M TRUE is the mass within thesecontour levels calculated with radiation from the ionizing star only, whereas M ⋆ TRUE is the mass within these contour levels calculated with radiation fromthe secondary sources included. Both, M TRUE and M ⋆ TRUE account only forthe mass of gas and dust and do not include embedded protostars; the totalmass of embedded protostars is listed in Table 1.
ACKNOWLEDGMENTS
We thank the anonymous referees for constructive comments onthe manuscript. SW and AW acknowledge the support of the Ma-rie Curie RTN CONSTELLATION. SW further thanks the GermanScience Foundation for their support in the framework of the ISMpriority programme 1573 and through SFB 956 on ’The conditi-ons and impact of star formation’. The work of TGB was fundedby STFC grant ST / J001511 /
1. RW acknowledges support by theCzech Science Foundation grant 209 / / Literatur
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