We study the photoevaporation of Jeans-unstable molecular clumps by isotropic FUV (6eV
keywords:
ISM: clouds, evolution, photodissociation region - methods: numerical
††pagerange: Photoevaporation of Jeans-unstable molecular clumps-C††pubyear: 2019\AtBeginShipout
=\AtBeginShipoutBox\AtBeginShipoutBox

Stars are known to form in clusters inside giant molecular clouds (GMCs), as a consequence of the gravitational collapse of overdense clumps and filaments (Bergin et al., 1996; Wong et al., 2008; Takahashi et al., 2013; Schneider et al., 2015; Sawada et al., 2018). The brightest (e.g. OB) stars have a strong impact on the surrounding interstellar medium (ISM), since their hard radiation field ionizes and heats the gas around them, increasing its thermal pressure. As a result, the structure of the GMC can be severely altered due to the feedback of newly formed stars residing inside the cloud, with the subsequent dispersal of low density regions. Collapse can then occur only in dense regions able to self-shield from impinging radiation (Dale et al., 2005, 2012a, 2012b; Walch et al., 2012).

The ISM within the Strömgren sphere around a star-forming region is completely ionized by the extreme ultra-violet (EUV) radiation, with energy above the ionization potential of hydrogen (hν>13.6ℎݜˆ13.6h\nu>13.6italic_h italic_ν >13.6 eV). The typical average densities of H IIII\scriptstyle\rm II\ roman_IIregions are ⟨n⟩≃100cm-3similar-to-or-equalsdelimited-⟨⟩ݑ›100superscriptcm3\langle n\rangle\simeq 100\,\,\rmcm^-3⟨ italic_n ⟩ ≃ 100 roman_cm start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT: this results in ionization fractions xhii<10-4subscriptݑ¥hiisuperscript104x_\textschii<10^-4italic_x start_POSTSUBSCRIPT hii end_POSTSUBSCRIPT <10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT and a final gas temperature T>104-5ݑ‡superscript1045T>10^4-5italic_T >10 start_POSTSUPERSCRIPT 4 - 5 end_POSTSUPERSCRIPT K. Far-ultraviolet (FUV) radiation (photon energy 6eV
The effect of FUV radiation on clumps is twofold: (1) FUV radiation dissociates the molecular gas, which then escapes from the clump surface at high velocity (photoevaporation); (2) radiation drives a shock which induces the clump collapse (Sandford et al., 1982, radiation-driven implosion). The first effect reduces the clump molecular mass, hence decreasing the mass budget for star formation within the clump. Instead, the latter effect may promote star formation by triggering the clump collapse. Hence, the net effect of radiative feedback on dense clumps is not trivial and deserves a careful analysis.

The flow of gas from clumps immersed in a radiation field has been studied by early works both theoretically (Dyson, 1968; Mendis, 1968; Kahn, 1969; Dyson, 1973) and numerically (Tenorio-Tagle, 1977; Bedijn & Tenorio-Tagle, 1984). Bertoldi (1989) and Bertoldi & McKee (1990) developed semi-analytical models to describe the photoevaporation of atomic and molecular clouds induced by ionizing radiation. In their models they also include the effects of magnetic fields and self-gravity. They find that clumps settle in a stationary cometary phase after the radiation-driven implosion, with clump self-gravity being negligible when the magnetic pressure dominates with respect to the thermal pressure (i.e. B>6μGݐµ6ݜ‡GB>6\,\mu\mathrmGitalic_B >6 italic_μ roman_G), or when the clump mass is much smaller than a characteristic mass mch≃50M⊙similar-to-or-equalssubscriptݑšch50subscriptMdirect-productm_\mathrmch\simeq 50\,\rm M_\odotitalic_m start_POSTSUBSCRIPT roman_ch end_POSTSUBSCRIPT ≃ 50 roman_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT. They focused on gravitationally stable clumps, thus their results are not directly relevant for star formation.

