Evolution and Photoevaporation of Protoplanetary Disks in Clusters with Hot Background Temperatures

Our model begins with the gravitational collapse of pre-stellar cores. We assume that the pre-stellar core is an isothermal sphere (e.g., Ebert 1955; Bonnor 1956; Bacmann et al. 2000; Alves et al. 2001) that rotates uniformly (Goodman et al. 1993) with angular momentum,

$$\begin{eqnarray}&&J=\displaystyle \frac{{G}^{2}{\mu }^{2}}{18{{\mathfrak{R}}}^{2}}\displaystyle \frac{{M}_{\mathrm{cd}}^{3}\omega }{{T}_{\mathrm{cd}}^{2}},\end{eqnarray} \tag{ 1 }$$

where G is the gravitational constant, μ = 2.33 is the mean molecular weight, ${\mathfrak{R}}$ is the gas constant, and ω, T_cd, and M_cd are the angular velocity, temperature, and mass of a pre-stellar core, respectively, which represent the initial conditions for the protoplanetary disk formation. Shu (1977) found the self-similar solution for the collapse of an isothermal sphere and evaluated the mass infall rate onto the protostar+disk system, ${\dot{M}}_{\mathrm{cd}}=0.975{a}^{3}/G=(0.975/G)$ ${({\mathfrak{R}}/\mu )}^{3/2}{T}_{\mathrm{cd}}^{3/2}$, where a is the isothermal sound speed. So, the collapse time is

$$\begin{eqnarray}&&{t}_{\inf }=\displaystyle \frac{{M}_{\mathrm{cd}}}{{\dot{M}}_{\mathrm{cd}}}={\left(\displaystyle \frac{0.975}{G}\right)}^{-1}{\left(\displaystyle \frac{{\mathfrak{R}}}{\mu }\right)}^{-3/2}{M}_{\mathrm{cd}}{T}_{\mathrm{cd}}^{-3/2},\end{eqnarray} \tag{ 2 }$$

which is proportional to M_cd and inversely proportional to ${T}_{\mathrm{cd}}^{3/2}$. Nakamoto & Nakagawa (1994) expressed the mass influx, $S(R,t)$, onto the disk,

$$\begin{eqnarray}S(R,t)=\left\{\begin{array}{ll}\displaystyle \frac{{\dot{M}}_{\mathrm{cd}}}{4\pi {{RR}}_{{\rm{d}}}(t)}{\left[1-\displaystyle \frac{R}{{R}_{{\rm{d}}}(t)}\right]}^{-1/2} & \mathrm{if}\tfrac{R}{{R}_{{\rm{d}}}(t)}\lt 1;\\ 0 & \mathrm{otherwise},\end{array}\right.\end{eqnarray} \tag{ 3 }$$

where ${R}_{{\rm{d}}}(t)$ represents the maximum radius of the core mass collapsed:

$$\begin{eqnarray}{R}_{{\rm{d}}}(t) & = & \displaystyle \frac{1}{16}a{\omega }^{2}{t}^{3}\\ & = & 31{\left(\displaystyle \frac{\omega }{{10}^{-14}{{\rm{s}}}^{-1}}\right)}^{2}{\left(\displaystyle \frac{{T}_{\mathrm{cd}}}{10{\rm{K}}}\right)}^{1/2}{\left(\displaystyle \frac{t}{5\times {10}^{5}\mathrm{yr}}\right)}^{3}\ \mathrm{au}.\end{eqnarray} \tag{ 4 }$$

The observed range of ω is about 0.1–13 × 10⁻¹⁴ s⁻¹ (Goodman et al. 1993; Jijina et al. 1999; Caselli et al. 2002). As discussed in the previous section, stellar clusters may have hot background temperatures due to the presence of massive OB stars. We estimate that T_cd is about 40–100 K near the massive stars in these clusters (e.g., Hatchell et al. 2013; Lindberg et al. 2015, 2017; Rumble et al. 2015).

We notice that for a pre-stellar core with fixed ω and M_cd, J is relatively low if T_cd is high (Equation (1)). As a result, the core mass falling directly onto the disk is small (Equation (3)). In addition, when T_cd is high, t_inf becomes short (Equation (2)). According to Equation (4), the disk size is smaller during the core collapse. We expect that the characteristics of disks in clusters are very different from those in isolated star fields due to the difference in T_cd.

