An Investigation on the Morphological Evolution of Bright-Rimmed Clouds
aa r X i v : . [ a s t r o - ph ] O c t An Investigation on the Morphological Evolution ofBright-Rimmed Clouds(BRCs)
Jingqi Miao , Glenn, J. White , , M. A. Thompson , Richard P. Nelson ABSTRACT
A new Radiative Driven Implosion (RDI) model based on Smoothed ParticleHydrodynamics (SPH) technique is developed and applied to investigate the mor-phological evolutions of molecular clouds under the effect of ionising radiation.This model self-consistently includes the self-gravity of the cloud in the hydro-dynamical evolution, the UV radiation component in the radiation transferringequations, the relevant heating and cooling mechanisms in the energy evolutionand a comprehensive chemical network. The simulation results reveal that underthe effect of ionising radiation, a molecular cloud may evolve through differentevolutionary sequences. Dependent on its initial gravitational state, the evolutionof a molecular cloud does not necessarily follow a complete morphological evolu-tion sequence from type A → B → C, as described by previous RDI models. Whenconfronted with observations, the simulation results provide satisfactory physicalexplanations for a series of puzzles derived from Bright-Rimmed Clouds(BRCs)observations. The consistency of the modelling results with observations showsthat the self-gravity of a molecular cloud should not be neglected in any investi-gation on the dynamical evolution of molecular clouds when they are exposed toionising radiation.
Subject headings: star: formation – ISM: evolution – ISM: HII regions – ISM:kinematics and dynamics – radiative transfer. Centre for Astrophysics & Planetary Science, School of Physical Sciences, University of Kent, Canterbury,Kent CT2 7NR, UK, [email protected] Centre for Earth, Planetary, Space & Astronomical Research, The Open University, Walton Hall, MiltonKeynes, MK7 6AA Space Physics Division, Space Science & Technology Division, CCLRC Rutherford Appleton Laboratory,Chilton, Didcot, Oxfordshire, OX11 0QX, UK School of Physics Astronomy & Maths, University of Hertfordshire, College Lane, Hatfield, AL10 9AB,UK School of Mathematical Sciences, Queen Mary College, University of London, Mile End Road, LondonE1 4NS, UK
1. Introduction
Bright-rimmed clouds (BRCs) found in and around HII regions are sites of ongoing starformation due to compression by ionisation/shock fronts, and provide an excellent labora-tory to study the influence of UV radiation from nearby massive stars on the evolutionaryprocess of molecular clouds. Numerous molecular line, millimetre/sub-millimetre contin-uum and mid-IR surveys have revealed very detailed structures and physical properties ofBRCs in various astrophysical environments (Sugitani et al 1991; Sugitani & Ogura 1994;Thompson et al 2004a,b; Thompson & White 2004; Urquhart et al 2006).One of the most intriguing characteristics of the observed BRCs is the diversity of theirmorphologies. The observed BRCs appear in different morphologies even when they are insimilar stellar environments. Sugitani et al (1991, 1994) classified 89 BRCs from a wholesky survey into three types depending on the curvature of the rim of BRCs. The three typesare categorised with type A cloud with a rim displaying moderate curvature, type B cloudwith a rim of a high degree of curvature which sometimes is also described as an elephanttrunk morphology, and finally type C cloud with a tightly curved rim and a tail, which isalso called cometary globule.Lefloch & Lazareff (1994) presented a 2-dimensional numerical simulation of the effect ofUV radiation on the dynamical evolution of a molecular cloud based on the Radiation DrivenImplosion (RDI) model. Although their hydrodynamical modelling successfully created acomplete sequence of the morphological evolution of a molecular cloud from type A to Band then to C, it seems difficult to provide physical explanations for the following questionsderived from BRC observations:1) Why are there so many more type A BRCs? The first whole sky survey by Sugitani etal (1991,1994) revealed that 61% of the observed 89 BRCs have type A morphology. If allof the clouds followed the same evolutionary sequence revealed by Lofloch’s modelling, fromtype A to type C morphology, there should be a balanced number of BRCs with differentmorphologies to be observed.2) Why do some of observed BRCs evolve to a quasi-stable type A rimmed morphology andshow signs of star formation at the heads of their structures? If all of the clouds should evolveto type C morphology and finally were completely photoevaporated as described by Lelfloch’smodelling, no star formation should be triggered in type A BRCs. The recent radio contin-uum and molecular line observations on BRCs in the catalogue of SFO (Sugitani et al 1991;Sugitani & Ogura 1994) presented by Urquhart et al (2006) and many other observations inmolecular lines and submilimeter lines (Lee et al 2005; Karr & Martin 2003; Thompson et al2004b) provided strong evidences of ongoing star formations in the condensed head of type 3 –A rimmed clouds where a pressure equilibrium at the boundary between the clouds and itsenvironment has been reached. Therefore it seems that for some of the molecular clouds,further evolution to type B or C rimmed morphologies is not necessary, i.e., not every BRC’smorphological evolution follows the sequence from type A to B then to C rimmed morphol-ogy, it may terminate their morphological evolution at any one of the three morphologies.3) What causes the spatial morphological distribution of BRCs in some of molecular cloudclusters? The remnant molecular clouds in the Ori OB 1 associations (Ogura & Sugitani1998) presented a spatial sequence from type A → B → C rimmed BRCs with their distancefrom the ionising star. It was once suggested that some molecular clouds which were fartheraway may directly evolve to type C rimmed morphology (Ogura & Sugitani 1998), but thereis no physical foundation for this speculation.For the molecular clumps very close to an ionising star, the ionised gas cannot expandfreely from the surface of the clump, because their expansion is constrained by a materialflow accompanying with the ionisation radiation flux. Lefloch’s simulation showed that azero radial velocity boundary condition, (the extreme case of flow effect) favours the type ABRC formation (Lefloch & Lazareff 1994). This result may partly explain the predominanceof type A BRCs at the border of the HII regions. However question remains for the type ABRC structures found at the outer location of the HII regions or type B/C BRCs found atthe border of HII regions (Sugitani et al 1991; Sugitani & Ogura 1994) and there are must besome other physical mechanisms which play dominant roles in the morphological evolutionof molecular clouds.Therefore searching for satisfactory solutions to the above puzzles requires a more com-prehensive model which should include as many physical processes as possible so that reliablephysical explanations on what observations revealed can be derived, hence to greatly improveour understanding on how the intensive UV radiation from nearby stars affects the dynamicalevolution of the surrounding molecular clouds.The fact that IRAS sources and bipolar outflows in cloud globules are very often foundat the edges of BRCs strongly suggests that the self-gravity of a molecular cloud can playan active role even in the early stages of evolution in molecular clouds (Lefloch & Lazareff1994). Nevertheless the self-gravity of a molecular cloud was neglected in Lefloch’s model,because the initiative of the model was to investigate photo-evaporation effect of molecularclouds, for which self-gravity does not play a dominant role. However, as pointed out byLefloch & Lazareff (1994) at the end of the discussions of their modelling results, inclusion ofthe self-gravitation of a molecular cloud is necessary in order to build a appropriate model forinvestigation on the dynamical evolution of globule clouds under the effect of UV radiation. 4 –Recently, Kessel-Deynet & Burkert (2000, 2003) have developed the first 3-D SPH RDImodel for investigation of the effect of UV radiation on the dynamical evolution of molecularclouds. Their model includes both self-gravity and the hydrogen ionisation of the molecularcloud in the dynamical evolution equations and revealed further features of molecular cloudevolution under the effect of ionising radiation. However a self-consistent treatment for theenergy evolution was not included which sets a barrier for a rigorous study on the evolutionof the physical properties of a cloud, so that a direct confrontation with observations isdifficult.Therefore it is our intention to build a comprehensive RDI model in order to obtain aconsistent description about the effect of the UV radiation on the dynamical evolution of amolecular cloud and to reveal the physical origin for the observed characteristics of BRCs’morphology formation.In the following sections we first give an outline for our 3-D RDI model, then brieflypresent an analytical solution which uses some observable physical properties of a condensedglobule and the UV radiation flux to define a maximum mass a stable condensed globule couldhave to against the gravitational collapse. Next the dynamical evolution of the cloud will bedetailed and the corresponding kinematics will be analysed based on numerical simulationresults. Finally we will apply the derived knowledge to confront with BRCs’ observationsand the conclusions are drawn at the end of the paper.