Later, the problem was tackled by Lefloch & Lazareff (1994), who performed numerical simulations which however only included the effect of thermal pressure on clump dynamics. Gravity was then added for the first time by Kessel-Deynet & Burkert (2003). For an initially gravitationally stable clump of 40 M⊙subscriptMdirect-product\rm M_\odotroman_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT, they find that the collapse can be triggered by the radiation-driven implosion (RDI); nevertheless, they notice that the collapse does not take place if a sufficient amount of turbulence is injected (vrms≃0.1km/ssimilar-to-or-equalssubscriptݑ£rms0.1kmsv_\rm rms\simeq 0.1\,\rm km/sitalic_v start_POSTSUBSCRIPT roman_rms end_POSTSUBSCRIPT ≃ 0.1 roman_km / roman_s). Bisbas et al. (2011) also ran simulations of photoevaporating clumps, with the specific goal of probing triggered star formation. They find that star formation occurs only when the intensity of the impinging flux is within a specific range (109cm-2s-1<Φeuv<3×1011cm-2s-1superscript109superscriptcm2superscripts1subscriptΦeuv3superscript1011superscriptcm2superscripts110^9\,\rm cm^-2\rm s^-1<10 start_POSTSUPERSCRIPT 9 end_POSTSUPERSCRIPT roman_cm start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT
In a previous work (Decataldo et al., 2017, hereafter D17), we have constructed a 1D numerical procedure to study the evolution of a molecular clump, under the effects of both FUV and EUV radiation. We have followed the time evolution of the structure of the iPDR (ionization-photodissociation region) and we have computed the photoevaporation time for a range of initial clump masses and intensity of impinging fluxes. However, since D17 did not account for gravity, H2subscriptH2\mathrmH_2roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT dissociation is unphysically accelerated during the expansion phase following RDI. Those results have been compared with the analytical prescriptions by Gorti & Hollenbach (2002), finding photoevaporation times in agreement within a factor 2, although different simplifying assumptions where made in modelling the clump dynamics.

The same setup by D17 has been used by Nakatani & Yoshida (2018) to run 3D simulations with on-the-fly radiative transfer and a chemical network including H+superscriptH\mathrmH^{+}roman_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, H2subscriptH2\mathrmH_2roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, H+superscriptH\mathrmH^{+}roman_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, OO\mathrmOroman_O, COCO\mathrmCOroman_CO and e-superscripte\mathrme^{-}roman_e start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT. Without the inclusion of gravity in their simulations, they find that the clump is confined in a stable cometary phase after the RDI, which lasts until all the gas is dissociated and flows away from the clump surface. Nevertheless, they point out that self-gravity may affect the clump evolution when photoevaporation is driven by a FUV-only flux, while the EUV radiation produces very strong photoevaporative flows which cannot be suppressed by gravity.

In the current paper, we attempt to draw a realistic picture of clump photoevaporation by running 3D hydrodynamical simulations with gravity, a non-equilibrium chemical network including formation and photo-dissociation of H2subscriptH2\mathrmH_2roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, and an accurate radiative transfer scheme for the propagation of FUV photons. We focus on the effect of radiation on Jeans-unstable clumps, in order to understand whether their collapse is favoured or suppressed by the presence of nearby stars emitting in the FUV range.

The paper is organized as follows. In Sec. 2, we describe the numerical scheme used for the simulations, and, in Sec. 3, the initial conditions for the gas and radiation. We analyse the evolution of the clump for different radiative fluxes and masses is Sec. 4 and Sec. 5, respectively. A cohesive picture of the photoevaporation process is given in Sec. 6. Our conclusions are finally summarised in Sec. 7.

2 Numerical scheme

Our simulations are carried out with ramses-rt333https://bitbucket.org/rteyssie/ramses, an adaptive mesh refinement (AMR) code featuring on-the-fly radiative transfer (RT) (Teyssier, 2002; Rosdahl et al., 2013). RT is performed with a momentum-based approach, using a first-order Godunov solver and the M1 closure relation for the Eddington tensor. The basic version of ramses-rt thermochemistry module accounts only for the photoionization of hydrogen and (first and second) photoionization of helium. We have used the chemistry package krome 444https://bitbucket.org/tgrassi/krome (Grassi et al., 2014) to implement a complete network of H and He reactions, including neutral and ionized states of He, H and H2subscriptH2\mathrmH_2roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT.

