Structure and Expansion Law of HII Regions in structured Molecular Clouds
Manuel Zamora-Avilés, Enrique Vázquez-Semadeni, Ricardo F. González, José Franco, Steven N. Shore, Lee W. Hartmann, Javier Ballesteros-Paredes, Robi Banerjee, Bastian Körtgen
MMNRAS , 1–16 (2018) Preprint 21 May 2019 Compiled using MNRAS L A TEX style file v3.0
Structure and Expansion Law of H II Regions in structuredMolecular Clouds
Manuel Zamora-Avil´es, , , (cid:63) Enrique V´azquez-Semadeni, Ricardo F. Gonz´alez, Jos´e Franco, Steven N. Shore, , , Lee W. Hartmann, Javier Ballesteros-Paredes, Robi Banerjee, Bastian K¨ortgen CONACYT-Instituto Nacional de Astrof´ısica, ´Optica y Electr´onica, Luis E. Erro 1, 72840 Tonantzintla, Puebla, M´exico Instituto de Radioastronom´ıa y Astrof´ısica, Universidad Nacional Aut´onoma de M´exico, Apdo. Postal 72-3 (Xangari), Morelia,Michoc´an 58089, M´exico Department of Astronomy, University of Michigan, 500 Church Street, Ann Arbor, MI 48105, USA Instituto de Astronom´ıa, Universidad Nacional Aut´onoma de M´exico, AP 70-264, CDMX, C.P. 04510, M´exico Dipartimento di Fisica “Enrico Fermi”, Universita’ di Pisa, Italy INFN Pisa, Largo B. Pontecorvo, 56127, PI, Italy Astronomical Institute, Charles University in Prague, V Holeˇsoviˇck´ach 2, 180 00, Praha 8, Czech Republic Hamburger Sternwarte, Universit¨at Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We present radiation-magnetohydrodynamic simulations aimed at studying evolution-ary properties of H II regions in turbulent, magnetised, and collapsing molecular cloudsformed by converging flows in the warm neutral medium. We focus on the structure,dynamics and expansion laws of these regions. Once a massive star forms in our highlystructured clouds, its ionising radiation eventually stops the accretion (through fila-ments) toward the massive star-forming regions. The new over-pressured H II regionspush away the dense gas, thus disrupting the more massive collapse centres. Also,because of the complex density structure in the cloud, the H II regions expand in ahybrid manner: they virtually do not expand toward the densest regions (cores), whilethey expand according to the classical analytical result towards the rest of the cloud,and in an accelerated way, as a blister region, towards the diffuse medium. Thus, theionised regions grow anisotropically, and the ionising stars generally appear off-centreof the regions. Finally, we find that the hypotheses assumed in standard H II -regionexpansion models (fully embedded region, blister-type, or expansion in a density gra-dient) apply simultaneously in different parts of our simulated H II regions, producinga net expansion law ( R ∝ t α , with α in the range of 0.93-1.47 and a mean value of1 . ± .
17) that differs from any of those of the standard models.
Key words: turbulence, magnetic fields – stars: formation –ISM: clouds –ISM: struc-ture –ISM: kinematics and dynamics – methods: numerical, magnetohydrodynamics,turbulence
Massive stars play a key role in the evolution of galaxies.Through a combination of massive outflows, expanding
Hii regions, and supernova explosions, they shape and providean important input of energy to the interstellar medium(ISM; e.g., Mac Low & Klessen 2004). In particular, theyerode and disperse their parent molecular clouds (MCs), di-rectly affecting the star formation activity within the clouds (cid:63)
E-mail:[email protected] (see, e.g., Krumholz et al. 2014; V´azquez-Semadeni 2015, forrecent reviews).It is generally thought that the negative feedback through blister-type Hii regions (or “champagne” flows; By negative feedback we refer to the suppression of star forma-tion by erosion of the dense regions where massive stars form.Similarly, we will refer to the promotion of star formation by ex-panding Hii regions as positive feedback , as in the classical collectand collapse scenario (Elmegreen & Lada 1977).c (cid:13) a r X i v : . [ a s t r o - ph . GA ] M a y Zamora-Avil´es et al.
Franco et al. 1990) is efficient in eroding and dispersingMCs on timescales of few tens of Myr (Blitz & Shu 1980;Matzner 2002). Idealised analytical (e.g., Whitworth 1979;Franco et al. 1994, hereafter, we will refer to the latter workas FST94) and numerical (e.g. Bodenheimer et al. 1979;Tenorio-Tagle 1979) works have shown that blister
Hii re-gions are also able to reduce the star formation efficiency(SFE) of MCs to the low observed values of (cid:46)
10% (e.g., My-ers et al. 1986). However, all these simplified models with-out self-gravity, with plane-parallel geometry and/or uni-form density fields are far from the complex morphologyand dynamics observed in MCs (see, e.g., Andr´e et al. 2013,for a recent review), which includes turbulence, magneticfields and anisotropic and hierarchical collapse, leading tothe formation of filamentary structures that funnel gas to thestar-forming sites (e.g., G´omez & V´azquez-Semadeni 2014;Smith et al. 2016; V´azquez-Semadeni et al. 2017, althoughsee Matzner & Jumper 2015 for an exception).Furthermore, the dynamics (free-fall motions or accre-tion) and the high-density environment of the birthplaces ofmassive stars could strongly attenuate the disruptive effectof the massive stars (Yorke et al. 1989; Dale et al. 2005;Peters et al. 2010). This picture gets more complicated ifwe take into account that MCs could be in global collapse,as recent and growing evidence shows (see, e.g., Hartmannet al. 2001; Burkert & Hartmann 2004; Hartmann & Burk-ert 2007; Peretto et al. 2007; V´azquez-Semadeni et al. 2007,2009; Galv´an-Madrid et al. 2009; Schneider et al. 2010; Csen-geri et al. 2011; Ballesteros-Paredes et al. 2011; Hartmannet al. 2012; Ballesteros-Paredes et al. 2015; Peretto et al.2014; Ju´arez et al. 2017).In a medium with power law density stratification r − p ,with p > /
2, Franco et al. (1990) showed analytically that
Hii regions expand in an accelerated way. Arthur & Hoare(2006), using numerical simulations of
Hii regions expandingin a stratified medium that decreases exponentially, showedthat a very weak shock develops toward the densest partof the cloud. This result is crucial in interpreting analyticalmodels in the literature (see App. A).At scales of MCs, recent numerical simulations (regard-less of the initial conditions or setup) of highly-structuredclouds have shown that negative feedback is able to reducethe SFE of MCs to values consistent with the observations(e.g., Dale et al. 2005; Walch et al. 2012; Col´ın et al. 2013;Geen et al. 2015). The effect of the ionising feedback dependsstrongly on the clouds’ masses and sizes, being weaker formore massive clouds (Dale et al. 2012). However, simulationsinvestigating this effect on clouds of various masses generallytend to assume idealised, spherical initial mass distributions,which are not necessarily realistic, since it is known thatclouds tend to be sheet-like and filamentary (e.g., Bally et al.1987; Andr´e et al. 2013). Thus, it is important to investigatethe effect of stellar feedback in realistically-shaped clouds,whose morphology is dictated self-consistently by their evo-lution since their formation.In contrast, several observational studies have at-tempted to infer empirical correlations between physicalproperties of
Hii regions, such as density and size. For ex-ample, Hunt & Hirashita (2009) studied extragalactic
Hii regions and compiled data from both Galactic (Garay &Lizano 1999; Kim & Koo 2001; Mart´ın-Hern´andez et al.2003; Dopita et al. 2006) and extragalactic (Kennicutt 1984; Gilbert & Graham 2007) samples, concluding that the en-tire sample follows a size ( R i ) versus ionised gas density( n i ) trend of the form n i ∝ R − , although with a consid-erable dispersion. Interestingly, combining this relation andthe Larson correlation ( n H ∝ R − . ; where n H and R H are the density and size of a given MC) these authors suggestthat the star formation is a scale-free process. However, therehave been claims in the literature that the Larson density-size correlation is only the result of a selection effect due tothe criteria used for defining a MC and their rapidly decay-ing column density PDFs (Ballesteros-Paredes et al. (2012),see also Kegel (1989); Scalo (1990); V´azquez-Semadeni et al.(1997); Ballesteros-Paredes & Mac Low (2002); Heyer et al.(2009); Camacho et al. (2016) for other possibilities).In order to study the evolutionary properties ofindividual Hii regions embedded in realistically struc-tured MCs, we present in this contribution radiation-magnetohydrodynamic (RMHD) simulations of MCs formedby converging flows, that evolve self-consistently, from theirformation to their destruction by ionising radiation. Becauseof this self-consistent evolution, the clouds also have a real-istic spatial structure, which, rather than spherical, is closerto being sheet-like, and highly inhomogeneous. In this work,we also study the time evolution of the size-density relationof the ionised gas.We organise the paper as follows. In §