2. The model
In this section, we present a brief description of the basic equations and various physicalprocesses included in our model. The hydrodynamical equations are solved with SmoothedParticle Hydrodynamical(SPH) technique. For the detailed description of the SPH theoryand the corresponding technique, readers are directed to Nelson & Langer (1999).
The continuity, momentum, and energy equations for a compressible fluid can be writtenas: dρdt + ρ ∇ · v = (1) d v dt = − ρ ∇ P − ∇ Φ + S visc (2) 5 – d U dt + Pρ ∇ · v = γ − Λ ρ (3)and the chemical rate equations take the general form: dX i dt = nK i (4)where d/dt = ∂/dt + v · ∇ denotes the convective derivative, ρ is the density; v is thevelocity; P is the pressure; S visc represents the viscous forces; U is the internal energy perunit mass; and Φ is the gravitational potential. γ and Λ represent the non adiabatic heatingand cooling functions respectively, which we will deal with later; The fractional abundanceof main chemical species X i in Equation (4) are for CO, CI, CII, HCO + , OI, He + , H +3 , OH x ,CH x , M + and electrons; K i is the associated chemical reaction rate and n is the total numberdensity. The molecular cloud in our model is under the effects of both UV and FUV radiationfields. The UV radiation is from a nearby star and interacts with the molecular cloud throughthe top layer of the surface facing the star (front surface). The FUV radiation onto the frontsurface is from the nearby star and that onto the rear surface is from the background starlight (Nelson & Langer 1999). Then the FUV radiation field around a molecular cloud canbe approximated as a spherical radiation field (Gorti & Hollenbach 2002). The the effectof the FUV field on the molecular cloud is mainly through the photoelectric emission ofelectrons from grains and is treated in the same way as that in Nelson & Langer (1999).In the following part of this subsection we only need discuss how to deal with the ionisingradiation (UV) field from the nearby star.Although Helium ionisation was included in the chemical network, we could safely ne-glect it when dealing with the ionisation radiation transfer, ionisation heating and cooling forsimplicity, because of the much lower abundance of the Helium compared to hydrogen atoms(Dyson & Williams 1997). The implementation of ionising radiation from nearby stars intothe above SPH code is based on solving the following ionisation rate and radiative transferequations, dn e dt = I − R (5) dJdz = − σn (1 − x ) J (6)where I = σn (1 − x ) J is the ionisation rate with σ being the effective ionisation cross section; R = n e α B = x n α B is the recombination rate with α B being the effective recombination 6 –coefficient under the assumption of the ’on the spot’ approximation (Dyson & Williams1997). In its original definition, the recombination coefficient α = X i α i includes all of the individual recombination coefficients α i to the hydrogen atomic energylevel i . In the ’on the spot’ assumption, recombinations into the ground level ( i = 1) do notlead to any net effect on the change in ionisation rate, since the photons released from thisrecombination process are able to re-ionise other hydrogen atoms on the spot. Therefore α can be neglected and the resulting net recombination coefficient can be written as α B = ∞ X i =2 α i For a planner infall of ionising photons from a distant source on the boarder with a flux J as the Lyman continuum photons per unit time and square area, the solution of Eq.(6) is J ( z ) = J exp [ − τ ( z )]where τ ( z ) is the optical depth for ionising photons along the line of sight parallel to the infalldirection of the photons, and z is the distance from the border of the integration volumealong the line of sight. Neglecting the absorption effect of dust on ionising radiation, τ ( z )can be expressed as (Kessel-Deynet & Burkert 2000) τ ( z ) = Z zR n H ( z ′ ) ¯ σ dz ′ (7)with n H = n (1 − x ) and ¯ σ being the mean of σ v over frequency, weighted by the spectrumof the source (Kessel-Deynet & Burkert 2000).Eq.(6) can be written in terms of the ionisation ratio x , i.e., dxdt = 1 n dn e dt = I n − x nα B (8)Solving Eq.(8) for the ionising ratio x includes three steps: i.e., calculate the optical path τ ( z ); calculate the ionising rate I and then finally integrate Eq.(8) for x . We employ themethod introduced by Kessel-Deynet & Burkert (2000) in the above three-step work. Wejust give a very short description about it here. The interested reader can find more detaileddescription in Kessel-Deynet & Burkert (2000).For any a simulated particle i with its number density n and its nearest neighbourlist being given by the dynamical simulation at each time step, calculating τ ( z ) is carried 7 –through a) finding the evaluation points on the path toward source by smallest angel criteriaand then b)summing up all the optical path elements on the path τ ( z i ) = Σ mk =0 ¯ σ n H,k ∆ z k where m is the total number of the evaluation points from the boundary of the volume tothe position located one effective radius ( a i = 0 . M i /ρ i ) / ) before the point z i which isderived from a); n H,k is the number density of the hydrogen atoms at the evaluation point k , M i and ρ i are the mass and mass density of the particle i respectively.The ionisation rate for the particle i is then calculated by I i = J a i n i exp [ − τ ( z i )] [1 − exp ( − a i n H,i )]Finally, the first-order discretisation of Eq. (8) over a time interval ∆ t for particle i is givenby x j +1 i = x ji + ∆ t [ I j +1 i n j +1 i − n j +1 i ( x i ) j +1 α B ] (9)where the indexes j and j + 1 denote the values at the beginning and the end of the actualtime-step δt respectively. All of the values on the right-hand side are known from advancingthe particles by the SPH formalism, except I j +1 i , which can be approximated by I j +1 i = I ji − exp [ − ¯ σn j +1 H,i a j +1 i ]1 − exp [ − ¯ σa j +1 i n jH,i ] (10) γ The heating function in the thermal evolution is mainly provided by a) the photoelectricemission of electrons from grains γ pe caused by the incident FUV component (6 . < hν < . γ ion produced by the illumination of UV component ( hν > . E is ionised following the absorption of onephoton of frequency ν and energy E = hν , it releases one electron which carries the excesskinetic energy E ν − E which will be transfered to the gas through collisions with other gasparticles. The resulting heating rate is described by the following equation. γ ion = n (1 − x ) σJ kT ∗ (11) 8 –where k is the Boltzmann constant and T ∗ is expressed as (Cant´o et al 1998) T ∗ = T eff x + 4 x + 6 x + 2 x + 2 (12)where T eff is the stellar temperature, and x = ( hν ) / ( kT eff ) with ν being the frequencyof the Lyman limit.Heating of the gas is also affected by cosmic ray heating, H formation heating andgas-dust thermal exchange, for which the same formulae as that in (Nelson & Langer 1999)are used in our model. ΛThe cooling function Λ is affected by recombination of the electrons with ions, thecollisional excitation of OII lines and by CO, CI, CII and OI line emission.When a free electron in the plasma is captured by a proton, a photon is emitted and anamount of energy E + m e v e / rec = β B n x kT (13)where β B = α B × (1 + 0 . t e ).Another important cooling resource comes from the collisional excitation of low-lyingenergy levels of OII ions in spite of their low abundance, since OII has energy levels withexcitation potentials of the order of kT . We use the following simplified formula derived byRaga et al (2002) to calculate the cooling rate due to the collisional excitation of OII,Λ colli = Λ colli (1) + Λ colli (2) (14)with log [ Λ colli (1) n e n OII ] = 7 . t − . log [ Λ colli (2) n e n OII ] = 1 . t | t | . − . t = 1 − /T and t = 1 − × K /T and n OII = xn OI . The above formulais valid under the limit of low electron density n e < cm − , which is true for the BRCs’structure.The cooling affects Λ line by CO, CI, CII and OI line emission are same as those in theNelson & Langer (1999). 