According to mass and angular momentum conservation (Jin & Sui 2010; Zhu et al. 2010), the evolution of the disk is governed by Xiao & Chang (2018),

$$\begin{eqnarray}\displaystyle \frac{\partial {\rm{\Sigma }}(R,t)}{\partial t} & = & \displaystyle \frac{3}{R}\displaystyle \frac{\partial }{\partial R}\left[{R}^{1/2}\displaystyle \frac{\partial }{\partial R}({\rm{\Sigma }}\nu {R}^{1/2})\right]+S(R,t)\\ & & +\,S(R,t)\left\{2-3{\left[\displaystyle \frac{R}{{R}_{{\rm{d}}}(t)}\right]}^{1/2}\right.\\ & & \left.+\,\displaystyle \frac{R/{R}_{{\rm{d}}}(t)}{1+{[R/{R}_{{\rm{d}}}(t)]}^{1/2}}\right\}-{\dot{{\rm{\Sigma }}}}_{{\rm{w}}}(R),\end{eqnarray} \tag{ 5 }$$

where Σ is the disk surface density, ${\dot{{\rm{\Sigma }}}}_{{\rm{w}}}(R)$ is the loss of surface density due to photoevaporation, and ν is the effective viscosity, which is expressed as $\nu =\alpha {c}_{{\rm{s}}}h=\alpha {c}_{{\rm{s}}}^{2}/{\rm{\Omega }}$ (Shakura & Sunyaev 1973), where h is the half thickness of the disk, α ≲ 1 is a dimensionless parameter reflecting the viscosity strength, and c_s is the sound speed,

$$\begin{eqnarray}&&{c}_{{\rm{s}}}={\left(\displaystyle \frac{{\mathfrak{R}}{T}_{m}}{\mu }\right)}^{1/2},\end{eqnarray} \tag{ 6 }$$

where T_m is the disk midplane temperature.

The radiative diffusion approximation is adopted to calculate T_m according to the disk surface temperature T_s (Nakamoto & Nakagawa 1994). By considering the thermal equilibrium between the cooling and heating fluxes in the disk, T_s and T_m are expressed as

$$\begin{eqnarray}&&\sigma {T}_{s}^{4}=\displaystyle \frac{1}{2}\left(1+\displaystyle \frac{1}{2{\tau }_{p}}\right)(\dot{{E}_{v}}+\dot{{E}_{s}})+\sigma {T}_{\mathrm{ir}}^{4}+\sigma {T}_{\mathrm{cd}}^{4}\end{eqnarray} \tag{ 7 }$$

and

$$\begin{eqnarray}\sigma {T}_{m}^{4} & = & \displaystyle \frac{1}{2}\left[\left(\displaystyle \frac{3}{8}{\tau }_{{\rm{R}}}+\displaystyle \frac{1}{2{\tau }_{p}}\right)\dot{{E}_{v}}+\left(1+\displaystyle \frac{1}{2{\tau }_{p}}\right)\dot{{E}_{s}}\right]\\ & & +\,\sigma {T}_{\mathrm{ir}}^{4}+\sigma {T}_{\mathrm{cd}}^{4},\end{eqnarray} \tag{ 8 }$$

where σ is the Stefan–Boltzmann constant, τ_p = κ_pΣ is the Planck mean optical depth, κ_p is the Planck mean opacity, $\dot{{E}_{v}}$ is the viscous dissipation rate, $\dot{{E}_{s}}$ is the energy generation rate by shock heating, T_ir is the effective temperature of the irradiation suffered by the disk (Hueso & Guillot 2005), τ_R = κ_RΣ is the Rosseland mean optical depth, and κ_p = 2.39κ_R is the Rosseland mean opacity (Nakamoto & Nakagawa 1994; Bell et al. 1997). The details of calculating T_s and T_m are presented in Nakamoto & Nakagawa (1994) and Jin & Sui (2010).