2, we describethe numerical model. In § §
5. Finally, the summary andconclusions are presented in § With the goal of investigating the effect of magnetic fields inthe formation and evolution of MCs, in Zamora-Avil´es et al.(2018, hereafter Paper I) we presented three-dimensional,self gravitating, MHD simulations of MCs formed by twoWNM colliding flows, including heating and cooling pro-cesses. These simulations were carried out using the Eule-rian adaptive mesh refinement
FLASH (v2.5) code (Fryxellet al. 2000).In this contribution we consider one of those models (la-belled B3), in which the magnitude of the magnetic field (ini-tially uniform) is 3 µ G in our WNM-like initial conditions.This value is consistent with the observed mean value mag-netic field of the uniform component in the Galaxy (Beck2001). In addition, we present an additional simulation in-cluding radiative transfer, in order to model the effects ofUV feedback from massive stars. For this, we use the ra-diation scheme introduced by Rijkhorst et al. (2006) andimproved by Peters et al. (2010). This implementation hassuccessfully passed several tests. It accurately follows thevelocity propagation for R-type (in a cosmological context;Iliev et al. 2006) and D-type (corresponding to the analyt-ical Spitzer solution; Peters et al. 2010) ionisation fronts.In this section, we focus on the description of the radiationmodule and for further information we refer the reader toPaper I. See also Buntemeyer et al. (2016). MNRAS000
FLASH (v2.5) code (Fryxellet al. 2000).In this contribution we consider one of those models (la-belled B3), in which the magnitude of the magnetic field (ini-tially uniform) is 3 µ G in our WNM-like initial conditions.This value is consistent with the observed mean value mag-netic field of the uniform component in the Galaxy (Beck2001). In addition, we present an additional simulation in-cluding radiative transfer, in order to model the effects ofUV feedback from massive stars. For this, we use the ra-diation scheme introduced by Rijkhorst et al. (2006) andimproved by Peters et al. (2010). This implementation hassuccessfully passed several tests. It accurately follows thevelocity propagation for R-type (in a cosmological context;Iliev et al. 2006) and D-type (corresponding to the analyt-ical Spitzer solution; Peters et al. 2010) ionisation fronts.In this section, we focus on the description of the radiationmodule and for further information we refer the reader toPaper I. See also Buntemeyer et al. (2016). MNRAS000 , 1–16 (2018) xpansion of H II Regions in structured MCs As in Paper I, we use the so called constant mass crite-rion in order to follow the evolution of high density regions.According to this criterion, the grid size scales with den-sity as ∆ x ∝ ρ − / , and so we refine once the cell densityis eight times larger than in the previous level to guaran-tee that the mass of each cell is preserved. At the maxi-mum level of refinement a sink particle can be formed whenthe density in this cell exceeds a threshold number density, n thr (cid:39) . × cm − , among other standard sink-formationtests. Once the sink is formed, it can accrete mass from itssurroundings (Federrath et al. 2010).Note that this criterion is not standard given that itdoes not fulfil the Jeans criterion (Truelove et al. 1997),which states that artificial fragmentation can be avoided ifthe Jeans length is resolved with at least four grid cells.However, in Paper I we showed that the only effect of usingthe Jeans criterion rather than the constant mass one is aslight delay in the onset of star formation, leaving unchangedthe sink mass distribution, which determines the intensity ofthe UV feedback sources. The reason we chose the constantmass criterion over the Jeans one is that the refinement isconcentrated in the densest gas, which allows us to speed upthe calculations. Given the size of our numerical box and the maximum reso-lution we can achieve, the sink particles rapidly reach hun-dreds of solar masses via accretion, and therefore we mustnot treat them as single stars but rather as groups of stars.Thus, we assume a standard initial mass function (IMF) andwe estimate the most massive star that the sink can host,which dominates its UV flux. The sink radiates accordingthis flux. We use a Kroupa (2001)-type IMF, which reads χ ( m ) ∝ m − α i , (1)where α is a piecewise constant, and dN = χ ( m ) dm is thenumber of single stars in the mass interval m to m + dm .We normalise this IMF as (cid:90) M (cid:12) . M (cid:12) m χ ( m ) dm = M Sink , (2)where M sink represents the individual mass of the sink par-ticles. We take the standard lower and upper limits of 0 . M (cid:12) , respectively. We then integrate dN over bins of1 M (cid:12) to obtain the number of stars, ∆ N , in each mass bin.The centre of the last bin ( m ∗ ) satisfying ∆ N (cid:38) m ∗ , and we assume that this star dom-inates the emission of ionising photons from the sink. Inorder to save computational time, we assume that only sinks We do not take into account that massive stars have frequentlya companion of similar mass so this would have an effect on the
Hii region evolution. containing stars with masses (cid:38) M (cid:12) emit ionising radia-tion, since stars with lower masses ( < M (cid:12) ) do not emitsignificant amounts of photoionising photons. We allow themassive stars to radiate for 5 Myr. We use an adapted version of the hybrid characteristic ray-tracing module in the
FLASH code (Rijkhorst et al. 2006;Peters et al. 2010). The method can be summarised as fol-lows. To calculate the flux of ionising photons arriving ateach cell, the column density is calculated by interpolation(grid mapping) along rays from the point sources to everycell. Then the ionisation fractions and temperature can becomputed through an iterative process (under the assump-tion of radiative equilibrium), taking advantage of the an-alytic solution to the rate equation for the ionisation frac-tions. Furthermore, the heating/cooling can be iterated toconvergence (see Sec. 2.4), so that the only restriction onthe time-step comes from the MHD module. The MHD andionisation calculations are coupled through operator split-ting (see also Frank & Mellema 1994; Mellema & Lundqvist2002). Throughout this paper, we assume solar metallic-ity/abundances. For a detailed description about the feed-back implementation in FLASH code we refer the reader toPeters (2009); Peters et al. (2010).
Following Krumholz et al. (2007b), we calculate the heatingand cooling rates by breaking them into heating and cool-ing associated with ionisation of hydrogen atoms by pointsources (the sink particles), and other relevant sources ofcooling and heating. For the former, the photoionisation rateis (Osterbrock 1989)Γ ph = n HI (cid:90) ∞ ν T πJ ν hν σ ν h ( ν − ν T ) dν, (3)where n HI is the number density of atomic hydrogen, ν isthe frequency and ν T is the ionisation threshold frequency(at 13 . σ ν is the absorption cross section of atomichydrogen, and h is the Planck constant. The specific meanintensity, J ν , of a point source/star of radius r star and effec-tive temperature T star (assuming a blackbody spectrum) isgiven by (Rijkhorst et al. 2006; Peters et al. 2010) J ν ( r ) = (cid:18) r star r (cid:19) c hν exp( hν/k B T star ) − − τ ν ( r )) . (4)with τ ν ( r ) the optical depth at position r computed directlyfrom the column density, N ( r ). We also take into accountthe dust heating term (Γ d ) by non-ionising radiation (e.g.,Krumholz et al. 2007a; Peters et al. 2010). To counterbal-ance the photoionisation heating rate, Γ ph , we consider thecollisional cooling (ions-electrons), Λ col , which is the mainmechanism for energy loss in partially ionised gas (see, e.g.,Dalgarno & McCray 1972). which correspond to F UV (cid:39) . s − for a cluster of (cid:39) M (cid:12) . Note, however, that this period does not affect our results sincewe are interested on the properties of
Hii regions at the initialstages of evolution.MNRAS , 1–16 (2018)
Zamora-Avil´es et al.
Figure 1.
Sketch illustrating the initial conditions used in thiswork, which consist of two cylindrical streams colliding at the cen-tre of the numerical box. Figure adapted from K¨ortgen & Banerjee(2015).