9 – The molecule cloud is assumed to have a spherical symmetry and its mass is uniformlydistributed through a sphere of radius R . It is embedded in surrounding hot medium with aconstant density (HII region) (Kessel-Deynet & Burkert 2003). The external pressure by thesurrounding hot medium provides an inwardly directed pressure force whose characteristicmagnitude is similar to that expected from a HII region. The value of the assumed bound-ary pressure P ext we adopted here is equivalent to an ionised gas with n e = 100cm − and T = 10 K(Lefloch et al 2002), which resembles to Lefloch’s zero radial velocity boundarycondition.An outflow condition is imposed at a fixed radius equal to several times of the originalradius of the cloud. Material that expand beyond this radius is simply removed from thecalculation. By this time, these particles represent very low density material and so providelittle UV extinction or other influence in the simulations.The initial turbulent velocity field was set up in the usual manner, as described byMac Low et al (1998), with perturbations set up with a flat power spectrum P ( k ), and witha minimum wavenumber k min corresponding to a wavelength equal to the cloud diameter.Eight wave numbers were used to set up the velocity perturbations with the maximum, k max = 8 k min . The velocity perturbations were drawn from a Gaussian random field de-termined by its power spectrum in Fourier space. For each three dimensional wavenumber k , we randomly select an amplitude from a Gaussian distribution centred on zero, and ofheight P ( k ), and selected a random phase uniformly distributed between 0 and 2 π . The fieldis then transformed back into real space to get the real velocity for each particle, and thenmultiplied by the amplitude needed to obtain the required total kinetic energy for the cloud.This procedure is repeated for each velocity component independently to get the full threedimensional velocity field. The amplitude of the turbulent motion was normalised such thatthe total kinetic energy of the clouds equals their initial gravitational potential energy.The initial geometry of the cloud and radiation field configurations are shown in Figure1. Since the distance of the cloud to the nearby illuminating star is assumed to be largecompared to the radius of the cloud, the UV radiations from the star can then be treatedas a plane-parallel radiation onto the front surface of the cloud (Lefloch & Lazareff 1994;Kessel-Deynet & Burkert 2000; Willams et al. 2001) while the FUV radiation is isotropic tothe cloud (Gorti & Hollenbach 2002; Nelson & Langer 1999)The radiation flux distribution at the boundary is 10 – J ( r = R, θ ) = (cid:26) J (Lyman) + J (FUV) 0 o ≤ θ ≤ o ) J (FUV) 90 o ≤ θ ≤ o )where J (Lyman) is the value of the Lyman continuum flux at the surface of the cloud forwhich we take the typical value of 2 × cm − s − (Miao et al 2006; Willams et al. 2001); J (FUV) is the value of the interstellar radiation at the surface of the cloud which has thevalue of 100 × G , where G is the mean interstellar medium radiation flux and traditionallyknown as the ’Habing flux’ (Habing 1968); θ is the azimuthal angle in a spherical coordinate. Lefloch & Lazareff (1994) classified the initial physical properties of a cloud and UV ra-diation field in a two dimensional parameter space (∆ , Γ), which are defined by the followingequations(Lefloch & Lazareff 1994),∆ = n i n = c i ηα B n R {− s ηα B RJ ( Lyman ) c i } (17)Γ = ηα B n i Rc i = 12 {− s ηα B RJ ( Lyman ) c i } (18)where ∆ relates the overpressure of the ionised gas to the undisturbed neutral gas and Γis the ratio of recombination rate to the ionisation rate, n i = xn = n e is the ionised gasdensity around the cloud, n is the initial density of the hydrogen in the cloud. η ≈ . c i = √R T ∗ is the isothermal sound speed in the ionised gas with R being the gasconstant. The 2-D parameter space can be divided into five different regions according tothe effect of the UV radiation on the evolution of a cloud.Region I is defined by Γ∆ ≤ . × − , where the opacity of the cloud is very low( ≪ ≤ c n c i ∼ − − − with c n = √R T n being the isothermal sound speed in the neutral gasof temperature T n , in which the effect of the ionisation is too weak to produce any noticeabledynamical effect. Therefore clouds starting in region I and II are too trivial to be discussed.On the other end, when ∆ is high, i.e., 2 ≤ ∆ ≤ √
10, which defines the region III,the cloud is entirely photo-ionised in an ionisation flash by Lefloch’s model in which theself-gravity of the cloud is not taken into account. Region IV is defined by ∆ < J ( Lyman ) n c i (1+Γ) , 11 –where the whole dynamical evolution of the cloud will be governed by the propagation of aD type ionisation front preceded by an isothermal shock in the cloud. Finally ∆ > J ( Lyman ) n c i (1+Γ) defines the region V, where an initial weak R ionisation front would gradually become a Dtype ionisation front propagating toward the interior of the cloud.For the clouds starting in regions III - V, although the detailed evolutionary featuresof the cloud can only be obtained through the numerical simulations described above, anexisting simplified analytical result derived by Kahn (1969) could help astronomers predictthe final evolution of the globule under observation and also provides us a way to validateour numerical results. Therefore we briefly present it in the following subsection.
3. A preview: A simplified analytical result
In the simplified analytical result, only the effect of UV component is considered since itis much stronger than FUV radiation as given in section 2.5. The viscosity term in Equation(2) will be neglected because of its smaller magnitude than the other terms.After an initial ionising stage by the UV radiation flux, the stream of the ionised hy-drogen forms a warm halo around the globule which has a radius r i , at which the globule isbounded by an ionisation front while an isothermal shock propagates toward the interior ofthe globule. Assuming a steady ionised gas flow from the surface of the globule which canbe approximated as a spherical isothermal sphere and a balanced ionisation-recombinationin the halo, Kahn (1969) derived a necessary condition for the globule to be stable againstthe gravitational collapse. Based on the analytical solutions of Equation (1-2) in the ionisedgas halo and in the neutral globule, this critical condition is expressed by the current massof the neutral globule M ≤ M max = 1 . α B c n π m H c i J (L yman ) G ) / (19)where G is the gravitational constant.The importance of this result is that the quantities such as T n ( c n ), T i ( c i ) and J (Lyman)can be estimated by the observational data, then Equation(19) can be used to estimate themaximum mass for the globule to contain in order not to go collapse if the structures of boththe globule and halo can be regarded as steady. On the other hand if the current mass M of the globule derived from the observational data is higher than M max , the globule underthe investigation would finally collapse to form new star at the end of the evolution, viceverse, the globule may stay stable until it is completely evaporated by the UV radiation.The advantage of the Equation(19) is that the criteria for globule’s stability is in terms of 12 –the current mass of the globule instead of the initial mass of the cloud, the latter is hardlyknown by observations.In the following, we will discuss the detailed evolutionary features of clouds, based onthe numerical simulations. In this paper we will mainly study the morphological evolutionof BRCs and try to find solutions for the questions raised in the beginning of the paper. Asystematic investigation on the ionisation radiation triggered star formation in BRCs will bepresented in our next paper.