Xiao & Chang (2018) considered that α varied with R and t,

$$\begin{eqnarray}&&\alpha =\max [{\alpha }_{\mathrm{GI}}+{\alpha }_{\mathrm{MRI}},{\alpha }_{\min }],\end{eqnarray} \tag{ 9 }$$

where α_GI and α_MRI are the viscosity parameters caused by the gravitational instability (GI; Laughlin & Bodenheimer 1994; Laughlin & Rozyczka 1996; Laughlin et al. 1997, 1998) and the magnetorotational instability (MRI; (Balbus & Hawley 1991, 1998), respectively, and α_min is assumed to be the minimum value for the viscosity parameter. We use the model developed by Kratter et al. (2008) to calculate α_GI, which is ${\alpha }_{\mathrm{GI}}={({\alpha }_{\mathrm{short}}^{2}+{\alpha }_{\mathrm{long}}^{2})}^{1/2}$, where ${\alpha }_{\mathrm{short}}=\max [0.14({1.3}^{2}/{Q}^{2}-1){(1-u)}^{1.15},0]$, ${\alpha }_{\mathrm{long}}\,=\max [1.4\times {10}^{-3}(2-Q)/({u}^{1.25}{Q}^{0.5}),0]$, u is the ratio of the disk mass to the mass of the protostar+disk, and Q is the Toomre parameter (Toomre 1964),

$$\begin{eqnarray}&&Q=\displaystyle \frac{{c}_{{\rm{s}}}{\rm{\Omega }}}{\pi G{\rm{\Sigma }}}.\end{eqnarray} \tag{ 10 }$$

The value of α_MRI is determined according to the numerical results of Fleming & Stone (2003). If both GI and MRI cannot survive, we consider that α = α_min = 5 × 10⁻⁴, which is the mean value of Dubrulle (1993), Klahr & Bodenheimer (2003), and Chambers (2006). To summarize, during the early evolution of the disk, GI provides the turbulent viscosity, and when the disk becomes gravitationally stable, MRI or hydrodynamic processes provide the turbulent viscosity. The details of calculating α are presented in Jin & Sui (2010) and Xiao & Chang (2018).

We notice that ν and Q are the functions of c_s, which is determined by T_m. Equation (8) shows that the disk temperature is higher in clusters with massive stars than that in isolated star fields. So, for disks in clusters, the transports of mass and angular momentum are more efficient due to high ν (Shakura & Sunyaev 1973). The disk lifetimes should be shorter. In addition, Q is large for the disk in clusters due to a high disk temperature (Equation (10)). The GI may be difficult to operate.

In this work, radiation from both the host star and external stars provides the photoevaporation of protoplanetary disks in clusters, which means ${\dot{{\rm{\Sigma }}}}_{{\rm{w}}}(R)={\dot{{\rm{\Sigma }}}}_{{\rm{X}}}(R)+{\dot{{\rm{\Sigma }}}}_{{\rm{F}}}(R)$ (Anderson et al. 2013; Xiao & Chang 2018). For internal X-ray photoevaporation, ${\dot{{\rm{\Sigma }}}}_{{\rm{X}}}(R)$, Owen et al. (2012) showed that ${\dot{{\rm{\Sigma }}}}_{{\rm{X}}}(R)=A{\dot{{\rm{\Sigma }}}}_{{\rm{X}},\mathrm{scale}}$, where ${\dot{{\rm{\Sigma }}}}_{{\rm{X}},\mathrm{scale}}$ was presented in their Appendix B. The scale coefficient, A, is determined by

$$\begin{eqnarray}&&{\dot{M}}_{{\rm{X}}}=A\int 2\pi R{\dot{{\rm{\Sigma }}}}_{{\rm{X}},\mathrm{scale}}{dR},\end{eqnarray} \tag{ 11 }$$

where ${\dot{M}}_{{\rm{X}}}$ is the total mass-loss rate due to X-ray photoevaporation (Owen et al. 2011, 2012),

$$\begin{eqnarray}{\dot{M}}_{{\rm{X}}} & = & 6.25\times {10}^{-9}{\left(\displaystyle \frac{{M}_{* }}{1{M}_{\odot }}\right)}^{-0.068}\\ & & \times \,{\left(\displaystyle \frac{{L}_{{\rm{X}}}}{{10}^{30}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{1.14}{M}_{\odot }\,{\mathrm{yr}}^{-1},\end{eqnarray} \tag{ 12 }$$

where M_* is the central star mass and L_X is the X-ray luminosity. Bae et al. (2013) considered that L_X is a function of M_* for a low-mass star $({M}_{* }\leqslant 2\ {M}_{\odot })$:

$$\begin{eqnarray}&&\mathrm{log}({L}_{{\rm{X}}}[\mathrm{erg}\,{{\rm{s}}}^{-1}])\\ &&\,=\,30.37(\pm 0.06)+1.44(\pm 0.10)\mathrm{log}({M}_{* }/{M}_{\odot }).\,\end{eqnarray} \tag{ 13 }$$