For heating and cooling that are not directly due toionisation from the sink particles, we use the analytic fitsby Koyama & Inutsuka (2002) . for the heating (Γ KI ) andcooling (Λ KI ) functions,Γ KI = 2 . × − erg s − (5)Λ KI ( T )Γ KI = 10 exp − . × T + 1000 +1 . × − √ T exp − T cm , (6)which are based on the thermal and chemical calculationsconsidered by Wolfire et al. (1995); Koyama & Inutsuka(2000), including photoelectric heating from small grainsand PAHs, heating and ionisation by X-rays, cosmic rays,and H formation/destruction. Cooling processes includeatomic line emission from C II, O I, hydrogen Ly α , rota-tion/vibration line cooling from CO and H , and atomicand molecular collisions with dust grains.Thus, the net heating and cooling rates areΓ = Γ ph + Γ d + Γ KI , Λ = n e n HII Λ col + n Λ KI (7)where n HI , n e , and n HII refers to the number density ofneutral gas, electrons, and ionised gas, respectively.
We use the setup pictured in Fig. 1 for the initial conditions,which are as follows. The numerical periodic box, of sizes L x = 256 pc and L y = L z = 128 pc, is initially filled with See also V´azquez-Semadeni et al. (2007) for corrections to ty-pographical errors in the original source paper warm neutral gas at uniform density of 2 cm − and constanttemperature of 1450 K. We impose an initial background velocity field, whichcorresponds to a moderate turbulence with a power spec-trum of k − and Mach number of M rms (cid:39) .
7, whose mainrole is to trigger instabilities in the converging-flow we set upon top of this random field. This setup consists of two cylin-drical streams entirely contained in the numerical domain(see Fig. 1), each of radius R flow = 32 pc and length L flow =112 pc, moving in opposite directions at a moderately su-personic velocity of v flow = 7 . − in the x -direction.Thus, the inflow Mach number is M flow = 2 .
42 and the cor-responding dynamical time is t dyn = L flow /v flow = 14 . ∼ . × M (cid:12) ,whereas the mass contained in the cylinders is ∼ . × M (cid:12) (assuming a mean molecular weight µ = 1 . µ G along the x -direction. Thus, the correspondingmass-to-flux ratio in the cylinders is 1 .
59 times the criticalvalue, so that the cloud formed by the colliding flows even-tually will become magnetically supercritical once it accreteenough mass. The initial plasma beta parameter (i.e., thethermal to magnetic pressure ratio) is β th ≡ P th /P mag (cid:39) . M A = 1 .
3. We achieve a max-imum resolution of ∆ = 0 .
03 pc in all the three dimensions.With this setup we ran model B3 in Paper I (
Feedback-Off model hereafter). In this work, we restart this simula-tion to include ionising feedback from the point when thefirst massive star appears ( t ∼ . Feedback-On hereafter) and let it evolve for ∼ Hii regionsas well as the global feedback effects on the parent MC.
Henceforth, we focus our discussion on the simulation withfeedback (model Feedback-On). For a detailed descriptionabout the formation and evolution of the control simulation(model Feedback-Off) we refer the reader to Paper I.
In general, the clouds formed in our colliding flows simu-lations start as a thin cylindrical sheet of cold atomic gasproduced by the thermal instability (see, e.g., Hennebelle& P´erault 1999; Koyama & Inutsuka 2000, 2002; Walder &Folini 2000; V´azquez-Semadeni et al. 2006). The inflows nat-urally inject turbulence to the cloud through various dynam-ical instabilities (see, e.g., Hunter et al. 1986; Vishniac 1994;Koyama & Inutsuka 2002; Heitsch et al. 2005; V´azquez-Semadeni et al. 2006). The cloud continues accumulatingmass via accretion and eventually becomes gravitationallyunstable and begins to contract gravitationally, entering aregime of hierarchical, multi-scale collapse (Hartmann et al.2001; Heitsch & Hartmann 2008; V´azquez-Semadeni et al. This temperature corresponds to the thermal equilibrium andimplies an isothermal sound speed of c s (cid:39) . − . We used this model since its initial magnetic field (3 µG ) ismore consistent with the uniform component strength observedin the Galaxy (Beck 2001). MNRAS000
In general, the clouds formed in our colliding flows simu-lations start as a thin cylindrical sheet of cold atomic gasproduced by the thermal instability (see, e.g., Hennebelle& P´erault 1999; Koyama & Inutsuka 2000, 2002; Walder &Folini 2000; V´azquez-Semadeni et al. 2006). The inflows nat-urally inject turbulence to the cloud through various dynam-ical instabilities (see, e.g., Hunter et al. 1986; Vishniac 1994;Koyama & Inutsuka 2002; Heitsch et al. 2005; V´azquez-Semadeni et al. 2006). The cloud continues accumulatingmass via accretion and eventually becomes gravitationallyunstable and begins to contract gravitationally, entering aregime of hierarchical, multi-scale collapse (Hartmann et al.2001; Heitsch & Hartmann 2008; V´azquez-Semadeni et al. This temperature corresponds to the thermal equilibrium andimplies an isothermal sound speed of c s (cid:39) . − . We used this model since its initial magnetic field (3 µG ) ismore consistent with the uniform component strength observedin the Galaxy (Beck 2001). MNRAS000 , 1–16 (2018) xpansion of H II Regions in structured MCs Figure 2.
Column density maps in evolutionary sequence for the “central cloud” in the simulation with feedback (Feedback-On model).The upper and lower panels show face-on and edge-on views, respectively. The dots represent the projected position of the sink particles,where the white ones host massive stars. In panel 3 we annotate the name of the four
Hii regions we analyse in Sec. 3.2. The projectionscorrespond to the 80 pc central sub-box. See an animation of this figure in the supplementary material. t ∼ . t ∼ . ∼ . M (cid:12) ) and starts to radiate at t ∼ . t ∼ . t ∼ . Hii regions, such as pillars, elephanttrunks, and champagne flows (see, e.g., Hester et al. 1996).In Figs. 3 and 4 we show slices of the number density, tem-perature, pressure, and ionisation fraction for the first
Hii region that appears (region labeled “R1” in panel 3 of Fig.2) in two different projections and for three different earlytimes ( t = 13 .
2, 13.7, and 14.2 Myr). It is worth noting thatthe HII regions in these filamentary structures immersed ina WNM are far from spherical and properties such as sizeare strongly projection-dependent.In Fig. 5 we show 30-pc slices at t = 14 . y − z , z − x , and x − y , re-spectively. The first feature worth noting in the simulation without feedback (upper panels) is that the filament con-taining the massive star is perpendicular to the magneticfield lines (see blue streamlines in panels 2 and 3; see alsoG´omez et al. 2018). This is probably a consequence of theinitial conditions, however, it could be also the result of themagnetic field being oriented by the accretion flow from thecloud onto the filament (e.g., Zamora-Avil´es et al. 2017),rather than the field guiding the flow, as such morphologyis often interpreted.In contrast, in the simulation with feedback, the ionisedgas is over-pressured and escapes in a champagne flow to theWNM at velocities of tens of km s − (see panels 5 and 6 inFig. 5). Note that in the nearest 10 pc around the mas-sive star, the magnetic field lines are highly disorganised.However, beyond that distance, the magnetic field is quiteordered and tends to be aligned with the velocity field ofthe outflowing gas. This outflowing material is mostly com-posed of fully ionised gas, as the ionisation fraction contoursshow in panels 4-6 of Fig. 5 (purple, light-blue, and yellowcontours denote ionisation fractions of 0.1, 0.5, and 0.9, re-spectively). Although we have been discussing mostly themorphology and dynamics of region R1 (see panel 3 of Fig.2), the other regions have quite similar characteristics.Finally, it is worth mentioning that magnetic fields mayplay only a minor role in the evolution of Hii regions sincethe ram pressure dominates the dynamics of the ionised gas,as was found by Arthur et al. (2011). However, magneticfields are probably important for the dense gas surrounding
Hii regions, but this investigation is beyond the scope of thisstudy (see, e.g., Krumholz & Federrath 2019).
MNRAS , 1–16 (2018)
Zamora-Avil´es et al.
Figure 3.