4. Results and discussions
The most important difference between our newly developed RDI model and Lefloch’smodel is the inclusion of the self-gravity of a molecular cloud, therefore we will firstly in-vestigate the role of the self-gravity of a molecular cloud in its dynamical evolution whenit is under the effect of UV radiations, so that we can explore the possibility to relax theextreme zero velocity boundary condition in order to derive the predominance of type ABRCs, and discuss the possible modification to the classification of Lefloch’s 2-D parameterspace caused by the inclusion of the self-gravity of the molecular cloud.We then will explore the effects of initial thermal state of a cloud, the strength of UVradiation flux, and also initial turbulent state (non-thermal motion) on the morphologicalevolution of molecular clouds.The molecular cloud is presented by 20,000 SPH particles. In the following analysis tothe numerical results, we will concentrate on the effect of the UV radiation falling onto thesurface of the front hemisphere, since the heating effect of photoelectric emission of electronsfrom dust grains by the isotropic FUV radiation is much weaker than that of hydrogenionisation by Lyman continuum radiation which only affects those gas particles at the frontsurface layer.
The gravitational potential energy of a spherically uniform molecular cloud of mass M and radius R is Ω = − GM R . In order to explore the role of the initial self-gravity of amolecular cloud in its morphological evolution, we perform two different sets of simulations,one set of the three clouds having a same mass of 35M ⊙ but different radii, the other havinga same radius of 0.5 pc but different masses. Therefore our simulations cover different casesin which the initial self-gravity of a molecular cloud changes. 13 –The initial temperature of the molecular clouds is all set to 60 K. These chosen initialconditions guarantee that the simulated molecular cloud will not collapse by self-gravity ifthere is no external radiations. ⊙ Due to the change of the radii of clouds, the initial ionisation/recombination stateschange as well. The first six columns in Table 1. list the parameters of the first set cloudscalled A, B, C and their initial locations in the ( ∆ , Γ ) parameter space respectively. Thefeatures of cloud D in the list will be discussed later. The ratio of the initial gravitationalenergy of the three cloud A, B, and C is Ω A : Ω B : Ω C = 1 : 0 .
38 : 0 .
26 and the ratioof gravitational forces of the clouds A, B and C onto a mass element at their surfaces arecorrespondingly F A : F B : F C = 1 : 0 .
14 : 0 .
07, which defines significantly different initialself-gravity in the three molecular clouds.According to Lefloch’s model, cloud A would evolve into a cometary globule after passinga morphological sequence from type A to type B then to type C, and finally it will becompletely evaporated in a few Mys, while cloud B and C would be evaporated in a so called’ionisation flash’. It is our interest to see how these clouds would evolve differently if theself-gravity of a cloud is taken into account.
I: CLOUD A
The images in Figure 2 reveal the morphological evolution of cloud A by number densityof hydrogen nuclei and the line plots in Figure 3 describe how the UV radiation induced shockfront leads the ionisation front propagating into the interior of the cloud and finally resultsin the collapse of the cloud. Figure 4 is the corresponding velocity field evolution diagram,which gives the kinematic explanation on the formation of the morphologies of the cloud atdifferent evolutionary stages.The image in the top-left panel of Figure 2 and the solid line in the top-left panel ofFigure 3 show the number density distribution of cloud A at an early stage ( t = 700 years).It is seen that the density profile of the whole cloud has not yet changed much from theinitial uniform distribution. The top-left panel in Figure 4 shows a rather random velocitydistribution which is very similar to the initial random velocity distribution of the cloud.However the dashed line in the top-left panel of Figure 3 tells that an ionisation fronthas been built up at the front surface of the upper hemisphere of the cloud after the intensiveLyman continuum flux falls onto the front surface. The gas in the top layer is fully ionised 14 –( x ∼
1) and the resulted ionisation heating rises its temperature to 10 K. When t = 0 . × years, the ionised gas at the front surface of the cloud evaporate radially away from the frontsurface as shown by the velocity profile in the middle panel of Figure 4 and by the furry gasabove the front surface in the top-middle panel of Figure 2. The falling tail beyond z = 0 . − .On the other hand, the high temperature in the front top layer produces a high pressure,i.e., an isothermal shock driven into the cloud. The neutral gas ahead the shock front is thencompressed. The contour lines around the front surface layer of the cloud in the top-middlepanel in Figure 2 and the small peak (if we define it as the shock front) around z = 0 . v c ( θ ) at the front surface is approximatelygiven by Lefloch & Lazareff (1994), V c ( θ ) = V c (0)( cos ( θ )) / (20)where θ is the angle of a surface point from z axis and 0 ≤ θ ≤ π/ V c (0) is the shock velocity at the the point r = R, θ = 0 and V c (0) ∼ [ J (Lyman) /n ] / (Bertoldi 1989). The direction of V c is along the normal of the point (R, θ ). The resultedvelocity distribution V c ( θ ) in the shocked gas is obviously seen in the top-middle panel ofFigure 4.At time t = 2 . × years, the shock front leads the ionisation front moving towardthe rear hemisphere as shown by the solid and dashed lines in the top-right panel of Figure3. The ionisation ratio x has decreased a lot in the shocked part of the cloud, because therecombinations of electrons with their parent ions in the surrounding envelop have consumedmost of Lyman photons. On the other hand the shocked gas at the front surface around θ ∼ V c so it moves faster toward the rear hemisphere than that in theperipheral part of the compressed layer. Consequently, the faster progression of the centralparts of the compressed layer at the front surface causes a deviation of the morphologyof the front surface from a hemisphere, i.e, the front surface of the hemisphere becomesseriously squashed so that a type A rim forms, which can been seen from the top-right andthe subsequent panels of Figure 2.At the same time, the strongest compression of the gas located at θ ∼ a G hence a velocity component V G as shown 15 –in Figure 5. The surrounding gas then gradually collapse toward the core centred at G, asshown in the panels from the top-right to the bottom-right panels in Figures 3 and 4.Due to the distortion of the front surface, the velocity of the shocked gas in the outermostparts of the front hemisphere ( θ ∼ π/ θ ∼ π/ V T of the gas particle at the sample pointin the ’ear-like’ structure directs to the C’ point on the z axis, instead of C point, whichwould be the converging point if the self-gravity of the globule was not taken into account.Hence the ’ear-like’ structures converges to z axis to form a tail structure which overshootsthe rear surface of the cloud and is well described by the solid line beyond z < − . t = 0.351 My, the type A BRC structure comes to a thermally quasi-static state (witha pressure P/k ∼ Kcm − close to the boundary between the neutral gas and the ionisedenvelop) after an initial drastic and transient evolutionary period. As shown in Figure 3,the whole structure has shown the typical stratification of an observed BRC (Lefloch et al2002) from the centre to outside:i) An isothermal neutral core of a radius of 0.15 pc ( − . < z < cm − and an average temperature of T n = 28 Kii) A very thin layer of PDR (Photon-Dominated Region: 0 < z < .