For external FUV photoevaporation, ${\dot{{\rm{\Sigma }}}}_{{\rm{F}}}(R)$, it is expressed as (Anderson et al. 2013; Xiao & Chang 2018)

$$\begin{eqnarray}&&{\dot{{\rm{\Sigma }}}}_{{\rm{F}}}(R)=\displaystyle \frac{B}{4\pi }{\left(\displaystyle \frac{{R}_{g}}{R}\right)}^{3/2}\left[1+\displaystyle \frac{{R}_{g}}{R}\right]{e}^{-{R}_{g}/2R},\end{eqnarray} \tag{ 14 }$$

where the parameter $B={{Cn}}_{d}\langle \mu \rangle {a}_{s}$, which is estimated by fitting the numerical results shown in Figure 4 of Adams et al. (2004) with Equation (A3) in their paper, and R_g is the gravitational radius (Anderson et al. 2013),

$$\begin{eqnarray}&&{R}_{g}=\displaystyle \frac{{{GM}}_{* }}{{a}_{s}^{2}}=\displaystyle \frac{{{GM}}_{* }\langle \mu \rangle }{{{kT}}_{\mathrm{FUV}}}\ \mathrm{au},\end{eqnarray} \tag{ 15 }$$

where k is the gas constant, and T_FUV is the function of the FUV flux G₀ (Kalyaan et al. 2015),

$$\begin{eqnarray}&&{T}_{\mathrm{FUV}}=250{\left(\displaystyle \frac{{G}_{0}}{3000}\right)}^{1/2}\ {\rm{K}}.\end{eqnarray} \tag{ 16 }$$

For details of the external photoevaporation model, see Adams et al. (2004), Anderson et al. (2013), and Xiao & Chang (2018).

As discussed in the Introduction, the massive OB stars in clusters may heat the nearby clumps, for example, the northern (IRAS 6/8) region of NGC 1333 and the Serpens WMC 297 region (Hatchell et al. 2013; Rumble et al. 2015). The mean temperature of these regions is about 40 K. Figure 1 shows the evolution of Σ and Q of a photoevaporating disk in clusters for G₀ = 3000 Habings, with core properties ω = 13 × 10⁻¹⁴ s⁻¹, T_cd = 40 K, and M_cd = 1.0 M_☉. The disk evolution is dramatically different from that shown in Section 3.2 in Xiao & Chang (2018). They found that in clusters with low T_cd (T_cd = 10 K), when ω > 5.0 × 10⁻¹⁴ s⁻¹, the falling material from the pre-stellar core was evaporated immediately due to the powerful external FUV photoevaporation. According to Equation (2), t_inf decreases with increasing T_cd. Since J is smaller for the high-T_cd case with T_cd = 40 K (Equation (1)), most of the core material falls onto the disk close to the central star and then is accreted (Equation (4)). Therefore, the gravitational potential well of the disk has been deep during the core collapse. External FUV photoevaporation is suppressed (Adams et al. 2004). The mass acquired from the pre-stellar core exceeds the mass loss accreted and evaporated, so the disk expands outward, as shown in Figure 1(a). Meanwhile, the minimum value of Q in the outer radius of the disk gradually decreases and becomes smaller than 1.3 (Figure 1(c)). Since Σ increases and T_m is low in the outer region of the disk, the disk is gravitationally unstable during the core-collapse stage. The duration of the unstable state is short due to the short t_inf. After the mass supply stops, Σ decreases with t due to the disk viscosity caused by the MHD turbulence or hydrodynamic processes (Figure 1(b)). The disk becomes gravitationally stable (Figure 1(d)). Its lifetime is about 4.50 × 10⁶ yr. We show that the evolutionary features displayed in Figure 1 are similar to those shown in Figure 2 of Xiao & Chang (2018) with ω = 1.0 × 10⁻¹⁴ s⁻¹ and T_cd = 10 K, because the total angular momenta (J; Equation (1)) are roughly the same for these two cores. The relatively low J means that most of the core mass is acquired by the central star. External photoevaporation is moderate. The disk evolution in these two cases is determined mainly by their own physical properties, such as MRI or hydrodynamic processes (e.g., Jin & Sui 2010).