Slices of the number density, temperature, pressure, and ionisation fraction (from top to bottom rows) in the y − z plane(face-on view) for the first Hii region (R1) at different times. Each slice is 30 pc wide and is centred at the position of the massive star.The contours in all panels delineate ionisation fractions of 0.5 (purple) and 0.9 (yellow). MNRAS000
Slices of the number density, temperature, pressure, and ionisation fraction (from top to bottom rows) in the y − z plane(face-on view) for the first Hii region (R1) at different times. Each slice is 30 pc wide and is centred at the position of the massive star.The contours in all panels delineate ionisation fractions of 0.5 (purple) and 0.9 (yellow). MNRAS000 , 1–16 (2018) xpansion of H II Regions in structured MCs Figure 4.
Same as Fig. 3 but in the x − y plane (edge-on view).MNRAS , 1–16 (2018) Zamora-Avil´es et al.
Figure 5.
Number density slices of the region “R1” at t = 14 . x , y , and z (left, middle, andright panels, respectively). The upper/lower panels correspond to the simulation without/with feedback (i.e., Feedback-Off/Feedback-Onmodels). The black arrows represent the projected velocity field (the vector scale is in the bottom left corner). Blue lines representmagnetic field streamlines. In the lower panels (Feedback-On model), the yellow, light-blue, and purple contours correspond to ionisingfractions of 0.9, 0.5, and 0.1, respectively. Figure 6.
Evolution of the ratio of minimum and maximum axes( µ ) for each region. Lower values of this ratio mean regions highlyanisotropic, while values tending to one correspond to more sym-metric regions. Table 1.
Properties of the analysed
Hii regions (see panel 3 ofFig. 2). Here, t is the time at which each massive star, of mass M ∗ , starts radiating; R , (cid:104) n (cid:105) , M I , and v rms are the radius, meandensity, ionised gas mass, and velocity dispersion of each Hii re-gion after ∼ Region M ∗ a t R (cid:104) n (cid:105) M I v rms name ( M (cid:12) ) (Myr) (pc) (cm − ) ( M (cid:12) ) (km s − )R1 15.2 12.8 14.5 0.9 358.5 11.2R2 10.1 13.1 3.2 5.3 23.6 14.8R3 9.9 13.4 3.8 2.5 17.4 16.0R4 12.7 14.3 3.7 8.5 38.1 12.4a Note that a once massive stars reach 8 M (cid:12) it starts to radi-ate. However, the radiation does not stop the mass accretionimmediately and the mass of the sink/massive star continuegrowing. Hii regions
We define an
Hii region around a massive star as the con-nected region around the ionising source where the ioni-sation fraction is greater than 0.5. We then calculate theevolutionary properties of individual
Hii regions as long asthey remain isolated and powered by a single massive star,which corresponds to roughly 1 Myr. After this time, the
MNRAS000
MNRAS000 , 1–16 (2018) xpansion of H II Regions in structured MCs Figure 7.
Time evolution of the radii of the four
Hii regions marked in panel 3 of Fig. 2. Grey dashed lines represent the best power-lawfit (the slopes are in the figure label). The solid red lines correspond to the classical Spitzer solution (eq. (A7) in the Appendix) forexpanding
Hii regions immersed in a uniform medium of density 10 , 10 , and 10 cm − , respectively, whereas the dashed red linestrace the blister-type solution (Eq. (9) in FST94; see also eq. (A3) in the Appendix), assuming the same cloud densities that in theSpitzer solution. Finally, The blue-shaded region represents the size range between the extremes of the distance between the star andthe ionisation front for each region. regions merge with each other or another massive star ap-pears. Thus, we study four Hii regions, which are markedin panel 3 of Fig. 2 as R1, R2, R3, and R4. In Table 1 welist the main properties of these regions after ∼ K, 12 km s − , and 1 µ G, respectively, for all re-gions.We also quantify the anisotropy level by computing thesize of the three principal axis. First we calculate the inertiamatrix I i,j = (cid:80) x i x j ∆ m , where x i and x j are the coor-dinates to every cell belonging to the Hii region. We thencalculate the eigenvectors of the inertia matrix, I i , whichrepresent the inertia momentum. The principal axes can becalculated as µ i = (cid:112) I i /M i , where M i is the ionised mass. In Fig. 6 we plot the ratio of minimum and maximum axes( µ = µ ) /µ ), and we can see that in general our regionsgrow anisotropically (with µ (cid:46) . Hii region (star symbols), which tends to be more symmetricafter ∼ . With the above definition of an
Hii region, and in orderto estimate a characteristic size of the
Hii regions, we firstcalculate the total volume ( V ) and then calculate the radiusas R i = V / . In Fig. 7 we show the evolution of the radiusof our four Hii regions. We have chosen the zero of time asthe moment when each massive star starts to radiate ( t inTable 1).In the same figure, we have over-plotted the Spitzersolution as well as the blister-type (FST94) estimation for MNRAS , 1–16 (2018) Zamora-Avil´es et al. the radius evolution of blister-type HII regions (solid anddashed red lines, respectively; see Appendix A). Note thatboth the Spitzer and the FST94 solutions describe only thegrowth of the cavity within the dense cloud, while our calcu-lations include the ionised gas that escapes into the diffuseintercloud medium. Thus, the analytic solutions should beconsidered only as guidelines. The actual flow is more similarto the “champagne” flow described by Franco et al. (1990),although without a unique expansion law because our cloudis filamentary and highly irregular rather than exhibiting asmooth stratification as assumed by those authors.Also, in Fig. 7, the blue-shaded regions represent thesize range between the minimum and maximum values ofthe distance between the star and the ionisation front forthe four
Hii regions studied. It is worth noting that theminimum values of this star-front distance occur in the di-rection toward the densest parts of the cloud (see, for exam-ple, panel 1 in Fig. 3). In this direction, the ionisation frontremains stationary, and very close to the star ( R i < ∼ . Hii regions we analysed. This imply that the simulated
Hii regions grow anisotropically, and thus the stars appearoff-centre of the regions during their evolution.In Fig. 7 we also show grey dashed lines that representthe best power-law fit to the mean radius of the regions, R i ,as a function of time. We find that R i ∝ t . ± . , where theuncertainty range represents the standard deviation amongour four regions. This implies that the expansion velocityand acceleration are proportional to ∼ t . and ∼ t − . ,respectively. Thus, once an ionised region breaks out from itshost filament, a blister-type flow is generated which expandsin an accelerated way towards the low density parts of theMC and the diffuse medium. This accelerated behaviour hasbeen reported by Franco et al. (1990) and Arthur & Hoare(2006) for Hii regions expanding in a stratified medium.