05 pc) with anaverage density 10 cm − , an temperature ∼ n e <
10 cm − with the ionisation fraction 10 − < x < − .The mass included in the isothermal core including the PDR (with density from 10 toa few 10 cm − as observed) is 15 M ⊙ .iii) A photoionised envelope (Bright rim: 0 . < z < . to a few 10 cm − and the ionisation ratio 10 − < x ≤
1, i.e., n e distributionbetween 10 and 10 cm − . The average temperature T i is about 10 K.iv) The evaporated ionised tenuous gas into the HII region ( z > . − and x = 1, but a very high average temperature of 5 × K.Now the most concerned question on its further evolution is that whether the wholeglobule is stable against the gravitational collapse or not. Before revealing the numericalsolution to this question, we can employ Kahn’s analytical result to predict this globule’sfinal evolution and then use the result to validate the numerical solution. Substituting all of 16 –the values into Equation (19), we obtain the maximum mass for the isothermal core to bestable is M max = 3 M ⊙ . Therefore this type A BRC should finally collapse since the currentmass of the core is 15 M ⊙ , much higher than M max .The numerical results after t = 0.35 My as shown by the bottom middle/right panelsin Figures 2, 3 and 4 reveal that the globule quickly collapse toward its G centre underthe effect of the enhanced self-gravity. The central density of the core reached 2 × cm − when t = 0.391My and 5 × cm − when t = 0.394My. Therefore we could conclude thatthe radiation triggered star formation in cloud A is expected to occur because the centraldensity has reached such a high value (Nelson & Langer 1999), which is consistent with theanalytical result. II: CLOUD B
The morphological evolution of cloud B is shown in Figure 6, while Figure 7 describeshow the shock front leads the ionisation front propagating into the cloud and Figure 8displays the corresponding evolutionary sequence of its velocity field.During the first 0 . t = 0 . z axis as shown by the solid lines beyond z < − t = 0 .
351 My is formed with an isothermal condensed coreof a radius of 0.2 pc at the head of the globule as shown in the bottom-left panel of Figure7. The core has a central density slightly lower than 10 cm − , and an average temperature T n = 31 K, which is surrounded by a thin layer of photo-ionised envelope of a temperature T i = 10 K. The core including the PDR contains a mass about 13.5 M ⊙ which is higherthan the M max =3.6 M ⊙ given by Equation(19). Therefore the core should be expected tocollapse. The simulation results presented in the bottom middle/right panels of Figure 6 and 17 –Figure 7 show that the core continuously gets denser, and the central density has reacheda few × cm − at t = 0.79 My and a few × cm − at t = 0.81 My, the sign of thetriggered star formation which is consistent with the analytical prediction. III: CLOUD C
The evolutionary sequences of density, ionisation ratio and the velocity field of cloud Care shown in Figures 9, 10 and 11 respectively. In the first 0.86 My, the dynamical evolutionof the cloud C displays similar features to that of cloud B. A type A rim firstly forms after t = 0 .
365 My, and then a type B rimmed morphology develops when t = 0 .
86 My, as shownin the corresponding panel in Figure 9. The relevant panel in Figure 10 shows the formationof an isothermal core with a central density slightly lower than 10 cm − and a thin layer ofPDR surrounding the core. The average temperature T n is 35 K in the core and T i = 1000in the photo-ionised bright layer. The thermal pressure P/k in the different regions is ∼ .The mass included within the isothermal core (PDR included) is 11.1 M ⊙ , which is higherthan the 5 M ⊙ estimated by Equation (19) Therefore, the core would collapse finally.However, the gas particles at the two sides of the rear hemisphere continue their move-ment to z axis to form a longer tail as shown in the bottom-middle panels of Figures 9 and11 so that the morphology of the globule becomes a type C BRC at t = 1 .
034 My.The last two panels in Figure 10 shows that the gas particles in the core region collapseto the G centre, so the central density reaches to 10 cm − , which means the beginning ofthe triggered star formation at t = 1 .
12 My.In table 1. the final morphologies of the simulated clouds are list in the 7th column andthe 8th and 9th columns are the isothermal Core Formation Time (C.F.T.) and the CoreCollapse Time(C.C.T.) respectively.In order to find the mechanism responsible for the differences in the morphologicalevolution of cloud A, B and C, a further investigation on the kinematics of the shocked gasin the front hemisphere is carried in the following part of the paper.
IV: The Kinematics of morphological evolution
Figure 5 shows that the direction of the total velocity V T of the shocked gas particleswithin two ’ear-like’ structures determines their converging points on the z axis, hence thelength of the tail or say it is the magnitudes of V C and V G which determine the morphologicalevolution of a BRC.The ratio of the gravitational accelerations of cloud A, B and C on the gas particlesin the ’ear-like’ structure is a A : a B : a C = 1 : 0 .
14 : 0 .
07. Hence the relevant velocity 18 –components obey the inequality V G ( A ) > V G ( B ) > V G ( C ). On the other hand, as statedin Equation(20), we have V C ( θ ) ∝ V C (0) ∼ n − / for fixed J and θ , hence for the particlesin the the ’ear-like’ structures in cloud A, B and C, the velocity component V C follows V C ( A ) < V C ( B ) < V C ( C ). Now it is not difficult to understand that the angle δ of a gasparticle in the ’ear-like’ structure in cloud A, B, C obeys the inequality: δ ( A ) > δ ( B ) > δ ( C ).The distance d between the converging point C’ and the new gravitational centre G in cloudsA, B and C obeys d A < d B < d C , therefore the three clouds evolved to type A, B, and CBRCs respectively.On the other hand, the escape velocity of a particle at the surface of a cloud is v es = q GMR . The ratio of the escaping velocities of three clouds is v A ( es ) : v B ( es ) : v C ( es ) = 1 : 0 .
37 :0 .
26. The easiness for the mass particles at the surface of the cloud to be photo-evaporatedis in the order of cloud C, B, and A, which explains why the masses left in the isothermalcores are 11.1, 13.5 and 15 M ⊙ for clouds C, B, and A respectively. V: Further increment of the radius
We further increase the initial radius of the molecular cloud so that the cloud is initiallymuch less gravitationally bounded than cloud C. The basic features in the evolutionarysequences of the molecular clouds are not much different from that of the cloud C if
R < . R = 4 . cm − when t = 0 . Table 2. lists the parameters of the second set clouds called A, A ′ , A ′′ respectively withtheir initial locations in ( ∆ , Γ ) parameter space and the information on their morphologicalevolutions.For cloud A, the evolutionary features has been fully discussed before. and are re-listedin Table 2 for comparison. With the decrement of the initial mass of the cloud, the densityof the cloud decreases accordingly, the morphological evolution follows a similar pattern tothat in the first set of clouds. The dynamical evolutions of the formation of type A, B andC BRCs exhibit similar sequences to that in the first set clouds. Therefore in order to savespace, we don’t present the sequential figures on their dynamical evolutions as we did for setone clouds, but only provide a summary on the relevant key features which are partly listedin the Table 2.When the initial mass of a cloud decreased to 12 M ⊙ , cloud A ′ evolved from type A toB morphology in a very similar way to cloud B in set one simulation. An isothermal coreformed in 0.42 My, and collapsed in 0.8 My, with the central density up to 10 cm − and afinal mass of 6.9 M ⊙ in a collapsed core of 0.1 pc in radius. When the initial mass furtherdecreased to 8 M ⊙ , cloud A ′′ evolved through a sequence of from type A to B then to Cmorphologies, with formation of an isothermal core at t = 0.53 My, which collapsed in 1.5My with a central density of 10 cm − and a mass of 4.5 M ⊙ in a core of radius of 0.1 pc.Finally when the initial cloud mass further decreased to 1 M ⊙ , cloud A ′′′ evolved quicklythrough type A and B morphologies with a condensed core of a central density of 10 cm − .After evolving to a cometary globule morphology at t = 0.3 My, it re-expands and then istotally evaporated when t = 0.9 My.In the second set simulations, the initial recombination parameter Γ is same for cloudA, A ′ and A ′′ because they have same radius. The ratio of the initial ionisation parameters∆( A ) : ∆( A ′ ) : ∆( A ′′ ) = 1 : 2 .