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Figure 2(a) shows the evolution of M_disk and M_* for the disk with parameters shown in Figure 1. As discussed in the previous paragraph, the mass supply from the core exceeds the mass loss due to photoevaporation. So, both M_disk and M_* increase in the core-collapse stage. Although ω = 13 × 10⁻¹⁴ s⁻¹ is large, T_cd = 40 K is relatively high. As a result, the central star gets a large amount of material from the pre-stellar core due to the low J (Equation (1)). When the collapse stops, M_* is about 0.58 M_☉. The gravitational potential well is so deep that external FUV photoevaporation is weak. After the collapse stops, M_disk decreases from 0.43 to 0 M_☉ within 4.4 × 10⁶ yr due to the disk viscosity (Jin & Sui 2010) and photoevaporation (Xiao & Chang 2018). Because the efficiency of external photoevaporation is relatively low, about one half of M_disk is accreted onto the central star (M_* ∼ 0.82 M_☉).

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In Figure 2(b), we show the evolution of R_disk for the disk with the parameters shown in Figure 1. Since M_* grows quickly, as shown in Figure 2(a) during the core collapse, R_g increases rapidly to 270 au within ∼10⁵ yr. Due to the mass supply from the pre-stellar core, R_disk also increases quickly at the same time (R_disk ∼ 110 au). After the collapse stops, external FUV photoevaporation coupled with viscous dissipation makes R_disk shrink from 110 to 0 au within 4.4 × 10⁶ yr, although the external photoevaporation becomes more and more weak.

Next, we present the effects of ω on the evolution of photoevaporating disks in warm clusters. Our calculations have indicated that when ω is lower than a critical value, no disk forms, and all of the core mass is captured by the central star (Xiao & Chang 2018). In this work, we show that the critical value of ω, ω_crit, increases with T_cd. When T_cd = 40 K, ω_crit is about 2.0 × 10⁻¹⁴ s⁻¹, which is larger than that shown in Xiao & Chang (2018; ω_crit ∼ 0.3 × 10⁻¹⁴ s⁻¹ for T_cd = 10 K). Figure 3 shows the evolution of Σ and Q for a photoevaporating disk (G₀ = 3000 Habings) in clusters for the low-ω case, with core properties ω = 2.0 × 10⁻¹⁴ s⁻¹, T_cd = 40 K, and M_cd = 1.0 M_☉. The evolutionary trend of Σ is similar to the case with ω = 13 × 10⁻¹⁴ s⁻¹ shown in Figure 1. The main differences are that the disk lifetime is shorter (∼1.35 × 10⁶ yr) and R_disk is smaller (Figure 4). In addition, the gravitational potential well is deeper, which results in the more ineffective external photoevaporation. Therefore, the disk evolves fast due to its viscosity. As shown in Figures 3(c) and (d), Q is high within the disk lifetime, with a minimum value larger than 6. The disk is always gravitationally stable.

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In Figure 4(a), we show the evolution of M_disk and M_* for the disk with parameters shown in Figure 3 (ω = 2.0 × 10⁻¹⁴ s⁻¹, T_cd = 40 K, and M_cd = 1.0 M_☉) and external FUV fields (G₀ = 3000 Habings). The evolution of M_disk and M_* is similar to that shown in Figure 2. As discussed in the previous paragraph, the low J of the core originating from the low ω causes M_* to be higher and M_disk to be much lower than that for the case in Figure 2. When the collapse ends, M_* is about 0.97 M_☉, and M_disk is only about 0.02 M_☉. Since the gravitational potential well is very deep, the external FUV photoevaporation is very weak. Finally, the central star acquires almost the entire core mass within the disk lifetime (M_* ∼ 0.99 M_☉). Figure 4(b) shows the evolution of R_disk for the disk with parameters shown in Figure 3. The low ω means that most core mass falls directly onto the central star or the disk near the star. Therefore, R_g grows quickly to its final value of ∼394 au, and R_disk is very small (<20 au). The disk evolution is primarily determined by the core properties (e.g., ω) rather than external photoevaporation (Xiao & Chang 2018).