The left panel of Fig. 8 shows the mean density, (cid:104) n i (cid:105) , asa function of time of the ionised gas. The initial density ishigh (cid:38) − cm − , but the Hii regions deflate quickly asthey reach (and expand through) the WNM, lowering theirdensity by roughly three orders of magnitude after ∼ ∼ − − . As expected, the ionised mass, M i , of the Hii regions de-pends on the mass and luminosity of the ionising star (seeFig. 8; right panel). Region R1 is produced by the mostmassive star in our sample (15 . M (cid:12) ; see Table 1), and itionises a gas mass of 400 M (cid:12) , whereas the stars with massesaround 10 M (cid:12) (regions R2 and R3) only ionise ∼ M (cid:12) af-ter ∼ As mentioned in the introduction,
Hii regions observation-ally show a clear correlation between size ( R i ) and the elec-tronic density ( n i ) of the form n i ∝ R p i , with p (cid:39) −
1, over a wide dynamical range (see, e.g., Hunt & Hirashita 2009), al-though with a considerable dispersion of ∼ Hii regions in this size-densitydiagram. We find a tight correlation, although with a steeperslope, p (cid:39) −
2, for all our
Hii regions throughout their evolu-tion. Some possible explanations for this discrepancy in theslope are the lack of other feedback mechanisms (winds andsupernova) in our models, or the fact that our
Hii regionsare produced for a single massive star, whereas probablymost of the regions in the Hunt & Hirashita (2009) samplecontain more than one OB star.Note also that, although both the evolution of the sizeand density depend on the mass of the ionising star, thedensity-size relations do not. Indeed, all the
Hii regions areseen to occupy the same region in this diagram (see Fig. 9).This suggests that the position of a given
Hii region in thisdiagram can be explained in terms of its evolutionary statealone. On the other hand, this relation also implies that themass of the
Hii regions roughly depends linearly on theirsize; that is, M i ∝ R i , assuming constant density. Hii
REGIONS
Our simulations, including a physically reasonable treat-ment of the radiative transfer, provide an ideal means for in-vestigating the structure of the
Hii regions and their bound-aries, especially since there exist various models in the lit-erature for the expansion of these regions which are basedon different assumptions of their boundary conditions andinternal structure.The expansion of
Hii regions into their parent MCs de-pends on whether they are fully embedded in the MC, con-stituting the “canonical”
Hii regions (e.g., Spitzer 1978), orbreak out of the dense clouds and begin expanding into thediffuse medium, constituting the so-called “blister” type of
Hii regions (e.g., Whitworth (1979); Franco et al. (1990);FST94). As summarised in Appendix A, during their dy-namical expansion phase, canonical
Hii regions are over-pressured relative to the cloud material, and are thereforebounded by a shock front, followed downstream by an ioni-sation front, with a shocked, dense layer in between the twofronts. Furthermore, as a consequence of the overpressure inthe region and the dynamical expansion it produces, whichreduces the region’s internal density, the region follows anexpansion law of the form R ∼ t / .Instead, blister-type regions are characterised by a lossof pressure by means of the “release valve” provided by thedischarge to the diffuse medium. Therefore, FST94 arguedthat these regions are essentially at the same pressure astheir parent cloud and thus do not produce a shock frontupstream of the ionisation front. This in turn implies thatthe region is bounded, on its interface with the cloud, bya single ionisation front, with no preceding shock front andno compressed neutral layer upstream of it. Also, because inthis case the region is not overpressured, its growth is con-trolled only by the rate at which the mass inflow through the This is analogous to standard thermal stellar winds or jetstreams (de Laval nozzle), which get accelerated to supersonicvelocities at the critical breakout point (the nozzle ”throat”) ifthe pressure is not balanced. MNRAS000
Hii regions are over-pressured relative to the cloud material, and are thereforebounded by a shock front, followed downstream by an ioni-sation front, with a shocked, dense layer in between the twofronts. Furthermore, as a consequence of the overpressure inthe region and the dynamical expansion it produces, whichreduces the region’s internal density, the region follows anexpansion law of the form R ∼ t / .Instead, blister-type regions are characterised by a lossof pressure by means of the “release valve” provided by thedischarge to the diffuse medium. Therefore, FST94 arguedthat these regions are essentially at the same pressure astheir parent cloud and thus do not produce a shock frontupstream of the ionisation front. This in turn implies thatthe region is bounded, on its interface with the cloud, bya single ionisation front, with no preceding shock front andno compressed neutral layer upstream of it. Also, because inthis case the region is not overpressured, its growth is con-trolled only by the rate at which the mass inflow through the This is analogous to standard thermal stellar winds or jetstreams (de Laval nozzle), which get accelerated to supersonicvelocities at the critical breakout point (the nozzle ”throat”) ifthe pressure is not balanced. MNRAS000 , 1–16 (2018) xpansion of H II Regions in structured MCs Figure 8.
Time evolution of the mean number density of ionised gas (left panel) and ionised gas mass (right panel) for the four
Hii regions.
Figure 9.
Size-density diagram for the four simulated
Hii
Re-gions. The blue shadow brackets the radius extremes for the re-gion R1 at each time (see upper-left panel of Fig. 7). The averageof the slopes is 2.0, i.e., n i ∝ R − . The red symbols correspondto the time at which each Hii region turns on (at higher densityand smaller sizes). ionisation front balances the mass loss to the diffuse medium(see Fig. 1 of FST94), resulting in a slower expansion rateinto the cloud of the form R ∼ t / .This conclusion, together with the associated massphoto-evaporation rate of the cloud predicted by FST94has been questioned by Matzner & Jumper (2015), who ar-gue that FST94 “incorrectly associate swept-up mass withionised mass.” This is because, according to Matzner (2002),most of the mass in the Hii region is in the shocked, com-pressed layer upstream of the ionisation front. Moreover,the criticism by Matzner & Jumper (2015) indirectly af-fects the accuracy of a model we have presented in pre-vious papers for the collapse of MCs and their SF activ- ity (Zamora-Avil´es et al. 2012; Zamora-Avil´es & V´azquez-Semadeni 2014), which was based on the prescription byFST94. However, this criticism only applies if blister
Hii re-gions develop a shock at the interface with the MC. If noshock is present, then there is no swept-up, dense layer, andso all the material in the region is indeed ionised.Our results, described in the previous sections, allowus to investigate what is the actual nature of the boundarybetween the
Hii region and the cloud, since the radiativetransfer and the cooling rates are followed self-consistently,and thus the ionisation fraction and the gas structure are re-alistic, including the presence and position of the ionisationfront and the shock. In Figs. 3 and 4 we show the contourswhere the ionisation fraction χ = 0 . χ = 0 . Hii regions, we plot in Fig. 10 profiles of thedensity, temperature, pressure, velocity ( y -component), andionization fraction for the R1 and for three different timesin the frame of reference of the massive star. No evidence ofa shock upstream of the ionization front is seen at the edgesof the R1 region where it meets with the densest gas, unlessit is too weak to be detected. Note that this behaviour (a Note that we have enough resolution to detect a shock.In the context of stellar winds, for a moderate shock, with ashock velocity v s ≤
80 km s − , the shock width can be approxi-mated by 1 . v − . s, n − , AU (Hartigan et al. 1987; Gonz´alez2002), where v s, is the shock velocity in units of 100 km s − and n pre , the preshock density in units of 100 cm − (see alsoGonz´alez et al. 2004). According to this approximation and usingtypical values of v s = 10 km s − and n pre = 10 cm − we expecta shock width of 0.02 pc, which is comparable to our resolution.MNRAS , 1–16 (2018) Zamora-Avil´es et al.
Figure 10.
Density, temperature, pressure, velocity, and ionisation fraction profiles (from top to bottom) for the R1 in the y -axis attimes 12.9, 13, and 13.1 Myrs (left, middle, and right panels, respectively). The zero point corresponds to the position of the massivestar (vertical thin black line). The dotted black and red vertical lines represent the shock and ionisation fronts, respectively. very weak shock toward the densest part) is also reportedby Arthur & Hoare (2006) in a stratified density field.On the other hand, we detect a clear shock toward thegeneral cloud (marked with vertical dotted black lines inFig. 10) ahead of the ionisation front (right vertical dottedred lines) due to the hydrodynamical expansion of the Hii re-gion. Thus, our
Hii regions expand in a hybrid way: towardsthe general cloud we detect a moderate shock, whereas wedo not detect any shock signature (or it is too weak to bedetected) toward the dense clump.Furthermore, this region also exhibits features of theblister type, as it has ample sections where it connects di-rectly to the warm, diffuse gas. In these sections, it expandsin an accelerated manner, in qualitative agreement with theanalytical predictions of Franco et al. (1990), although, dueto the non-unique density stratification, our region does notexhibit a unique acceleration law. Thus, we conclude thatneither the classical t / nor the FST94 t / expansion lawsapply to our regions, which instead expand in an accelerated I.e., toward the rest of the cloud, except dense region wherethe massive star was born. way as R i ∼ t . , in qualitative agreement with Franco et al.(1990). Hunt & Hirashita (2009) interpreted the observed size-density relation for
Hii regions ( n i ∝ R − ) as being a conse-quence of the observed Larson relation for molecular clouds( n H ∝ R − H ), suggesting that Hii regions retain an imprintfrom the molecular environment in which they form. How-ever, this Larson relation has been repeatedly questioned,since it may be the result of various selection effects, ratherthan real property of MCs (e.g., Kegel 1989; Scalo 1990;V´azquez-Semadeni et al. 1997; Ballesteros-Paredes & MacLow 2002; Ballesteros-Paredes et al. 2012). Indeed, numeri-cal simulations of MC formation and evolution do not showany evidence for the appearance of such a density-size scal-ing appearing in the clouds unless a selection criterion ofroughly constant column density is imposed (Ballesteros-Paredes & Mac Low 2002; Camacho et al. 2016). On the
MNRAS000
MNRAS000 , 1–16 (2018) xpansion of H II Regions in structured MCs other hand, our results strongly suggest that individual Hii regions do follow an evolutionary path in the n i − R i dia-gram. This evidence suggests that, the size-density scalingof Hii regions may be a real effect, although, contrary to theinterpretation of Hunt & Hirashita (2009), unrelated to thedensity structure of the parent MC.