85 : 4, i.e., cloud A ′′ has the highest value of ∆ which is 4times the lowest value for cloud A. On the other hand, the ratio of the initial gravitationalenergies of the three clouds is Ω A : Ω A ′ : Ω A ′′ = 1 : 0 .
12 : 0 . ′′ .It is obvious the difference in the initial self-gravitational energies among the three clouds 20 –is much more distinctive than the difference in their initial ionisation states. Therefore,we can safely conclude that the different morphological evolutions of the three clouds arecaused by their different initial self-gravitational states. The role of the initial self-gravityin determining the morphological evolutions of the BRCs is further confirmed. ∆ , Γ ) parameter space From the morphological evolution paths of the 7 clouds, we see that for clouds startingfrom region V such as cloud A and A’, their dynamical evolutions are driven initially by aweak-R ionisation front which quickly changes into a D-type ionisation front, just as whatLefloch described.However for clouds starting from region III such as cloud B,C,and A”, they are notat all entirely photo-ionised by an R-weak front and permeated by an ionisation flash likewhat Lefloch & Lazareff (1994) described, their dynamical evolutions are driven by a similarmechanism to that of cloud A and A’.For clouds A ′′′ and D, which also start from region III but with a high initial ionisationstate ∆ ≥
23, its dynamical evolution is driven by a weak R-front and the whole structureget permeated by an ionisation flash and evaporated quickly just as what Lefloch & Lazareff(1994) described for region III cloud.Hence it is clear that in term of the dynamical classification of (∆ , Γ) parameter space,the boundary for region III has been pushed greatly from ∆ < √
10 to ∆ <
23 due to theinclusion of the self-gravity of the cloud.
In order to further support our argument on the role of self-gravity of the cloud in themorphologic evolution process, we take off the self-gravity of the three clouds A, B and Cand re-run the simulations. The result reveals very similar evolutionary sequence to that ofLefloch & Lazareff (1994). The evolutionary sequence passes type A to B morphologies andthen form a type C BRC. Then it re-expands and finally totally evaporated between 1 to 2My. To some stage of the evolution of the molecular cloud, the ionisation flux onto the frontsurface of the cloud is perturbed somehow due to the recombinations of the electrons ontothe ionic atoms, which causes instability of the front surface of the cloud as described in 21 –the lefloch’s simulations (Lefloch & Lazareff 1994). We found from our simulations that thescale of the instability is closely related to the initial ionisation state of the cloud. For allof the 7 clouds we simulated, except cloud D, their initial ionisation state are not very highwhen compared to that of the cloud D, very small scale perturbations (with wavelength λ < . R << R )) at the front surface appeared, as seen from the up-middle panels of Fig.2,6 and 9, but they are quickly smoothed out.For cloud D, due to its extremely high initial ionisation state, the surrounding of thefront surface quickly becomes recombination dominant region, so large scale perturbations(with wavelength 0 . R < λ < R ) grow at t = 0 .
06 My from the front surface of the cloudwhich can be seen from the up-left panel of Figure 12. However the large scale perturbationis stable because it is gradually smoothed out with the core further being compressed by theisothermal shock. At t = 0 . In order to see how the three clouds would evolve if they are only initially thermallysupported, i.e., if the cloud A, B and C start their evolutions without the initial turbulence.We re-run the simulations for cloud A, B and C respectively by setting a zero initial velocityfield to each of them. Since the general features of the morphological evolutions of thethree molecular clouds are qualitatively similar to that presented in last section, we mainlydescribe the different features in their evolutions.Cloud A evolved into a type A BRC with a condensed core of a central density of a few10 cm − within 0.37 My. Cloud B evolved into a type B BRC with a condensed core of acentral density 10 cm − within 0.74 My. Cloud C reached a type C BRC structure with a 22 –core density of 10 cm − within 0.98 My. Compared with the corresponding results in Section4.1.1, the clouds without initial turbulence take less time to collapse. Further comparing thevelocity field evolutions of the two group molecular clouds, we find the initially turbulentfield takes 0.02 My in cloud A, 0.07 My in cloud B and 0.14 My in cloud C to dissipate,which shows the initially turbulent clouds is just delayed to form a highly condensed core.Therefore we conclude that the initial turbulence does not effect a cloud’s final morphologytype. The results is consistent with that of an investigation by Nelson & Langer (1999) onthe dynamical evolutions of Bok globules affected by FUV radiation. All of the above simulations are based on an initially gravitationally stable and uniformcloud. Although large degrees of central condensation should not be expected in the ini-tially gravitationally stable molecular cloud, we are still interested in examining the effectof the initial central condensation of a molecular cloud on its morphological evolution. Thefollowing initial density distribution is used, n ( r ) = (cid:26) n (0 ≤ r ≤ r ) n ( r r ) ( r < r ≤ R )which represents for a condensed uniform core with a radius of r and density of n sur-rounded by an envelop of a density profile proportional to r − . For a given r , R and M , n can be found from the equation 4 πm H R R n ( r ) r dr = M .When we start with r = 0 . R , the evolutionary sequences of clouds A, B and C withan initial core-halo mass distribution do not show significant change from that of initiallyuniform clouds. The basic features of type A, B and C are kept, except that the main body ofthe formed BRC structure are all smaller than that formed in initially uniform clouds. Thenwe decrease the radius of the central condensation to r = 0 . R , the morphological evolutionshow obvious changes. Cloud A ends its evolution in type B as shown in Figure 13, and cloudB in type C as shown in Figure 14, while cloud C developed into a very narrow shaped typeC at a much early time ( t = 0 .
67 My ) as shown in Figure 15. Their morphologies are allelongated than that of their corresponding uniform clouds. This morphological change canbe well explained by the illustration in Figure (5), the gas particles in the halo has a lowerdensity therefore a higher V C component, which makes them converge to a farther point C’from point G on the z axis.The mass left at the collapsed core of 0.15 pc in cloud A, B, and C are 10.5, 7.2, 5.6 M ⊙ respectively, which are less than that in their corresponding initially uniform clouds. This is 23 –because the density in the initial halo of a cloud is lower than that in the same part of theinitially uniform cloud hence the halo in the front hemisphere is at a higher initial ionisationstate, in which particles are easier to be ionised and then to be photo-evaporated.In summary, the initial mass condensation state does affect a cloud’s morphologicalevolutions because it changes the initial self-gravitational state of a cloud. As stated in Section 2.5, we apply a constant pressure boundary condition, which re-sembles the physical conditions of a very hot HII region. In order to examine the effectof the boundary condition on the dynamical evolution of a cloud we decrease the value ofthe surrounding pressure P ext /k to mimic a less violent environment. For each of threeclouds in the first set, we re-run the simulation for two different values of the external pres-sure, i) P ext /k = 10 Kcm − which corresponds to a warm interstellar medium environment(Nelson & Langer 1999) and ii) P ext /k = 0 which describes a vacuum environment so thatthe ionised gas is able to freely expand from the surface of the cloud.It is found that with decreasing external pressure, although cloud A still evolves to a typeA BRCs under the boundary condition i) and ii), the curvature of the front surface increasesand the shape of the final condensed core becomes elongated. Cloud B with boundarycondition i) evolves to a slightly elongated type B BRC but evolves to a type C BRC at t = 1 . According to observations, the temperatures of molecular clouds are generally in therange of 10 −
100 K (Bertoldi 1989). We then carried out two more sets of simulations foreach of the three molecular clouds A, B and C with the initial temperatures T = 20 K and T = 100 K respectively, while all of the others physical conditions are kept the same as thatin Section 4.1.1.The evolutionary sequence of each simulation is qualitatively similar to its correspondingcloud with an initial temperature of 60 K. 24 –When the initial temperature T = 20 K, the evolutionary sequences of cloud A, B andC basically exhibit no significant differences from that described in Section 4.1.1. Cloud Aevolves to a quasi-stable type A rimmed morphology with a condensed core at the head ofits front hemisphere. The central density of the core reaches a value of 10 cm − in 0.37 Mywhen it finally collapses. Cloud B evolves from type A to B morphology with a condensedcore at the head of its front hemisphere in 0.6 My and the core reaches a central density of10 cm − at t = 0 .