We track the time evolution of the following 9 species: H, H+superscriptH\mathrmH^{+}roman_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, H-superscriptH\mathrmH^{-}roman_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT, H2subscriptH2\mathrmH_2roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, H2+superscriptsubscriptH2\mathrmH_2^{+}roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, He, He+superscriptHe\mathrmHe^{+}roman_He start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, He++superscriptHeabsent\mathrmHe^{++}roman_He start_POSTSUPERSCRIPT + + end_POSTSUPERSCRIPT and free electrons. Our chemical network includes 46 reactions in total555The included reactions and the respective rates are taken from Bovino et al. (2016): reactions 1 to 31, 53, 54 and from 58 to 61 in Tab. B.1 and B.2, photoreactions P1 to P9 in Tab. 2., and features neutral-neutral reactions, charge-exchange reactions, collisional dissociation and ionization, radiative association reactions and cosmic ray-induced reactions (we consider a cosmic ray ionization rate ζH=3×10-17s-1subscriptݜH3superscript1017superscripts1\zeta_\mathrmH=3\times 10^-17\,\rm s^-1italic_ζ start_POSTSUBSCRIPT roman_H end_POSTSUBSCRIPT = 3 × 10 start_POSTSUPERSCRIPT - 17 end_POSTSUPERSCRIPT roman_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, the reference value in the Milky Way (Webber, 1998). We follow H2subscriptH2\mathrmH_2roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT formation on dust, adopting the Jura rate at solar metallicity Rf(H2)=3.5×10-17nHntotsubscriptݑ…ݑ“subscriptH23.5superscript1017subscriptݑ›Hsubscriptݑ›totR_f(\mathrmH_2)=3.5\times 10^-17n_\mathrmHn_\mathrmtotitalic_R start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ( roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 3.5 × 10 start_POSTSUPERSCRIPT - 17 end_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT roman_H end_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT roman_tot end_POSTSUBSCRIPT (Jura, 1975). There are 9 reactions involving photons, listed in Tab. 1: photoionization of H, He, He+superscriptHe\mathrmHe^{+}roman_He start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, H-superscriptH\mathrmH^{-}roman_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT and H2subscriptH2\mathrmH_2roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT to H2+superscriptsubscriptH2\mathrmH_2^{+}roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, direct photodissociation of H2+superscriptsubscriptH2\mathrmH_2^{+}roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT and H2subscriptH2\mathrmH_2roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and the two-step Solomon process (Draine & Bertoldi, 1996). The Solomon process rate is usually taken to be proportional to the total flux at 12.87 eV (Glover & Jappsen, 2007; Bovino et al., 2016), but this is correct only if the flux is approximately constant in the Lyman-Werner band (11.2 - 13.6 eV), as pointed out by Richings et al. (2014a). These authors find that the most general way to parametrize the dissociation rate is

ΓH2=7.5×10-11nγ(12.24-13.51eV)2.256×104cm-3s-1,subscriptΓsubscriptH27.5superscript1011subscriptݑ›ݛ¾12.2413.51eV2.256superscript104superscriptcm3superscripts1\Gamma_\mathrmH_2=7.5\times 10^-11\dfracn_\gamma(12.24-13.51\,% \mathrmeV)2.256\times 10^4\rm cm^-3\,\mathrms^-1\,,roman_Γ start_POSTSUBSCRIPT roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 7.5 × 10 start_POSTSUPERSCRIPT - 11 end_POSTSUPERSCRIPT divide start_ARG italic_n start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ( 12.24 - 13.51 roman_eV ) end_ARG start_ARG 2.256 × 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT roman_cm start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT end_ARG roman_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT , (1)
where nγ(ΔEbin)subscriptݑ›ݛ¾Δsubscriptݐ¸binn_\gamma(\Delta E_\mathrmbin)italic_n start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ( roman_Δ italic_E start_POSTSUBSCRIPT roman_bin end_POSTSUBSCRIPT ) is the photon density in the energy interval ΔEbinΔsubscriptݐ¸bin\Delta E_\mathrmbinroman_Δ italic_E start_POSTSUBSCRIPT roman_bin end_POSTSUBSCRIPT. We work in the on-the-spot approximation, hence photons emitted by recombination processes are neglected.

Given the chemical network and the included reactions, the code can in principle be used with an arbitrary number of photon energy bins. In the particular context of photoevaporating clumps, we decided to make only use of two bins with energies in the FUV (far ultra-violet) domain, i.e. [6.0 eV, 11.2 eV] and [11.2 eV, 13.6 eV]. As we consider molecular clumps located outside stellar H IIII\scriptstyle\rm II\ roman_IIregions, we expect that EUV radiation does not reach the surface of the clump. On the other hand, we neglect photons with energies <6.0absent6.0<6.0< 6.0 eV since they do not take part in any chemical reactions of interest in our case.