Analytical models of blister-type expansion of
Hii regionsare based on different hypotheses. While FST94 assume thatthe ionization front advances with no shock towards thedense region of the cloud, Matzner (2002) assumes that it isthrough this shock that the
Hii region incorporates most ofits mass (see App. A). However, in the present study we haveshown that both of these hypotheses are satisfied at the sametime in different parts of the
Hii regions. The assumptionby FST94 is valid toward the dense gas (where the massivestar was born), while the assumption of Matzner (2002) canbe applied toward the rest of the cloud. On the other hand,the
Hii regions we analysed expand in an accelerated waytoward the low density gas (WNM) as predicted by Francoet al. (1990) in a stratified medium (see also Arthur & Hoare2006).Although the structure and dynamics of massive star-forming regions are quite complicated and far from theidealised geometries assumed in analytical models, an ap-proximate comparison is possible. The size-density relation( n i ∝ R − . ) implies that the ionised gas mass, M i ∝ R i , orequivalently, ˙ M i ∝ ˙ R i . As we found in § Hii regionradius depends roughly on time as R i ∝ t . . Therefore,the mass ionisation rate is ˙ M i ∝ t . . Note that this depen-dence is quite close to ˙ M i ∝ t / predicted by FST94 (Eq.A4) for blister type Hii regions. In turn, this implies thatthe blister-
Hii region approximation used in the analyticalmodel by Zamora-Avil´es et al. (2012) is justified.On the numerical side, several works have included ion-ising feedback (e.g., Dale et al. 2005; Col´ın et al. 2013; Geenet al. 2015), although they only study global properties ofMCs, rather than the internal structure of the
Hii regions.On pc scales, Arthur et al. (2011) studied the evolution of in-dividual
Hii regions, albeit only during the embedded phase,so we cannot compare directly our findings with these works.In any case, they showed that the presence of a magneticfield does not modify the
Hii region structure in a funda-mental way. Finally, although we have investigated only onesimulation, our results can be considered robust since oursimulated MC is highly chaotic, due to the turbulence in-jected in a self-consistent way, and therefore the idealisedinitial conditions are quickly forgotten. Moreover, we haveinvestigated a sample of four
Hii regions that span a range ofbehaviours, from which we have extracted some meaningfulaverages.
As we are interested in studying the effect of the ionisingfeedback, in this study we have neglected other feedbackprocesses, such as winds, supernova explosions and radia-tion pressure, which can help disperse the cloud and furtherreduce both the SFR and the SFE. However, these feedback effects can also enhance the positive feedback through theinjection of mechanical energy, and can possibly bring theslope of the size-density relation of our
Hii regions closer tothe observed value. Thus, the net effect of these feedbackmechanisms is so far unclear, and more numerical researchis necessary in order to study their relative importance (see,e.g., K¨ortgen et al. 2016; Wareing et al. 2017).Also, our spatial and temporal output resolutions areinsufficient to resolve the stages of hyper-compact/compact Hii regions. Particularly, we do not resolve the initialStr¨omgren radius in our star formation regions. However,this is not an issue since we are interested in studying theexpansion of
Hii regions at the scale of the MCs, as a meansof destroying them. Although our numerical simulations in-clude magnetic fields, it is still necessary to assess its detailedinfluence on the feedback processes through comparison withnon-magnetic simulations, a task we defer to a future con-tribution.Finally, the FLASH radiative transport module doesnot take into account the absorption of UV photons by dustgrains inside the
Hii regions. Also, models by Hosokawa &Omukai (2009) suggest that the ZAMS model we use canoverestimate the ionising luminosity. Both effects tend tooverestimate the number of UV photons emitted by a mas-sive star, so properties of
Hii regions such as size shouldbe taken as upper limits (see Peters et al. 2010, for a de-tailed discussion). However, the power law index of the ra-dius growth should remain unchanged, as well as their as-sociated relations. In addition, the effects of molecular hy-drogen destruction by photodissociation from massive andintermediate-mass stars is not considered here, and, as dis-cussed by Diaz-Miller et al. (1998), this effect makes thecloud evaporation process more effective.
In this paper, we have studied the evolution of the physicalproperties of
Hii regions in radiation-MHD simulations ofMCs formed by diffuse converging flows in the presence ofmagnetic fields and massive-star ionisation feedback.Our simulations reproduce the rich morphology ob-served around
Hii regions, such as elephant trunks,cloudlets, champagne flows, etc. Due to the highly com-plex structure of our filamentary clouds, the
Hii regionsgrow anisotropically, causing the massive stars to appearoff-centre of the ionised regions. Our simulated
Hii regionsexpand in a hybrid way in our filamentary clouds: towardsthe intermediate-density regions in the cloud, the
Hii regionsexpand according to the classical theory, developing a shockahead of the ionisation front, while towards the densest partsof the cloud, no shock is apparent, and the ionisation frontstalls in 3 out of 4 cases, and advances according to classicaltheory in the other case. Finally, towards the diffuse, warmgas, the regions expand roughly according to the “blister” Corresponding to 0.03 pc and 0.1 Myr, respectively. The tem-poral resolution refers to the time interval between output snap-shots. The typical sizes and life times of compact
Hii regions are ≤ . ∼ , 1–16 (2018) Zamora-Avil´es et al. case, at an accelerated pace. On average, the radius of theregions is dominated by this latter mode, so that the averageradius grows at an accelerated pace, roughly as R i ∼ t . .Our Hii regions exhibit a tight relation between size andaverage density, (cid:104) n i (cid:105) , of the form (cid:104) n i (cid:105) ∝ R − . This impliesthat, on average, the mass ionisation rate is ˙ M i ∝ t . ,which is in good agreement with the analytical predictionby FST94 ( ˙ M i ∝ t . ). Therefore, we conclude that the an-alytic prescription for the rate of mass ionisation used inthe model by Zamora-Avil´es et al. (2012) is adequate, con-trary to Matzner & Jumper (2015). Interestingly, the elec-tron density-size relation we observe in our Hii regions isthe result of the expansion mechanism itself, rather thanan imprint of a Larson-type relation in their environment,which, in addition, is not observed in general in this kind ofsimulations.
ACKNOWLEDGEMENTS
We thank the anonymous referee for helpful comments andsuggestions, which helped to improve this manuscript. Wealso gratefully acknowledge useful comments from GuillermoTenorio-Tagle. MZA and EVS acknowledge financial sup-port from CONACYT grant number 255295 to EVS. RGacknowledges UNAM-PAPIIT grant number IN112718. Theresearch of LH was supported in part by NASA grantNNX16AB46G. JBP acknowledges UNAM-PAPIIT grantnumber IN110816. The visualization was carried out withthe yt software (Turk et al. 2011). The FLASH code usedin this work was in part developed by the DOE NNSA-ASCOASCR Flash Center at the University of Chicago. The au-thors thankfully acknowledge computer resources, technicaladvise and support provided by Laboratorio Nacional de Su-perc´omputo del Sureste de M´exico (LNS), a member of theCONACYT network of national laboratories.