79 My. Similarly cloud C evolves to a type C BRC after passing typeA and B morphologies. It collapse at t = 1 .
01 My when the central density of the core is2 × cm − .When the initial temperature rises to 100 K, cloud A still evolves to a type A rimmedmorphology having a condensed core which collapses after the central density is up to a few10 cm − in the front hemisphere at t = 0 .
42 My. Cloud B evolves to a type B rimmedmorphology and the condensed core collapse with the central density being up to 10 cm − at t = 0 .
93 My. Cloud C evolves to a type C rimmed BRC plus a small core of centre densityof 10 cm − in 1.32 My.In summary, the variation of the initial temperature of molecular clouds between 10 - 100K does not significantly change the morphological evolution of a cloud, which is physicallyunderstandable since the thermal pressure of a cloud is not important in the dynamicalevolution in BRCs (Bertoldi 1989; Lefloch & Lazareff 1994; White et al. 1999). Higher initialtemperature in a cloud delays its collapsing time, for the initial thermal motions of gasparticles dissipate over a period of time. The morphological evolution of a cloud is mainlydecided by the dynamical behaviour of the gas particles within the two sides of the cloud( θ ∼ π/ We also carried out a set of simulations on the influence of the strength of an ionisingflux on the evolutionary process of BRCs. The change in the strength of ionising flux incidenton the front surface of a molecular cloud corresponds to two different cases: i) a cloud can beat different distances from an ionising star; ii) an ionising star in different types radiates withdifferent flux strength. This set of simulations is to see whether the molecular clouds within 25 –one giant cloud cluster would exhibit essential difference in their morphological evolutionsdue to their different distances from the central star.Since the ionising flux J (Lyman) ∝ /D , where D is the distance of a cloud from theionising star. For each of cloud A, B and C, we re-run the simulations twice by setting twodifferent J (Lyman) flux values. If we assume cloud A, B and C in Section 4.1.1 are in adistance of D from the ionising star, then J (Lyman) = J (Lyman) / D = 2 D . If we set J (Lyman) = 4 × J (Lyman) which meansthe original cloud is moved to half distance closer, i.e, D = D /
2. The spatial range of 0.5 -2 D from the ionising star should be large enough to include most molecular clouds in onecloud cluster.The simulation results revealed that for all of the three molecular cloud A, B, and C,their morphological evolutions are not very sensitive to the strength of the ionising flux inthe above specified range and they follow very similar evolutionary sequences to that whenionising flux is J ( Lyman ).The result looks a bit puzzling at the first glance, since the velocity component V C of the ionised gas in the top layer of the cloud increases with the ionising flux, as shownin Figure 5, so that we should expect the angle δ decreases with the ionising flux, whichwill favour the formation of type C rim morphology. However, if we consider the effectof the enhanced shock effect due to increased ionising flux, the compression to the core inthe front hemisphere will be stronger. Therefore the core will exert a stronger attractiveforce on its peripheral gas particles which then results in a higher V G so that angle δ ofthe velocity remains basically unchanged although the magnitude of the total velocity mayincrease, which would decrease the time needed for the gas particles in the ear-like structuresto collapse onto the symmetrical axis. The same consideration can explain the insensitivityof the morphological evolution of the cloud to the decreased radiation flux and the timeneeded for the gas particles to collapse onto the symmetrical axis is found to be elongated. The simulations on a set of molecular clouds reveal that the morphological evolution ofa molecular cloud under the effect of the UV radiation does not necessarily go through all ofthe three morphologies and then end up at cometary type C morphology as described by theprevious models. Depending on its initial gravitational state, a cloud could evolve to any oneof the three type BRCs. The diversity of the evolutionary sequence of BRCs revealed fromour modelling provides a good prospect for us to explore the possible explanations for the 26 –questions raised from BRCs’ observations and presented in the introduction of this paper.The modelling results for the molecular cloud A, B, C under the same initial temper-ature and ionising flux tell us that the evolutionary paths of all molecular clouds wouldfirstly pass the type A morphology. However some of molecular clouds would only evolveto a quasi-equilibrium type A morphology plus a highly condensed core at the head of itsfront hemisphere, if their initial self-gravitational energy is high enough. Therefore we shallstatistically be able to observe many more type A BRCs than type B or C BRCs in a randomobservation.Our modelling results also shows that the UV radiation triggered collapse of a cloud ontothe centre core in its front hemisphere could occur with any one of the three morphologies,so it is not surprising at all that the UV radiation triggered star formation occurring in typeA BRCs could be observed.Finally, we come to understand the spacial sequence of type A-B-C BRCs with theirdistance from the centre star in the Ori OB 1 association (Ogura & Sugitani 1998). We mightthink this morphological distribution of BRCs as the result of decreasing illuminating fluxwith distance. However as shown in the argument in Section 4.6, the morphological evolutionof the cloud is insensitive to the change of the UV radiation flux in the range discussed. Basedon the results derived from the simulations, we can understand this observation from twoaspects.First, we know that where a new star is formed inside a giant molecular cloud as aresult of local molecular gas collapse onto a central point, the density of the surroundinggas can be reasonably assumed to decrease with the distance from the central star. It isalso known that giant molecular clouds are very clump (Tatematsu et al 1991). Consideringthe above two points we can assume that the densities of molecular clumps surrounding anew star decrease with their distance D from the central star. If we further assume thatthe clumps in a giant molecular cloud are of similar radius R , then the initial gravitationalpotential energy of a clump decrease with their distance from the centre of the star for thegravitational potential energy can be written as Ω ∼ M /R ∼ R n , where n is the initialnumber density of the clumps.The molecular clumps closest to the central star could possibly evolve to a quasi-equilibrium type A BRC because they have highest initial self-gravitational potential en-ergies, while for those clumps farther away, they evolve from type A to a quasi-equilibriumtype B BRC because of their moderate initial self-gravitational potential energies. For thosefarthest away from the central star, they evolve to pass type A and B morphologies, then to acometary type C morphology due to their lowest initial self-gravitational potential energies. 27 –Secondly, we already showed in Section 4.4 that at the borders of HII region, type ABRCs are favoured.Therefore the spatial sequence of type A - B - C BRCs with their distance to the centralstar can be seen as a manifestation of the mass density distribution of the very clumpy giantmolecular clouds.