Comparing with the studies of Xiao & Chang (2018), we find that the external photoevaporation is suppressed in warm clusters with T_cd ∼ 40 K. The effectiveness of external photoevaporation is mainly determined by the initial angular momentum of the progenitor core of the disk. Therefore, the core properties should be considered when investigating the evolution of a photoevaporating disk. The external photoevaporation becomes important only when the disk can expand to a few tens of au (Anderson et al. 2013; Kalyaan et al. 2015). In this case, the gravitational potential well of the disk in the outer edge is shallow, and the disk gas can easily escape. For a warm cloud core, J is relatively small. A large amount of collapsing core mass concentrates near the central star, most of which is accreted onto the star because of the disk viscosity originating from the MHD turbulence or hydrodynamic processes (Jin & Sui 2010). As a result, the effect of external photoevaporation is weakened due to the small size of R_disk. Observations of molecular cloud cores suggested that the most frequent value of ω was about 1–2 × 10⁻¹⁴ s⁻¹ (e.g., Goodman et al. 1993; Jijina et al. 1999; Caselli et al. 2002). Our calculations imply that for the core with M_cd ∼ 1 M_☉ in warm clusters, the lifetime of the inherited disk is ∼10⁶ yr, and only a few disks can last as long as ∼5 × 10⁶ yr. It is very different from that shown in Xiao & Chang (2018), who suggested that disk lifetimes in cold clusters are >5 × 10⁶ yr.

Simulations of molecular clouds have suggested that radiative heating of a cloud could suppress it to fragment and collapse (e.g., Bate 2009; Offner et al. 2009; Hennebelle & Chabrier 2011). However, observations found evidence of the presence of disks in the hot regions of stellar clusters (e.g., Stolte et al. 2010; Richert et al. 2015). In this work, we assumed that the pre-stellar core is able to collapse even in hot environments. Figure 5 shows the evolution of Σ and Q for a photoevaporating disk (G₀ = 3000 Habings) in the hot cluster regions, with core properties ω = 13 × 10⁻¹⁴ s⁻¹, T_cd = 100 K, and M_cd = 1.0 M_☉. Since T_cd is high, the collapse time, t_inf, is very short (∼2 × 10⁴ yr). Although ω is large in this case, the J of the core is still very low due to the high T_cd (Equation (1)). Most of the core material falls directly onto the central star during the very short collapse stage. The external photoevaporation is too ineffective to influence the disk evolution. Under the action of disk viscosity, Σ decreases with t (Figure 5). In this case, almost all of the core mass is captured by the central star, and M_disk is very low (Figure 6(a)). Figure 6(b) shows that R_disk is rather small (<17 au), so the disk is gravitationally stable all the time (Figure 5). The disk lifetime is short (∼1.2 × 10⁶ yr).

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We finally present the disk evolution with higher G₀. Figure 7 shows the evolution of Σ and Q for a photoevaporating disk (G₀ = 30,000 Habings) in the hot cluster regions, with core properties ω = 13 × 10⁻¹⁴ s⁻¹, T_cd = 100 K, and M_cd = 1.0 M_☉. The evolution of M_disk, M_*, and R_disk is shown in Figure 8. We find that the evolution of the disk with G₀ = 30,000 Habings is the same as the case of G₀ = 3000 Habings with the same core properties adopted, which illustrates that the disk properties in the hot environments are determined by their progenitor core properties rather than the radiative intensity of external FUV fields. We show that in the hot regions of clusters (T_cd ∼ 100 K), J is low for the core with M_cd ∼ 1 M_☉, which causes R_disk to be small with R_disk < 20 au, and M_disk is also low, with M_disk < 0.02 M_☉. The disk lifetime is very short (<10⁶ yr).