Fig. 1 summarises the approach we used to couple RT module in ramses with the non-equilibrium network adopted via krome (as also done in Pallottini et al., 2019). At each timestep (ΔtΔݑ¡\Delta troman_Δ italic_t), photons are first propagated from each cell to the nearest ones by ramses-rt (A). Then, the gas-radiation interaction step (B) is executed, sub-cycling in absorption (B1) and chemical evolution (B2) steps with a timestep Δti<ΔtΔsubscriptݑ¡ݑ–Δݑ¡\Delta t_iroman_Δ italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT
In step B1, we account for (1) photons that take part in chemical reactions, (2) H2subscriptH2\mathrmH_2roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT self-shielding and (3) dust absorption. The optical depth of a cell in the radiation bin iݑ–iitalic_i (excluding the Solomon process) is computed by summing over all photo-reactions:

τi=∑jnjΔxcellσij,subscriptݜݑ–subscriptݑ—subscriptݑ›ݑ—Δsubscriptݑ¥cellsubscriptݜŽݑ–ݑ—\tau_i=\sum_jn_j\Delta x_\mathrmcell\sigma_ij\,\,,italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT roman_Δ italic_x start_POSTSUBSCRIPT roman_cell end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT , (2)
where njsubscriptݑ›ݑ—n_jitalic_n start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT is the number density of the photo-ionized/dissociated species in the reaction jݑ—jitalic_j, ΔxcellΔsubscriptݑ¥cell\Delta x_\rm cellroman_Δ italic_x start_POSTSUBSCRIPT roman_cell end_POSTSUBSCRIPT is the size of the cell, and σijsubscriptݜŽݑ–ݑ—\sigma_ijitalic_σ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT is the average cross section of the reaction jݑ—jitalic_j in the bin iݑ–iitalic_i. For the Solomon process, the self-shielding factor SselfH2superscriptsubscriptݑ†selfsubscriptH2S_\mathrmself^\mathrmH_2italic_S start_POSTSUBSCRIPT roman_self end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT is taken from Richings et al. (2014b), and it is related to the optical depth by τselfH2=-log(SselfH2)/ΔxcellsuperscriptsubscriptݜselfsubscriptH2superscriptsubscriptݑ†selfsubscriptH2Δsubscriptݑ¥cell\tau_\mathrmself^\mathrmH_2=-\log(S_\mathrmself^\mathrmH_2% )/\Delta x_\mathrmcellitalic_τ start_POSTSUBSCRIPT roman_self end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT = - roman_log ( italic_S start_POSTSUBSCRIPT roman_self end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) / roman_Δ italic_x start_POSTSUBSCRIPT roman_cell end_POSTSUBSCRIPT. Absorption from dust is included, with opacities taken from Weingartner & Draine (2001). We have used the Milky Way size distribution for visual extinction-to-reddening ratio RV=3.1subscriptݑ…ݑ‰3.1R_V=3.1italic_R start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT = 3.1, with carbon abundance (per H nucleus) bC=60subscriptݑݐ¶60b_C=60italic_b start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT = 60 ppm in the log-normal populations666https://www.astro.princeton.edu/~draine/dust/dustmix.html.

After every absorption substep, krome is called in each cell (step B2): photon densities in each bin are passed as an input, together with the current chemical abundances and the gas temperature in the cell, and krome computes the new abundances after a timestep accordingly.

In App. A, we show two successful tests that we performed to validate our scheme for the coupling between ramses-rt and krome:

A.1
An ionized region, comparing the results with the analytical solution;

A.2
The structure of H2subscriptH2\rmH_2\,\,roman_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPTin a PDR, compared with the standard benchmarks of Röllig et al. (2007).