APPENDIX A: DYNAMICAL EVOLUTION OF
Hii
REGIONS
The different phases of the evolution of an
Hii region havebeen studied in several previous works (see, e.g., Spitzer1978; Whitworth 1979; Dyson & Williams 1980) At first,a
Hii region expands until the ionisation balance is reached,when the ionisation front is located at the initial Str¨omgrenradius R . Afterwards, the Hii region enters into a secondstage of dynamical evolution as long as the ionised gas hasa higher pressure than the neutral ambient medium. In theparticular case of a massive star located near the surface of amolecular cloud, the
Hii region is radiation-bounded on theinner part of the cloud, and density-bounded on the outerpart (e.g., Whitworth 1979). Consequently, a cometary
Hii region is produced, in which the ionised gas flows into thelow-density medium.FST94 estimated the maximum number of massive starsthat can form within a molecular cloud. These authorspointed out that the most efficient destruction mechanismis the evaporation of the cloud by stars located near thecloud’s boundary. In that case, the growth of a
Hii regioninside the cloud is due to the mass flux ( a blister-type massloss) that expands into the external low-density medium n at a velocity equal to the sound speed c I in the Hii region.In their model, it is assumed that the mass loss into the en-vironment is equal to the mass gained by the expansion ofthe
Hii region inside the cloud, that is, πR i m p (cid:104) n i (cid:105) c s = 2 πR m p n ˙ R i , (A1)being R i is the position of the ionisation front, (cid:104) n i (cid:105) the meandensity of the ionised gas, c s the initial expansion speed ofthe ionised gas into the external low-density medium (whichis assumed equal to the sound speed in the ionised gas), and m p the proton mass.Consequently, the velocity of the ionisation front intothe cloud is given by,˙ R i (cid:39) (cid:104) n i (cid:105) n c s , (A2)Considering that the total ionised mass remains con-stant, it follows that the position of the ionisation front ata time t within the cloud is obtained by, R i (cid:39) R (cid:32) c s tR (cid:33) / . (A3)For this, these authors neglected the effects of the weakshock of the expanding Hii region due to the mass loss fromthe blister, and pointed out that both the mass and ionisa-tion balance determines the evolution.Assuming a constant luminosity during the main-sequence stage of the star, the cloud evaporation rate in-duced by a single star located near the cloud’s boundarycalculated by these authors is given by,˙ M i ( t ) (cid:39) π R m p c s n (cid:32) c s tR (cid:33) / . (A4)On the other hand, Matzner & Jumper (2015) arguedthat this photoevaporation rate incorrectly associates theswept-up mass with ionised mass. In another work, Matzner(2002) presented a simple treatment of the momentum gen-eration by an Hii region. This author takes into accountthe inertia of the dense shell produced by the shock mov-ing into the cloud. In this stage of evolution ( R i > R ), theexpansion of the ionisation front is caused by the pressuregradient between the Hii region and the environment of neu-tral gas. It is assumed by the author that nearly all of themass originally located inside the radius R i remains withinthe shocked shell. According to this author, if the ionisedgas is effectively isothermal in blister regions, the ionisationfront tends to a D-critical case, for which, u II − ˙ r II = − c II (A5)where u II is the velocity of the ionised gas relative to the It is worth to mention that this assumption represents an im-portant difference with respect to the model developed by FST94in which the effects of the compressive effects are neglected dueto the mass loss from the blister. Hereafter, we follow the notation used in Matzner (2002).MNRAS000
Hii region inside the cloud, that is, πR i m p (cid:104) n i (cid:105) c s = 2 πR m p n ˙ R i , (A1)being R i is the position of the ionisation front, (cid:104) n i (cid:105) the meandensity of the ionised gas, c s the initial expansion speed ofthe ionised gas into the external low-density medium (whichis assumed equal to the sound speed in the ionised gas), and m p the proton mass.Consequently, the velocity of the ionisation front intothe cloud is given by,˙ R i (cid:39) (cid:104) n i (cid:105) n c s , (A2)Considering that the total ionised mass remains con-stant, it follows that the position of the ionisation front ata time t within the cloud is obtained by, R i (cid:39) R (cid:32) c s tR (cid:33) / . (A3)For this, these authors neglected the effects of the weakshock of the expanding Hii region due to the mass loss fromthe blister, and pointed out that both the mass and ionisa-tion balance determines the evolution.Assuming a constant luminosity during the main-sequence stage of the star, the cloud evaporation rate in-duced by a single star located near the cloud’s boundarycalculated by these authors is given by,˙ M i ( t ) (cid:39) π R m p c s n (cid:32) c s tR (cid:33) / . (A4)On the other hand, Matzner & Jumper (2015) arguedthat this photoevaporation rate incorrectly associates theswept-up mass with ionised mass. In another work, Matzner(2002) presented a simple treatment of the momentum gen-eration by an Hii region. This author takes into accountthe inertia of the dense shell produced by the shock mov-ing into the cloud. In this stage of evolution ( R i > R ), theexpansion of the ionisation front is caused by the pressuregradient between the Hii region and the environment of neu-tral gas. It is assumed by the author that nearly all of themass originally located inside the radius R i remains withinthe shocked shell. According to this author, if the ionisedgas is effectively isothermal in blister regions, the ionisationfront tends to a D-critical case, for which, u II − ˙ r II = − c II (A5)where u II is the velocity of the ionised gas relative to the It is worth to mention that this assumption represents an im-portant difference with respect to the model developed by FST94in which the effects of the compressive effects are neglected dueto the mass loss from the blister. Hereafter, we follow the notation used in Matzner (2002).MNRAS000 , 1–16 (2018) xpansion of H II Regions in structured MCs cloud, ˙ r II is the expansion speed of the ionisation front, and c II is the sound speed of the ionised gas. Consequently, therate at which mass is ionised is given by,˙ M II = ρ II ( ˙ r II − u II ) 2 πr II (A6)where ρ II is the density within the Hii region.Since the
Hii region expands supersonically with respectto the molecular gas, it is bounded by a thin shocked layerthat is located near the ionisation front. Assuming a radialexpansion, Matzner (2002) found a self-similar solution for r II (cid:29) R St,0 (with R St,0 the initial Str¨omgren radius) givenby, r II = 496 R / St,0 ( c II t ) / (A7)Considering the momentum of the radial motion of the ex-panding shell, the mass evaporated for a blister region canbe estimated by, δM d est = 1 . × (cid:32) t . yr (cid:33) / (cid:32) N H , . (cid:33) − / (A8) × M / cl,6 S / M (cid:12) , where N H , is the mean hydrogen column density in unitsof 10 cm − , M cl,6 is the mass of the cloud in units of 10 M (cid:12) , and S is the ionising photons’ rate in units of 10 photons per second. According to this author, equation [A8]for the mass evaporated agrees with predictions by Williams& McKee (1997) who argued that only 10% of the mass ofa GMC becomes stellar, within 1%. REFERENCES