5. Conclusions
A three dimensional and comprehensive RDI model has been developed with the SPHtechnique, which self-consistently includes the hydrodynamical evolution, the self-gravity ofthe cloud, the energy evolution, radiation transferring and a basic chemical network. TThe investigation on the morphological evolutions of BRCs are carried out by employingthe newly developed model in order to understand a series of puzzles coming from BRCsobservations. It is found that the morphological evolution of a molecular cloud is much moresensitive to its initial self-gravitational state than to other physical conditions such as theinitial temperature between 20 and 100 K, or the ionising radiation flux within one orderof magnitude higher or lower than J (Lyman). Our modelling results have revealed thatthere are three different evolutionary prospects when a molecular cloud is under the effectsof ionisation radiations.For a molecular cloud of an initial mass 35 M ⊙ and initial temperature 60 K underthe effects of UV and FUV radiation fields specified above, the simulation results revealed:a) if its initial radius is 0.5 pc, it will evolve to a quasi-stable type A BRC; b) when itsinitial radius increases to 1.3 pc, its evolution will firstly pass a type A morphology andcontinue its journey to a quasi-stable type B rimmed morphology; c) when the initial radiusfurther increases to 1.9 pc, its evolutionary process follows a sequence of type A → B → Cmorphologies. d) Further increasing the cloud’s radius to 4.7 pc, the evolution of the cloudalso follows the sequence of type A → B → C, but the finally formed type C BRC will betotally photoevaporated.Further simulation results also show that the initial central condensation of the mass in acloud could change its morphological evolution in term of its initially uniform correspondence.At the border of HII regions, the formation of type A BRCs are favoured. When a molecularcloud is set to a zero initial self-gravitational potential energy or with a low initial density,its morphological evolution sequence is consistent with that by Lefloch’s model which is a2-D model and didn’t include the self-gravity of the cloud.With the knowledge obtained from the modelling, we are now able to confront with the 28 –questions raised by observations. Since all of the clouds evolve to pass a type A morphologyand some of the clouds with enough high initial gravitational potential energy would justevolve to a quasi-stable type A morphology, statistically many more type A BRCs should beobserved over a random observation. For those BRCs who evolve to a quasi-stable type Arimmed morphology, if the core mass contained in their front hemisphere is high enough forit to collapse, new star could be triggered to form. Star forming signal should be observed.Furthermore if the initial densities of the molecular clumps in Ori OB 1 associations decreasewith their distances from the centre of the ionising star, a spacial distribution of type A-B-CBRCs with distance from the central star could exist.It is obvious that the inclusion of the self-gravitation in our model revealed a diversityof the evolutionary sequence of a molecular cloud under the effects of UV radiations. Thesatisfactory explanations based on the modelling results to the observational puzzles statethat the self-gravity of a molecular cloud indeed plays an important role in controllingthe evolutionary destiny of a molecular cloud and should not be neglected especially whenwe try to achieve a whole picture of the dynamical evolutions of molecular clouds. This isbecause the radiation induced shock dramatically enhances the effect of the self-gravity of theclouds. Consequently, some modifications on the region classification in Lefloch’s parameterspace have been confidently derived. The boundary between the D-type ionisation front and’ionising flush’ is extended from δ = √
10 to 23 as a result of the inclusion of the self-gravityof the molecular cloud.
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31 –Fig. 1.— The initial geometry of a molecular cloud and the configuration of the radiationfields. The heavy arrow lines above the upper hemisphere represent the sum of the strongUV (Lyman continuum) flux from the massive stars above the molecular cloud and theisotropic FUV flux from the background star radiation and the light arrow lines below thelower hemisphere represent the isotropic FUV radiation ( hν < . y = 0 from t = 700 years to t = 0 .
394 My. The top grey bar shows thedensity scale in logarithm for the upper row snapshots and the bottom bar for the secondrow snapshots. 33 –Fig. 3.— The distributions of the number density (solid line) and ionisation ratio x (dot-dashe line) of the simulated molecular cloud A along the symmetrical axis z from t = 700years to t = 0 .
394 My. 34 –Fig. 4.— The evolution of the velocity field for the same cloud as in Figure 2 over a periodof 0.394 My. The length of the arrow indicates the magnitude of velocity and the maximummagnitude of the velocities at different stages of the evolution have been written inside thecorresponding panels. 35 –Fig. 5.— An illustration about the effect of the gravity of the main body of the cloud on thevelocity of the shocked gas particle in the ’ear-like’ structures at the two sides of the cloud.The central solid lines are their structure symmetrical line z . 36 –Fig. 6.— The evolutionary snapshots of the number density of the simulated molecular cloudB at the cross section y = 0 from t = 1800 years to t = 0 .
81 My. The top grey bar shows thedensity scale in logarithm for the upper row snapshots and the bottom bar for the secondrow snapshots. 37 –Fig. 7.— The distributions of the number density (solid line) and ionisation ratio x (dot-dashe line) of the simulated molecular cloud B along the symmetrical axis z from t = 1800years to t = 0 .
81 My. 38 –Fig. 8.— The evolution of the velocity field for the same cloud as in Figure 6 over a periodof time of 0.81 My. The length of the arrow indicates the magnitude of velocity and themaximum magnitude of the velocities at different stages of the evolution have been writteninside the corresponding panels. 39 –Fig. 9.— The evolutionary snapshots of the number density of the simulated molecular cloudC at the cross section y = 0 over a period of 1 .
12 My. The top grey bar shows the densityscale in logarithm for the upper row snapshots and the bottom bar for the snapshots in thebottom row. 40 –Fig. 10.— The distributions of the number density (solid line) and ionisation ratio x (dot-dashe line) of the simulated molecular cloud C along the symmetrical axis x = 0 over aperiod of 1.12 My. 41 –Fig. 11.— The evolution of the velocity field for the same cloud as in Figure 9 over thesame period of time. The length of the arrows indicates the magnitude of velocity and thelongest arrow in each panel represents the maximum velocity in the corresponding velocityfield distribution at different stages of the evolution. 42 –Fig. 12.— The growth and smooth processes of the large-scale surface instability over theevolutionary process in cloud D. 43 –Fig. 13.— The evolutionary snapshots of the number density of the simulated molecularcloud A with an initial core-halo distribution at the cross section y = 0. The top grey barshows the density scale in logarithm for the upper row snapshots and the bottom bar for thesecond row snapshots. 44 –Fig. 14.— The evolutionary snapshots of the number density of the simulated molecularcloud B with an initial core-halo distributionat the cross section y = 0. The top grey barshows the density scale in logarithm for the upper row snapshots and the bottom bar for thesecond row snapshots. 45 –Fig. 15.— The evolutionary snapshots of the number density of the simulated molecularcloud C with an initial core-halo distributionat the cross section y = 0. The top grey barshows the density scale in logarithm for the upper row snapshots and the bottom bar for thesecond row snapshots. 46 –Fig. 16.— The evolutionary snapshots of the number density of the simulated molecularcloud B with a zero external pressure boundary at the cross section y = 0. The top grey barshows the density scale in logarithm for the upper row snapshots and the bottom bar for thesnapshots in the bottom row. 47 – Morphology type Core formation time in My Core collapse time in My
Table 1. Parameters of set one cloudsCloud ∆ Γ n ( cm − ) R(pc) Region M type CFT CCT A 0.68 55 2672 0.5 V A 0.35 0.39B 7.3 89.1 152 1.3 III B 0.63 0.6C 18 108 49 1.9 III C 0.86 1.12D 182 169 3.2 4.7 III C 1.0 N/A 48 –
They have the same meanings as that in Table 1.
Table 2. Parameters of second set cloudsCloud ∆ Γ n ( cm − ) M(M ⊙ ) Region M type CFT CCT3