3 Simulation setup

3.1 Gas

The computational box is filled with molecular gas777The gas has helium relative mass abundance XHe=25%subscriptݑ‹Hepercent25X_\rm He=25\%italic_X start_POSTSUBSCRIPT roman_He end_POSTSUBSCRIPT = 25 %. of density n=100cm-3ݑ›100superscriptcm3n=100\,\,\rmcm^-3italic_n = 100 roman_cm start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT and metallicity Z=Z⊙ݑsubscriptZdirect-productZ=\rm Z_\odotitalic_Z = roman_Z start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT. A dense clump (n=103-104cm-3ݑ›superscript103superscript104superscriptcm3n=10^3-10^4\,\,\rmcm^-3italic_n = 10 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT - 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT roman_cm start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT) is then located at the centre of the domain, with the same initial composition of the surrounding gas (Fig. 2).

Clumps in GMCs are self-gravitating overdensities. Observations of GMCs of different sizes and masses (Hobson, 1992; Howe et al., 2000; Lis & Schilke, 2003; Minamidani et al., 2011; Parsons et al., 2012; Liu et al., 2018; Barnes et al., 2018) show that clumps have a wide range of physical properties: radii range from 0.1-100.1100.1-100.1 - 10 pc, densities can be 10-10410superscript10410-10^410 - 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT times the average density of the GMC; typical masses range from few solar masses to few hundreds M⊙subscriptMdirect-product\rm M_\odotroman_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT.

The density distribution of clumps in GMCs has been studied with numerical simulations of supersonic magnetohydrodynamic turbulence (Padoan & Nordlund, 2002; Krumholz & McKee, 2005; Padoan & Nordlund, 2011; Federrath & Klessen, 2013), yielding a log-normal PDF (Probability Distribution Function):

g(s)=1(2πσ2)1/2exp[-12(s-s0σ)],ݑ”ݑ 1superscript2ݜ‹superscriptݜŽ21212ݑ subscriptݑ 0ݜŽg(s)=\dfrac1(2\pi\sigma^2)^1/2\,\exp\left[-\dfrac12\,\left(\dfrac% s-s_0\sigma\right)\right]\,,italic_g ( italic_s ) = divide start_ARG 1 end_ARG start_ARG ( 2 italic_π italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT end_ARG roman_exp [ - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( divide start_ARG italic_s - italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG italic_σ end_ARG ) ] , (3a)

where s=ln(n/n0)ݑ ݑ›subscriptݑ›0s=\ln(n/n_0)italic_s = roman_ln ( italic_n / italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ), with n0subscriptݑ›0n_0italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT being the mean GMC density, s0=-σ2/2subscriptݑ 0superscriptݜŽ22s_0=-\sigma^2/2italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = - italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2, and σݜŽ\sigmaitalic_σ a parameter quantifying the pressure support. Turbulent and magnetic contribution to gas pressure can be parametrized by the Mach number ℳℳ\mathcalMcaligraphic_M and the thermal-to-magnetic pressure ratio βݛ½\betaitalic_β, respectively; then σݜŽ\sigmaitalic_σ is given by

σ2=ln(1+b2ℳ2ββ+1),superscriptݜŽ21superscriptݑ2superscriptℳ2ݛ½ݛ½1\sigma^2=\ln\left(1+b^2\mathcalM^2\dfrac\beta\beta+1\right)\,,italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = roman_ln ( 1 + italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT caligraphic_M start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_β end_ARG start_ARG italic_β + 1 end_ARG ) , (3b)
where b≃0.3-1similar-to-or-equalsݑ0.31b\simeq 0.3-1italic_b ≃ 0.3 - 1 is a factor taking into account the kinetic energy injection mechanism which is driving the turbulence (Molina et al., 2012). Including self-gravity, the high-density end of the PDF is modified with a power-law tail g(n)∼n-κsimilar-toݑ”ݑ›superscriptݑ›ݜ…g(n)\sim n^-\kappaitalic_g ( italic_n ) ∼ italic_n start_POSTSUPERSCRIPT - italic_κ end_POSTSUPERSCRIPT, with κ∼1.5-–2.5similar-toݜ…1.5italic-–2.5\kappa\sim 1.5-–2.5italic_κ ∼ 1.5 - italic_⠀ “ 2.5 (Krumholz & McKee, 2005; Padoan & Nordlund, 2011; Federrath & Klessen, 2013; Schneider et al., 2015).