Andr´e P., Di Francesco J., Ward-Thompson D., Inutsuka S.-i.,Pudritz R. E., Pineda J., 2013, preprint, ( arXiv:1312.6232 )Arthur S. J., Hoare M. G., 2006, The Astrophysical Journal Sup-plement Series, 165, 283Arthur S. J., Henney W. J., Mellema G., de Colle F., V´azquez-Semadeni E., 2011, MNRAS, 414, 1747Ballesteros-Paredes J., Mac Low M.-M., 2002, ApJ, 570, 734Ballesteros-Paredes J., Hartmann L. W., V´azquez-Semadeni E.,Heitsch F., Zamora-Avil´es M. A., 2011, MNRAS, 411, 65Ballesteros-Paredes J., D’Alessio P., Hartmann L., 2012, MNRAS,427, 2562Ballesteros-Paredes J., Hartmann L. W., P´erez- Goytia N.,Kuznetsova A., 2015, MNRAS, 452, 566Bally J., Langer W. D., Stark A. A., Wilson R. W., 1987, ApJL,312, L45Beck R., 2001, Space Sci. Rev., 99, 243Blitz L., Shu F. H., 1980, ApJ, 238, 148Bodenheimer P., Tenorio-Tagle G., Yorke H. W., 1979, ApJ, 233,85Buntemeyer L., Banerjee R., Peters T., Klassen M., Pudritz R. E.,2016, New Astron., 43, 49Burkert A., Hartmann L., 2004, ApJ, 616, 288Camacho V., V´azquez-Semadeni E., Ballesteros-Paredes J.,G´omez G. C., Fall S. M., Mata-Ch´avez M. D., 2016, ApJ,833, 113Col´ın P., V´azquez-Semadeni E., G´omez G. C., 2013, MNRAS,435, 1701Csengeri T., Bontemps S., Schneider N., Motte F., Dib S., 2011,A& A, 527, A135 Dale J. E., Bonnell I. A., Clarke C. J., Bate M. R., 2005, MNRAS,358, 291Dale J. E., Ercolano B., Bonnell I. A., 2012, MNRAS, 424, 377Dalgarno A., McCray R. A., 1972, ARA&A, 10, 375Diaz-Miller R. I., Franco J., Shore S. N., 1998, ApJ, 501, 192Dopita M. A., et al., 2006, ApJ, 639, 788Dyson J. E., Williams D. A., 1980, Physics of the interstellarmediumElmegreen B. G., Lada C. J., 1977, ApJ, 214, 725Federrath C., Banerjee R., Clark P. C., Klessen R. S., 2010, ApJ,713, 269Franco J., Tenorio-Tagle G., Bodenheimer P., 1990, ApJ, 349, 126Franco J., Shore S. N., Tenorio-Tagle G., 1994, ApJ, 436, 795Frank A., Mellema G., 1994, A& A, 289, 937Fryxell B., et al., 2000, ApJS, 131, 273Galv´an-Madrid R., Keto E., Zhang Q., Kurtz S., Rodr´ıguez L. F.,Ho P. T. P., 2009, ApJ, 706, 1036Garay G., Lizano S., 1999, PASP, 111, 1049Geen S., Hennebelle P., Tremblin P., Rosdahl J., 2015, MNRAS,454, 4484Gilbert A. M., Graham J. R., 2007, ApJ, 668, 168G´omez G. C., V´azquez-Semadeni E., 2014, ApJ, 791, 124G´omez G. C., V´azquez-Semadeni E., Zamora-Avil´es M., 2018,MNRAS, p. 1924Gonz´alez R., 2002, PhD thesis, Universidad Nacional Aut´onomade M´exicoGonz´alez R. F., de Gouveia Dal Pino E. M., Raga A. C., Vel´azquezP. F., 2004, ApJ, 616, 976Hartigan P., Raymond J., Hartmann L., 1987, ApJ, 316, 323Hartmann L., Burkert A., 2007, ApJ, 654, 988Hartmann L., Ballesteros-Paredes J., Bergin E. A., 2001, ApJ,562, 852Hartmann L., Ballesteros-Paredes J., Heitsch F., 2012, MNRAS,420, 1457Heitsch F., Hartmann L., 2008, ApJ, 689, 290Heitsch F., Burkert A., Hartmann L. W., Slyz A. D., DevriendtJ. E. G., 2005, ApJL, 633, L113Hennebelle P., P´erault M., 1999, A& A, 351, 309Hester J. J., et al., 1996, AJ, 111, 2349Heyer M., Krawczyk C., Duval J., Jackson J. M., 2009, ApJ, 699,1092Hosokawa T., Omukai K., 2009, ApJ, 703, 1810Hunt L. K., Hirashita H., 2009, A& A, 507, 1327Hunter Jr. J. H., Sandford II M. T., Whitaker R. W., Klein R. I.,1986, ApJ, 305, 309Iliev I. T., et al., 2006, MNRAS, 371, 1057Ju´arez C., et al., 2017, ApJ, 844, 44Kegel W. H., 1989, A& A, 225, 517Kennicutt Jr. R. C., 1984, ApJ, 287, 116Kim K.-T., Koo B.-C., 2001, ApJ, 549, 979K¨ortgen B., Banerjee R., 2015, MNRAS, 451, 3340K¨ortgen B., Seifried D., Banerjee R., V´azquez-Semadeni E.,Zamora-Avil´es M., 2016, MNRAS, 459, 3460Koyama H., Inutsuka S. I., 2000, ApJ, 532, 980Koyama H., Inutsuka S. I., 2002, ApJL, 564, L97Kroupa P., 2001, MNRAS, 322, 231Krumholz M. R., Federrath C., 2019, Frontiers in Astronomy andSpace Sciences, 6, 7Krumholz M. R., Klein R. I., McKee C. F., 2007a, ApJ, 656, 959Krumholz M. R., Stone J. M., Gardiner T. A., 2007b, ApJ, 671,518Krumholz M. R., et al., 2014, Protostars and Planets VI, pp 243–266Mac Low M.-M., Klessen R. S., 2004, Reviews of Modern Physics,76, 125Mart´ın-Hern´andez N. L., van der Hulst J. M., Tielens A. G. G. M.,2003, A& A, 407, 957Matzner C. D., 2002, ApJ, 566, 302MNRAS , 1–16 (2018) Zamora-Avil´es et al.
Matzner C. D., Jumper P. H., 2015, ApJ, 815, 68Mellema G., Lundqvist P., 2002, A& A, 394, 901Myers P. C., Dame T. M., Thaddeus P., Cohen R. S., SilverbergR. F., Dwek E., Hauser M. G., 1986, ApJ, 301, 398Osterbrock D. E., 1989, Astrophysics of gaseous nebulae and ac-tive galactic nucleiPaxton B., 2004, PASP, 116, 699Peretto N., Hennebelle P., Andr´e P., 2007, A& A, 464, 983Peretto N., et al., 2014, A& A, 561, A83Peters T., 2009, PhD thesis, University of HeidelbergPeters T., Banerjee R., Klessen R. S., Mac Low M.-M., Galv´an-Madrid R., Keto E. R., 2010, ApJ, 711, 1017Rijkhorst E.-J., Plewa T., Dubey A., Mellema G., 2006, A& A,452, 907Scalo J., 1990, in Capuzzo-Dolcetta R., Chiosi C., di Fazio A., eds,Astrophysics and Space Science Library Vol. 162, PhysicalProcesses in Fragmentation and Star Formation. pp 151–176Schneider N., Csengeri T., Bontemps S., Motte F., Simon R.,Hennebelle P., Federrath C., Klessen R., 2010, A& A, 520,A49Smith R. J., Glover S. C. O., Klessen R. S., 2014, MNRAS, 445,2900Smith R. J., Glover S. C. O., Klessen R. S., Fuller G. A., 2016,MNRAS, 455, 3640Spitzer L., 1978, Physical processes in the interstellar medium,doi:10.1002/9783527617722.Tenorio-Tagle G., 1979, A& A, 71, 59Truelove J. K., Klein R. I., McKee C. F., Holliman II J. H., HowellL. H., Greenough J. A., 1997, ApJL, 489, L179Turk M. J., Smith B. D., Oishi J. S., Skory S., Skillman S. W.,Abel T., Norman M. L., 2011, ApJS, 192, 9V´azquez-Semadeni E., 2015, in Lazarian A., de Gouveia Dal PinoE. M., Melioli C., eds, Astrophysics and Space Science Li-brary Vol. 407, Magnetic Fields in Diffuse Media. p. 401( arXiv:1208.4132 ), doi:10.1007/978-3-662-44625-6˙14V´azquez-Semadeni E., Ballesteros-Paredes J., Rodr´ıguez L. F.,1997, ApJ, 474, 292V´azquez-Semadeni E., Ryu D., Passot T., Gonz´alez R. F., GazolA., 2006, ApJ, 643, 245V´azquez-Semadeni E., G´omez G. C., Jappsen A. K., Ballesteros-Paredes J., Gonz´alez R. F., Klessen R. S., 2007, ApJ, 657,870V´azquez-Semadeni E., G´omez G. C., Jappsen A. K., Ballesteros-Paredes J., Klessen R. S., 2009, ApJ, 707, 1023V´azquez-Semadeni E., Gonz´alez-Samaniego A., Col´ın P., 2017,MNRAS, 467, 1313V´azquez-Semadeni E., Palau A., Ballesteros-Paredes J., G´omezG. C., Zamora-Avil´es M., 2019, arXiv e-prints, p.arXiv:1903.11247Vishniac E. T., 1994, ApJ, 428, 186Walch S. K., Whitworth A. P., Bisbas T., W¨unsch R., HubberD., 2012, MNRAS, 427, 625Walder R., Folini D., 2000, Ap& SS, 274, 343Wareing C. J., Pittard J. M., Falle S. A. E. G., 2017, MNRAS,465, 2757Whitworth A., 1979, MNRAS, 186, 59Williams J. P., McKee C. F., 1997, ApJ, 476, 166Wolfire M. G., Hollenbach D., McKee C. F., Tielens A. G. G. M.,Bakes E. L. O., 1995, ApJ, 443, 152Yorke H. W., Tenorio-Tagle G., Bodenheimer P., Rozyczka M.,1989, A& A, 216, 207Zamora-Avil´es M., V´azquez-Semadeni E., 2014, ApJ, 793, 84Zamora-Avil´es M., V´azquez-Semadeni E., Col´ın P., 2012, ApJ,751, 77Zamora-Avil´es M., Ballesteros-Paredes J., Hartmann L. W., 2017,MNRAS, 472, 647Zamora-Avil´es M., V´azquez-Semadeni E., K¨ortgen B., BanerjeeR., Hartmann L., 2018, MNRAS, 474, 4824 This paper has been typeset from a TEX/L A TEX file prepared bythe author. MNRAS000