The bottom panel of Fig. 9 shows the mass accretion histories for Z = 10−2 (-2Sol), 10−1, (-1Sol_from-2Sol), and Z⊙ (Sol_from-2Sol), starting from the envelope structure of Z = 10−2 Z⊙. In the early phase of M* ≲ 100 M⊙, while radiative feedback is still weak, the accretion histories are mostly determined by the initial condition and very similar to each other. At later time, radiative feedback starts operating and the accretion history depends on the metallicity in a way that the final mass is lower, or the feedback is stronger, at lower metallicities. This metallicity dependence comes from the dust attenuation of ionizing photons, which can be seen in the following analytic argument. The photoionization feedback quenches the accretion by the photoevaporation of an accretion disc, which proceeds most efficiently near the gravitational radius
(19)$$\begin{eqnarray*}
r_{\rm g} &amp;=&amp; \frac{GM_{*}}{c_{\rm i}^2} \nonumber \\
&amp;\simeq &amp; 1.1 \times 10^3 ~{\rm au} \left(\frac{M_*}{500~\mathrm{ M}_{\odot }} \right)
\end{eqnarray*}$$(e.g. Hollenbach et al. 1994; McKee & Tan 2008), where ci is the sound speed of the ionized gas for which we assume T = 3 × 104 K. Equations (15) and (19) indicate that the gravitational radius is located outside the dust destruction front and thus the ionizing photons around the gravitational radius is attenuated by dust grains. Using the disc surface density at r ∼ rg set by the ionization balance,
(20)$$\begin{eqnarray*}
n_0 &amp;=&amp; \left(\frac{3 S_*}{4 \pi r_{\rm g}^3 \alpha _{\rm B}} \right)^{1/2} \nonumber \\
&amp;\simeq &amp; 2 \times 10^{7} ~{\rm cm^{-3}} \left(\frac{S_*}{10^{51} ~{\rm s^{-1}}} \right)^{1/2} \left(\frac{M_*}{500~\mathrm{ M}_{\odot }} \right)^{-3/2},
\end{eqnarray*}$$where S* is the stellar emissivity of ionizing photons, we find the optical depth from the star to the gravitational radius,
(21)$$\begin{eqnarray*}
\tau _{\rm d} &amp;=&amp; \rho \kappa _{\rm d} r_{\rm g} \nonumber \\
&amp;=&amp; 3.0 \left(\frac{S_*}{10^{51} ~{\rm s^{-1}}} \right)^{1/2} \left(\frac{M_*}{200~M_{\odot }} \right)^{-1/2} \left(\frac{\kappa _{\rm d}}{350~{\rm cm^2 g^{-1}}} \right) \nonumber\\
&amp;&amp;\times \,\left(\frac{Z}{10^{-2}~Z_{\odot }} \right),
\end{eqnarray*}$$larger than unity for Z ≳ 10−2 Z⊙, where κd is the dust opacity for Lyman-limit frequency at the solar metallicity, meaning that the photoionizing radiation is substantially attenuated by the dust, especially at higher metallicity. This explains the trend of weaker feedback at higher metallicity seen in Fig. 9. According to recent studies (Tanaka, Tan & Zhang 2017; Kuiper & Hosokawa 2018; Tanaka et al. 2018), the impact of photoionization is significantly diminished by the dust attenuation at the solar metallicity. In fact, in our cases with Z = 0.1 (-1Sol_from-2Sol) and 1 Z⊙ (Sol_from-2Sol), the photoionization effect being disabled by this process, the accretion continues until the star becomes very massive with >300 M⊙ and the radiation-pressure feedback finally shuts off the inflow. In this process, the radiation force by the infrared dust re-emission also contributes in terminating the accretion (see equation 17). In particular, it can also repel the flow coming from the shade of the disc, for which the radiation force by the stellar direct light is ineffective (e.g. Yorke & Bodenheimer 1999; Kuiper et al. 2010b; Kuiper & Yorke 2013). Our results suggest that the metallicity dependence of the radiation-pressure effect is weaker than that of the photoionization owing to the dust attenuation in metal-enriched environments. As a result, the final stellar mass increases with increasing metallicity for Z > 10−2 Z⊙.
Citation: McKee & Tan 2008
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