If a value nݑ›nitalic_n of the density is drawn from the PDF g(s)ݑ”ݑ g(s)italic_g ( italic_s ), the corresponding radius of the clump can be estimated with the turbulent Jeans length (Federrath & Klessen, 2012):

R=12λj=12πσ2+36πcs2GL2mpμn+π2σ46GLmpμnݑ…12subscriptݜ†j12ݜ‹superscriptݜŽ236ݜ‹superscriptsubscriptݑݑ 2ݐºsuperscriptݐ¿2subscriptݑšݑݜ‡ݑ›superscriptݜ‹2superscriptݜŽ46ݐºݐ¿subscriptݑšݑݜ‡ݑ›R=\dfrac12\lambda_\textscj=\dfrac12\dfrac\pi\sigma^2+\sqrt36% \pi c_s^2GL^2m_p\mu n+\pi^2\sigma^46GLm_p\mu nitalic_R = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_λ start_POSTSUBSCRIPT j end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG divide start_ARG italic_π italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + square-root start_ARG 36 italic_π italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_G italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT italic_μ italic_n + italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG end_ARG start_ARG 6 italic_G italic_L italic_m start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT italic_μ italic_n end_ARG (4)
where cssubscriptݑݑ c_sitalic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT is the isothermal sound speed, Lݐ¿Litalic_L is the size of the GMC, mpsubscriptݑšݑm_pitalic_m start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT the proton mass and μݜ‡\muitalic_μ the mean molecular weight. The corresponding clump mass is estimated by assuming a spherical shape and uniform density.

In Fig. 3 the clump radius is plotted as a function of the number density, for different Mach numbers ℳℳ\mathcalMcaligraphic_M, GMC size fixed at L=25ݐ¿25L=25italic_L = 25 pc and cssubscriptݑݑ c_sitalic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT computed with a temperature T=10ݑ‡10T=10italic_T = 10 K and molecular gas (μ=2.5ݜ‡2.5\mu=2.5italic_μ = 2.5). The coloured points correspond to the position of clumps with different masses M=10-200M⊙ݑ€10200subscriptMdirect-productM=10-200\,\rm M_\odotitalic_M = 10 - 200 roman_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT in the Rݑ…Ritalic_R-nݑ›nitalic_n diagram. In GMCs with the same Mach number, clumps with larger mass are less dense: indeed, as a rough approximation, we have that R∝n-1proportional-toݑ…superscriptݑ›1R\propto n^-1italic_R ∝ italic_n start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT and M∼mpμnR3∝n-2similar-toݑ€subscriptݑšݑݜ‡ݑ›superscriptݑ…3proportional-tosuperscriptݑ›2M\sim m_p\mu nR^3\propto n^-2italic_M ∼ italic_m start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT italic_μ italic_n italic_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ∝ italic_n start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT. It is also interesting to check how the properties of clumps with the same mass vary for different parent GMCs. Considering the 10 M⊙subscriptMdirect-product\rm M_\odotroman_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT clump (red point), we notice that decreasing ℳℳ\mathcalMcaligraphic_M, its position in the Rݑ…Ritalic_R-nݑ›nitalic_n diagram shifts towards lower density and larger radius. Thus, in a ℳ=1ℳ1\mathcalM=1caligraphic_M = 1 cloud, clumps with mass higher than 10 M⊙subscriptMdirect-product\rm M_\odotroman_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT would have a density as low as the average cloud density, implying that clumps more massive than 10 M⊙subscriptMdirect-product\rm M_\odotroman_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT do not exist at all in such a cloud.

Here we consider clumps residing in a GMC with size L=25ݐ¿25L=25italic_L = 25 pc, average temperature T=10ݑ‡10T=10italic_T = 10 K and Mach number ℳ=15ℳ15\mathcalM=15caligraphic_M = 15. In particular, we explore the range of masses represented in Fig. 3, i.e. M=10,50,100,200M⊙ݑ€1050100200subscriptMdirect-productM=10,50,100,200\,\rm M_\odotitalic_M = 10 , 50 , 100 , 200 roman_M start_POSTSUBSCRIPT ⊙ end_POSTSUBSCRIPT. Masses, radii and average densities of these clumps are summarised in Tab. 2. Clumps are modelled as spheres located at the centre of the computational box; their initial density profile is constant up to half of the radius, and then falls as a power law:

n={ncrwhere for each clump the core density ncsubscriptݑ›ݑn_{c}italic_n start_POSTSUBSCRI

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