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We have shown that for the core with M_cd ∼ 1.0 M_☉, R_disk is expected to be ≲100 au when the background temperature is ≳40 K. According to Equation (1), the J of the core is also the function of M_cd. In this subsection, we investigate whether a disk with large R_disk in warm clusters could form from a massive cloud core. The evolution of Σ and Q of a photoevaporating disk (G₀ = 3000 Habings) in clusters, with core properties ω = 13 × 10⁻¹⁴ s⁻¹, T_cd = 40 K, and M_cd = 3.0 M_☉, are shown in Figure 9. Obviously, when M_cd increases, Σ becomes higher compared with the case shown in Figure 1. Therefore, M_disk and M_* are both higher with M_disk ∼ 1.1 M_☉ and M_* ∼ 1.0 M_☉ when the core collapse ends (Figure 10). However, J also becomes large with increasing M_cd (Equation (1)). As a result, R_disk is larger with R_disk ∼ 340 au after the core collapse ends. In this case, the external photoevaporation is effective. In addition, the internal X-ray photoevaporation rate is also high due to the high M_* (Equation (12)). So, the decreasing of M_disk and R_disk are both more rapid than the case shown in Figure 2. The final M_* is about 1.2 M_☉, which means that about 60% of M_cd is evaporated by the internal and external photoevaporation.

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Figure 11 shows the evolution of Σ and Q of a photoevaporating disk (G₀ = 3000 Habings) in the hot cluster regions, with core properties ω = 13 × 10⁻¹⁴ s⁻¹, T_cd = 100 K, and M_cd = 3.0 M_☉. The evolution of M_disk, M_*, and R_disk is shown in Figure 12. As discussed in Section 3.2, J is lower for hot molecular cloud cores. So, M_disk is lower (<1.0 M_☉) and R_disk is smaller (∼100 au) compared with the case shown in Figure 10. Most of M_cd is accreted onto the central star (Figure 12). External photoevaporation is still suppressed, although M_cd is larger (M_cd = 3.0 M_☉).

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Our studies simulate the disk formation from the collapse of a molecular cloud core, which shows that R_disk never exceeded R_g. Adams et al. (2004) explained how the external photoevaporation operated in these subcritical disks. Due to the external FUV radiation, some of the gas near R_disk expands to a larger distance. When the gas temperature is high enough, it will escape from the disk and form the outflow. Adams et al. (2004) derived the analytic approximations for the mass-loss rates from the disk outer edge and surface, which are both dominated by the exponential term (Equations (A7) and (A8) in Adams et al. 2004). They found that the mass-loss rate from the outer edge was much higher than that from the disk surface. So, many papers that investigate the viscous evolution of disks in clusters include the external photoevaporation from the disk outer edge only (e.g., Clarke 2007; Kalyaan et al. 2015; Rosotti et al. 2017; Haworth et al. 2018). In this study, we adopt the method as reported in Anderson et al. (2013) to handle the external photoevaporation. Since the mass-loss rate is dominated by the exponential term, Equation (14) can capture the features of the external photoevaporation discussed above. To quantitatively illustrate how disk evolution changes between the two methods mentioned above, we adopt Equation (24) from Kalyaan et al. (2015) to simulate disk evolution with the same core properties as shown in Figure 1. We show that the evolution of Σ, M_disk, and R_disk in Figures 13–15 is very similar to that shown in Figures 1 and 2. Therefore, the formula for mass-loss rate adopted in this study does not change the features of external photoevaporation in subcritical disks. Since R_disk ≪ R_g, the effect of external photoevaporation is weak, and the disk viscosity drives mass accretion and transport of angular momentum. The main differences are that the maximum value of R_disk is larger with the ratio of the maximum R_disk for the two methods about 1.2, R_disk shrinks more slowly, and the disk lifetime (∼4.82 × 10⁶ yr) is slightly longer with the ratio of the disk lifetimes for the two methods about 1.1, as shown in Figure 15.

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We notice that many studies of external photoevaporating disks do not consider the disk formation. For example, Anderson et al. (2013) and Kalyaan et al. (2015) set the initial values of M_* = 1.0 M_☉, M_disk = 0.1 M_☉, and R_disk = 30–100 au. In this case, the external photoevaporation is moderate. Most of the disk mass is accreted onto the central star due to the disk viscosity. The effects of external photoevaporation are truncating R_disk and dissipating the disk rapidly from the outside in when the disk approaches its lifetime. In this study, we show that the evolution of a external photoevaporating disk may be very different for different core properties. The initial value of J of the core determines the effectiveness of the external photoevaporation. If J is low (for example, T_cd = 100 K and M_cd = 1.0 M_☉ or T_cd = 40 K, M_cd = 1.0 M_☉, and ω = 2 × 10⁻¹⁴ s⁻¹), most of core mass collapses onto the central star, and R_disk is very small (<20 au). The external photoevaporation is severely suppressed. In this case, for a core with M_cd ≳ 1.0 M_☉, M_* should be high (M_* > 1.0 M_☉), but M_disk should be low (M_disk/M_* ≪ 1). If J is very high (for example, T_cd = 10 K and ω > 5 × 10⁻¹⁴ s⁻¹), as discussed in Xiao & Chang (2018), most of core mass falls onto the disk far from the central star. The external photoevaporation is very strong, and M_disk is severely evaporated. In this case, both M_disk and M_* are very low. However, when J is moderate (for example, T_cd = 40 K and ω = 13 × 10⁻¹⁴ s⁻¹), both M_disk and M_* are relatively high, and R_disk becomes large after the core collapse ends. The external photoevaporation becomes important when the evolution of the disk approaches its lifetime. In this case, the disk evolution is similar to that shown in Anderson et al. (2013) and Kalyaan et al. (2015).

Theoretical studies of external photoevaporation in clusters have predicted that the diversity of the M_disk − R_disk relation originates from the diverse G₀ and α (Anderson et al. 2013). However, the diversity of protoplanetary disks near massive stars in clusters (such as the core of Orion; e.g., Stolte et al. 2010; Richert et al. 2015; Gorti et al. 2016) may not be explained by this mechanism, because these disks are expected to have the same G₀ and α. We notice that temperatures in the vicinity of massive stars may be relatively high, ∼35–70 K (e.g., Hogerheijde et al. 1995; Arab et al. 2012; Goicoechea et al. 2017). Since the background temperature (T_cd) is relatively high, the external photoevaporation is suppressed (Figure 7). The disk properties are determined by the initial cloud core properties. If ω or M_cd is large, M_disk and R_disk are both large. Otherwise, if ω and M_cd are small, M_disk and R_disk are both small. The diversity of core properties leads to the diversity of disk properties. The role of external photoevaporation is to prevent disk spreading and remove it from the outside in.

NGC 1333 is a very young stellar cluster (1–2 × 10⁶ yr old) in the Perseus molecular cloud. Dodds et al. (2015) found that the disk mass in this cluster was very low. They discovered eight disks with one having a mass of about 0.009 M_☉ and the others having only 0.002–0.004 M_☉. Since the stellar density of NGC 1333 is low with <100 pc⁻³ (Huff & Stahler 2006; Scholz et al. 2013), dynamical interactions of stellar encounters may not work (Rosotti et al. 2014). Hatchell et al. (2013) confirmed that there were only B stars near NGC 1333, which implied that the effect of UV radiation on the disks was weaker than that in the ONC. However, Hatchell et al. (2013) found that the B star SVS3 was heating the part region of NGC 1333 and the temperature of this region was about ∼40 K. Our calculations in Figure 3 suggest that the disks found in NGC 1333 may form in the mild UV radiation environments from the pre-stellar core with warm background temperature ∼40 K and low ω (ω ∼ 2.0 × 10⁻¹⁴ s⁻¹). As discussed in Xiao & Chang (2018), the low ω leads to the lower angular momentum of the cloud core. As a result, the disk is less massive, and its size is small. The disk lifetime is short, ∼1.35 × 10⁶ yr.

We finally try to understand the formation of the 114–426 disk found in the ONC. Observations have shown that its diameter is as large as 950 au (McCaughrean et al. 1998; Bally et al. 2015). Bally et al. (2015) provided the details of the disk properties: the disk mass was relatively low, ∼3 × 10⁻³ M_☉, and the central star had a mass of ∼1–1.5 M_☉. In addition, they also found that this disk might be an evolved and depleted disk because of the absence of dense-gas tracers. Considering the background temperature is as high as 50–80 K, the formation of the disk is very interesting. The external photoevaporation model predicted that R_disk should be very small (<100 au) because external photoevaporation truncated the disk and dissipated it from the outside in (e.g., Adams et al. 2004; Anderson et al. 2013; Kalyaan et al. 2015). Our calculations showed that only in the early evolution of the disk, R_disk might be larger than ∼300 au. However, at this time, the disk is much higher than 0.1 M_☉. The failure in explaining the formation of the 114–426 disk implies that the external photoevaporation may be very weak. The disk evolution is not determined by the core properties. This means that UV radiation is prohibited from irradiating the disk, or the disk is simply far from the UV radiation sources (Bally et al. 2015).